Learning and Computation of $Φ$-Equilibria at the Frontier of Tractability
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866914198787719168 |
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| author | Zhang, Brian Hu Anagnostides, Ioannis Tewolde, Emanuel Berker, Ratip Emin Farina, Gabriele Conitzer, Vincent Sandholm, Tuomas |
| author_facet | Zhang, Brian Hu Anagnostides, Ioannis Tewolde, Emanuel Berker, Ratip Emin Farina, Gabriele Conitzer, Vincent Sandholm, Tuomas |
| contents | $Φ$-equilibria -- and the associated notion of $Φ$-regret -- are a powerful and flexible framework at the heart of online learning and game theory, whereby enriching the set of deviations $Φ$ begets stronger notions of rationality. Recently, Daskalakis, Farina, Fishelson, Pipis, and Schneider (STOC '24) -- abbreviated as DFFPS -- settled the existence of efficient algorithms when $Φ$ contains only linear maps under a general, $d$-dimensional convex constraint set $\mathcal{X}$. In this paper, we significantly extend their work by resolving the case where $Φ$ is $k$-dimensional; degree-$\ell$ polynomials constitute a canonical such example with $k = d^{O(\ell)}$. In particular, positing only oracle access to $\mathcal{X}$, we obtain two main positive results: i) a $\text{poly}(n, d, k, \text{log}(1/ε))$-time algorithm for computing $ε$-approximate $Φ$-equilibria in $n$-player multilinear games, and ii) an efficient online algorithm that incurs average $Φ$-regret at most $ε$ using $\text{poly}(d, k)/ε^2$ rounds.
We also show nearly matching lower bounds in the online learning setting, thereby obtaining for the first time a family of deviations that captures the learnability of $Φ$-regret.
From a technical standpoint, we extend the framework of DFFPS from linear maps to the more challenging case of maps with polynomial dimension. At the heart of our approach is a polynomial-time algorithm for computing an expected fixed point of any $ϕ: \mathcal{X} \to \mathcal{X}$ based on the ellipsoid against hope (EAH) algorithm of Papadimitriou and Roughgarden (JACM '08). In particular, our algorithm for computing $Φ$-equilibria is based on executing EAH in a nested fashion -- each step of EAH itself being implemented by invoking a separate call to EAH. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learning and Computation of $Φ$-Equilibria at the Frontier of Tractability Zhang, Brian Hu Anagnostides, Ioannis Tewolde, Emanuel Berker, Ratip Emin Farina, Gabriele Conitzer, Vincent Sandholm, Tuomas Machine Learning Computer Science and Game Theory $Φ$-equilibria -- and the associated notion of $Φ$-regret -- are a powerful and flexible framework at the heart of online learning and game theory, whereby enriching the set of deviations $Φ$ begets stronger notions of rationality. Recently, Daskalakis, Farina, Fishelson, Pipis, and Schneider (STOC '24) -- abbreviated as DFFPS -- settled the existence of efficient algorithms when $Φ$ contains only linear maps under a general, $d$-dimensional convex constraint set $\mathcal{X}$. In this paper, we significantly extend their work by resolving the case where $Φ$ is $k$-dimensional; degree-$\ell$ polynomials constitute a canonical such example with $k = d^{O(\ell)}$. In particular, positing only oracle access to $\mathcal{X}$, we obtain two main positive results: i) a $\text{poly}(n, d, k, \text{log}(1/ε))$-time algorithm for computing $ε$-approximate $Φ$-equilibria in $n$-player multilinear games, and ii) an efficient online algorithm that incurs average $Φ$-regret at most $ε$ using $\text{poly}(d, k)/ε^2$ rounds. We also show nearly matching lower bounds in the online learning setting, thereby obtaining for the first time a family of deviations that captures the learnability of $Φ$-regret. From a technical standpoint, we extend the framework of DFFPS from linear maps to the more challenging case of maps with polynomial dimension. At the heart of our approach is a polynomial-time algorithm for computing an expected fixed point of any $ϕ: \mathcal{X} \to \mathcal{X}$ based on the ellipsoid against hope (EAH) algorithm of Papadimitriou and Roughgarden (JACM '08). In particular, our algorithm for computing $Φ$-equilibria is based on executing EAH in a nested fashion -- each step of EAH itself being implemented by invoking a separate call to EAH. |
| title | Learning and Computation of $Φ$-Equilibria at the Frontier of Tractability |
| topic | Machine Learning Computer Science and Game Theory |
| url | https://arxiv.org/abs/2502.18582 |