Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

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Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

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machine learning, deep learning, neural networks

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Kellian Cottart 1 Théo Ballet 1 Djohan Bonnet 1 2 Damien Querlioz 1

arXiv:2605.30198v1 [cs.LG] 28 May 2026

Abstract

are even more attractive because they quantify epistemic uncertainty beyond point predictions, enabling reliability monitoring and OOD detection in safety- or mission-critical deployments.

Always-on edge systems must keep learning as conditions change under tight compute budgets and must detect unreliable predictions. Bayesian binary neural networks are attractive in this setting, but mean-field Bernoulli posteriors can saturate on long non-stationary streams, wiping out epistemic uncertainty and freezing plasticity. We propose BiMU, derived from a bounded-memory variational objective that balances stability, plasticity, and forgetting. BiMU combines a data term with controlled relaxation toward the prior and an uncertainty-dependent step size that prevents saturation and sustains informative uncertainty. This non-degenerate posterior enables fully online, buffer-free active querying via Monte Carlo disagreement, reducing label queries and backpropagation updates under imbalance. BiMU sustains learning and strong OOD detection on 1000tasks Permuted-MNIST, and on OpenLORISObject achieves up to 32× label/update savings at matched accuracy under class imbalance and feature compression.

However, Bayesian BiNNs trained with mean-field Bernoulli posteriors face a critical failure mode on long, non-stationary streams: the Bernoulli parameters saturate toward 0 or 1 as evidence accumulates. Equivalently, the natural parameters grow in magnitude, posterior samples become nearly deterministic, epistemic uncertainty collapses, and weight sign changes become exceedingly unlikely. This posterior saturation (often observed as “frozen synapses”) has been reported in continual-learning practice for binary networks (Laborieux et al., 2021) and prevents sustained online adaptation. At the edge, this issue is compounded by two practical constraints: weight updates are expensive, and data streams are strongly imbalanced, so most samples are redundant while rare events and distribution shifts carry most of the learning signal. In this work, we introduce Binary Metaplasticity from Uncertainty (BiMU), a Bayesian continual-learning rule for mean-field Bernoulli synapses that prevents posterior degeneracy while maintaining usable epistemic uncertainty over long streams. BiMU is derived from the bounded-memory Bayesian learning-and-forgetting objective of (Bonnet et al., 2025), and we specialize it here to Bernoulli synapses to avoid saturation in binary posteriors. The update combines (i) a data-driven gradient term with (ii) an uncertainty-gated relaxation toward the prior that counteracts saturation, together with (iii) a bounded, state-dependent (metaplastic) step size that stabilizes consolidated synapses while keeping uncertain ones plastic. This yields a fully online, bufferfree Bayesian update for continual learning without taskboundary signals.

1. Introduction Always-on edge systems face a core tension: they must run inference continuously under tight energy budgets, yet still adapt online as conditions drift (new users, sensor aging, changing environments) and detect when predictions are unreliable (out-of-distribution inputs, rare events). Binary neural networks (BiNNs) (Hubara et al., 2016) are a natural fit: constraining activations and weights to {−1, +1} reduces memory traffic and replaces multiply-accumulate operations with cheap bitwise arithmetic, which is critical when inference dominates an always-on device’s lifetime compute. Bayesian BiNNs (Meng et al., 2020; Khan & Rue, 2023)

Preserving epistemic uncertainty is not only diagnostic: it enables fully online active continual learning. BiMU supports cheap Monte Carlo sampling of binary weights, and we use a one-pass threshold rule: an incoming example is labeled (and triggers backpropagation) only if a disagreement score exceeds a fixed threshold. This directly reduces label requests and parameter updates and is particularly effective under class imbalance, where most samples are uninformative, but rare events are critical. More broadly, BiMU

1 Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, Palaiseau, France 2 Forschungszentrum Jülich, Germany. Correspondence to: Damien Querlioz <[email protected]>.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

treats epistemic uncertainty as a finite resource that must be preserved to support future adaptation.

accumulates parameter-importance scores online (Zenke et al., 2017). Online variants (e.g., Online EWC) introduce discounting to limit unbounded accumulation of importance (Schwarz et al., 2018). However, over very long horizons these methods can still become increasingly rigid as evidence/importance accumulates and plasticity decreases. Moreover, many such approaches require storing additional per-parameter state (e.g., Fisher/importance and/or reference parameters), which is non-trivial under tight edgememory budgets.

We evaluate BiMU in three regimes: (i) 1000-tasks Permuted-MNIST to stress long-horizon non-stationarity and OOD detection; (ii) OpenLORIS-Object continual recognition under distribution shifts with a frozen ImageNet feature extractor to isolate online adaptation under constrained trainable capacity; and (iii) imbalanced streaming scenarios with online querying to quantify label/update efficiency, including per-class effects. Contributions.

2.2. Bounded-memory Bayesian learning and controlled forgetting

• We derive BiMU: a bounded-memory variational update for mean-field Bernoulli synapses (Eq. (6)) with uncertainty-gated relaxation and a bounded metaplastic step size (Eq. (7)) that prevents posterior saturation.

A direct way to prevent long-horizon rigidity is to explicitly bound the amount of retained information. Bayesian learning and forgetting (Bonnet et al., 2025) targets a boundedmemory posterior that adds evidence from the current batch while removing the contribution of the oldest batch in an effective window of size N :

• We couple BiMU with one-pass Monte Carlo disagreement to perform buffer-free online querying, so only informative samples are labeled and updated, reducing annotation and backprop cost under imbalance.

p(ω|Dt−N :t ) =

• We demonstrate sustained learning and useful uncertainty (OOD detection and querying) on 1000tasks Permuted-MNIST, and OpenLORIS-Object with frozen/compressed features, achieving 32× fewer label/update steps at matched accuracy.

p(Dt |ω) p(ω|Dt−N −1:t−1 ) p(Dt ) p(Dt−N −1 ) × . p(Dt−N −1 |ω)

(1)

This yields controlled information decay with memory independent of stream length and provides an explicit stabilityplasticity-forgetting trade-off. In the Gaussian case, this principle leads to MESU (Bonnet et al., 2025), which sustains adaptation while preserving meaningful uncertainty over long streams.

2. Related work 2.1. Bayesian inference, uncertainty, and continual learning

2.3. Bayesian binary networks and posterior saturation

Bayesian neural networks (BNNs) treat weights as random variables and quantify epistemic uncertainty through a posterior over parameters. In practice, BNNs are commonly trained with variational inference (VI), approximating the posterior with a tractable distribution q(ω) by minimizing a KL divergence or maximizing an ELBO (Jordan et al., 1999; Graves, 2011; Kingma et al., 2015; Blundell et al., 2015). This provides uncertainty estimates that are useful for reliability monitoring and OOD detection (Kendall & Gal, 2017; Gal et al., 2017).

Our work builds on the same bounded-memory principle as MESU but targets binary weights with mean-field Bernoulli posteriors. In this setting, a key long-horizon failure mode is Bernoulli posterior saturation: probabilities collapse toward 0 or 1, destroying epistemic uncertainty and making weight sign changes exceedingly unlikely. This freezing effect is observed in continual-learning practice for binary networks (Laborieux et al., 2021) and also arises when applying mean-field Bernoulli BNN training rules (e.g., BayesBiNN (Meng et al., 2020; Khan & Rue, 2023)) naı̈vely in long non-stationary streams without explicit forgetting.

Continual learning (CL) considers sequentially arriving data where past samples are typically unavailable, so naı̈ve fine-tuning leads to catastrophic forgetting. A classical Bayesian approach reuses the previous approximate posterior as the next prior, yielding online/streaming VI updates (Broderick et al., 2013; Nguyen et al., 2018; Zeno et al., 2021). Regularization-based CL methods can also be interpreted through a Bayesian lens: Elastic Weight Consolidation (EWC) constrains important parameters via a Fisher-based quadratic penalty (Pascanu & Bengio, 2014; Kirkpatrick et al., 2017), and Synaptic Intelligence (SI)

2.4. Active learning under streaming and continual shift Active learning (AL) aims to reduce annotation and update cost by querying informative samples, which is particularly valuable under class imbalance and rare-event streams (MacKay, 1996; Khan et al., 2019; Ngartera et al., 2024). Bayesian models naturally support AL through uncertainty-based acquisition functions such as mutual information (BALD) (Houlsby et al., 2011) or predictive en2

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

tropy (Shannon, 1948). However, much of the AL literature assumes pool-based storage and ranking of unlabeled data, which is a poor match for always-on streaming systems with strict memory and latency constraints.

projection:

Fully online AL instead uses one-pass decision rules that query immediately based on an uncertainty score (Liu et al., 2015; Rajendran et al., 2023). Disagreement-based measures such as the variation ratio (Freeman, 1965) are attractive in this setting because they depend only on Monte Carlo predicted labels, avoiding information-theoretic computations given the same MC predictions. Existing continual AL methods often add replay buffers, task-boundary assumptions, or consolidation state (Perkonigg et al., 2021; Ash et al., 2021; Das et al., 2023; Vu et al., 2024; Park et al., 2025), increasing overhead.

Using the variational decomposition (Prop. A.2), the objective is the sum of a data term L(λ, Dt ) and KL regularizers. Under the mean-field Bernoulli approximation, the KL terms decouple across synapses (while the data term still couples parameters through the network). This yields a coordinate-wise update for each λ(i) : h (i) λt = arg min L(λ, Dt ) + N1 DKL q(ω (i) |λ(i) )∥p(ω (i) ) λ(i) i (i) + 1 − N1 DKL q(ω (i) |λ(i) )∥q(ω (i) |λt−1 ) . (5)

Taken together, these works highlight two requirements for practical online continual AL with binary posteriors: (i) preventing Bernoulli posterior saturation so epistemic uncertainty remains informative over long horizons, and (ii) exposing a cheap uncertainty signal that supports one-pass thresholding without buffering. BiMU addresses (i) through bounded-memory Bernoulli updates and metaplastic step sizing, and leverages (ii) via Monte Carlo disagreement thresholding to reduce both label queries and backpropagation updates in streaming continual learning.

This form makes the roles explicit (Fig. 1): the data term drives plasticity, the KL-to-previous term provides stability, and the KL-to-prior term implements controlled forgetting, preventing unbounded evidence accumulation in long streams.

λt = arg min DKL (q(ω|λ) ∥ p(ω | Dt−N :t )) . λ

The window size N should be interpreted as a forgetting horizon: smaller N relaxes faster toward the prior, while larger N approaches cumulative learning and can induce rigidity (Appendix G).

3. Forgetting in binary Bayesian neural networks

-1

Plasticity

1 1 1

-1

-1 1

-1 1 -1 1

-1

-1 1

3.1. Bernoulli variational parameterization

-1 1

-1

Stability

We approximate the posterior over s binary weights ω ∈ {−1, +1}s with a mean-field Bernoulli distribution parameterized by natural parameters λ ∈ Rs . For each synapse i ∈ [1, . . . , s], ω (i) ∼ 2 Ber σ(2λ(i) ) − 1. (2)

-1

Forgetting

-1 1 1 1

1 -1

-1

-1 1 -1

Figure 1. Schematic of the BiMU update. The next variational state is shaped jointly by the current loss (plasticity), the previous posterior (stability), and the prior (forgetting). Bars show Bernoulli probabilities for ω ∈ {−1, +1}.

Equivalently (Prop. A.1), s Y exp λ(i) ω (i) . q(ω|λ) = 2 cosh λ(i) i=1

(4)

(3) Eqs. (1)-(3) constitute the background that allows deriving BiMU. Combining the closed-form Bernoulli KL (Prop. A.3) with a second-order expansion around λt−1 yields a per-synapse update (Theorem A.4): " # (i) (i) λt−1 − λprior ∂L (i) (i) (i) λt = λt−1 − η(λt−1 ) + . ∂λ(i) λt−1 N cosh2 (λ(i) t−1 ) (6) The gradient term ∂L/∂λ(i) implements plasticity. The second term implements controlled forgetting via a relaxation toward the prior. This relaxation term is uncertainty-gated: since Varq (ω) = 1 − tanh2 (λ) = cosh−2 (λ), the pull

Here λ(i) = 0 corresponds to maximal uncertainty p(ω (i) = +1) = 1/2, while large |λ(i) | yields near-deterministic weights. We denote by λprior the natural parameters of the initialization prior p(ω). 3.2. Binary Metaplasticity from Uncertainty At time step t (one online update per batch Dt ), we seek an approximate posterior that retains only a bounded amount of information from the past, corresponding to a sliding window of roughly the last N updates. Following the controlledforgetting posterior (Eq. (1)), we compute λt by variational 3

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks 0.50

Derivative

0.25

The stability mechanism appears in the metaplastic step size η(·). In the exact second-order update (Theorem A.4), η(·) depends on a curvature term involving ∂ 2 L/∂(λ(i) )2 . Estimating this curvature online is challenging for Bernoulli synapses: unlike mean-field Gaussians, Bernoulli posteriors do not expose a continuous dispersion parameter that enables MESU’s Stein-based diagonal-Hessian estimator, and naive Fisher/Hessian approximations can be unstable in single-pass regimes. We therefore avoid explicit curvature estimation and use a bounded surrogate that preserves the qualitative effect of curvature while guaranteeing positivity and a maximum step size (Prop. A.5). Concretely, we use the element-wise learning rate η(λt−1 ) defined by: 1 (i) η(λt−1 )

1 (i) cosh (λt−1 ) 2

(i)

+ 2 tanh(λt−1 )

1.0 Consolidation regime

Deconsolidation regime

0.6

Uncertainty regime

0.00

0.8

Learning rate η

toward the prior is strongest for uncertain synapses and vanishes as |λ| → ∞. Accordingly, the mechanism primarily prevents runaway growth of |λ| that would otherwise drive Bernoulli probabilities toward 0 or 1 and collapse epistemic uncertainty.

0.4 -0.25 Deconsolidation regime

-0.50 -5.00

-2.50

0.00

λt(i−) 1

Consolidation regime

2.50

5.00

0.2 0.0

Figure 2. Color map of the learning rate η (αmax = 1) as a (i) function of the synaptic state λt−1 and the derivative ∂L/∂λ(i) .

∂L ∂λ(i) λt−1

∂L + +2 , αmax ∂λ(i) λt−1

For an incoming (possibly unlabeled) example xj , we draw K independent weight samples {ωk }K k=1 ∼ q(ω|λ) and K compute predictive distributions Vj = {p(y|xj , ωk )}k=1 . Given Vj , we compute an uncertainty score u(Vj ) ∈ R on-the-fly and use it for one-pass querying decisions.

(7)

The gradient-dependent terms make the step size asymmetric: consolidation steps that reinforce the current synaptic sign are full, whereas de-consolidation steps that push against the current sign are inhibited. This effect is seen in Fig. 2. Three regimes emerge: (i) Uncertain synapses (λ(i) ≈ 0): updates are conservatively bounded. (ii) Consolidation (λ(i) g (i) < 0): consistent gradients reinforce the current weight sign and η approaches its upper bound, enabling fast consolidation. (iii) De-consolidation (λ(i) g (i) > 0): gradients opposing the current sign reduce |λ(i) |, but η shrinks, making sign changes difficult unless such gradients persist. This consolidation/de-consolidation asymmetry is consistent with the qualitative metaplastic behavior reported in Laborieux et al. (2021), but here it emerges directly from the Bayesian formulation and the bounded learning-rate construction.

Predictive uncertainty decomposition. In Bayesian predictive models, epistemic uncertainty can be quantified as the mutual information between predictions and parameters (Gal et al., 2017), I(ω, y|xj ) = H[p(y|xj )] − Eq(ω|λ) [H[p(y|xj , ω)]] , (8) (H(·) denotes Shannon entropy.), i.e., predictive uncertainty minus expected predictive entropy (aleatoric). We use these scores primarily for analysis and OOD detection. For online querying, we prefer a cheaper disagreement score. Variation ratio. We adopt the variation ratio (VR) (Freeman, 1965), which measures disagreement among K Monte Carlo predictors. Let ŷk = arg maxc∈C p(yc |xj , ωk ) be the predicted PK class under sample k, and let fmode = maxc∈C k=1 1[ŷk = c] be the number of samples predicting the modal class. We define VR = 1 − fmode /K.

Together, Eqs. (6)-(7) define BiMU: a fully online update that mitigates Bernoulli posterior saturation while preserving actionable epistemic uncertainty over long, nonstationary streams.

When labels are available, we also report the oracle diagnosPK 1 tic VR-True = 1 − K k=1 1[ŷk = yj ].

3.3. Uncertainty Estimation and Online Querying

The unlabeled scores above (VR, predictive uncertainty, aleatoric uncertainty, and epistemic uncertainty) are computed before observing yj . Therefore, if the input distribution remains familiar but the task function changes, i.e., p(y|x) shifts while p(x) does not, unlabeled scores may fail to trigger a query because the input does not appear out-ofdistribution. VR-True is particularly useful for diagnosing and adapting to such pure labeling-function shifts, but it

BiMU can use posterior uncertainty as a control signal for fully online active continual learning: an example is labeled and triggers an update only when the current posterior has epistemic disagreement. Since q(ω|λ) is Bernoulli, uncertainty can be estimated by sampling binary networks and running Monte Carlo (MC) forward passes, with no buffer or revisiting past data. 4

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

requires labels for the corresponding training samples.

ing is already competitive in this small-network regime.

One-pass threshold querying. At time t, we query the label of xj iff u(Vj ) ≥ τ, with a fixed threshold τ ∈ R. Only queried samples trigger backpropagation and a BiMU update. Computing the query score requires K Monte Carlo forward passes, but these are particularly cheap in binary networks (bit-level sampling and low-cost binary arithmetic), whereas backpropagation and weight writes dominate ondevice training cost.

Table 1. 1000-tasks Permuted-MNIST (online; 100-unit MLP). Mean accuracy over the last 5 tasks; OOD AUC (Permuted MNIST vs. Fashion-MNIST); MMRR; Single-task accuracy. Mean±std over 5 runs.

M ETHOD

TASK MEAN ACC. BOUNDS 5 TASKS (%)

OOD DET. ACC . MMRR (AUC) 1 TASK (%)

Binary neural networks

4. Experiments We evaluate BiMU along three axes: long-horizon plasticity under sustained non-stationarity without task-boundary signals; uncertainty quality for OOD detection; and label/update efficiency via online, buffer-free active querying. Methodological details, including the hyperparameter tuning procedure, are provided in Appendix B. For BiMU, the main tuned hyperparameters are the memory window N , the maximum metaplastic step size αmax , and the scaling coefficients of the likelihood and KL terms; for active learning, the query threshold τ controls the realized label/update budget.

B I MU

90.30 ± 0.38 0.99 ± 0.00 139.47 94.67 ± 0.11

BAYES B I NN

YES

41.12 ± 1.62 0.57 ± 0.12

2.04

93.22 ± 0.09

S YN . M ETA .

YES

10.27 ± 0.01

1.64

71.40 ± 1.48

STE

29.35 ± 0.96 0.69 ± 0.04

9.32

77.56 ± 1.35

MESU

91.69 ± 0.58 0.95 ± 0.03 261.10 96.10 ± 0.18

EWC O.

YES

81.78 ± 0.82 0.66 ± 0.11

6.63

96.06 ± 0.11

YES

74.41 ± 1.19 0.66 ± 0.17

5.11

95.55 ± 0.28

SGD

66.64 ± 2.70 0.87 ± 0.05 43.50 96.03 ± 0.34

Real-valued neural networks

After training on all 1000 tasks, we report three quantities (Table 1). (i) MEAN ACC. (last 5 tasks) averages test accuracy over the final five tasks and directly measures latestream adaptability (ability to keep learning after a very long sequence of shifts). (ii) MMRR (Appendix D) summarizes rigidity accumulation by comparing late-stream performance to each method’s best observed per-task performance over the stream. (iii) OOD AUC evaluates uncertainty-based discrimination between in-distribution Permuted-MNIST and Fashion-MNIST (see Appendix B for the uncertainty score used per method). Appendix C also reports backward transfer (BWT) and explains why late-stream accuracy and MMRR are more informative than backward transfer for diagnosing loss of plasticity in very long non-stationary streams.

4.1. Gradient estimation for Bernoulli synapses BiMU provides a closed-form update for the stability and forgetting terms in the natural parameters λ (Eqs. (6)-(7)). In all experiments, we estimate stochastically ∇λ L(λ, Dt ) using a differentiable Bernoulli relaxation (Concrete / Gumbel-softmax reparameterization) (Maddison et al., 2017), following Algorithm 1 (Appendix O). Concretely, we draw relaxed weight samples, backpropagate through the relaxation, and average gradients over K MC samples. 4.2. 1000-tasks Permuted-MNIST: long-horizon continual learning

Among binary-weight methods, BiMU is the only approach that remains accurate at the end of the 1000-tasks stream: it reaches 90.30% mean accuracy on the last five tasks, whereas BayesBiNN, Synaptic Metaplasticity, and STE drop to 41.12%, 10.27%, and 29.35%, respectively. BiMU also has the highest MMRR among binary methods, indicating far less rigidity accumulation.

We stress-test long-horizon online continual learning on 1000-tasks Permuted-MNIST, where each task applies a fixed random pixel permutation to MNIST. Training is strictly online (one epoch per task; batch size 1; no replay). Methods that require explicit task-boundary signals (Table 1, Task bounds = yes) are provided the task boundaries; BiMU and all Task bounds = no baselines are run without them.

The gap to BayesBiNN reflects two mechanisms. First, BiMU implements bounded-memory forgetting. Second, BiMU uses the uncertainty-dependent metaplastic step size in Eq. (7), which changes how evidence is consolidated or de-consolidated online. The N ablation supports this distinction: even when N = 100,000, where BiMU approaches cumulative learning, it remains above BayesBiNN in the 100-tasks study (83.70% vs. 67.91% accuracy, 0.94 vs. 0.75 OOD AUC, MMRR 13.13 vs. 5.03; Appendix G). The same ablation study shows that the results of BiMU remain strong over a broad range of N values. N acts as an interpretable

We use a compact MLP with one hidden layer of 100 units to emphasize plasticity limits in a low-capacity binary regime. As a sanity check, the last column of Table 1 reports singletask accuracy. In this setting, BiMU and BayesBiNN (Meng et al., 2020; Khan & Rue, 2023) perform best, ahead of STE (Hubara et al., 2016) and Synaptic Metaplasticity (Laborieux et al., 2021), confirming that Bayesian binary train5

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks Table 2. OpenLORIS-Object (online; linear head on frozen VGG19). Mean accuracy over 12 tasks and OOD AUC (heldout toy); results for 1,024/8,192/25,088 features. Mean±std over 5 runs.

stability-plasticity-forgetting control parameter, analogous to a replay size or discount factor. STE, lacking a continual-learning mechanism, suffers both catastrophic forgetting and some rigidity from latent realvalued weight divergence over long training. Synaptic Metaplasticity has the strongest rigidity of all tested methods, by construction, due to its nearly irreversible metaplasticity mechanism.

MEAN ACC. A LEATORIC (%) (AUC)

E PISTEMIC (AUC)

M ETHOD

F EATURES

B I MU

1,024 8,192 25,088

73.61 ± 1.53 89.19 ± 0.19 90.62 ± 0.22

BAYES B I NN

1,024 8,192 25,088

72.01 ± 1.69 0.93 ± 0.01 1.00 ± 0.00 86.93 ± 0.41 0.99 ± 0.00 1.00 ± 0.00 89.37 ± 0.77 0.92 ± 0.00 0.90 ± 0.01

S YN . M ETA .

1,024 8,192 25,088

62.82 ± 2.31 0.72 ± 0.04 88.03 ± 0.38 0.63 ± 0.03 86.72 ± 0.34 0.55 ± 0.00

– – –

STE

1,024 8,192 25,088

52.88 ± 3.39 0.73 ± 0.02 79.12 ± 1.39 0.61 ± 0.05 83.79 ± 1.13 0.55 ± 0.00

– – –

Binary neural networks

Among real-valued baselines, MESU achieves the best latestream accuracy (91.69%), narrowly above BiMU, but relies on a higher-dimensional posterior parameterization (µ, σ). EWC Online and Synaptic Intelligence reach intermediate performance (81.78% and 74.41%), while SGD degrades substantially at the tail of the stream (66.64%). Finally, BiMU yields the strongest OOD detection (0.99 AUC), ahead of MESU (0.95) and SGD (0.87). We do not report OOD AUC for Synaptic Metaplasticity because its near-chance late-stream accuracy makes its predictive uncertainty uninformative for in-vs-out discrimination in this setting. We further find that the Reverse Binary Gate activation sharpens disagreement between posterior samples on out-of-distribution inputs, improving OOD separability (Appendix H). BiMU’s gains persist with increased network capacity (Appendix E) while retaining a minimal, constant training-state memory footprint (Appendix F).

0.96 ± 0.01 0.99 ± 0.00 0.93 ± 0.00

1.00 ± 0.00 1.00 ± 0.00 0.90 ± 0.00

Real-valued neural networks MESU

1,024 8,192 25,088

80.82 ± 0.94 0.90 ± 0.01 0.99 ± 0.00 87.01 ± 0.69 1.00 ± 0.00 0.98 ± 0.00 87.84 ± 0.11 0.83 ± 0.00 0.86 ± 0.00

EWC O NLINE

1,024 8,192 25,088

75.72 ± 0.87 0.97 ± 0.01 87.18 ± 0.58 1.00 ± 0.00 88.23 ± 0.05 0.79 ± 0.00

– – –

1,024 8,192 25,088

70.75 ± 0.57 0.95 ± 0.03 86.44 ± 0.64 1.00 ± 0.00 88.04 ± 0.03 0.83 ± 0.00

– – –

SGD

1,024 8,192 25,088

62.31 ± 1.80 0.50 ± 0.00 86.27 ± 0.51 1.00 ± 0.00 88.04 ± 0.03 0.83 ± 0.00

– – –

4.3. OpenLORIS-Object: lifelong learning under nuisance-factor shifts lines (SGD, SI), while remaining below real-valued MESU and EWC Online in this low-dimensional regime. As the feature dimension increases, BiMU scales smoothly to 89.19% at 8,192 features and 90.62% at 25,088 features, slightly above the real-valued baselines in this raw-feature setting. BiMU also remains consistently ahead of BayesBiNN, with a smaller gap than on 1000-tasks Permuted-MNIST, consistent with the shorter horizon (12 tasks) where Bernoulli posterior saturation is less pronounced.

We next evaluate BiMU on OpenLORIS-Object, a lifelong recognition benchmark designed around robotic nuisance factors. We follow the sequential factors analysis protocol (12 tasks: illumination, occlusion, pixel corruption, and clutter at three difficulty levels) without active querying. To isolate online adaptation under constrained trainable capacity, we freeze an ImageNet-pretrained VGG19 feature extractor and train only an online linear classifier on top of its 512 × 7 × 7 = 25,088-dimensional representation. To emulate edge-like feature-compression constraints, we additionally train the head on a fixed random subset of 8,192 and 1,024 features (≈ 3× and 25× compression).

Uncertainty remains informative under compression: BiMU attains near-perfect epistemic OOD separability (AUC = 1.00) at 1,024 and 8,192 features. At full dimensionality, epistemic AUC drops to 0.90 (a trend shared by BayesBiNN), indicating reduced uncertainty-based separability in this linear-head setting as capacity increases. When the extracted features are standardized using OpenLORIS statistics (offline control), real-valued methods regain their advantage (Appendix J), highlighting BiMU’s robustness to unnormalized features in the streaming regime.

After training on all 12 tasks in a single pass (no replay), we report the mean accuracy averaged across the 12 task test sets (Table 2). For OOD evaluation, we remove the toy class during training and treat it as out-of-distribution at test time. We report ROC-AUC using aleatoric and epistemic uncertainty estimates. Under strong compression (1,024 features), BiMU reaches 73.61% mean accuracy, improving substantially over STE, Synaptic Metaplasticity, and non-Bayesian real-valued base-

4.4. Animals: one-pass active learning under class imbalance

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

We first isolate the active learning component (without continual distribution shift) in a strongly imbalanced imageclassification stream built from a 20-class subset of the Animals dataset (Jana, 2023) (Appendix B.5). We designate as low-frequency the classes with fewer than 200 training images and group the remaining classes as high-frequency. Images are encoded with a frozen ImageNet-pretrained VGG19 feature extractor (25,088-dim), and we train only an online linear classifier in a single pass (batch size 1; no replay buffer). Active learning follows our one-pass thresholding protocol: for each incoming sample, we compute an uncertainty score and query its label (and perform backpropagation) only if the score exceeds a fixed threshold; otherwise the example is skipped (no label request, no update). Therefore, the queried-label fraction directly measures both the annotation budget and the update budget. We compare random querying to predictive, aleatoric, epistemic uncertainty, and variation ratio (VR), averaging over five runs (with K = 10 MC predictors for Bayesian methods). We additionally report an oracle diagnostic, VR-True, which compares sampled predictions to the ground-truth label; it is not deployable because it assumes labels for all samples.

(a)

100

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Aleatoric Epistemic Predictive Random VR VR-True

Accuracy (%)

60 40 100 (b)

100

101 Low frequency

102 (c) 100

High frequency

75 80

50 25 0 100

60 101

102

100

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Fig. 3 shows final test accuracy as a function of the queriedlabel fraction, evaluated on a balanced test set (50 images per class). Selective querying consistently improves over random querying at matched budgets, indicating that onepass active learning is effective in this low-label regime. The horizontal line shows 100% update baseline under the same one-pass protocol (i.e., always querying/updating on every sample), reaching 86.28%. VR is the most effective criterion, outperforming entropy scores at comparable budgets: it reaches 84.46% accuracy while querying only 11% of labels, and reaches 87.12% at 18% labels, slightly exceeding the 100% update baseline, consistent with updates being concentrated on informative and under-represented examples rather than redundant majority-class samples.

Figure 3. Imbalanced Animals active learning (frozen VGG19 features). Accuracy vs. queried-label fraction: (a) overall, (b) high-frequency classes, (c) low-frequency classes. Horizontal line: 100% update baseline. Mean over five runs.

Figs. 3(b,c) break down results by class frequency. Gains are driven primarily by low-frequency classes while highfrequency accuracy is largely preserved. At the 11% labeling budget, VR improves low-frequency accuracy by 41 absolute points over random querying, yielding substantially better balanced performance under severe imbalance.

100

Accuracy (%)

STE MESU (Standardized) BiMU MESU BayesBiNN

60 40 20

Fig. 4 compares VR thresholding across Bayesian methods; for the deterministic STE baseline we use aleatoric uncertainty for querying. In this stationary setting, inherent class confusability makes entropy-based querying already useful, but posterior sampling further improves the accuracy-label trade-off when the disagreement signal is well calibrated.

0 100

101

Data used for training (%)

102

Figure 4. Variation ratio-based and STE aleatoric-based active learning on the Animals dataset.

Among Bayesian baselines, MESU is sensitive to feature scaling on raw VGG19 features; per-dimension standardization substantially improves MESU and can be competitive over a narrow range of query budgets (Appendix I). 7

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

In contrast, BiMU remains consistently strong without feature standardization, indicating that its disagreement signal remains informative under the native VGG19 feature distribution, a practical advantage for edge-like streams where reliable normalization may be unavailable or drift over time.

(a)

100

Total

Aleatoric Epistemic Predictive Random VR VR-True

90 80

Accuracy (%)

BayesBiNN performs markedly worse under VR querying, requiring substantially more labels to reach a given accuracy and often not exceeding random querying (Fig. 8, Appendix I). Empirically, its posterior-sample disagreement is less aligned with learning progress in this single-pass, imbalanced regime, causing VR to select less informative samples at matched budgets.

70 100 (b)

100

101

102 (c)

Low frequency

100

High frequency

80 90 60

4.5. OpenLORIS-Object: active continual learning under shift and imbalance

40 100

101

102

101

100

102

Data used for training (%)

We finally test whether BiMU’s one-pass active-learning gains persist under continual nuisance-factor shifts. We use OpenLORIS-Object under the sequential factors analysis protocol (12 tasks) with a frozen VGG19 feature extractor and an online linear head, using a fixed random subset of 8,192 features from the original 25,088-dimensional representation. To induce a long-tailed stream, we define low-frequency classes as those with fewer than 700 training examples and randomly remove 50-80% of their training images. This makes rare categories appear more sparsely over time and increases the share of redundant majorityclass samples. Active learning follows the same one-pass rule as in Sec. 4.4: for each incoming sample, we compute an uncertainty score and query its label only if it exceeds a fixed threshold. Only queried samples trigger labeling and backpropagation updates, so the queried fraction directly measures both annotation and update budgets.

Figure 5. Active continual learning on OpenLORIS-Object under class imbalance (sequential factor analysis; frozen VGG19 features subsampled to 8,192 dims; online linear head). Accuracy vs. queried-label fraction for (a) overall accuracy, (b) frequent classes, and (c) rare classes. Horizontal line: 100% update baseline. Mean over five runs.

while remaining non-zero later in each task.

100

Accuracy (%)

STE MESU BiMU MESU (Standardized) BayesBiNN

60 40 20 0 100

Fig. 5 reports accuracy versus queried-label fraction (overall, low frequency, and high frequency classes). With VR thresholding (computed with K = 10 posterior-sampled predictors unless stated otherwise), BiMU reaches 88.70% while updating on only 3.1% of the stream, corresponding to a 32× reduction in labeled samples and gradient updates relative to the 100% update baseline (87.76%, equivalent to training on the full stream in this one-pass setting). With a 4.0% update budget, accuracy increases to 90.91% (a 25× reduction). The improvement over the 100% update baseline is consistent with the imbalanced stream: skipping redundant majority-class samples implicitly reweights updates toward informative and under-represented examples. The gains in the low-label regime are indeed driven primarily by improved performance on low-frequency categories (Fig. 5c), while avoiding redundant updates on frequent classes (Fig. 5b), yielding higher overall accuracy than random querying at matched budgets. Appendix M further shows that VR queries concentrate shortly after task shifts

101

Data used for training (%)

102

Figure 6. OpenLORIS-Object active continual learning (8,192 frozen VGG19 features): accuracy vs. queried-label fraction for BiMU, MESU, and BayesBiNN (VR thresholding) and STE (aleatoric thresholding). Results averaged over five runs.

Fig. 6 compares VR thresholding across Bayesian methods at matched queried-label fractions. BiMU achieves higher accuracy than MESU and BayesBiNN, indicating that its posterior-sample disagreement remains more useful for onepass querying under feature compression and continual shift. Appendix K provides additional details for other algorithms. Appendix L further shows that small K already recovers most of the label/update savings, supporting low-overhead deployment. Overall, BiMU enables effective active continual learning without replay buffers and without requiring task-identity/boundary signals by sustaining informative epistemic uncertainty in a binary-weight model. 8

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

5. Conclusion

gation is still comparatively expensive; BiMU’s one-pass querying mitigates this by triggering learning only on samples that are predicted to be informative under the current posterior. This makes the average training-time budget compatible with always-on operation, while retaining calibrated uncertainty for both OOD monitoring and selective supervision.

We introduced BiMU, a Bayesian continual-learning rule for binary neural networks that prevents mean-field Bernoulli posteriors from saturating while preserving informative epistemic uncertainty on long, non-stationary streams. Starting from a bounded-memory Bayesian learning-and-forgetting objective, BiMU yields a fully online update in the natural parameters that combines a data-driven term with controlled relaxation toward the prior and a bounded, uncertaintydependent (metaplastic) step size. This sustains plasticity without replay buffers and without requiring task-identity or task-boundary signals.

Overall, BiMU shows that binary Bayesian models can combine edge-efficient inference with sustained plasticity and actionable uncertainty, providing a practical route to reliable, data-efficient lifelong learning under tight memory, compute, and annotation budgets.

Empirically, BiMU maintains long-horizon adaptation where existing binary baselines lose plasticity. On 1000tasks Permuted-MNIST, it achieves high final accuracy and near-perfect OOD detection, while BayesBiNN, Synaptic Metaplasticity, and STE degrade substantially as training proceeds. On OpenLORIS-Object with frozen VGG19 features and aggressive dimensionality reduction, BiMU remains competitive in the streaming setting while retaining a compact binary posterior and useful uncertainty estimates. Preserving epistemic uncertainty is also operationally beneficial: BiMU enables fully online active continual learning via one-pass Monte Carlo disagreement (variation ratio) thresholding. On OpenLORIS-Object under class imbalance, BiMU reaches 88.70% accuracy while updating on only 3.1% of samples (a 32× reduction in labels and gradient updates), with gains driven primarily by improved performance on under-represented classes while avoiding redundant updates on frequent ones.

Code availability The code to reproduce our experiments is available at https://github.com/kellian-cottart/ active-continual-learning-bayesianbinn.

Acknowledgments This work was supported by the Horizon Europe program (EIC Pathfinder METASPIN, grant number 326101098651). It also benefited from a France 2030 government grant managed by the French National Research Agency (ANR-23PEIA-0002). The authors would like to thank Emre Neftci for discussion and invaluable feedback.

Impact Statement BiMU targets continual learning on resource-constrained devices by enabling fully online adaptation with binary Bayesian neural networks while preserving epistemic uncertainty over long, non-stationary streams. This can reduce energy use, labeling effort, and unnecessary gradient updates, and it can improve reliability through uncertaintyaware monitoring (e.g., selective supervision and out-ofdistribution detection) in applications such as robotics and embedded vision.

Appendix N further shows that the fixed threshold τ can be replaced by a fully online budget-driven controller that targets a desired query/update fraction. Appendix P quantifies the inference-update trade-off on an STM32 microcontroller unit and shows that active BiMU reduces expected compute despite the extra Monte Carlo forward passes. BiMU makes uncertainty evaluation practical on energyconstrained edge platforms because its Monte Carlo inference is intrinsically cheaper than in real-valued Bayesian networks. Sampling Bernoulli synapses can be implemented with simple bit-level draws and comparisons, and each sampled forward pass leverages binary arithmetic (e.g., XNOR/popcount or bitwise accumulations) rather than real-valued multiplications, with substantially reduced data movement due to compact weight representations. This shifts the cost profile of uncertainty estimation toward operations that are well matched to microcontroller-class deployments (Cerutti et al., 2020) and becomes even more compelling on dedicated hardware (Conti et al., 2018), where emerging devices and circuits can provide stochastic sampling primitives at very low overhead (Querlioz & Vianello, 2025). When an update is required, however, backpropa-

At the same time, continual adaptation can amplify biases present in streaming and imbalanced data, and uncertainty estimates may become unreliable under unmodeled shifts or adversarial conditions. Practical deployments should include privacy and data-governance protections, ongoing evaluation across subpopulations, and human oversight, especially in safety-critical settings. Our contribution is methodological; its societal impact depends on how the approach is integrated, validated, and governed in downstream systems.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

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Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

A. Mathematical derivations Proposition A.1 (Exponential family product of Bernoulli). The joint distribution of ω can be written in exponential form as s Y exp λ(i) ω (i) , (9) q(ω|λ) = 2 cosh λ(i) i=1 where the natural parameters λ(i) ∈ R are related to the Bernoulli parameters by p(i) = σ 2λ(i) .

(10)

Proof. We begin by using a change of variable that maps the standard Bernoulli support {0, 1} to {−1, 1}. Let v (i) ∼ Bernoulli(p(i) ) with v (i) ∈ {0, 1}, and define ω (i) = 2v (i) − 1. (11) This transformation ensures that ω (i) ∈ {−1, 1} and preserves the Bernoulli probability, with P(ω (i) = 1) = p(i) and P(ω (i) = −1) = 1 − p(i) . For a single synapse ω (i) ∈ {−1, 1}, the Bernoulli probability mass function can be written as q(ω (i) ) = p(i)

1+ω2 (i)

1 − p(i)

1−ω2 (i)

(12)

Taking the logarithm yields i 1h (1 + ω (i) ) log p(i) + (1 − ω (i) ) log(1 − p(i) ) 2 (i) i 1h p (i) (i) (i) = log p (1 − p ) + ω log . 2 1 − p(i)

log q(ω (i) ) =

(13) (14)

Defining the natural parameter λ

(i)

(i) p 1 = log , 2 1 − p(i)

(15)

The Bernoulli parameter p(i) can be recovered by inverting the definition of the natural parameter: (i) (i) p(i) 1 p (i) ⇐⇒ = e2λ λ = log 2 1 − p(i) 1 − p(i) (i) e2λ (i) = σ 2λ , ⇐⇒ p(i) = 1 + e2λ(i) where σ(·) denotes the logistic sigmoid function. The distribution is rewritten as 1 (i) (i) (i) (i) (i) q(ω ) = exp λ ω + log p (1 − p ) . 2

(16) (17)

(18)

Using the identity p(i) (1 − p(i) ) =

1 2 cosh(λ(i) )

2 ,

(19)

the remaining log term simplifies to 1 1 (i) (i) exp log p (1 − p ) = . 2 2 cosh(λ(i) )

(20)

exp λ(i) ω (i) q(ω ) = . 2 cosh(λ(i) )

(21)

Hence, for each variable, (i)

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Finally, independence of the synapses implies that the joint distribution factorizes as s s Y Y exp λ(i) ω (i) (i) q(ω|λ) = , q(ω ) = 2 cosh(λ(i) ) i=1 i=1

(22)

which completes the proof. Proposition A.2 (Variational free-energy decomposition under controlled forgetting). Let p(ω) be the prior distribution over binary synaptic variables ω, and let {Dt }t>0 denote a stream of data batches. Assume that all past batches Dt−N −1 , . . . , Dt−1 have equal marginal likelihood. Then, approximating the posterior p(ω|Dt−N :t ) with a variational distribution q(ω|λ), the variational free energy at time t decomposes as 1 1 DKL q(ω|λ) p(ω|Dt−N :t ) ≈ 1 − DKL q(ω|λ) q(ω|λt−1 ) + DKL q(ω|λ) p(ω) + L(λ, Dt ), (23) N N where L(λ, Dt ) = − Eq(ω|λ) [log p(Dt |ω)]

(24)

is the negative log-likelihood of the current batch. Proof. By Bayes’ rule with controlled forgetting, the posterior over the window Dt−N :t satisfies N1 p(ω) p(ω|Dt−N :t ) ∝ p(Dt |ω) p(ω|Dt−N −1:t−1 ) , p(ω|Dt−N −1:t−1 )

(25)

where the equal-marginal-likelihood assumption implies t−1 Y

p(Dt−N −1:t−1 |ω) =

N p(Di |ω) = p(Dt−N −1 |ω) .

(26)

i=t−N −1

(27)

(28)

(29)

Rearranging terms, we obtain DKL q(ω|λ) p(ω|Dt−N :t ) =

1 1− DKL q(ω|λ) p(ω|Dt−N −1:t−1 ) N 1 + DKL q(ω|λ) p(ω) − Eq [log p(Dt |ω)]. N

(30)

Finally, since the true posterior p(ω|Dt−N −1:t−1 ) is intractable, we approximate it by the previous variational solution q(ω|λt−1 ), assuming it minimized the divergence at the previous time step. Defining L(λ, Dt ) = − Eq(ω|λ) [log p(Dt |ω)] , the stated decomposition follows. 13

(31)

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Proposition A.3 (Closed-form KL divergence for Bernoulli variables). Let q(ω|θ) and q(ω|ξ) be Bernoulli distributions in exponential-family form with support X = {−1, 1}, ω ∈ X : q(ω|θ) =

exp(θ ω) , 2 cosh(θ)

q(ω|ξ) =

exp(ξ ω) . 2 cosh(ξ)

(32)

cosh(θ) . cosh(ξ)

(33)

Proof. By definition, the KL divergence is given by X q(ω|θ) . DKL q(ω|θ) ∥ q(ω|ξ) := q(ω|θ) log q(ω|ξ)

(34)

Substituting the exponential-family forms: q(ω|θ) log

exp(θ ω) exp(θ ω)/2 cosh(θ) q(ω|θ) = log q(ω|ξ) 2 cosh(θ) exp(ξ ω)/2 cosh(ξ) cosh(θ) i exp(θ ω) h (θ − ξ) ω − log . = 2 cosh(θ) cosh(ξ)

(35)

Summing over X : exp(θ) h cosh(θ) i DKL q(ω|θ) ∥ q(ω|ξ) = (θ − ξ) − log 2 cosh(θ) cosh(ξ) exp(−θ) h cosh(θ) i + − (θ − ξ) − log 2 cosh(θ) cosh(ξ) exp(θ) − exp(−θ) cosh(θ) = (θ − ξ) − log 2 cosh(θ) cosh(ξ) cosh(θ) . = (θ − ξ) tanh(θ) − log cosh(ξ)

(36)

This completes the derivation of the closed-form. Theorem A.4 (Memory-limited second-order asymmetric update for binary synapses). Consider a Bayesian neural (1) (s) s network with synaptic weights Qs ω = (ω , . . . , ω ) ∈ X , with X = {−1, 1} Assume a mean-field Bernoulli variational approximation q(ω|λ) = i=1 q(ω (i) |λ(i) ), where each synapse follows the exponential-family form exp λ(i) ω (i) (i) (i) q(ω |λ ) = . (37) 2 cosh(λ(i) ) Let Dt denote the current batch of data and L(λ, Dt ) = −Eq(ω|λ) [log p(Dt |ω)] the negative log-likelihood. Assume a (i)

memory window of size N and a prior parameter λprior . The second-order expansion of the variational free energy yields the following asymmetric optimal update rule: (i)

(i)

λt = λt−1 −

(i)

λt−1 − λprior ∂L(λ, Dt ) + (i) ∂λ(i) N cosh2 (λt−1 ) λ=λt−1 1

(i)

cosh2 (λt−1 )

+ 2 tanh(λt−1 )

∂ 2 L(λ, Dt ) ∂L(λ, Dt ) + (i) ∂λ ∂λ(i)2 λ=λt−1 λ=λt−1

(38)

This update corresponds to a memory-modulated second-order step with an asymmetric, state-dependent learning rate. 14

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Proof. We derive the optimal update for the variational parameter λ(i) by minimizing the variational free energy associated with synapse i under the memory-limited Bayesian continual learning objective. Variational objective. Following the derivation of the memory-limited posterior in Prop. A.2, the variational free energy for a single synapse i is given by 1 1 (i) F(λ , Dt ) = 1 − DKL q(ω (i) |λ(i) ) ∥ q(ω (i) |λt−1 ) + DKL q(ω (i) |λ(i) ) ∥ p(ω (i) ) + L(λ, Dt ), N N (i)

(39)

where L(λ, Dt ) = −Eq(ω|λ) [log p(Dt |ω)] is the negative log-likelihood. Exponential-family form. Following Prop. A.1, each synapse follows a Bernoulli distribution over X , written in exponentialfamily form as q(ω (i) |λ(i) ) =

exp(λ(i) ω (i) ) . 2 cosh(λ(i) )

(40)

(i)

The prior p(ω (i) ) is of the same form with natural parameter λprior . Closed-form KL divergence. Prop. A.3 states that for any Bernoulli distributions over ω ∈ X in exponential-family form q(ω|θ) and q(ω|ξ): exp(θ ω) exp(ξ ω) q(ω|θ) = , q(ω|ξ) = , (41) 2 cosh(θ) 2 cosh(ξ) the KL divergence between these two distributions admits the closed-form expression: " # q(ω|θ) DKL q(ω|θ) ∥ q(ω|ξ) = Eq(ω|θ) log q(ω|ξ) = (θ − ξ) tanh(θ) − log

cosh(θ) . cosh(ξ)

(42)

Closed-form free-energy. Substituting the closed-form KL divergence of Bernoulli distributions from (Eq. (42)) into the single-synapse variational free energy (39), we obtain: cosh(λ(i) ) i 1 h (i) (i) F(λ , Dt ) = 1 − (λ − λt−1 ) tanh(λ(i) ) − log (i) N cosh(λt−1 ) 1 h (i) cosh(λ(i) ) i (i) + (λ − λprior ) tanh(λ(i) ) − log + L(λ, Dt ). (i) N cosh(λprior ) (i)

(i)

(43)

(i)

We reorganize the terms to isolate contributions from previous parameter values λt−1 and the prior λprior . First, note that (i)

(i)

(λ(i) − λprior ) = (λ(i) − λt−1 ) + (λt−1 − λprior ), so that 1 (i) 1 1 (i) (i) (i) (i) (λ − λprior ) tanh(λ(i) ) = (λ(i) − λt−1 ) tanh(λ(i) ) + (λt−1 − λprior ) tanh(λ(i) ). N N N

(44)

Similarly, the logarithmic term can be split: (i)

−

cosh(λt−1 ) 1 cosh(λ(i) ) 1 cosh(λ(i) ) 1 log = − log − log . (i) (i) (i) N N cosh(λ ) cosh(λ ) N cosh(λ ) t−1

prior

(45)

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Combining these expansions with the first KL term weighted by 1 − 1/N , we factor the contributions of (λ(i) − (i) (i) λt−1 ) tanh(λ(i) ) and − log(cosh(λ(i) )/ cosh(λt−1 )): (i)

F(λ(i) , Dt ) = (λ(i) − λt−1 ) tanh(λ(i) ) − log

cosh(λ(i) ) (i)

cosh(λt−1 ) (i)

cosh(λt−1 ) i 1 h (i) (i) (λt−1 − λprior ) tanh(λ(i) ) − log (i) N cosh(λ ) prior

+ L(λ, Dt ).

(46)

First-order optimality condition. Differentiating (46) with respect to λ(i) yields " # (i) (i) λt−1 − λprior ∂F 1 ∂L(λ, Dt ) (i) (i) = (λ − λt−1 ) + + . 2 (i) (i) N ∂λ ∂λ(i) cosh (λ )

(47)

(i)

The optimal update λt satisfies (i)

(i)

∂L(λ, Dt ) λt−1 − λprior ∂F (i) = 0 ⇐⇒ λ(i) − λt−1 = − cosh2 (λ(i) ) − (i) N ∂λ ∂λ(i)

(48)

(i)

Second-order expansion. To retrieve terms in λt−1 , we apply a second-order Taylor expansion around λt−1 for t) , with L twice differentiable, and cosh2 (λ(i) ) differentiable. cosh2 (λ(i) ) ∂L(λ,D ∂λ(i) (i)

cosh2 (λt )

∂L(λ, Dt ) (i) ∂L(λ, Dt ) ≈ cosh2 (λt−1 ) λ=λt−1 (i) ∂λ(i) ∂λ " 2 (i) (i) (i) ∂ L(λ, Dt ) + (λt − λt−1 ) cosh2 (λt−1 ) ∂(λ(i) )2 λ=λt−1 (i) (i) ∂L(λ, Dt ) + 2 sinh(λt−1 ) cosh(λt−1 ) λ=λt−1 (i)

# (49)

∂(λ )

Solving for the update. Using the identity tanh(x) = sinh(x)/ cosh(x), substituting (49) into the stationary condition (48) and collecting terms (i) (i) linear in ∆λ(i) = λt − λt−1 yields (i)

(i)

λt−1 − λprior ∂L(λ, Dt ) + λ=λt−1 (i) ∂λ(i) N cosh2 (λt−1 )

(i)

λt = λt−1 −

(i)

cosh2 (λt−1 )

+ 2 tanh(λt−1 )

∂L(λ, Dt ) ∂ 2 L(λ, Dt ) + λ=λ (i) t−1 ∂λ ∂(λ(i) )2 λ=λt−1

(50)

This concludes the proof. Proposition A.5 (Bounded Learning Rate with a Curvature Surrogate). Consider the binary Bayesian update rule with learning rate 1

(i)

η(λt−1 ) =

2 L(λ,D ) (i) 1 t) t + 2 tanh(λt−1 ) ∂L(λ,D + ∂ ∂(λ (i) (i) )2 ∂λ(i) λ=λt−1 λ=λt−1 cosh2 (λt−1 )

(51)

In large-scale neural networks, the exact evaluation of the second derivative term ∂ 2 L/∂(λ(i) )2 is computationally prohibitive. Let f : R → R be a surrogate curvature function that replaces this second derivative in the denominator. 16

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Fix a maximum learning rate αmax > 0. If f satisfies 1 1 (i) ∂L(λ, Dt ) , − − 2 tanh(λt−1 ) 2 (i) αmax ∂λ(i) cosh (λ ) λ=λt−1

(i)

f (λt−1 ) ≥

(52)

t−1

then the resulting learning rate satisfies (i)

(i)

0 < η(λt−1 ) ≤ αmax

∀λt−1 ∈ R,

and therefore defines a valid bounded gradient descent step. In particular, the choice ∂L(λ, Dt ) 1 +2 αmax ∂λ(i) λ=λt−1

(i)

f (λt−1 ) =

(53)

is a computationally efficient surrogate that guarantees positivity and boundedness of the learning rate without requiring second-order derivatives. Proof. Fix αmax > 0. By definition of the update rule, replacing the second derivative with a surrogate curvature function f yields 1 (i) η(λt−1 )

(i)

(i) cosh (λt−1 ) 2

+ 2 tanh(λt−1 )

∂L(λ, Dt ) (i) + f (λt−1 ). ∂λ(i) λ=λt−1

(54)

(i)

The constraint 0 < η(λt−1 ) ≤ αmax is equivalent to requiring 1 (i) η(λt−1 )

≥

1 αmax

(55)

Thus, a sufficient condition is 1

(i)

(i) cosh (λt−1 ) 2

+ 2 tanh(λt−1 )

1 ∂L(λ, Dt ) (i) + f (λt−1 ) ≥ . (i) αmax ∂λ λ=λt−1

(56)

We now derive a uniform lower bound for the terms that do not involve f . Since 1/ cosh2 (x) ≥ 0 and | tanh(x)| ≤ 1 for all x ∈ R, we obtain (i)

2 tanh(λt−1 )

∂L(λ, Dt ) ∂L(λ, Dt ) . ≥ −2 ∂λ(i) ∂λ(i) λ=λt−1 λ=λt−1

(57)

Consequently, 1 (i) cosh (λt−1 ) 2

(i)

+ 2 tanh(λt−1 )

∂L(λ, Dt ) ∂L(λ, Dt ) ≥ −2 . (i) ∂λ ∂λ(i) λ=λt−1 λ=λt−1

(58)

Substituting (58) into (56), it suffices to define the surrogate curvature term as (i)

f (λt−1 ) ≥

1 ∂L(λ, Dt ) +2 . αmax ∂λ(i) λ=λt−1

(59)

This choice of f depends only on first-order information and can be evaluated efficiently even in large-scale neural networks. (i) With this surrogate, the denominator in (54) is strictly positive and uniformly lower bounded by 1/αmax for all λt−1 ∈ R, (i) ensuring 0 < η(λt−1 ) ≤ αmax . 17

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

B. Main results: Experiments methodology B.1. Implemented Objective In practice, we optimize a scaled bounded-memory variational objective for BiMU, which explicitly balances data fitting, stability to previous parameters, and forgetting toward the prior. Given an online batch Dt , we minimize 1 1 St (λ) = βL L(λ, Dt ) + βKL 1 − KL(q(ω | λ) ∥ q(ω | λt−1 )) + KL(q(ω | λ) ∥ p(ω)) , (60) N N where λ denotes the Bernoulli natural parameters, L(λ, Dt ) = −Eq(ω|λ) [log p(Dt | ω)] is the negative log-likelihood, p(ω) is the prior, N controls the effective memory window, likelihood coefficient βL controls the scale of the asymmetry, and KL coefficient βKL allows for regularization strength. The second-order approximation of the optimal update of St yields the metaplastic learning rate η(λt−1 ) =

1 , −1 βKL 1 − tanh2 (λt−1 ) + 2βL tanh(λt−1 ) g + 2βL |g| + αmax

(61)

where g = ∂L(λ,Dt )/∂λ|λ=λt−1 and αmax bounds the maximum step size. We prefer 1 − tanh2 (λt−1 ) to cosh−2 (λt−1 ), which is mathematically equivalent but allows factoring the computation of tanh(λt−1 ). The resulting online update of the natural parameters is βKL (λt−1 − λprior ) 1 − tanh2 (λt−1 ) , λt = λt−1 − η(λt−1 ) ⊙ γ βL g + (62) N where γ is an added learning rate scaling the gradient, and ⊙ denotes element-wise multiplication. This formulation is used in all experiments unless stated otherwise. B.2. Hyperparameters tuning Hyperparameter optimization (HPO) is performed using Optuna (Akiba et al., 2019), with 200 optimization trials per method. We define a task-dependent objective H, evaluated at each task, as Ht = w1 · Acc0 + w2 · Acc + w3 · Acct ,

(63)

where Acc0 is the accuracy on the first task, Acc denotes the mean accuracy over all tasks learned so far, and Acct is the accuracy on the current task. The objective H jointly captures initial-task retention, overall performance, and adaptability to the most recent task, with their relative importance controlled by weights (w1 , w2 , w3 ). Maximizing H therefore promotes balanced hyperparameter configurations that neither over-emphasize stability nor plasticity, preventing HPO from artificially amplifying catastrophic forgetting or catastrophic remembering in the different algorithms. B.3. 1000-tasks Permuted MNIST Experimental setup. We evaluate all methods on the Permuted MNIST benchmark with 1000 sequential tasks, where each task applies a distinct random pixel permutation to the MNIST images, inducing continual distribution shifts. Training follows a fully online protocol: each sample is observed once and no replay is used. All approaches share a common continual learning setup. We use a compact MLP with a single hidden layer of 100 neurons, train for one epoch per task with batch size 1, and evaluate with batch size 100. Inputs are standardized. Monte Carlo sampling is used for Bayesian methods. BiMU and BayesBiNN employ 5 posterior samples for both inference and gradient estimation, while MESU uses 10 samples. Out-of-distribution detection is evaluated using Fashion-MNIST (Xiao et al., 2017) as OOD data. Performance is measured by ROC-AUC, computed over 1000 decision thresholds to discriminate in-distribution from OOD samples. Hyperparameter optimization. Hyperparameters are computed on 10 tasks of Permuted MNIST with different permutations as the ones presented in the main paper as validation. Hyperparameters are obtained by maximizing the hyperparameter tuning cost function H (see Appendix B.2). 18

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks Table 3. Hyperparameter configurations for all evaluated methods on the 1000-tasks Permuted MNIST benchmark with 100 neurons. M ETHOD

ACTIVATION

L EARNING R ATE ( S )

A DDITIONAL PARAMETERS

MC SAMPLES

B I MU

R EVERSE B INARY G ATE

γ = 4.9

αMAX = 0.0023, βL = 161.3, βKL = 3.76, N = 700

BAYES B I NN

R EVERSE B INARY G ATE

0.77

P RIOR STRENGTH = 1.25 × 10−5

S IGN

3.7 × 10−5

M ETAPLASTICITY = 7.2, WEIGHT DECAY = 2.9 × 10−9

S IGN

−4

1 × 10

W EIGHT DECAY = 2.2 × 10−9

Binary Neural Networks

S YNAPTIC M ETAPLASTICITY STE (BASELINE ) Real-Valued Neural Networks MESU

R E LU

αµ = 1.6, ασ = 2.1

σPRIOR = 0.3, N = 600,000, CLAMP GRAD = 0.36

O NLINE EWC

R E LU

0.003

I MPORTANCE = 4.1, DOWNWEIGHTING = 0.7

S YNAPTIC I NTELLIGENCE

R E LU

0.003

C OEFF . = 1 × 10−4 , DAMPING = 1.0

SGD (BASELINE )

R E LU

9 × 10−4

–

B.4. OpenLORIS-Object Experimental setup. We evaluate all methods on the OpenLORIS-Object dataset (She et al., 2020) using the sequential factor analysis protocol. The benchmark comprises 12 sequential tasks defined by four nuisance factors (illumination, occlusion, pixel corruption, and clutter) each evaluated at three difficulty levels, covering 19 object classes. Learning follows a fully online continual setting, where each sample is processed once without replay. We adopt a frozen ImageNet-pretrained VGG19 (Deng et al., 2009; Simonyan & Zisserman, 2014) network as a feature extractor. Images are preprocessed using the standard VGG19 pipeline, and features are taken from the final convolutional block after removing the classifier, yielding 512×7×7 representations. To study adaptation under constrained capacity, we randomly subsample these features to 8,192 and 1,024 dimensions when specified. All methods share a unified continual learning configuration. A linear classifier is trained online with no hidden layers, for one epoch per task, using batch size 1 for training and batch size 4 for evaluation. No input normalization is applied. Bayesian methods (MESU, BiMU, and BayesBiNN) use 10 Monte Carlo samples for inference and gradient estimation. B.4.1. C ONTINUAL L EARNING Additional experimental setup. We evaluate all algorithms on 18 classes instead of 19. The ‘toy’ class is removed from the dataset. We evaluate out-ofdistribution performance using images taken from the ‘toy’ class. The ROC-AUC is computed through 1000 threshold points to distinguish between in and out-of-distribution data. Hyperparameter optimization. Hyperparameters are computed on the validation set of OpenLORIS-Object. Hyperparameters are obtained by maximizing the hyperparameter tuning cost function H (see Appendix B.2).

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Table 4. Hyperparameter configurations for OpenLORIS-Object for 25,088, 8,192, and 1,024 input features. M ETHOD

L EARNING R ATE ( S )

A DDITIONAL PARAMETERS

MC SAMPLES

γ = 2.16

αMAX = 0.06, βL = 164.9, βKL = 0.156, N = 6900

0.011

P RIOR STRENGTH = 1.3 × 10−6

3 × 10−4

M ETAPLASTICITY = 5.6, WEIGHT DECAY = 3.3 × 10−12

1 × 10

–

αµ = 71.91, ασ = 0.015

σPRIOR = 0.69, N = 449,000, CLAMP GRAD = 0.61

3.9 × 10−4

I MPORTANCE = 0.12, DOWNWEIGHTING = 0.03

S YNAPTIC I NTELLIGENCE

0.002

C OEFF . = 7 × 10−5 , DAMPING = 12.6

SGD (BASELINE )

0.002

–

γ = 33.2

αMAX = 0.08, βL = 187.8, βKL = 1.5, N = 6700

0.86

P RIOR STRENGTH = 1 × 10−6

6 × 10−4

M ETAPLASTICITY = 6, WEIGHT DECAY = 6.4 × 10−12

0.003

–

25,088 INPUT FEATURES Binary Neural Networks B I MU BAYES B I NN S YNAPTIC M ETAPLASTICITY STE (BASELINE )

−4

Real-Valued Neural Networks MESU O NLINE EWC

8,192 INPUT FEATURES Binary Neural Networks B I MU BAYES B I NN S YNAPTIC M ETAPLASTICITY STE (BASELINE ) Real-Valued Neural Networks αµ = 16.34, ασ = 0.04

σPRIOR = 0.9, N = 460,000, CLAMP GRAD = 0.9

O NLINE EWC

0.001

I MPORTANCE = 0.069, DOWNWEIGHTING = 0.2

S YNAPTIC I NTELLIGENCE

0.001

C OEFF . = 0.025, DAMPING = 0.15

SGD (BASELINE )

0.01

–

γ = 0.87

αMAX = 0.092, βL = 27.26, βKL = 1.6, N = 8900

0.3

P RIOR STRENGTH = 6.25 × 10−6

S YNAPTIC M ETAPLASTICITY

6 × 10−4

M ETAPLASTICITY = 2.15, WEIGHT DECAY = 5.22 × 10−6

STE (BASELINE )

0.0035

–

MESU

1,024 INPUT FEATURES Binary Neural Networks B I MU BAYES B I NN

Real-Valued Neural Networks αµ = 2.96, ασ = 11.27

σPRIOR = 0.92, N = 275,000, CLAMP GRAD = 0.34

O NLINE EWC

0.005

I MPORTANCE = 0.028, DOWNWEIGHTING = 0.6

S YNAPTIC I NTELLIGENCE

0.006

C OEFF . = 0.003, DAMPING = 0.13

SGD (BASELINE )

89.5

–

MESU

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

B.4.2. I MBALANCED ACTIVE C ONTINUAL L EARNING Additional experimental setup. From the 19 classes of the OpenLORIS-Object dataset (She et al., 2020) under the sequential factor analysis protocol, we designate as low frequency the classes containing fewer than 700 training samples. To induce further class imbalance, we randomly remove between 50% and 80% of the training images from each low frequency class. For evaluation, the test set is balanced by subsampling all classes to the minimum class cardinality, enabling separate and fair assessment of performance on high and low frequency classes. In the main results, we perform an active learning ablation study, comparing epistemic, aleatoric, predictive, variation ratio (VR), and oracle VR-True strategies across a range of uncertainty thresholds and multiple algorithms. Hyperparameter optimization. Hyperparameter optimization is conducted without active learning, and the resulting performance is reported as the baseline. Hyperparameters are computed on the validation set of OpenLORIS-Object. They are obtained by maximizing the hyperparameter tuning cost function H (see Appendix B.2). Table 5. Hyperparameters for algorithms on the OpenLORIS-Object dataset with an imbalanced training set with 8,192 input features. M ETHOD

L EARNING R ATE ( S )

A DDITIONAL PARAMETERS

MC SAMPLES

Binary Neural Networks γ = 48.7

αMAX = 0.065, βL = 16.7, βKL = 0.53, N = 1600

BAYES B I NN

0.066

P RIOR STRENGTH = 4.6 × 10−5

STE

0.005

B I MU

WEIGHT DECAY = 5 × 10

−7

Real-valued Neural Networks µPRIOR = 0, σPRIOR = 0.36, N = 328,000, CLAMP GRAD = 0.7

MESU ( STANDARDIZED ) αµ = 20.2, ασ = 0.005 µPRIOR = 0, σPRIOR = 0.61, N = 485,000, CLAMP GRAD = 0.05

MESU

αµ = 60.16, ασ = 2.95

B.5. Animals Detection: Imbalanced active learning In the main text, we evaluate and compare active learning strategies on a subset of the Animals Detection dataset (Jana, 2023). From the 80 available classes, we select 20 classes exhibiting strong class imbalance (see Table 6). Classes with fewer than 200 training samples are considered low frequency. For fair evaluation, we construct a balanced test set by uniformly sampling 50 examples per class, allowing us to explicitly assess the impact of class imbalance on low and high frequency classes. Images are processed using a frozen VGG19 network (Simonyan & Zisserman, 2014) pretrained on ImageNet (Deng et al., 2009). Standard VGG19 preprocessing is applied, and features are extracted from the final convolutional block after removing the classifier, yielding 512×7×7 representations. All methods share a common experimental configuration. A linear classifier is trained online with no hidden layers, for a single epoch, using batch size 1 for training and batch size 50 for evaluation. No input normalization is applied. Bayesian methods (MESU, BiMU, and BayesBiNN) use 10 Monte Carlo samples for inference and gradient estimation.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks Table 6. Train/test splits for selected animal classes. Classes under 200 training examples are considered as being under-represented classes.

C LASS

N UMBER OF TRAIN EXAMPLES N UMBER OF TEST EXAMPLES

B UTTERFLY L IZARD F ISH M ONKEY S PIDER E AGLE F ROG J ELLYFISH P ENGUIN W HALE Z EBRA C ROCODILE L EOPARD S HEEP R ACCOON R AVEN PANDA LYNX B ULL S CORPION

1997 1412 1404 1043 1015 849 617 501 390 291 164 136 132 125 106 91 62 66 72 76

50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

Hyperparameter optimization. To decouple model selection from the active-learning mechanism, we tune all training hyperparameters once in a fully supervised streaming run (i.e., the model updates on every incoming example; no querying is applied during HPO). In the active-learning runs, we do not retune hyperparameters for any acquisition function or labeling budget: accuracy-budget curves are obtained solely by sweeping the uncertainty threshold τ with these fixed settings. This protocol ensures that differences between acquisition strategies reflect the querying criterion rather than strategy-specific hyperparameter tuning. Table 7. Baseline hyperparameter configurations for the Animals Detection dataset with imbalanced classes. M ETHOD

L EARNING R ATE ( S )

A DDITIONAL PARAMETERS

MC SAMPLES

γ = 5.8

αMAX = 0.054, βL = 14, βKL = 0.16, N = 4600

P RIOR STRENGTH = 1 × 10−6

−9

Binary Neural Networks B I MU BAYES B I NN

WEIGHT DECAY = 2.4 × 10

0.0001

STE Real-Valued Neural Networks MESU

αµ = 42.4, ασ = 97.7

σPRIOR = 0.56, N = 61,000, CLAMP GRAD = 0.1

MESU ( STANDARDIZED )

αµ = 65.6, ασ = 4.7

σPRIOR = 0.86, N = 397,000, CLAMP GRAD = 0.41

C. Backward transfer clarifications Backward transfer (BWT) (Lopez-Paz & Ranzato, 2017) is a standard metric for quantifying forgetting in continual learning. If at,i denotes the test accuracy on task i after training up to task t, a common definition is BWT =

T −1 1 X (aT,i − ai,i ) . T − 1 i=1

(64)

BWT measures how much performance on previously learned tasks changes after subsequent training. A negative value indicates stronger forgetting, while a value close to zero is usually interpreted as better backward stability. Positive values indicate positive transfer. 22

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

The goal of our work is to both mitigate catastrophic forgetting of old tasks, as well as progressive loss of plasticity due to posterior saturation on new tasks. In Binary networks, the latent variable (or natural parameter) can grow in magnitude over a long stream, driving synapses to saturation. Once this happens, weights are nearly deterministic and sign changes become increasingly unlikely. BWT is conditional on the amount of task-specific performance that was acquired in the first place. A method that learns a task well and later forgets it will have a negative BWT. In contrast, a method that fails to learn, or that becomes rigid and remains uniformly poor, can obtain a near-zero BWT simply because there is little performance left to lose. Therefore, on very long non-stationary streams, BWT can confound two qualitatively different regimes: useful stability after learning, and apparent stability caused by loss of plasticity. Table 8. BWT on the 1000-tasks Permuted-MNIST stream. Synaptic Metaplasticity has the least negative BWT, but it reaches only chance-level accuracy. BiMU has a more negative BWT, but remains highly plastic and accurate after 1000 tasks.

M ETHOD

M EAN ACC . 5 TASKS (%)

MMRR

BWT

B I MU BAYES B I NN S YNAPTIC M ETAPLASTICITY STE

90.30 ± 0.38 41.12 ± 1.62 10.27 ± 0.01 29.35 ± 0.96

139.47 2.04 1.64 9.32

−0.8187 ± 0.0014 −0.3897 ± 0.0021 −0.0009 ± 0.0001 −0.5689 ± 0.0105

In the main paper, we report the mean accuracy on the last five tasks and Maximum Memory Rigidity Resilience (MMRR) instead of BWT, which appears in Table 8. Late-stream accuracy directly measures whether the model can still acquire new tasks after a long history of distribution shifts. MMRR complements this by measuring how much of the method’s best attainable performance is preserved at the end of the stream. Together, these quantities distinguish a model that remains both stable and plastic from one that stops changing. Specifically, Synaptic Metaplasticity has the least negative BWT among the binary methods. However, it is the weakest method at the end of the stream, with only 10.27% mean accuracy over the last five tasks. This value is close to chance for MNIST classification, showing that the near-zero BWT neither reflects good performance in continual learning nor how much loss of plasticity occurred in the model. This supplementary analysis emphasizes that a better BWT is not necessarily a good sign of well-performing continual learning, specifically for our use-case. When posterior saturation or synaptic rigidity prevents new learning, BWT can become artificially favorable. To solve the challenges we presented in the introduction, the relevant question is whether the method can still learn after hundreds or thousands of distribution shifts. Late-stream accuracy and MMRR directly answer this question, while BWT only measures one aspect of backward stability. The OpenLORIS-Object results provide a complementary case, due to not facing the same rigidity as for the 1000-tasks Permuted MNIST. As shown in Table 9, BiMU achieves the highest mean accuracy, 90.62%, while having BWT comparable to BayesBiNN, Synaptic Metaplasticity and within range of other continual learning methods. STE is worse both in accuracy and in BWT.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks Table 9. OpenLORIS-Object with 25,088 frozen VGG19 features. We report the mean accuracy over all tasks and backward transfer (BWT). M ETHOD

M EAN ACC . OVER ALL TASKS (%)

BWT

90.62 ± 0.22 89.37 ± 0.77 86.72 ± 0.34 83.79 ± 1.13

−0.0525 ± 0.0027 −0.0399 ± 0.0078 −0.0485 ± 0.0021 −0.0790 ± 0.0157

87.84 ± 0.11 88.23 ± 0.05 88.04 ± 0.03 88.04 ± 0.03

−0.0515 ± 0.0012 −0.0452 ± 0.0003 −0.0517 ± 0.0004 −0.0517 ± 0.0004

Binary Neural Networks B I MU BAYES B I NN S YNAPTIC M ETAPLASTICITY STE (BASELINE ) Real-Valued Neural Networks MESU O NLINE EWC S YNAPTIC I NTELLIGENCE SGD (BASELINE )

So far, this is consistent with our main claim: BiMU deliberately forgets part of the accumulated posterior information in order to avoid rigidity and preserve plasticity. If the information is not useful currently and lies outside of the memory window N , it is discarded. A loss in backward transfer is exchanged for higher overall accuracy on late stream: it tends to indicate that the method discards non-essential or outdated representations.

D. Maximum Memory Rigidity Resilience (MMRR): A measure of plasticity loss In long-horizon continual learning, models may progressively lose plasticity as evidence accumulates, becoming increasingly resistant to parameter updates. This rigidity can degrade performance on new tasks even in the absence of explicit forgetting. Standard metrics such as average accuracy (ACC), Forward Transfer (FWT), or Backward Transfer (BWT) (Lopez-Paz & Ranzato, 2017) do not explicitly capture this phenomenon, as they focus on task-to-task retention rather than long-term adaptability under prolonged non-stationarity. To quantify plasticity loss, we build on the Memory Rigidity Resilience (MRR) metric of (Bonnet et al., 2025) and introduce Maximum Memory Rigidity Resilience (MMRR), a scalar diagnostic that measures performance degradation relative to the model’s own best observed performance. Let at ∈ [0, 1] denote the accuracy of the model evaluated on task t, and let amax = max at′ ′ t ≤T

be the maximum accuracy achieved by the model on any task over the full training horizon T . We define the Maximum Memory Rigidity Resilience (MMRR) at time t as MMRRt =

1 , amax − at + ε

where ε > 0 ensures numerical stability. Larger values indicate smaller performance degradation and thus higher resistance to rigidity. MMRR captures the inability of a model to sustain its peak performance as new tasks arrive, independently of explicit forgetting. Unlike the original MRR, which anchors rigidity to early-task performance and can be misleading when performance improves over time, MMRR evaluates each model relative to its maximum attainable accuracy over the stream. This ensures stability and interpretability in long-horizon settings. In our experiments, MMRR clearly separates methods that maintain long-term adaptability from those that become rigid: BayesBiNN, Synaptic Metaplasticity, and STE exhibit rapid MMRR collapse due to posterior saturation, while BiMU maintains high MMRR over extended non-stationary streams, indicating sustained plasticity. 24

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

E. 1000-tasks Permuted MNIST: Effect of Network Capacity This appendix investigates whether BiMU’s advantages persist when network capacity is substantially increased. While the main paper focuses on compact architectures to emphasize bounded-memory effects, scaling the model allows us to disentangle representational limitations from loss of plasticity and synaptic rigidity. We repeat the 1000-tasks Permuted MNIST experiment using a single-hidden-layer network with 2000 neurons, keeping the online continual learning protocol unchanged. Training is strictly online, with each sample observed once. Inputs are standardized, training uses batch size 1, evaluation uses batch size 100, and out-of-distribution (OOD) detection is assessed on Fashion-MNIST using ROC-AUC over 1000 thresholds. Bayesian methods employ Monte Carlo sampling during inference: BiMU and BayesBiNN use 5 samples, while MESU uses 10. All other methods are deterministic. Table 10 reports results averaged over the last five tasks after training on all 1000 permutations. Increasing capacity improves accuracy for all methods, confirming that additional representational capacity partly mitigates long-horizon continual learning challenges. BiMU benefits most from scaling, achieving a mean accuracy of 95.20%, the best performance among binary-weight models and higher than all real-valued baselines. In contrast, other binary approaches fail to fully exploit the additional capacity: BayesBiNN improves relative to the low-capacity setting but remains constrained by progressive rigidity, STE shows only moderate gains, and Synaptic Metaplasticity collapses to near-chance performance. This indicates that capacity alone is insufficient without mechanisms that actively regulate plasticity. Among real-valued models, scaling preserves the relative performance trends observed in smaller networks. MESU achieves strong results (92.99%), followed by EWC Online and Synaptic Intelligence. Notably, BiMU surpasses all real-valued baselines despite operating under a more constrained synaptic posterior, highlighting that its advantage stems from synapse-level control of learning dynamics rather than architectural flexibility. The Maximum Memory Rigidity Resilience (MMRR) metric further emphasizes these differences. BiMU achieves the highest resilience (862.09), indicating minimal degradation between early and late tasks and sustained plasticity across the entire stream. All other methods exhibit substantially lower resilience, confirming that they progressively lose adaptability even with increased capacity. Scaling also accentuates differences in uncertainty estimation. BiMU achieves perfect OOD discrimination on FashionMNIST (AUC = 1.00), while MESU remains strong but inferior (AUC = 0.86), and other baselines lag behind. This demonstrates that BiMU preserves meaningful epistemic uncertainty over extremely long task horizons, even as capacity increases.

Table 10. Benchmark on 1000 tasks Permuted MNIST in the Online Continual Learning context (2000 neurons). The mean accuracy is taken after training on all tasks, computing the average of the accuracy on the last five tasks test datasets. Memory rigidity resilience is defined as the inverse of the absolute differential in accuracy between the first and the last task. Mean ± standard deviation are given over five runs. Methods are grouped by whether they use binary weights or not.

M ETHOD

MEAN ACC. OOD DET. MMRR 5 TASKS (%) (AUC)

B I MU

95.20 ± 0.26 1.00 ± 0.00 862.09

BAYES B I NN

86.61 ± 0.27

0.80 ± 0.10

9.86

S YN . M ETA .

10.34 ± 0.19

0.00 ± 0.04

1.33

47.39 ± 1.35

0.61 ± 0.10

39.12

Binary neural networks

STE

Real-valued neural networks MESU

92.99 ± 0.71 0.86 ± 0.02

171.82

EWC O.

91.47 ± 0.31

139.66

92.82 ± 0.49

0.96 ± 0.04

89.77

SGD

90.61 ± 1.32

0.98 ± 0.02

117.65

0.44 ± 0.14

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Hyperparameter optimization. Hyperparameters are computed on 10 tasks of Permuted MNIST with different permutations as the ones presented in the main paper as validation. They are obtained by maximizing the hyperparameter tuning cost function H (see Appendix B.2). Table 11. Hyperparameter configurations for all evaluated methods on the 1000-tasks Permuted MNIST benchmark with 2000 neurons. M ETHOD

ACTIVATION

L EARNING R ATE ( S )

A DDITIONAL PARAMETERS

MC SAMPLES

Binary Neural Networks B I MU

R EVERSE B INARY G ATE

αMAX = 0.005, βL = 280.7, βKL = 6.2, N = 900

BAYES B I NN

R EVERSE B INARY G ATE

0.05

P RIOR STRENGTH = 1.1 × 10−6

S YNAPTIC M ETAPLASTICITY

S IGN

1 × 10−5

M ETAPLASTICITY = 6.3, WEIGHT DECAY = 1.46 × 10−6

STE (BASELINE )

S IGN

1 × 10−4

W EIGHT DECAY = 9.7 × 10−5

Real-Valued Neural Networks αµ = 2.2, ασ = 0.47 σPRIOR = 0.46, N = 1.6 × 106 , CLAMP GRAD = 0.83

MESU

R E LU

O NLINE EWC

R E LU

0.006

I MPORTANCE = 4.0, DOWNWEIGHTING = 0.4

S YNAPTIC I NTELLIGENCE

R E LU

0.0015

C OEFF . = 1 × 10−4 , DAMPING = 1.0

SGD (BASELINE )

R E LU

0.001

–

F. 1000-tasks Permuted MNIST: Training-State Memory Overhead Table 12 reports the training-state memory overhead of all methods evaluated on the 1000-tasks Permuted MNIST benchmark. We measure the total persistent memory required during training, including model parameters and all auxiliary variables needed to perform parameter updates, excluding activations, capturing the dominant constraints in long-horizon learning scenarios. BiMU is the only continual learning method among the others that is fully online, not requiring stored past parameters, task-specific statistics, replay buffers, or optimizer momentum. Its updates rely solely on the current Bernoulli posterior parameters and the incoming batch, so the training-state memory footprint is identical to the inference-state memory footprint and remains constant over time. In the binary setting, BayesBiNN stores both Bernoulli parameters and associated the posterior of the previous task, resulting in a larger memory overhead than BiMU. STE and Synaptic Metaplasticity are trained with Adam, which requires storing two additional momentum matrices per parameter. In addition, Synaptic Metaplasticity maintains task-specific Batch Normalization parameters, which accumulate linearly with the number of tasks and dominate the memory overhead in long-horizon regimes. In the real-valued setting, MESU attaches an explicit variance parameter to each synapse, increasing the memory footprint relative to SGD. EWC Online and Synaptic Intelligence require storing both an importance matrix and a copy of the parameters from the previous task, resulting in persistent additional storage even in their online variants.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks Table 12. Training-state memory overhead on the 1000-tasks Permuted MNIST benchmark. Reported values correspond to the total persistent memory required during training, including model parameters and all auxiliary update variables, but excluding transient activations.

M ETHOD

T RAINING - STATE MEMORY OVERHEAD (MB) Binary neural networks 0.32 0.64 1.84 0.95

B I MU BAYES B I NN S YN . M ETA . STE

Real-valued neural networks 0.64 0.95 0.95 0.32

MESU EWC O. SI SGD

G. 100-tasks Permuted MNIST: Effect of the memory window on stability and forgetting G.1. Forgetting, rigidity, controlled regimes (i)

Consider BiMU’s update of the natural parameters λt defined in Eq. (6), obtained from the bounded-memory Bayesian objective with memory window N . For a fixed data stream and learning-rate bound αmax , the memory window N explicitly controls the forgetting-stability trade-off as follows. Small memory window (low N ): forgetting-dominated regime. When N is small, the forgetting term (i)

(i)

λt−1 − λprior (i)

N cosh2 (λt−1 ) rapidly relaxes the parameters toward the prior. Past evidence is, therefore, quickly discarded, preventing sustained accumulation of information in λ(i) and leading to excessive forgetting. Large memory window (high N ): stability-dominated regime. As N increases, the forgetting term vanishes, and updates approach cumulative Bayesian learning. In this regime, the magnitude of λ(i) grows monotonically as evidence accumulates, driving Bernoulli probabilities toward deterministic values. This induces synaptic degeneracy and increases memory rigidity, reflected by low MMRR values, inducing a situation akin to the one observed on BayesBiNN. Intermediate memory window: controlled forgetting. For intermediate values of N , forgetting counterbalances evidence accumulation, bounding the magnitude of λ(i) over time. This prevents synaptic degeneracy while preserving sufficient information for learning, yielding a favorable trade-off between plasticity and stability. Overall, the memory window N acts as a direct control parameter for forgetting, regulating both the long-term magnitude of the synaptic parameters and the degree of rigidity in the learned representation. G.2. Memory window ablation study Table 13 quantitatively illustrates the effect of the memory window N on forgetting, memory rigidity, and uncertainty in 100-tasks Permuted-MNIST. For very small N = 100, the model falls into the forgetting-dominated regime: final accuracy is only 27.09%, OOD detection remains near-perfect (0.99), and MMRR is high (216.45), indicating rapid weight fluctuations induced by forgetting, unstable posterior retention circumventing stability. Increasing N to 400 allows more evidence accumulation, improving accuracy to 73.46%, with OOD AUC stable (1.00) and a high MMRR (574.72), reflecting higher retention than N = 100 without accumulating rigidity. Intermediate windows (N = 700–1000) achieve the best balance: accuracy peaks at 90.29% (N = 700) while OOD detection remains strong (0.99–0.98) and MMRR is moderate (215.52–46.30), consistent with controlled forgetting that stabilizes synapses without compromising plasticity. For larger windows (N = 1300–1900), final accuracy slightly declines (86.55%–87.67%) while MMRR steadily decreases to 28.36–21.01, indicating increased rigidity, and slower adaptation to 27

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

recent tasks. Finally, a large memory window (N = 100,000) leads to further drops in accuracy, at about 83.70% and OOD AUC (0.94), with minimal MMRR (13.13), consistent with synaptic degeneracy. Overall, Table 13 confirms that N acts as a direct control mechanism over forgetting and rigidity: small N causes rapid forgetting, very large N induces rigidity and synaptic degeneracy, and intermediate N preserves a favorable stabilityplasticity trade-off while maintaining informative epistemic uncertainty. Contrary to MESU (Bonnet et al., 2025) that suggests using N as the number of batches, here the exponential dependency prevents from asserting a similar statement. Table 13. Effect of the memory window N of BiMU, and of BayesBiNN on forgetting, rigidity, and uncertainty in 100-tasks PermutedMNIST (OCL; 100-hidden-unit MLP). MEAN ACC. (last 5) is averaged over the final five tasks. OOD AUC is ROC-AUC for Permuted MNIST vs Fashion-MNIST. MMRR: Maximum Memory Rigidity Resilience (see main text). Mean±std over 5 runs.

M EMORY WINDOW N

MEAN ACC. 5 TASKS (%)

OOD DET. (AUC)

MMRR

100 400 700 1000 1300 1600 1900 100000

27.09 ± 0.57 73.46 ± 2.59 90.29 ± 0.24 89.07 ± 0.28 87.67 ± 0.33 86.94 ± 0.29 86.55 ± 0.24 83.70 ± 0.30

0.99 ± 0.00 1.00 ± 0.00 0.99 ± 0.01 0.98 ± 0.01 0.97 ± 0.02 0.96 ± 0.02 0.96 ± 0.02 0.94 ± 0.05

216.45 574.72 215.52 46.30 28.36 24.91 21.01 13.13

BAYES B I NN

67.91 ± 0.96

0.75 ± 0.20

5.03

We further illustrate the effect of the memory window N on synaptic weight distributions at the end of training. Figure 7 shows the histograms of the probability of each synapse for all values of N . Small N results in broad, low-confidence distributions, reflecting rapid forgetting without stability. Intermediate N produces well-formed, moderately peaked distributions, indicating a favorable balance between plasticity and stability. Very large N leads to sharply peaked distributions near 0 or 1, highlighting distribution degeneracy and loss of plasticity, a behaviour also observed on BayesBiNN due to the absence of forgetting for the algorithm.

0.4

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Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

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Figure 7. Histograms of the synaptic probabilities p = sigmoid(2λ) at the end of training for different memory windows N and BayesBiNN in 100-tasks Permuted-MNIST (OCL; 100-hidden-unit MLP). Small N shows broad distributions indicative of forgettingdominated dynamics, intermediate N yields moderate peaks reflecting a balanced stability-plasticity-forgetting trade-off, and very large N exhibits sharp peaks near 0 and 1, consistent with synaptic degeneracy and rigidity, highlighting too high stability, akin to BayesBiNN’s behaviour.

H. 100-tasks Permuted MNIST: Activation function ablation study Binary Neural Networks (BiNNs) traditionally rely on the sign activation function (Hubara et al., 2016; Laborieux et al., 2021), together with a surrogate hardtanh during backpropagation to overcome the non-differentiability of the binary non-linearity:     if x > 0, if x > 1, 1 1 sign(x) = −1 if x < 0, hardtanh(x) = x (65) if − 1 ≤ x ≤ 1,     0 otherwise, −1 if x < −1. In this setting, neurons are activated according to the sign of their pre-activations. Combined with instance normalization (Ulyanov et al., 2016), gradients are only propagated through neurons whose pre-activations lie in the narrow neutral region [−1, 1]. Consequently, learning is dominated by the most frequent pre-activation values, which can be suboptimal in continual learning scenarios where selective and stable parameter updates are required. Reverse Binary Gate. We introduce the Reverse Binary Gate (RBG) activation function and its surrogate sRBG. Unlike the sign function, RBG selects neurons based on the magnitude of their pre-activations rather than their sign:

( 0 if |x| < a2 , RBG(x, a) = 1 otherwise,

 a −x if − 3a  2 < x < −2,   x if a2 < x < 3a 2 , sRBG(x, a) = a  0 if |x| < 2 ,    1 otherwise.

(66)

This design inverts the usual gating logic: neurons with the most frequent activations are suppressed, while only the moderate to low activations contribute to the forward and backward passes. As a result, a majority of synapses are explicitly 29

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

deactivated, promoting sparse and selective updates. The RBG function can be interpreted as a reversed version of the Elephant activation (Lan & Mahmood, 2024) in the limit case. Disentangling activation functions from continual learning. Table 14 reports an ablation study on 100-tasks Permuted MNIST that isolates the effect of the activation function from the continual learning mechanism. By comparing sign-based and RBG-based variants of the same algorithms, we disentangle the respective contributions of activation design and learning rules for the Permuted MNIST experiment. Table 14. Disentangling the effect of the activation function on continual learning performance (100-tasks Permuted MNIST). Mean accuracy is computed over the last five tasks; OOD detection is ROC-AUC for Permuted MNIST vs Fashion-MNIST. Maximum Resilience corresponds to the Maximum Memory Rigidity Resilience (MMRR). Mean±std over 5 runs. MEAN ACC. OOD DET. ACC . MMRR 5 TASKS (%) (AUC) 1 TASK (%)

M ETHOD

ACTIVATION

BAYES B I NN BAYES B I NN

S IGN RBG

66.40 ± 0.97 0.54 ± 0.19 67.41 ± 1.03 0.76 ± 0.17

5.85 4.99

90.27 ± 0.32 93.22 ± 0.09

B I MU B I MU

S IGN RBG

81.78 ± 0.58 0.76 ± 0.08 90.29 ± 0.24 0.99 ± 0.01

22.25 215.52

92.89 ± 0.22 94.67 ± 0.11

MESU MESU

R E LU RBG

93.51 ± 0.18 0.91 ± 0.03 92.35 ± 0.21 0.80 ± 0.05

500.02 943.44

96.10 ± 0.18 93.01 ± 0.20

S YN . M ETA . S YN . M ETA .

S IGN RBG

10.22 ± 0.05 11.35 ± 0.00

9.92 26.29

71.40 ± 1.48 13.20 ± 1.38

To isolate the impact of the activation function from long-horizon continual learning effects, we provide an ablation study conducted on 100-tasks Permuted MNIST. This reduced setting compared to the main enables a clearer separation between optimization effects induced by the activation function and degradation caused by prolonged memory saturation. The results in Table 14 reveal heterogeneous sensitivities to the choice of activation across methods. For BayesBiNN, replacing the sign activation with the Reverse Binary Gate yields a small but consistent improvement in continual performance: mean accuracy over the last five tasks increases from 66.40% to 67.41%, while OOD detection improves markedly from 0.54 to 0.76 ROC-AUC. One-task accuracy also increases from 90.27% to 93.22%, indicating that RBG provides a stronger single-task accuracy. However, memory resilience slightly decreases (5.85 to 4.99), suggesting that while magnitude-based gating improves representation quality, it does not fundamentally resolve the rigidity and plasticity limitations of BayesBiNN. In contrast, BiMU benefits substantially from the Reverse Binary Gate. Switching from sign to RBG increases the mean accuracy from 81.78% to 90.29%, improves OOD detection from 0.76 to 0.99 ROC-AUC, and dramatically boosts MMRR from 22.25 to 215.52. Importantly, one-task accuracy also improves from 92.89% to 94.67%, confirming that the gains are not solely due to reduced forgetting but to which neurons are going to be activated during the forward and backward passes. These results indicate that selecting neurons based on activation magnitude rather than sign leads to more stable Bayesian updates and mitigates interference between tasks. For MESU, the activation change does not provide a clear advantage. While RBG substantially increases memory rigidity (500.02 to 943.44), it degrades OOD detection (0.91 to 0.80 ROC-AUC), slightly reduces mean continual accuracy (93.51% to 92.35%), and lowers one-task accuracy from 96.10% to 93.01%. This indicates that the Reverse Binary Gate increases rigidity in real-valued networks, harming both generalization and predictive confidence. As a result, RBG does not replace the role of structured Bayesian updates in MESU. Finally, for Synaptic Metaplasticity, one-task accuracy collapses from 71.40% to 13.20%. This highlights a fundamental incompatibility between the method and magnitude-based gating, exacerbating known sensitivities to network size, batching, and training regime. Overall, this ablation demonstrates that selecting synapses based on activation magnitude rather than sign can substantially improve both single-task performance and continual learning robustness in Bayesian binary neural networks, particularly for BiMU. The effect is strongly method-dependent: it is moderate for BayesBiNN, ineffective for MESU and Synaptic Metaplasticity. While it improves performance, it does not prevent memory rigidity nor does it work better than a ReLU for real-valued networks. 30

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

I. Animals Imbalanced Active Learning: Complementary Figures Complementary to the main text, we provide additional figures illustrating the behavior of each acquisition strategy on the Animals benchmark under imbalanced active learning. These figures correspond to the aggregated results reported in Fig. 4 and are presented here to facilitate a more detailed, per-method analysis. Certain strategies are not appearing on figures due to the threshold not being activated, very small measures of uncertainty or points overlapping. Fig. 8 reports the performance of BayesBiNN on the Animals dataset. All acquisition strategies perform below random sampling, indicating that the uncertainty estimates are not sufficiently informative to support effective active learning. Unlike settings with a very large number of tasks such as Permuted MNIST, where weight divergence naturally emerges, the Animals dataset contains relatively few training examples per task (10,549 images in total, corresponding to roughly one sixth of a MNIST task). As a result, the model’s weights do not have sufficient opportunity to diverge across tasks. Moreover, in the absence of a metaplastic learning rate mechanism, BayesBiNN is unable to learn efficiently enough from these limited examples to consolidate its weights. This prevents the model from reliably distinguishing between low-frequency and high-frequency samples. While this leads to high overall uncertainty and consequently strong out-of-distribution detection, the quality of the uncertainty estimates remains poor, as predictions are primarily influenced by the most recently observed samples rather than by the dataset as a whole, in contrast to methods incorporating metaplasticity such as BiMU.

100

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25 0 100

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Figure 8. Animals dataset under imbalanced active learning using BayesBiNN. Classification accuracy as a function of the fraction of queried samples, highlighting the behavior of Bayesian binary weights under uncertainty-driven querying.

Fig. 9 and Fig. 10 show the results for MESU, without and with input standardization, respectively. Without standardization, all querying strategies other than VR and VR-True collapse early in training, reflecting the inability to learn with low amount of data and poorly conditioned features. While both VR variants benefit from uncertainty-driven sampling, standardization markedly improves performance, yielding lower data used for training and more balanced exploration of minority classes, consistently outperforming random acquisition. Among all criteria, epistemic uncertainty achieves the strongest trade-off between data and accuracy, reaching the random-query baseline with the smallest fraction of labeled samples. This confirms that when uncertainty estimates are well calibrated, they enable highly selective supervision, and further highlights the sensitivity of real-valued Bayesian methods such as MESU to input preprocessing.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

100

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Data used for training (%) Figure 10. Animals dataset under imbalanced active learning using MESU with feature standardisation. Standardisation improves stability and performance across labeling budgets.

Finally, Fig. 11 illustrates the behavior of STE. STE achieves competitive early performance, its acquisition strategy exhibits higher sensitivity to class imbalance at low percentage of labeled data, resulting in consistent drops in accuracy when threshold are increased, leading to a poor accuracy/labeled data ratio compared to BiMU and MESU.

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Data used for training (%) Figure 11. Animals dataset under imbalanced active learning using STE. Accuracy as a function of the queried-label fraction when acquisition is driven by aleatoric uncertainty in binary weights setup.

Overall, these complementary figures highlight qualitative differences in acquisition dynamics that are not fully captured by summary metrics alone, especially for BayesBiNN, which, while having reliable OOD detection estimates, does not capture 32

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

as well what examples are important and which are not.

J. OpenLORIS-Object: Effect of standardization on the output features of VGG19 The goal of this appendix experiment is to quantify how much of the performance gap observed in the main results between Bayesian binary algorithms and real-valued algorithms is attributable to input preprocessing rather than to the learning algorithms themselves. To this end, we repeat the OpenLORIS-Object experiment using standardized 8,192-dimensional VGG19 features, where each feature is normalized using the mean and variance estimated from the training data. This contrasts with the streaming setting of Table 2, where no such normalization is applied. Table 15. OpenLORIS-Object (OCL; linear head) with standardized 8,192-dimensional VGG19 features. Mean accuracy is averaged over the 12 tasks after training on all tasks. OOD AUC is computed using the held-out toy class. Results are mean±std over 5 runs.

MEAN ACC. A LEATORIC E PISTEMIC (%) (AUC) (AUC)

M ETHOD

Binary neural networks B I MU BAYES B I NN STE S YNAPTIC M ETAPLASTICITY

90.98 ± 0.38 90.20 ± 0.13 77.69 ± 2.29 79.28 ± 0.85

1.00 ± 0.00 1.00 ± 0.00 – –

Real-valued neural networks MESU EWC O NLINE S YNAPTIC I NTELLIGENCE SGD

92.55 ± 0.24 92.34 ± 0.32 92.23 ± 0.30 92.31 ± 0.31

0.99 ± 0.00 1.00 ± 0.00 1.00 ± 0.00 1.00 ± 0.00

0.99 ± 0.00 – – –

Standardization consistently improves performance across all methods, but the gains are markedly larger for real-valued neural networks. In particular, MESU improves from 87.01% to 92.55% mean accuracy, while EWC Online and Synaptic Intelligence increase from 87.18% and 86.44% to 92.34% and 92.23%, respectively. These substantial gains indicate that real-valued continual learning methods strongly benefit from well-conditioned inputs and rely on stable feature statistics induced by mean–variance normalization. Their optimization dynamics and regularization mechanisms are therefore closely tied to offline preprocessing assumptions. Binary Bayesian methods also benefit from standardization, but to a lesser extent. BiMU improves from 89.19% to 90.98% accuracy, and BayesBiNN from 86.93% to 90.20%, while maintaining near-perfect epistemic uncertainty separation (ROC-AUC ≈ 1.00). The more modest performance increase suggests that binary models are comparatively less sensitive to feature scaling. When input statistics are known in advance and preprocessing based on the full training distribution is feasible, real-valued methods achieve higher final accuracy than binary approaches. However, the main results focus on realistic continual learning scenarios for edge and robotic systems, where data arrive as an unbounded stream and reliable estimation of global feature statistics is not possible online. In this regime, the robustness of binary Bayesian methods – and in particular BiMU – to unnormalized inputs and shifting distributions provides a practical advantage, despite their more constrained parameterization. Hyperparameter optimization. Hyperparameters are computed on the validation set of OpenLORIS-Object. They are computed through the custom hyperparameter tuning cost function H (see Appendix B.2).

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks Table 16. Hyperparameter configurations for all evaluated methods on the OpenLORIS dataset (standardized, 8192 features). M ETHOD

L EARNING R ATE ( S )

A DDITIONAL PARAMETERS

MC SAMPLES

B I MU

1.2

αMAX = 0.07, βL = 184, βKL = 0.14, N = 7,900

BAYES B I NN

1.59

P RIOR STRENGTH = 1.2 × 10−6

S YNAPTIC M ETAPLASTICITY

4.5 × 10−5

M ETAPLASTICITY = 13.7, WEIGHT DECAY = 8 × 10−12

STE (BASELINE )

1 × 10−4

–

Binary Neural Networks

Real-Valued Neural Networks αµ = 4.0, ασ = 5.0

σPRIOR = 0.76, N = 5,000,000, CLAMP GRAD = 0.53

O NLINE EWC

4 × 10−4

I MPORTANCE = 0.124, DOWNWEIGHTING = 0.7

S YNAPTIC I NTELLIGENCE

4 × 10−4

C OEFF . = 4.5 × 10−5 , DAMPING = 0.05

SGD (BASELINE )

3 × 10−4

–

MESU

K. OpenLORIS-Object Imbalanced Continual Active Learning: Complementary Figures This section provides additional results on the OpenLORIS-Object benchmark, focusing on the imbalanced continual active learning setting. The figures presented here complement the main results and allow for a detailed inspection of each method’s behavior over time and across acquisition steps. Fig. 12 presents the results for BayesBiNN on the OpenLORIS dataset in imbalanced continual active learning setting. As in the Animals setting, all active learning strategies perform below random acquisition, highlighting the limited usefulness of the uncertainty estimates produced by the model. Although OpenLORIS contains a larger total number of training examples (442,194 images), this corresponds to only approximately seven Permuted MNIST–scale tasks. This short task horizon limits the degree of weight divergence that emerge during training. In the absence of a metaplastic learning rate mechanism, BayesBiNN is therefore unable to leverage these examples efficiently enough to consolidate task-relevant weights. As a consequence, the model struggles to differentiate between low-frequency and high-frequency samples. While this results in high overall uncertainty and yields strong out-of-distribution detection, the resulting uncertainty remains poorly structured, as predictions are dominated by the most recently observed samples rather than reflecting statistics accumulated over the full dataset, unlike approaches that explicitly incorporate metaplasticity such as BiMU.

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Data used for training (%) Figure 12. OpenLORIS-Object (OCL; linear head) with 8,192-dimensional VGG19 features under imbalanced continual active learning using BayesBiNN. Evolution of classification accuracy across labeling budgets in the presence of distribution shifts for Bayesian binary weights without forgetting.

Fig. 13 and Fig. 14 report MESU’s performance with and without input standardization. In contrast to the Animals benchmark, performance degradation without standardization is less severe, owing to the substantially larger number of samples available to accumulate information about the importance of each parameter, leading to more reliable uncertainty estimates. Nevertheless, standardization consistently improves robustness and reduces variability across acquisition steps. In particular, standardization yields an accuracy gain of roughly 10 percentage points at the 3% labeling budget for VR. Under standardized features, aleatoric, epistemic, and predictive uncertainty criteria exhibit similar behavior, indicating reliable 34

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

discrimination of informative samples throughout the stream leading to strong efficiency of uncertainty estimates to select which data to integrate to increase accuracy. Low frequency

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Fig. 15 shows the results obtained with STE. STE’s performance degrades more noticeably when confronted with successive task shifts, yielding poor aleatoric measurements that are unable to capture efficient examples to learn on.

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Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

These complementary figures further support the main conclusions of the paper: that reliable epistemic uncertainty is critical for effective active continual learning under class imbalance, that its quality is strongly influenced by input conditioning in real-valued Bayesian methods, and that preserving uncertainty over long, non-stationary streams enables substantial reductions in labeling and update costs without sacrificing accuracy.

L. OpenLORIS-Object Imbalanced Continual Active Learning: Number of predictors for variation ratio In this appendix, we investigate the performance of variation ratio when limiting or increasing the number of predictors associated with the disagreement in the Online Continual Learning setting. Even when using few samples, variation ratio achieves strong predictive performance. For instance, using only 2 samples with 3.3% of the data achieves 89.3% ± 0.88 accuracy, while 3 samples with 3.87% of the data reach 90.61% ± 0.53. Low-sample regimes already provide high accuracy even without a large range of thresholds, denoting the ability of BiMU to distinguish well useful samples from non-informative ones. Increasing the number of samples generally improves accuracy and allows finer control via the threshold parameter. For example, with 10 samples and 3.97% of the data, accuracy reaches 90.91% ± 0.98, and with 25 samples using 4.81–5.63% of the data, accuracy increases up to 91.74% ± 0.23. These higher-sample regimes provide additional thresholds for flexible performance trade-offs, but require more forward passes to estimate disagreement, leading to more computational overhead. High number of predictors also induce a higher variance at low data usage due to the difference of training examples selected through seeds.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Table 17. Ablation study of the number of samples used to compute the variation ratio uncertainty measure on OpenLORIS-Object (OCL; linear head) with 8,192-dimensional VGG19 features. Rows with identical samples, accuracy, and data usage are merged, and the thresholds column lists all thresholds for that configuration. Low sample regimes achieve high accuracy under strong compression. Additional samples yield better accuracy and offer more thresholds at the cost of more forward passes. Data used now shows mean ± standard deviation across final updates. Results are averaged over five runs.

S AMPLES ACCURACY (%) DATA USED (%) T HRESHOLD 2

89.30 ± 0.88

3.30 ± 0.04

0.50

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3.87 ± 0.05 1.62 ± 0.03

0.33 0.66

5 5 5

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2.91 ± 0.16 1.47 ± 0.16 0.34 ± 0.19

0.20 0.40 – 0.60 0.80

10 10 10 10 10 10

90.91 ± 0.98 88.70 ± 1.94 79.27 ± 4.00 44.02 ± 32.02 9.37 ± 8.22 5.26 ± 0.00

3.97 ± 0.03 3.10 ± 0.31 1.37 ± 0.24 0.42 ± 0.39 0.02 ± 0.04 0.00 ± 0.00

0.10 0.20 – 0.30 0.40 – 0.50 0.60 0.70 0.80 – 0.90

25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25

91.34 ± 0.50 91.11 ± 0.44 90.93 ± 0.70 90.23 ± 0.68 89.00 ± 0.64 87.16 ± 1.88 86.40 ± 2.68 79.14 ± 13.18 66.41 ± 30.90 51.53 ± 37.93 34.82 ± 36.25 19.95 ± 29.37 19.16 ± 27.79 18.48 ± 26.44 17.01 ± 23.49 16.07 ± 21.61 5.26 ± 0.00

5.63 ± 0.09 4.40 ± 0.04 4.03 ± 0.05 3.65 ± 0.14 3.27 ± 0.23 2.93 ± 0.36 2.61 ± 0.44 1.96 ± 0.78 1.48 ± 0.84 1.03 ± 0.88 0.52 ± 0.65 0.23 ± 0.45 0.17 ± 0.34 0.13 ± 0.26 0.10 ± 0.19 0.07 ± 0.14 0.00 ± 0.00

0.04 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 > 0.72

BASELINE

87.76 ± 0.19

100

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

M. OpenLORIS-Object Imbalanced Continual Active Learning: Querying dynamics and temporal distribution of active updates In this appendix, we further analyze when active updates occur during OpenLORIS-Object. We want to assert whether variation ratio querying reacts to distributional changes or if it selects samples uniformly throughout each task. For each task, we divide the online stream into 100 parts, with three temporal regions: the first quarter (0–24), the middle part of the task (25–74), and the last quarter (74–99). As labels are queried only when the variation ratio score exceeds the fixed threshold, the number of queried samples is also the number of backpropagation updates. Table 18 reports the accuracy at several checkpoints within each task, together with the percentage of updates occurring in each temporal region. The main trend is consistent across tasks: most updates occur shortly after the distribution shift. On average, 45.85% of all updates are performed in the first quarter of a task, while only 12.56% occur in the last quarter. Querying is concentrated when the stream changes then progressively decreases as the model adapts to the current factor. Importantly, the update rate does not collapse to zero at the end of the task. The last quarter still accounts for 12.56% of the updates on average, indicating that the posterior remains sufficiently uncertain to select informative samples after the initial adaptation phase. The model reacts strongly after a shift and does not become completely rigid once the current task has been partially learned. Table 18. Temporal distribution of variation ratio updates on OpenLORIS-Object (OCL; linear head) with 8,192-dimensional VGG19 features. Accuracy is reported at fixed checkpoints within each task. Update percentages indicate the fraction of updates occurring in the first quarter, middle part, and last quarter of the task. Most updates occur immediately after the task shift, while a smaller but non-zero fraction remains available later in the task. ( A ) ACCURACY CHECKPOINTS TASK

ACC @0 (%)

ACC @25 (%)

ACC @50 (%)

ACC @75 (%)

ACC @99 (%)

TASK 1 TASK 2 TASK 3 TASK 4 TASK 5 TASK 6 TASK 7 TASK 8 TASK 9 TASK 10 TASK 11 TASK 12

5.26 ± 0.00 5.26 ± 0.00 8.73 ± 6.93 8.90 ± 7.28 12.68 ± 9.56 11.45 ± 10.22 18.60 ± 17.69 18.25 ± 16.71 12.32 ± 8.70 19.78 ± 17.86 21.32 ± 19.67 14.87 ± 11.93

86.40 ± 12.42 92.72 ± 3.96 94.21 ± 1.19 75.53 ± 3.06 55.18 ± 2.04 37.85 ± 3.13 66.10 ± 4.36 68.33 ± 2.48 39.61 ± 2.42 68.82 ± 3.38 64.61 ± 5.02 42.11 ± 2.14

90.53 ± 6.55 92.50 ± 4.16 89.82 ± 0.58 94.61 ± 2.59 95.44 ± 0.93 93.51 ± 1.74 80.88 ± 3.11 80.44 ± 2.69 53.25 ± 2.05 70.79 ± 2.00 72.06 ± 2.27 41.58 ± 2.92

92.54 ± 4.96 93.03 ± 2.93 89.47 ± 1.68 95.96 ± 1.19 94.65 ± 0.97 91.54 ± 2.68 91.89 ± 0.31 96.45 ± 0.95 91.54 ± 1.36 83.25 ± 1.57 76.18 ± 1.36 42.37 ± 1.25

90.26 ± 4.30 91.01 ± 2.79 88.11 ± 1.78 91.93 ± 1.32 90.00 ± 1.28 81.14 ± 1.21 90.31 ± 0.78 93.60 ± 0.97 80.96 ± 1.06 89.78 ± 1.40 92.37 ± 1.08 94.25 ± 2.00

AVERAGE

13.12 ± 5.29

65.96 ± 18.73

79.62 ± 16.68

86.57 ± 14.40

89.48 ± 4.10

( B ) T EMPORAL DISTRIBUTION OF UPDATES TASK

U PDATES 0–24 (%)

U PDATES 25–73 (%)

U PDATES 74–99 (%)

TASK 1 TASK 2 TASK 3 TASK 4 TASK 5 TASK 6 TASK 7 TASK 8 TASK 9 TASK 10 TASK 11 TASK 12

25.43 ± 31.15 41.00 ± 20.55 53.47 ± 1.72 58.65 ± 1.75 61.35 ± 1.65 56.65 ± 3.22 42.37 ± 2.68 41.28 ± 4.12 42.83 ± 2.84 40.73 ± 1.84 39.53 ± 2.58 46.93 ± 2.25

40.19 ± 29.00 41.82 ± 11.01 33.85 ± 1.05 29.72 ± 1.47 29.07 ± 1.63 32.93 ± 2.51 40.33 ± 2.30 42.43 ± 3.22 40.16 ± 1.31 43.60 ± 1.14 42.64 ± 1.02 39.01 ± 1.72

13.08 ± 12.57 15.66 ± 9.49 10.95 ± 1.35 9.81 ± 1.37 7.67 ± 1.26 8.26 ± 1.02 14.96 ± 0.79 14.60 ± 1.55 14.76 ± 2.30 13.65 ± 2.15 15.13 ± 2.02 12.19 ± 1.45

AVERAGE

45.85 ± 9.70

37.98 ± 4.95

12.56 ± 2.66

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

N. Budget-driven adaptive thresholding for variation ratio In the main experiments, active continual learning uses a fixed one-pass threshold: an incoming example is queried if its uncertainty score exceeds τ . While a fixed threshold is simple and hardware-friendly, deployment may require tighter control over the realized query rate as we do not know in advance the budget of updates that variation ratio will use. In this appendix, we propose a fully online budget-driven controller that adapts τ from the discrepancy between the current query rate and a target budget, set in advance as a hyperparameter. Let B ∈ [0, 1] denote the target fraction of samples to query, and let Qrate (t) be the fraction of samples queried up to time t. The signed budget error is ϵt = Qrate (t) − B. (67) A positive value indicates that the current stream has exceeded the target budget, whereas a negative value indicates that the controller is below budget. We compute a continuous threshold γ

τcont (t) = B + sign(ϵt ) |ϵt | ,

(68)

where γ > 0 controls the adaptation speed. When Qrate (t) > B, the threshold is increased, making the controller more selective. Conversely, when Qrate (t) < B, the threshold is decreased, making the controller more permissive. For variation ratio querying, the uncertainty score is discrete due to being computed from a finite number of Monte Carlo predictors. With K posterior samples, variation ratio can only take values in a finite set. We quantize the controller output onto the same set: nt = round(Kτcont (t)) ,

(69)

nt = min(max(nt , 0), K) , nt τt = . K

(70) (71)

The controller is memory-free; it only requires the current cumulative query rate, the target budget, and the finite-K quantization level. We evaluate this controller on OpenLORIS-Object under the same active continual learning setting as in Sec. 4.5: sequential factor analysis, frozen VGG19 features, an online linear head, 8,192 randomly selected features, class imbalance, and one-pass training without replay. We use variation ratio querying with K = 10 posterior samples. The queried fraction is reported as the percentage of samples that trigger both a label request and a parameter update. Adaptive thresholding can recover the performance of the best fixed-threshold variation ratio while explicitly targeting an update budget. With a 3% target budget and γ = 0.50, the controller reaches 90.79 ± 0.61% accuracy while querying 3.96 ± 0.03% of the stream. This is close to the best fixed-threshold variation ratio result, which reaches 90.91% accuracy at 3.97% queried data. Budget-driven adaptive thresholding is feasible in the fully online setting. However, the controller is sensitive to the operating regime. At very low budgets, performance degrades sharply for small γ. For example, at B = 0.5% and γ = 0.01, the controller queries exactly 0.50% of the stream but reaches only 52.68 ± 2.29% accuracy. Here, the budget is too limited and variation ratio stays at its minimal threshold and avoids taking updates. Hence, matching the target budget does not allow to preserve enough queried samples to learn the task correctly. To sum up, we proposed an algorithm enabling online adaptation of the querying threshold with a constraint parameter setting how tight the budget should be kept and demonstrated results on par to static thresholding.

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

Table 19. Budget-driven adaptive variation ratio thresholding on OpenLORIS-Object (OCL; linear head) with 8,192-dimensional VGG19 features. The target budget B is the desired queried fraction. The exponent γ controls how aggressively the threshold reacts to the signed budget error. Queried samples are also the only samples used for backpropagation updates. Results are averaged over five runs. M ETHOD

C ONTROLLER SETTING

DATA USED (%)

ACCURACY (%)

R ANDOM QUERYING

F ULL STREAM

100

87.76 ± 0.19

B UDGETED VR B UDGETED VR B UDGETED VR

B = 0.5%, γ = 0.01 B = 0.5%, γ = 0.10 B = 0.5%, γ = 0.50

0.50 ± 0.00 1.11 ± 0.03 3.34 ± 0.05

52.68 ± 2.29 79.84 ± 0.65 89.62 ± 0.43

B UDGETED VR B UDGETED VR B UDGETED VR

B = 1.0%, γ = 0.01 B = 1.0%, γ = 0.10 B = 1.0%, γ = 0.50

1.00 ± 0.00 1.22 ± 0.01 3.34 ± 0.05

61.95 ± 1.95 79.53 ± 1.40 89.88 ± 0.57

B UDGETED VR B UDGETED VR B UDGETED VR

B = 2.0%, γ = 0.01 B = 2.0%, γ = 0.10 B = 2.0%, γ = 0.50

2.00 ± 0.01 2.03 ± 0.01 3.70 ± 0.02

70.20 ± 1.72 85.64 ± 1.21 89.97 ± 0.74

B UDGETED VR B UDGETED VR B UDGETED VR

B = 3.0%, γ = 0.01 B = 3.0%, γ = 0.10 B = 3.0%, γ = 0.50

3.00 ± 0.01 3.00 ± 0.01 3.96 ± 0.03

73.90 ± 0.60 86.79 ± 0.71 90.79 ± 0.61

B UDGETED VR B UDGETED VR B UDGETED VR

B = 4.0%, γ = 0.01 B = 4.0%, γ = 0.10 B = 4.0%, γ = 0.50

4.00 ± 0.01 4.00 ± 0.01 4.23 ± 0.03

76.97 ± 0.76 87.94 ± 0.30 90.56 ± 0.30

B UDGETED VR B UDGETED VR B UDGETED VR

B = 5.0%, γ = 0.01 B = 5.0%, γ = 0.10 B = 5.0%, γ = 0.50

5.00 ± 0.01 5.00 ± 0.01 5.01 ± 0.01

78.34 ± 0.60 87.66 ± 0.85 90.87 ± 0.51

O. Algorithm: Binary Metaplasticity from Uncertainty Algorithm 1 Binary Metaplasticity from Uncertainty Input: M Batches {Dt }M t=1 , Monte Carlo samples K, number of synapses s, temperature T , prior parameter λprior , memory window N , maximum learning rate αmax for t = 1 to M do for k = 1 to K do (1) (s) i.i.d. Sample ϵk , . . . , ϵk ∼ U(0, 1) δk = 12 (log(ϵk ) − log(1 − ϵk )) Compute the Gumbel-softmax trick ωk = tanh T1 (λt−1 + δk ) Compute loss L(ωk , Dt ) k ,Dt ) Compute gradient of the loss function ∂L(ω ∂ωk Compute gradient over λt−1 ∂L(ωk , Dt ) ∂L(ωk , Dt ) 1 − ωk2 ← ⊙ ∂λt−1 ∂ωk T end for Estimate expected gradients: K ∂L(λ, Dt ) 1 X ∂L(ωk , Dt ) ← ∂λ K ∂λt−1 λ=λt−1 k=1 Compute adaptative learning rate: 1 η(λt−1 ) = 1 1 t) t) + 2 tanh(λt−1 ) ∂L(λ,D + αmax + 2 ∂L(λ,D ∂λ ∂λ cosh2 (λt−1 ) λ=λt−1

λ=λt−1

Update parameters: λt = λt−1 − η(λt−1 ) ⊙

(λt−1 − λprior ) ∂L(λ, Dt ) + ∂λ λ=λt−1 N · cosh2 (λt−1 )

end for 40

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

P. Computational Cost Evaluation The proposed active learning strategy presented in this paper relies on the fact that in binary neural networks, inference is substantially cheaper than gradient-based updates. BiMU exploits this asymmetrical cost by using repeated low-cost Bayesian inference passes to estimate uncertainty, while triggering expensive backpropagation updates only for informative samples. In this appendix, we ask whether the overhead induced by Monte Carlo Bayesian sampling remains smaller than the compute saved by avoiding most training updates. We evaluate this trade-off directly on an embedded device. We measure the inference and training costs of BiMU, BayesBiNN, STE, and Synaptic Metaplasticity on a microcontroller unit (MCU) implementation, and analyze the effective compute cost of active continual learning under realistic querying rates. More specifically, we address two claims: (i) binary inference is substantially cheaper than online training updates, even in Bayesian settings requiring multiple stochastic forward passes; (ii) BiMU reduces overall compute despite the overhead of Monte Carlo samples to estimate uncertainty. We implemented all methods on an STMicroelectronics STM32 NUCLEO-64 MCU board running at 216 MHz. Forward passes were implemented using 32-bit signed integers, while backward passes used 32-bit floating-point operations due to gradient estimation requiring non-integer values. The measurements in Table 20 were obtained on a reference layer with Nin = 512 and Nout = 16. To estimate the runtime of the OpenLORIS experiments, we project these measurements to the architecture used in the main benchmark, with Nin = 8192 and Nout = 19. This corresponds to a 19× increase in the number of layer operations relative to the measured reference layer. As all compared methods use the same classifier structure and the same MCU implementation constraints, this projection preserves the relative compute ratios between inference and training while reflecting the substantially larger dimensionality of the OpenLORIS feature space. We note that binary inference benefits from sub-byte operations and popcount instructions, whereas training remains dominated by floating-point computations and memory writes. Consequently, the gap measured here between inference and training cost in this appendix is likely conservative with respect to dedicated binary hardware accelerators. Table 20. NUCLEO-64 benchmark results (216 MHz). Latency and CPU cycles correspond to the mean ± standard deviation over 400 iterations for a layer with Nin = 512 and Nout = 16.

I NFERENCE

T RAINING

A LGORITHM

T IME (µs)

C YCLES

T IME (µs)

C YCLES

B I MU BAYES B I NN STE S YN . M ETA .

717.93 ± 0.37 718.01 ± 0.37 310.85 ± 0.83 310.91 ± 0.81

155,170 155,195 67,233 67,243

13991.27 ± 3.72 13117.61 ± 5.27 3643.02 ± 2.69 7723.78 ± 3.82

3,022,211 2,833,522 786,999 1,668,432

Table 20 supports claim (i) by showing that, for BiMU specifically, the forward pass requires only Cinference = 0.72 ms, whereas the backward update requires Ctrain = 14.0 ms, making inference approximately 19.4× cheaper than training. Similar trends are observed across all binary methods, confirming that the dominant computational cost arises from parameter updates rather than inference itself. Table 21. Projected compute cost on the OpenLORIS-8192 architecture (Nin = 8192, Nout = 19).

A LGORITHM

I NFERENCE (ms)

T RAINING (ms)

E XPECTED COMPUTE PER DATA POINT (ms)

ACCURACY (%)

B I MU (ACTIVE L EARNING )

13.7

266.0

45.0

89.30

B I MU BAYES B I NN STE S YN . M ETA .

13.7 13.6 5.9 5.9

266.0 249.2 62.2 146.7

559.4 525.6 62.2 146.7

90.98 84.18 77.69 79.28

Active Continual Learning with Metaplastic Binary Bayesian Neural Networks

We next evaluate whether the reduction in update frequency enabled by active learning compensates for the additional Monte Carlo forward passes required for uncertainty estimation. We consider the OpenLORIS-8192 setting used in the main text. Table 17 shows that BiMU reaches 89.30% accuracy using only K = 2 Monte Carlo samples while querying and updating on only Brate = 3.3% of the stream. We also consider K = 2 Monte Carlo samples for the backward pass of BiMU and BayesBiNN. Under this setting, the expected compute cost per incoming data point for BiMU with active learning is CAL = K × (Cinference + Brate Ctrain ) ,

(72)

where K is the number of Monte Carlo predictors and Brate the fraction of queried samples triggering backpropagation. By contrast, BiMU and BayesBiNN full online training performs one inference and one update for every Monte Carlo predictor: CB = K × (Cinference + Ctrain ).

(73)

Finally, STE and Synaptic Metaplasticity only require their backward cost. Table 21 supports claim (ii). Despite performing multiple stochastic forward passes per data point, BiMU with active learning reduces the expected compute cost to 45.0 ms, compared to 559.4 ms for full BiMU training and 525.6 ms for BayesBiNN. This corresponds to approximately a 12.4× reduction relative to full BiMU training while maintaining comparable accuracy (89.30% versus 90.98%). STE remains computationally very cheap because it does not provide uncertainty estimates: this comes at a substantial accuracy degradation in accuracy (77.69%). Overall, the results show that the low cost of binary inference is sufficient to amortize the overhead of Bayesian uncertainty estimation when active querying significantly reduces the number of gradient updates, supporting claims (i) and (ii).

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