Data-Free Client Contribution Estimation via Logit Maximization for Federated Learning Asim Ukaye1 , Nurbek Tastan1 , Mubarak Abdu-Aguye1 , and Karthik Nandakumar1,2 MBZUAI, Abu Dhabi, UAE Michigan State University, Michigan, USA
arXiv:2605.18892v1 [cs.LG] 17 May 2026
1
2
Abstract. Federated learning (FL) enables collaborative learning of computer vision models, where privacy and regulatory constraints prevent centralizing data across devices or organizations. However, practical FL deployments often exhibit severe class imbalance and label skew, causing standard aggregation protocols to overfit dominant clients and degrade minority-class performance. We propose a data-free, classwise contribution estimation and aggregation framework based on logit maximization (CELM) that does not require sharing raw data, client metadata, or auxiliary public datasets. The FL server probes client updates to obtain class-wise evidence scores and assembles a cross-client evidence matrix, which quantifies both per-class competence and class coverage. Using this matrix, we compute contribution weights that upweight clients providing strong, discriminative evidence for underrepresented classes. The resulting aggregation is stable due to simplex constraints and momentum smoothing, and it remains compatible with standard FL training pipelines. We evaluate the approach on representative vision benchmarks under controlled non-IID and pathological label splits, demonstrating that CELM-based aggregation improves robustness to imbalance and statistical heterogeneity, while yielding better performance without requiring any additional data exchange. Keywords: Federated Learning · Collaborative Fairness · Logit Maximization
1
Introduction
Federated learning (FL) enables collaborative model training across distributed clients without sharing raw data [15]. This paradigm is especially attractive for computer vision systems deployed across organizations, edge devices, and geographically distributed silos [29]. However, practical FL deployments are rarely independent and identically distributed (non-IID): clients differ in dataset size, class support, and label frequency [10,11,23,24]. Under such heterogeneity, standard aggregation can over-emphasize dominant clients and under-represent minority classes, leading to unstable optimization and degraded fairness across classes [7].
2
A.Ukaye et al.
Fig. 1: Overview of CELM. During the initial warm-up communication rounds, the server probes each client update using class-wise logit maximization, constructs a debiased client–class evidence matrix, and converts normalized class-wise evidence into contribution weights for aggregation. After warm-up, these contribution weights are frozen and reused for the remaining rounds to stabilize training and reduce overhead.
A core bottleneck is how to estimate client contribution without relying on selfreported metadata or server-side validation data. Metadata-driven weighting can be unreliable or strategically manipulated, while validation-based scoring may be unavailable, biased, or costly to maintain. Recent data-free methods partially address this issue, but many are still anchored to similarity-to-average assumptions in gradient/update space, which can undervalue informative clients that carry rare classes or atypical yet useful signals. In this work, we propose a data-free, class-wise contribution estimation mechanism based on logit maximization. During an initial warm-up window, the server probes each client model class-by-class by optimizing noisy inputs to maximize class logits. These probes yield a client-class evidence matrix that captures relative per-class evidence across clients. We then de-bias evidence using a globalmodel reference, normalize class-wise shares, and aggregate them into client contribution scores used for weighted model aggregation. This design offers three main advantages. First, it does not require any raw data exchange, client-reported metadata, or auxiliary validation set. Second, classwise normalization improves robustness under label skew by valuing relative class evidence rather than dominant-class overlap. Third, warm-up estimation followed by score freezing and EMA smoothing yields stable aggregation with bounded server overhead. These properties make the method effective not only for standard non-IID splits, but also for extreme cases such as Maverick clients (informative rare-class holders) and free-riders (non-informative contributors). We evaluate the method on representative vision benchmarks, including FashionMNIST, CIFAR-10, and FedISIC, under controlled non-IID partitions and
Data-free Contribution Estimation via Logit Maximization
3
stress-test settings. Beyond predictive performance, we analyze three additional aspects: (i) performance with rare-class holding clients (Mavericks), (ii) free-rider detection from contribution estimates, and (iii) fidelity of estimated client/global class distributions compared to the true distribution. Together, these studies show that logit-maximization-based scoring is both useful for aggregation and semantically meaningful as a proxy for client distributional contribution. The key contributions of this paper are: – We introduce a data-free, class-wise contribution estimator for FL based on server-side logit maximization. – We propose an evidence construction and class-share scoring rule that emphasizes informative rare-class clients. – We design a warm-up-and-freeze aggregation strategy with EMA smoothing that improves stability and limits overhead. – We provide extensive empirical evidence across non-IID, Maverick/free-rider, and distribution-fidelity evaluations, showing consistent gains in robustness and minority-class behavior without data exchange.
2
Related Work
Contribution Estimation in Federated Learning. A central challenge in FL is to fairly weight or reward clients based on their actual contribution to the global model. Early work such as FedAvg [15] uses the number of local samples as a proxy for client weight, implicitly assuming honest self-reporting. However, when incentives are tied to contribution, this assumption may not hold [21]. More recent approaches rely on an auxiliary validation set to assess each client’s update. For example, CFFL [12] and FedCE [6] evaluate client models on server-side validation data to estimate their goodness. Yet, validation sets may be unavailable or biased relative to client data distributions, limiting the reliability of these metrics [8–11, 23]. Data-Free and Non-Self-Reported Methods. To avoid privacy risks or reliance on client metadata, several works propose data-free scoring of client updates [20–22,26]. CGSV [26] computes the cosine similarity between the gradient of each client and the average gradient of the cohort, rewarding updates that are better aligned with the average of the group. ShapFed [20] improves upon the gradient alignment strategy by proposing class-wise gradient alignment for the final layer of the model. This allows for better estimates of client contributions in scenarios with a high skew in class labels within each client. These methods highlight a growing interest in server-side evaluation of client updates without any private or auxiliary data. Free-riders and Mavericks in FL. Recent FL works highlight two challenging client types under heterogeneity: Mavericks [4, 5, 27], which may hold rare but highly informative classes, and free-riders [2, 3], which contribute weak or
4
A.Ukaye et al.
non-informative updates while still benefiting from global training. Similarity-toaverage weighting can suppress Maverick clients because their updates are intentionally atypical, while naive sample-count or uniform weighting can over-credit free-riders. Prior fairness and reputation-oriented methods [12, 21, 22, 25, 26] address parts of this issue by differentiating client rewards or reducing the impact of low-quality contributors, but most operate primarily in update space rather than explicitly class-wise evidence space. Our method complements this line by estimating class-wise evidence directly from client model behavior, which helps preserve rare-class signal and downweight non-informative clients during aggregation. Logit/Activation Maximization. Activation maximization (AM), including class logit maximization objectives, has been used to probe class-selective behavior and internal representations of deep networks by optimizing inputs to increase a target neuron or class logit [1, 19, 28]. Subsequent work emphasized practical priors and optimization choices to improve interpretability and reduce high-frequency artifacts, including feature-visualization refinements [17] and reconstruction-based analyses that motivate smoothness-inducing regularization [13]. In contrast to interpretability-only use, our method repurposes classwise logit maximization as a server-side probing mechanism in federated learning: we probe each client model, derive debiased class evidence scores, and convert them into aggregation weights without requiring raw data sharing. In summary, our work differs from prior methods by (i) requiring no auxiliary validation data, (ii) avoiding similarity-to-average assumptions, (iii) offering an inexpensive proxy for client contribution and (iv) explicitly supporting Maverick/free-rider stress settings through class-wise evidence modeling.
3
Preliminaries
3.1
Federated Learning Setup
We consider a cross-silo federated learning (FL) system with N clients indexed by i ∈ N , where N = {1, . . . , N }. Each client i has a private dataset Di = i {(xi,j , yi,j )}nj=1 , where xi,j ∈ X denotes an image with class label yi,j ∈ Y, X represents the image space, Y = {1, . . . , K} represents the label space, K is the number of classes, and ni is the size of the training set of client i. The overarching goal of the system is to enable clients to collaboratively learn a multi-class classifier gw : X → Y without sharing their private data, where g represents the model architecture and w ∈ Rd denotes the model parameters. Let Fi (w; (x, y)) = Li (gw (x), y) be the per-sample loss function at client i. The global optimization objective of the FL system is: \min _{w\in \mathbb {R}^d} f(w), \quad f(w)=\frac {1}{N}\sum _{i=1}^{N} f_i(w), \quad f_i(w)=\mathbb {E}_{(\mathbf {x},y)\sim \mathcal {D}_i}\big [F_i(w;(\mathbf {x},y))\big ]. \label {eq:fl-objective}
(1)
Data-free Contribution Estimation via Logit Maximization
5
(0)
The server initializes the global model parameters wg . At communication round (t−1) t ∈ [1, . . . , T ], the server broadcasts the global parameters wg . Each client performs local optimization based on this initialization and returns updated (t) weights wi . The server then aggregates the client models using client-specific (t) (t) weights c(t) = [c1 , . . . , cN ]⊤ : w_g^{(t)} = \sum _{i=1}^{N} c_i^{(t)} w_i^{(t)}, \qquad c_i^{(t)}\ge 0, \qquad \sum _{i=1}^{N} c_i^{(t)}=1. \label {eq:weighted-agg}
(2)
In the classical FedAvg algorithm [15], clients are assigned weights based on their (t) self-reported sample sizes, i.e., ci = PNni n . Unlike FedAvg, we aim to estimate j=1
j
c(t) from class-wise evidence obtained by server-side probing of the client models, without using any raw client data or self-reported metadata. 3.2
Activation Maximization
Activation maximization (AM) is a well-known approach [1] that attempts to find an input that maximizes a target functional T (x) under a regularization prior R(x): \mathbf {x}^{\star }=\arg \max _{\mathbf {x}\in \mathcal {X}}\; \mathcal {T}(\mathbf {x})-\lambda \mathcal {R}(\mathbf {x}), \label {eq:am-general}
(3)
where λ ≥ 0 controls the activation-regularization trade-off. Previous works in the AM literature have used total variation (TV) regularization to suppress high-frequency artifacts, especially for image inputs [13, 28].
4
Proposed Methodology
We present a data-free contribution assessment method for federated learning called Contribution Estimation from Logit Maximization (CELM). The method has three stages: (i) extract class-specific evidence with logit maximization, (ii) convert that evidence into stable aggregation weights, and (iii) reuse frozen weights after warm-up to reduce overhead. We assume an honest-but-curious server with no additional privacy guarantees beyond standard FL. 4.1
Logit Maximization Framework
In this work, we utilize the class-specific logit activation as the target functional in AM. Let sc (x; w) be the pre-softmax logit for class c ∈ Y output by the classifier gw based on input x. Given only a classification model gw , the logit maximization (LM) framework attempts to find an input x that maximizes the activation of the class-specific logits, i.e. T (x) = sc (x; w). For a given model gw and class c, we solve the following problem.
q_{c}=s_{c}(\mathbf {x}_{c}^{\star };w). \label {eq:probe-score}
(5)
This optimized logit score qc can be used as a proxy for the class-specific evidence. The intuition behind this choice is as follows. If the model gw is trained on sufficient samples from a specific class c, it can be expected to learn the attributes of this class well. Consequently, when probed using the LM framework, it should be possible to obtain an input that results in a high logit value qc . In contrast, if the model gw is trained on a negligible number of samples from class c, it cannot learn the characteristics of this class well, and consequently, it will fail to produce a high logit score qc under LM probing. 4.2
Class-specific Client Contribution Estimation in FL
Now, consider the FL setup described in Sec. 3.1. At communication round t, (t) after receiving client models {wi }N i=1 , the server performs class-wise probing for each client model. For each client i ∈ N and class c ∈ Y, the server randomly initializes an input x and runs L gradient ascent steps with step size η to obtain: \mathbf {x}_{i,c}^{\star ,(t)} = \arg \max _{\mathbf {x}} \; s_{c}(\mathbf {x}; w_i^{(t)})-\lambda \lVert \mathbf {x}\rVert _2^2. \label {eq:celm_probe_obj}
(6)
The above objective drives the synthetic input toward class-specific evidence while regularization controls degenerate solutions. We then compute the raw client-class logit as: \tilde {q}_{i,c}^{(t)} = s_{c}(\mathbf {x}_{i,c}^{\star ,(t)}; w_i^{(t)}). \label {eq:celm_q_raw}
(7)
To de-bias this value, we run the same logit-maximization step in Eq. (6) on (t−1) the global model from the previous round wg for class c to obtain the corre⋆,(t−1) sponding optimized input xg,c . The bias is then estimated as: b^{(t-1)} = \frac {1}{K}\sum _{c=1}^{K} s_{c}\!\left (\mathbf {x}_{g,c}^{\star ,(t-1)}; w_g^{(t-1)}\right ). \label {eq:celm_baseline}
(8)
This baseline captures the global model’s generic confidence level and helps remove shared calibration effects across clients. The final evidence score applies baseline subtraction followed by ReLU suppression: q_{i,c}^{(t)} = \max \!\left (0,\; \tilde {q}_{i,c}^{(t)} - b^{(t-1)}\right ). \label {eq:celm_q}
(9)
Data-free Contribution Estimation via Logit Maximization
7
(t)
Stacking all qi,c forms the evidence matrix Q(t) ∈ RN ×K . In practice, this matrix is the key intermediate object that summarizes class-wise client utility for round t. Note that each communication round in FL aggregates the client models to obtain a global model, which is then used as the initialization for local training in the next round. Therefore, after the initial communication rounds, the global model will acquire knowledge about all the classes as long as one or more clients have sufficient samples of each class. Consequently, it becomes increasingly difficult to obtain class-specific evidence from the client models. Hence, we apply the LM framework only when t ≤ Tw , where Tw is the warm-up horizon. To capture relative class expertise, CELM normalizes evidence across clients for each class: r_{i,c}^{(t)} = \frac {q_{i,c}^{(t)}}{\sum _{j=1}^{N} q_{j,c}^{(t)} + \epsilon }, \label {eq:celm_rel}
(10)
where ϵ > 0 ensures numerical stability. This class-wise normalization is important as it prevents globally dominant clients from trivially dominating all classes. We will highlight the importance of this normalization in the later sections. The client contribution score is then the average relative share over classes: \hat {c}_i^{(t)} = \frac {1}{K}\sum _{c=1}^{K} r_{i,c}^{(t)}. \label {eq:celm_avg}
(11)
This averaging step gives a single interpretable score per client while still preserving class-aware evidence in the computation. Next, we compute instantaneous simplex-normalized scores as follows: \bar {c}_i^{(t)} = \frac {\hat {c}_i^{(t)}}{\sum _{j=1}^{N} \hat {c}_j^{(t)}}, \qquad \bar {c}_i^{(t)}\ge 0,\; \sum _{i=1}^{N}\bar {c}_i^{(t)}=1. \label {eq:celm_simplex}
(12)
Subsequently, we apply exponential moving average (EMA) smoothing with factor β ∈ [0, 1): c_i^{(t)} = \beta c_i^{(t-1)} + (1-\beta )\bar {c}_i^{(t)}. \label {eq:celm_ema}
(13)
EMA dampens the round-to-round volatility in estimated contributions and improves aggregation stability when client updates are noisy. CELM computes c(t) only during the initial Tw communication rounds. During this warm-up phase, the server computes the full global model by aggregating the client models, but only shares the global backbone with the clients. The clients retain their local classifier layer; empirically, this preserves a stronger class-discriminative signal in the probed logits. After the warm-up phase, the final score vector is frozen: \mathbf {c}^{(t)} = \mathbf {c}^{(T_w)}, \qquad \forall ~ t > T_w. \label {eq:celm_freeze}
(14)
A.Ukaye et al.
4
4
3
3
3
2 1
Clients
4
Clients
Clients
8
2 1
0
1
0 0 1 2 3 4 5 6 7 8 9
Classes
(a) Dirichlet (α = 0.05)
2
0 0 1 2 3 4 5 6 7 8 9
Classes
0 1 2 3 4 5 6 7 8 9
Classes
(b) Pure Label Skew (PLS) (c) Step Label Skew (SLS)
Fig. 2: Visualization of the non-IID split settings used in our experiments. Each bubble plot encodes the client-class label allocation matrix: the bubble size is proportional to the number of samples of a given class held by a given client. The top marginal summarizes per-class totals across all clients, while the right marginal summarizes perclient totals across all classes.
After Tw , the server broadcasts the full aggregated model (including classifier head) in each subsequent round. This design keeps per-round overhead low after early calibration while preserving class-aware contribution signals. Algorithm 1 in the appendix summarizes the complete procedure.
5
Experimental Setup
Datasets and Models. We evaluate CELM on FashionMNIST, CIFAR-10, and FedISIC. For FashionMNIST, we use a 4-layer MLP. For CIFAR-10, we use a randomly initialized 5-layer CNN. For FedISIC, we use an ImageNet-1k pretrained ViT-B/16. Training Hyperparameters. We simulate 5 clients for FashionMNIST and CIFAR-10, and 6 clients for FedISIC. Each client performs one local epoch per communication round. We run 100 rounds for FashionMNIST, and 200 rounds for CIFAR-10 and FedISIC. Client-side optimization uses vanilla SGD with crossentropy loss and batch size 128. The initial learning rate is 0.1 for FashionMNIST and CIFAR-10, and 0.01 for FedISIC, with a step schedule that decays the learning rate by a factor of 0.1 after 50 rounds. The hyperparameters chosen are tuned on the FedAvg algorithm to achieve optimal performance with the fewest rounds. LM and Contribution Estimation Hyperparameters. For logit-maximization probing, we use Adam with an LM learning rate 0.01 and 200 optimization steps per class-client probe. The ℓ2 -regularization coefficient λ is set to 0.001. The images for LM are initialized with random Gaussian noise from the standard normal distribution for the first round. For subsequent rounds the final image from the previous round is used as the starting point. The EMA factor for contribution smoothing is set to β = 0.5. The warm-up horizon is set to 5% of total communication rounds, i.e., Tw = 0.05T .
Data-free Contribution Estimation via Logit Maximization
9
Data Splits. For FashionMNIST and CIFAR-10, we consider three non-IID regimes, illustrated in Fig. 2. – Pure Label Skew (PLS) assigns different numbers of classes to different clients while keeping the total number of samples per client fixed, isolating the effect of label-support imbalance. – Step Label Skew (SLS) increases both the number of classes and the sample count across clients in a step-wise manner, creating a coupled labeland-quantity heterogeneity pattern. – Dirichlet splits use class-wise Dirichlet sampling with concentration parameter α ∈ {0.01, 0.05, 0.1} to generate randomized non-IID partitions; smaller α yields stronger skew and larger cross-client imbalance. For FedISIC, we use the naturally provided client partition. Baselines and comparison protocol. We compare CELM against FedAvg, CFFL, CGSV, and ShapFed, which together represent strong and commonly used baselines for contribution-aware FL under heterogeneous data. Among these, CGSV and ShapFed are data-free and do not rely on client metadata, making them the most directly comparable non-metadata baselines to CELM. CFFL, in contrast, uses client-side validation information to score updates and therefore does not follow the strict no-metadata/no-validation constraint adopted by CELM. To keep the comparison principled under our no client-metadata setting, we report FedAvg with uniform client weighting (instead of sample-count weighting), so no client-reported metadata is used in aggregation. All performance results are reported as mean±std. over three random seeds.
6
Results
Tab. 1 shows that CELM is consistently competitive and often the best across heterogeneous settings, with especially clear gains under stronger non-IID skew. On FashionMNIST, CELM achieves the top score in four of the five reported splits: Dirichlet(0.01), Dirichlet(0.05), Dirichlet(0.10), and PLS. The improvements over FedAvg are most pronounced in challenging skewed regimes, e.g., PLS (83.70 vs. 80.62) and Dirichlet(0.05) (83.64 vs. 81.63), indicating that classwise contribution weighting provides meaningful robustness when label support is uneven across clients. On CIFAR-10, CELM is the strongest method in all five reported splits. The margins are again largest in highly heterogeneous settings: ++2.29 points over FedAvg in PLS (59.11 vs. 56.82), +8.8 over CFFL in Dirichlet(0.1), and +2.05 over ShapFed in SLS. Even in milder skew, CELM remains consistently ahead, suggesting that warm-up probing and debiased evidence construction produce stable aggregation weights rather than overfitting to any single partition pattern. FedISIC (natural split), CELM yields the best balanced accuracy (70.18). , substantially outperforming the next best method ShapFed (62.31) by a margin of +7.87. This large gap on a naturally partitioned clinical dataset supports the
10
A.Ukaye et al.
Table 1: Global predictive performance of the methods across datasets and splits. †We report the balanced accuracy for FedISIC to account for the label imbalance in its test set. The best results are shown in bold. Second-best results are underlined. Dataset Split
FedAvg
CFFL
CGSV ShapFed
CELM
Dir. (0.01) 80.64±2.25 78.89±2.93 47.40±7.82 81.15±1.83 81.76±1.85 Dir. (0.05) 81.63±2.69 81.89±0.45 46.30±3.34 81.97±2.61 83.64±0.42 F.MNIST Dir. (0.10) 84.83±0.62 84.58±0.64 63.37±3.71 85.14±0.65 85.34±0.09 PLS 80.62±0.52 80.95±1.81 49.41±2.17 81.80±0.56 83.70±0.19 SLS 83.13±2.61 88.13±0.35 55.39±5.06 84.17±2.07 87.00±1.14 Dir. (0.01) 63.41±1.42 53.59±6.74 18.25±6.09 63.12±1.39 64.37±0.98 Dir. (0.05) 65.12±1.89 55.92±9.31 24.48±1.57 65.83±1.71 67.16±1.29 CIFAR-10 Dir. (0.10) 68.98±0.67 60.41±4.85 29.03±3.16 68.98±0.42 69.21±0.76 PLS 56.82±2.96 49.93±6.28 28.75±4.92 56.80±2.80 59.11±1.96 SLS 67.40±0.95 70.47±2.10 33.58±1.76 69.91±0.43 71.96±0.08 FedISIC† Natural
61.25±0.04 62.60±6.34 26.17±0.59 62.31±0.16 70.18±0.58
practical utility of CELM beyond synthetic splits, where both label-distribution mismatch and client variability are intrinsic. Overall, Tab. 1 highlights three trends: (i) CELM is most beneficial when heterogeneity is severe, (ii) performance gains are consistent across different skewgeneration mechanisms (PLS, SLS, and Dirichlet), and (iii) the approach transfers from benchmark-style partitions to real-world natural partitions. These observations align with our design objective of class-aware, data-free contribution estimation for robust federated aggregation. 6.1 Performance on clients with unique classes (Mavericks) We next evaluate a Maverick split, where one or more clients hold distinctive or rare classes that are weakly represented in the rest of the federation. This setting is important because conventional averaging can underweight such clients, even when they contain critical class information for global generalization. As illustrated in Fig. 3, Mavericks are structurally different from free-rider-like clients. Even when both possess few labels, a free-rider mainly carries labels that are globally well-represented. As a contrast to this, mavericks carry unique classes that are absent globally. A classic real-world example of this scenario is hospitals with few labels for rare diseases. CELM is naturally suited to this regime because its scoring rule is class-aware by construction. For each class, CELM computes relative class share across clients and then aggregates these class-wise shares into a final contribution score. As a result, a client with strong evidence on a rare class is not drowned out by clients that dominate only frequent classes. This behavior is what we want under Maverick splits, i.e. to reward informative rarity rather than majority overlap. Tab. 2 confirms this hypothesis quantitatively. On FashionMNIST, CELM achieves the best balanced accuracy (87.32) and the strongest rare-class accuracy (90.77),
4
4
3
3
3
2 1
Clients
4
Clients
Clients
Data-free Contribution Estimation via Logit Maximization
2 1
0
11
2 1
0
0
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
(a) Maverick Client
(b) Free-rider Client
(c) Maverick and Free-rider
Classes
Classes
Classes
Fig. 3: Maverick versus free-rider client patterns. Bubble area is proportional to class sample count for each client–class pair. (a) A Maverick client contributes distinctive class evidence; (b) a free-rider client contributes weak or non-informative class signal; (c) the mixed setting used to evaluate whether aggregation methods can emphasize informative rare-class clients while suppressing non-contributors. Table 2: Predictive performance under the Maverick split (clients with unique/rare class support). We report balanced accuracy and rare-class accuracy to assess whether each method correctly values informative minority-class clients. The best results are shown in bold. Second-best results are underlined. Dataset
FedAvg
CFFL
CGSV ShapFed
CELM
F.MNIST
Balanced Acc. 84.79±0.24 84.39±2.43 50.74±0.23 84.87±0.32 87.32±0.10 Rare Class Acc. 81.76±0.68 82.99±8.56 0.00±0.00 82.02±0.81 90.77±0.24
CIFAR-10
Balanced Acc. 64.75±0.29 62.10±1.15 42.12±4.17 64.14±0.27 68.60±0.53 Rare Class Acc. 39.95±1.06 44.69±18.06 0.00±0.00 38.16±1.13 61.21±0.59
improving substantially over FedAvg and ShapFed on the rare classes. On CIFAR10, CELM again leads both balanced accuracy (68.60) and rare-class accuracy (61.21), with a particularly large gain on rare classes compared to all baselines. Notably, CGSV collapses to near-zero rare-class performance in both datasets, highlighting the limitation of similarity-to-average scoring when rare informative updates are present. Overall, these results show that CELM not only identifies Maverick clients but also translates that identification into better global performance on underrepresented classes. 6.2
Detecting Free Riders
We next evaluate free-rider detection using the synthetic client patterns shown earlier in Fig. 3. In particular, we consider two settings: (i) FR, where a single free-rider client is present, and (ii) FRM, where a free-rider co-exists with a Maverick client that carries rare but informative classes. This second setting is intentionally more challenging because a robust detector should suppress freeriders without mistakenly downweighting informative outlier clients.
12
A.Ukaye et al. FPR for Free-rider with Maverick split
0.4 0.3 0.2 0.1 0.0
0.5
False Positive Rate
False Positive Rate
FPR for Free-rider split
CFFL CGSV ShapFed CELM
0.5
CFFL CGSV ShapFed CELM
0.4 0.3 0.2 0.1 0.0
0.0
0.2
0.4
0.6
0.8
Threshold
1.0
0.0
(a) FashionMNIST
0.2
0.4
0.6
Threshold
0.8
1.0
(b) CIFAR-10
Fig. 4: FPR-versus-threshold curves for z-score-based free-rider detection. A client is flagged as a free-rider when its standardized contribution score falls below a threshold. (a) FashionMNIST under the FRM setting (free-rider + Maverick), and (b) CIFAR-10 under the FR setting (single free-rider). Lower curves indicate more robust detection over a broad threshold range. Table 3: Free-rider detection performance using z-score thresholding over contribution estimates. FR denotes a single free-rider split, and FRM denotes a free-rider plus Maverick split. AUROC is averaged across communication rounds. FPR is reported as the average false-positive rate over multiple detection thresholds. Higher AUROC and lower FPR indicate better discrimination between free-rider and informative clients. AUROC (↑) FPR(↓) CFFL CGSV ShapFed CELM CFFL CGSV ShapFed CELM
Dataset
Split
F.MNIST
0.52 FR FRM 0.67
1.00 0.79
1.00 0.97
1.00 1.00
0.38 0.29
0.00 0.24
0.01 0.23
0.00 0.08
CIFAR-10
FR 0.76 FRM 0.67
0.96 0.08
1.00 0.97
1.00 1.00
0.31 0.32
0.08 0.15
0.01 0.07
0.03 0.11
Our detection rule is built directly on each algorithm’s contribution estimates. For every communication round, we standardize client contributions into a zscore, then flag clients with z-scores below a threshold as free-riders. Sweeping the threshold produces a threshold–FPR curve and enables threshold-agnostic evaluation. Tab. 3 reports average AUROC across rounds for FR and FRM, while the FPR numbers are computed as the average false-positive rate across multiple threshold values. The results show that CELM maintains strong separability between informative and non-informative clients, especially in the more realistic FRM regime. Fig. 4 illustrates representative FPR-versus-threshold behavior for FashionMNIST (FRM) and CIFAR-10 (FR), where CELM remains competitive across a wide threshold range and avoids the instability seen in methods that rely on weaker class-aware signals.
Data-free Contribution Estimation via Logit Maximization
13
Table 4: Distribution-modelling fidelity of CELM. We report distances between the true class distribution and either (i) a uniform reference distribution or (ii) the CELMestimated distribution. Lower is better (↓). JSD: Jensen-Shannon Divergence, EMD: Earth Mover Distance, and Hellinger: Hellinger distance. Across settings, lower CELM distances indicate that CELM captures the global class distribution and client-level distribution structure more accurately than the uninformed uniform baseline.
6.3
Dataset
Split
JSD (↓) EMD(↓) Hellinger(↓) Uniform CELM Uniform CELM Uniform CELM
F.MNIST
Dir. (0.05) PLS
0.613 0.508
0.294 0.231
0.114 0.077
0.079 0.053
0.693 0.586
0.306 0.241
CIFAR-10
Dir. (0.01) SLS
0.646 0.433
0.145 0.200
0.124 0.066
0.036 0.046
0.735 0.500
0.151 0.201
FedISIC
Natural
0.535
0.265
0.097
0.051
0.583
0.286
Estimation Fidelity
Fig. 5 and Tab. 4 jointly evaluate how accurately CELM recovers client-level and cohort-level class structure without direct data access. The central question is whether class-wise evidence (qi,c ) extracted through logit-maximization probes can serve as a reliable proxy for the underlying label distributions across clients. Fig. 5 provides a qualitative view: panel (a) shows the true client distribution, panel (b) shows the CELM-estimated distribution, and panels (c)–(d) compare client marginals and global class marginals, respectively. Visually, the estimated distributions track the dominant classes and relative mass allocation patterns in the true distributions, indicating that CELM captures not only marginal trends but also global client-class distribution heterogeneity. Tab. 4 complements this visual evidence with quantitative distribution-distance metrics between true and estimated distributions. We report Jensen–Shannon Divergence (JSD) [14], Earth Mover Distance (EMD) [18], and Hellinger distance [16], and include a uniform-distribution baseline for context. Across all reported datasets/splits, CELM yields lower distances than the uniform baseline, showing that CELM estimates are meaningfully closer to the true datagenerating distributions. For example, on CIFAR-10 Dirichlet(0.01), JSD drops from 0.646 (uniform) to 0.145 (CELM), EMD from 0.124 to 0.036, and Hellinger from 0.735 to 0.151. Similar improvements appear on FashionMNIST (e.g., PLS: JSD 0.508 → 0.231) and FedISIC natural split (JSD 0.535 → 0.265). Importantly, these results indicate that CELM captures the global class distribution and not just isolated client signals. The reduction in EMD suggests better alignment in mass transport across class bins, while the consistent gains in JSD and Hellinger indicate improved probabilistic overlap with the true distribution. Together, Fig. 5 and Tab. 4 support the claim that data-free logit probing can recover distributional structure well enough to inform robust aggregation, which helps explain CELM’s performance advantages under non-IID training in Tab. 1.
14
A.Ukaye et al.
Total Data Share
4
Clients
True Estimated
0.5
5
3 2 1
0.4 0.3 0.2 0.1
0 0
1
2
3
4
Classes
5
6
0.0
7
0
(a) True Distribution
1
2
3
Client
0.6
Total Data Share
Clients
True Class share Estimated Class Share
0.5
3 2
5
(c) Client Marginals
5 4
4
1
0.4 0.3 0.2 0.1
0 0
1
2
3
4
Classes
5
6
7
(b) Estimated Distribution
0.0
0
1
2
3
4
Class Label
5
6
7
(d) Global Class Marginals
Fig. 5: Estimation fidelity of CELM on FedISIC dataset. Panels (a) and (b) compare the true and CELM-estimated client class distributions, while panels (c) and (d) compare client-level marginals and the aggregated global class marginals. The close visual agreement indicates that CELM preserves both per-client class tendencies and cohortlevel class prevalence without accessing raw client data.
7
Conclusion & Limitations
Our results consistently show that, class-wise, data-free contribution estimation can improve federated aggregation under realistic non-IID conditions. Across synthetic and natural splits, CELM shows strong gains in both overall and minority-sensitive metrics, indicating that class-aware weighting is more robust than purely size-based or similarity-to-average aggregation. The Maverick and free-rider experiments further strengthen the conclusion that CELM not only improves global predictive performance but also better distinguishes informative outlier clients from non-contributing clients. Together with the estimationfidelity analyses, these results suggest that logit probing captures a meaningful distributional structure that is directly useful for server-side decision making. At the same time, CELM introduces additional server-side compute during warm-up due to repeated logit-maximization probes across clients and classes. Although this overhead is bounded by the freeze-after-warm-up design, scaling to very large numbers of clients or classes may require subsampling or adaptive class selection. Another practical consideration is that detection quality depends
Data-free Contribution Estimation via Logit Maximization
15
on model confidence calibration. Highly miscalibrated client models can distort relative evidence estimates. Future work can therefore explore calibration-aware probing, adaptive warm-up schedules, and theoretical guarantees on contribution identifiability under extreme heterogeneity and adversarial participation.
References 1. Erhan, D., Bengio, Y., Courville, A., Vincent, P.: Visualizing higher-layer features of a deep network. University of Montreal 1341(3), 1 (2009) 2. Fraboni, Y., Vidal, R., Lorenzi, M.: Free-rider attacks on model aggregation in federated learning. In: International conference on artificial intelligence and statistics. pp. 1846–1854. PMLR (2021) 3. He, J., Wu, L., Zhang, Z., Lu, N., Wei, X.: Client evaluation and revision in federated learning: Towards defending free-riders and promoting fairness. In: AsiaPacific Web (APWeb) and Web-Age Information Management (WAIM) Joint International Conference on Web and Big Data. pp. 100–118. Springer (2024) 4. Huang, J., Hong, C., Liu, Y., Chen, L.Y., Roos, S.: Tackling mavericks in federated learning via adaptive client selection strategy. In: AAAI. vol. 2023 (2022) 5. Huang, J., Hong, C., Liu, Y., Chen, L.Y., Roos, S.: Maverick matters: Client contribution and selection in federated learning. In: Pacific-Asia Conference on Knowledge Discovery and Data Mining. pp. 269–282. Springer (2023) 6. Jiang, M., Roth, H.R., Li, W., Yang, D., Zhao, C., Nath, V., Xu, D., Dou, Q., Xu, Z.: Fair federated medical image segmentation via client contribution estimation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 16302–16311 (2023) 7. Kairouz, P., McMahan, H.B., Avent, B., Bellet, A., Bennis, M., Bhagoji, A.N., Bonawitz, K., Charles, Z., Cormode, G., Cummings, R., et al.: Advances and open problems in federated learning. Foundations and trends® in machine learning 14(1–2), 1–210 (2021) 8. Karimireddy, S.P., Kale, S., Mohri, M., Reddi, S., Stich, S., Suresh, A.T.: Scaffold: Stochastic controlled averaging for federated learning. In: International conference on machine learning. pp. 5132–5143. PMLR (2020) 9. Li, Q., He, B., Song, D.: Model-contrastive federated learning. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 10713– 10722 (2021) 10. Li, T., Sahu, A.K., Zaheer, M., Sanjabi, M., Talwalkar, A., Smith, V.: Federated optimization in heterogeneous networks. In: Dhillon, I., Papailiopoulos, D., Sze, V. (eds.) Proceedings of Machine Learning and Systems. vol. 2, pp. 429– 450 (2020), https://proceedings.mlsys.org/paper_files/paper/2020/file/ 1f5fe83998a09396ebe6477d9475ba0c-Paper.pdf 11. Li, X., Huang, K., Yang, W., Wang, S., Zhang, Z.: On the convergence of fedavg on non-iid data. In: International Conference on Learning Representations (2020), https://openreview.net/forum?id=HJxNAnVtDS 12. Lyu, L., Xu, X., Wang, Q.: Collaborative Fairness in Federated Learning (Aug 2020). https://doi.org/10.48550/arXiv.2008.12161 13. Mahendran, A., Vedaldi, A.: Visualizing deep convolutional neural networks using natural pre-images. International Journal of Computer Vision 120(3), 233–255 (2016)
16
A.Ukaye et al.
14. Manning, C., Schutze, H.: Foundations of statistical natural language processing. MIT press (1999) 15. McMahan, B., Moore, E., Ramage, D., Hampson, S., y Arcas, B.A.: Communication-efficient learning of deep networks from decentralized data. In: Artificial intelligence and statistics. pp. 1273–1282. PMLR (2017) 16. Nikulin, M.S., et al.: Hellinger distance. Encyclopedia of mathematics 78 (2001) 17. Olah, C., Mordvintsev, A., Schubert, L.: Feature visualization. Distill 2(11), e7 (2017) 18. Rubner, Y., Tomasi, C., Guibas, L.J.: A metric for distributions with applications to image databases. In: Sixth international conference on computer vision (IEEE Cat. No. 98CH36271). pp. 59–66. IEEE (1998) 19. Simonyan, K., Vedaldi, A., Zisserman, A.: Visualising image classification models and saliency maps. Deep Inside Convolutional Networks 2(2) (2014) 20. Tastan, N., Fares, S., Aremu, T., Horváth, S., Nandakumar, K.: Redefining Contributions: Shapley-Driven Federated Learning. In: Larson, K. (ed.) Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, IJCAI-24. pp. 5009–5017. International Joint Conferences on Artificial Intelligence Organization (8 2024). https://doi.org/10.24963/ijcai.2024/554, https://doi.org/10.24963/ijcai.2024/554, main Track 21. Tastan, N., Horváth, S., Nandakumar, K.: Aequa: Fair Model Rewards in Collaborative Learning via Slimmable Networks. In: Singh, A., Fazel, M., Hsu, D., Lacoste-Julien, S., Berkenkamp, F., Maharaj, T., Wagstaff, K., Zhu, J. (eds.) Proceedings of the 42nd International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 267, pp. 59210–59236. PMLR (13–19 Jul 2025), https://proceedings.mlr.press/v267/tastan25a.html 22. Tastan, N., Horváth, S., Nandakumar, K.: CYCle: Choosing Your Collaborators Wisely to Enhance Collaborative Fairness in Decentralized Learning. Transactions on Machine Learning Research (2025), https://openreview.net/forum? id=ygqNiLQqfH 23. Tastan, N., Horváth, S., Takáč, M., Nandakumar, K.: Fedpews: Personalized warmup via subnetworks for enhanced heterogeneous federated learning. In: Chen, B., Liu, S., Pilanci, M., Su, W., Sulam, J., Wang, Y., Zhu, Z. (eds.) Conference on Parsimony and Learning. Proceedings of Machine Learning Research, vol. 280, pp. 462–483. PMLR (24–27 Mar 2025), https://proceedings.mlr.press/v280/ tastan25a.html 24. Wang, J., Liu, Q., Liang, H., Joshi, G., Poor, H.V.: Tackling the objective inconsistency problem in heterogeneous federated optimization. Advances in neural information processing systems 33, 7611–7623 (2020) 25. Xu, X., Lyu, L.: A reputation mechanism is all you need: Collaborative fairness and adversarial robustness in federated learning. In: International Workshop on Federated Learning for User Privacy and Data Confidentiality in Conjunction with ICML 2021 (FL-ICML’21) (2021) 26. Xu, X., Lyu, L., Ma, X., Miao, C., Foo, C.S., Low, B.K.H.: Gradient Driven Rewards to Guarantee Fairness in Collaborative Machine Learning. In: Advances in Neural Information Processing Systems. vol. 34, pp. 16104–16117. Curran Associates, Inc. (2021) 27. Yang, M., Buyukates, B., Markopoulou, A.: Rewarding the rare: Maverick-aware shapley valuation in federated learning. Transactions on Machine Learning Research 28. Yosinski, J., Clune, J., Nguyen, A., Fuchs, T., Lipson, H.: Understanding neural networks through deep visualization. arXiv preprint arXiv:1506.06579 (2015)
Data-free Contribution Estimation via Logit Maximization
17
29. Zhuang, W., Xu, J., Chen, C., Li, J., Lyu, L.: COALA: A practical and visioncentric federated learning platform. In: Forty-first International Conference on Machine Learning (2024), https://openreview.net/forum?id=ATRnM8PyQX
18
A
A.Ukaye et al.
Algorithm
Algorithm 1 CELM: Contribution Estimation from Logit Maximization (0)
Input: Initial global model wg , clients N = {1, . . . , N }, classes Y = {1, . . . , K}, total rounds T , warm-up rounds Tw , LM steps L, LM step size η, regularization weight λ, EMA factor β 1: function LMProbe(w, Xinit , Y, L, η, λ) 2: for all class c ∈ Y in parallel do (0) 3: xc ← Xinit c 4: for ℓ = 0 to L − 1 do (ℓ) (ℓ) (ℓ) 5: gc ← ∇x sc (xc ; w) − λ∥xc ∥22 (ℓ+1) (ℓ) (ℓ) 6: xc ← Adam(xc , gc , η, β1 = 0.9, β2 = 0.999) 7: end for (L) (L) 8: uc ← sc (xc ; w), Xfinal ← xc c 9: end for final 10: return {uc }K c=1 , X 11: end function (0) 12: Initialize ci ← 1/N, ∀ i ∈ N (0) (0) 13: Initialize Xg and Xi ∀i ∈ N from Standard Gaussian Noise 14: for t = 1 to T do 15: if t ≤ Tw then (t−1) 16: Broadcast global backbone wg ; clients retain local heads 17: else (t−1) 18: Broadcast full global model wg 19: end if (t) 20: Clients perform local training and return {wi }N i=1 21: if t ≤ Tw then (t) (t) (t−1) (t−1) 22: {ug,c }K , Xg , Y, L, η, λ) c=1 , Xg ← LMProbe(wg PK (t) (t−1) 1 23: b ← K c=1 ug,c 24: for each client i ∈ N do (t−1) (t) (t) (t) , Y, L, η, λ) ← LMProbe(wi , Xi 25: {ui,c }K c=1 , Xi 26: for each class c ∈ Y do (t) (t) 27: qi,c ← max 0, ui,c − b(t−1) 28: end for 29: end for (t) (t) P (t) 30: Compute ri,c ← qi,c /( N j=1 qj,c ) P (t) (t) K 1 31: Compute ĉi ← K c=1 ri,c (t) (t) P (t) 32: Instantaneous score: c̄i ← ĉi / N j=1 ĉj (t)
(t−1)
(t)
33: EMA smoothing: ci ← βci + (1 − β)c̄i 34: else (t) (T ) 35: Freeze weights: ci ← ci w 36: end if P (t+1) (t) (t) 37: Aggregate global model: wg ← N i=1 ci wi 38: end for
Data-free Contribution Estimation via Logit Maximization
B
Additional Results
B.1
Results on Homogeneous Partition
19
Tab. 5 reports performance under homogeneous (IID) client distributions. We simulate homogeneous behavior by setting a high value for the skew parameter (α = 100) for the Dirichlet split. In this regime, all methods operate near their upper-bound behavior because there is little cross-client distribution shift to correct. CELM remains competitive with all the baselines on both datasets, indicating that class-wise contribution weighting does not introduce a penalty when client data are already balanced. Table 5: Global predictive performance under homogeneous (IID) client distributions. We report test accuracy (%,mean±std. ) on FashionMNIST and CIFAR-10. This setting serves as a sanity check showing that CELM remains competitive when non-IID skew is minimal. Dataset Split FedAvg
B.2
CFFL
CGSV ShapFed CELM
F.MNIST IID
89.23±0.09 89.04±0.04 86.57±0.25 89.19±0.07 89.16±0.05
CIFAR-10 IID
76.76±0.46 76.40±0.75 62.65±1.41 76.51±0.52 76.15±0.47
Number of Clients
Tab. 6 studies scalability of CELM to larger client size (N = 20) under Dirichlet non-IID splits. As the number of clients increases, each client typically sees fewer samples and stronger local skew, making contribution estimation harder. CELM is consistently best or tied-best across the splits, with the largest gains on CIFAR-10 under stronger heterogeneity (Dirichlet, α = 0.01 ).
Table 6: Global predictive performance for 20 clients under non-IID partitions. We report test accuracy (%,mean±std. ); best values per row are shown in bold. CELM shows consistent improvements, especially under stronger skew. Dataset Split
FedAvg
CFFL
CGSV
ShapFed
CELM
Dir. (0.01) 77.20±1.41 71.76±1.61 24.48±6.74 77.29±1.48 78.71±1.22 F.MNIST Dir. (0.05) 80.28±1.05 78.76±2.36 45.50±7.84 80.39±1.17 81.70±1.05 Dir. (0.1) 82.81±1.06 77.51±3.81 59.22±3.66 82.88±0.96 82.88±1.27 Dir. (0.01) 36.17±4.36 28.57±5.63 15.47±2.51 36.62±4.79 44.85±1.78 CIFAR-10 Dir. (0.05) 48.33±3.28 39.57±8.49 25.09±4.29 48.65±3.34 51.00±2.96 Dir. (0.1) 53.91±1.62 51.03±4.61 28.38±3.08 54.05±1.84 54.58±2.16
20
A.Ukaye et al.
B.3
Effect of number of classes
Tab. 7 compares the performance of CELM on the EMNIST dataset with a higher number of labels (47 classes), across multiple non-IID partitioning schemes. We simulate this split with 5 clients. Compared with the lower-class-count benchmarks, this setting is more sensitive to noisy client evidence. CELM remains competitive across most splits, albeit with smaller improvement margins than in the low-label-count setting. Table 7: Global predictive performance on a larger-class benchmark (EMNIST) across Dirichlet, PLS, and SLS non-IID splits. We report test accuracy (%,mean±std. ), with best values per row in bold. Dataset Split
FedAvg
CFFL
CGSV
ShapFed
CELM
Dir. (0.01) 82.12±0.15 81.67±0.63 15.02±2.89 82.23±0.20 82.42±0.32 Dir. (0.05) 82.99±0.77 83.09±0.35 22.12±7.11 82.99±0.79 83.15±0.49 EMNIST Dir. (0.1) 84.37±0.44 85.00±0.12 33.89±3.31 84.45±0.49 84.35±0.28 PLS 82.02±0.92 81.98±0.93 35.74±1.72 82.16±0.85 82.41±0.89 SLS 83.47±0.34 86.18±0.45 31.34±14.28 83.91±0.37 85.25±0.26
C
Ablation study
C.1
Warm-up rounds
Tab. 8 studies the warm-up horizon Tw as a fraction of total communication rounds. Choosing Tw creates a trade-off between estimation quality and efficiency. A longer warm-up provides more rounds to estimate class-wise evidence before freezing contributions, but it also increases server-side probing costs and can slow global convergence because classifier heads are not shared during this stage. A shorter warm-up reduces computation and speeds up full-model synchronization, but it may produce noisier estimates of the class-evidence matrix (Q), limiting the benefit of contribution-aware weighting. This trend is reflected in the results: for Dirichlet(α = 0.05), a shorter warm-up performs best and a longer warm-up leads to saturation or mild decline; for Pure Label Skew, a Table 8: Warm-up horizon ablation for CELM. We report test accuracy (%,mean±std. ) for different warm-up fractions Tw /T . Smaller Tw reduces probing cost, while larger Tw can help in better estimation of the class-evidence matrix. Dataset Split
5%
Tw as % of total rounds T 10 % 15 % 20 %
30 %
Dir. (0.05) 83.64±0.42 83.31±0.99 83.07±0.95 83.06±0.75 82.70±0.91 F.MNIST PLS 83.70 83.57 83.73 83.99 84.14 ±0.19
±0.45
±0.34
±0.10
±0.51
Dir. (0.05) 67.16±1.29 66.88±1.54 66.68±1.49 66.54±1.43 66.43±1.49 CIFAR-10 PLS 59.11 59.03 59.36 59.44 59.86 ±1.96
±2.32
±1.42
±0.95
±0.57
Data-free Contribution Estimation via Logit Maximization
0.001 0.003 0.01 0.03 LM Learning rate ( )
0.1
67 66 65 64
53.65 56.48 56.51 55.59 55.89 53.07 53.67 56.11 56.42 56.33 54.39 53.75 56.60 56.24 55.79 53.68 53.49 54.23 56.39 56.38 54.15 53.89 53.87 53.77 56.26
63 62
0.1
0.001 0.003 0.01 0.03 LM Learning rate ( )
0.1
400 LM steps L 100 200 50
85.19 86.32 87.11 87.70 87.57
25
400 LM steps L 100 200 50
0.001 0.003 0.01 0.03 LM Learning rate ( )
82
85.85 86.14 86.81 87.66 87.93
81
0.001 0.003 0.01 0.03 LM Learning rate ( )
57 56 55 54
85.83 86.59 87.70 87.64 87.63
0.1
400
65.16 64.73 64.90 65.32 65.20
80
83
86.12 87.60 87.87 87.65 87.80
70.87 71.86 72.05 72.32 72.72
LM steps L 100 200
25
65.01 64.91 65.19 64.91 65.16
65.59 65.19 64.98 65.37 65.25
81.88 82.11 82.98 82.83 83.71
81
84
86.86 87.80 87.61 87.73 87.67
71.00 71.52 71.95 72.21 71.82 71.22 71.12 71.13 72.39 72.01
50
65.40 64.99 65.18 65.00 65.74
82.03 82.54 82.64 83.95 83.76
85
71.29 71.51 71.27 71.63 72.07
25
400
65.47 65.23 64.99 64.95 65.11
LM steps L 50 100 200
0.1
82.32 82.88 84.19 83.77 83.86
25
83.18 83.25 83.09 83.07 83.13
82
82.49 82.96 83.57 83.66 83.81
400
83.29 83.09 83.00 83.22 83.24
83
82.69 84.10 83.65 83.90 83.91
LM steps L 100 200
50
83.24 82.84 83.45 83.33 83.52
84
Step Label Skew
50
LM steps L 100 200
83.11 82.89 83.32 83.48 83.77
85
25
400
83.01 83.60 83.37 83.28 83.22
0.001 0.003 0.01 0.03 LM Learning rate ( )
CIFAR-10
Pure Label Skew
25
FashionMNIST
Dirichlet ( = 0.05)
21
71.48 71.33 70.37 71.33 72.47
53 52
0.001 0.003 0.01 0.03 LM Learning rate ( )
0.1
89 88 87 86 85 84
74 73 72 71 70
Fig. 6: Sensitivity of CELM to LM optimization steps L and LM learning rate η. Each heatmap cell reports final test accuracy (%) for a specific (L, η) pair across representative non-IID splits on FashionMNIST and CIFAR-10.
longer warm-up improves performance, consistent with stronger class asymmetry requiring more evidence collection. To balance accuracy and computational overhead, we set Tw = 0.05T in the main experiments. C.2
Ablation on LM steps and LM learning rate
Fig. 6 presents a grid search over LM optimization steps L and LM learning rate η on representative non-IID splits of FashionMNIST and CIFAR-10. Very small learning rates (e.g., 0.001) and shallow probes (e.g., L = 25) under-optimize class evidence, while very large L provides diminishing returns. Across settings, a broad high-performing region appears for η ∈ [0.01, 0.1] and L ∈ [100, 400], indicating that CELM is reasonably robust to moderate hyperparameter variation. In the main experiments, we use η = 0.01 and L = 200 as a stable operating point that balances accuracy and probing cost. When compute is constrained, high η, with smaller L configurations are viable alternatives.
D
Compute Complexity
Let N denote the number of clients, K the number of classes, Tw the warm-up rounds, and L the LM optimization steps. Let CLM be the cost of one forwardbackward Logit Maximization step with respect to the input for a single class and model. This cost depends on the model size and the input image dimensions. The per-model compute cost for running all LM steps is O K L CLM . Importantly, the per-class LM loop is embarrassingly parallel. In our implementation, we opti mize all classes simultaneously. The per-model latency then becomes O L CLM .
22
A.Ukaye et al.
CELM runs LM on the global model and all client models for Tw rounds, leading to a total server-side complexity is O Tw (N + 1) L CLM . The additional memory overhead is dominated by cached warm-start initial images and class-wise evidence tensors, i.e., O((N + 1)K|x| + N K) ∼ O(N K|x|). After warm-up (t > Tw ), contribution weights are frozen, and CELM reduces to standard weighted aggregation with cost comparable to FedAvg, while communication overhead remains unchanged. Since LM is independent across models as well, there is further scope for reducing latency at the cost of additional memory.
E
Optimized LM image samples Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Class 7
Class 8
Class 9
Client 4
Client 3
Client 2
Client 1
Client 0
Class 0
Fig. 7: Final warm-up LM probe images (Xfinal ) on FashionMNIST (Pure Label Skew). Each row is a client, and each column is a target class; red outlines indicate classes present in that client’s local data. Present-class cells often exhibit more distinct response patterns, while the images remain abstract and non-semantic, indicating useful class evidence with no visual privacy leakage.
Fig. 7 provides a qualitative view of the final LM images used by CELM after the warm-up phase. Rows correspond to clients and columns correspond to target classes; red outlines mark classes present in a client’s local data. In most cases, cells aligned with present classes show clearer and more distinct activation patterns with higher contrasts than absent-class cells, supporting the role of LM as class-selective evidence for contribution estimation. At the same time, these LM outputs remain highly synthetic and do not show semantic value. They do not resemble recognizable training samples and do not reveal client-specific content in a human-interpretable way. This is expected because CELM optimizes random Gaussian initializations for class-logit response (with only l2 regularization), and not for data reconstruction. Therefore, while LM outputs are useful for relative class evidence scoring, they appear to carry no direct privacy leakage risk.