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Extracting Dual Solutions via Primal Optimizers

Yair Carmon, Arun Jambulapati, Liam O'Carroll, Aaron Sidford

TL;DR

This work presents a general dual-extraction framework that converts a black-box primal solver for regularized convex-concave minimax problems into an algorithm for solving the associated dual maximin problem. By introducing recursive dual regularization and dual-best-response steps, the framework achieves epsilon-optimal dual solutions at essentially the same cost as obtaining epsilon-optimal primal solutions (up to logarithmic factors). The authors instantiate the framework to derive state-of-the-art runtimes for bilinear matrix games and CVaR-level DRO, and they also recover optimal query complexity for finding stationary points of smooth convex functions via a Fenchel-game formulation. The results broaden the applicability of black-box reductions in minimax optimization and provide practically impactful reductions in computational complexity across several canonical problems.

Abstract

We provide a general method to convert a "primal" black-box algorithm for solving regularized convex-concave minimax optimization problems into an algorithm for solving the associated dual maximin optimization problem. Our method adds recursive regularization over a logarithmic number of rounds where each round consists of an approximate regularized primal optimization followed by the computation of a dual best response. We apply this result to obtain new state-of-the-art runtimes for solving matrix games in specific parameter regimes, obtain improved query complexity for solving the dual of the CVaR distributionally robust optimization (DRO) problem, and recover the optimal query complexity for finding a stationary point of a convex function.

Extracting Dual Solutions via Primal Optimizers

TL;DR

This work presents a general dual-extraction framework that converts a black-box primal solver for regularized convex-concave minimax problems into an algorithm for solving the associated dual maximin problem. By introducing recursive dual regularization and dual-best-response steps, the framework achieves epsilon-optimal dual solutions at essentially the same cost as obtaining epsilon-optimal primal solutions (up to logarithmic factors). The authors instantiate the framework to derive state-of-the-art runtimes for bilinear matrix games and CVaR-level DRO, and they also recover optimal query complexity for finding stationary points of smooth convex functions via a Fenchel-game formulation. The results broaden the applicability of black-box reductions in minimax optimization and provide practically impactful reductions in computational complexity across several canonical problems.

Abstract

We provide a general method to convert a "primal" black-box algorithm for solving regularized convex-concave minimax optimization problems into an algorithm for solving the associated dual maximin optimization problem. Our method adds recursive regularization over a logarithmic number of rounds where each round consists of an approximate regularized primal optimization followed by the computation of a dual best response. We apply this result to obtain new state-of-the-art runtimes for solving matrix games in specific parameter regimes, obtain improved query complexity for solving the dual of the CVaR distributionally robust optimization (DRO) problem, and recover the optimal query complexity for finding a stationary point of a convex function.

Paper Structure

This paper contains 51 sections, 43 theorems, 113 equations, 1 figure, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. From primal algorithms to dual optimization.
  4. Application 1: Bilinear matrix games.
  5. Application 2: CVaR at level $\alpha$ DRO.
  6. Application 3: Obtaining stationary points of convex functions.
  7. Overview of the framework and analysis
  8. A first try.
  9. Recursive regularization and the dual-extraction framework.
  10. Analysis of Algorithm \ref{['alg:dual-extraction-framework-sketch']}.
  11. Related work
  12. Black-box reductions.
  13. Bilinear matrix games.
  14. CVaR at level $\alpha$ distributionally robust optimization (DRO).
  15. Stationary point computation.
  16. ...and 36 more sections

Key Result

Lemma 1

For a given $x \in \mathcal{X}$, suppose $-\psi(x, \cdot)$ is $\mu$-strongly convex relative to the dgf $r$ for some $\mu > 0$. Then $y_x \coloneqq \mathop{\rm argmax}_{y \in \mathcal{Y}} \psi(x, y)$ satisfies

Figures (1)

  • Figure 1: An example to give intuition behind Lemma \ref{['lem:bound-dual-div-primal-subopt-informal']}. Here, $\psi(x, y) = xy - 0.8 y^2$, $(x^\star, y^\star) = (0, 0)$, $x = 0.8$, and $y_x = 0.5$. To see why it is possible to bound $|y^\star - y_x|$ in terms of the primal suboptimality $f(x) - f(x^\star)$, note that by the strong concavity of $\psi(x, \cdot)$ and the fact that $y_x$ is the maximizer of $\psi(x, \cdot)$ over $\mathcal{Y}$, we can upper bound $|y^\star - y_x|$ in terms of $\psi(x, y_x) - \psi(x, y^\star)$ (the vertical drop over the green line) via a standard strong-concavity inequality. In turn, $\psi(x, y_x) - \psi(x, y^\star)$ can be upper bounded by $\psi(x, y_x) - \psi(x^\star, y^\star) = f(x) - f(x^\star)$ (the vertical drop over the green line plus the vertical drop over the red line) due to the fact that $\psi(x^\star, y^\star) \le \psi(x, y^\star)$ by the optimality of $x^\star$.

Theorems & Definitions (90)

  • Lemma 1: Lemma \ref{['lem:dual-div-bound-exact-best-response']} specialized
  • Theorem 1: Theorem \ref{['thm:main-framework-guarantee']} specialized
  • proof
  • Remark 1: Picking the parameters for Theorem \ref{['thm:main-framework-result-informal']}
  • Corollary 2: Corollary \ref{['cor:example-schedules-wlog']} specialized
  • Remark 2
  • Corollary 3: Corollary \ref{['cor:example-schedules-no-log']} specialized
  • Remark 3
  • Definition 1: dgf setup
  • Definition 2: Dual-extraction setup
  • ...and 80 more