Convergence of $\text{log}(1/ε)$ for Gradient-Based Algorithms in Zero-Sum Games without the Condition Number: A Smoothed Analysis
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912090866843648 |
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| author | Anagnostides, Ioannis Sandholm, Tuomas |
| author_facet | Anagnostides, Ioannis Sandholm, Tuomas |
| contents | Gradient-based algorithms have shown great promise in solving large (two-player) zero-sum games. However, their success has been mostly confined to the low-precision regime since the number of iterations grows polynomially in $1/ε$, where $ε> 0$ is the duality gap. While it has been well-documented that linear convergence -- an iteration complexity scaling as $\textsf{log}(1/ε)$ -- can be attained even with gradient-based algorithms, that comes at the cost of introducing a dependency on certain condition number-like quantities which can be exponentially large in the description of the game.
To address this shortcoming, we examine the iteration complexity of several gradient-based algorithms in the celebrated framework of smoothed analysis, and we show that they have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, $\textsf{log}(1/ε)$, and $1/σ$, where $σ$ measures the magnitude of the smoothing perturbation. Our result applies to optimistic gradient and extra-gradient descent/ascent, as well as a certain iterative variant of Nesterov's smoothing technique. From a technical standpoint, the proof proceeds by characterizing and performing a smoothed analysis of a certain error bound, the key ingredient driving linear convergence in zero-sum games. En route, our characterization also makes a natural connection between the convergence rate of such algorithms and perturbation-stability properties of the equilibrium, which is of interest beyond the model of smoothed complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of $\text{log}(1/ε)$ for Gradient-Based Algorithms in Zero-Sum Games without the Condition Number: A Smoothed Analysis Anagnostides, Ioannis Sandholm, Tuomas Computer Science and Game Theory Gradient-based algorithms have shown great promise in solving large (two-player) zero-sum games. However, their success has been mostly confined to the low-precision regime since the number of iterations grows polynomially in $1/ε$, where $ε> 0$ is the duality gap. While it has been well-documented that linear convergence -- an iteration complexity scaling as $\textsf{log}(1/ε)$ -- can be attained even with gradient-based algorithms, that comes at the cost of introducing a dependency on certain condition number-like quantities which can be exponentially large in the description of the game. To address this shortcoming, we examine the iteration complexity of several gradient-based algorithms in the celebrated framework of smoothed analysis, and we show that they have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, $\textsf{log}(1/ε)$, and $1/σ$, where $σ$ measures the magnitude of the smoothing perturbation. Our result applies to optimistic gradient and extra-gradient descent/ascent, as well as a certain iterative variant of Nesterov's smoothing technique. From a technical standpoint, the proof proceeds by characterizing and performing a smoothed analysis of a certain error bound, the key ingredient driving linear convergence in zero-sum games. En route, our characterization also makes a natural connection between the convergence rate of such algorithms and perturbation-stability properties of the equilibrium, which is of interest beyond the model of smoothed complexity. |
| title | Convergence of $\text{log}(1/ε)$ for Gradient-Based Algorithms in Zero-Sum Games without the Condition Number: A Smoothed Analysis |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2410.21636 |