Efficient Learning and Computation of Linear Correlated Equilibrium in General Convex Games

TL;DR

The paper develops a framework to efficiently compute and learn linear correlated equilibria in general convex games, where strategy spaces can be exponentially large. It introduces a semi-separation approach and shell-based optimization (ShellEllipsoid, ShellGradientDescent, ShellProjection) to overcome the intractability of optimizing over all linear endomorphisms, achieving no-regret dynamics with $O( ext{poly}(d)\, oot 2{T})$ swap regret and polynomial-time computation of $oldsymbol{ ext{ε}}$-approximate equilibria. A generalized Ellipsoid Against Hope framework is provided to handle general convex sets and weak oracles, enabling a polynomial-time Correlator–Deviator meta-game to produce an $oldsymbol{ ext{ε}}$-approximate linear correlated equilibrium, expressed as a mixture of product distributions. The results unify and extend previous tractable equilibria notions across normal-, extensive-, Bayesian-, and other convex games, offering practically implementable procedures under oracle access without requiring full endomorphism separation. Overall, the work broadens the frontier of efficiently computable and learnable equilibria in complex multi-agent settings, with potential applications to routing, resource allocation, and strategic decision-making in large-scale systems.

Abstract

We propose efficient no-regret learning dynamics and ellipsoid-based methods for computing linear correlated equilibria$\unicode{x2014}$a relaxation of correlated equilibria and a strengthening of coarse correlated equilibria$\unicode{x2014}$in general convex games. These are games where the number of pure strategies is potentially exponential in the natural representation of the game, such as extensive-form games. Our work identifies linear correlated equilibria as the tightest known notion of equilibrium that is computable in polynomial time and is efficiently learnable for general convex games. Our results are enabled by a generalization of the seminal framework of of Gordon et al. [2008] for $Φ$-regret minimization, providing extensions to this framework that can be used even when the set of deviations $Φ$ is intractable to separate/optimize over. Our polynomial-time algorithms are similarly enabled by extending the Ellipsoid-Against-Hope approach of Papadimitriou and Roughgarden [2008] and its generalization to games of non-polynomial type proposed by Farina and Pipis [2024a]. We provide an extension to these approaches when we do not have access to the separation oracles required by these works for the dual player.

Figures (3)

• Figure 1: Illustration of our semi-separation oracle for the set of linear endomorphisms $\Phi$ of the feasible set $\mathcal{P}$. Given a candidate linear transformation $\phi$, the oracle returns a fixed point of $\phi$ in the set $\mathcal{P}$, if one exists (this is the case of the two points marked with ✓), or a hyperplane separating $\phi$ from $\Phi$ (this is the case of the point marked ✗). In the figure, we denoted with $\Phi_\text{FP}$ the set of linear transformations that admit a fixed point in $\mathcal{P}$. In general, $\Phi_\text{FP}$ is a strict superset of $\Phi$, and is not a convex set. Building a separation oracle for $\Phi$ is generally computationally intractable.
• Figure 2: Illustration of the behavior of the ShellEllipsoid subroutine depending on the input convex set ${\mathcal{F}}$. $\Phi$ is the set of linear endomorphism on $\mathcal{P}$, and $\Phi_\text{FP}$ is the set of linear transformations with a fixed point in $\mathcal{P}$.
• Figure 3: Visual depiction of a generic step of the ShellProj$(\phi)$, \ref{['algo:noisyproj-strong']}.

Theorems & Definitions (78)

• Theorem 1.1: Informal; formal version given as Theorems \ref{['thm:linswap-regret-main-algo-strong']} and \ref{['thm:eah-equil-computation-strong']}
• Definition 2.1: convex game
• Definition 2.2: linear correlated equilibrium
• Lemma 2.2
• proof
• Definition 2.3: Isotropic Position
• Lemma 3.1
• proof
• Lemma 3.2
• proof