An Accelerated Method for Derivative-Free Smooth Stochastic Convex Optimization

TL;DR

This paper introduces accelerated and non-accelerated derivative-free algorithms for smooth stochastic convex optimization with noisy observations, achieving near-optimal complexity bounds with improvements for sparse solutions.

Contribution

It presents novel accelerated and non-accelerated derivative-free algorithms with complexity bounds close to gradient-based methods, including enhancements for sparse solutions using 1-norm proximal setups.

Findings

Accelerated algorithm's complexity is only √n worse than gradient-based methods.

Non-accelerated algorithm matches stochastic-gradient bounds, with logarithmic factors.

Proposed methods perform better for sparse solutions using 1-norm proximal setup.

Abstract

We consider an unconstrained problem of minimizing a smooth convex function which is only available through noisy observations of its values, the noise consisting of two parts. Similar to stochastic optimization problems, the first part is of stochastic nature. The second part is additive noise of unknown nature, but bounded in absolute value. In the two-point feedback setting, i.e. when pairs of function values are available, we propose an accelerated derivative-free algorithm together with its complexity analysis. The complexity bound of our derivative-free algorithm is only by a factor of $\sqrt{n}$ larger than the bound for accelerated gradient-based algorithms, where $n$ is the dimension of the decision variable. We also propose a non-accelerated derivative-free algorithm with a complexity bound similar to the stochastic-gradient-based algorithm, that is, our bound does not have any dimension-dependent factor except logarithmic. Notably, if the difference between the starting point and the solution is a sparse vector, for both our algorithms, we obtain a better complexity bound if the algorithm uses an $1$-norm proximal setup, rather than the Euclidean proximal setup, which is a standard choice for unconstrained problems