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Expected Variational Inequalities

Brian Hu Zhang, Ioannis Anagnostides, Emanuel Tewolde, Ratip Emin Berker, Gabriele Farina, Vincent Conitzer, Tuomas Sandholm

TL;DR

This work introduces expected variational inequalities (EVIs), a relaxation of variational inequalities that seeks a distribution over decision variables so that the VI constraint holds in expectation for a broad class of deviations. By leveraging ellipsoid-based methods (EAH) and regret minimization, the authors show polynomial-time solvability when the deviation set $\Phi$ consists of linear endomorphisms, and they connect EVIs to correlated equilibria, including refined notions like anonymous linear correlated equilibria (ALCE). They establish existence results under mild conditions, provide hardness results for unrestricted $\Phi$, and develop scalable algorithms that work even with coupled constraints and nonconcave or nonsmooth utilities. The framework extends the reach of correlated equilibria beyond games and yields performance guarantees for underlying objectives via a generalized smoothness notion, offering a principled, tractable alternative to solving VIs in many settings. Overall, EVIs offer a principled balance between expressivity and tractability with potential impact across optimization, game theory, and economic modeling.

Abstract

Variational inequalities (VIs) encompass many fundamental problems in diverse areas ranging from engineering to economics and machine learning. However, their considerable expressivity comes at the cost of computational intractability. In this paper, we introduce and analyze a natural relaxation -- which we refer to as expected variational inequalities (EVIs) -- where the goal is to find a distribution that satisfies the VI constraint in expectation. By adapting recent techniques from game theory, we show that, unlike VIs, EVIs can be solved in polynomial time under general (nonmonotone) operators. EVIs capture the seminal notion of correlated equilibria, but enjoy a greater reach beyond games. We also employ our framework to capture and generalize several existing disparate results, including from settings such as smooth games, and games with coupled constraints or nonconcave utilities.

Expected Variational Inequalities

TL;DR

This work introduces expected variational inequalities (EVIs), a relaxation of variational inequalities that seeks a distribution over decision variables so that the VI constraint holds in expectation for a broad class of deviations. By leveraging ellipsoid-based methods (EAH) and regret minimization, the authors show polynomial-time solvability when the deviation set consists of linear endomorphisms, and they connect EVIs to correlated equilibria, including refined notions like anonymous linear correlated equilibria (ALCE). They establish existence results under mild conditions, provide hardness results for unrestricted , and develop scalable algorithms that work even with coupled constraints and nonconcave or nonsmooth utilities. The framework extends the reach of correlated equilibria beyond games and yields performance guarantees for underlying objectives via a generalized smoothness notion, offering a principled, tractable alternative to solving VIs in many settings. Overall, EVIs offer a principled balance between expressivity and tractability with potential impact across optimization, game theory, and economic modeling.

Abstract

Variational inequalities (VIs) encompass many fundamental problems in diverse areas ranging from engineering to economics and machine learning. However, their considerable expressivity comes at the cost of computational intractability. In this paper, we introduce and analyze a natural relaxation -- which we refer to as expected variational inequalities (EVIs) -- where the goal is to find a distribution that satisfies the VI constraint in expectation. By adapting recent techniques from game theory, we show that, unlike VIs, EVIs can be solved in polynomial time under general (nonmonotone) operators. EVIs capture the seminal notion of correlated equilibria, but enjoy a greater reach beyond games. We also employ our framework to capture and generalize several existing disparate results, including from settings such as smooth games, and games with coupled constraints or nonconcave utilities.

Paper Structure

This paper contains 36 sections, 34 theorems, 67 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution: the expected VI problem
  3. Algorithms and complexity for $\Phi$-EVIs
  4. Connection to other solution concepts
  5. Further properties
  6. Further related work
  7. Preliminaries
  8. Notation
  9. Ellipsoid against hope
  10. Existence and Complexity Barriers
  11. Complexity
  12. Efficient Computation with Linear Endomorphisms
  13. Regret minimization for EVIs on polytopes
  14. Game Theory Applications of EVIs
  15. Refining correlated equilibria
  16. ...and 21 more sections

Key Result

Theorem 2.1

Given a $\mathtt{GER}$ oracle and a separation oracle ($\mathtt{SEP}$) for ${\mathcal{Y}}$, $\mathtt{EAH}$ runs in time $\mathrm{poly}(d, m, \log(1/\epsilon))$ and returns an $\epsilon$-approximate solution to eq:eah.

Figures (2)

  • Figure 1: Marginals of the set of correlated equilibria (CE) and of the set of solutions to $\Phi_\mathsf{LIN}$-EVI in the simple $2 \times 2$ game "Bach or Stravinsky." The x- and y-axes show the probability with which the two players select the first action (Bach). The set of marginals of $\Phi_\mathsf{LIN}$-EVI solutions appears to have a curved boundary corresponding, we believe, to the hyperbola $10 x^2 - 25 xy + 10 y^2 - 6x+11y=0$.
  • Figure 2: Function $u$, defined in \ref{['eq:u-smooth']}, for $p \in \{1, 2, 4, 8\}$.

Theorems & Definitions (64)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1: Generalized form of $\mathtt{EAH}$; Farina24:Polynomial
  • Theorem 3.1
  • Corollary 3.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.3
  • Corollary 3.4
  • ...and 54 more