arXiv subjects

Nikolas Patris

Publications and source records attributed to Nikolas Patris.

4 recordsLinked to original sources

arXiv2026

(Doubly) Exponential Lower Bounds for Follow the Regularized Leader in Potential Games

Follow the regularized leader FTRL is the premier algorithm for online optimization. However, despite decades of research on its convergence in constrained optimization -- and potential games in particular -- its behavior remained hitherto poorly understood. In this paper, we establish that FTRL can take exponential time to converge to a Nash equilibrium in two-player potential games for any (permutation-invariant) regularizer and potentially vanishing learning rate. By known equivalences, this translates to an exponential lower bound for certain mirror descent counterparts, most notably multiplicative weights update. On the positive side, we establish the potential property for FTRL and obtain an exponential upper bound $\exp(O_{\epsilon}(1/\epsilon^2))$ for any no-regret dynamics executed in a lazy, alternating fashion, matching our lower bound up to factors in the exponent. Finally, in multi-player potential games, we show that fictitious play -- the extreme version of FTRL -- can take doubly exponential time to reach a Nash equilibrium. This constitutes an exponentially stronger lower bound for the foundational learning algorithm in games.

cs.GT↗

arXiv2024

Computing Nash Equilibria in Potential Games with Private Uncoupled Constraints

We consider the problem of computing Nash equilibria in potential games where each player's strategy set is subject to private uncoupled constraints. This scenario is frequently encountered in real-world applications like road network congestion games where individual drivers adhere to personal budget and fuel limitations. Despite the plethora of algorithms that efficiently compute Nash equilibria (NE) in potential games, the domain of constrained potential games remains largely unexplored. We introduce an algorithm that leverages the Lagrangian formulation of NE. The algorithm is implemented independently by each player and runs in polynomial time with respect to the approximation error, the sum of the size of the action-spaces, and the game's inherent parameters.

cs.GT↗

arXiv2023

Exponential Lower Bounds for Fictitious Play in Potential Games

Fictitious Play (FP) is a simple and natural dynamic for repeated play with many applications in game theory and multi-agent reinforcement learning. It was introduced by Brown (1949,1951) and its convergence properties for two-player zero-sum games was established later by Robinson (1951). Potential games Monderer and Shapley (1996b) is another class of games which exhibit the FP property (Monderer and Shapley (1996a)), i.e., FP dynamics converges to a Nash equilibrium if all agents follows it. Nevertheless, except for two-player zero-sum games and for specific instances of payoff matrices (Abernethy et al. (2021)) or for adversarial tie-breaking rules (Daskalakis and Pan (2014)), the convergence rate of FP is unknown. In this work, we focus on the rate of convergence of FP when applied to potential games and more specifically identical payoff games. We prove that FP can take exponential time (in the number of strategies) to reach a Nash equilibrium, even if the game is restricted to two agents and for arbitrary tie-breaking rules. To prove this, we recursively construct a two-player coordination game with a unique Nash equilibrium. Moreover, every approximate Nash equilibrium in the constructed game must be close to the pure Nash equilibrium in $\ell_1$-distance.

cs.GT↗

arXiv2021

Improved Bounds for Online Facility Location with Predictions

We consider Online Facility Location in the framework of learning-augmented online algorithms. In Online Facility Location (OFL), demands arrive one-by-one in a metric space and must be (irrevocably) assigned to an open facility upon arrival, without any knowledge about future demands. We focus on uniform facility opening costs and present an online algorithm for OFL that exploits potentially imperfect predictions on the locations of the optimal facilities. We prove that the competitive ratio decreases from sublogarithmic in the number of demands $n$ to constant as the so-called $\eta_1$ error, i.e., the sum of distances of the predicted locations to the optimal facility locations, decreases. E.g., our analysis implies that if for some $\varepsilon > 0$, $\eta_1 = \mathrm{OPT} / n^\varepsilon$, where $\mathrm{OPT}$ is the cost of the optimal solution, the competitive ratio becomes $O(1/\varepsilon)$. We complement our analysis with a matching lower bound establishing that the dependence of the algorithm's competitive ratio on the $\eta_1$ error is optimal, up to constant factors. Finally, we evaluate our algorithm on real world data and compare the performance of our learning-augmented approach against the performance of the best known algorithm for OFL without predictions.

cs.DS↗