Learning correlated equilibria in games with compact sets of strategies

Stoltz, Gilles; Lugosi, Gábor

doi:10.1016/j.geb.2006.04.007

Cited by 66 publications

(69 citation statements)

References 23 publications

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“…This guarantee rules out precisely the anomalous behavior described above. Stoltz and Lugosi (2007) generalize these results to convex games. Extending the framework of Greenwald and Jafari (2003) for matrix games, they define a continuum of regret measures called Φ-regret, as well as corresponding Φ-equilibria, for convex games.…”

Section: Introductionmentioning

confidence: 68%

No-regret learning in convex games

Gordon

Greenwald

Marks

2008

Proceedings of the 25th International Conference on Machine Learning - ICML '08

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Quite a bit is known about minimizing different kinds of regret in experts problems, and how these regret types relate to types of equilibria in the multiagent setting of repeated matrix games. Much less is known about the possible kinds of regret in online convex programming problems (OCPs), or about equilibria in the analogous multiagent setting of repeated convex games. This gap is unfortunate, since convex games are much more expressive than matrix games, and since many important machine learning problems can be expressed as OCPs. In this paper, we work to close this gap: we analyze a spectrum of regret types which lie between external and swap regret, along with their corresponding equilibria, which lie between coarse correlated and correlated equilibrium. We also analyze algorithms for minimizing these regret types. As examples of our framework, we derive algorithms for learning correlated equilibria in polyhedral convex games and extensive-form correlated equilibria in extensive-form games. The former is exponentially more efficient than previous algorithms, and the latter is the first of its type.

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Section: Introductionmentioning

confidence: 68%

No-regret learning in convex games

Gordon

Greenwald

Marks

2008

Proceedings of the 25th International Conference on Machine Learning - ICML '08

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“…Of course, the players may well use these signals when making their strategic choices. To date, there are several efficient algorithms [8,14,15,3,4,23,24,2,25,13] that, in all games, converge fast to (approximate) correlated equilibria.…”

Section: Introductionmentioning

confidence: 99%

The communication complexity of uncoupled nash equilibrium procedures

Hart

Mansour

2007

Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing

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We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players. General Terms

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“…Naturally, that finite noncooperative games with greater numbers of pure strategies at their players (three and more) are significantly hard to solve them [10], [37], [38]. Moreover, often an admissible player's action is described with a series of its continuous parameters, constituting thus an infinite (continuous) set of pure strategies [1], [6], [7], [12], [39], [40]. If this continuous set is compact then it is easy to find an isomorphic game to the initial one, that the set of every player's pure strategies would be Euclidean finite-dimensional subspace [6], [10], [12], [23], [41].…”

Section: Solving Noncooperative Gamesmentioning

confidence: 99%

“…Moreover, often an admissible player's action is described with a series of its continuous parameters, constituting thus an infinite (continuous) set of pure strategies [1], [6], [7], [12], [39], [40]. If this continuous set is compact then it is easy to find an isomorphic game to the initial one, that the set of every player's pure strategies would be Euclidean finite-dimensional subspace [6], [10], [12], [23], [41]. Normally, the spoken subspace may be a unit cube of the appropriate dimension [6], [10], [11].…”

Section: Solving Noncooperative Gamesmentioning

confidence: 99%

Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

Romanuke

2015

Electrical, Control and Communication Engineering

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-The paper suggests a method of obtaining an approximate solution of the infinite noncooperative game on the unit hypercube. The method is based on sampling uniformly the players' payoff functions with the constant step along each of the hypercube dimensions. The author states the conditions for a sufficiently accurate sampling and suggests the method of reshaping the multidimensional matrix of the player's payoff values, being the former player's payoff function before its sampling, into a matrix with minimally possible number of dimensions, where also maintenance of one-to-one indexing has been provided. Requirements for finite NE-strategy from NE (Nash equilibrium) solution of the finite game as the initial infinite game approximation are given as definitions of the approximate solution consistency. The approximate solution consistency ensures its relative independence upon the sampling step within its minimal neighborhood or the minimally decreased sampling step. The ultimate reshaping of multidimensional matrices of players' payoff values to the minimal number of dimensions, being equal to the number of players, stimulates shortened computations.

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Learning correlated equilibria in games with compact sets of strategies

Cited by 66 publications

References 23 publications

No-regret learning in convex games

No-regret learning in convex games

The communication complexity of uncoupled nash equilibrium procedures

Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

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