To establish last-iterate convergence for Counterfactual Regret Minimization (CFR) algorithms in learning a Nash equilibrium (NE) of extensive-form games (EFGs), recent studies reformulate learning an NE of the original EFG as learning the NEs of a sequence of (perturbed) regularized EFGs. Hence, proving last-iterate convergence in solving the original EFG reduces to proving last-iterate convergence in solving (perturbed) regularized EFGs. However, these studies only establish last-iterate convergence for Online Mirror Descent (OMD)-based CFR algorithms instead of Regret Matching (RM)-based CFR algorithms in solving perturbed regularized EFGs, resulting in a poor empirical convergence rate, as RM-based CFR algorithms typically outperform OMD-based CFR algorithms. In addition, as solving multiple perturbed regularized EFGs is required, fine-tuning across multiple perturbed regularized EFGs is infeasible, making parameter-free algorithms highly desirable. This paper show that CFR+, a classical parameter-free RM-based CFR algorithm, achieves last-iterate convergence in learning an NE of perturbed regularized EFGs. This is the first parameter-free last-iterate convergence for RM-based CFR algorithms in perturbed regularized EFGs. Leveraging CFR+ to solve perturbed regularized EFGs, we get Reward Transformation CFR+ (RTCFR+). Importantly, we extend prior work on the parameter-free property of CFR+, enhancing its stability, which is vital for the empirical convergence of RTCFR+. Experiments show that RTCFR+ exhibits a significantly faster empirical convergence rate than existing algorithms that achieve theoretical last-iterate convergence.

The design of revenue-maximizing combinatorial auctions, i.e. multi-item auctions over bundles of goods, is one of the most fundamental problems in computational economics, unsolved even for two bidders and two items for sale. In the traditional economic models, it is assumed that the bidders' valuations are drawn from an underlying distribution and that the auction designer has perfect knowledge of this distribution. Despite this strong and oftentimes unrealistic assumption, it is remarkable that the revenue-maximizing combinatorial auction remains unknown. In recent years, automated mechanism design has emerged as one of the most practical and promising approaches to designing high-revenue combinatorial auctions. The most scalable automated mechanism design algorithms take as input samples from the bidders' valuation distribution and then search for a high-revenue auction in a rich auction class. In this work, we provide the first sample complexity analysis for the standard hierarchy of deterministic combinatorial auction classes used in automated mechanism design. In particular, we provide tight sample complexity bounds on the number of samples needed to guarantee that the empirical revenue of the designed mechanism on the samples is close to its expected revenue on the underlying, unknown distribution over bidder valuations, for each of the auction classes in the hierarchy. In addition to helping set automated mechanism design on firm foundations, our results also push the boundaries of learning theory. In particular, the hypothesis functions used in our contexts are defined through multi-stage combinatorial optimization procedures, rather than simple decision boundaries, as are common in machine learning.

CMWU achieves constant regret ofln(m)/η in all normal-form games with m actions and fixed step-sizesη.

There has been tremendous recent progress on equilibrium-finding algorithms for zero-sum imperfect-information extensive-form games, but there has been a puzzling gap between theory and practice. First-order methods have significantly better theoretical convergence rates than any counterfactual-regret minimization (CFR) variant. Despite this, CFR variants have been favored in practice. Experiments with first-order methods have only been conducted on small-and medium-sized games because those methods are complicated to implement in this setting, and because CFR variants have been enhanced extensively for over a decade they perform well in practice. In this paper we show that a particular first-order method, a state-ofthe-art variant of the excessive gap technique--instantiated with the dilated entropy distance function--can efficiently solve large real-world problems competitively with CFR and its variants. We show this on large endgames encountered by the Libratus poker AI, which recently beat top human poker specialist professionals at no-limit Texas hold'em. We show experimental results on our variant of the excessive gap technique as well as a prior version. We introduce a numerically friendly implementation of the smoothed best response computation associated with first-order methods for extensive-form game solving.

In'00: Revised the Second International Conferenceon Computersand Games333-345, 2000.

Aconceptually appealing approach forlearning Extensive-FormGames (EFGs) is to convert them to Normal-Form Games (NFGs).