No quantum speedup over gradient descent for non-smooth convex optimization

TL;DR

This paper demonstrates that quantum algorithms do not provide a speedup over classical gradient descent in black-box first-order convex optimization for non-smooth functions, establishing matching lower bounds for both.

Contribution

It proves that quantum algorithms cannot outperform classical gradient descent in this setting, confirming the optimality of classical methods even with quantum resources.

Findings

Quantum algorithms solve certain hard instances with O(GR/ε) queries.

Classical gradient descent requires O((GR/ε)^2) queries.

No quantum speedup exists for general non-smooth convex optimization.

Abstract

We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function $f:\mathbb{R}^n \to \mathbb{R}$ and its (sub)gradient. Our goal is to find an $\epsilon$-approximate minimum of $f$ starting from a point that is distance at most $R$ from the true minimum. If $f$ is $G$-Lipschitz, then the classic gradient descent algorithm solves this problem with $O((GR/\epsilon)^{2})$ queries. Importantly, the number of queries is independent of the dimension $n$ and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension $n$. In this paper we reprove the randomized lower bound of $\Omega((GR/\epsilon)^{2})$ using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using $O(GR/\epsilon)$ quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need $\Omega((GR/\epsilon)^2)$ queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.