# Fast Convergence of Regularized Learning in Games

Annual Conference on Neural Information Processing Systems, (2015): 2989-2997

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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 ...More

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Introduction

- What happens when players in a game interact with one another, all of them acting independently and selfishly to maximize their own utilities? If they are smart, the authors intuitively expect their utilities — both individually and as a group — to grow, perhaps even to approach the best possible.
- When played against one another, it can be shown that the sum of utilities approaches an approximate optimum [2, 18], and the player strategies converge to an equilibrium under appropriate conditions [6, 1, 8], at rates governed by the regret bounds.
- For two-player zero-sum games, they developed a decentralized variant of Nesterov’s accelerated saddle point algorithm [15] and showed that each player’s average regret converges at the remarkable rate of

Highlights

- What happens when players in a game interact with one another, all of them acting independently and selfishly to maximize their own utilities? If they are smart, we intuitively expect their utilities — both individually and as a group — to grow, perhaps even to approach the best possible
- When played against one another, it can be shown that the sum of utilities approaches an approximate optimum [2, 18], and the player strategies converge to an equilibrium under appropriate conditions [6, 1, 8], at rates governed by the regret bounds
- We present our main theoretical results characterizing a class of no-regret dynamics which lead to faster convergence in smooth games
- We have shown simple dynamics with rapid convergence properties in favorable environments when each player in the game uses an algorithm with the Regret bounded by Variation in Utilities (RVU) property
- There are a number of interesting questions and directions for future research which are suggested by our results, including the following: Convergence rates for vanilla Hedge: The fast rates of our paper do not apply to algorithms such as Hedge without modification
- Convergence of players’ strategies: The optimistic follow the regularized leader (OFTRL) algorithm often produces much more stable trajectories empirically, as the players converge to an equilibrium, as opposed to say Hedge

Results

- The authors simulate a 4-bidder simultaneous auction game, and compare the optimistic algorithms against Hedge [7] in terms of utilities, regrets and convergence to equilibria.
- For some natural classes of no-regret algorithms the average welfare converges at the much faster rate of log(d)/T
- The authors' main result in this sect=io1n bounds this sum when each player uses dynamics with the RVU property.
- The theorem does not require that all players use the same no-regret algorithm unlike previous results [4, 17], as long as each player’s algorithm satisfies the RVU property with a common bound on the constants.
- Rakhlin and Sridharan [16] analyze an FTRL variant, but require a self-concordant barrier for the constraint set as opposed to an arbitrary strongly convex regularizer, and their bound is missing the crucial negative terms of the RVU property which are essential for obtaining Theorem 4.
- The authors have shown simple dynamics with rapid convergence properties in favorable environments when each player in the game uses an algorithm with the RVU property.
- Rakhlin and Sridharan [17] address this concern by modifying the OMD algorithm with additional smoothing and adapptive step-sizes so as to preserve the fast rates in the favorable case while still guaranteeing regret for each player, no matter how the opponents play.

Conclusion

- The authors present a generic way of transforming an algorithm which satisfies the RVU property so that it pretains the fast convergence in favorable settings, but always guarantees a worst-case regret of
- The key reason is the following: the proof that no-regret dynamics are approximately efficient (Proposition 2) only relies on the fact that each player does not have regret against the strategy s⇤i used in the definition of a smooth game.
- There are a number of interesting questions and directions for future research which are suggested by the results, including the following: Convergence rates for vanilla Hedge: The fast rates of the paper do not apply to algorithms such as Hedge without modification.

Reference

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