In reinforcement learning (RL), agents sequentially interact with a changing environment, aiming to collect as much reward as possible.

Counterfactual regret minimization is a family of algorithms of no-regret learning dynamics capable of solving large-scale imperfect information games. We propose implementing this algorithm as a series of dense and sparse matrix and vector operations, thereby making it highly parallelizable for a graphical processing unit, at a cost of higher memory usages. Our experiments show that our implementation performs up to about 352.5 times faster than OpenSpiel's Python implementation and up to about 22.2 times faster than OpenSpiel's C++ implementation and the speedup becomes more pronounced as the size of the game being solved grows. Counterfactual regret minimization (CFR) (Zinkevich et al., 2007) is a family of algorithms of noregret learning dynamics capable of solving large-scale imperfect information games. Its variants dominated the development of AI agents for large imperfect information games like Poker (Tammelin et al., 2015; Moravčík et al., 2017; Brown & Sandholm, 2018; 2019b) and The Resistance: Avalon (Serrino et al., 2019) and were components of ReBeL (Brown et al., 2020) and student of games (Schmid et al., 2023).

In general, the interest lies in obtaining informations about the central characteristics of the underlying lifetime distribution (mean lifetime or survival probabilities for instance), often with the objective of comparing results between different conditions under which the lifetime data are acquired. In this work, we will address the problem of inferring about the (upper) tail of the lifetime distribution, for data subject both to random (right) censoring and competing risks. Suppose indeed that we are interested in the lifetimes of n individuals or items, which are subject to K different causes of death or failure, and to random censorship (from the right) as well. We are particularly interested in one of these causes (this main cause will be considered as cause number k thereafter, where k P t1,..., Ku), and we suppose that all causes are exclusive and are likely to be dependent on the others. The censoring time is assumed to be independent of the different causes of death or failure and of the observed lifetime itself.

We introduce a new discrepancy score between two distributions that gives an indication on their similarity. While much research has been done to determine if two samples come from exactly the same distribution, much less research considered the problem of determining if two finite samples come from similar distributions. The new score gives an intuitive interpretation of similarity; it optimally perturbs the distributions so that they best fit each other. The score is defined between distributions, and can be efficiently estimated from samples. We provide convergence bounds of the estimated score, and develop hypothesis testing procedures that test if two data sets come from similar distributions. The statistical power of this procedures is presented in simulations. We also compare the score's capacity to detect similarity with that of other known measures on real data.