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Quantum Algorithms for Non-smooth Non-convex Optimization

Chengchang Liu, Chaowen Guan, Jianhao He, John C. S. Lui

Abstract

This paper considers the problem for finding the $(δ,ε)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of $\tilde{\mathcal{O}}(d^{3/2}δ^{-1}ε^{-3})$ on the stochastic function value oracle, where $d$ is the dimension of the problem. We also enhance the query complexity to $\tilde{\mathcal{O}}(d^{3/2}δ^{-1}ε^{-7/3})$ by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of $ε$ by a factor of $ε^{-2/3}$.

Quantum Algorithms for Non-smooth Non-convex Optimization

Abstract

This paper considers the problem for finding the -Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of on the stochastic function value oracle, where is the dimension of the problem. We also enhance the query complexity to by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of by a factor of .

Paper Structure

This paper contains 16 sections, 9 theorems, 62 equations, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries for Quantum Computing Model
  4. Preliminaries for Non-convex Non-smooth Optimization
  5. Zeroth-order Based Stochastic Quantum Estimator
  6. Quantum Estimators via Quantum Stochastic Function Value Oracle
  7. Mini-batch Quantum Estimators via Quantum Mean Estimation
  8. Quantum Algorithms for Finding the Goldstein Stationary Point
  9. Conclusion and Future Work
  10. Explicit Construction of Quantum Sampling Oracles
  11. The Proof of Lemma \ref{['lm:O_g_delta']}
  12. The Proof of Corollary \ref{['cor:est_deltag']}
  13. The Proof of Theorem \ref{['thm:mini-batch']}
  14. The Proof of Theorem \ref{['thm:GFQM']}
  15. The Proof of Theorem \ref{['thm:FGFQM']}
  16. ...and 1 more sections

Key Result

Proposition 2.1

If $f(\cdot)$ satisfies Assumption ass:lip, its smoothed surrogate $f_{\delta}(\cdot)$ satisfies that:

Theorems & Definitions (33)

  • Definition 2.1: Clarke sub-differential
  • Definition 2.2: Goldstein sub-differential
  • Definition 2.3: $(\delta,\epsilon)$-Goldstein stationary point
  • Definition 2.4: $\delta$-smoothed surrogate
  • Proposition 2.1
  • Remark 2.2
  • Definition 3.1: Quantum stochastic function value oracle
  • Proposition 3.1: chen2023faster
  • Definition 3.2: Quantum sampling oracle
  • Definition 3.3: Quantum $\delta$-estimated stochastic gradient oracle
  • ...and 23 more