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LightTune: Lightweight Forward-Only Online Fine-Tuning with Applications to Link Adaptation

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arXiv CS · Papers · License: Open Access · 2026

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networking, internet, protocols, distributed systems

LightTune: Lightweight Forward-Only Online Fine-Tuning with Applications to Link Adaptation

arXiv:2604.12406v1 [cs.NI] 14 Apr 2026

Ramy E. Ali, Member, IEEE, and Federico Penna, Member, IEEE

Abstract—Deploying machine learning (ML) algorithms on mobile phones is bottlenecked by performance degradation under dynamic, real-world conditions that differ from the ofﬂine training conditions. While continual learning and adaptation are essential to mitigate this distributional shift, conventional online learning methods are often computationally prohibitive for resource-constrained devices. In this paper, we propose LightTune, a lightweight, backpropagation-free online ﬁne-tuning framework with provable convergence guarantees. LightTune opportunistically reﬁnes ML models using live test-time data only when performance falls below a predeﬁned threshold, ensuring minimal computational overhead and efﬁcient responsiveness. As a practical demonstration, we integrate LightTune into a block error rate (BLER) prediction algorithm for 6G mobile systems. This integration enables the ML BLER prediction model to dynamically adapt to previously unseen channel conditions in real time. Simulation results show a substantial reduction in the average BLER prediction error by up to 48.8% with online ﬁne-tuning. Furthermore, we leverage this BLER prediction algorithm for link adaptation and demonstrate average throughput improvements by up to 15.5% compared to a conventional tablebased outer loop link adaptation (OLLA) algorithm.

reﬁne deployed ML models using inference-time observations. LightTune is speciﬁcally designed for prediction tasks where

ground-truth metrics, such as link-level throughput or block error rate (BLER), become available after a brief delay. Such metrics are inherently available in wireless communications and networked systems. By leveraging this delayed groundtruth data, LightTune performs targeted updates to continuously improve model accuracy in a direct, supervised manner. Unlike reinforcement learning (RL)-based approaches, which rely on reward signals and unsupervised exploration [7], [8], LightTune utilizes direct ground-truth labels to ensure convergence. Crucially, while standard adaptation techniques rely on computationally expensive backpropagation [4], LightTune is entirely backpropagation-free, making it ideally suited for a resource-limited UE, particularly wireless modems. Our main contributions are as follows:

1) We develop a ﬁne-tuning algorithm termed LightTune that is opportunistically triggered when the performance of the ML model, initially trained ofﬂine, is not as desired. Index Terms—6G, ﬁne-tuning, backpropagation-free, forwardforward, online learning, link adaptation. The performance of the ML model is monitored in terms of the prediction error and once it reaches a predeﬁned threshold δ, the ﬁne-tuning procedure is triggered. Our I. I NTRODUCTION proposed online ﬁne-tuning algorithm offers Achine learning (ML)-based wireless algorithms are a) backpropagation-free online ﬁne-tuning by leveraging emerging as pivotal enablers for 6G applications, inthe forward-forward (FF) algorithm [9], enhanced by a cluding channel state information (CSI) prediction, compresnewly proposed smooth loss function with closed-form sion, and beam management [2]. However, a major bottleneck gradients, which enables low-complexity online gradient in realizing their full potential is the inevitable training-test computation, and distribution mismatch. These ML models are typically trained b) buffer-less ﬁne-tuning through a proposed thresholdofﬂine using synthetic datasets that fail to completely capture based update policy that decides which samples are used the complexity and variability of real-world environments. on a sample-by-sample basis, without storing them in an Even when empirical data is utilized during training, the experience replay buffer ﬁrst [10]–[12]. highly dynamic nature of wireless channels frequently causes To the best of our knowledge, LightTune is the ﬁrst signiﬁcant post-deployment performance degradation [3], [4]. application of the FF algorithm in cellular wireless To mitigate this training-test mismatch, ML models must communications. continuously adapt to dynamic deployment environments 2) We provide ﬁnite-time and asymptotic convergence guaranthrough online learning [5], [6]. However, conventional online tees for LightTune showing that, under training-test distrilearning is often infeasible for resource-constrained devices, bution mismatch, the average frequency of prediction errors such as user equipment (UE) and edge devices, which typically reaching or exceeding any ﬁxed threshold δ converges to lack the computational and memory capacities required to 0 as the number of ﬁne-tuning steps increases. support real-time gradient computations and backpropagation. 3) To demonstrate the practical utility of LightTune, we In this paper, we propose LightTune, a lightweight online evaluate its performance within a short-term BLER preﬁne-tuning framework that enables the UE to incrementally diction framework for 6G systems. Simulation results inThis article was presented in part at the proceedings of the 2026 Indicate that LightTune achieves substantial improvements ternational Conference on Communications (ICC) [1]. The authors are in prediction accuracy, speciﬁcally reducing the average with Samsung Device Solutions Research America, Samsung SemiconducBLER prediction error by up to 48.8% through online ﬁnetor, Inc., San Diego, CA 92121 USA (e-mails: [email protected], [email protected]). tuning. Furthermore, integrating this BLER predictor into

the link adaptation process yields signiﬁcant throughput gains reaching up to 15.5% relative to conventional tablebased outer-loop link adaptation (OLLA) baselines [13]. The rest of this paper is organized as follows. We discuss background concepts and prior works in Section II. LightTune is then presented in Section III and the convergence results are provided in Section IV. LightTune is then leveraged for BLER prediction and link adaptation in Section V. We present our extensive simulation results in Section VI. Finally, we discuss concluding remarks and future work in Section VII. Notation

Description

x ∈ Rd

Input feature vector Ground-truth (positive) label Incorrect (negative) label Augmented input vector, [x⊤ , y]⊤ Positive augmented input vector, [x⊤ , y+ ]⊤ Negative augmented input vector, [x⊤ , y- ]⊤ Activation vector at layer l Positive activation vector at layer l Negative activation vector at layer l ⊤ Augmented activation vector at layer l, [h⊤ l , 1] Positive augmented activation vector at layer l, ⊤ [h⊤ +,l , 1] Negative augmented activation vector at layer l, ⊤ [h⊤ -,l , 1] Positive goodness vector for layer l Negative goodness vector for layer l Weight matrix and bias vector for layer l Augmented parameter matrix for layer l Goodness function for a feature-label pair (x, y) Positive goodness value of neuron j in layer l Negative goodness value of neuron j in layer l Predeﬁned threshold for the goodness function Loss of layer l Number of layers Number of neurons (width) in layer l

y+ ∈ Y y- ∈ Y \ {y+ } z ∈ Rd+1 z+ ∈ Rd+1 z- ∈ Rd+1 hl ∈ RMl h+,l ∈ RMl h-,l ∈ RMl h̃l ∈ RMl +1 h̃+,l ∈ RMl +1 h̃-,l

∈ RMl +1

g+,l g-,l Wl , bl Θl Gθ (x, y) g+,l [j] g-,l [j] T Ll L Ml

Y = {y1 , . . . , yC } is the corresponding true label. The positive pass operates on the positive data and adjusts the model parameters with the goal of increasing a goodness value in every layer above a predeﬁned threshold T . 2) Forward Pass with Negative Data Samples A negative data sample is deﬁned as the tuple (x, y- ), where y- ∈ Y \{y+ } is an incorrect label drawn from a distribution over incorrect labels (e.g., uniformly at random). The negative pass operates on negative data and adjusts the model parameters to decrease a goodness value in every layer below the threshold T . • Label Encoding. While label encoding can be implemented via various methods [9], the standard approach for imagebased tasks involves embedding a one-hot representation of the label into a speciﬁc subset of input pixels. This “overlay” ensures that label and spatial features are processed jointly from the initial layers of the network. Given that LightTune operates on one-dimensional feature vectors, we adopt a concatenation-based encoding. Speciﬁcally, the label y ∈ Y is appended to the feature vector x ∈ Rd to form the augmented input vector z = [x⊤ , y]⊤ ∈ Rd+1 . •

(L)

hL := Fθ (z) = (fL ◦ fL−1 ◦ · · · ◦ f1 )(z),

TABLE I: Notations

from the initial augmented input space to the activation space of the l-th layer. By convention, the base case is (0) deﬁned as h0 = Fθ (z) = z ∈ Rd+1 and M0 = d + 1.

•

Goodness Function. The compatibility between the features x and the candidate label y is quantiﬁed by a goodness function Gθ (x, y), deﬁned as the squared ℓ2 -norm of the activations at the terminal layer as follows (L)

Gθ (x, y) = khL k22 = kFθ ([x⊤ , y]⊤ )k22 .

A. The Forward-Forward (FF) Algorithm In contrast to the usual ML training using backpropagation (BP), the FF algorithm trains each layer locally sequentially without backpropagation. This offers the advantage of avoiding the complex operations of propagating the error derivatives backward through the layers. Hence, it is appealing when used for resource-constrained devices such as UE that may not support BP. The FF algorithm is primarily used for image classiﬁcation problems and is based on two forward passes [9]. 1) Forward Pass with Positive Data Samples A positive data sample is deﬁned as the tuple (x, y+ ), where x ∈ Rd is the input d-dimensional feature vector and y+ ∈

(1)

(l) where Fθ : Rd+1 → RMl denotes the cumulative mapping

II. BACKGROUND AND R ELATED W ORK We begin by providing an overview of the FF algorithm, with our notations provided in Table I, followed by a brief discussion of the experience replay mechanisms. Finally, we review closely related works to situate our proposed framework within the existing literature.

End-to-end Mapping. We denote the model parameters as ⊤ ⊤ θ = [θ1⊤ , θ2⊤ , . . . , θL ] , where θl ∈ Rdl represents the parameter vector of layer l ∈ {1, 2, . . . , L}. Each layer implements a transformation fl (·; θl ) that maps an input representation to a subsequent hidden state. For a given feature vector x ∈ Rd and a candidate scalar label y ∈ Y, we construct the augmented input z = [x⊤ , y]⊤ ∈ Rd+1 , which serves as the initial activation h0 . The model’s forward pass is deﬁned by the recursive composition of these layer-wise functions, hl = fl (hl−1 ; θl ), such that the ﬁnal activation vector hL is expressed as

(2)

The learning objective is to maximize Gθ (x, y) for positive samples (ground-truth pairings) and minimize it for negative samples (incorrect or corrupted pairings). By optimizing the goodness, the ML model learns to map the high-dimensional feature-label relationship into a latent space where correct associations (i.e., ground-truth pairings) exhibit a higher magnitude of activation. •

(i)

Softplus Loss Function. Let h+,l and h-,l ∈ RMl denote the activation vectors of layer l for the i-th positive and negative samples, respectively. The per-neuron positive and negative goodness values are then deﬁned as (i)

(i)

g+,l [j] = (h+,l [j])2 , g-,l [j] = (h-,l [j])2 ,

(3)

for each neuron j ∈ {1, 2, . . . , Ml }, where Ml repre- experiences that have high prediction error to accelerate consents the number of neurons in layer l. The positive vergence [11]. and negative hgoodness vectors for layer il are then given LightTune is inspired by the prioritized selection, focusing ⊤ (i) (i) (i) (i) (i) its computational resources on high-error samples to drive and g-,l = as g+,l = g+,l [1], g+,l [2], . . . , g+,l [Ml ] online adaptation. However, unlike traditional approaches that h i⊤ (i) (i) (i) require memory-intensive replay buffers to store and re-sample g-,l [1], g-,l [2], . . . , g-,l [Ml ] . It is worth noting that the goodness function Gθ (x, y) de- data, LightTune operates in a buffer-less manner. By dyﬁned in (2) corresponds to the sum of these per-neuron good- namically identifying informative samples on a per-sample ness values at the terminal layer L, such that Gθ (x, y) = basis using a threshold-based policy, LightTune performs P ML 2 immediate ﬁne-tuning on-the-ﬂy. This eliminates the storage j=1 gL [j], where gL [j] = (hL [j]) . For a set of positive samples S+ and negative samples S- , overhead associated with experience replay, making the framework uniquely suited for the stringent memory and latency the Softplus loss for layer l is expressed as constraints of wireless modems and edge devices. Ml XX (i) 1 −(g+,l [j]−T ) LSoftplus,l = ln 1 + e |S+ |Ml C. Related Work i∈S+ j=1 M l The problem of adapting ML-based wireless algorithms (i) 1 XX ln 1 + eg-,l [j]−T , (4) after deployment under real-world practical constraints such + |S- |Ml i∈S- j=1 as avoiding backpropagation has received limited attention in where T is a ﬁxed threshold. This loss function encourages the literature. We review the closely-related prior works with the goodness values of individual neurons to stay above T focus on the link adaptation problem. for positive samples and below T for negative samples. 1) Ofﬂine Training. Several prior works adopt an ofﬂine training paradigm, where the ML model is trained once • Inference. At the inference phase, the predicted label ŷ and its parameters remain ﬁxed after deployment [14]–[16]. for a given feature vector x is obtained by performing an In this ofﬂine phase, the primary objective is to construct exhaustive search over the candidate label space Y. The a training dataset that broadly anticipates and represents model evaluates the goodness function for each potential all potential deployment scenarios which is an exhaustive pairing and selects the label that maximizes the goodness process that is inherently time-consuming. Furthermore, deas spite this signiﬁcant upfront complex training process, this ŷ = arg max Gθ (x, y) static approach suffers from training-test mismatches, often y∈Y resulting in signiﬁcantly degraded inference performance 2 (5) = arg max (fL ◦ · · · ◦ f1 ) [x⊤ , y]⊤ 2 . within dynamic environments. y∈Y 2) Ofﬂine Training with Online Calibration/Adaptation. Remark 1 (Output layer in the FF algorithm). Unlike Several calibration techniques attempt to mitigate trainingconventional classiﬁers, where the output layer’s dimension test mismatches by adjusting the model’s output rather than is equal to the number of classes C, the output layer in the its internal parameters [16]–[18]. These approaches rely FF algorithm may have a number of neurons not equal to C. on acknowledgment (ACK) and negative acknowledgment This is because the FF algorithm predicts the label based on (NACK) signals obtained from the cyclic redundancy check goodness, as illustrated in Fig. 1. (CRC) validation of physical downlink shared channel (PDSCH) transmissions. By tracking these signals, an empirical metric, such as the short-term BLER, is com(x, y1 ) Gθ (x, y1 ) puted to dynamically adjust or scale the ML predictions. However, because the underlying ML model parameters ŷ = arg max Gθ (x, y) x (x, y2 ) Gθ (x, y2 ) y∈Y remain frozen, these methods act as a superﬁcial outdated . . . . correction rather than enabling the model to inherently . . adapt to the deployment environment. (x, yC ) Gθ (x, yC ) 3) Reinforcement Learning. Although deep RL has been Fig. 1: Inference process in the FF algorithm: the input x is paired with each widely explored for continuous adaptation of ML-based candidate label y ∈ Y and the label with highest goodness is selected. wireless algorithms [4], [7], [8], such solutions typically overlook the strict hardware limitations of wireless modems. Speciﬁcally, continuous RL training demands B. Experience Replays substantial memory for experience replay buffers, alongside Experience replay (ER) is widely utilized in RL and continsigniﬁcant computational power for ongoing backpropagaual learning to decouple temporal correlations and mitigate tion and gradient evaluations, making it largely prohibitive catastrophic forgetting by storing and replaying past obserfor resource-constrained devices. Furthermore, by operatvations [10]–[12]. While uniform experience replay methods ing purely on indirect reward signals, RL methodologies treat all stored experiences with equal importance [10], priorifail to exploit a critical structural advantage in wireless tized experience replay (PER) selectively samples “surprising” systems: the inherent availability of delayed ground-truth

labels (e.g., ACKs/NACKs). Rather than relying on trialand-error exploration, these delayed labels can be utilized to enable highly efﬁcient, targeted supervised ﬁne-tuning. 4) Table-based Baselines. Prior to the proliferation of MLdriven techniques, table-based algorithms served as the de facto standard for wireless communications. Traditional link adaptation maps estimated channel conditions, such as signal-to-interference-plus-noise ratio (SINR) or mean mutual information per bit (MMIB), to a channel quality indicator (CQI) index using predeﬁned lookup tables [13]. To adapt to dynamic environments, outer loop link adaptation (OLLA) algorithms utilize ACK/NACK feedback to continuously adjust an offset margin. While recent advancements, such as the exponential decay mechanism [19], aim to accelerate OLLA convergence, a fundamental structural limitation remains. Speciﬁcally, conventional OLLA updates are typically isolated to the speciﬁc CQI currently scheduled for transmission. Because the remainder of the lookup table is not jointly updated, transitioning to a different CQI relies on stale, uncalibrated mappings. This severely bottlenecks the overall adaptation process and frequently results in the selection of CQIs with high shortterm BLER, as pointed out in our earlier work [1]. 5) Parametric Adaptation. Concurrent [20] and subsequent research [21] tackle the link adaptation problem by modeling the mapping between signal quality and BLER using differentiable sigmoid functions. These approaches propose an online convex optimization framework, implemented at the base station which applies continuous gradient descent to minimize the cross-entropy loss between ACK/NACK feedback and predicted BLER. While these methods provide smoother margin adjustments than conventional OLLA, they rely on one-dimensional parametric heuristics and require continuous gradient evaluations. Consequently, they remain agnostic to high-dimensional UE-side channel dynamics, such as Doppler and delay spreads, and lack the representational capacity required to proactively adapt to the complex non-stationarities of 6G networks. Our Work: To overcome the limitations of the prior approaches, LightTune opportunistically ﬁne-tunes an expressive ML model capable of processing multi-dimensional feature set. By employing a backpropagation-free online ﬁnetuning strategy based on closed-form local gradients, LightTune adapts to shifting environmental distributions.

A. Proposed Loss: Quadratic Softplus Approximation Loss The gradient computation for the standard Softplus loss function requires evaluating exponentiations and divisions. These operations introduce undesirable computational latency and hardware complexity for resource-constrained wireless modems. To circumvent this overhead, we propose an alternative smooth quadratic loss function that closely approximates the Softplus objective using only basic arithmetic operations. For layer l, the proposed loss is deﬁned as Ml XX 1 (i) (i) (g+,l [j] − T )2 − 4(g+,l [j] − T ) LProp,l = |S+ |Ml j=1 i∈S+

M 1 X Xl (i) (i) (g-,l [j] − T )2 + 4(g-,l [j] − T ) . |S- |Ml j=1 i∈S-

(6)

Similar to the Softplus loss, this quadratic formulation serves as a soft thresholding mechanism that encourages the goodness values to exceed the threshold T for positive samples and fall below the threshold T for negative samples. In the balanced case where the number of positive and negative samples is equal, denoted as |S+ | = |S- | := S, this loss function simpliﬁes to  M 1  X Xl (i) (i) LProp,l = (g+,l [j] − T )2 − 4g+,l [j] SMl i∈S+ j=1  Ml XX (i) (i) (g-,l [j] − T )2 + 4g-,l [j]  . (7) + i∈S- j=1

We provide the derivation and experimental validation of the proposed loss function in Appendix A, demonstrating that it achieves accuracy comparable to the standard Softplus loss. Throughout the remainder of this paper, the proposed loss function for layer l is denoted by Ll for simplicity. B. Closed-Form Gradient Computations

To optimize the ML model using the proposed quadratic loss, we compute the local closed-form gradients for each layer l of the multilayer perceptron (MLP). Let Wl and bl denote the weight matrix and bias vector of layer l, respectively. We deﬁne the augmented parameter matrix Θl = [Wl , bl ] ∈ RMl ×(Ml−1 +1) . This matrix formulation is equivalent to the ﬂattened parameter vector θl ∈ Rdl introduced previously, III. P ROPOSED F INE - TUNING A LGORITHM : LightTune where the total number of layer parameters is dl := Ml (Ml−1 + We propose a closed-loop learning framework that enables 1). Correspondingly, we absorb the bias term into the forward adaptive reﬁnement of a deployed ML model, initially trained pass by deﬁning the augmented input activations for the ofﬂine, when its performance deviates from desired behavior. positive and negative samples as h̃+,l−1 = [h⊤ , 1]⊤ and +,l−1 Initially, the FF algorithm is used to train a baseline ML h̃ ⊤ ⊤ -,l−1 = [h-,l−1 , 1] , respectively. model ofﬂine. Subsequently, the FF algorithm is employed for In LightTune, each update step utilizes a single positive online ﬁne-tuning of the baseline ML model. This mechanism sample z = [x⊤ , y ]⊤ and a single negative sample z = + + is feasible in scenarios where ground-truth labels can be [x⊤ , y ]⊤ , where y ∈ Y \ {y }. Consequently, we have |S | = + + inferred at a later time. In wireless communication systems, |S | = 1. The derivative of the proposed layer-wise loss with such ground-truth labels can be derived from key performance respect to the augmented parameter matrix is computed as indicators (KPIs), including BLER, bit error rate (BER) and throughput. ∇Θl Ll = ∇Θl L+,l + ∇Θl L-,l , (8)

where ∇Θl L+,l and ∇Θl L-,l denote the gradient components Algorithm 1 Proposed Fine-tuning Algorithm: LightTune corresponding to the positive and negative samples, respec- Input: Current model parameters θ(t) , ﬁne-tuning threshold δ, tively. Assuming the use of rectiﬁed linear unit (ReLU) activaAdam optimizer parameters (tA , β1 , β2 , ǫ), ﬁne-tuning learning rate αf tions, these gradients can be derived using the chain rule and Output: ML prediction ŷ (t) , ﬁne-tuned model parameters θ(t+1) expressed in a closed-form matrix notation as (t) (t) 4 (g+,l − T − 2) ⊙ h+,l ⊙ 1(h+,l > 0) h̃⊤ +,l−1 , Ml 4 (g-,l − T + 2) ⊙ h-,l ⊙ 1(h-,l > 0) h̃⊤ ∇Θl L-,l = -,l−1 , Ml (9)

∇Θl L+,l =

where ⊙ denotes the element-wise Hadamard product and 1(·) is the indicator function applied element-wise. Since the online ﬁne-tuning process utilizes exactly one positive and one negative sample, both the gradient computation and the parameter update are performed only once at each ﬁne-tuning step. Crucially, each parameter update is purely local to its respective layer. By requiring only the current layer’s activations and the forward-propagated inputs from the preceding layer, this localized approach eliminates the memory and computational overhead associated with backpropagation.

1: Extract feature vector x

and retrieve actual metric y+

2: Compute model prediction: ŷ (t) = arg maxy∈Y G x(t) , y

(t)

3: if e(ŷ (t) , y+ ) ≥ δ then 4: 5: 6: 7: 8: 9: 10: 11: 12:

⊤

(t)

Construct positive sample: z+ = [x(t) , y+ ]⊤ ⊤ Construct negative sample: z-(t) = [x(t) , y-(t) ]⊤ (t) Compute gradients ∇θ L(t) using z+ and z-(t) (as in (9)) if Standard Adam variant then tA ← tA + 1 else if One-step variant then tA ← 1, m0 ← 0, v0 ← 0 end if Compute updated model parameters as follows [24]: mtA = β1 mtA −1 + (1 − β1 )∇θ L(t) , vtA = β2 vtA −1 + (1 − β2 )(∇θ L(t) )2 , ∆θ(t) = q

mtA /(1 − β1tA )

vtA /(1 − β2tA ) + ǫ

θ(t+1) ← θ(t) − αf ∆θ(t) .

C. Threshold-based Fine-tuning Without Replay Buffer To entirely eliminate the memory overhead associated with experience replay buffers at the UE, LightTune employs a threshold-based update policy, enabling the model to be ﬁne-tuned in an opportunistic, sample-by-sample manner. The surprise of an observation at time t is quantiﬁed by the (t) prediction error, denoted as e(ŷ (t) , y+ ), between the model’s (t) (t) predicted label ŷ and the delayed ground-truth label y+ . When this error meets or exceeds a predeﬁned threshold δ, the model identiﬁes a distribution shift and triggers a local update. ⊤ (t) (t) For this update, the true observation z+ = [x(t) , y+ ]⊤ is utilized as the positive training pair. Simultaneously, a ⊤ corresponding negative sample z-(t) = [x(t) , y-(t) ]⊤ is constructed by pairing the current input feature vector x(t) with (t) an incorrect label y-(t) ∈ Y \ {y+ }. This lightweight, onthe-ﬂy ﬁne-tuning mechanism is detailed in Algorithm 1 and illustrated in Fig. 2. Negative Sampling. We consider two strategies for selecting the negative label y-(t) used during the ﬁne-tuning process. 1) Uniform Negative Sampling. In this scheme, the negative label y-(t) is sampled uniformly at random from the set of (t) incorrect candidates Y \ {y+ }. This stochastic approach is widely adopted in the FF literature [9], [22]. 2) Hard Negative Sampling. The negative label is set to the model’s current erroneous prediction: y-(t) = ŷ (t) . Inspired by hard triplet mining and adversarial training [23], this selection forces the model to speciﬁcally penalize its most conﬁdent errors, thereby sharpening the decision boundaries more effectively during ﬁne-tuning.

13: else 14: θ(t+1) ← θ(t) 15: end if x(t) Parameter Update θ (t+1) = θ (t) − αf ∆θ (t)

θ ← θ (t+1)

ML Model θ ŷ (t) = arg maxy∈Y G(x(t) , y)

∆θ (t) (t)

Compute Gradients

e(ŷ (t) , y+ ) ≥ δ

Error Computation (t) e(ŷ (t) , y+ )

Fig. 2: The proposed ﬁne-tuning algorithm LightTune uses the delayed true (t) label y+ to compute the prediction error and ﬁne-tune the model if needed.

based on Adam optimizer [24]. Adam is a widely used optimizer that adapts each parameter by maintaining two running averages (moments): the ﬁrst moment (mean of gradients) at time t denoted by mt and the second moment (variance of gradients) at time t denoted by vt . 1) Standard Adam Update: In this variant, the Adam optimizer maintains an internal counter tA ∈ {1, 2, . . . } that exclusively tracks the number of executed ﬁne-tuning updates. Speciﬁcally, tA is incremented only when the ﬁnetuning condition e(t) ≥ δ is triggered. This ensures the bias correction mechanism accurately reﬂects the true number of parameter updates. The updates reuse the moment estimates from the previous ﬁne-tuning instance, following the standard recursive form mtA = β1 mtA −1 + (1 − β1 )∇θ L(t) ,

vtA = β2 vtA −1 + (1 − β2 )(∇θ L(t) )2 ,

D. Fine-tuning Variants While LightTune is compatible with stochastic gradient descent (SGD) and any optimizer, we focus on two variants

(t)

(10)

where β1 and β2 are the exponential decay rates for the ﬁrst and second moments, respectively, and (·)2 is applied element-wise.

2) One-step Update: To entirely eliminate the need for storing the previous moments, this variant resets the Adam internal counter to tA = 1 at every ﬁne-tuning instance. Consequently, the historical moments are effectively reinitialized to zero (m0 = 0 and v0 = 0), and the update simpliﬁes to m1 = (1 − β1 )∇θ L(t) ,

v1 = (1 − β2 )(∇θ L(t) )2 .

(11)

(t)

where e(t) (ŷ (t) , y+ ) denotes the prediction error. (t)

Hence, E[Iδ ] = Pr(e(t) ≥ δ). Deﬁnition 2 (Filtration). The ﬁltration Ft represents the history of the stochastic process up to time t and is deﬁned as the σ-algebra generated by the ofﬂine model parameters and the sequence of positive samples observed up to time t: (t) (1) Ft = σ θ (1) , z+ , . . . , z+ .

We now state our assumptions. This variant is memory-efﬁcient because the UE is not Assumption 1 (ReLU MLPs). The underlying ML model of required to store or maintain the high-dimensional ﬁrst and LightTune is assumed to be a ReLU MLP. second moment vectors across different ﬁne-tuning steps, thereby signiﬁcantly reducing the dynamic RAM footprint. Assumption 2 (Bounded Input and Model). There exist positive constants Bz , Bθ such that for all time steps t and Remark 2 (Rationale for One-step Update). In the online, all layers l: sparse-update setting of LightTune, “surprising” samples ⊤ (i) kz (t) k2 ≤ Bz , where z (t) = [x(t) , y (t) ]⊤ ∈ Rd+1 , frequently indicate a fundamental distribution shift in the (t) underlying wireless channel. By resetting the internal Adam (ii) kθl,j k2 ≤ Bθ for all neurons j in layer l, moments, the one-step variant ensures that the parameter where θ (t) denote the parameters of neuron j in layer l. l,j update relies strictly on the current gradient. Consequently, Next, to model the training-test mismatch problem, we the update simpliﬁes to a normalized gradient step given by consider Assumption 3. ∇θ L(t) (t+1) (t) . (12) Assumption 3 (Training and Online Data Distributions). θ ← θ − αf · p (∇θ L(t) )2 + ǫ The ofﬂine training data are i.i.d. from distribution D1 , and When ǫ = 0, this one-step update rule reduces to the online ﬁne-tuning data {z (t) }t≥1 are i.i.d. from a (possibly different) distribution D2 . (13) θ (t+1) ← θ (t) − αf · sgn ∇θ L(t) , Assumption 4 (Gradient Lower Bound). For any ﬁxed δ > 0, where sgn(·) denotes the element-wise sign function. This there exists a constant γ2 (δ) > 0 such that whenever e(t) ≥ δ, gradient update is advantageous for a resource-limited UE, the expected squared norm of the gradient (with respect to the as it replaces computationally expensive divisions with highly random negative sample) under D2 satisﬁes1 i h efﬁcient bit-sign extractions. Furthermore, by eliminating the (t) (t) 2 (t) )k (θ F , e ≥ δ ≥ γ2 (δ). (15) L k∇ E (t) t θL L y2 L inertia of historical gradients, this variant not only minimizes the memory footprint of the UE but also enables real-time This assumption is analogous to the gradient dominance or adaptation to channel variations. Polyak–Łojasiewicz (PL) conditions frequently invoked in non-convex optimization literature [25]. It ensures that the IV. C ONVERGENCE OF LightTune “learning signal” remains strictly bounded away from zero We now provide the convergence guarantee for LightTune, whenever the model’s performance is not as desired (i.e., (t) showing that the average frequency of prediction errors reach- e ≥ δ). In effect, this condition guarantees that the stochastic ing or exceeding any ﬁxed threshold δ > 0 converges to nature of the negative label selection does not lead to vanishing zero as the number of ﬁne-tuning steps increases, despite the gradients during the online ﬁne-tuning process. training-test distribution mismatch. The core intuition is that We provide an informal intuitive justiﬁcation for this assumpwhenever the prediction error reaches or exceeds the threshold tion in Appendix D. δ, the model update reduces the loss by a guaranteed amount. Since the loss is bounded below, the total number of such B. Preliminary Lemmas improvements is ﬁnite. Consequently, large errors can occur We now present the foundational lemmas that we build on only ﬁnitely many times, and their frequency must decay to to prove our convergence theorem. While detailed proofs are zero. deferred to Appendix D, we provide their statements here. We We start with deﬁnitions, assumptions and useful lemmas that ﬁrst establish the boundedness of activations and gradients, we build on to derive our main convergence result. followed by the Lipschitz smoothness of the loss function. A. Deﬁnitions and Assumptions Deﬁnition 1 (Error-driven Update Indicator). For an error tolerance δ > 0, the update at step t is controlled by a binary (t) random variable Iδ , deﬁned as (t)

Iδ := 1{e(t) ≥ δ},

(14)

Lemma 1 (Bounded Activations). Under Assumption 2 (i.e., bounded input and model), there exists a positive constant Bh (t) such that for all time steps t and all layers l, khl k2 ≤ Bh . In particular, one can take √ Bh = max0≤l≤L Bl , where B0 = Bz and for l ≥ 1, Bl = Ml Bθ (Bl−1 + 1). 1 The subscript in γ means that this is under the distribution D . 2 2

Lemma 2 (Bounded Gradient). Under Assumption 2 (i.e., bounded input and model), for all time steps t, layers l, and neurons j, the gradient of the layer loss with respect to the parameters of neuron j is bounded by (t)

(t)

k∇θl,j Ll (θl )k2 ≤

8(Bh2 + T + 2)Bh (Bh + 1) , Ml

(t)

(16) (t)

where ∇θl,j Ll (θl ) denotes the partial derivative of Ll with respect to θl,j .

yields the following upper bound on the total expected number of large errors up to time N ≥ 1: N

αf γ2 (δ) X (t) (1) (1) E[Iδ ] ≤ E[LL (θL )] − L∗L , 2 t=1 (t)

where L∗L = inf θ ED2 [LL (θ)] PN (t) 4) Consequences. Rearranging gives N1 ≥ t=1 PrD2 (e δ) ≤ O(1/N ) when the ofﬂine and online distributions are identical. In that case, Pr(e(t) ≥ δ) → 0. 5) Distribution shift. When p D1 6= D2 , Pinsker’s inequality introduces an extra term 2DKL (D2 kD1 )/N , and the average frequency still tends to zero.

Lemma 3 (Bounded Loss). Under Assumption 2 (i.e., bounded input and model), there exists a constant M > 0 (t) (t) such that the loss is bounded as supt≥1 LL (θL ) ≤ M, where M = (Bh2 + T + 2)2 + 4(Bh2 + T + 2). Theorem 1 (Convergence under Distribution Shift). SupLemma 4 (Smoothness). Under Assumption 2 (i.e., bounded pose Assumptions 1, 2, 4, and 3 hold. For a ﬁxed error (t) input and model), for each layer l, the loss function Ll (θ) tolerance δ > 0 and a learning rate αf ∈ (0, 1/ρL ), LightTune with SGD satisﬁes for any N ≥ 1: is ρl -smooth with i h (1) (1) ∗ N )] − L (θ [L 2 E X D 8 3Bh2 + T + 2 (Bh + 1)2 1 L L L 1 ρl = . (17) Pr e(t) ≥ δ ≤ Ml N t=1 D2 αf γ2 (δ)N r ′ dl That is, for any θl , θl ∈ R , 2DKL (D2 kD1 ) 2M + , (22) α γ (δ) N (t) (t) ′ f 2 k∇Ll (θl ) − ∇Ll (θl )k2 ≤ ρl kθl − θl′ k2 . (18) (t)

Next, we consider the descent lemma [26, Lemma 1.2.3]. (t)

Lemma 5 (Descent Lemma). For any ρl -smooth function Ll which satisﬁes that for any θl , θl′ (t)

(t)

k∇Ll (θl ) − ∇Ll (θl′ )k2 ≤ ρl kθl − θl′ k2 ,

(19)

we have (t)

Ll (θl′ ) ≤ Ll (θl ) + ∇Ll (θl )⊤ (θl′ − θl ) +

ρl ′ kθ − θl k22 . 2 l (20)

Finally, to handle the training-test distribution mismatch, we leverage Pinsker’s inequality. Lemma 6 (Pinsker’s Inequality [27]). Let P and Q be two probability distributions on a measurable space. For any measurable function f with kf k∞ = sup |f | ≤ M , we have r 1 |EP [f ] − EQ [f ]| ≤ M DKL (P kQ). (21) 2

where L∗L = inf θ ED2 [LL (θ)] is the minimum achievable expected loss under D2 , the constant M satisﬁes (t) (t) supt≥1 |LL (θL )| ≤ M , and γ2 (δ) is the gradient lower bound from Assumption 4.

Corollary 1 (Asymptotic Convergence of Average Error Probability). Under the conditions of Theorem 1, for any ﬁxed error tolerance δ > 0, the average probability of a signiﬁcant prediction error satisﬁes N

1 X Pr e(t) ≥ δ = 0. D2 N →∞ N t=1 lim

(23)

Corollary 2 (Pointwise Convergence under Identical Distributions). If D P1∞= D2 , then under the conditions of Theorem 1, the series t=1 Pr(e(t) ≥ δ) converges, and consequently lim Pr e(t) ≥ δ = 0. (24) t→∞

V. A PPLICATIONS OF LightTune

C. Convergence Theorem

In this section, we leverage LightTune for short-term BLER prediction and link adaptation.

The proof follows a non-convex optimization approach adapted to the threshold-based updates of LightTune. The key steps are:

A. BLER Prediction

(t)

1) Local decrease. When a large error occurs (i.e., Iδ = 1), α (t) the gradient step reduces the loss by at least 2f k∇LL k2 (by the smoothness and the descent lemmas). 2) Gradient lower bound. Assumption 4 guarantees that whenever the error is large, the expected squared gradient norm is at least γ2 (δ) > 0. Thus, each large error α γ (δ) produces an expected decrease of at least f 22 . 3) Telescoping sum. Summing these expected decreases over time and using stationarity of the data distribution

We consider the downlink short-term BLER prediction problem as a practical application of LightTune. In this context, “short-term” refers to a prediction horizon spanning a few PDSCH transmission slots, typically corresponding to the periodicity of the CSI reports. Such short-term BLER prediction at the UE side is essential in advanced cellular systems (e.g., 5G/6G) for enabling accurate and timely CSI reporting. We begin with a brief background. The CSI reference signal (CSI-RS) period denotes the interval between consecutive

reference signals used for CSI acquisition. This interval de- Remark 3 (Using the FF Algorithm for Regression). The termines how frequently the receiver (i.e., UE) can update FF algorithm was originally designed for classiﬁcation tasks. its channel estimates and report CSI to the transmitter (i.e., To adapt it for regression, we discretize the continuous target next-generation NodeB (gNB)). CSI-RS transmissions can variable (i.e., BLER) into a ﬁnite set of discrete classes deﬁned be conﬁgured as either periodic or aperiodic. In periodic by P = {p1 , p2 , . . . , pC }. This introduces a fundamental mode, reference signals are transmitted at regular intervals. trade-off. On one hand, the prediction accuracy is inherently In contrast, aperiodic mode involves dynamically triggered bounded by the granularity of this discretization. On the other transmissions, where the interval between reference signals is hand, increasing the number of classes C to improve resolution variable. correspondingly increases the computational complexity and inference latency. However, as discussed in Sec. V-B, our PBLER P̂BLER link adaptation framework only requires a coarse estimate to Actual BLER Computed BLER Prediction reliably determine whether the short-term BLER exceeds a Time predeﬁned threshold, fully justifying this quantized approach. Start of CSI-RS period

End of CSI-RS period e(P̂BLER , PBLER ) = P̂BLER − PBLER CSI-RS Period

Fig. 3: Timeline showing BLER prediction at the start and actual BLER computation at the end of a CSI-RS period

Algorithm 2 BLER Prediction Algorithm: BLER-P REDICT Input: Current parameters θ(t) , feature vector x(t) , quantized BLER classes P = {p1 , p2 , · · · , pC } Output: Predicted short-term BLER P̂BLER 1: Gmax ← −1 2: P̂BLER ← 0 3: for each candidate BLER p ∈ P do 4: Compute the goodness G = Gθ (t) (x(t) , p) 5: if G > Gmax then 6: Gmax ← G 7: P̂BLER ← p 8: end if 9: end for 10: return P̂BLER

The core inference mechanism for estimating the short-term BLER is detailed in Algorithm 2. At the beginning of the t-th CSI-RS period, the ML model predicts the short-term BLER, (t) denoted as P̂BLER , utilizing input features such as the CSIRS signal-to-noise ratio (SNR). Following the physical layer transmission, the true empirical BLER at the end of the period, (t) denoted as PBLER , is computed as (t)

nNACK

(t)

PBLER = (t)

(t)

nNACK + nACK

(25)

(t)

where nACK and nNACK denote the total number of PDSCH ACKs and NACKs recorded during the t-th CSI-RS period, respectively. Consequently, the UE evaluates the absolute BLER prediction error as (t) (t) (t) (t) (26) e(t) P̂BLER , PBLER = P̂BLER − PBLER . Within the overarching link adaptation framework, the opportunistic ﬁne-tuning process is triggered if and only if this prediction error meets or exceeds a predeﬁned threshold δ. This condition ensures that the model undergoes parameter updates exclusively when its predictions deviate signiﬁcantly from the empirical ground truth, enabling computationally efﬁcient and targeted adaptation at the UE.

B. Channel Quality Indicator (CQI) Selection In 5G, the UE reports a channel quality indicator (CQI) and a rank indicator (RI) to the next-generation NodeB (gNB), which reﬂect the perceived downlink channel conditions. Based on the reported CQI, the gNB dynamically selects an appropriate modulation and coding scheme (MCS) to optimize throughput while maintaining reliable communication. The online BLER prediction algorithm of Section V-A can be integrated with a conventional table-based CQI selection algorithm such as the algorithm of [13] as a backoff mechanism. Traditional CQI selection methods rely on look-up tables that map the mean mutual information per bit (MMIB) or SNR to a CQI index. However, these methods may select a CQI index that results in an excessively high BLER, particularly initially when the algorithm has not yet converged. To mitigate this high short-term BLER problem, we propose an adaptive, ML-guided back-off strategy driven by the online BLER prediction framework. Speciﬁcally, if the CQI selected by the conventional table-based algorithm yields a predicted BLER that meets or exceeds a predeﬁned BLER threshold τBLER (e.g., 0.9), the algorithm iteratively decrements the CQI index until the predicted short-term BLER falls below this threshold provided that the CQI does not drop below a predeﬁned minimum allowable bound, CQImin (e.g., CQI(r) − 1). Crucially, the candidate CQI is explicitly embedded as a feature within the input feature vector x(t) . This allows BLERPredi t to predict the BLER of that CQI. It is important to clarify that the loop in Algorithm 3 operates within a single CSI-RS reporting period. That is, the CQI is reported at the end of the loop. We also note that CQI-Tune does not change the RI selected by the table-based algorithm. Remark 4 (False Alarm and Missed Detection Effect on CQI Selection.). It is worth noting that CQI-Tune only requires a coarse estimate of the BLER. Speciﬁcally, the goal of the underlying BLER prediction algorithm in CQI-Tune is to determine if the short-term BLER is below the threshold τBLER or not. That is, CQI-Tune does not need to predict the short-term BLER accurately, but rather needs to predict if it is below the threshold τBLER or not. Hence, it is critical to analyze the false alarm (FA) and the missed detection (MD) probabilities of the underlying BLER prediction algorithm

Algorithm 3 CQI Back-off Algorithm: CQI-Tune Input: Current parameters θ(t) , table-based CQI: CQI(r) , feature vector x(t) , minimum CQI: CQImin , BLER threshold τBLER and ﬁne-tuning threshold δ (r) Output: Adjusted CQI: CQIL , updated model parameters θ(t+1) (r) 1: CQIL ← CQI(r) (r) 2: while CQIL > CQImin do (r) 3: Update the CQI feature in x(t) to CQIL (t) (t) 4: p̂ ← BLER-Predi t (θ , x ) 5: if p̂ < τBLER then 6: break 7: end if (r) (r) 8: CQIL ← CQIL − 1 9: end while (r) 10: Update the CQI feature in x(t) to CQIL (r) 11: Transmit PDSCH utilizing the ﬁnal adjusted CQIL (t) (t) 12: Compute nACK and nNACK at the end of the CSI-RS Period 13: Calculate the empirical ground-truth BLER: 14:

(t)

PBLER ←

(t) nNACK (t) (t) nNACK +nACK

(t)

15: P̂BLER , θ(t+1) ← LightTune (r)

(t)

θ(t) , x(t) , PBLER , δ, . . .

where rmax denotes the maximum supported rank and SE(i, CQI(i) ) denotes the estimated SE per-layer when RI = i and CQI(i) are selected. The per-layer SE is deﬁned as (i) · R(i) , SE(i, CQI(i) ) = Qm (i)

where Qm and R(i) denote the modulation order and code rate, respectively, corresponding to the selected CQI(i) . Finally, the UE reports the optimal rank RI = r and the associated CQI = CQI(r) to the gNB. Our algorithm, RI-CQI-Tune, builds on CQI-Tune to jointly select the RI and the CQI that maximize the estimated spectral efﬁciency while avoiding the excessively high BLER typically associated with conventional algorithms. Speciﬁcally, RI-CQI-Tune applies CQI-Tune for each candidate RI = i to reﬁne the baseline CQI selected by the table-based algorithm, denoted as CQI(i) , to a more robust, possibly lower CQI (i) denoted as CQIL . Then, we select the rank rL as follows2 rL , arg

16: return CQIL , θ(t+1)

(28)

max

i∈{1,...,rmax }

(i)

{i · SE(i, CQIL )}.

(29)

(r )

The UE then reports rL and CQIL L to the gNB. Algorithm 4 RI-CQI Selection Algorithm: RI-CQI-Tune

denoted as PFA and PMD , respectively. The FA and MD events are deﬁned as EFA := {P̂BLER ≥ τBLER | PBLER < τBLER },

EMD := {P̂BLER < τBLER | PBLER ≥ τBLER }.

EFA is speciﬁcally critical as it leads to changing the decision of the table-based algorithm. EMD is also an important event, but it does not change the decision of the table-based algorithm. C. Rank Indicator (RI) and Channel Quality Indicator (CQI) Selection In 5G New Radio (NR) systems, the RI determines the number of spatial data streams (i.e., layers) transmitted in parallel over the MIMO channel. While the underlying physical channel may support a higher rank according to singular value decomposition (SVD), the lack of ideal waterﬁlling across layers often creates an inherent optimization gap between theoretical capacity and practical throughput. Consequently, the UE may strategically report a lower RI to facilitate a higherorder MCS (e.g., 256-QAM) on the dominant layers, yielding better aggregate throughput than a full-rank transmission that is bottlenecked by the weakest layers forcing a lower-order modulation (e.g., 64-QAM). We begin with a brief overview of the RI selection problem. The conventional table-based baseline algorithms typically ﬁrst estimate the best CQI for each possible RI. Then, they select the RI that maximizes the expected spectral efﬁciency (SE). Speciﬁcally, denoting the CQI selected by the table-based algorithm for RI = i as CQI(i) , the RI selected by the conventional table-based algorithm is given by r , arg

max

i∈{1,··· ,rmax }

{i · SE(i, CQI(i) )},

(27)

Input: Current parameters θ(t) , table-based rank r and table-based max , feature vector x(t) , minimum CQI for each rank {CQI(i) }ri=1 CQI: CQImin , BLER threshold τBLER , maximum rank rmax , ﬁnetuning threshold δ (r ) Output: Adjusted rank rL , adjusted CQI denoted by CQIL L , (t+1) updated model parameters θ 1: rlow ← ⌈r/2⌉ 2: rhigh ← min(rlow + 2, rmax ) 3: SEmax ← −1 (r ) 4: rL ← r, CQIL L ← CQI(r) 5: for i = rlow to rhigh do 6: CQItest ← CQI(i) 7: while CQItest > CQImin do 8: Update rank to i and CQI to CQItest in x(t) 9: if BLER-Predi t (θ(t) , x(t) ) < τBLER then 10: break 11: end if 12: CQItest ← CQItest − 1 13: end while 14: SE ← i · SE(i, CQItest ) 15: if SE > SEmax then 16: SEmax ← SE (r ) 17: rL ← i, CQIL L ← CQItest 18: end if 19: end for (r ) 20: Transmit PDSCH utilizing ﬁnal adjusted rL and CQIL L (t) (t) (t) 21: Compute nACK and nNACK to calculate empirical PBLER (r ) 22: Update rank to rL and CQI toCQIL L in x(t) (t) (t) (t) (t+1) 23: P̂BLER , θ ← LightTune θ , x(t) , PBLER , δ, . . . (r )

24: return rL , CQIL L , θ(t+1) RI-CQI-Tune yields superior throughput gains compared to CQI-Tune by jointly optimizing the RI and CQI, albeit at the cost of increased computational complexity. To ensure that the overhead of RI-CQI-Tune remains independent of 2 The subscript L denotes parameters optimized via the proposed Light-

Tune framework.

the maximum supported rank rmax (e.g., 4 or 8), we utilize the search bounds rlow and rhigh . Speciﬁcally, for any rmax ≥ 4, the search window covers at most 3 candidate ranks (i.e., rhigh − rlow ≤ 2). Consequently, the computational complexity of RI-CQI-Tune is strictly three times that of CQI-Tune, regardless of the system’s maximum rank capability

Standard Backpropagation LightTune (FF-based) P L Peak RAM O O (maxl Ml ) l=1 Ml Control Logic Complex Autodiff Local Training Feature

TABLE II: Complexity analysis of BP vs. FF for an MLP with L layers, where Ml is the width of the l-th layer.

Remark 5 (Motivation for RI-CQI-Tune). RI-CQI-Tune inference PL procedure over the quantized classes. We deﬁne is motivated by empirical observations in live network deQ = l=1 Ml Ml−1 as the number of multiply-accumulate ployments, where it is noted that while the gNB typically (MAC) operations required for a single forward pass through adheres to the RI reported by the UE, it frequently overrides the MLP. Since BLER-Predi t evaluates C discrete classes, the recommended CQI. This behavior is consistent with the ﬂexibility granted to the gNB in 3GPP speciﬁcations [28], the total inference complexity is exactly CQ MAC operations. Complexity of CQI-Tune. The complexity of the CQIwhich allow the network to determine the ﬁnal transmission Tune back-off algorithm (Alg. 3) is divided across the CQI parameters based on its own scheduling constraints and load. selection and the post-transmission ﬁne-tuning. The maximum By performing joint optimization, RI-CQI-Tune provides a number of pre-transmission search iterations is bounded by robust feedback pair that better aligns with these practical (r) (r) scheduling behaviors, reducing the likelihood of high-BLER K := CQI − CQImin . By setting CQImin = CQI − 1, the worst-case pre-transmission overhead is strictly limited to events caused by stale table-based mappings. Furthermore, a key distinction is that while CQI-Tune is exactly one BLER prediction (i.e., CQ MACs). Combined restricted to lowering the CQI selected by the table-based with the two post-transmission forward passes (one for the algorithm (representing a conservative back-off), RI-CQI- positive sample and one for the negative sample) required for ﬁne-tuning, the worst-case total complexity per CSI-RS period Tune has the ﬂexibility to re-optimize the RI. This transition occurs when the CQI of the RI selected by the table-based is exactly (C + 2)Q MAC operations. algorithm reduces such that this speciﬁc rank is no longer the Complexity of RI-CQI-Tune. The joint RI-CQI-Tune selection framework (Alg. 4) introduces an outer search over a best estimated rank. In such cases, RI-CQI-Tune can shift localized rank window [rlow , rhigh ] covering at most 3 candidate the selection to a different rank that offers a higher SE. ranks. Since each candidate rank invokes at most one pretransmission BLER prediction (given K = 1), the joint search D. Complexity Analysis performs a maximum of 3 predictions. P L We ﬁrst discuss the potential beneﬁts of the FF algorithm Fine-tuning Complexity. Let N total = l=1 Ml (Ml−1 + 1) compared to the backpropagation algorithm as summarized denote the total number of model parameters, including biases. in Table II. Then, we analyze the complexity of the BLER The gradient computation in (9) requires two outer products prediction algorithm. Finally, we compare the complexity of (one positive, one negative). Since each outer product genthe CQI-Tune algorithm and the RI-CQI-Tune algorithm erates a matrix of size Ml × (Ml−1 + 1), the computation as summarized in Table III. requires 2Ntotal operations. Including the summation of these Complexity of the FF and the BP algorithms. A primary gradients and the parameter update, the total ﬁne-tuning cost is motivation for adopting the FF algorithm is its signiﬁcant approximately 4N total operations. Crucially, when utilizing the advantages for resource-constrained devices [22], [29], [30]. one-step sign-update variant in (13), the ﬁnal parameter update We discuss these aspects next. requires 0 additional multiplications and Ntotal additions (or 1) Dynamic Memory (RAM) Footprint. Standard back- bit-sign extractions). propagation requires storing all activations during the Finally, as discussed in Remark 4, the number of quantized backward pass; hence, the required RAM P scales with the BLER classes could potentially be restricted to C = 2, further network depth and layer sizes (i.e., l Ml for an MLP). minimizing the computational overhead of the FF inference. In contrast, LightTune trains each layer locally, which eliminates the necessity to store intermediate activations FF Inferences Update Cost Algorithm Total MACs across the entire depth. Consequently, the peak memory Inference Fine-tuning One-step Update footprint scales only with the size of the largest layer (i.e., BLER-Predi t C·Q 0 0 C·Q maxl Ml for an MLP). CQI-Tune C·Q 2Q 4Ntotal (C + 2)Q + 4Ntotal 2) Implementation Simplicity. Standard backpropagation RI-CQI-Tune 3C · Q 2Q 4Ntotal (3C + 2)Q + 4Ntotal necessitates a complex automatic differentiation (AutodPL Worst-case complexity analysis where Q = l=1 Ml Ml−1 and iff) engine to build a computational graph and execute TABLE III: PL the chain rule. This typically requires a full deep-learning Ntotal = l=1 Ml (Ml−1 + 1). framework (e.g., Torch, TensorFlow). LightTune avoids this overhead entirely by leveraging the FF algorithm VI. S IMULATION R ESULTS which utilizes local sequential update rules that do not require global graph tracking. We now evaluate the performance of the proposed ﬁneBLER Prediction Complexity. The computational cost of tuning algorithm for BLER prediction and link adaptation. We a single short-term BLER prediction is governed by the FF begin with our training and test settings.

Parameter

Training

Channels

Test

TDL-A30

TDL-A10, TDL-A30, TDL-B50, TDL-B100, TDL-B200, TDLC200 SNR Low (0–12 dB) Low/Medium/High (0–40 dB) Delay Proﬁle Low Delay Low/High Delay Doppler Frequency Low (10 Hz) Low/Medium (10–50 Hz) Antenna Correlation Low Low/Medium/High CSI-RS Period 80 ms 10, 40 or 80 ms TABLE IV: Training and testing conﬁgurations. Correlation

gNB Correlation (α)

UE Correlation (β)

Low Medium High

0 0.3 0.9

0 0.9 0.9

TABLE V: Antenna Correlation Scenarios [32], where α and β represent the gNB and the UE antenna correlation coefﬁcients, respectively

A. Training and Test Settings We train and test our model on tapped delay line (TDL) channels [31], as summarized in Table IV. In TDL channels, the letter (A, B, C) indicates the channel proﬁle, while the number (10, 30, 50, 100, 200) speciﬁes the root mean square delay spread (RMS DS) in nanoseconds. We consider various 3GPP antenna correlation scenarios as provided in Table V [31]. While the low and the medium antenna correlation scenarios are more common in practice, we also test our algorithms under the high correlation scenario. For both the training and the test settings, the bandwidth is 100 MHz, the sub-carrier spacing is 30 KHz and the maximum supported RI rmax is 4. Our BLER threshold is set as τBLER = 0.9 and the minimum CQI of CQI-Tune is set as CQImin = min(CQI(r) − 1, 1), where CQI(r) is the table-based selection. B. BLER Prediction Our goal is to simulate scenarios in which the ML model is trained in an environment but evaluated in a different one to mimic the training-test mismatch problem. We use 12 features for the BLER prediction ML model, hence the input size is 13 since a label is attached to the feature vector. Features. We utilize features that capture both channel conditions and transmission parameters. These include the CSIRS SNR and the CSI-RS capacity computed using the best precoding matrix indicator (PMI) for the reported RI, which helps characterize antenna correlation levels. We also incorporate the delay spread and Doppler frequency estimates to distinguish between different channel proﬁles. Additionally, we use the instantaneous PDSCH SNR along with the PDSCH SNR values from the three most recent transmissions, which aid in predicting the likelihood of NACKs. The feature set is completed by the current RI and CQI, the number of allocated resource blocks (RBs), and the number of demodulation reference signal (DMRS) symbols used in the transmission. BLER Prediction error with and without Fine-tuning. We investigate which ﬁne-tuning variant is better. Our results show that the one-step update approach is the best in terms

Parameter

Value

Neural Network Size Activation Function Ofﬂine Learning Rate α Online Learning Rate αf Fine-tuning Threshold δ Training Optimizer Threshold T Epochs Training Samples BLER Classes

[13, 32, 32] ReLU 0.03 0.03 0.3 Adam (β1 = 0.9, β2 = 0.999, ǫ = 10−8 ) 9 22,000 83,200 P = {0, 0.1, 0.2, · · · , 0.9}

TABLE VI: Hyperparameters for the BLER prediction algorithm

of the BLER prediction error as shown in Fig. 4 and the FA probability as shown in Fig. 5. • BLER Prediction for TDL-A30 with SNR Mismatch. Fig. 4a shows for SNR 0 to 12 dB that the average BLER prediction error is almost the same with or without online ﬁne-tuning since the baseline ofﬂine model has been trained with TDL-A30 data from 0 to 12 dB. But above 12 dB, we see the signiﬁcant decrease in the BLER prediction error due to enabling online ﬁne-tuning. Speciﬁcally, the average BLER prediction error over all SNRs is reduced by 43.5% using LightTune with one-step update. • BLER Prediction for TDL-B50 and TDL-B100 with SNR and Channel Proﬁle Mismatch. Fig. 4b and Fig. 4c show a more signiﬁcant decrease in the BLER prediction error as a result of enabling online ﬁne-tuning. This is because TDL-B50 and TDL-B100 have not been used in the ofﬂine training. Hence, the training-test mismatch leads to low performance of the baseline ofﬂine ML model. But with LightTune, the average BLER prediction error decreased by 36% for TDL-B50 and up to 48.8% for TDL-B100 for the one-step update variant. Since our BLER prediction results show that the one-step variant of LightTune leads to a lower BLER prediction error in most cases, we only consider this variant for CQI selection and for joint RI and CQI selection. C. CQI Selection We ﬁrst show the effect of LightTune on the FA probability in Fig. 5 under channel and SNR mismatch. As we discussed in Remark 4, the FA probability is crucial in selection of the CQI. Since the ML BLER prediction model has been trained ofﬂine within the low SNR range, we see that the FA probability is low even without ﬁne-tuning in the low SNR range. But in the other SNR ranges, ﬁne-tuning is essential to keep the FA probability as low as possible. We next compare the throughput performance of three approaches with CQI reporting based on: a table-based outer loop link adaptation (OLLA) method similar to the algorithm of [13], the ofﬂine ML-based method [17] with calibration/adaptation described in Appendix C, and the table-based OLLA method augmented with the proposed backoff mechanism described in Algorithm 3. We provide a comparison in Fig. 6. We show the throughput gains in the medium SNR range (16 to 24 dB) and the high SNR range (28 to 40 dB). • CQI Selection for TDL-B50. CQI-Tune provides medium throughput gains of 12.1% and 9.0% using uniform and hard

Without Online fine-tuning LightTune with Standard Adam Update LightTune with One-step Update

0.8

0.6

0.4

0.2

0.8

0.6

0.4

0.2

0 4

0.8

0.6

0.4

0.2

0 0

Without Online fine-tuning LightTune with Standard Adam Update LightTune with One-step Update

Average BLER Prediction Error

0 0

SNR (dB)

(a) TDL-A30 (Doppler freq. = 10 Hz)

SNR (dB)

(b) TDL-B50 (Doppler freq. = 30 Hz)

Fig. 4: BLER prediction error with and without online ﬁne-tuning, uniform sampling, under low antenna correlation with CSI-RS period = 80 ms.

Without Online fine-tuning LightTune with Standard Adam Update LightTune with One-step Update

medium and high SNR regimes are summarized in Table VII. We note that the ofﬂine ML method [17] shows degraded performance due to a training-test mismatch, having been trained exclusively on TDL-A30 low SNR data. This occurs despite using ACKs/NACKs to calibrate the model output, highlighting the need for an online learning mechanism to help ofﬂine ML models generalize to the unseen conditions.

Without Online fine-tuning LightTune with Standard Adam Update LightTune with One-step Update

0.6

0.5

0.8

0.4 0.6 0.3 0.4 0.2 0.2

0.1 0

0 0

SNR (dB)

(a) TDL-A30 (Dop. freq. = 10 Hz).

D. RI and CQI Selection

(b) TDL-B50 (Dop. freq. = 30 Hz)

Fig. 5: False alarm (FA) probability with and without online ﬁne-tuning under low antenna correlation with CSI-RS period = 80 ms, where τBLER = 0.9. Table-based OLLA Baseline CQI-Tune [Uniform] (Med.: 12.1%, High: 2.0%) CQI-Tune [Hard] (Med.: 9.0%, High: 2.1%) Offline ML with Calibration [Baknina et al.]

Table-based OLLA Baseline CQI-Tune [Uniform] (Med.: 9.1%, High: 1.3%) CQI-Tune [Hard] (Med.: 12.4%, High: 12.2%) Offline ML with Calibration [Baknina et al.]

100

Normalized Throughput (%)

100

0 0

SNR (dB)

(a) TDL-B50 (Dop. freq. = 30 Hz).

(b) TDL-C200 (Dop. freq. = 50 Hz)

Fig. 6: Throughput of CQI-Tune, low correlation, CSI-RS period = 80 ms.

sampling, respectively. In the high SNR regime, CQI-Tune achieves gains of approximately 2.0% for both strategies. • CQI Selection for TDL-C200. CQI-Tune achieves medium SNR throughput gains of 9.1% and 1.3% with uniform and hard sampling, respectively. In the high SNR regime, the hard sampling strategy signiﬁcantly outperforms the uniform scheme, achieving a gain of 12.2% compared to 1.3%. Medium SNR Gain

High SNR Gain

Channel

Uniform

Hard

Uniform

Hard

TDL-B50, 30 Hz TDL-C200, 50 Hz

12.1% 9.1%

9.0% 12.4%

2.0% 1.3%

2.1% 12.2%

TABLE VII: Throughput gains of CQI-Tune with uniform and hard sampling over the table-based OLLA baseline, low correlation, CSI-RS period = 80 ms.

The throughput gains for both sampling strategies across the

We compare RI-CQI-Tune against CQI-Tune with uniform sampling, since it has favorable convergence guarantees as discussed in Appendix D, under different antenna correlation scenarios and different CSI-RS periods. Low antenna correlation. Our results in Fig. 7 under low antenna correlation indicate that CQI-Tune is highly effective in the medium SNR range, while RI-CQI-Tune demonstrates higher gains in the high SNR range. These ﬁndings suggest an adaptive selection strategy: leveraging CQI-Tune for mediumrange SNR, while enabling RI-CQI-Tune in the high SNR range to optimize peak spectral efﬁciency through rank adaptation. We summarize the gains under low antenna correlation in Table VIII. Medium, high correlations and shorter CSI-RS periods. Next, we consider different antenna correlation scenarios and shorter CSI-RS periods as summarized in Table IX and Table X. For a CSI-RS period of 40 ms, CQI-Tune achieves throughput gains by up to 2.4%, whereas RI-CQI-Tune yields gains as high as 11.5%. When the CSI-RS period is reduced to 10 ms, the maximum gains for CQI-Tune and RICQI-Tune moderate to 2.2% and 7.7%, respectively. Notably, the relative performance enhancement scales inversely with the CSI-RS frequency. As the reference signal period decreases, the baseline algorithm more effectively tracks channel variations and mitigates CSI aging, thereby reducing the available headroom for further optimization via ﬁne-tuning. VII. C ONCLUSION This paper presented LightTune, an efﬁcient online ﬁnetuning framework that enables post-deployment model adaptation through a backpropagation-free, FF mechanism. By leveraging a buffer-less, opportunistic update strategy, LightTune enables real-time reﬁnement of ML models on resource-

0 0

Table-based OLLA CQI-Tune [Med.: 7.0%, High: 0.7%] RI-CQI-Tune [Med.: 6.3%, High: 11.0%]

100

Normalized Throughput (%)

Table-based OLLA CQI-Tune [Med.: 12.1%, High: 2.0%] RI-CQI-Tune [Med.: 8.1%, High: 1.1%]

100

0 0

SNR (dB)

(a) TDL-A10 (Dop. freq. = 20 Hz).

0 0

SNR (dB)

(b) TDL-B50 (Dop. freq. = 30 Hz).

Table-based OLLA CQI-Tune [Med.: 9.1%, High: 1.3%] RI-CQI-Tune [Med.: 8.5%, High: 10.9%]

100

Normalized Throughput (%)

Table-based OLLA CQI-Tune [Med.: 5.3%, High: 1.3%] RI-CQI-Tune [Med.: 2.6%, High: 2.6%]

100

SNR (dB)

(d) TDL-C200 (Dop. freq. = 50 Hz).

Fig. 7: Throughput of RI-CQI-Tune and CQI-Tune with uniform sampling under low antenna correlation with CSI-RS period = 80 ms.

Channel TDL-A10, 20 Hz TDL-B50, 30 Hz TDL-B200, 50 Hz TDL-C200, 50 Hz

Medium SNR Gain

High SNR Gain

CQI-Tune RI-CQI-Tune

5.3% 12.1% 7% 9.1%

2.6% 2% 0.7% 1.3%

1.3% 8.1% 6.3% 8.5%

2.6% 1.1% 11% 10.9%

TABLE VIII: Throughput gains, low correlation, CSI-RS period = 80 ms. Medium SNR Gain Channel

High SNR Gain

CQI-Tune RI-CQI-Tune CQI-Tune RI-CQI-Tune

TDL-B50, 30 Hz (Med. Corr.) TDL-B50, 30 Hz (High Corr.) TDL-C200, 50 Hz (Low Corr.) TDL-C200, 50 Hz (High Corr.)

0.3% 0.5% 3% 0.2%

1.6% 2% 6.7% 5.2%

0.4% 2.4% 0.7% 0.6%

5.2% 4.2% 9.1% 11.5%

TABLE IX: Throughput gains with CSI-RS period = 40 ms.

constrained devices. When applied to link adaptation, LightTune signiﬁcantly improved throughput compared to the con-

ventional outer-loop table-based baseline. While we utilized the FF algorithm for a lightweight UE-side implementation, the core principles of this opportunistic, buffer-less framework could be extended to backpropagation-based architectures. Extending LightTune to other 6G physical layer tasks, such as PMI and beam selection, is an area for future investigation. ACKNOWLEDGMENT The authors thank Carlos Feres for his review of the convergence proof, and Junho Lee and Seungjin Choi for their insightful discussions and assistance with the implementation. R EFERENCES [1] R. E. Ali and F. Penna, “LightTune: Lightweight Online Fine-Tuning for 6G,” in IEEE International Conference on Communications (ICC), 2026. [2] 3rd Generation Partnership Project (3GPP), “Study on Artiﬁcial Intelligence (AI)/Machine Learning (ML) for NR Air Interface,” 3GPP, Technical Report 38.843, Release 18, 2023.

Medium SNR Gain Channel

CQI-Tune

TDL-B50, 30 Hz (Low Corr.) TDL-C200, 50 Hz (Low Corr.) TDL-A10, 20 Hz (Med. Corr.) TDL-C200, 50 Hz (Med. Corr.)

2% 0.7% 0.6% 0%

High SNR Gain

RI-CQI-Tune CQI-Tune 1% 0.1% 0.2% 3.1%

0.2% −0.2% 0.2% 2.2%

RI-CQI-Tune 1.8% 7.7% 3% 6.1%

TABLE X: Throughput gains with CSI-RS period = 10 ms.

[3] P. Kaswan et al., “Statistical AI/ML model monitoring for 5G/6G: Interference prediction case study,” in IEEE International Conference on Communications Workshops (ICC Workshops), 2024. [4] J. Xu et al., “Learning to estimate: A real-time online learning framework for MIMO-OFDM channel estimation,” IEEE Transactions on Wireless Communications, 2024. [5] ——, “Learning at the speed of wireless: Online real-time learning for AI-enabled MIMO in NextG,” IEEE Communications Magazine, 2024. [6] Samsung, “AI/ML Use Cases and Framework for 6GR,” 3GPP TSG RAN1 Meeting #122, Bengaluru, India, R1-2505588, Aug. 2025. [7] V. Saxena, H. Tullberg, and J. Jaldén, “Reinforcement learning for efﬁcient and tuning-free link adaptation,” IEEE Transactions on Wireless Communications, vol. 21, no. 2, 2021. [8] Q. An et al., “DRAGON: A DRL-based MIMO Layer and MCS Adapter in Open RAN 5G Networks,” in Proceedings of the 30th Annual International Conference on Mobile Computing and Networking, 2024. [9] G. Hinton, “The Forward-Forward Algorithm: Some Preliminary Investigations,” arXiv preprint arXiv:2212.13345, 2022. [10] L.-J. Lin, “Self-improving reactive agents based on reinforcement learning, planning and teaching,” Machine learning, vol. 8, no. 3, 1992. [11] T. Schaul et al., “Prioritized Experience Replay,” ICLR, 2016. [12] D. Rolnick et al., “Experience Replay for Continual Learning,” Advances in neural information processing systems, vol. 32, 2019. [13] E. Peralta et al., “Outer loop link adaptation enhancements for ultra reliable low latency communications in 5G,” in IEEE 95th Vehicular Technology Conference:(VTC-Spring), 2022. [14] Z. Dong et al., “Machine learning based link adaptation method for MIMO system,” in IEEE 29th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), 2018. [15] T. Van Le and K. Lee, “Machine-learning-aided link-performance prediction for coded MIMO systems,” IEEE Transactions on Vehicular Technology, vol. 71, no. 3, 2021. [16] R. E. Ali and H. Kwon, “Online Adaptation and ML-Non-ML Combining for Improved Wireless Link Adaptation,” US Patent, 2026. [17] A. Baknina and H. Kwon, “Adaptive CQI and RI Estimation for 5G NR: A Shallow Reinforcement Learning Approach,” in IEEE Global Communications Conference (GLOBECOM), 2020. [18] Y. Huang, Y. T. Hou, and W. Lou, “DELUXE: A DL-based link adaptation for URLLC/eMBB multiplexing in 5G NR,” IEEE Journal on Selected Areas in Communications, vol. 40, no. 1, 2021. [19] A. Mazumdar, S. Paris, A. Amiri, K. I. Pedersen, and R. Adeogun, “Enhancing olla via exponential decay for efﬁcient link adaptation in emerging 6g trafﬁc,” IEEE Access, vol. 14, pp. 5764–5776, 2026. [20] R. Wiesmayr, L. Maggi, S. Cammerer, J. Hoydis, F. A. Aoudia, and A. Keller, “Salad: Self-adaptive link adaptation,” arXiv preprint arXiv:2510.05784, 2025. [21] L. Maggi, B. Bonev, R. Wiesmayr, S. Cammerer, and A. Keller, “Sinr estimation under limited feedback via online convex optimization,” arXiv preprint arXiv:2603.02061, 2026. [22] M. O. Torres, M. Lange, and A. P. Raulf, “On Advancements of the Forward-Forward Algorithm,” arXiv preprint arXiv:2504.21662, 2025. [23] F. Schroff, D. Kalenichenko, and J. Philbin, “Facenet: A uniﬁed embedding for face recognition and clustering,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015. [24] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in International Conference on Learning Representations (ICLR), 2015. [25] H. Karimi, J. Nutini, and M. Schmidt, “Linear convergence of gradient and proximal-gradient methods under the polyak-łojasiewicz condition,”

in Joint European conference on machine learning and knowledge discovery in databases. Springer, 2016, pp. 795–811. [26] Y. Nesterov, Introductory lectures on convex optimization: A basic course. Springer Science & Business Media, 2013, vol. 87. [27] M. S. Pinsker, “Information and information stability of random variables and processes,” Holden-Day, 1964. [28] 3GPP, “Physical layer procedures for data (release 16),” Technical Speciﬁcation (TS) 38.214, 2021. [29] B. Huang and A. Aminifar, “TinyFoA: Memory efﬁcient forwardonly algorithm for on-device learning,” in Proceedings of the AAAI Conference on Artiﬁcial Intelligence, 2025. [30] F. De Vita et al., “µ-FF: on-device forward-forward training algorithm for microcontrollers,” in IEEE Conference on Smart Computing, 2023. [31] 3GPP, “Study on channel model for frequencies from 0.5 to 100 GHz,” Tech. Rep. TR 38.901 V14.0.0, July 2017. [32] 3GPP, “User equipment (UE) radio transmission and reception,” Tech. Rep. TS 36.101, 2024. [33] Y. LeCun et al., “Gradient-based learning applied to document recognition,” Proc. IEEE, vol. 86, no. 11, 1998. [34] M. Pezeshki, “Implementation of Forward-Forward (FF) training algorithm,” https://github.com/mpezeshki/pytorch forward forward, 2023. [35] R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge university press, 2012.

Parameter

Value

Network Architecture Activation Function Optimizer Learning Rate α Threshold T Epochs

[784, 500, 500] ReLU Adam (β1 = 0.9, β2 = 0.999, ǫ = 10−8 ) 0.03 2 1,000

TABLE XI: MNIST training conﬁguration Loss Function

Softplus

Proposed

Test Accuracy

93.15%

93.74%

TABLE XII: Test accuracy on MNIST using Softplus loss vs. proposed loss

A PPENDIX B P ERFORMANCE UNDER DIFFERENT A NTENNA C ORRELATIONS AND CSI-RS PERIODS

In this appendix, we ﬁrst provide the throughput results under different antenna correlations. The throughput gains of our proposed schemes under medium and high correlation are A PPENDIX A up to 12.6% as summarized in Table XIII. Such gains are T HEORETICAL I NSIGHTS AND E XPERIMENTAL particularly noteworthy given that the underlying ofﬂine ML VALIDATION OF THE P ROPOSED L OSS F UNCTION model was trained exclusively on TDL-A30 data at low SNR We derive our alternative quadratic loss from the second- with low antenna correlation. The ability of the framework order Taylor expansion of the function f (x) = ln(1 + ex ), to adapt to unseen spatial correlation scenarios underscores its viability for practical 6G deployments where real-world centered at x = 0 that is given as channel statistics often deviate from training environments. 1 1 f (x) = ln 2 + x + x2 + R2 (x), (30) While CQI-Tune achieves signiﬁcant throughput improve2 8 ments in the vast majority of scenarios, marginal losses where R2 (x) denotes the Lagrange remainder term, deﬁned as (≤ 0.6%) were observed in a few cases due to the FA ξ (3) (1−eξ ) 3 events discussed in Remark 4. In contrast, the proposed joint R2 (x) = f 3!(ξ) x3 = 16 e(1+e for some ξ ∈ [0, x]. ξ )3 x We apply this expansion to the Softplus loss while discarding framework, RI-CQI-Tune, exhibited no such performance the higher-order remainder R2 (x). Because the optimization degradation across any of the evaluated scenarios. landscape is invariant to constant shifts, we discard the conMedium SNR Gain High SNR Gain stant ln 2 term. Finally, to eliminate the fractional coefﬁcients Correlation Channel CQI-Tune RI-CQI-Tune CQI-Tune RI-CQI-Tune and simplify the hardware arithmetic, we apply a uniform TDL-B50, 30 Hz 1.1% 1.7% 0.8% 0.9% Medium scaling factor of 8. This yields our proposed loss function TDL-B200, 50 Hz 1.5% 6.2% 3.4% 15.5%

i∈S+

1 |S- |Ml

Ml XX

i∈S- j=1

(i) (i) (g-,l [j] − T )2 + 4(g-,l [j] − T ) .

To validate the proposed loss function, we compare the test accuracy with that of the Softplus loss function on MNIST dataset [33]. The settings of this experiment are summarized in Table XI. Since the size of an MNIST image is 28 × 28, then the input layer in our MLP is of size 784 as the image is ﬂattened ﬁrst. In this experiment, positive and negative samples are constructed by replacing the ﬁrst 10 pixels of an image by the one-hot encoding of the true label y+ for positive samples and by the one-hot encoding of an incorrect label yfor negative samples [9]. The details and code are available in [34]. Our results in Table XII demonstrate that the proposed loss function achieves comparable accuracy while offering enhanced computational efﬁciency.

0.6% −0.6%

TDL-B50, 30 Hz TDL-B200, 50 Hz

High

1.4% 1.4%

0.1% 0.7%

2.1% 12.6%

TABLE XIII: Throughput gains under medium and high antenna correlation with CSI-RS period = 80 ms.

We also provide the performance under shorter CSI-RS periods as provided in Fig. 11 and Fig. 10.

Table-based OLLA CQI-Tune [Med.: 1.1%, High: 0.8%] RI-CQI-Tune [Med.: 1.7%, High: 0.9%]

100

Table-based OLLA CQI-Tune [Med.: 1.5%, High: 3.4%] RI-CQI-Tune [Med.: 6.2%, High: 15.5%]

100

Normalized Throughput (%)

Ml XX 1 (i) (i) (g+,l [j] − T )2 − 4(g+,l [j] − T ) |S+ |Ml j=1

Normalized Throughput (%)

LProp,l =

0 0

SNR (dB)

(a) TDL-B50 (Dop. freq. = 30 Hz).

SNR (dB)

(b) TDL-B200 (Dop. freq. = 50 Hz).

Fig. 8: Throughput of RI-CQI-Tune and CQI-Tune with uniform sampling under medium antenna correlation with CSI-RS period = 80 ms.

Table-based OLLA CQI-Tune [Med.: 0.6%, High: 0.1%] RI-CQI-Tune [Med.: 1.4%, High: 2.1%]

Table-based OLLA CQI-Tune [Med.: -0.6%, High: 0.7%] RI-CQI-Tune [Med.: 1.4%, High: 12.6%]

100

Normalized Throughput (%)

100

0 0

SNR (dB)

(a) TDL-B50 (Dop. freq. = 30 Hz).

(b) TDL-B200 (Dop. freq. = 50 Hz).

Fig. 9: Throughput of RI-CQI-Tune and CQI-Tune with uniform sampling under high antenna correlation with CSI-RS period = 80 ms. Parameter

Value

Neural Network Size Training Learning Rate α Learning Rate Schedule

12 × 64 × 64 × 1 0.001 initially Decays over 10 steps, then restarts with cycles Peak learning rate remains constant Minimum rate is 1 × 10−5 ReLU 250 83,200 0.0909

Activation Function Epochs Training Samples η

TABLE XIV: Hyperparameters of the ofﬂine ML-based approach in [16], [17]

A PPENDIX C O FFLINE ML WITH C ALIBRATION

(1)

(t)

Ft = σ θ (1) , z+ , . . . , z+

(32)

i.e., the σ-algebra generated by all random variables up to time t, containing all information available at that time. Note that y-(t) is not included because it is used only to construct θ (t+1) and is not needed for conditioning. When we write Ey-(t) [· | Ft ], we mean expectation with respect to the random draw of y-(t) given the past and the current sample. We denote by ED2 [·] expectation under the online data distribution, and by ED1 [·] expectation under the ofﬂine training data distribution. The unconditional expectation E[·] without a subscript denotes expectation over all sources of randomness; when needed we indicate the measure explicitly.

B. Justiﬁcation of Assumptions

The ofﬂine ML with calibration method was proposed in [17] and subsequently extended in [16]. We brieﬂy describe the scheme of [16], which employs an MLP to predict the spectral efﬁciency (SE) for all possible RI and CQI candidate pairs, and subsequently selects the pair that maximizes the estimated SE. To mitigate training-test mismatch, the recorded ACKs/NACKs are leveraged to compute an empirical SE estimate used to reﬁne the ML output. Speciﬁcally, the calibrated SE estimate for a given pair (RI = i, CQI = q) is deﬁned as Ŝ(i, q) = (1 − η) · SML (i, q) + η · S(i, q),

are independent of the online data stream. The online data (t) (t) {z+ = (x(t) , y+ )}∞ t=1 are i.i.d. from D2 . At each time t, a negative label y-(t) is drawn independently from a conditional (t) (t) distribution R(· | y+ ) over Y \ {y+ }; for example, R could be uniform over the other labels. The exact form of R is not needed beyond the existence of a gradient lower bound (Assumption 4). The algorithm updates the parameters according to the rule of LightTune, producing a sequence θ (t) . The ﬁltration Ft is deﬁned as

(31)

where η ∈ [0, 1] is a hyperparameter, SML (·) is the ML estimate of the SE, and S(·) is the empirical SE derived from inﬁnite impulse response (IIR) ﬁltering of the ACK/NACK history. We provide the hyperparameters used in Table XIV. We set η 1 η = 11 ≈ 0.0909 to achieve a scaling ratio of 1−η = 0.1. This is since maximizing the calibrated objective Ŝ is equivalent to η S, thereby assigning a maximizing Ŝ/(1 − η) = SML + 1−η relative weight of exactly 0.1 to the empirical SE. A PPENDIX D C ONVERGENCE A NALYSIS OF L IGHT T UNE This appendix provides the convergence proof of LightTune. We begin with our notations.

A. Probability Space and Notations All random variables are deﬁned on a common probability space (Ω, F , P). The initial parameters θ (1) are obtained from ofﬂine training on data drawn i.i.d. from D1 and

We provide an intuitive rationale for our assumptions. While these justiﬁcations are not intended as formal proofs, they clarify the logic of our convergence analysis. We ﬁrst note that Algorithm 1 does not explicitly enforce the parameter bound in Assumption 2(ii) via projection or weight decay. This is a standard assumption in the analysis of non-convex stochastic optimization, and in our experiments we empirically observe that the iterates remain bounded throughout ﬁne-tuning. Next, we note that Assumption 4 is motivated by the observation that a vanishing expected squared norm would necessitate a perfect cancellation between the positive and negative gradients in expectation. In the context of the FF algorithm, such systematic cancellation is highly improbable as long as the model is driven by non-negligible prediction errors. In particular, if the negative label y-(t) is drawn from a distribution that assigns positive probability to every incorrect (t) label (e.g., uniform distribution over Y \ {y+ }), then (t) Ey-(t) k∇LL k22 = 0 =⇒ kp+nc k2 = 0 (t)

(t)

for every c 6= y+ ,

where p = ∇L+,l and nc = ∇L-,l for label c. Hence nc = −p for all c. This would mean that the gradient does not depend on the label c, implying that the network’s response to different incorrect labels is identical in the sense of its gradient. When the prediction error is large, the true label and the predicted label differ, consequently, the gradients for those labels cannot all be the same. Therefore, for any reasonable sampling scheme that covers all incorrect labels (such as uniform sampling), the expected squared gradient norm must be bounded below by a positive constant γ2 (δ) > 0. This justiﬁes Assumption 4.

0 0

Table-based OLLA CQI-Tune [Med.: 0.6%, High: 0.2%] RI-CQI-Tune [Med.: 0.2%, High: 3.0%]

100

Normalized Throughput (%)

Table-based OLLA CQI-Tune [Med.: 0.7%, High: -0.2%] RI-CQI-Tune [Med.: 0.1%, High: 7.7%]

100

0 0

SNR (dB)

0 0

SNR (dB)

(a) TDL-B50, Low correlation (Dop. freq. = 30 Hz).

Table-based OLLA CQI-Tune [Med.: -0.0%, High: 2.2%] RI-CQI-Tune [Med.: 3.1%, High: 6.1%]

100

Normalized Throughput (%)

Table-based OLLA CQI-Tune [Med.: 2.0%, High: 0.2%] RI-CQI-Tune [Med.: 1.0%, High: 1.8%]

100

SNR (dB)

(b) TDL-C200, Low correlation (Dop. freq. = 50 Hz).

SNR (dB)

(d) TDL-C200, Medium correlation (Dop. freq. = 50 Hz).

Fig. 10: Throughput of RI-CQI-Tune and CQI-Tune with uniform sampling with CSI-RS period = 10 ms.

SNR (dB)

(a) TDL-B50, Medium correlation (Dop. freq. = 30 Hz).

(b) TDL-B50, High correlation (Dop. freq. = 30 Hz).

SNR (dB)

Table-based OLLA CQI-Tune [Med.: 0.2%, High: 0.6%] RI-CQI-Tune [Med.: 5.2%, High: 11.5%]

100

0 0

Table-based OLLA CQI-Tune [Med.: 3.0%, High: 0.7%] RI-CQI-Tune [Med.: 6.7%, High: 9.1%]

100

Normalized Throughput (%)

Table-based OLLA CQI-Tune [Med.: 0.5%, High: 0.4%] RI-CQI-Tune [Med.: 2.4%, High: 4.2%]

100

Normalized Throughput (%)

Table-based OLLA CQI-Tune [Med.: 0.3%, High: 0.4%] RI-CQI-Tune [Med.: 1.6%, High: 5.2%]

100

0 0

SNR (dB)

(d) TDL-C200, High correlation (Dop. freq. = 50 Hz).

Fig. 11: Throughput of RI-CQI-Tune and CQI-Tune with uniform sampling with CSI-RS period = 40 ms.

C. Preliminary Lemmas We recall and introduce some notations. The input to layer (t) (t) l is denoted as h+,l−1 for a positive sample and by h-,l−1 for (t) a negative sample. We use the augmented notation h̃+,l−1 = (t) (t) (t) [h+,l−1 , 1]⊤ and h̃-,l−1 = [h-,l−1 , 1]⊤ for positive and negative samples, respectively. The parameters of neuron j in layer l are θl,j = [wl,j , bl,j ]⊤ . Hence, the positive pre-activation is (t) ⊤ (t) given as p+,l [j] = θl,j h̃+,l−1 , and the negative pre-activation (t) ⊤ (t) is given as p-,l [j] = θl,j h̃-,l−1 . We recall from (6), the loss function for a single neuron j in layer l combines positive and negative contributions as follows (t)

(t)

(t) − 4 (h+,l [j])2 − T 2 (t) (t) + (h-,l [j])2 − T + 4 (h-,l [j])2 − T , (33)

Ll [j](θl,j ) = (h+,l [j])2 − T

(t)

where h+,l [j] = max(0, p+,l [j]) and h-,l [j] = max(0, p-,l [j]). The total loss for layer l is the average over all neurons:

and the gradient of the negative branch is given similarly as h i (t) (t) ∇θl,j L-,l = 4 (h-,l [j])2 − T + 2 (t)

(t)

1 Xl (t) L [j](θl,j ). Ml j=1 l

Proof. We prove by induction on the layer index l. (t)

Base case l = 0: By deﬁnition, h0 = z (t) , and Assump(t) tion 2(i) gives kz (t) k2 ≤ Bz . Hence kh0 k2 ≤ Bz =: B0 . Inductive step: Assume that for some l ≥ 1, we have (t) khl−1 k2 ≤ Bl−1 . Consider layer l. For any neuron j in this layer, the pre-activation is (t)

(t)

⊤ pl [j] = θl,j h̃l−1 , (t)

(34)

(t)

where h̃l−1 = [hl−1 , 1]⊤ is the augmented input. By the Cauchy–Schwarz inequality and Assumption 2(ii), (t)

(t)

|pl [j]| ≤ kθl,j k2 kh̃l−1 k2 ≤ Bθ (khl−1 k2 + 1) (t)

(t)

(38)

(t)

The activation is hl [j] = max(0, pl [j]), so |hl [j]| ≤ (t) |pl [j]|. Therefore, v u Ml uX (t) (t) khl k2 = t (hl [j])2 j=1

p ≤ Ml Bθ (Bl−1 + 1) =: Bl .

h i (t) (t) ∇θl,j L+,l = 4 (h+,l [j])2 − T − 2 (t)

(37)

≤ Bθ (Bl−1 + 1).

As derived in (9), the gradient of the positive branch with respect to its parameter vector θl,j for a neuron j in layer l is given as

× h+,l [j] · 1{p+,l [j] > 0} · h̃+,l−1 ,

(36)

We begin with the proof of Lemma 1 which shows that the activations are bounded.

(t) Ll (θ) =

(t)

× h-,l [j] · 1{p-,l [j] > 0} · h̃-,l−1 .

(39)

This completes the induction. (35)

Finally, since the network has a ﬁnite number of layers L, the

set {B0 , B1 , . . . , BL } is ﬁnite. Taking Bh = max0≤l≤L Bl gives a uniform bound valid for all layers and all time steps.

Finally, we prove Lemma 4, which demonstrates that the proposed loss function is smooth. This smoothness property is a key enabler for our main convergence result.

Next, we provide the proof of Lemma 2 which shows that the gradients are also bounded.

Proof. Let dl be the total number of parameters in layer l. The parameter vector θl ∈ Rdl can be written as the concatenation of the parameter vectors for each neuron:

Proof. The gradient for neuron j is given as 1 (t) (t) (t) ∇θl,j L+,l [j] + ∇θl,j L-,l [j] , ∇θl,j Ll (θl ) = Ml

(40)

where all quantities on the right-hand side are evaluated at the (t) current parameters θl,j and the ﬁxed inputs from the previous layer. We ﬁrst bound each branch separately using the bounds from Assumption 2 and Lemma 1. For the positive branch, the (t) derivative of the scalar loss component is 2(g+,l [j] − T ) − 4 = (t) (t) 2(g+,l [j] − T − 2). Applying the chain rule with g+,l [j] = (t) (t) (h+,l [j])2 yields an additional factor of 2h+,l [j]. Thus, we can bound each term in the gradient as follows: (t)

|4((h+,l [j])2 − T − 2)| ≤ 4(Bh2 + T + 2), (t)

|h+,l [j]| ≤ Bh , (t)

kh̃+,l−1 k2 ≤ Bh + 1, (t)

|1{p+,l [j] > 0}| ≤ 1. Multiplying these bounds (since the gradient is the product of these terms) yields (t)

k∇θl,j L+,l [j]k2 ≤ 4(Bh2 + T + 2)Bh (Bh + 1).

⊤ ⊤ ⊤ θl = [θl,1 , θl,2 , . . . , θl,M ]⊤ ∈ R d l , l

(43)

where θl,j ∈ Rdn are the parameters of neuron j (each neuron has the same number of parameters, including bias). The loss function for layer l is the average over neurons: M

(t) Ll (θl ) =

1 Xl (t) L [j](θl,j ), Ml j=1 l

(44)

(t)

where each Ll [j](θl,j ) depends only on θl,j and not on the parameters of other neurons. a) Gradient structure.: Because the loss separates over neurons, the gradient with respect to θl is the concatenation of the per-neuron gradients:   (t) ∇θl,1 Ll [1](θl,1 )   (t) ∇θl,2 Ll [2](θl,2 )  1  (t) .  (45) ∇Ll (θl ) = ..  Ml  .   (t) ∇θl,Ml Ll [Ml ](θl,Ml )

b) Hessian structure.: Differentiating again, the Hessian (t) matrix ∇2 Ll (θl ) ∈ Rdl ×dl is block-diagonal: Applying the same reasoning to the negative branch gives the 1 (t) (t) (t) identical bound: ∇2 Ll (θl ) = diag ∇2θl,1 Ll [1], . . . , ∇2θl,M Ll [Ml ] . l Ml (t) (46) k∇θl,j L-,l [j]k2 ≤ 4(Bh2 + T + 2)Bh (Bh + 1). (42) 2 This is because cross-partial derivatives ∂θl,i∂∂θl,j for i 6= j Finally, using the triangle inequality and the expression for the are zero, as each term in the sum depends only on its own (t) total gradient evaluated at θl , we obtain neuron’s parameters. c) Spectral norm of a block-diagonal matrix.: For a 1 (t) (t) (t) (t) k∇θl,j Ll (θl )k2 ≤ k∇θl,j L+,l [j]k2 + k∇θl,j L-,l [j]k2 block-diagonal matrix, the spectral norm equals the maximum Ml 2 of the spectral norms of the individual blocks [35]. Therefore, 8(Bh + T + 2)Bh (Bh + 1) . ≤ Ml 1 (t) (t) k∇2 Ll (θl )k2 = max ∇2 L [j](θl,j ) 2 . (47) Ml 1≤j≤Ml θl,j l (41)

Next, we prove Lemma 3 showing that our proposed loss function is bounded. Proof. Recall that for any neuron, g = h2 . From Lemma 1, we have |h| ≤ Bh , so 0 ≤ g ≤ Bh2 . Since g is bounded by Bh2 , we have |(g − T )2 − 4(g − T )| ≤ (Bh2 + T )2 + 4(Bh2 + T )

≤ (Bh2 + T + 2)2 + 4(Bh2 + T + 2),

and similarly for the negative branch term. Adding the two bounds and noting that the loss for the layer is the average over neurons, we obtain (t)

|LL (θL )| ≤ (Bh2 + T + 2)2 + 4(Bh2 + T + 2) =: M.

Thus it sufﬁces to bound the Hessian of a single neuron; the full Hessian norm will be at most that bound divided by Ml . d) Bounding the per-neuron Hessian.: Fix a neuron j and time t, and drop the indices l, j, t for brevity. When p+ > 0 (neuron active), we have (48) ∇L+ = 4p3+ − 4(T + 2)p+ h̃+ .

+ Differentiating again with respect to θ (using ∂p ∂θ = h̃+ ) yields the Hessian (49) ∇2 L+ = 12p2+ − 4(T + 2) h̃+ h̃⊤ + .

When p+ ≤ 0, the neuron is inactive and the Hessian is zero. Similarly, when p- > 0, (50) ∇2 L- = 12p2- − 4(T + 2) h̃- h̃⊤ - ,

and zero otherwise. By Lemma 1, kh̃+ k2 ≤ Bh + 1 and kh̃- k2 ≤ Bh + 1. Moreover, when p+ > 0, we have p+ = h+ ≤ Bh , so |p+ | ≤ Bh . Hence k∇2 L+ k2 ≤ (12Bh2 + 4T + 8)(Bh + 1)2 .

(51)

The same bound holds for k∇2 L- k2 . By the triangle inequality, k∇2 L+ + ∇2 L- k2 ≤ 2(12Bh2 + 4T + 8)(Bh + 1)2 = 8(3Bh2 + T + 2)(Bh + 1)2 := ρ̄l .

(52)

Thus we have shown that for any neuron j, (t) k∇2θl,j Ll [j](θl,j )k2 ≤ ρ̄l = 8 3Bh2 + T + 2 (Bh + 1)2 . (53)

Using the block-diagonal structure and the fact that the spectral norm of a block-diagonal matrix is the maximum of the block norms scaled by the overall factor 1/Ml , we obtain

1 ρ̄l (t) max k∇2θl,j Ll [j]k2 ≤ =: ρl . Ml j Ml (54) e) From Hessian bound to Lipschitz gradient.: Now we prove that this bound on the Hessian implies the gradient is ρl -Lipschitz everywhere. Take any θl , θl′ ∈ Rdl and consider the line segment (t)

k∇2 Ll (θl )k2 ≤

θl (s) = θl + s(θl′ − θl ),

s ∈ [0, 1].

(55)

contributions are polynomials in the parameters, and the parameters are afﬁne in s). Hence it is inﬁnitely differentiable on the open interval and extends continuously to the endpoints. 4) Derivative on a smooth subinterval. On any subinterval (t) where ∇Ll (θ(s)) is C 1 , we can differentiate: d (t) (t) ∇Ll (θl (s)) = ∇2 Ll (θl (s)) (θl′ − θl ), (58) ds where the Hessian exists everywhere on the interval because the activation pattern is constant. From the bound on (t) the Hessian, we have k∇2 Ll (θl (s))k2 ≤ ρl , so d (t) ∇Ll (θl (s)) ≤ ρl kθl′ − θl k2 . ds 2

(59)

5) Integration over each subinterval. Apply the fundamental theorem of calculus on each subinterval. Because (t) ∇Ll (θ(s)) is continuously differentiable on the open interval and continuous up to the endpoints, we have (t)

(t)

∇Ll (θl (si+1 )) − ∇Ll (θl (si )) Z si+1 (t) ∇2 Ll (θl (s)) (θl′ − θl ) ds. = si

Summing these equalities from i = 0 to m (with s0 = 0, sm+1 = 1) telescopes the left-hand side, giving

1) Points where the gradient may not be differentiable. For each neuron k in layer l, its pre-activation along the segment is (56) pk (s) = θl,k (s)⊤ h̃l−1 ,

∇Ll (θl′ ) − ∇Ll (θl ) m Z si+1 X (t) ∇2 Ll (θl (s)) (θl′ − θl ) ds. =

where θl,k (s) is the part of θ(s) corresponding to neuron k, and h̃l−1 is ﬁxed (it comes from the sample at time t and does not depend on s). This is an afﬁne function of s, i.e., pk (s) = ak s + bk for some constants ak , bk .

6) Norm estimate. Taking norms and using the triangle inequality,

2) Zeros of afﬁne functions are isolated. For a ﬁxed k, the equation pk (s) = 0 is linear in s. Hence it has either: 1) no solution (if ak = 0 and bk 6= 0), 2) exactly one solution s∗k (if ak 6= 0) or 3) the whole interval (if ak = 0 and bk = 0, which would mean the pre-activation is identically zero; this degenerate case occurs on a set of measure zero and can be ignored). Thus each neuron contributes at most one point where pk (s) = 0. 3) The exceptional set is ﬁnite. Since there are ﬁnitely many neurons, the set S0 = {s ∈ [0, 1] : ∃k such that pk (s) = 0}

(57)

is ﬁnite. Order its elements as 0 ≤ s1 < · · · < sm ≤ 1. Remove these points to obtain a partition of [0, 1] into subintervals [0, s1 ], [s1 , s2 ], . . . , [sm , 1]. On each such subinterval, no pre-activation changes sign, so the activation pattern (which neurons are active) remains ﬁxed. Consequently, on each subinterval, the gradient (t) ∇Ll (θ(s)) is a polynomial in s (because the per-neuron

(t)

i=0

(t)

k∇Ll (θl′ ) − ∇Ll (θl )k2 m Z si+1 X (t) k∇2 Ll (θl (s))k2 kθl′ − θl k2 ds ≤ si

i=0

≤ ρl kθl′ − θl k2 (t)

Thus, Ll

m X i=0

(si+1 − si ) = ρl kθl′ − θl k2 .

is ρl -smooth.

D. Convergence Theorem We now provide the proof of Theorem 1. Proof. We proceed in steps as follows. (t) 1) Local decrease. For any t, if Iδ = 1, the algorithm performs a gradient update: (t+1)

θL

(t)

= θL − αf ∇LL (θL ).

(60)

(t)

Because LL is ρL -smooth (Lemma 4), we can apply the (t) (t+1) descent lemma (Lemma 5) with θ = θL and θ ′ = θL : (t)

(t+1)

LL (θL

(t)

(t+1)

) ≤ LL (θL ) + ∇LL (θL )⊤ (θL ρL (t+1) (t) + − θL k22 . kθ 2 L

(t)

− θL ) (61)

(t+1)

Substituting the update θL gives

(t)

− θL = −αf ∇LL (θL )

(t+1)

The same inequality holds for θL upper bounds into (67) yields

. Substituting these

(t) (t) (t) (t) αf γ2 (δ) (t) (1) (t) (1) (t+1) ) ≤ LL (θL ) − αf k∇LL (θL )k22 ED2 [Iδ ]. ED1 [LL (θL )] ≤ ED1 [LL (θL )] − 2 2 ρL αf (72) (t) (t) k∇LL (θL )k22 (62) + 2 ρL αf 5) Summation and telescoping. Summing (72) from t = 1 (t) (t) (t) (t) = LL (θL ) − αf 1 − k∇LL (θL )k22 . to N , 2 (63) N αf γ2 (δ) X (1) (1) (t) ρL αf ρL αf 1 1 ED2 [Iδ ] ≤ ED1 [LL (θL )]− Since αf < 1/ρL , we have 2 < 2 , hence 1− 2 > 2 . 2 t=1 Therefore, (1) (N +1) )]. (73) ED1 [LL (θL αf (t) (t+1) (t) (t) (t) (t) LL (θL ) ≤ LL (θL ) − k∇LL (θL )k22 . (64) 2 (1) 6) Bounding the ﬁnal term. Let L∗L = inf θ ED2 [LL (θ)]. (t) (t) (t) (t) (t+1) If Iδ = 0, no update occurs, so LL (θL ) = LL (θL ). Applying Lemma 6 again, Combining both cases yields q (1) (N +1) (1) (N +1) 1 (θ )] − M [L (θ )] ≥ E [L E D D αf 2 1 L L L L (t) (t) (t) (t+1) (t) (t) (t) 2 DKL LL (θL ) ≤ LL (θL ) − k∇LL (θL )k22 Iδ . (65) q 2 ≥ L∗L − M 12 DKL . (74) 2) Conditional expectation under D2 . Conditioning on Ft (t) (t) Hence (which ﬁxes θL , x(t) , y+ ) and using the gradient lower (1) (N +1) (1) (1) bound (Assumption 4), )] ED1 [LL (θL )] − ED1 [LL (θL q (t) (t+1) (1) (1) Ey-(t) LL (θL ) | Ft ≤ ED1 [LL (θL )] − L∗L + M 12 DKL . (75) αf (t) (t) (t) 2 (t) (t) (t) I Ey-(t) k∇LL (θL )k2 | Ft , Iδ = 1 7) Final bound. Combining and dividing by N yields our ≤ LL (θL ) − 2 δ bound αf γ2 (δ) (t) (t) (t) ≤ LL (θL ) − Iδ . (66) (1) (1) N 2 2 ED1 [LL (θL )] − L∗L 1 X (t) ED2 [Iδ ] ≤ N t=1 αf γ2 (δ)N 3) Total expectation under D2 . Taking expectation under D2 , r αf γ2 (δ) (t) (t) (t) (t) (t+1) 2DKL (D2 kD1 ) 2M ED2 [Iδ ]. ED2 [LL (θL )] ≤ ED2 [LL (θL )] − + . (76) 2 αf γ2 (δ) N (67) (t) Recalling that ED2 [Iδ ] = PrD2 (e(t) ≥ δ) completes the 4) Relating to D1 via Pinsker. For any ﬁxed θ, by Lemma proof. (t) 6 applied with P = D2 , Q = D1 , and f = LL (θ), r 1 (t) (t) Next, we provide the proof of Corollary 1. |ED2 [LL (θ)] − ED1 [LL (θ)]| ≤ M DKL (D2 kD1 ). 2 (68) Proof. From Theorem 1, we have for every N ≥ 1, (t) Since θL is independent of the sample at time t, we can N 1 X B A (t) condition on θL and integrate: +√ , Pr (e(t) ≥ δ) ≤ N t=1 D2 N N (t) (t) (t) (t) ED2 [LL (θL )] = E ED2 [LL (θ) | θ = θL ] (1) (1) 2[ED1 [LL (θL )]−L∗ q L] and B = where A = (t) (t) α γ (δ) 2 1 f p ≤ E ED1 [LL (θ) | θ = θL ] + M 2 DKL 2M 2DKL (D2 kD1 ) are constants independent of q αf γ2 (δ) √ (t) (t) 1 = ED1 [LL (θL )] + M 2 DKL . (69) N . As N → ∞, both A/N → 0 and B/ N → 0. By the squeeze theorem, the left-hand side also tends to 0. By Assumption 3, the ofﬂine training data are i.i.d. from (1) (t) Finally, we provide the proof of Corollary 2. D1 , so for any ﬁxed θ, ED1 [LL (θ)] = ED1 [LL (θ)]. (t) Moreover, because θL is independent of the sample at Proof. When D1 = D2 , we have DKL (D2 kD1 ) = 0. Substitime t under D1 as well, tuting into Theorem 1 yields (t) (t) (t) (t) N ED1 [LL (θL )] = E ED1 [LL (θ) | θ = θL ] 1 X A Pr(e(t) ≥ δ) ≤ , (1) (t) N t=1 N = E ED1 [LL (θ) | θ = θL ] (t)

(t+1)

LL (θL

(1)

(t)

= ED1 [LL (θL )].

(70)

Thus (t)

(t)

(1)

(t)

ED2 [LL (θL )] ≤ ED1 [LL (θL )] + M

1 2 DKL .

(71)

with A as deﬁned above. Multiplying by N , we obtain for every N ≥ 1, N X Pr(e(t) ≥ δ) ≤ A. t=1

The right-hand side A is a constant independent of N , and all terms are non-negative. Hence, the partial sums are uniformly bounded and nondecreasing, so they converge to a ﬁnite limit as N → ∞. Therefore, ∞ X t=1

Pr(e(t) ≥ δ) < ∞.

A necessary condition for a convergent series is that its terms tend to zero; thus Pr(e(t) ≥ δ) → 0 as t → ∞.

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