A One-bit, Comparison-Based Gradient Estimator
This work addresses optimization problems where function evaluations are unavailable, which is incremental but relevant for scenarios with limited access to objective functions.
The paper tackles zeroth-order optimization for convex functions without direct function evaluations, using only a comparison oracle that indicates which of two points has a larger function value. It proposes the SCOBO algorithm, which leverages one-bit compressed sensing to estimate gradients and shows improved query complexity over state-of-the-art methods when exploiting low-dimensional structure, as verified by numerical experiments.
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.