We propose a new approach to value function approximation which combines linear temporal difference reinforcement learning with subspace identification. In practical applications, reinforcement learning (RL) is complicated by the fact that state is either high-dimensional or partially observable. Therefore, RL methods are designed to work with features of state rather than state itself, and the success or failure of learning is often determined by the suitability of the selected features. By comparison, subspace identification (SSID) methods are designed to select a feature set which preserves as much information as possible about state. In this paper we connect the two approaches, looking at the problem of reinforcement learning with a large set of features, each of which may only be marginally useful for value function approximation. We introduce a new algorithm for this situation, called Predictive State Temporal Difference (PSTD) learning. As in SSID for predictive state representations, PSTD finds a linear compression operator that projects a large set of features down to a small set that preserves the maximum amount of predictive information. As in RL, PSTD then uses a Bellman recursion to estimate a value function. We discuss the connection between PSTD and prior approaches in RL and SSID. We prove that PSTD is statistically consistent, perform several experiments that illustrate its properties, and demonstrate its potential on a difficult optimal stopping problem.
Just as showed for the two simpler methods, we show that our two improved methods (model-free and model-based) are equivalent. This result yields some appealing theoretical benefits: for example, PSTD features can be explicitly interpreted as a statistically consistent estimate of the true underlying system state. And, the feasibility of finding the true value function can be shown to depend on the of the dynamical system, or equivalently, the dimensionality of the predictive state representation--- on the cardinality of the POMDP state space. Therefore our representation is naturally compressed'' in the sense of @cite_15 , speeding up convergence.
@cite_15
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When designing optimal controllers for any system, it is often the case that the true state of the system is unknown to the controller. Imperfect state information must be taken into account in the controller’s design in order to preserve its optimality. The same is true when performing reachability calculations. To estimate the probability that the state of a stochastic system reaches, or stays within, some set of interest in a given time horizon, it is necessary to find a controller that drives the system to that set with maximum probability, given the controller’s knowledge of the true state of the system. To date, little work has been done on stochastic reachability calculations with partially observable states. The work that has been done relies on converting the reachability optimization problem to one with an additive cost function, for which theoretical results are well known. Our approach is to preserve the multiplicative cost structure when deriving a sufficient statistic that reduces the problem to one of perfect state information. Our transformation includes a change of measure that simplifies the distribution of the sufficient statistic conditioned on its previous value. We develop a dynamic programming recursion for the solution of the equivalent perfect information problem, proving that the recursion is valid, an optimal solution exists, and results in the same solution as to the original problem. We also show that our results are equivalent to those for the reformulated additive cost problem, and so such a reformulation is not required. Sparsity or parsimony of statistical models is crucial for their proper interpretations, as in sciences and social sciences. Model selection is a commonly used method to find such models, but usually involves a computationally heavy combinatorial search. Lasso (Tibshirani, 1996) is now being used as a computationally feasible alternative to model selection. Therefore it is important to study Lasso for model selection purposes. In this paper, we prove that a single condition, which we call the Irrepresentable Condition, is almost necessary and sufficient for Lasso to select the true model both in the classical fixed p setting and in the large p setting as the sample size n gets large. Based on these results, sufficient conditions that are verifiable in practice are given to relate to previous works and help applications of Lasso for feature selection and sparse representation. This Irrepresentable Condition, which depends mainly on the covariance of the predictor variables, states that Lasso selects the true model consistently if and (almost) only if the predictors that are not in the true model are "irrepresentable" (in a sense to be clarified) by predictors that are in the true model. Furthermore, simulations are carried out to provide insights and understanding of this result. One key challenge in reinforcement learning is the ability to generalize knowledge in control problems. While deep learning methods have been successfully combined with model-free reinforcement-learning algorithms, how to perform model-based reinforcement learning in the presence of approximation errors still remains an open problem. Using successor features, a feature representation that predicts a temporal constraint, this paper presents three contributions: First, it shows how learning successor features is equivalent to model-free learning. Then, it shows how successor features encode model reductions that compress the state space by creating state partitions of bisimilar states. Using this representation, an intelligent agent is guaranteed to accurately predict future reward outcomes, a key property of model-based reinforcement-learning algorithms. Lastly, it presents a loss objective and prediction error bounds showing that accurately predicting value functions and reward sequences is possible with an approximation of successor features. On finite control problems, we illustrate how minimizing this loss objective results in approximate bisimulations. The results presented in this paper provide a novel understanding of representations that can support model-free and model-based reinforcement learning. Predictive state representations (PSRs) model dynamical systems using appropriately chosen predictions about future observations as a representation of the current state. In contrast to the hidden states posited by HMMs or RNNs, PSR states are directly observable in the training data; this gives rise to a moment-matching spectral algorithm for learning PSRs that is computationally efficient and statistically consistent when the model complexity matches that of the true system generating the data. In practice, however, model mismatch is inevitable and while spectral learning remains appealingly fast and simple it may fail to find optimal models. To address this problem, we investigate the use of gradient methods for improving spectrally-learned PSRs. We show that only a small amount of additional gradient optimization can lead to significant performance gains, and moreover that initializing gradient methods with the spectral learning solution yields better models in significantly less time than starting from scratch.