UCT, a state-of-the art algorithm for Monte Carlo tree search (MCTS) in games and Markov decision processes, is based on UCB1, a sampling policy for the Multi-armed Bandit problem (MAB) that minimizes the cumulative regret. However, search differs from MAB in that in MCTS it is usually only the final "arm pull" (the actual move selection) that collects a reward, rather than all "arm pulls". In this paper, an MCTS sampling policy based on Value of Information (VOI) estimates of rollouts is suggested. Empirical evaluation of the policy and comparison to UCB1 and UCT is performed on random MAB instances as well as on Computer Go.
A different scheme for control of sampling can use the principles of bounded rationality @cite_6 and rational metareasoning @cite_9 @cite_0 . In search, one maintains a current best action @math , and finds the expected gain from finding another action @math to be better than the current best.
@cite_0 @cite_9 @cite_6
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We consider the problem of learning an unknown Markov Decision Process (MDP) that is weakly communicating in the infinite horizon setting. We propose a Thompson Sampling-based reinforcement learning algorithm with dynamic episodes (TSDE). At the beginning of each episode, the algorithm generates a sample from the posterior distribution over the unknown model parameters. It then follows the optimal stationary policy for the sampled model for the rest of the episode. The duration of each episode is dynamically determined by two stopping criteria. The first stopping criterion controls the growth rate of episode length. The second stopping criterion happens when the number of visits to any state-action pair is doubled. We establish @math bounds on expected regret under a Bayesian setting, where @math and @math are the sizes of the state and action spaces, @math is time, and @math is the bound of the span. This regret bound matches the best available bound for weakly communicating MDPs. Numerical results show it to perform better than existing algorithms for infinite horizon MDPs. Abstract In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. For the class of Daubechies wavelets we derive the exact constant of proportionality. Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of so-called perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential ill-conditioning. Thus generalized sampling provides a nearly-optimal solution to this problem. One popular method for dealing with large-scale data sets is sampling. For example, by using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows columns of data matrices to reduce the data size before performing computations on the subproblem. This method has been successful in improving computational efficiency of algorithms for matrix problems such as least-squares approximation, least absolute deviations approximation, and low-rank matrix approximation. Existing work has focused on algorithmic issues such as worst-case running times and numerical issues associated with providing high-quality implementations, but none of it addresses statistical aspects of this method. In this paper, we provide a simple yet effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model with a fixed number of predictors. In particular, for several versions of leverage-based sampling, we derive results for the bias and variance, both conditional and unconditional on the observed data. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms: one constructs a smaller least-squares problem with "shrinkage" leverage scores (SLEV), and the other solves a smaller and unweighted (or biased) least-squares problem (LEVUNW). A detailed empirical evaluation of existing leverage-based methods as well as these two new methods is carried out on both synthetic and real data sets. The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance. For example, with the same computation reduction as in the original algorithmic leveraging approach, our proposed SLEV typically leads to improved biases and variances both unconditionally and conditionally (on the observed data), and our proposed LEVUNW typically yields improved unconditional biases and variances. We consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multidimensional heat equation with a periodic terminal state constraint. This problem has a unique optimal solution, which can be characterized by an optimality system derived from the Pontryagin maximum principle. We define two approximations of this optimal control problem. The first one is an impulse approximation and consists of considering a system of linear heat equations with impulse control. The second one is obtained by the sample-and-hold procedure applied to the control, resulting in a sampled-data approximation of the controlled heat equation. We prove that both problems have a unique optimal solution, and we establish precise error estimates for the optimal controls and optimal states of the initial problem with respect to its impulse and sampled-data approximations.