A One-bit, Comparison-Based Gradient Estimator

TL;DR

This paper introduces SCOBO, a novel zeroth-order optimization algorithm that estimates gradients using only comparison-based one-bit feedback, outperforming existing methods when the function has low-dimensional structure.

Contribution

The paper develops a gradient estimator from comparison oracle data using one-bit compressed sensing, and proposes SCOBO, a new algorithm that leverages this estimator for efficient optimization.

Findings

SCOBO outperforms state-of-the-art methods in low-dimensional structured settings.

The gradient estimator is robust and reliable despite limited one-bit comparison data.

Extensive experiments verify the theoretical advantages of SCOBO.

Abstract

We study zeroth-order optimization for convex functions where we further assume that function evaluations are unavailable. Instead, one only has access to a $\textit{comparison oracle}$, which given two points $x$ and $y$ returns a single bit of information indicating which point has larger function value, $f(x)$ or $f(y)$. By treating the gradient as an unknown signal to be recovered, we show how one can use tools from one-bit compressed sensing to construct a robust and reliable estimator of the normalized gradient. We then propose an algorithm, coined SCOBO, that uses this estimator within a gradient descent scheme. We show that when $f(x)$ has some low dimensional structure that can be exploited, SCOBO outperforms the state-of-the-art in terms of query complexity. Our theoretical claims are verified by extensive numerical experiments.