Sylvain Gelly

Combining online and offline knowledge in UCT

The UCT algorithm learns a value function online using sample-based search. The TD(λ) algorithm can learn a value function offline for the on-policy distribution. We consider three approaches for combining offline and online value functions in the UCT algorithm. First, the offline value function is used as a default policy during Monte-Carlo simulation. Second, the UCT value function is combined with a rapid online estimate of action values. Third, the offline value function is used as prior knowledge in the UCT search tree. We evaluate these algorithms in 9 x 9 Go against GnuGo 3.7.10. The first algorithm performs better than UCT with a random simulation policy, but surprisingly, worse than UCT with a weaker, handcrafted simulation policy. The second algorithm outperforms UCT altogether. The third algorithm outperforms UCT with handcrafted prior knowledge. We combine these algorithms in MoGo, the worlds strongest 9 x 9 Go program. Each technique significantly improves MoGos playing strength.

Monte-Carlo tree search and rapid action value estimation in computer Go

A new paradigm for search, based on Monte-Carlo simulation, has revolutionised the performance of computer Go programs. In this article we describe two extensions to the Monte-Carlo tree search algorithm, which significantly improve the effectiveness of the basic algorithm. When we applied these two extensions to the Go program MoGo, it became the first program to achieve dan (master) level in 9x9 Go. In this article we survey the Monte-Carlo revolution in computer Go, outline the key ideas that led to the success of MoGo and subsequent Go programs, and provide for the first time a comprehensive description, in theory and in practice, of this extended framework for Monte-Carlo tree search.

Modifications of UCT and sequence-like simulations for Monte-Carlo Go

Algorithm UCB1 for multi-armed bandit problem has already been extended to algorithm UCT which works for minimax tree search. We have developed a Monte-Carlo program, MoGo, which is the first computer Go program using UCT. We explain our modification of UCT for Go application and also the sequence-like random simulation with patterns which has improved significantly the performance of MoGo. UCT combined with pruning techniques for large Go board is discussed, as well as parallelization of UCT. MoGo is now a top-level computer-Go program on 9 times 9 Go board

The grand challenge of computer Go: Monte Carlo tree search and extensions

The ancient oriental game of Go has long been considered a grand challenge for artificial intelligence. For decades, computer Go has defied the classical methods in game tree search that worked so successfully for chess and checkers. However, recent play in computer Go has been transformed by a new paradigm for tree search based on Monte-Carlo methods. Programs based on Monte-Carlo tree search now play at human-master levels and are beginning to challenge top professional players. In this paper, we describe the leading algorithms for Monte-Carlo tree search and explain how they have advanced the state of the art in computer Go.

General lower bounds for evolutionary algorithms

Evolutionary optimization, among which genetic optimization, is a general framework for optimization. It is known (i) easy to use (ii) robust (iii) derivative-free (iv) unfortunately slow. Recent work [8] in particular show that the convergence rate of some widely used evolution strategies (evolutionary optimization for continuous domains) can not be faster than linear (i.e. the logarithm of the distance to the optimum can not decrease faster than linearly), and that the constant in the linear convergence (i.e. the constant C such that the distance to the optimum after n steps is upper bounded by Cn) unfortunately converges quickly to 1 as the dimension increases to ∞. We here show a very wide generalization of this result: all comparison-based algorithms have such a limitation. Note that our result also concerns methods like the Hooke & Jeeves algorithm, the simplex method, or any direct search method that only compares the values to previously seen values of the fitness. But it does not cover methods that use the value of the fitness (see [5] for cases in which the fitness-values are used), even if these methods do not use gradients. The former results deal with convergence with respect to the number of comparisons performed, and also include a very wide family of algorithms with respect to the number of function-evaluations. However, there is still place for faster convergence rates, for more original algorithms using the full ranking information of the population and not only selections among the population. We prove that, at least in some particular cases, using the full ranking information can improve these lower bounds, and ultimately provide superlinear convergence results.

Comparison-based algorithms are robust and randomized algorithms are anytime

Randomized search heuristics (e.g., evolutionary algorithms, simulated annealing etc.) are very appealing to practitioners, they are easy to implement and usually provide good performance. The theoretical analysis of these algorithms usually focuses on convergence rates. This paper presents a mathematical study of randomized search heuristics which use comparison based selection mechanism. The two main results are that comparison-based algorithms are the best algorithms for some robustness criteria and that introducing randomness in the choice of offspring improves the anytime behavior of the algorithm. An original Estimation of Distribution Algorithm combining both results is proposed and successfully experimented.

On the ultimate convergence rates for isotropic algorithms and the best choices among various forms of isotropy

In this paper, we show universal lower bounds for isotropic algorithms, that hold for any algorithm such that each new point is the sum of one already visited point plus one random isotropic direction multiplied by any step size (whenever the step size is chosen by an oracle with arbitrarily high computational power). The bound is 1–O(1/d) for the constant in the linear convergence (i.e. the constant C such that the distance to the optimum after n steps is upper bounded by Cn), as already seen for some families of evolution strategies in [19,12], in contrast with 1–O(1) for the reverse case of a random step size and a direction chosen by an oracle with arbitrary high computational power. We then recall that isotropy does not uniquely determine the distribution of a sample on the sphere and show that the convergence rate in isotropic algorithms is improved by using stratified or antithetic isotropy instead of naive isotropy. We show at the end of the paper that beyond the mathematical proof, the result holds on experiments. We conclude that one should use antithetic-isotropy or stratified-isotropy, and never standard-isotropy.

A Statistical Learning Perspective of Genetic Programming

This paper proposes a theoretical analysis of Genetic Programming (GP) from the perspective of statistical learning theory, a well grounded mathematical toolbox for machine learning. By computing the Vapnik-Chervonenkis dimension of the family of programs that can be inferred by a specific setting of GP, it is proved that a parsimonious fitness ensures universal consistency. This means that the empirical error minimization allows convergence to the best possible error when the number of test cases goes to infinity. However, it is also proved that the standard method consisting in putting a hard limit on the program size still results in programs of infinitely increasing size in function of their accuracy. It is also shown that cross-validation or hold-out for choosing the complexity level that optimizes the error rate in generalization also leads to bloat. So a more complicated modification of the fitness is proposed in order to avoid unnecessary bloat while nevertheless preserving universal consistency.

A statistical learning theory approach of bloat

Code bloat, the excessive increase of code size, is an important issue in Genetic Programming (GP). This paper proposes a theoretical analysis of code bloat in the framework of symbolic regression in GP, from the viewpoint of Statistical Learning Theory, a well grounded mathematical toolbox for Machine Learning. Two kinds of bloat must be distinguished in that context, depending whether the target function lies in the search space or not. Then, important mathematical results are proved using classical results from Statistical Learning. Namely, the Vapnik-Chervonenkis dimension of programs is computed, and further results from Statistical Learning allow to prove that a parsimonious fitness ensures Universal Consistency (the solution minimizing the empirical error does converge to the best possible error when the number of examples goes to infinity). However, it is proved that the standard method consisting in choosing a maximal program size depending on the number of examples might still result in programs of infinitely increasing size with their accuracy; a more complicated modification of the fitness is proposed that theoretically avoids unnecessary bloat while nevertheless preserving the Universal Consistency.

Universal consistency and bloat in GP : Some theoretical considerations about genetic programming from a statistical learning theory viewpoint

In this paper, we provide an analysis of Genetic Programming (GP) from the Statistical Learning Theory viewpoint in the scope of symbolic regression. Firstly, we are interested in Universal Consistency, i.e. the fact that the solution minimizing the empirical error does converge to the best possible error when the number of examples goes to infinity, and secondly, we focus our attention on the uncontrolled growth of program length (i.e. bloat), which is a well-known problem in GP. Results show that (1) several kinds of code bloats may be identified and that (2) Universal consistency can be obtained as well as avoiding bloat under some conditions. We conclude by describing an ad hoc method that makes it possible simultaneously to avoid bloat and to ensure universal consistency.