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Federated Learning Meets Multi-objective Optimization

Zeou Hu, Kiarash Shaloudegi, Guojun Zhang, Yaoliang Yu

arXiv:2006.11489v2cs.LGstat.ML

TL;DR

Federated learning must balance private, heterogeneous user data with fairness and robustness against malicious manipulation. The paper formulates this setting as multi-objective optimization and proposes FedMGDA+, which is guaranteed to converge to Pareto stationary solutions and compares favorably across experiments.

  • Problem

    Federated learning needs to preserve on-device privacy while addressing fairness among users and robustness against malicious manipulations.

  • Method

    The paper formulates federated learning as multi-objective optimization and proposes FedMGDA+ with a proximal-average perspective on existing federated-learning practices.

  • Results

    FedMGDA+ is guaranteed to converge to Pareto stationary solutions and compares favorably against state-of-the-art across varied datasets and metrics.

  • Takeaways & Limitations

    FedMGDA+ complements existing federated-learning systems while explicitly addressing fairness and robustness without sacrificing participating users’ performance.

  • Takeaways & Limitations

    Formal quantification of the tradeoff induced by multiple local updates and privacy guarantees remain future work.

Abstract

from arXiv · show

Federated learning has emerged as a promising, massively distributed way to train a joint deep model over large amounts of edge devices while keeping private user data strictly on device. In this work, motivated from ensuring fairness among users and robustness against malicious adversaries, we formulate federated learning as multi-objective optimization and propose a new algorithm FedMGDA+ that is guaranteed to converge to Pareto stationary solutions. FedMGDA+ is simple to implement, has fewer hyperparameters to tune, and refrains from sacrificing the performance of any participating user. We establish the convergence properties of FedMGDA+ and point out its connections to existing approaches. Extensive experiments on a variety of datasets confirm that FedMGDA+ compares favorably against state-of-the-art.

1 Introduction

Federated learning addresses privacy-sensitive, heterogeneous edge-device training but must also contend with non-IID data, limited communication, fairness, and malicious manipulation. The paper formulates these concerns through multi-objective optimization and proposes FedMGDA+ with convergence guarantees and favorable empirical performance.

  • Motivation: Federated learning enables on-device training while addressing privacy-sensitive data and heterogeneous edge-device constraints.Relevant challenges include device capacity, power, data, availability, communication, and memory.
  • Motivation: Non-IID data, limited communication, and privacy complicate the formulation and execution of federated learning.User distributions can be unbalanced, communication is constrained, and private data cannot be shared.
  • Motivation: Fairness matters because it influences participation and provides some robustness against malicious user manipulation, while adversarial attacks expose vulnerabilities in federated systems.The paper treats fairness and robustness as central design motivations.
  • Contributions: FedMGDA+ uses multi-objective optimization to incorporate fairness and robustness, with fewer hyperparameters and a guarantee of convergence to Pareto stationary solutions.The algorithm is presented as complementing existing federated-learning systems.
  • Contributions: The paper unifies existing federated-learning practices through a proximal-average interpretation and connects federated learning with multi-objective optimization.These perspectives are developed as conceptual contributions alongside the proposed algorithm.
  • Contributions: Experiments evaluate FedMGDA+ across desirable metrics and compare its competitiveness with alternative algorithms.The introduction reports extensive experiments and discusses future directions and released code.

2 Related Work

Related work situates FedMGDA+ among federated averaging, adaptive, fairness-oriented, and neural-network aggregation methods. Prior work addresses convergence, non-IID data, fairness, and parameter matching, while benchmark standardization remains an active concern.

  • Federated optimization: FedAvg sends a global model to selected users, aggregates their locally updated models, and serves as the foundation for several federated extensions.FedProx adds proximal regularization for non-IID data and reduces to FedAvg when regularization vanishes.
  • Federated optimization: FedAvg analysis is complicated by flexible updates, partial participation, and non-IID client data, especially when datasets are unbalanced.Prior theory studies sampling, averaging schemes, and convergence under convex settings.
  • Federated optimization: Adaptive federated optimizers decouple server and client optimization, while FedSplit targets fixed-point issues identified for FedAvg and FedProx.The cited discussion notes unresolved convergence questions for FedSplit under asynchronous and stochastic updates.
  • Fairness: Agnostic federated learning addresses user fairness by optimizing against a worst-case mixture of user distributions.This formulation leads to a saddle-point optimization problem.
  • Model aggregation: Neural-network aggregation methods such as PFNM address limitations of coordinate-wise model averaging caused by parameter permutation invariance.PFNM is described as applicable only to fully connected feed-forward networks.
  • Evaluation: The paper follows emerging federated-learning benchmarking practices by reporting common datasets, releasing code, and including experimental details.Benchmark, protocol, and evaluation standardization is identified as an area of significant interest.

3 Problem Setup

The problem setup models federated learning as distributed optimization over private user objectives and explains FedAvg through proximal averages. This perspective clarifies how local-update counts interpolate between global averaging and individual-user minimization, while motivating alternatives to fixed arithmetic averaging.

  • Problem formulation: Federated learning assigns each of m users a private-data-dependent objective f_i over a shared model parameter w.The framework seeks to optimize the user objectives collectively and efficiently.
  • FedAvg: FedAvg optimizes an arithmetic average of individual user functions using sampled local gradient updates followed by server-side model averaging.The weights may reflect dataset size, user importance, or uniform participation.
  • FedAvg: The number of local epochs controls the optimization regime: k = 1 corresponds to gradient descent, whereas k = ∞ averages local minimizers.Intermediate k values motivate the proximal-average interpretation.
  • Proximal average: The proximal average is defined for convex user functions with positive weights summing to one, and its proximal map equals the weighted average of individual proximal maps.This provides the mathematical basis for parallel user-side computation.
  • Proximal average: FedAvg with one local step and infinitely many local steps matches proximal-average minimization at η = 0 and η = ∞, respectively.Intermediate local-step counts correspond to intermediate proximal-average parameters.
  • Proximal average: FedProx is described as a randomized version of the proximal-map update, and FedAvg and FedProx approximate an arithmetic average for small step size.The proximal-map formulation avoids directly evaluating the complicated proximal-average expression.
  • Motivation: Fixed arithmetic weights are problematic under highly non-IID, unbalanced data and are vulnerable to malicious loss manipulation.The setup motivates fairness- and robustness-oriented multi-objective formulations.

4 Multi-objective Minimization (MoM)

Multi-objective minimization represents federated learning as simultaneous optimization of potentially incompatible user objectives. Pareto stationarity supplies a tractable first-order target, while weighted, constrained, and minimax scalarizations recover connections to FedAvg and AFL.

  • Definition: Multi-objective minimization jointly minimizes multiple potentially incompatible scalar objectives represented by a vector-valued function.The ordering is component-wise, so objective vectors may be incomparable.
  • Definition: A Pareto-optimal solution cannot improve one objective without compromising another, whereas a weakly Pareto-optimal solution cannot improve all objectives simultaneously.The set of Pareto-optimal solutions can contain infinitely many incomparable points.
  • Pareto stationarity: For continuously differentiable, possibly nonconvex objectives, Pareto stationarity requires a convex combination of objective gradients to vanish.This first-order condition accommodates deep models and serves as the practical target.
  • Pareto stationarity: Any Pareto-optimal solution is Pareto stationary, and for convex objectives any Pareto-stationary solution is weakly Pareto optimal.These relationships connect the tractable stationarity condition to Pareto optimality.
  • Scalarization methods: The weighted approach fixes simplex weights and includes FedAvg, with stationary points of the weighted scalar problem being Pareto stationary.Different fixed weights can produce different Pareto-stationary solutions.
  • Scalarization methods: The ε-constraint approach fixes m − 1 parameters, while the Chebyshev approach uses a fixed lower-bound vector and is associated with AFL.Both approaches produce weakly Pareto-optimal global minimizers under their stated conditions.

5 FL as Multi-objective Minimization

The paper treats each federated user function as a separate objective, using multi-objective optimization to pursue fairness and robustness without sacrificing participating users. It extends MGDA into FedMGDA+, combines adaptive weighting, normalization, and subsampling, and establishes convergence results under stated assumptions.

  • FedMGDA+ treats each user function as a separate objective, so Pareto optimality and stationarity directly encode fairness by discouraging improvements for some users that sacrifice others.The method extends multiple gradient descent to federated learning and targets simultaneous optimization of user objectives.
  • Balancing user average performance and fairness: MGDA adaptively tunes user weights and produces a common descent direction, whereas fixed-weight FedAvg can trade one participating user's performance for another's.An intermediate weighting parameter can balance average performance and fairness between FedAvg and unrestricted MGDA.
  • Robustness against malicious users through normalization: Normalization preserves a direction descending for all participating objectives and counters multiplicative malicious inflation, complementing MGDA's robustness to additive inflation.The paper presents normalization as a robustness extension within FedMGDA+ rather than as a separate aggregation strategy.
  • Subsampling to alleviate non-iid and enhance throughput: Subsampling supports heterogeneous federated participation, offers users an incentive to participate because participating objectives decrease, and avoids a high-participation degeneracy in MGDA.The degeneracy arises when the number of participating users exceeds the model dimension and MGDA reaches Pareto stationarity in one iteration.
  • Convergence: The convergence analysis relates stochastic-direction variance and approximation errors to user subsampling, heterogeneity, local updates, and the decay of global step sizes.The stated results cover one-step stochastic updates and deterministic updates with arbitrary numbers of local updates under their respective assumptions.
  • Convergence: For convex objectives, the paper gives a finer result when each round includes a strongly convex user function with weight bounded away from zero, while thorough analysis for multiple local updates remains future work.The paper also notes that for r, k > 1, FedMGDA+ is expected to optimize a proxy problem such as the proximal average.

6 Experiments

Experiments evaluate FedMGDA+ across several datasets, models, baselines, and fairness, robustness, and interpolation settings. Results examine recovery of FedAvg behavior, resistance to adversarial loss manipulation, and whether participating users avoid performance degradation.

  • FedMGDA+ is evaluated on CIFAR-10, F-MNIST, Federated EMNIST, Shakespeare, and Adult against FedAvg, FedProx, q-FedAvg, AFL4, and related extensions.
  • 6.1 Experimental setups: The experiments use medium-sized models and tune server and local learning rates, with exponential server learning-rate decay selected for the reported evaluations.CIFAR-10 and Fashion MNIST use non-iid shard assignments, while Federated EMNIST uses naturally heterogeneous and unbalanced data.
  • 6.2.2 Robustness: Robustness experiments test additive-bias and multiplicative-scaling attacks from a malicious user on Adult and CIFAR-10.The Adult bias-attack setup uses the PhD domain as the underrepresented user and compares results with training only on that domain.
  • 6.2.3 Fairness: FedMGDA+ consistently outperforms other algorithms in the percentage of users whose training loss improves after each participation.With full local batches and decaying step sizes, this percentage approaches 100% during training.

7 Conclusion

The paper concludes that FedMGDA+ uses multi-objective optimization to target Pareto-stationary solutions while addressing fairness and robustness in federated learning.

  • FedMGDA+ is designed to be robust against additive and multiplicative adversarial manipulations while ensuring fairness among participating users.
  • The algorithm is simple to implement, uses fewer hyperparameters, and complements existing federated-learning systems.
  • The authors establish preliminary convergence guarantees, connect FedMGDA+ to recent federated-learning algorithms, and evaluate its effectiveness experimentally.

A Proofs

The proofs analyze FedMGDA+ under smoothness, continuity, convexity, stochasticity, and local-update assumptions, establishing convergence toward Pareto-stationary behavior under suitable conditions.

  • The analysis assumes user losses are Lipschitz smooth and Lipschitz continuous, with global step sizes restricted to a specified interval.
  • For stochastic updates, convergence bounds depend on the variance of the stochastic common direction and vanish when the relevant variance and step-size conditions hold.
  • With local updates, accumulated user updates are compared with true gradients, and their discrepancy is bounded through the local optimization steps.
  • Under strong convexity and participation conditions, the iterates converge almost surely to the Pareto-stationarity set.
  • The appendix includes supplementary interpolation and training-loss plots for F-MNIST and CIFAR-10 under iid and non-iid settings.

B.2 Robustness full results: bias attack on Adult dataset

This appendix reports the full Adult-dataset robustness experiment, comparing state-of-the-art methods under adversarial bias added to the PhD domain loss.

  • Table 7 provides the full results corresponding to the robustness figure, while the listed hyperparameters include settings for AFL, q-FedAvg, and FedMGDA+.
  • The Adult-dataset experiment evaluates test accuracy under multiple adversarial-bias scales and compares methods with training only on the PhD domain as baseline.
  • The algorithms run for 500 rounds, with results averaged across 5 runs using different random seeds.
  • The appendix identifies different bias-scale conventions for AFL and q-FedAvg because AFL uses averaged loss while q-FedAvg uses non-averaged total loss.

B.3 Fairness full results: first experiment on CIFAR-10

This appendix gives full CIFAR-10 fairness results across local batch sizes and user-participation fractions, with tables reporting user test accuracy under specified training settings.

  • Tables 8–11 report the full results of the CIFAR-10 fairness experiment across different batch sizes and fractions of participating users.
  • For local batch size b = 10, results use participation fractions p = 0.1 or p = 0.2, learning rate η = 0.01, and 2000 communication rounds.
  • For local batch size b = 400, results use participation fractions p = 0.1 or p = 0.2, learning rate η = 0.1, and 3000 communication rounds.
  • The reported CIFAR-10 test accuracies are averaged across 4 runs with different random seeds.

B.4 Fairness full results: first experiment on Federated EMNIST

This section reports full results for the first Federated EMNIST fairness experiment across different batch sizes, including local batch size 20 and full-batch training.

  • Full results for the Table 6 experiment are reported across different batch sizes.
  • With local batch size b = 20, the experiment uses 10 users per round, η = 0.1, and 1500 communication rounds.
  • The full-batch setting likewise uses 10 users per round, η = 0.1, and 1500 communication rounds.

B.5 Fairness full results: second experiment

This section presents full results for the second fairness experiment across batch-size settings and reports user accuracies on Shakespeare under multiple learning-rate configurations.

  • Full results for the Figure 4 experiment are shown for different batch sizes.
  • Figures 8 and 9 evaluate improved-user percentages for training loss and global test accuracy on CIFAR-10 with b = 10 and b = 400.
  • The Shakespeare evaluation reports average client and data-point-weighted accuracies with batch size 10, full participation, and multiple local learning rates.
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