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Subgame Perfect Methods in Nonsmooth Convex Optimization

Benjamin Grimmer, Alex L. Wang

TL;DR

This work introduces subgame perfection for nonsmooth convex optimization under two oracle models: subgradient and proximal. It proves that the Kelley-Like Method (KLM) is subgame perfect in the subgradient setting by constructing a history-consistent dynamic lower bound, and it designs a new Subgame Perfect Proximal Point Algorithm (SPPPA) for the proximal oracle setting, with a planning subproblem solved at each iteration that adapts the inductive hypothesis. The analysis shows that both methods achieve guarantees never worse than minimax-optimal bounds and often substantially better, by exploiting information revealed during execution. The results establish a principled framework for adaptive, history-aware optimization in high dimensions and pave the way for extending subgame perfection to additional optimization settings.

Abstract

This paper considers nonsmooth convex optimization with either a subgradient or proximal operator oracle. In both settings, we identify algorithms that achieve the recently introduced game-theoretic optimality notion for algorithms known as subgame perfection. Subgame perfect algorithms meet a more stringent requirement than just minimax optimality. Not only must they provide optimal uniform guarantees on the entire problem class, but also on any subclass determined by information revealed during the execution of the algorithm. In the setting of nonsmooth convex optimization with a subgradient oracle, we show that the Kelley cutting plane-Like Method due to Drori and Teboulle [1] is subgame perfect. For nonsmooth convex optimization with a proximal operator oracle, we develop a new algorithm, the Subgame Perfect Proximal Point Algorithm, and establish that it is subgame perfect. Both of these methods solve a history-aware second-order cone program within each iteration, independent of the ambient problem dimension, to plan their next steps. This yields performance guarantees that are never worse than the minimax optimal guarantees and often substantially better.

Subgame Perfect Methods in Nonsmooth Convex Optimization

TL;DR

This work introduces subgame perfection for nonsmooth convex optimization under two oracle models: subgradient and proximal. It proves that the Kelley-Like Method (KLM) is subgame perfect in the subgradient setting by constructing a history-consistent dynamic lower bound, and it designs a new Subgame Perfect Proximal Point Algorithm (SPPPA) for the proximal oracle setting, with a planning subproblem solved at each iteration that adapts the inductive hypothesis. The analysis shows that both methods achieve guarantees never worse than minimax-optimal bounds and often substantially better, by exploiting information revealed during execution. The results establish a principled framework for adaptive, history-aware optimization in high dimensions and pave the way for extending subgame perfection to additional optimization settings.

Abstract

This paper considers nonsmooth convex optimization with either a subgradient or proximal operator oracle. In both settings, we identify algorithms that achieve the recently introduced game-theoretic optimality notion for algorithms known as subgame perfection. Subgame perfect algorithms meet a more stringent requirement than just minimax optimality. Not only must they provide optimal uniform guarantees on the entire problem class, but also on any subclass determined by information revealed during the execution of the algorithm. In the setting of nonsmooth convex optimization with a subgradient oracle, we show that the Kelley cutting plane-Like Method due to Drori and Teboulle [1] is subgame perfect. For nonsmooth convex optimization with a proximal operator oracle, we develop a new algorithm, the Subgame Perfect Proximal Point Algorithm, and establish that it is subgame perfect. Both of these methods solve a history-aware second-order cone program within each iteration, independent of the ambient problem dimension, to plan their next steps. This yields performance guarantees that are never worse than the minimax optimal guarantees and often substantially better.

Paper Structure

This paper contains 22 sections, 16 theorems, 101 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and organization.
  3. Subgame Perfect Subgradient Method
  4. The Kelley-Like Method of Drori and Teboulle drori2016Kelley
  5. Proof of Theorem \ref{['thm:Kelley-perfection']}
  6. The Subgame Perfect Proximal Point Algorithm (SPPPA)
  7. The Existing Optimized Proximal Point Algorithm (OPPA)
  8. Design of SPPPA
  9. The Planning Subproblem
  10. Subgame Perfect Proximal Point Algorithm and its Guarantees
  11. Subgame Perfection of SPPPA
  12. Dynamic Construction of Candidate Hard Problem Instance
  13. Dual of the Planning Subproblem
  14. Construction of Our Dynamic Hard Instance
  15. Properties of the Candidate Hard Instance
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $\{(x_i, f_i,g_i)\}_{i\in{\cal I}_\star}\subseteq {\mathbb{R}}^d\times{\mathbb{R}}\times{\mathbb{R}}^d$. The quantities $Q_{i,j}$ and $S_i$ are nonnegative for all $i,j\in{\cal I}_\star$ if and only if there exists an $M$-Lipschitz convex function $f$ satisfying

Theorems & Definitions (27)

  • Lemma 2.1: taylor2017CompositePEP
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm:Kelley-perfection']}
  • Lemma 3.1
  • proof
  • ...and 17 more