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Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods

Ruichen Jiang, Aryan Mokhtari, Francisco Patitucci

TL;DR

The paper tackles finding an $\varepsilon$-FOSP for smooth nonconvex functions using only gradient information. It introduces a two-level online-learning framework with an optimistic quasi-Newton update that leverages a matrix Hessian proxy updated via online learning in the matrix space, all while avoiding Hessian-vector products. The main result is a gradient complexity of $O(d^{1/4}\varepsilon^{-13/8})$, improving upon the state-of-the-art when $d = O(\varepsilon^{-1/2})$, and a detailed analysis of the associated computational costs through a projection-free trust-region subproblem solver and a Hessian-update oracle. The work also provides the first theoretical guarantee that quasi-Newton updates can outperform gradient-descent-type methods in nonconvex settings and discusses lower-bound implications and future directions for dimension-independent results.

Abstract

We study the problem of finding an $ε$-first-order stationary point (FOSP) of a smooth function, given access only to gradient information. The best-known gradient query complexity for this task, assuming both the gradient and Hessian of the objective function are Lipschitz continuous, is ${O}(ε^{-7/4})$. In this work, we propose a method with a gradient complexity of ${O}(d^{1/4}ε^{-13/8})$, where $d$ is the problem dimension, leading to an improved complexity when $d = {O}(ε^{-1/2})$. To achieve this result, we design an optimization algorithm that, underneath, involves solving two online learning problems. Specifically, we first reformulate the task of finding a stationary point for a nonconvex problem as minimizing the regret in an online convex optimization problem, where the loss is determined by the gradient of the objective function. Then, we introduce a novel optimistic quasi-Newton method to solve this online learning problem, with the Hessian approximation update itself framed as an online learning problem in the space of matrices. Beyond improving the complexity bound for achieving an $ε$-FOSP using a gradient oracle, our result provides the first guarantee suggesting that quasi-Newton methods can potentially outperform gradient descent-type methods in nonconvex settings.

Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods

TL;DR

The paper tackles finding an -FOSP for smooth nonconvex functions using only gradient information. It introduces a two-level online-learning framework with an optimistic quasi-Newton update that leverages a matrix Hessian proxy updated via online learning in the matrix space, all while avoiding Hessian-vector products. The main result is a gradient complexity of , improving upon the state-of-the-art when , and a detailed analysis of the associated computational costs through a projection-free trust-region subproblem solver and a Hessian-update oracle. The work also provides the first theoretical guarantee that quasi-Newton updates can outperform gradient-descent-type methods in nonconvex settings and discusses lower-bound implications and future directions for dimension-independent results.

Abstract

We study the problem of finding an -first-order stationary point (FOSP) of a smooth function, given access only to gradient information. The best-known gradient query complexity for this task, assuming both the gradient and Hessian of the objective function are Lipschitz continuous, is . In this work, we propose a method with a gradient complexity of , where is the problem dimension, leading to an improved complexity when . To achieve this result, we design an optimization algorithm that, underneath, involves solving two online learning problems. Specifically, we first reformulate the task of finding a stationary point for a nonconvex problem as minimizing the regret in an online convex optimization problem, where the loss is determined by the gradient of the objective function. Then, we introduce a novel optimistic quasi-Newton method to solve this online learning problem, with the Hessian approximation update itself framed as an online learning problem in the space of matrices. Beyond improving the complexity bound for achieving an -FOSP using a gradient oracle, our result provides the first guarantee suggesting that quasi-Newton methods can potentially outperform gradient descent-type methods in nonconvex settings.

Paper Structure

This paper contains 28 sections, 25 theorems, 94 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Additional Related Work
  3. Preliminaries and Background
  4. Online-to-nonconvex Conversion
  5. Proposed Algorithm
  6. Learning Update Directions: Optimistic Quasi-Newton Algorithm
  7. Efficient Subroutine for OQN Update
  8. Hessian approximation update via online learning
  9. Complexity Analysis
  10. Convergence Rate
  11. Characterizing the Computational Cost
  12. Discussion on the Lower Bound and Concluding Remarks
  13. Proofs for Online-to-non-convex Conversion
  14. Proof of Lemma 2.1
  15. Proof of Lemma 2.2
  16. ...and 13 more sections

Key Result

Lemma 2.1

Consider ${\mathbf{x}}_{n}={\mathbf{x}}_{n-1} + \bm{\Delta}_n$ where $\|\bm{\Delta}_n\| \leq D$. Further, define ${\mathbf{g}}_n=\nabla f({\mathbf{w}}_n)$ where ${\mathbf{w}}_n=\frac{1}{2}({\mathbf{x}}_{n-1}+{\mathbf{x}}_{n})$. If Assumption asm:L2 holds, then $f({\mathbf{x}}_{n-1}) - f({\mathbf{x}}

Theorems & Definitions (42)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Definition 3.1
  • Lemma 3.2
  • Remark 3.1
  • Remark 3.2
  • Definition 3.2
  • Remark 3.3
  • ...and 32 more