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.
Recently, looked at the problem of value function estimation from the perspective of both model-free and model-based reinforcement learning @cite_33 . The model-free approach estimates a value function directly from sample trajectories, i.e., from sequences of feature vectors of visited states. The model-based approach, by contrast, first learns a model and then computes the value function from the learned model. compared LSTD (a model-free method) to a model-based method in which we first learn a linear model by viewing features as a proxy for state (leading to a linear transition matrix that predicts future features from past features), and then compute a value function from this approximate model. demonstrated that these two approaches compute exactly the same value function @cite_33 , formalizing a fact that has been recognized to some degree before @cite_37 .
@cite_37 @cite_33
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The goal of reinforcement learning (RL) is to let an agent learn an optimal control policy in an unknown environment so that future expected rewards are maximized. The model-free RL approach directly learns the policy based on data samples. Although using many samples tends to improve the accuracy of policy learning, collecting a large number of samples is often expensive in practice. On the other hand, the model-based RL approach first estimates the transition model of the environment and then learns the policy based on the estimated transition model. Thus, if the transition model is accurately learned from a small amount of data, the model-based approach is a promising alternative to the model-free approach. In this paper, we propose a novel model-based RL method by combining a recently proposed model-free policy search method called policy gradients with parameter-based exploration and the state-of-the-art transition model estimator called least-squares conditional density estimation. Through experiments, we demonstrate the practical usefulness of the proposed method. We propose a reinforcement learning based approach to tackle the cost-sensitive learning problem where each input feature has a specific cost. The acquisition process is handled through a stochastic policy which allows features to be acquired in an adaptive way. The general architecture of our approach relies on representation learning to enable performing prediction on any partially observed sample, whatever the set of its observed features are. The resulting model is an original mix of representation learning and of reinforcement learning ideas. It is learned with policy gradient techniques to minimize a budgeted inference cost. We demonstrate the effectiveness of our proposed method with several experiments on a variety of datasets for the sparse prediction problem where all features have the same cost, but also for some cost-sensitive settings. Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual @math Learning method in light of the linear duality between the value and policy. The @math learning method is model-free and makes primal-dual updates to the policy and value vectors as new data are revealed. For infinite-horizon undiscounted Markov decision process with finite state space @math and finite action space @math , the @math learning method finds an @math -optimal policy using the following number of sample transitions @math where @math is an upper bound of mixing times across all policies and @math is a parameter characterizing the range of stationary distributions across policies. The @math learning method also applies to the computational problem of MDP where the transition probabilities and rewards are explicitly given as the input. In the case where each state transition can be sampled in @math time, the @math learning method gives a sublinear-time algorithm for solving the averaged-reward MDP. We consider the problem of multi-task reinforcement learning where the learner is provided with a set of tasks, for which only a small number of samples can be generated for any given policy. As the number of samples may not be enough to learn an accurate evaluation of the policy, it would be necessary to identify classes of tasks with similar structure and to learn them jointly. We consider the case where the tasks share structure in their value functions, and model this by assuming that the value functions are all sampled from a common prior. We adopt the Gaussian process temporal-difference value function model and use a hierarchical Bayesian approach to model the distribution over the value functions. We study two cases, where all the value functions belong to the same class and where they belong to an undefined number of classes. For each case, we present a hierarchical Bayesian model, and derive inference algorithms for (i) joint learning of the value functions, and (ii) efficient transfer of the information gained in (i) to assist learning the value function of a newly observed task. 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. Reinforcement Learning is divided in two main paradigms: model-free and model-based. Each of these two paradigms has strengths and limitations, and has been successfully applied to real world domains that are appropriate to its corresponding strengths. In this paper, we present a new approach aimed at bridging the gap between these two paradigms. We aim to take the best of the two paradigms and combine them in an approach that is at the same time data-efficient and cost-savvy. We do so by learning a probabilistic dynamics model and leveraging it as a prior for the intertwined model-free optimization. As a result, our approach can exploit the generality and structure of the dynamics model, but is also capable of ignoring its inevitable inaccuracies, by directly incorporating the evidence provided by the direct observation of the cost. Preliminary results demonstrate that our approach outperforms purely model-based and model-free approaches, as well as the approach of simply switching from a model-based to a model-free setting. Recent approaches to Reinforcement Learning (RL) with function approximation include Neural Fitted Q Iteration and the use of Gaussian Processes. They belong to the class of fitted value iteration algorithms, which use a set of support points to fit the value-function in a batch iterative process. These techniques make efficient use of a reduced number of samples by reusing them as needed, and are appropriate for applications where the cost of experiencing a new sample is higher than storing and reusing it, but this is at the expense of increasing the computational effort, since these algorithms are not incremental. On the other hand, non-parametric models for function approximation, like Gaussian Processes, are preferred against parametric ones, due to their greater flexibility. A further advantage of using Gaussian Processes for function approximation is that they allow to quantify the uncertainty of the estimation at each point. In this paper, we propose a new approach for RL in continuous domains based on Probability Density Estimations. Our method combines the best features of the previous methods: it is non-parametric and provides an estimation of the variance of the approximated function at any point of the domain. In addition, our method is simple, incremental, and computationally efficient. All these features make this approach more appealing than Gaussian Processes and fitted value iteration algorithms in general. Model-free reinforcement learning algorithms, such as Q-learning, perform poorly in the early stages of learning in noisy environments, because much effort is spent unlearning biased estimates of the state-action value function. The bias results from selecting, among several noisy estimates, the apparent optimum, which may actually be suboptimal. We propose G-learning, a new off-policy learning algorithm that regularizes the value estimates by penalizing deterministic policies in the beginning of the learning process. We show that this method reduces the bias of the value-function estimation, leading to faster convergence to the optimal value and the optimal policy. Moreover, G-learning enables the natural incorporation of prior domain knowledge, when available. The stochastic nature of G-learning also makes it avoid some exploration costs, a property usually attributed only to on-policy algorithms. We illustrate these ideas in several examples, where G-learning results in significant improvements of the convergence rate and the cost of the learning process. Learning robust value functions given raw observations and rewards is now possible with model-free and model-based deep reinforcement learning algorithms. There is a third alternative, called Successor Representations (SR), which decomposes the value function into two components -- a reward predictor and a successor map. The successor map represents the expected future state occupancy from any given state and the reward predictor maps states to scalar rewards. The value function of a state can be computed as the inner product between the successor map and the reward weights. In this paper, we present DSR, which generalizes SR within an end-to-end deep reinforcement learning framework. DSR has several appealing properties including: increased sensitivity to distal reward changes due to factorization of reward and world dynamics, and the ability to extract bottleneck states (subgoals) given successor maps trained under a random policy. We show the efficacy of our approach on two diverse environments given raw pixel observations -- simple grid-world domains (MazeBase) and the Doom game engine. Specifying a numeric reward function for reinforcement learning typically requires a lot of hand-tuning from a human expert. In contrast, preference-based reinforcement learning (PBRL) utilizes only pairwise comparisons between trajectories as a feedback signal, which are often more intuitive to specify. Currently available approaches to PBRL for control problems with continuous state action spaces require a known or estimated model, which is often not available and hard to learn. In this paper, we integrate preference-based estimation of the reward function into a model-free reinforcement learning (RL) algorithm, resulting in a model-free PBRL algorithm. Our new algorithm is based on Relative Entropy Policy Search (REPS), enabling us to utilize stochastic policies and to directly control the greediness of the policy update. REPS decreases exploration of the policy slowly by limiting the relative entropy of the policy update, which ensures that the algorithm is provided with a versatile set of trajectories, and consequently with informative preferences. The preference-based estimation is computed using a sample-based Bayesian method, which can also estimate the uncertainty of the utility. Additionally, we also compare to a linear solvable approximation, based on inverse RL. We show that both approaches perform favourably to the current state-of-the-art. The overall result is an algorithm that can learn non-parametric continuous action policies from a small number of preferences. Model-free reinforcement learning has been successfully applied to a range of challenging problems, and has recently been extended to handle large neural network policies and value functions. However, the sample complexity of modelfree algorithms, particularly when using high-dimensional function approximators, tends to limit their applicability to physical systems. In this paper, we explore algorithms and representations to reduce the sample complexity of deep reinforcement learning for continuous control tasks. We propose two complementary techniques for improving the efficiency of such algorithms. First, we derive a continuous variant of the Q-learning algorithm, which we call normalized advantage functions (NAF), as an alternative to the more commonly used policy gradient and actor-critic methods. NAF representation allows us to apply Q-learning with experience replay to continuous tasks, and substantially improves performance on a set of simulated robotic control tasks. To further improve the efficiency of our approach, we explore the use of learned models for accelerating model-free reinforcement learning. We show that iteratively refitted local linear models are especially effective for this, and demonstrate substantially faster learning on domains where such models are applicable. Model-free episodic reinforcement learning problems define the environment reward with functions that often provide only sparse information throughout the task. Consequently, agents are not given enough feedback about the fitness of their actions until the task ends with success or failure. Previous work addresses this problem with reward shaping. In this paper we introduce a novel approach to improve model-free reinforcement learning agents' performance with a three step approach. Specifically, we collect demonstration data, use the data to recover a linear function using inverse reinforcement learning and we use the recovered function for potential-based reward shaping. Our approach is model-free and scalable to high dimensional domains. To show the scalability of our approach we present two sets of experiments in a two dimensional Maze domain, and the 27 dimensional Mario AI domain. We compare the performance of our algorithm to previously introduced reinforcement learning from demonstration algorithms. Our experiments show that our approach outperforms the state-of-the-art in cumulative reward, learning rate and asymptotic performance. Model-free reinforcement learning has been successfully applied to a range of challenging problems, and has recently been extended to handle large neural network policies and value functions. However, the sample complexity of model-free algorithms, particularly when using high-dimensional function approximators, tends to limit their applicability to physical systems. In this paper, we explore algorithms and representations to reduce the sample complexity of deep reinforcement learning for continuous control tasks. We propose two complementary techniques for improving the efficiency of such algorithms. First, we derive a continuous variant of the Q-learning algorithm, which we call normalized adantage functions (NAF), as an alternative to the more commonly used policy gradient and actor-critic methods. NAF representation allows us to apply Q-learning with experience replay to continuous tasks, and substantially improves performance on a set of simulated robotic control tasks. To further improve the efficiency of our approach, we explore the use of learned models for accelerating model-free reinforcement learning. We show that iteratively refitted local linear models are especially effective for this, and demonstrate substantially faster learning on domains where such models are applicable. Model-free reinforcement learning (RL) has been proven to be a powerful, general tool for learning complex behaviors. However, its sample efficiency is often impractically large for solving challenging real-world problems, even for off-policy algorithms such as Q-learning. A limiting factor in classic model-free RL is that the learning signal consists only of scalar rewards, ignoring much of the rich information contained in state transition tuples. Model-based RL uses this information, by training a predictive model, but often does not achieve the same asymptotic performance as model-free RL due to model bias. We introduce temporal difference models (TDMs), a family of goal-conditioned value functions that can be trained with model-free learning and used for model-based control. TDMs combine the benefits of model-free and model-based RL: they leverage the rich information in state transitions to learn very efficiently, while still attaining asymptotic performance that exceeds that of direct model-based RL methods. Our experimental results show that, on a range of continuous control tasks, TDMs provide a substantial improvement in efficiency compared to state-of-the-art model-based and model-free methods. This thesis investigates risk-sensitive sequential decision-making problems in an uncertain environment. We rst introduce the axiomatic concept of valuation functions that generalize known concepts of risk measures in mathematical nance to cover most of the existing risk related models in various elds, in particular, behavioral economics and cognitive neuroscience. By applying this concept to Markov processes, we construct valuation maps and develop thereby a uni ed framework for incorporating risk into Markov decision processes on general spaces. Within the framework, we study mainly two types of in nite-horizon risk-sensitive criteria, discounted and average valuations, and solve the associated optimization problems by value iteration. For the discounted case, we propose a new discount scheme, which is di erent from the conventional form but consistent with existing literature, while for the average criterion, we state Lyapunov-type stability conditions that generalize known conditions for Markov chains to ensure the existence of solutions to the optimality equation and a geometric convergence rate for the value iteration. Applying a set of valuation functions, called utility-based shortfall, we derive a family of model-free risk-sensitive reinforcement learning algorithms for solving the optimization problems corresponding to risk-sensitive valuations. In addition, we nd that when appropriate utility functions are chosen, agents’ behaviors express key features of human behavior as predicted by prospect theory, for example, di erent risk preferences for gains and losses, as well as the shape of subjective probability curves. As a proof of principle for the applicability of the new algorithms, we apply them to two tasks, 1) to quantify human behavior in a sequential investment task and 2) to perform risk control in simulated algorithmic trading of stocks. In the rst task, the risk-sensitive variant provides a signi cantly better t to the behavioral data and it leads to an interpretation of the subject’s responses which is indeed consistent with prospect theory. The analysis of simultaneously measured fMRI signals show a signi cant correlation of the risk-sensitive temporal di erence error with BOLD signal change in the ventral striatum. In the second task, our algorithm outperforms the risk-neutral reinforcement learning algorithm by keeping the trading cost at a substantially low level at the spot when the 2010 Flash Crash happened, and signi cantly reducing the risk over the whole test period. The goal of reinforcement learning algorithms is to estimate and or optimise the value function. However, unlike supervised learning, no teacher or oracle is available to provide the true value function. Instead, the majority of reinforcement learning algorithms estimate and or optimise a proxy for the value function. This proxy is typically based on a sampled and bootstrapped approximation to the true value function, known as a . The particular choice of return is one of the chief components determining the nature of the algorithm: the rate at which future rewards are discounted; when and how values should be bootstrapped; or even the nature of the rewards themselves. It is well-known that these decisions are crucial to the overall success of RL algorithms. We introduce a novel, gradient-based meta-learning algorithm that is able to adapt the nature of the return, online, whilst interacting and learning from the environment. When applied to 57 games on the Atari 2600 environment over 200 million frames, our algorithm achieved a new state-of-the-art. We investigate the dual approach to dynamic programming and reinforcement learning, based on maintaining an explicit representation of stationary distributions as opposed to value functions. A significant advantage of the dual approach is that it allows one to exploit well developed techniques for representing, approximating and estimating probability distributions, without running the risks associated with divergent value function estimation. A second advantage is that some distinct algorithms for the average reward and discounted reward case in the primal become unified under the dual. In this paper, we present a modified dual of the standard linear program that guarantees a globally normalized state visit distribution is obtained. With this reformulation, we then derive novel dual forms of dynamic programming, including policy evaluation, policy iteration and value iteration. Moreover, we derive dual formulations of temporal difference learning to obtain new forms of Sarsa and Q-learning. Finally, we scale these techniques up to large domains by introducing approximation, and develop new approximate off-policy learning algorithms that avoid the divergence problems associated with the primal approach. We show that the dual view yields a viable alternative to standard value function based techniques and opens new avenues for solving dynamic programming and reinforcement learning problems The goal of reinforcement learning algorithms is to estimate and or optimise the value function. However, unlike supervised learning, no teacher or oracle is available to provide the true value function. Instead, the majority of reinforcement learning algorithms estimate and or optimise a proxy for the value function. This proxy is typically based on a sampled and bootstrapped approximation to the true value function, known as a return. The particular choice of return is one of the chief components determining the nature of the algorithm: the rate at which future rewards are discounted; when and how values should be bootstrapped; or even the nature of the rewards themselves. It is well-known that these decisions are crucial to the overall success of RL algorithms. We discuss a gradient-based meta-learning algorithm that is able to adapt the nature of the return, online, whilst interacting and learning from the environment. When applied to 57 games on the Atari 2600 environment over 200 million frames, our algorithm achieved a new state-of-the-art performance. In the model-based policy search approach to reinforcement learning (RL), policies are found using a model (or "simulator") of the Markov decision process. However, for high-dimensional continuous-state tasks, it can be extremely difficult to build an accurate model, and thus often the algorithm returns a policy that works in simulation but not in real-life. The other extreme, model-free RL, tends to require infeasibly large numbers of real-life trials. In this paper, we present a hybrid algorithm that requires only an approximate model, and only a small number of real-life trials. The key idea is to successively "ground" the policy evaluations using real-life trials, but to rely on the approximate model to suggest local changes. Our theoretical results show that this algorithm achieves near-optimal performance in the real system, even when the model is only approximate. Empirical results also demonstrate that---when given only a crude model and a small number of real-life trials---our algorithm can obtain near-optimal performance in the real system. Recent model-free reinforcement learning algorithms have proposed incorporating learned dynamics models as a source of additional data with the intention of reducing sample complexity. Such methods hold the promise of incorporating imagined data coupled with a notion of model uncertainty to accelerate the learning of continuous control tasks. Unfortunately, they rely on heuristics that limit usage of the dynamics model. We present model-based value expansion, which controls for uncertainty in the model by only allowing imagination to fixed depth. By enabling wider use of learned dynamics models within a model-free reinforcement learning algorithm, we improve value estimation, which, in turn, reduces the sample complexity of learning.