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Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss

Hao Wang, Chenyi Zhang, Tongyang Li

TL;DR

The paper analyzes quantum algorithms for minimizing the maximum of N convex Lipschitz functions by introducing a quantum zeroth-order framework and a Ball Regularized Optimization Oracle (BROO) for the softmax surrogate F_{smax,\epsilon}. It achieves a quantum upper bound of \tilde{O}(\sqrt{N}\epsilon^{-5/3}+\epsilon^{-8/3}) oracle queries and proves a near-optimal quantum lower bound of \tilde{\Omega}(\sqrt{N}\epsilon^{-2/3}), demonstrating a quadratic speedup in the dependence on N relative to classical methods. The approach leverages quantum Gibbs-sampling to speed up the crucial sampling step, enabling a quantum-accelerated ball optimization within a Monteiro-Svaiter-type framework, and employs a quantum progress-control method to establish lower bounds. The results illuminate the potential and limits of quantum optimization for robust, max-loss problems and open questions about exploiting smoothness to close remaining gaps. Overall, the work advances quantum optimization by combining BROO with Gibbs-sampled softmax and provides foundational lower bounds for this problem class.

Abstract

The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(Nε^{-2/3} + ε^{-8/3})$ queries to a first-order oracle to compute an $ε$-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of $\tilde{O}(\sqrt{N}ε^{-5/3} + ε^{-8/3})$. On the other hand, we prove that quantum algorithms must take $\tildeΩ(\sqrt{N}ε^{-2/3})$ queries to a first order quantum oracle, showing that our dependence on $N$ is optimal up to poly-logarithmic factors.

Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss

TL;DR

The paper analyzes quantum algorithms for minimizing the maximum of N convex Lipschitz functions by introducing a quantum zeroth-order framework and a Ball Regularized Optimization Oracle (BROO) for the softmax surrogate F_{smax,\epsilon}. It achieves a quantum upper bound of \tilde{O}(\sqrt{N}\epsilon^{-5/3}+\epsilon^{-8/3}) oracle queries and proves a near-optimal quantum lower bound of \tilde{\Omega}(\sqrt{N}\epsilon^{-2/3}), demonstrating a quadratic speedup in the dependence on N relative to classical methods. The approach leverages quantum Gibbs-sampling to speed up the crucial sampling step, enabling a quantum-accelerated ball optimization within a Monteiro-Svaiter-type framework, and employs a quantum progress-control method to establish lower bounds. The results illuminate the potential and limits of quantum optimization for robust, max-loss problems and open questions about exploiting smoothness to close remaining gaps. Overall, the work advances quantum optimization by combining BROO with Gibbs-sampled softmax and provides foundational lower bounds for this problem class.

Abstract

The problem of minimizing the maximum of convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring queries to a first-order oracle to compute an -suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of . On the other hand, we prove that quantum algorithms must take queries to a first order quantum oracle, showing that our dependence on is optimal up to poly-logarithmic factors.
Paper Structure (18 sections, 24 theorems, 95 equations, 1 figure, 1 table)

This paper contains 18 sections, 24 theorems, 95 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Techniques.
  4. Open questions.
  5. Preliminaries
  6. Basic notations in quantum computing.
  7. Basic quantum algorithms.
  8. Quantum Algorithm for Minimizing the Maximal Loss
  9. Quantum BROO implementation of $\mathop{\mathrm{\mathit{F}_{smax,\epsilon}}}\nolimits$
  10. Quantum query complexity of minimizing the maximal loss
  11. Quantum lower bound for minimizing the maximum loss
  12. Progress control argument for quantum algorithms
  13. The lower bound statement
  14. Proof details of our quantum upper bound
  15. Proof details of our quantum lower bound
  16. ...and 3 more sections

Key Result

Theorem 1

There is a quantum algorithm (algo:innerloop-SGD) that outputs an $x_\star$ satisfying Eq. ( ) with probability at least $2/3$ using $\tilde{O}(\sqrt{N}\epsilon^{-5/3} + \epsilon^{-8/3})$ queries to the $O_{f}$ in Eq. ( ). On the other hand, such quantum algorithms must take $\tilde{\Omega}(\sqrt{N}\epsilon^{-2/3})$ queries to $O_{f}$ (thm:lower).

Figures (1)

  • Figure 1: Circuit $\mathcal{C}$ built from $\mathcal{D}$, in which $O_{\bar{x}}$ is a quantum oracle built from $O_f$ such that $O_{\bar{x}}\ket{i}\ket{0} = \ket{i}\ket{e^{f_i(\bar{x})/\epsilon'}}$.

Theorems & Definitions (42)

  • Theorem 1: Main theorem
  • Proposition 1: Top-K Maximum Finding - durr2006maxfinding
  • Proposition 2: State Preparation - zalka1998simulatinggrover2002creatingphillip2001preparing
  • Proposition 3: Amplitude Amplification - brassard2002amplitude
  • Proposition 4: Quantum Gradient Estimation - jordan2005fast
  • Definition 1: carmon2021thinking
  • Proposition 5: BROO Acceleration - Rephrased from carmon2021thinking
  • Theorem 2
  • Lemma 1
  • Lemma 2: hazan2014beyond
  • ...and 32 more