Decentralized machine learning often relies on outsourcing computations, such as gradient evaluations, to untrusted worker nodes. Existing robust aggregation methods can mitigate malicious behavior under honest-majority assumptions, but may fail when adversaries control a majority of the workers. We study this adversary-dominated setting through an incentive-oriented framework in which reports are accepted and rewarded only when they are mutually consistent up to a threshold. This turns the adversary from a pure saboteur into a rational agent that trades off increasing estimation error against the risk of rejection and loss of reward. We consider iterative optimization under this model. Unlike one-shot computation, iterative learning requires long-horizon decisions: permissive acceptance rules enable faster early progress but admit more adversarial corruption, while strict rules improve estimation accuracy but cause frequent rejections. We propose \mathsf{VISTA}, an adaptive algorithm that tunes the acceptance threshold using the optimization history. Numerical results show that \mathsf{VISTA} improves convergence over static thresholds. We also provide a rigorous convergence analysis showing that, with suitable incentive-aware adaptation, adversary-dominated decentralized learning can retain the asymptotic convergence behavior of standard SGD without relying on an honest majority.
We propose FAR-SIGN (Fully Asynchronous Robust optimization via SIGNed directional projections) for adversary-resilient learning in parameter-server--worker systems. FAR-SIGN achieves robustness through sign-based updates along carefully designed directions and mitigates the resulting bias via a two-timescale mechanism. It admits both first-order and zeroth-order implementations and enables fully asynchronous execution without requiring a private reference dataset at the server. We establish almost-sure convergence of FAR-SIGN to the set of stationary points for smooth, nonconvex objectives. Moreover, we prove the near-optimal rate of O(n−1/4+ε) in the first-order setting and the standard O(n−1/6+ε) in the zeroth-order setting, where n is the iteration count and ε>0 can be chosen arbitrarily small. Experiments on MNIST show that FAR-SIGN outperforms robust aggregation-based methods in both accuracy and wall-clock time.
Decentralized training is often regarded as inferior to centralized training because the consensus errors between workers are thought to undermine convergence and generalization. This work challenges this view by introducing decentralized SGD with Adaptive Consensus (DSGD-AC), which uses a time-dependent scaling mechanism to maintain consensus errors throughout the training. We show that adaptive consensus changes the stationary variance of disagreement modes by balancing two effects: it preserves consensus-error magnitude through weaker graph damping while still allowing curvature-dependent damping to shape the disagreement directions. This balance can produce a stronger Hessian-weighted loss-envelope penalty around the deployed model, even when normalized Hessian alignment is weaker than in standard DSGD. Empirical results on image classification show that DSGD-AC reaches flatter solutions and higher test accuracy than standard DSGD and even centralized SGD. Together, these results support consensus errors as a useful implicit regularizer and open a new perspective on the design of decentralized learning algorithms.
Privacy concerns in distributed learning often lead clients to return intentionally altered gradient information. We consider the problem of learning convex and L-smooth functions under adversarial gradient perturbation, where a client's gradient reply to a server query can deviate arbitrarily from the true gradient subject to a distance bound. Our study focuses on two fundamental questions: (i) what is the smallest achievable sub-optimality gap (i.e., excess error in optimization) under such responses, and (ii) how many queries are sufficient to guarantee a given sub-optimality gap? We establish tight feasibility thresholds on the sub-optimality gap and provide algorithms that achieve these thresholds with provable query complexity guarantees.