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CovFedAvg: Local Low-Loss Curves and Federated Model Selection

Federico Lucchetti, Marcus Völp

arXiv:2610.39902v1cs.LGcs.AI

TL;DR

Clients may fit locally compatible models that behave incompatibly outside their observed support. CovFedAvg fits local paths toward a selected shared endpoint using local records and aggregate messages, then selects with distributed losses. Results show modest gains over FedAvg but do not establish a curve-specific advantage over equally costly optimization.

  • Problem

    Locally equivalent models can behave incompatibly outside client-observed support, motivating a search for compatible shared models.

    Source: §1 · p. 2

  • Method

    CovFedAvg fits local paths to a shared candidate using local training records and aggregate messages, then selects a model using distributed losses without centralized calibration data.

    Source: §7 · p. 15

  • Results

    At width 16, participating-Gas macro balanced accuracy is 89.84% for CovFedAvg versus FedAvg’s 88.88%, while matched direct fine-tuning reaches 91.65%; no reliable advantage over matched direct optimization is established.

    Source: §8.3 · p. 19, §11 · p. 31

  • Takeaways & Limitations

    The results support feasibility and modest gains over FedAvg, while leaving open whether curve constraints improve optimization over equally costly alternatives.

    Source: §11 · p. 31

    The numerical experiments use exact aggregate-only interfaces without cryptographic aggregation or differential privacy, so they are not an end-to-end privacy guarantee.

    Source: §7.3 · p. 17

Abstract

from arXiv · show

Clients observing different parts of a prediction task may learn locally equivalent models with incompatible behavior outside their observed support. We study whether redundant models permit local low-loss paths toward a compatible shared model. CovFedAvg retains local models and curve controls on clients, fits a common endpoint through aggregated gradient contributions, and selects an actual candidate model using distributed training losses. No centralized calibration examples or client-to-client sample exchange are required. Exact selection guarantees no higher evaluated training loss than the previous model and the current-round FedAvg candidate; this is not a global-optimality, stationarity, generalization, or differential-privacy guarantee. We state conditional client-compatibility and approximate-selection bounds and retain corrected transformation identities and a separate full-gradient safeguard theorem for reference algorithms. A complementary input-support example recovers the shared absolute-value regression, while matched direct fine-tuning also solves it. Across a synthetic support task, recorded activity-recognition subjects, and participating and future gas-sensor batches, we evaluate five methods, five seeds, two network widths, and 20 rounds without calibration data. At width 16, CovFedAvg raises participating-Gas macro balanced accuracy from FedAvg's 88.88% to 89.84%; matched distributed fine-tuning reaches 91.65%. Results support feasibility and modest gains over FedAvg, but do not establish a curve-specific advantage over equally costly optimization. Exact-sum numerical experiments establish data flow; end-to-end privacy requires additional mechanisms and analysis.

1 Introduction

Feature heterogeneity can leave locally equivalent models with incompatible behavior outside clients’ observed support. CovFedAvg uses local low-loss paths and a selected shared endpoint, with conditional rather than universal guarantees.

1 Introduction

  • Feature heterogeneity changes input distributions without changing label meaning, unlike changes in class proportions or arbitrary changes in the labeling rule.FedAvg’s behavior also depends on local steps, learning rate, model, and client objectives.Source: §1 · p. 1
  • Locally unidentifiable model directions can let a redundant model incorporate another client’s information while preserving its original predictions, but compatible local low-loss regions need not intersect.Parameter symmetries that preserve a function everywhere do not add knowledge about an unobserved domain.Source: §1 · p. 2
  • CovFedAvg keeps local models and path controls on clients, fits a common endpoint through local losses and aggregate messages, and selects a model without central calibration data or client-to-client sample exchange.Its aggregate-only simulation is not a formal privacy guarantee, and endpoint fitting adds gradient costs that should be compared with direct optimization.Source: §1 · p. 2
  • The contribution is an executable local-curve fitting and model-selection protocol with conditional compatibility and exact or bounded-error selection statements, evaluated against matched fine-tuning and centroid controls.Corrected transformation identities and a separate full-client-gradient safeguard theorem are retained as reference components; that theorem does not automatically apply to CovFedAvg’s selector.Source: §1 · p. 2
  • Statistical heterogeneity alone does not imply poor FedAvg performance: one full-gradient step matches a centralized gradient step, and the paper claims neither universal recovery nor invention of geometric aggregation.The evaluation preserves both favorable FedAvg results and failures.Source: §1 · p. 2

2 Related work

The paper situates its approach among established federated optimization and feature-heterogeneity methods, while distinguishing its local-curve protocol from prior geometric aggregation and calibration-based constructions.

2 Related work

  • FedBN retains client-specific batch-normalization parameters and statistics during aggregation, whereas FRAug trains a shared representation generator with client-specific transformation networks [11] [4].Numerical comparisons with FedBN and FRAug remain open.Source: §2 · pp. 2–3
  • The reference centroid construction uses public labeled calibration examples, while current CovFedAvg fits local curves through aggregate messages without central calibration data.The two calibration proposal methods receive the same labeled examples and gradient-evaluation budget; full-client risk and gradient queries for safeguards are reported separately.Source: §2 · pp. 2–3
  • Mode connectivity motivates low-loss paths and stochastic weight averaging provides precedent for averaging nearby weights, but neither guarantees low loss at an arbitrary arithmetic centroid [5] [6].Git Re-Basin studies permutation alignment before merging networks [1]; weight-space connections alone do not identify feature transformations or quantify distributional heterogeneity.Source: §2 · p. 3
  • FedGMCC already combines output probing, path connectivity, and aggregation of connected client models within clusters; the paper therefore claims no novelty for geometric federated learning or path aggregation.FedProx controls local drift with a proximal penalty, while SCAFFOLD uses persistent client and server control variates [8].Source: §2 · pp. 2–3

3 Problem and transformation model

The section defines a sample-weighted federated objective and a conditional model for label-preserving transformed client domains. Risk covariance requires compatible transformations and does not imply universal compatibility or cover every form of heterogeneity.

3.1 Federated objective

  • Client objectives are combined in a sample-weighted mixture, so a global optimizer need not minimize every client objective when their minimizers differ.The global mixture is P = Σ_k α_k P_k, with α_k determined by client sample sizes unless another target population is chosen.Source: §3.1 · p. 3

3.2 Label-preserving domains are an assumption

  • Pushforwards are defined for measurable maps, but inverse-density formulas require a diffeomorphism with nonsingular Jacobian; invertibility is a modeling restriction, not an automatic consequence of naming a map.Cropping, clipping, and quantization need not satisfy that restriction, and interpolating invertible maps can pass through a singular map.Source: §3.2 · pp. 3–4
  • The transformed-domain model assumes clients observe transformed inputs with labels preserved; a visual deformation alone establishes neither label preservation nor an invertible map.Figure 1 illustrates intensity changes, spatial changes, and deformation, but its digit interpolation may change class.Source: §3.2 · p. 4, Figure 1 · p. 4
  • The transformed-domain model is not synonymous with classical covariate shift and does not claim to cover every form of statistical heterogeneity.Equation (2) preserves the label marginal but does not generally preserve the labeling conditional in observed coordinates.Source: §3.2 · p. 5

3.3 What risk covariance does and does not mean

  • When the model class admits a compatible parameter action, transforming both data and predictor can preserve risk, but a single predictor on an unknown mixture may not attain all client-specific risks.Compatible transformations, sufficient model capacity, and informative observations remain substantive assumptions; a weight-space curve does not identify a distributional distance.Source: §3.3 · p. 5

4 Reference construction: public-data curve centroids

The reference construction explores paths from a shared anchor to client models, then evaluates a normalized centroid using public calibration data. Its empirical guard and computational costs are limited to the stated validation and calibration procedures.

4 Reference construction: public-data curve centroids

  • The section distinguishes earlier public-data centroid proposals from the current Section 7 local-curve and model-selection construction, which eliminates the server calibration requirement.These are different algorithms and access models.Source: §4 · p. 5

4.1 Geometric rationale

  • A curved low-loss region can make a straight parameter average higher-loss, motivating path exploration, but the schematic does not establish that a suitable path exists for every client pair.The implemented reference construction follows paths from the FedAvg anchor to client models and explicitly evaluates their normalized centroid; a low-loss path alone does not certify that centroid.Source: §4.1 · p. 5, Figure 2 · p. 6

4.2 Access model

  • The reference access model uses labeled public calibration and disjoint validation sets; client-loss and gradient queries are an alternative, while the current construction instead uses local training losses and aggregate messages.The public sets are additional information, not something recovered from client weights.Source: §4.2 · p. 7

4.3 Continuous paths with a normalized centroid

  • Each client’s polygonal path joins the common anchor to its local model through a trainable control, and the weighted path centroid recovers weighted FedAvg exactly when fitting is disabled.The midpoint control initialization yields the anchor centroid for any number of clients, positive sample sizes, and model dimension.Source: §4.3 · p. 7

4.4 Empirical guard and computation

  • The empirical guard accepts a candidate only when its validation loss is no larger than the current anchor’s, which does not guarantee monotone loss across rounds or population improvement.Each round adds JKQ calibration gradient evaluations and two validation evaluations; matching the calibration baseline’s gradient-call budget helps separate update geometry from extra data and computation.Source: §4.4 · p. 8

5 Mathematical guarantees and limits

The section gives conditional results about when transformations preserve risk, how local training can drift, and what path aggregation or descent guarantees do—and do not—establish.

5.1 Feature transformations require a compatible model action

  • A feature transformation preserves risk only when the model family has a compatible parameter action; label preservation alone does not imply covariate shift or a common optimizer.With a bijective action, minimizers correspond; general nonlinear transformations need not admit a compatible action within a fixed architecture.Source: §5.1 · p. 9

5.2 A deterministic bound on local-training drift

  • Local-training drift is bounded over one communication block under smoothness and gradient-heterogeneity assumptions, but the bound does not establish convergence across rounds.A distributional sufficient condition uses shared class probabilities, an input-Lipschitz sample gradient, and Wasserstein distance between complete input-vector distributions.Source: §5.2 · pp. 10–11

5.3 What integrating a fitted path guarantees

  • Integrating fitted paths gives a centroid-risk bound under smoothness, but low-risk paths alone do not guarantee a low-risk centroid.Initializing each control at the midpoint between the weighted endpoint and local endpoint makes every path straight and recovers the FedAvg centroid exactly.Source: §5.3 · pp. 11–12

5.4 A conditional descent guarantee

  • A relative-gradient-error condition supports descent and vanishing gradient norms, but the path construction and anchor comparison do not establish that condition for CovFedAvg.A validation guard certifies only the stated comparison on that validation set, not population-risk descent or unconditional convergence.Source: §5.4 · pp. 12–13

6 Reference extension: affine adapters and a descent safeguard

This reference extension uses affine adapters to align a specific common-latent setting and a full-gradient safeguard to certify descent of a fixed empirical objective.

6.1 What an affine adapter can establish

  • When clients observe invertible affine transformations of a shared latent input, known adapters give them the same adapted joint distribution and a compatible first-layer transformation.For approximate adapters, an input-Lipschitz sample gradient yields a conditional bound on gradient differences; empirical standardization does not establish the common-latent model.Source: §6.1 · pp. 13–14

6.2 An empirical objective with a certified fallback

  • With fixed adapters, training sets, and regularization, an exact full-gradient backtracking safeguard gives a stationarity bound for the regularized empirical objective.The objective's coercivity and smoothness support the theorem; its guarantee is not test accuracy, population risk, privacy, or superiority over a separate baseline.Source: §6.2 · pp. 14–15
  • The safeguard requires additional full-client gradient, candidate-loss, and line-search queries beyond ordinary FedAvg costs.The stronger global-minimizer conclusion applies only to a fixed-feature linear softmax head, not the trained nonlinear networks.Source: §6.2 · p. 15

7 Local curves with a selected shared endpoint

CovFedAvg fits local paths toward a shared endpoint, then selects among actual candidates using distributed training losses rather than returning a curve centroid.

7.1 Local objectives, private controls, and a common endpoint

  • Clients retain their local models and path controls, contribute endpoint gradients through aggregation, and fit a common endpoint in the full parameter space.The protocol uses local training records and does not require client-to-client record sharing or a server-held labeled calibration set.Source: §7.1 · pp. 15–16, §7 · p. 15, §7.3 · p. 17

7.2 Selection and the limits of compatibility certificates

  • With exact evaluations, selection guarantees training loss no higher than the previous model or the current-round FedAvg candidate.Compatibility is conditional on the selected endpoint meeting each client's loss tolerance; sampled path points do not certify the entire path or population risk.Source: §7.2 · p. 16
  • Noisy candidate-loss estimates with error at most δ yield a selected model whose true loss is at most 2δ above the best candidate's loss.Repeated adaptive queries require appropriate privacy and statistical analysis; exact monotonicity no longer follows from privacy noise.Source: §7.2 · p. 16

7.3 Data locality and matched controls

  • The main accuracy experiments emulate exact aggregate-only interfaces and use neither cryptographic aggregation nor differential privacy.Matched direct training uses the same anchor and matched gradient work, while the centroid control has fewer proposal candidates, so these controls do not isolate parameter location alone.Source: §7.3 · p. 17

8 Evaluation without server calibration data

CovFedAvg fits and selects a shared endpoint from client-local models and curve controls without server-held calibration data. Across the reported benchmarks it improves macro balanced accuracy over FedAvg, while matched direct fine-tuning performs better in every setting.

8 Evaluation without server calibration data

  • CovFedAvg keeps each client’s local model and path control local, fits a common endpoint using local supervised losses and aggregate messages, and selects a candidate without centralized calibration or sample sharing.The evaluated exact aggregate interface establishes optimization and data flow, not deployment security.Source: §8 · p. 17

8.1 Complementary support: an identifiable motivating example

  • In the complementary-support example, local fits average to (1, 1, 0.3), losing half of each learned slope, while the selected endpoint recovers both branches; matched direct fine-tuning also solves the task.Selecting the shared endpoint gives MSE 3.24 × 10^-9, while matched direct fine-tuning gives 2.13 × 10^-30; this illustrates feasibility, not an advantage over that control.Source: Figure 3 · p. 18, §8.1 · p. 18
  • The soft path penalty is not an exact constraint solver: a 101-node path diagnostic has weighted maximum violation 0.00361 at zero tolerance.The reported near-correct endpoint does not establish exact preservation along the entire curve.Source: §8.1 · p. 18

8.2 Fixed protocol and controls

  • The fixed protocol compares five methods over four tasks, five seeds, two widths, and 20 rounds; the matched direct control matches gradient evaluations and processed examples, but not communication.It uses the same five-candidate selection budget as CovFedAvg, while the curve-centroid ablation has fewer selection opportunities and follows a different trajectory.Source: §8.2 · pp. 18–19
  • The synthetic support task is a controlled covariate-shift diagnostic with six clients and disjoint first-coordinate intervals, rather than a naturally collected dataset.Recorded activity-recognition subjects and Gas acquisition batches, including a future-batch split, are also evaluated without reassigning reserved calibration or validation records.Source: §8.2 · p. 19

8.3 Results and limits

  • All 3,200 selected-method rounds have no greater evaluated training objective than either the previous model or that round’s FedAvg candidate, but this does not guarantee test improvement or preserve every client.Finite soft-penalty optimization also does not guarantee that every client’s tolerance is met.Source: §8.3 · p. 19
  • Matched direct fine-tuning has higher mean balanced accuracy than CovFedAvg in all eight settings, so the results do not establish a curve-specific advantage over equally costly optimization.The methods differ in communication: direct fine-tuning sends 15 gradient updates per outer round, versus three endpoint-gradient updates for the curve method; this does not imply fivefold lower total communication.Source: §8.3 · pp. 19–20
  • 0.14–1.12 percentage points: CovFedAvg improves mean macro balanced accuracy over FedAvg in all eight task–width settings; at width 16 on participating Gas, scores are 88.88% for FedAvg, 89.84% for CovFedAvg, and 91.65% for matched direct fine-tuning.These are descriptive comparisons on recorded benchmarks, not a statistical significance claim or independent replications across populations.Source: §8.3 · p. 19, Figure 4 · p. 22

9 Reference evaluation of public-calibration variants

The reference evaluation compares public-calibration variants on recorded activity-recognition subjects and gas-sensor batches, using fixed task splits and matched training settings. Results vary by task and target domain: curve variants do not consistently outperform stronger comparators, and adapter behavior differs between participating and future domains.

9.1 Tasks, native domains, and split provenance

  • HAR uses subject-based roles with nine official test subjects held out; participating Gas holds out records within each batch, while future Gas tests on batches 9–10.The HAR split assigns source-training subjects to training, calibration, and unused validation roles; the participating-Gas task retains native batches, missing classes, and unequal client sizes.Source: §9.1 · p. 23

9.2 Training, representation, and access-matched controls

  • The comparison fixes shared model and optimization settings across methods, while calibrated methods use the same 128-record labeled sample and matched public-gradient calls for curve fitting or direct calibration.The study includes FedAvg, FedProx, SCAFFOLD, safeguard and calibration variants, adapter variants, and a pooled Centralized reference; fixed settings do not establish best-tuned baseline performance.Source: §9.2 · pp. 23–24

9.3 IID control, diagnostics, and reported quantities

  • At E = 5, native Gas FedAvg macro balanced accuracy rises from 88.88% to 95.48% under IID reshuffling of the same source-training records, while HAR changes from 91.58% to 91.87%.On Gas, worst-domain balanced accuracy rises from 56.59% to 85.08%; the section also reports feature, label-prior, and gradient-disagreement diagnostics, which describe selected summaries rather than full joint distributions.Source: §9.4 · pp. 25–26, §9.3 · p. 25

9.4 Results and scope

  • 89.01 ± 0.55% macro balanced accuracy for CovFedAvg (calibration) on native Gas at E = 5 compares with 88.88 ± 0.58% for FedAvg, 90.78 ± 1.43% for SafeCal, 94.75 ± 0.27% for SCAFFOLD, and 96.82 ± 0.54% for the pooled reference.The 0.13-point gain over FedAvg is smaller than direct calibration’s improvement and does not close the native–IID gap.Source: §9.4 · p. 26, Table 2 · p. 27
  • 91.71 ± 0.39% macro balanced accuracy for CovFedAvg (calibration) on HAR at E = 5 is close to FedAvg’s 91.58 ± 0.60%, SafeCal’s 91.78 ± 0.59%, and SCAFFOLD’s 91.83 ± 0.72%.At E = 10, CovFedAvg (calibration) is slightly below FedAvg, and the small mixed differences provide no consistent curve or safeguard advantage.Source: §9.4 · p. 26, Table 2 · p. 27
  • 69.85% versus 57.14% worst-domain balanced accuracy on participating Gas accompanies adapter CovFedAvg’s 66.95% macro balanced accuracy on future Gas.On future Gas, FedAvg, CovFedAvg (calibration), SCAFFOLD, and the pooled reference score 76.51%, 76.59%, 76.36%, and 77.08%, respectively.Source: §9.4 · p. 26, Table 2 · p. 27
  • All 4,600 safeguarded rounds satisfy their recorded exact source-objective comparisons, but the method adds substantial gradient and candidate-loss work and does not establish better generalization.The evaluation uses recorded domains and does not validate an invertible relationship between natural domains or guarantee that curves improve generalization.Source: §9.4 · pp. 26, 28

10 Interpretation, limits, and next tests

CovFedAvg shows modest gains over FedAvg on natural domains, but matched direct optimization remains stronger in participating Gas, and the results do not establish a curve-specific generalization advantage. The discussion bounds that conclusion with safeguard, centroid, adapter, evaluation-scope, and privacy limitations.

10 Interpretation, limits, and next tests

  • CovFedAvg’s gains over FedAvg are modest, while matched direct distributed optimization remains stronger on participating Gas; the results do not establish a curve-specific generalization advantage.A training-loss selector can still permit worse held-out or worst-domain performance.Source: §10 · p. 30
  • The safeguard’s stationarity proof applies to its full-client-gradient fallback, not CovFedAvg’s selector; a low-loss path’s centroid can still have high loss.The centroid bound therefore retains a dispersion term, and the implementation evaluates the actual candidate.Source: §10 · p. 30, Figure 7 · p. 31
  • Calibration-only controls and SCAFFOLD do not support a general curve-specific advantage, and these methods were not reproduced for comparative superiority claims.Further tests include calibration scarcity and domain coverage, larger models, and incomplete participation; five seeds do not replicate independent sensor arrays or acquisition-site populations.Source: §10 · p. 30
  • Fixed per-client moment adapters can remove class-dependent information when class proportions differ, and no target-domain adapter can be inferred without target information.Matching unconditional moments does not establish the specified affine measurement model; the future-Gas deterioration illustrates this risk.Source: §10 · p. 30
  • Natural HAR subject and Gas acquisition-period domains extend beyond the supplementary imposed-shift pilot, while hospital and cross-style domains remain proposed future tests.The assumed transformations need not fit natural domains, and NLP effectiveness has not been tested.Source: §10 · p. 30
  • Keeping examples local is a data-flow property, not a formal privacy guarantee; private deployment requires a threat model and compatible mechanisms.Client weights and additional training-risk, gradient, and preprocessing summaries can leak information; the paper makes no privacy or clinical-utility claim.Source: §10 · pp. 30–31

11 Conclusion

CovFedAvg uses local feature coverage to explore compatible shared candidates and selects among them using distributed losses. The paper reports feasibility and modest gains over FedAvg, but no reliable advantage over matched direct optimization.

11 Conclusion

  • Local feature coverage can leave multiple parameter configurations compatible with one client’s observations.CovFedAvg explores that freedom through local paths to a shared candidate and selects using distributed losses, without centralized calibration.Source: §11 · p. 31
  • Exact candidate selection empirically compares evaluated losses, while preservation of every client remains conditional and stationarity requires additional descent conditions.Correct transformation identities, centroid counterexamples, and a separate safeguarded stationarity theorem delimit the claims.Source: §11 · p. 31
  • The motivating regression is solved, and naturally separated domains show modest gains over FedAvg but no reliable advantage over matched direct optimization.The paper identifies establishing when path constraints improve optimization or generalization, and developing an end-to-end protected communication protocol, as further work.Source: §11 · p. 31

A Controlled-shift feasibility pilot

The supplementary pilot tests a corrected construction on imposed shifts of public records. Its scope is a small federated simulation, not evidence of invariance, uniformly low-loss paths, or universal client-risk improvement.

A Controlled-shift feasibility pilot

  • This pilot is supplementary evidence using client shifts imposed as transformations of public records.The main-text natural-domain evaluation instead uses actual subject and acquisition-period identities.Source: A Controlled-shift feasibility pilot · p. 31
  • The pilot tests whether the corrected construction can be executed and whether its benefit exceeds using public calibration data directly.It is a small federated simulation using synthetic measurement domains on two real benchmark datasets and one synthetic process.Source: A Controlled-shift feasibility pilot · p. 31
  • The pilot does not establish invariance, transformation identification, uniformly low-loss paths, or improvement in every client’s risk.Source: A Controlled-shift feasibility pilot · p. 31

A.1 Tasks, domains, and held-out data

This controlled-shift pilot uses synthetic measurement domains, matched training and calibration budgets, and held-out evaluation. CovFedAvg slightly exceeds FedAvg in macro accuracy across the four settings, but direct calibration matches or exceeds it in shifted settings.

A.1 Tasks, domains, and held-out data

  • The pilot covers digits, cancer, and a generated sensor task, using four clients and invertible, label-preserving synthetic feature transformations.The synthetic sensor task has 2,400 records; transformed domains are simulated rather than actual sites.Source: §A.1 · p. 32
  • Each seed uses disjoint calibration, validation, and test splits, while the fixed pilot trains a 32-unit MLP for 20 rounds with three seeds.All methods share initialization, minibatch permutations, and fixed hyperparameters; the pilot does not optimize each baseline’s hyperparameters.Source: §A.1 · p. 32, §A.2 · p. 32

A.2 Matched training and calibration budgets

  • CovFedAvg’s curve controls use three simultaneous gradient steps per round, matching FedAvg+cal’s 36 calibration gradient evaluations but not its effective update sizes or objective.Both calibrated methods accept a candidate only when finite pooled validation cross entropy does not exceed the anchor; reusing validation across rounds is not an untouched population-risk certificate.Source: §A.2 · pp. 32–33
  • The calibrated methods add 720 server-gradient evaluations over 20 rounds, and the small cancer task processes 46,080 public examples.The pilot omits real multi-institution shifts, larger models, partial participation, privacy analysis, and calibration scarcity or mismatch; its three seeds and one configuration limit generalization.Source: §A.2 · p. 33, §A.3 · p. 35

A.3 Results and their limits

  • CovFedAvg’s final macro accuracy is slightly higher than FedAvg’s in all four settings under this fixed configuration.The pilot reports round-20 means and sample standard deviations across three seeds; these are not confidence intervals or significance tests.Source: §A.3 · p. 33, Table 3 · p. 33
  • Direct calibration equals or exceeds CovFedAvg’s macro accuracy in all three shifted settings and yields lower digit cross entropy: 0.256 versus 0.395 under shift.It also yields lower digit cross entropy in the IID control: 0.228 versus 0.311; these results do not establish a reliable advantage from curve aggregation rather than public calibration.Source: §A.3 · p. 33

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