Distributed Zero-Order Optimization under Adversarial Noise

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.