Function-free Optimization via Comparison Oracles

TL;DR

This paper introduces a framework for optimizing without explicit objective functions, relying solely on preference comparisons to find optimal solutions in complex, nonsmooth, or nonconvex settings.

Contribution

It develops geometric and algorithmic methods for function-free optimization using comparison oracles, including estimation of normal directions and adaptive schemes.

Findings

Comparison-based normal direction estimation matches lower bounds in complexity.

The descent method achieves near-optimal comparison complexity for convex preference relations.

Adaptive schemes effectively estimate directions and solve optimization problems with minimal comparisons.

Abstract

In this work, we study optimization specified only through a comparison oracle: given two points, it reports which one is preferred. We call it function-free optimization because we do not assume access to, nor the existence of, a canonical application-given objective function. The goal is to find the most preferred feasible point, which we call the optimal solution. This model arises in preference- and ranking-based settings where objective values and derivatives are unavailable or meaningless. Even when a representative function exists, it may be nonsmooth, nonconvex, or discontinuous. We develop an analytical and algorithmic framework based on the geometry of preference level sets, which remains well-defined from comparisons alone. We introduce the level-set optimality gap, the distance from a preference level set to the optimal solutions, and the regularity radius, a stationarity certificate. Under regularity of the preference relation in a $d$-dimensional Euclidean space, we estimate normal directions to accuracy $\epsilon$ using $O(d\log(d/\epsilon))$ comparisons, nearly matching a lower bound of $\Omega(d\log(1/\epsilon))$. Under convexity, regularity, and a local growth condition on the regularity radius, the resulting normal direction descent method reaches an $\epsilon$ level-set optimality gap using at most $\widetilde O(dD^2/\epsilon^2)$ comparisons over $O(D^2/\epsilon^2)$ normal direction estimation steps, where $D$ is the distance from the initial point to the optimal solutions. This number of steps matches the lower bound of $\Omega(D^2/\epsilon^2)$ for normal direction span-based methods. Since prior knowledge in practical applications is usually limited, we also develop adaptive schemes for estimating the normal direction and solving the optimization problem. They match the fixed-parameter complexity bounds up to logarithmic factors.