Minimax estimation of smooth optimal transport maps

Brenier's theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map $T$ between two probability distributions $P$ and $Q$ over $\mathbb{R}^d$ under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from $P$ and $Q$ under additional smoothness assumptions on $T$. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semi-dual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semi-dual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure

The minimax property in infinite two-person win-lose games

We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we prove that a game satisfies the minimax property along with all its subgames if and only if none of its subgames is isomorphic to the "larger number game." This generalizes a recent theorem of Hanneke, Livni and Moran. We also propose several applications of our results outside of game theory.Comment: 22 page

Best approximation in the space of continuous vector-valued functions

AbstractUsing a well-known characterization theorem for best approximations, direct proofs are given of some (generalizations of) recent results of Tanimoto who deduced them from a general minimax theorem that he first established

Efficiency and formalism of quantum games

We pursue a general theory of quantum games. We show that quantum games are more efficient than classical games, and provide a saturated upper bound for this efficiency. We demonstrate that the set of finite classical games is a strict subset of the set of finite quantum games. We also deduce the quantum version of the Minimax Theorem and the Nash Equilibrium Theorem.Comment: 10 pages. Efficiency is explicitly defined. More discussion on the connection of quantum and classical game

Program Equilibria and Discounted Computation Time

Tennenholtz (GEB 2004) developed Program Equilibrium to model play in a finite two-player game where each player can base their strategy on the other player's strategies. Tennenholtz's model allowed each player to produce a "loop-free" computer program that had access to the code for both players. He showed a folk theorem where any mixed-strategy individually rational play could be an equilibrium payo in this model even in a one-shot game. Kalai et al. gave a general folk theorem for correlated play in a more generic commitment model. We develop a new model of program equilibrium using general computational models and discounting the payos based on the computation time used. We give an even more general folk theorem giving correlated-strategy payoffs down to the pure minimax of each player. We also show equilibrium in other games not covered by the earlier work.brokers, applied mechanism design, linear commission fees, optimal indirect mechanisms, internet auctions, auction houses.