Distributed Zero-Order Optimization under Adversarial Noise

TL;DR

This paper introduces a distributed zero-order optimization algorithm capable of handling noisy function evaluations, with theoretical guarantees on regret and error bounds, improving upon existing methods especially in networked settings.

Contribution

The paper proposes a novel distributed zero-order projected gradient descent algorithm that works with noisy function values and improves theoretical bounds over prior work.

Findings

Upper bounds for regret and error are derived, showing dependence on network connectivity and noise.

The algorithm outperforms existing bounds in the standard non-distributed setting.

Lower bounds suggest the bounds are nearly optimal.

Abstract

We study the problem of distributed zero-order optimization for a class of strongly convex functions. They are formed by the average of local objectives, associated to different nodes in a prescribed network of connections. We propose a distributed zero-order projected gradient descent algorithm to solve this problem. Exchange of information within the network is permitted only between neighbouring nodes. A key feature of the algorithm is that it can query only function values, subject to a general noise model, that does not require zero mean or independent errors. We derive upper bounds for the average cumulative regret and optimization error of the algorithm which highlight the role played by a network connectivity parameter, the number of variables, the noise level, the strong convexity parameter of the global objective and certain smoothness properties of the local objectives. When the bound is specified to the standard undistributed setting, we obtain an improvement over the state-of-the-art bounds, due to the novel gradient estimation procedure proposed here. We also comment on lower bounds and observe that the dependency over certain function parameters in the bound is nearly optimal.