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Title: Infinite-horizon Off-Policy Policy Evaluation with Multiple Behavior Policies

Paper content:
We consider off-policy policy evaluation when the trajectory data are generated by multiple behavior policies.
Recent work has shown the key role played by the state or state-action stationary distribution corrections in the infinite horizon context for off-policy policy evaluation.
We propose estimated mixture policy (EMP), a novel class of partially policy-agnostic methods to accurately estimate those quantities.
With careful analysis, we show that EMP gives rise to estimates with reduced variance for estimating the state stationary distribution correction while it also offers a useful induction bias for estimating the state-action stationary distribution correction.
In extensive experiments with both continuous and discrete environments, we demonstrate that our algorithm offers significantly improved accuracy compared to the state-of-the-art methods.
In many real-world decision-making scenarios, evaluating a novel policy by directly executing it in the environment is generally costly and can even be downright risky.
Examples include evaluating a recommendation policy (Swaminathan et al., 2017; Zheng et al., 2018) , a treatment policy (Hirano et al., 2003; Murphy et al., 2001) , and a traffic light control policy ( Van der Pol & Oliehoek, 2016) .
Off-policy policy evaluation methods (OPPE) utilize a set of previously-collected trajectories (for example, website interaction logs, patient trajectories, or robot trajectories) to estimate the value of a novel decision-making policy without interacting with the environment (Precup et al., 2001; Dudík et al., 2011) .
For many reinforcement learning applications, the value of the decision is defined in a long-or infinite-horizon, which makes OPPE more challenging.
The state-of-the-art methods for infinite-horizon off-policy policy evaluation rely on learning (discounted) state stationary distribution corrections or ratios.
In particular, for each state in the environments, these methods estimate the likelihood ratio of the long-term probability measure for the state to be visited in a trajectory generated by the target policy, normalized by the probability measure generated by the behavior policy.
This approach can effectively avoid the exponentially high variance compared to the more classic importance sampling (IS) estimation methods (pre; Dudík et al., 2011; Hirano et al., 2003; Wang et al., 2017; Murphy et al., 2001) , especially for infinite-horizon policy evaluation (Liu et al., 2018; Nachum et al., 2019; Hallak & Mannor, 2017) .
However, learning state stationary distribution requires detailed information on distributions of the behavior policy, and we call them policy-aware methods.
As a consequence, policy-aware methods are difficult to apply when off-policy data are pre-generated by multiple behavior policies or when the behavior policy's form is unknown.
To address this issue, Nachum et al. (2019) proposes a policy-agnostic method, DualDice, which learns the joint state-action stationary distribution correction that is much higher dimension, and therefore needs more model parameters than the state stationary distribution.
Besides, there is no theoretic comparison between policy-aware and policy-agnostic methods.
In this paper, we propose a OPPE method with behavior policy learning, EMP (estimated mixture policy) for infinite-horizon off-policy policy evaluation with multiple known or unknown behavior policies.
We call EMP a partially policy-agnostic method in the sense that, EMP does not require any information on each"physical" behavior policy, instead, it utilizes some aggregated information of the behavior policies learned from data.
In detail, EMP includes a pre-estimation step using certain parametric model to learn a "virtual" policy (we call it the mixture policy and formally define it in Section 4).
Hence, its performance depends on the accuracy of mixture policy estimation.
Like the method in Liu et al. (2018) , EMP obtain OPPE also via learning the state stationary distribution correction, so it remains computationally cheap and is scalable in terms of the number of behavior policies.
Besides, inspired by Hanna et al. (2019) , we provide a theoretic guarantee that EMP yields smaller mean square error (MSE) than the policy-aware methods in stationary distribution corrections learning, even in the single-behavior policy setting.
On the other hand, compared to DualDice, EMP learns the state stationary distribution correction of smaller dimension, more importantly the estimation of the mixture policy can be considered as an inductive bias as far as the stationary distribution correction is concerned, and hence could achieve better performance when the pre-estimation is not expensive.
In addition, we propose an ad-hoc improvement of EMP, whose theoretical analysis is left for future studies.
EMP is compared with both policy-aware and policy-agnostic methods in a set of continuous and discrete control tasks and shows significant improvement.
In this paper, we advocate the viewpoint of partial policy-awareness and the benefits of estimating a "virtual" mixture policy for off-policy policy evaluation.
The theoretical results of reduced variance coupled with experimental results illustrate the power of this class of methods.
One key question that still remains is the following: if we are willing to estimate the individual behavior policies, can we further improve EMP by developing an efficient algorithm to compute the optimal weights?
The preliminary experiment results suggest that the answer would be yes, and we will leave this for future study.

assistant: text: An algorithm to solve infinite horizon off policy evaluation with multiple behavior policies by estimating a mixed policy under regression, and theoretical proof that an estimated policy ratio can reduce variance.