Rethinking Warm-Starts with Predictions: Learning Predictions Close to Sets of Optimal Solutions for Faster $\text{L}$-/$\text{L}^\natural$-Convex Function Minimization

TL;DR

This paper introduces a new framework for warm-starting algorithms in convex function minimization that accounts for multiple optimal solutions, providing provable bounds and a polynomial-time learning method.

Contribution

It presents the first polynomial-time method to learn predictions close to all optimal solutions, improving warm-start efficiency for L- and L-natural convex minimization.

Findings

Time complexity bounds depend on the distance to the set of all optimal solutions.

Proposes an online-gradient-descent-based method for learning predictions.

Achieves polynomial-time learnability for warm-start predictions in complex convex minimization.

Abstract

An emerging line of work has shown that machine-learned predictions are useful to warm-start algorithms for discrete optimization problems, such as bipartite matching. Previous studies have shown time complexity bounds proportional to some distance between a prediction and an optimal solution, which we can approximately minimize by learning predictions from past optimal solutions. However, such guarantees may not be meaningful when multiple optimal solutions exist. Indeed, the dual problem of bipartite matching and, more generally, $\text{L}$-/$\text{L}^\natural$-convex function minimization have arbitrarily many optimal solutions, making such prediction-dependent bounds arbitrarily large. To resolve this theoretically critical issue, we present a new warm-start-with-prediction framework for $\text{L}$-/$\text{L}^\natural$-convex function minimization. Our framework offers time complexity bounds proportional to the distance between a prediction and the set of all optimal solutions. The main technical difficulty lies in learning predictions that are provably close to sets of all optimal solutions, for which we present an online-gradient-descent-based method. We thus give the first polynomial-time learnability of predictions that can provably warm-start algorithms regardless of multiple optimal solutions.