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Multi Expression Programming (MEP) is an evolutionary technique that may be used for solving computationally difficult problems. MEP uses a linear solution representation. Each MEP individual is a string encoding complex expressions (computer programs). A MEP individual may encode multiple solutions of the current problem. In this paper MEP is used for evolving a Traveling Salesman Problem (TSP) heuristic for graphs satisfying triangle inequality. Evolved MEP heuristic is compared with Nearest Neighbor Heuristic (NN) and Minimum Spanning Tree Heuristic (MST) on some difficult problems in TSPLIB. For most of the considered problems the evolved MEP heuristic outperforms NN and MST. The obtained algorithm was tested against some problems in TSPLIB. The results emphasizes that evolved MEP heuristic is a powerful tool for solving difficult TSP instances.
Evolving TSP heuristics using Multi Expression Programming

The EM-algorithm is a general procedure to get maximum likelihood estimates if part of the observations on the variables of a network are missing. In this paper a stochastic version of the algorithm is adapted to probabilistic neural networks describing the associative dependency of variables. These networks have a proba­ bility distribution, which is a special case of the distribution generated by proba­ bilistic inference networks. Hence both types of networks can be combined al­ lowing to integrate probabilistic rules as well as unspecified associations in a sound way. The resulting network may have a number of interesting features including cycles of probabilistic rules, hidden 'un­ observable' variables, and uncertain and contradictory evidence.
Integrating Probabilistic Rules into Neural Networks: A Stochastic EM Learning Algorithm

This report presents Giraffe, a chess engine that uses self-play to discover all its domain-specific knowledge, with minimal hand-crafted knowledge given by the programmer. Unlike previous attempts using machine learning only to perform parametertuning on hand-crafted evaluation functions, Giraffe’s learning system also performs automatic feature extraction and pattern recognition. The trained evaluation function performs comparably to the evaluation functions of state-of-the-art chess engines all of which containing thousands of lines of carefully hand-crafted pattern recognizers, tuned over many years by both computer chess experts and human chess masters. Giraffe is the most successful attempt thus far at using end-to-end machine learning to play chess. We also investigated the possibility of using probability thresholds instead of depth to shape search trees. Depth-based searches form the backbone of virtually all chess engines in existence today, and is an algorithm that has become well-established over the past half century. Preliminary comparisons between a basic implementation of probability-based search and a basic implementation of depth-based search showed that our new probability-based approach performs moderately better than the established approach. There are also evidences suggesting that many successful ad-hoc add-ons to depth-based searches are generalized by switching to a probability-based search. We believe the probability-based search to be a more fundamentally correct way to perform minimax. Finally, we designed another machine learning system to shape search trees within the probability-based search framework. Given any position, this system estimates the probability of each of the moves being the best move without looking ahead. The system is highly effective the actual best move is within the top 3 ranked moves 70% of the time, out of an average of approximately 35 legal moves from each position. This also resulted in a significant increase in playing strength. With the move evaluator guiding a probability-based search using the learned evaluator, Giraffe plays at approximately the level of an FIDE International Master (top 2.2% of tournament chess players with an official rating)12. Fédération Internationale des Échecs, or the World Chess Federation, is the international organisation that governs all major international chess competitions. Please see Appendix A for a description of the Elo rating system.

Giraffe: Using Deep Reinforcement Learning to Play Chess