Impact of Redundancy on Resilience in Distributed Optimization and Learning

TL;DR

This paper explores how redundancy in local cost functions enhances the resilience of distributed optimization and learning systems against Byzantine faults and stragglers, providing theoretical and empirical evidence for improved approximation.

Contribution

It introduces the concepts of $(f, r; \epsilon)$-resilience and redundancy, establishing conditions under which approximate solutions remain viable despite faults and delays.

Findings

Redundancy improves approximation quality in resilient distributed optimization.

Theoretical bounds on resilience are validated through empirical experiments.

Higher redundancy enables better tolerance to Byzantine faults and stragglers.

Abstract

This report considers the problem of resilient distributed optimization and stochastic learning in a server-based architecture. The system comprises a server and multiple agents, where each agent has its own local cost function. The agents collaborate with the server to find a minimum of the aggregate of the local cost functions. In the context of stochastic learning, the local cost of an agent is the loss function computed over the data at that agent. In this report, we consider this problem in a system wherein some of the agents may be Byzantine faulty and some of the agents may be slow (also called stragglers). In this setting, we investigate the conditions under which it is possible to obtain an "approximate" solution to the above problem. In particular, we introduce the notion of $(f, r; \epsilon)$-resilience to characterize how well the true solution is approximated in the presence of up to $f$ Byzantine faulty agents, and up to $r$ slow agents (or stragglers) -- smaller $\epsilon$ represents a better approximation. We also introduce a measure named $(f, r; \epsilon)$-redundancy to characterize the redundancy in the cost functions of the agents. Greater redundancy allows for a better approximation when solving the problem of aggregate cost minimization. In this report, we constructively show (both theoretically and empirically) that $(f, r; \mathcal{O}(\epsilon))$-resilience can indeed be achieved in practice, given that the local cost functions are sufficiently redundant.