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### Towards Optimal Off-Policy Evaluation for Reinforcement Learning with Marginalized Importance Sampling

Motivated by the many real-world applications of reinforcement learning (RL) that require safe-policy iterations, we consider the problem of off-policy evaluation (OPE) --- the problem of evaluating a new policy using the historical data obtained by different behavior policies --- under the model of nonstationary episodic Markov Decision Processes (MDP) with a long horizon and a large action space. Existing importance sampling (IS) methods often suffer from large variance that depends exponentially on the RL horizon $H$. To solve this problem, we consider a marginalized importance sampling (MIS) estimator that recursively estimates the state marginal distribution for the target policy at every step. The result matches the Cramer-Rao lower bound in [Jiang and Li, 2016] up to a multiplicative factor of $H$. To the best of our knowledge, this is the first OPE estimation error bound with a polynomial dependence on $H$.

### Intrinsically Efficient, Stable, and Bounded Off-Policy Evaluation for Reinforcement Learning

Off-policy evaluation (OPE) in both contextual bandits and reinforcement learning allows one to evaluate novel decision policies without needing to conduct exploration, which is often costly or otherwise infeasible. The problem's importance has attracted many proposed solutions, including importance sampling (IS), self-normalized IS (SNIS), and doubly robust (DR) estimates. DR and its variants ensure semiparametric local efficiency if Q-functions are well-specified, but if they are not they can be worse than both IS and SNIS. It also does not enjoy SNIS's inherent stability and boundedness. We propose new estimators for OPE based on empirical likelihood that are always more efficient than IS, SNIS, and DR and satisfy the same stability and boundedness properties as SNIS.

### Doubly Robust Bias Reduction in Infinite Horizon Off-Policy Estimation

Infinite horizon off-policy policy evaluation is a highly challenging task due to the excessively large variance of typical importance sampling (IS) estimators. Recently, Liu et al. (2018a) proposed an approach that significantly reduces the variance of infinite-horizon off-policy evaluation by estimating the stationary density ratio, but at the cost of introducing potentially high biases due to the error in density ratio estimation. In this paper, we develop a bias-reduced augmentation of their method, which can take advantage of a learned value function to obtain higher accuracy. Our method is doubly robust in that the bias vanishes when either the density ratio or the value function estimation is perfect. In general, when either of them is accurate, the bias can also be reduced. Both theoretical and empirical results show that our method yields significant advantages over previous methods.

### Inverse Policy Evaluation for Value-based Sequential Decision-making

Value-based methods for reinforcement learning lack generally applicable ways to derive behavior from a value function. Many approaches involve approximate value iteration (e.g., $Q$-learning), and acting greedily with respect to the estimates with an arbitrary degree of entropy to ensure that the state-space is sufficiently explored. Behavior based on explicit greedification assumes that the values reflect those of \textit{some} policy, over which the greedy policy will be an improvement. However, value-iteration can produce value functions that do not correspond to \textit{any} policy. This is especially relevant in the function-approximation regime, when the true value function can't be perfectly represented. In this work, we explore the use of \textit{inverse policy evaluation}, the process of solving for a likely policy given a value function, for deriving behavior from a value function. We provide theoretical and empirical results to show that inverse policy evaluation, combined with an approximate value iteration algorithm, is a feasible method for value-based control.

### Using Options and Covariance Testing for Long Horizon Off-Policy Policy Evaluation

Evaluating a policy by deploying it in the real world can be risky and costly. Off-policy policy evaluation (OPE) algorithms use historical data collected from running a previous policy to evaluate a new policy, which provides a means for evaluating a policy without requiring it to ever be deployed. Importance sampling is a popular OPE method because it is robust to partial observability and works with continuous states and actions. However, the amount of historical data required by importance sampling can scale exponentially with the horizon of the problem: the number of sequential decisions that are made. We propose using policies over temporally extended actions, called options, and show that combining these policies with importance sampling can significantly improve performance for long-horizon problems. In addition, we can take advantage of special cases that arise due to options-based policies to further improve the performance of importance sampling. We further generalize these special cases to a general covariance testing rule that can be used to decide which weights to drop in an IS estimate, and derive a new IS algorithm called Incremental Importance Sampling that can provide significantly more accurate estimates for a broad class of domains.