Fast Convergence of Regularized Learning in Games (1507.00407v5)

Abstract: We show that natural classes of regularized learning algorithms with a form of recency bias achieve faster convergence rates to approximate efficiency and to coarse correlated equilibria in multiplayer normal form games. When each player in a game uses an algorithm from our class, their individual regret decays at $O(T^{-3/4})$, while the sum of utilities converges to an approximate optimum at $O(T^{-1})$--an improvement upon the worst case $O(T^{-1/2})$ rates. We show a black-box reduction for any algorithm in the class to achieve $\tilde{O}(T^{-1/2})$ rates against an adversary, while maintaining the faster rates against algorithms in the class. Our results extend those of [Rakhlin and Shridharan 2013] and [Daskalakis et al. 2014], who only analyzed two-player zero-sum games for specific algorithms.

• The paper demonstrates that integrating recency bias in regularized learning yields faster regret decay (O(T^-3/4)) and improved utility convergence (O(T^-1)) in multiplayer games.
• The paper extends algorithmic analysis beyond two-player contexts, using a black-box reduction to generalize faster convergence results in normal form games.
• Empirical simulations in dynamic auction settings validate that these methods stabilize and converge more rapidly than traditional approaches like Hedge.

Fast Convergence of Regularized Learning in Games

The paper explores the convergence dynamics of learning algorithms in multiplayer normal form games. It establishes that specific classes of regularized learning algorithms, incorporating recency bias, can achieve improved rates for convergence to approximate efficiency and coarse correlated equilibria when applied to these games. The central finding is that when each player employs a learning algorithm from the studied class, individual player regret declines at a rate of $O(T^{-3/4})$, and the sum of player utilities approaches an approximate optimum at a rate of $O(T^{-1})$, thus enhancing upon the previous worst-case rates of $O(T^{-1/2})$.

Key Results and Contributions

1. Improved Regret and Utility Convergence: The paper shows that the proposed class of algorithms allows regret to decay faster compared to standard no-regret algorithms. The convergence to coarse correlated equilibria is significantly improved.
2. Algorithmic Analysis: The research extends existing knowledge by generalizing beyond two-player zero-sum games to consider arbitrary multi-player normal form games. It offers detailed analysis on the dynamics of such games when players utilize regularized learning algorithms with recency bias.
3. Black-box Reduction: The authors introduce a transformation approach that allows the maintenance of lower regret bounds against adversaries while preserving the faster rates against algorithms in the studied class. This work universalizes modifications needed across algorithmic families to enjoy these properties.
4. Experimental Validation: Simulation of a dynamic auction setting demonstrates the robustness and speed of convergence of the studied optimistic algorithms. The results mark a stark contrast with classical algorithms like Hedge, which are slower to stabilize and converge.

Theoretical Implications

The paper advances the understanding of how learning dynamics in multiplayer games can be accelerated. By pinpointing the critical role of recency bias and stability in algorithm design, it contributes to the ongoing dialogue about optimizing efficient and strategic behaviors in decentralized systems. As algorithms capable of achieving these faster rates of convergence are further developed, there are strong implications for their application in economic frameworks, network routing, and beyond.

Practical Implications

In practice, these findings could facilitate more efficient computation and faster adaptation to equilibria in decentralized systems, such as markets and network platforms. The improved convergence rates offer potential to enhance the design of systems where strategic interactions are prevalent.

Future Directions

One of the promising avenues for further exploration is the investigation of the necessity versus sufficiency of the introduced properties for faster convergence. Furthermore, exploring the applicability of these techniques to games with different structure or information limitations might reveal broader insights.

Overall, this work presents a substantial advancement in the understanding of game dynamics under learning algorithms, paving the way for future research in achieving even more efficient strategic simulation and computation in complex multi-player settings.