The weighted majority algorithm
Information and Computation
A new polynomial-time algorithm for linear programming
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Excessive Gap Technique in Nonsmooth Convex Minimization
SIAM Journal on Optimization
Prediction, Learning, and Games
Prediction, Learning, and Games
First-order algorithm with O(ln(1/ε )) convergence for ε -equilibrium in two-person zero-sum games
AAAI'08 Proceedings of the 23rd national conference on Artificial intelligence - Volume 1
Mathematical Programming: Series A and B
On the rate of convergence of fictitious play
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
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We propose a new no-regret learning algorithm. When used against an adversary, our algorithm achieves average regret that scales as O (1/√T) with the number T of rounds. This regret bound is optimal but not rare, as there are a multitude of learning algorithms with this regret guarantee. However, when our algorithm is used by both players of a zero-sum game, their average regret scales as O (ln T/T), guaranteeing a near-linear rate of convergence to the value of the game. This represents an almost-quadratic improvement on the rate of convergence to the value of a game known to be achieved by any no-regret learning algorithm, and is essentially optimal as we show a lower bound of Ω (1/T). Moreover, the dynamics produced by our algorithm in the game setting are strongly-uncoupled in that each player is oblivious to the payoff matrix of the game and the number of strategies of the other player, has limited private storage, and is not allowed funny bit arithmetic that can trivialize the problem; instead he only observes the performance of his strategies against the actions of the other player and can use private storage to remember past played strategies and observed payoffs, or cumulative information thereof. Here, too, our rate of convergence is nearly-optimal and represents an almost-quadratic improvement over the best previously known strongly-uncoupled dynamics.