In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
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Question: In many combinatorial problems one may need to model the diversity or similarity of sets of assignments. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type we can use soft variants of the well known AllDifferent and AllEqual constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however bounds consistency can be achieved in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor of 12 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.

Answer: Soft Constraints of Difference and Equality

Question: We propose to prune a random forest (RF) for resource-constrained prediction. We first construct a RF and then prune it to optimize expected feature cost & accuracy. We pose pruning RFs as a novel 0-1 integer program with linear constraints that encourages feature re-use. We establish total unimodularity of the constraint set to prove that the corresponding LP relaxation solves the original integer program. We then exploit connections to combinatorial optimization and develop an efficient primal-dual algorithm, scalable to large datasets. In contrast to our bottom-up approach, which benefits from good RF initialization, conventional methods are top-down acquiring features based on their utility value and is generally intractable, requiring heuristics. Empirically, our pruning algorithm outperforms existing state-of-the-art resource-constrained algorithms.

Answer: Pruning Random Forests for Prediction on a Budget

Question: 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. Federation Internationale des Echecs, 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.

Answer:
Giraffe: Using Deep Reinforcement Learning to Play Chess