accountableoffpolicyevaluationwithkernelbellmanstatistics/full.md

Accountable Off-Policy Evaluation With Kernel Bellman Statistics

Yihao Feng $^{1}$ Tongzheng Ren $^{*1}$ Ziyang Tang $^{*1}$ Qiang Liu

Abstract

We consider off-policy evaluation (OPE), which evaluates the performance of a new policy from observed data collected from previous experiments, without requiring the execution of the new policy. This finds important applications in areas with high execution cost or safety concerns, such as medical diagnosis, recommendation systems and robotics. In practice, due to the limited information from off-policy data, it is highly desirable to construct rigorous confidence intervals, not just point estimation, for the policy performance. In this work, we propose a new variational framework which reduces the problem of calculating tight confidence bounds in OPE into an optimization problem on a feasible set that catches the true state-action value function with high probability. The feasible set is constructed by leveraging statistical properties of a recently proposed kernel Bellman loss (Feng et al., 2019). We design an efficient computational approach for calculating our bounds, and extend it to perform post-hoc diagnosis and correction for existing estimators. Empirical results show that our method yields tight confidence intervals in different settings.

1. Introduction

Reinforcement learning (Sutton & Barto, 1998) has achieved remarkable successes recently, but is still highly data demanding. An essential approach to improve data efficiency is to use off-policy methods, which leverage historical data collected from different behavior policies (a.k.a. off-policy data) to evaluate and optimize the target policies of interest. Using off-policy data is of critical importance in many real-world applications (e.g., Murphy et al., 2001; Li et al., 2011; Bottou et al., 2013; Thomas et al., 2017), especially in cases when it is infeasible or even risky to collect online

*Equal contribution ${}^{1}$ Department of Computer Science,The University of Texas at Austin. Correspondence to: Yihao Feng yihao@cs.utexas.edu, Tongzheng Ren tzren@cs.utexas.edu.

Proceedings of the $37^{th}$ International Conference on Machine Learning, Online, PMLR 119, 2020. Copyright 2020 by the author(s).

experimental data or build high-quality simulators, such as robotics, advertisement, online education, and medical treatment (e.g., Murphy et al., 2001; Li et al., 2011; Hirano et al., 2003; Liao et al., 2019).

This work concerns the problem of off-policy evaluation (OPE), which estimates the expected reward of a given target policy from off-policy data. Because the information in off-policy data is often limited, typical point estimation may suffer from large error in difficult cases. Therefore, for making high-stakes decisions in areas such as medical treatment, it is crucially important to provide reliable confidence bounds for quantifying the uncertainties in the estimation.

Importance sampling (IS) provides a basic principle for developing OPE methods (e.g., Dudík et al., 2011; Jiang & Li, 2016; Thomas & Brunskill, 2016; Liu et al., 2018; Nachum et al., 2019; Zhang et al., 2020), as well as doubly robust estimators (Dudík et al., 2011; Jiang & Li, 2016; Farajtabar et al., 2018; Kallus & Uehara, 2020; Tang et al., 2020), as well as constructing confidence bounds for OPE (e.g., Thomas et al., 2015a; Hanna et al., 2017). However, the typical IS-based methods suffer from a number of critical disadvantages:

• Requiring known behavior policies: The typical IS estimators require the observed data to be drawn from a known behavior policy. However, this is often not the case in practice. In practice, the policies from which the observed data is collected may be unknown, or even not exist (e.g., when the data is drawn from a mix of different policies).
• High variance and loose bounds: The variance of IS-based estimators can be excessively high when the target and behavior policies are very different from each other. As a consequence, concentration confidence bounds based on IS (e.g., Thomas et al., 2015a;b) could be very loose.
• Curse of long horizon: The variance of IS estimators are known to deteriorate quickly when the horizon the Markov decision processes becomes large. Although there are recent improved methods for long horizon problems (Liu et al., 2018; Tang et al., 2020; Mousavi et al., 2020), they introduce additional biases due to the estimation of the density ratios.

Our Method We propose a variational framework for constructing tight, provable confidence bounds which avoids both the requirement on known behavior policies and the

curse of long horizon. Instead of using importance sampling, we construct bounds based on the state-action value function (a.k.a. Q-function) specified by Bellman equation. The idea is to construct a high confidence feasible set of the true Q-function of the target policy, by leveraging the tail bound of the kernel Bellman loss (Feng et al., 2019), and derive the upper (resp. lower) bounds of the expected reward of interest by solving a maximization (resp. minimization) problem on the high confidence feasible set. Further, by constraining the optimization inside a reproducing kernel Hilbert space (RKHS) ball, we obtain a computationally efficient framework for providing confidence bounds of off-policy value evaluation.

Post-hoc Analysis In practice, we may already have an estimated Q-function from previous data/experiments, and we want to construct post-hoc confidence bounds around the existing estimation, or perform post-hoc correction if the estimation is detected to deviate significantly from the true value. In this work, we enable such post-hoc diagnosis by extending our main framework. We empirically evaluate our method on several control benchmarks; results show that our method can provide tight confidence bounds, and also reasonable post-hoc corrections for existing estimators.

2. Related Work

Off-policy value (point) estimation has been studied in various reinforcement learning settings, including contextual bandits (e.g., Dudík et al., 2011; Wang et al., 2017), finite horizon Markov decision processes (MDPs) (e.g., Thomas & Brunskill, 2016; Jiang & Li, 2016; Farajtabar et al., 2018; Xie et al., 2019), and infinite horizon RL MDPs (e.g., Liu et al., 2018; Nachum et al., 2019; Tang et al., 2020). Unlike our work, most of these works are not behavior agnostic and require to know the behavior policy, except a line of very recent works (e.g., Nachum et al., 2019; Zhang et al., 2020; Mousavi et al., 2020).

A smaller set of works focus on confidence bounds for various RL problems (e.g., White & White, 2010; Thomas et al., 2015a;b; Hanna et al., 2017; Dann et al., 2019; Jiang & Huang, 2020; Duan et al., 2020). A recent line of works (e.g., Dann et al., 2019; Jin et al., 2018; 2020) have utilized upper confidence bounds for explorations on tabular (or linear) MDPs. Thomas et al. (2015a); Hanna et al. (2017) give high concentration confidence bounds for off-policy estimators, based on importance sampling methods, which suffer from the issue of high variance of IS-based estimators.

Recently, Jiang & Huang (2020) proposes upper/ lower bounds for OPE based a minimax formulation of double robustness (e.g., Tang et al., 2020; Uehara et al., 2020), but their method does not take the randomness of empirical data (data uncertainty) into account to get high confidence

bounds. Concurrently, Duan et al. (2020) provides a data-dependent confidence bounds, based on regression-based Fitted Q iteration (FQI), by assuming the function class is linear and making additional assumptions on MDPs.

Compared with these existing works, our method provides tighter bounds that avoid the high variance issue of IS, work for behavior-agnostic OPE estimators, and can be directly applied to both continuous and discrete state and action MDPs once a kernel function can be properly defined.

Finally, the kernel method has been widely used in reinforcement learning, but is typically used to represent value functions or transition models (e.g., Xu et al., 2005; 2007; Jung & Polani, 2007; Taylor & Parr, 2009; Grunewalder et al., 2012; Farahmand et al., 2016). Our work admits a novel application of kernel in RL, leveraging it as a technical tool for both defining the kernel Bellman loss and constructing high confidence bounds.

3. Off-Policy Evaluation and Kernel Bellman Statistics

We start with introducing background on reinforcement learning and off-policy evaluation (OPE), then give a review on the kernel Bellman loss (Feng et al., 2019) and introduce the concentration bounds for it.

3.1. Preliminary

Denote by $M = \langle S, \mathcal{A}, \mathcal{P}, r, \gamma, \mu_0 \rangle$ a Markov decision process (MDP), where $S$ is the state space; $\mathcal{A}$ is the action space; $\mathcal{P}(s'|s, a)$ is the transition probability; $r(s, a)$ is the average immediate reward; $\mu_0 \in \Delta(S)$ is the initial state distribution, and $\gamma \in (0, 1)$ is the discount factor. We focus on the discounted case $(\gamma < 1)$ in the main paper, and discuss the extension to average reward case $(\gamma = 1)$ in Appendix C.

A policy $\pi$ specifies a distribution of actions given states, and $\pi(a|s)$ denotes the probability of selecting $a$ given $s$ . Given a target policy $\pi$ , we are interested in estimating the expected total discounted reward associated with $\pi$ :

$$\begin{array}{cccc}& {\eta }^{\pi }:=\underset{T\to \mathrm{\infty }}{\lim }{\mathbb{E}}_{\pi }\left[\sum _{t=0}^T{\gamma }^t{r}_t\right]={\mathbb{E}}_{\pi }\left[\sum _{t=0}^{\mathrm{\infty }}{\gamma }^t{r}_t\right],& & \text{(1)}\end{array}\backslash eta^\; \{\backslash pi\}\; :=\; \backslash lim\; \_\; \{T\; \backslash rightarrow\; \backslash infty\}\; \backslash mathbb\; \{E\}\; \_\; \{\backslash pi\}\; \backslash left[\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{T\}\; \backslash gamma^\; \{t\}\; r\; \_\; \{t\}\; \backslash right]\; =\; \backslash mathbb\; \{E\}\; \_\; \{\backslash pi\}\; \backslash left[\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{\backslash infty\}\; \backslash gamma^\; \{t\}\; r\; \_\; \{t\}\; \backslash right],\; \backslash tag\; \{1\}$$

where we start from an initial state $s_0 \sim \mu_0$ , and execute the policy $\pi$ in the MDP to generate trajectories; here $T$ is the trajectory length. We mainly consider the case with infinite horizon $(T = \infty)$ in this work.

Assume we are given an observed set of transition pairs $\mathcal{D} = (s_i, a_i, r_i, s_i')_{i=1}^n$ , where $r_i$ and $s_i'$ are the observed local reward and the next state following $(s_i, a_i)$ . The data is assumed to be off-policy in that they are collected from some arbitrary, unknown policy (a.k.a. behavior policy),

which is different from the target policy $\pi$ of interest. The goal of off-policy evaluation (OPE) is to provide interval estimation (upper and lower bounds) for the expected reward $\eta^{\pi}$ from the off-policy data $\mathcal{D}$ , without knowing the underlying behavior policy (i.e., behavior agnostic).

Value Estimation via Q-Function We approach this problem by approximating the expected reward $\eta^{\pi}$ through the Q-function. The Q-function (a.k.a state-action value function) $Q^{\pi}:\mathcal{S}\times \mathcal{A}\to \mathbb{R}$ of a policy $\pi$ is defined by

$$Q^{\pi }(s,a):={\mathbb{E}}_{\pi }[\sum _{t=0}^{\mathrm{\infty }}{\gamma }^{t}r(s_t,a_t)\mid s_0=s,a_0=a],Q\; ^\; \{\backslash pi\}\; (s,\; a)\; :=\; \backslash mathbb\; \{E\}\; \_\; \{\backslash pi\}\; \backslash left[\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{\backslash infty\}\; \backslash gamma^\; \{t\}\; r\; (s\; \_\; \{t\},\; a\; \_\; \{t\})\; \backslash mid\; s\; \_\; \{0\}\; =\; s,\; a\; \_\; \{0\}\; =\; a\; \backslash right],$$

which equals the expected long-term discounted reward when we execute policy $\pi$ starting from $(s,a)$ .

Obviously, $Q^{\pi}$ and $\eta^{\pi}$ are related via

$$\begin{array}{cccc}& {\eta }^{\pi }=\eta (Q^{\pi }):={\mathbb{E}}_{s_0\sim {\mu }_0,a_0\sim \pi (\cdot \mathrm{\mid }{s}_0)}[Q^{\pi }(s_0,a_0)],& & \text{(2)}\end{array}\backslash eta^\; \{\backslash pi\}\; =\; \backslash eta\; (Q\; ^\; \{\backslash pi\})\; :=\; \backslash mathbb\; \{E\}\; \_\; \{s\; \_\; \{0\}\; \backslash sim\; \backslash mu\_\; \{0\},\; a\; \_\; \{0\}\; \backslash sim\; \backslash pi\; (\backslash cdot\; |\; s\; \_\; \{0\})\}\; [\; Q\; ^\; \{\backslash pi\}\; (s\; \_\; \{0\},\; a\; \_\; \{0\})\; ],\; \backslash tag\; \{2\}$$

where $s_0$ is drawn from the fixed initial distribution $\mu_0$ . Therefore, given an empirical approximation $\widehat{Q}$ of $Q^{\pi}$ , we can estimate $\eta^{\pi}$ by $\eta (\widehat{Q})$ , which can be further approximated by drawing an i.i.d. sample $(s_{i,0},a_{i,0})_{i = 1}^{N}$ from $\mu_0\times \pi$ :

$$\begin{array}{cccc}& \eta (\widehat{Q})\approx \widehat{\eta }(\widehat{Q}):=\frac{1}{N}\sum _{i=1}^N\widehat{Q}(s_{i,0},a_{i,0})\mathrm{.}& & \text{(3)}\end{array}\backslash eta\; (\backslash widehat\; \{Q\})\; \backslash approx\; \backslash widehat\; \{\backslash eta\}\; (\backslash widehat\; \{Q\})\; :=\; \backslash frac\; \{1\}\{N\}\; \backslash sum\_\; \{i\; =\; 1\}\; ^\; \{N\}\; \backslash widehat\; \{Q\}\; \backslash left(s\; \_\; \{i,\; 0\},\; a\; \_\; \{i,\; 0\}\backslash right).\; \backslash tag\; \{3\}$$

Since $\mu_0$ and $\pi$ are assumed to be known, we can draw an i.i.d. sample with a very large size $N$ to get arbitrarily high accuracy in the Monte Carlo estimation.

Bellman Equation It is well-known that $Q = Q^{\pi}$ is the unique solution of the Bellman equation (Puterman, 1994): $Q = \mathcal{B}{\pi}Q$ , where $\mathcal{B}{\pi}$ is the Bellman evaluation operator, defined by

$${\mathcal{B}}_{\pi }Q(s,a):={\mathbb{E}}_{(s^{\mathrm{\prime }},a^{\mathrm{\prime }})}[r(s,a)+\gamma Q(s^{\mathrm{\prime }},a^{\mathrm{\prime }})\mid s,a]\mathrm{.}\backslash mathcal\; \{B\}\; \_\; \{\backslash pi\}\; Q\; (s,\; a)\; :=\; \backslash mathbb\; \{E\}\; \_\; \{(s\; ^\; \{\backslash prime\},\; a\; ^\; \{\backslash prime\})\}\; \backslash left[\; r\; (s,\; a)\; +\; \backslash gamma\; Q\; (s\; ^\; \{\backslash prime\},\; a\; ^\; \{\backslash prime\})\; \backslash mid\; s,\; a\; \backslash right].$$

where $(s', a')$ is drawn from $s' \sim \mathcal{P}(\cdot \mid s, a)$ and $a' \sim \pi(\cdot \mid s')$ . For simplicity, we can define the Bellman residual operator as $\mathcal{R}{\pi}Q = \mathcal{B}{\pi}Q - Q$ .

We can approximate the Bellman operator on a state-action pair $(s_i, a_i)$ by bootstrapping:

$${\widehat{\mathcal{B}}}_{\pi }Q(s_i,a_i):=r_i+\gamma {\mathbb{E}}_{a^{\mathrm{\prime }}\sim \pi (\cdot \mid s_i^{\mathrm{\prime }})}\left[Q(s_i^{\mathrm{\prime }},a^{\mathrm{\prime }})\right],\backslash widehat\; \{\backslash mathcal\; \{B\}\}\; \_\; \{\backslash pi\}\; Q\; \backslash left(s\; \_\; \{i\},\; a\; \_\; \{i\}\backslash right)\; :=\; r\; \_\; \{i\}\; +\; \backslash gamma\; \backslash mathbb\; \{E\}\; \_\; \{a\; ^\; \{\backslash prime\}\; \backslash sim\; \backslash pi\; \backslash left(\backslash cdot\; \backslash mid\; s\; \_\; \{i\}\; ^\; \{\backslash prime\}\backslash right)\}\; \backslash left[\; Q\; \backslash left(s\; \_\; \{i\}\; ^\; \{\backslash prime\},\; a\; ^\; \{\backslash prime\}\backslash right)\; \backslash right],$$

where $r_i, s_i'$ are the observed local reward and the next state of $(s_i, a_i)$ , and $a'$ is drawn from policy $\pi$ by Monte Carlo (same as expected Sarsa (Sutton & Barto, 1998)). Because the policy is known, we can approximate the expectation term $\mathbb{E}_{a' \sim \pi(\cdot | s_i')} [Q(s_i', a')]$ up to arbitrarily high accuracy by drawing a large sample from $\pi(\cdot | s_i')$ . We can also empirically estimate the Bellman residual by

$\widehat{\mathcal{R}}{\pi}Q(s_i,a_i)\coloneqq \widehat{B}{\pi}Q(s_i,a_i) - Q(s_i,a_i)$ , which provides an unbiased estimation of the Bellman residual, that is, $\mathbb{E}[\widehat{\mathcal{R}}{\pi}Q(s_i,a_i)\mid s_i,a_i] = \mathcal{R}{\pi}Q(s_i,a_i)$ . Note that $\widehat{B}_{\pi}Q(s_i,a_i)$ is a function of $(s_i,a_i,r_i,s_i')$ , but the dependency on $r_i,s_i'$ is dropped in notation for convenience.

The Double Sampling Problem A naive approach for estimating $Q^{\pi}$ is to minimize the empirical Bellman residual. However, a key barrier is the so-called "double sampling" problem. To illustrate, consider estimating $Q^{\pi}$ by minimizing the square norm of the empirical Bellman residual:

$${\widehat{L}}_2(Q)=\frac{1}{n}\sum _{i=1}^n\left[{\left({\widehat{\mathcal{R}}}_{\pi }Q(s_i,a_i)\right)}^2\right]\mathrm{.}\backslash widehat\; \{L\}\; \_\; \{2\}\; (Q)\; =\; \backslash frac\; \{1\}\{n\}\; \backslash sum\_\; \{i\; =\; 1\}\; ^\; \{n\}\; \backslash left[\; \backslash left(\backslash widehat\; \{\backslash mathcal\; \{R\}\}\; \_\; \{\backslash pi\}\; Q\; \backslash left(s\; \_\; \{i\},\; a\; \_\; \{i\}\backslash right)\backslash right)\; ^\; \{2\}\; \backslash right].$$

Unfortunately, minimizing $\widehat{L}2(Q)$ does not yield a consistent estimation of $Q^{\pi}$ , because $(\widehat{\mathcal{R}}{\pi}Q(s_i,a_i))^2$ does not correctly estimate $(\mathcal{R}{\pi}Q(s_i,a_i))^2$ due to the square function outside of the empirical Bellman residual (even though $\widehat{R}{\pi}Q(s_i,a_i)$ is an unbiased estimation of $\mathcal{R}{\pi}Q(s_i,a_i)$ ). One strategy for obtaining an unbiased estimation of the exact square $(\mathcal{R}{\pi}Q(s_i,a_i))^2$ is to multiply two copies of $\widehat{\mathcal{R}}_{\pi}Q(s_i,a_i)$ obtained from two independent copies of $r_i$ , $s_i'$ following the same $(s_i,a_i)$ , hence requiring double sampling (Baird, 1995). However, double sampling rarely happens in observed data collected from natural environment, and is impractical to use in real-world problems. As a result, many works on Q-function estimation resort to temporal difference (TD) based algorithms, which, however, suffer from instability and divergence issues, especially when nonlinear function approximation is used.

3.2. Kernel Bellman Statistics

Recently, Feng et al. (2019) proposed a kernel Bellman loss to address the double sampling problem in value function estimation. This objective function provides a consistent estimation of the derivation from a given $Q$ to the true $Q^{\pi}$ , and hence allows us to estimate $Q^{\pi}$ without requiring double sampling, nor risking the instability or divergence in TD-based algorithms. We now give a brief introduction to the kernel Bellman loss and the related U/V-statistics, and their concentration inequalities that are useful in our framework.

For notation, we use $x = (s,a)$ to denote a state-action pair, and $x' := (s',a')$ a successive state-action pair following $x = (s,a)$ collected under some behavior policy. We use $\tau = (s,a,r,s')$ to denote a transition pair.

Following Feng et al. (2019), let $K: (\mathcal{S} \times \mathcal{A}) \times (\mathcal{S} \times \mathcal{A}) \to \mathbb{R}$ be an integrally strictly positive definite (ISPD) kernel. The expected kernel Bellman loss is defined by

$$\begin{array}{cccc}& L_K(Q):={\mathbb{E}}_{x,\bar{x}\sim \mu }[{\mathcal{R}}_{\pi }Q(x)K(x,\bar{x}){\mathcal{R}}_{\pi }Q(\bar{x})],& & \text{(4)}\end{array}L\; \_\; \{K\}\; (Q)\; :=\; \backslash mathbb\; \{E\}\; \_\; \{x,\; \backslash bar\; \{x\}\; \backslash sim\; \backslash mu\}\; \backslash left[\; \backslash mathcal\; \{R\}\; \_\; \{\backslash pi\}\; Q\; (x)\; K\; (x,\; \backslash bar\; \{x\})\; \backslash mathcal\; \{R\}\; \_\; \{\backslash pi\}\; Q\; (\backslash bar\; \{x\})\; \backslash right],\; \backslash tag\; \{4\}$$

where $\bar{x} = (\bar{s},\bar{a})$ is an independent and identical copy of $x$ and $\mu$ is any distribution supported on $S\times \mathcal{A}$ . Note that $\mu$

can be either the on-policy visitation distribution induced by $\pi$ , or some other valid distributions defined by the observed data (the off-policy case).

As shown in Feng et al. (2019), the expected kernel Bellman loss fully characterizes the true $Q^{\pi}$ function in that $L_{K}(Q)\geq 0$ for any $Q$ , and

$$L_K(Q)=0\text{i f a n d o n l y i f}Q=Q^{\pi }\mathrm{.}L\; \_\; \{K\}\; (Q)\; =\; 0\; \backslash qquad\; \backslash text\; \{i\; f\; a\; n\; d\; o\; n\; l\; y\; i\; f\}\; \backslash qquad\; Q\; =\; Q\; ^\; \{\backslash pi\}.$$

Another key property of $L_{K}(Q)$ is that it can be easily estimated and optimized from observed transitions, without requiring double samples. Given a set of observed transition pairs $\mathcal{D} = {\tau_i}{i=1}^n$ , we can use the so-called V-statistics to estimate $L{K}(Q)$ :

$$\begin{array}{cccc}& {\widehat{L}}_K^V(Q):=\frac{1}{n^2}\sum _{i,j=1}^n{\mathrm{\ell }}_{\pi ,Q}({\tau }_i,{\tau }_j),& & \text{(5)}\end{array}\backslash widehat\; \{L\}\; \_\; \{K\}\; ^\; \{V\}\; (Q)\; :=\; \backslash frac\; \{1\}\{n\; ^\; \{2\}\}\; \backslash sum\_\; \{i,\; j\; =\; 1\}\; ^\; \{n\}\; \backslash ell\_\; \{\backslash pi\; ,\; Q\}\; \backslash left(\backslash tau\_\; \{i\},\; \backslash tau\_\; \{j\}\backslash right),\; \backslash tag\; \{5\}$$

where $\ell_{\pi ,Q}$ is defined by

$$\begin{array}{cccc}& {\mathrm{\ell }}_{\pi ,Q}({\tau }_i,{\tau }_j):={\widehat{\mathcal{R}}}_{\pi }Q\left(x_i\right)K(x_i,x_j){\widehat{\mathcal{R}}}_{\pi }Q\left(x_j\right)\mathrm{.}& & \text{(6)}\end{array}\backslash ell\_\; \{\backslash pi\; ,\; Q\}\; \backslash left(\backslash tau\_\; \{i\},\; \backslash tau\_\; \{j\}\backslash right)\; :=\; \backslash widehat\; \{\backslash mathcal\; \{R\}\}\; \_\; \{\backslash pi\}\; Q\; \backslash left(x\; \_\; \{i\}\backslash right)\; K\; \backslash left(x\; \_\; \{i\},\; x\; \_\; \{j\}\backslash right)\; \backslash widehat\; \{\backslash mathcal\; \{R\}\}\; \_\; \{\backslash pi\}\; Q\; \backslash left(x\; \_\; \{j\}\backslash right).\; \backslash tag\; \{6\}$$

Another way is to use the so-called $U$ -statistics, which removes the diagonal terms in (5):

$$\begin{array}{cccc}& {\widehat{L}}_K^U(Q):=\frac{1}{n(n-1)}\sum _{1\le i\mathrm{\ne }j\le n}{\mathrm{\ell }}_{\pi ,Q}({\tau }_i,{\tau }_j)\mathrm{.}& & \text{(7)}\end{array}\backslash widehat\; \{L\}\; \_\; \{K\}\; ^\; \{U\}\; (Q)\; :=\; \backslash frac\; \{1\}\{n\; (n\; -\; 1)\}\; \backslash sum\_\; \{1\; \backslash leq\; i\; \backslash neq\; j\; \backslash leq\; n\}\; \backslash ell\_\; \{\backslash pi\; ,\; Q\}\; \backslash left(\backslash tau\_\; \{i\},\; \backslash tau\_\; \{j\}\backslash right).\; \backslash tag\; \{7\}$$

We call $\widehat{L}_K^U (Q)$ and $\widehat{L}K^V (Q)$ the kernel Bellman U/V-statistics. Both the U/V-statistics can be shown to give consistent estimation of $L{K}(Q)$ . The U-statistics $\widehat{L}K^U (Q)$ is known to be an unbiased estimator of the kernel loss $L{K}(Q)$ , but can be negative and hence unstable when being used as an objective function in practice. In comparison, the V-statistics $\widehat{L}_K^V (Q)$ is always non-negative, that is, $\widehat{L}_K^V (Q)\geq 0$ for any $Q$ , and hence behaves more stably in practice. We mainly use the V-statistics in this work.

3.3. Concentration Bounds of Kernel Bellman Statistics

We now introduce concentration inequalities for the kernel Bellman statistics, which form an important tool in our framework. The inequalities here are classical results from Hoeffding (1963).

Proposition 3.1 (Concentration bound of U-/V-statistics). Consider a set of i.i.d. random transition pairs ${\tau_i}_{i=1}^n$ with $x_i \sim \mu$ . Assume

and $n$ is an even number, then we have the following concentration bound for $U$ -statistics,

$$\mathbb{P}[\mathrm{\mid }{\widehat{L}}_K^U(Q)-L_K(Q)\mathrm{\mid }\ge 2{\mathrm{\ell }}_{\max }\sqrt{\frac{\log \frac{2}{\delta }}{n}}]\le \delta ,\backslash mathbb\; \{P\}\; \backslash left[\; |\; \backslash widehat\; \{L\}\; \_\; \{K\}\; ^\; \{U\}\; (Q)\; -\; L\; \_\; \{K\}\; (Q)\; |\; \backslash geq\; 2\; \backslash ell\_\; \{\backslash max\}\; \backslash sqrt\; \{\backslash frac\; \{\backslash log\; \backslash frac\; \{2\}\{\backslash delta\}\}\{n\}\}\; \backslash right]\; \backslash leq\; \backslash delta\; ,$$

and for $V$ -statistics,

$$\mathbb{P}[\mathrm{\mid }{\widehat{L}}_K^V(Q)-L_K(Q)\mathrm{\mid }\ge 2{\mathrm{\ell }}_{\max }(\frac{n-1}{n}\sqrt{\frac{\log \frac{2}{\delta }}{n}}+\frac{1}{n})]\le \delta \mathrm{.}\backslash mathbb\; \{P\}\; \backslash left[\; |\; \backslash widehat\; \{L\}\; \_\; \{K\}\; ^\; \{V\}\; (Q)\; -\; L\; \_\; \{K\}\; (Q)\; |\; \backslash geq\; 2\; \backslash ell\_\; \{\backslash max\; \}\; \backslash left(\backslash frac\; \{n\; -\; 1\}\{n\}\; \backslash sqrt\; \{\backslash frac\; \{\backslash log\; \backslash frac\; \{2\}\{\backslash delta\}\}\{n\}\}\; +\; \backslash frac\; \{1\}\{n\}\backslash right)\; \backslash right]\; \backslash leq\; \backslash delta\; .$$

We include the proof in the Appendix A for completeness. In addition, in the case when $Q = Q^{\pi}$ , it can be shown that the i.i.d. assumption can be removed; see Appendix A.3.

The key quantity in the concentration inequalities is the upper bound $\ell_{\mathrm{max}}$ . The lemma below shows that it can be estimated easily in practice.

Lemma 3.1. Assume the reward function and the kernel function are bounded, i.e. $\sup_x |r(x)| \leq r_{\max}, \sup_{x,\bar{x}} |K(x,\bar{x})| \leq K_{\max}$ . Then we have

$$\underset{x}{\sup }\mathrm{\mid }{Q}^{\pi }(x)\mathrm{\mid }\le Q_{\max }:=\frac{r_{\max }}{1-\gamma },\backslash sup\; \_\; \{x\}\; |\; Q\; ^\; \{\backslash pi\}\; (x)\; |\; \backslash leq\; Q\; \_\; \{\backslash max\; \}\; :=\; \backslash frac\; \{r\; \_\; \{\backslash max\; \}\}\{1\; -\; \backslash gamma\},$$

$$\underset{\tau ,\bar{\tau }}{\sup }\mathrm{\mid }{\mathrm{\ell }}_{\pi ,Q^{\pi }}(\tau ,\bar{\tau })\mathrm{\mid }\le {\mathrm{\ell }}_{\max }:=\frac{4K_{\max }{r}_{\max }^{}}{(1-\gamma {)}^2}\mathrm{.}\backslash sup\; \_\; \{\backslash tau\; ,\; \backslash bar\; \{\backslash tau\}\}\; |\; \backslash ell\_\; \{\backslash pi\; ,\; Q\; ^\; \{\backslash pi\}\}\; (\backslash tau\; ,\; \backslash bar\; \{\backslash tau\})\; |\; \backslash leq\; \backslash ell\_\; \{\backslash max\; \}\; :=\; \backslash frac\; \{4\; K\; \_\; \{\backslash max\; \}\; r\; \_\; \{\backslash max\; \}\; ^\; \{2\}\}\{(1\; -\; \backslash gamma)\; ^\; \{2\}\}.$$

4. Accountable Off-Policy Evaluation

This section introduces our approach for accountable off-policy evaluation. We start with the main variational framework for providing confidence bounds in Section 4.1, and then discuss post-hoc diagnosis analysis and correction for existing estimators in Section 4.3 and 4.4.

4.1. Variational Confidence Bounds for Off-Policy Evaluation

We consider the problem of providing confidence bounds for the expected reward $\eta^{\pi}$ . To simplify the presentation, we focus on the upper bounds here, and one can derive the lower bounds with almost the same arguments.

Let $\mathcal{F}$ be a function class that contains $Q^{\pi}$ , that is, $Q^{\pi} \in \mathcal{F}$ . Then we can construct an upper bound of $\eta^{\pi}$ by solving the following variational (or functional) optimization on $\mathcal{F}$ :

$$\begin{array}{cccc}& {\eta }^{+}=\underset{Q\in \mathcal{F}}{\max }\{[\eta (Q)],\text{s . t .}{L}_K(Q)\le \lambda \},\mathrm{\forall }\lambda >0.& & \text{(8)}\end{array}\backslash eta^\; \{+\}\; =\; \backslash max\; \_\; \{Q\; \backslash in\; \backslash mathcal\; \{F\}\}\; \backslash left\backslash \{\backslash left[\; \backslash eta\; (Q)\; \backslash right],\; \backslash text\; \{s\; .\; t\; .\}\; L\; \_\; \{K\}\; (Q)\; \backslash leq\; \backslash lambda\; \backslash right\backslash \},\; \backslash quad\; \backslash forall\; \backslash lambda\; >\; 0.\; \backslash tag\; \{8\}$$

Here $\eta^{+}$ is an upper bound of $\eta^{\pi}$ because $Q^{\pi}$ satisfies the condition that $Q^{\pi} \in \mathcal{F}$ and $L_{K}(Q^{\pi}) = 0$ , and hence $\eta^{+} \geq \eta^{\pi}$ follows the definition of the max operator.

In practice, we can not exactly calculate $\eta^{+}$ , as it involves the exact kernel loss $L_{K}(Q)$ and the exact $\eta (Q)$ . However, if we replace $L_{K}(Q)$ and $\eta (Q)$ with proper empirical estimates, we can derive a computationally tractable high-probability upper bound of $\eta^{\pi}$ .

Algorithm 1 Confidence Bounds for Off-Policy Evaluation

Input: Off-policy data $\mathcal{D} = {x_i, r_i, s_i'}{i \leq i \leq n}$ ; maximum reward $r{\max}$ , discounted factor $\gamma$ , positive definite kernel $K$ , RKHS norm $\rho$ , random feature $\Phi(\cdot)$ , failure probability $\delta$ , Monte Carlo sample size $N$ .

Draw ${x_{i,0}}{i = 1}^{N}$ from the initial distribution $\mu_0\times \pi$ ; decide $\lambda_K$ and $\lambda{\eta}$ by concentration inequalities.

Calculate the upper bound $\widehat{\eta}^{+}$ by solving (14), and the lower bound $\widehat{\eta}^{-}$ by (15).

Output: $\widehat{\eta}^{+},\widehat{\eta}^{-}$

Proposition 4.1. Let $\widehat{\eta} (Q)$ and $\widehat{L}K(Q)$ be the estimators of $\eta (Q)$ and $L{K}(Q)$ , such that for $\delta \in (0,1)$

$$\begin{array}{cccc}& \mathbb{P}[\eta \left(Q^{\pi }\right)\le \widehat{\eta }\left(Q^{\pi }\right)+{\lambda }_{\eta }]\le 1-\delta ,& & \text{(9)}\end{array}\backslash mathbb\; \{P\}\; \backslash left[\; \backslash eta\; \backslash left(Q\; ^\; \{\backslash pi\}\backslash right)\; \backslash leq\; \backslash widehat\; \{\backslash eta\}\; \backslash left(Q\; ^\; \{\backslash pi\}\backslash right)\; +\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right]\; \backslash leq\; 1\; -\; \backslash delta\; ,\; \backslash tag\; \{9\}$$

$$\mathbb{P}[{\widehat{L}}_K\left(Q^{\pi }\right)\le {\lambda }_K]\le 1-\delta ,\backslash mathbb\; \{P\}\; \backslash left[\; \backslash widehat\; \{L\}\; \_\; \{K\}\; \backslash left(Q\; ^\; \{\backslash pi\}\backslash right)\; \backslash leq\; \backslash lambda\_\; \{K\}\; \backslash right]\; \backslash leq\; 1\; -\; \backslash delta\; ,$$

where $\lambda_{\eta}$ and $\lambda_{K}$ are two constants. Note that (9) only needs to hold for $Q = Q^{\pi}$ . Assume $Q^{\pi} \in \mathcal{F}$ . Define

$$\begin{array}{cccc}& {\widehat{\eta }}^{+}:=\underset{Q\in \mathcal{F}}{\max }\{[\widehat{\eta }(Q)+{\lambda }_{\eta }],\text{s . t .}{\widehat{L}}_K(Q)\le {\lambda }_K\}\mathrm{.}& & \text{(10)}\end{array}\backslash widehat\; \{\backslash eta\}\; ^\; \{+\}\; :=\; \backslash max\; \_\; \{Q\; \backslash in\; \backslash mathcal\; \{F\}\}\; \backslash left\backslash \{\backslash left[\; \backslash widehat\; \{\backslash eta\}\; (Q)\; +\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right],\; \backslash text\; \{s\; .\; t\; .\}\; \backslash widehat\; \{L\}\; \_\; \{K\}\; (Q)\; \backslash leq\; \backslash lambda\_\; \{K\}\; \backslash right\backslash \}.\; \backslash tag\; \{10\}$$

Then we have $\widehat{\eta}^+ \geq \eta^\pi$ with probability $1 - 2\delta$ .

Proof. From the assumption, with probability at least $1 - 2\delta$ , we have both $\eta(Q^{\pi}) \leq \widehat{\eta}(Q^{\pi}) + \lambda_{\eta}$ and $\widehat{L}_K(Q^{\pi}) \leq \lambda_K$ . In this case, $Q^{\pi}$ belongs to the feasible set of the optimization in (10), i.e., $Q^{\pi} \in \mathcal{F}$ , $\widehat{L}K(Q^{\pi}) \leq \lambda_K$ . Therefore, we have $\eta^{\pi} = \eta(Q^{\pi}) \leq \widehat{\eta}(Q^{\pi}) + \lambda{\eta} \leq \widehat{\eta}^{+}$ by the definition of the max operator.

We can use the kernel Bellman V-statistics in Proposition 3.1 to construct $\widehat{L}K(Q^\pi)$ and $\lambda_K$ . For $\eta(Q^\pi)$ , we use the Monte Carlo estimator $\widehat{\eta}(Q^\pi)$ in (3) and set $\lambda_\eta = Q{\max} \sqrt{2 \log(2/\delta)/N}$ by Hoeffding inequality.

4.2. Optimization in Reproducing Kernel Hilbert Space

To provide tight high confidence bounds, we need to choose $\mathcal{F}$ to be a function space that is both simple and rich enough to contain $Q^{\pi}$ . Here we choose $\mathcal{F}$ to be a reproducing kernel Hilbert space (RKHS) $\mathcal{H}{K_0}$ induced by a positive kernel $K{0}(x,\bar{x})$ . We should distinguish $K_{0}$ with the kernel $K$ used in kernel Bellman loss. Using RKHS allows us to incorporate a rich, infinite dimensional set of functions, and still obtain computationally tractable solution for the optimization in (10).

Proposition 4.2. For RKHS $\mathcal{H}{K_0}$ with kernel $K{0}$ , define

$$\begin{array}{cccc}& \mathcal{F}=\{Q\in {\mathcal{H}}_{K_0}:\mathrm{\parallel }Q{\mathrm{\parallel }}_{{\mathcal{H}}_{K_0}}^2\le \rho \},& & \text{(11)}\end{array}\backslash mathcal\; \{F\}\; =\; \backslash left\backslash \{Q\; \backslash in\; \backslash mathcal\; \{H\}\; \_\; \{K\; \_\; \{0\}\}:\; \backslash |\; Q\; \backslash |\; \_\; \{\backslash mathcal\; \{H\}\; \_\; \{K\; \_\; \{0\}\}\}\; ^\; \{2\}\; \backslash leq\; \backslash rho\; \backslash right\backslash \},\; \backslash tag\; \{11\}$$

where $\rho$ is a positive constant. Using the $\mathcal{F}$ in (11), the optimization solution of (10) can be expressed as

$$\begin{array}{cccc}& Q(\cdot )=\sum _{i=0}^n{\alpha }_i{f}_i(\cdot )\mathrm{.}& & \text{(12)}\end{array}Q\; (\backslash cdot)\; =\; \backslash sum\_\; \{i\; =\; 0\}\; ^\; \{n\}\; \backslash alpha\_\; \{i\}\; f\; \_\; \{i\}\; (\backslash cdot).\; \backslash tag\; \{12\}$$

Here $\alpha = [\alpha_i]{i=0}^n$ are the coefficients decided by the optimization problem, and $f_0(\cdot) = \mathbb{E}{x \sim \mu_0 \times \pi}[K_0(\cdot, x)]$ ,

$$f_i(\cdot )=K_0(\cdot ,x_i)-\gamma {\mathbb{E}}_{a_i^{\mathrm{\prime }}\sim \pi (\cdot \mathrm{\mid }{s}_i^{\mathrm{\prime }})}[K_0(\cdot ,x_i^{\mathrm{\prime }})],i=1,\dots ,n,f\; \_\; \{i\}\; (\backslash cdot)\; =\; K\; \_\; \{0\}\; (\backslash cdot\; ,\; x\; \_\; \{i\})\; -\; \backslash gamma\; \backslash mathbb\; \{E\}\; \_\; \{a\; \_\; \{i\}\; ^\; \{\backslash prime\}\; \backslash sim\; \backslash pi\; (\backslash cdot\; |\; s\; \_\; \{i\}\; ^\; \{\backslash prime\})\}\; [\; K\; \_\; \{0\}\; (\backslash cdot\; ,\; x\; \_\; \{i\}\; ^\; \{\backslash prime\})\; ],\; \backslash quad\; i\; =\; 1,\; \backslash ldots\; ,\; n,$$

where $x_{i}^{\prime} = (s_{i}^{\prime},a_{i}^{\prime})$ , with $s_i^\prime$ the observed next state following $x_{i} = (s_{i},a_{i})$ and $a_{i}^{\prime}$ randomly drawn from $\pi (\cdot |s_i')$ .

In addition, for $Q$ of form (12), $\widehat{\eta}(Q)$ is a linear function of $\alpha$ , and both $\widehat{L}K(Q)$ and $|Q|{\mathcal{H}{K_0}}^2$ are convex quadratic functions of $\alpha$ . In particular, we have $|Q|{\mathcal{H}{K_0}}^2 = \alpha^\top B\alpha$ , where $B = [B{ij}]$ is a $(n+1) \times (n+1)$ matrix with $B_{ij} = \langle f_i, f_j \rangle_{\mathcal{H}_{K_0}}$ . Therefore, the optimization in (10) reduces to an optimization on $\alpha$ with linear objective and convex quadratic constraints,

$$\underset{{\alpha }^{\mathrm{\top }}B\alpha \le {\rho }^2}{\max }\{[c^{\mathrm{\top }}\alpha +{\lambda }_{\eta }],s\mathrm{.}t\mathrm{.}(\alpha -b{)}^{\mathrm{\top }}A(\alpha -b)\le {\lambda }_K\},\backslash max\; \_\; \{\backslash alpha^\; \{\backslash top\}\; B\; \backslash alpha\; \backslash leq\; \backslash rho^\; \{2\}\}\; \backslash left\backslash \{\backslash left[\; c\; ^\; \{\backslash top\}\; \backslash alpha\; +\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right],\; \backslash quad\; s.\; t.\; \backslash quad\; (\backslash alpha\; -\; b)\; ^\; \{\backslash top\}\; A\; (\backslash alpha\; -\; b)\; \backslash leq\; \backslash lambda\_\; \{K\}\; \backslash right\backslash \},$$

where $A, B$ are two $(n + 1) \times (n + 1)$ positive definite matrices and $b, c$ two $(n + 1) \times 1$ vectors, whose definition can be found in Appendix E.

Random Feature Approximation Unfortunately, solving the programming in Proposition 4.2 requires an $O((n + 1)^3)$ time complexity and is too slow when the data size $n$ is very large. We can speed up the computation by using random feature approximations. The idea is that any positive definite kernel can be expressed as $K_{0}(x,y) = \int_{\mathcal{W}}\phi (x,w)\phi (y,w)d\mu (w)$ , where $\phi (x,w)$ denotes a feature map indexed by a parameter $w$ in some space $\mathcal{W}$ and $\mu$ is a measure on $\mathcal{W}$ . A typical example of this is the random Fourier expansion of stationary kernels by Bochner's theorem (Rahimi & Recht, 2007), in which $\phi (x,w) = \cos (w^{\top}[x,1])$ .

To speed up the computation, we draw i.i.d. ${w_i}_{i=1}^n$ from $\mu$ and take

$$\begin{array}{cccc}& K_0(x,y)=\frac{1}{m}\sum _{k=1}^m\varphi (x,w_k)\varphi (y,w_k)\mathrm{.}& & \text{(13)}\end{array}K\; \_\; \{0\}\; (x,\; y)\; =\; \backslash frac\; \{1\}\{m\}\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{m\}\; \backslash phi\; (x,\; w\; \_\; \{k\})\; \backslash phi\; (y,\; w\; \_\; \{k\}).\; \backslash tag\; \{13\}$$

Then one can show that any function $Q$ in the RKHS of $K_{0}$ in (13) can be expressed as

$$Q(x)={\theta }^{\mathrm{\top }}\mathrm{\Phi }(x),Q\; (x)\; =\; \backslash theta^\; \{\backslash top\}\; \backslash Phi\; (x),$$

Algorithm 2 Post-hoc Diagnosis for Existing Estimators

Input: Off-policy data $\mathcal{D} = {x_i, r_i, s_i'}{i \leq i \leq n}$ ; maximum reward $r{\mathrm{max}}$ , discounted factor $\gamma$ , positive definite kernel $K$ , RKHS norm $\rho$ , random feature $\Phi(\cdot)$ , fail probability $\delta$ , Monte Carlo sample size $N$ , existing estimation $\widehat{Q}$ .

Draw ${x_0^i}{i = 1}^N$ from the initial distribution $\mu_0\times \pi$ ; Decide $\lambda{\eta}$ and $\lambda_{K}$ by concentration inequalities.

Calculate the upper bound $\widehat{\eta}^{+}$ by solving (16), the lower bound $\widehat{\eta}^{-}$ by (17), and the debias function $Q_{\mathrm{debias}}$ by (18).

Output: The upper and lower bounds $\widehat{\eta}^{+}$ , $\widehat{\eta}^{-}$ , and $Q_{\mathrm{debias}}(x) = \theta^{\top}\Phi(x)$ .

where $\Phi (x) = [\phi (x,w_i)]{i = 1}^m$ with $| Q|{\mathcal{H}_{K_0}}^2 = m| \theta | _2^2$ where $| \cdot | 2$ is the typical $L{2}$ norm on $\mathbb{R}^m$

Given the off-policy dataset $\mathcal{D} = {x_i, r_i, s_i'}$ with $x_i = (s_i, a_i)$ , and an i.i.d. sample ${x_{i,0}}{i=1}^N$ with $x{i,0} = (s_{i,0}, a_{i,0})$ from the initial distribution $\mu_0 \times \pi$ . The optimization in (10) can be shown to reduce to

$$\begin{array}{cccc}& {\widehat{\eta }}^{+}=\underset{\mathrm{\parallel }\theta {\mathrm{\parallel }}_2^2\le \rho \mathrm{/}m}{\max }\{[c_0^{\mathrm{\top }}\theta +{\lambda }_{\eta }],& & \text{(14)}\end{array}\backslash widehat\; \{\backslash eta\}\; ^\; \{+\}\; =\; \backslash max\; \_\; \{\backslash |\; \backslash theta\; \backslash |\; \_\; \{2\}\; ^\; \{2\}\; \backslash leq\; \backslash rho\; /\; m\}\; \backslash left\backslash \{\backslash left[\; c\; \_\; \{0\}\; ^\; \{\backslash top\}\; \backslash theta\; +\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right],\; \backslash right.\; \backslash tag\; \{14\}$$

s.t. $(Z\theta - v)^\top M(Z\theta - v) \leq \lambda_K}$ ,

where $v = [r_i]{i=1}^n \in \mathbb{R}^{n \times 1}$ , $M \in \mathbb{R}^{n \times n}$ with $M{ij} = [K(x_i, x_j) / n^2]{ij}$ , $c_0 = \sum{i=1}^N \Phi(x_{i,0}) / N \in \mathbb{R}^{m \times 1}$ , and

$$Z={[\mathrm{\Phi }(x_i)-\gamma {\mathbb{E}}_{a_i^{\mathrm{\prime }}\sim \pi (\cdot \mathrm{\mid }{s}_i^{\mathrm{\prime }})}[\mathrm{\Phi }(x_i^{\mathrm{\prime }})]]}_{i=1}^n\in {\mathbb{R}}^{n\times m},Z\; =\; \backslash left[\; \backslash Phi\; (x\; \_\; \{i\})\; -\; \backslash gamma\; \backslash mathbb\; \{E\}\; \_\; \{a\; \_\; \{i\}\; ^\; \{\backslash prime\}\; \backslash sim\; \backslash pi\; (\backslash cdot\; |\; s\; \_\; \{i\}\; ^\; \{\backslash prime\})\}\; \backslash left[\; \backslash Phi\; (x\; \_\; \{i\}\; ^\; \{\backslash prime\})\; \backslash right]\; \backslash right]\; \_\; \{i\; =\; 1\}\; ^\; \{n\}\; \backslash in\; \backslash mathbb\; \{R\}\; ^\; \{n\; \backslash times\; m\},$$

with $x_{i}^{\prime} = [s_{i}^{\prime}, a_{i}^{\prime}]$ . The expectation in $Z$ can be approximated by Monte Carlo sampling from $\pi(\cdot | s_{i}^{\prime})$ .

Similarly, we can get the lower confidence bounds via

$$\begin{array}{cccc}& {\widehat{\eta }}^{-}=\underset{\mathrm{\parallel }\theta {\mathrm{\parallel }}_2^2\le \rho \mathrm{/}m}{\min }\{[c_0^{\mathrm{\top }}\theta -{\lambda }_{\eta }],& & \text{(15)}\end{array}\backslash widehat\; \{\backslash eta\}\; ^\; \{-\}\; =\; \backslash min\; \_\; \{\backslash |\; \backslash theta\; \backslash |\; \_\; \{2\}\; ^\; \{2\}\; \backslash leq\; \backslash rho\; /\; m\}\; \backslash left\backslash \{\backslash left[\; c\; \_\; \{0\}\; ^\; \{\backslash top\}\; \backslash theta\; -\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right],\; \backslash right.\; \backslash tag\; \{15\}$$

s.t. $(Z\theta - v)^\top M(Z\theta - v) \leq \lambda_K}$ .

Compared with the programming in Proposition (4.2), the optimization problems in (14)-(15) have lower dimensions and is hence much faster when $m \ll n$ , since the dimension of $\theta$ is $m$ , while that of $\alpha$ is $n + 1$ . We describe the detailed procedure for obtaining the upper and lower confidence bounds in Algorithm 1.

4.3. Post-hoc Confidence Bound of Existing Estimators

We extend our method to provide post-hoc confidence bounds around existing estimators provided by users. Given an existing estimator $\widehat{Q}$ of the Q-function, we denote by $Q_{\mathrm{res}}\coloneqq Q^{\pi} - \widehat{Q}$ the difference between the ground truth $Q^{\pi}$ and the prior estimate $\widehat{Q}$ . Assuming $Q_{\mathrm{res}}$ belongs to $\mathcal{F}$ , we obtain an upper bound by

$$\underset{Q_{\text{r e s}}\in \mathcal{F}}{\max }\{[\widehat{\eta }(Q_{\text{r e s}}+\widehat{Q})+{\lambda }_{\eta }],\text{s . t .}{\widehat{L}}_K(\widehat{Q}+Q_{\text{r e s}})\le {\lambda }_K\}\mathrm{.}\backslash max\; \_\; \{Q\; \_\; \{\backslash text\; \{r\; e\; s\}\}\; \backslash in\; \backslash mathcal\; \{F\}\}\; \backslash left\backslash \{\backslash left[\; \backslash widehat\; \{\backslash eta\}\; \backslash left(Q\; \_\; \{\backslash text\; \{r\; e\; s\}\}\; +\; \backslash widehat\; \{Q\}\backslash right)\; +\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right],\; \backslash text\; \{s\; .\; t\; .\}\; \backslash widehat\; \{L\}\; \_\; \{K\}\; \backslash left(\backslash widehat\; \{Q\}\; +\; Q\; \_\; \{\backslash text\; \{r\; e\; s\}\}\backslash right)\; \backslash leq\; \backslash lambda\_\; \{K\}\; \backslash right\backslash \}.$$

This can be viewed as a special case of (10) but with the function space anchored around the existing estimator $\widehat{Q}$ .

Similar to the earlier case, in the case of random feature approximation, the optimization reduces to

$${\widehat{\eta }}^{+}=\underset{\mathrm{\parallel }\theta {\mathrm{\parallel }}_2^2\le \rho \mathrm{/}m}{\max }\{[\widehat{\eta }(\widehat{Q})+c_0^{\mathrm{\top }}\theta +{\lambda }_{\eta }],\backslash widehat\; \{\backslash eta\}\; ^\; \{+\}\; =\; \backslash max\; \_\; \{\backslash |\; \backslash theta\; \backslash |\; \_\; \{2\}\; ^\; \{2\}\; \backslash leq\; \backslash rho\; /\; m\}\; \backslash left\backslash \{\backslash left[\; \backslash widehat\; \{\backslash eta\}\; (\backslash widehat\; \{Q\})\; +\; c\; \_\; \{0\}\; ^\; \{\backslash top\}\; \backslash theta\; +\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right]\; \backslash right.,$$

s.t. $(Z\theta -\zeta)^{\top}M(Z\theta -\zeta)\leq \lambda_{K}}$ (16)

where $c_{0},Z,M$ are defined as before, $\widehat{\eta} (\widehat{Q}) = \sum_{i = 1}^{N}\widehat{Q} (x_{i,0}) / N,$ and $\zeta \in \mathbb{R}^{n\times 1}$ is defined to be the TD error vector of $\widehat{Q} (x)$ evaluated at the dataset $\mathcal{D}$ ; that is, $\zeta = [\zeta_i]_{i = 1}^n$ with

$${\zeta }_i=r_i+\gamma {\mathbb{E}}_{a_i^{\mathrm{\prime }}\sim \pi (\cdot \mathrm{\mid }{s}_i^{\mathrm{\prime }})}\widehat{Q}(s_i^{\mathrm{\prime }},a_i^{\mathrm{\prime }})-\widehat{Q}(x_i)\mathrm{.}\backslash zeta\_\; \{i\}\; =\; r\; \_\; \{i\}\; +\; \backslash gamma\; \backslash mathbb\; \{E\}\; \_\; \{a\; \_\; \{i\}\; ^\; \{\backslash prime\}\; \backslash sim\; \backslash pi\; (·\; |\; s\; \_\; \{i\}\; ^\; \{\backslash prime\})\}\; \backslash widehat\; \{Q\}\; \backslash left(s\; \_\; \{i\}\; ^\; \{\backslash prime\},\; a\; \_\; \{i\}\; ^\; \{\backslash prime\}\backslash right)\; -\; \backslash widehat\; \{Q\}\; (x\; \_\; \{i\}).$$

The post-hoc lower bound follows a similar form:

$${\widehat{\eta }}^{-}=\underset{\mathrm{\parallel }\theta {\mathrm{\parallel }}_2^2\le \rho \mathrm{/}m}{\min }\{[\widehat{\eta }(\widehat{Q})+c_0^{\mathrm{\top }}\theta -{\lambda }_{\eta }],\backslash widehat\; \{\backslash eta\}\; ^\; \{-\}\; =\; \backslash min\; \_\; \{\backslash |\; \backslash theta\; \backslash |\; \_\; \{2\}\; ^\; \{2\}\; \backslash leq\; \backslash rho\; /\; m\}\; \backslash left\backslash \{\backslash left[\; \backslash widehat\; \{\backslash eta\}\; (\backslash widehat\; \{Q\})\; +\; c\; \_\; \{0\}\; ^\; \{\backslash top\}\; \backslash theta\; -\; \backslash lambda\_\; \{\backslash eta\}\; \backslash right]\; \backslash right.,$$

s.t. $(Z\theta -\zeta)^{\top}M(Z\theta -\zeta)\leq \lambda_{K}}$ (17)

4.4. Post-hoc Correction of Existing Estimators

In addition to providing confidence bounds around the existing estimator $\widehat{Q}$ , we may want to further correct $\widehat{Q}$ when $\widehat{Q}$ is identified to admit large error. The post-hoc correction should ensure the corrected estimation falls into the confidence bounds provided earlier, while keeping the information in $\widehat{Q}$ as much as possible.

Our idea is to correct $\widehat{Q}$ by adding a debiasing term $Q_{\mathrm{debias}}$ such that $\widehat{L}K^V (\widehat{Q} +Q{\mathrm{debias}})\leq \lambda_K$ , while keeping $Q_{\mathrm{debias}}$ as small as possible. This is framed into

$$\underset{Q_{\text{d e b i a s}}\in \mathcal{F}}{\min }\{\mathrm{\parallel }{Q}_{\text{d e b i a s}}{\mathrm{\parallel }}_{{\mathcal{H}}_{K_0}}^2\text{s . t .}{\widehat{L}}_K(Q+Q_{\text{d e b i a s}})\le {\lambda }_K\}\mathrm{.}\backslash min\; \_\; \{Q\; \_\; \{\backslash text\; \{d\; e\; b\; i\; a\; s\}\}\; \backslash in\; \backslash mathcal\; \{F\}\}\; \backslash left\backslash \{\backslash |\; Q\; \_\; \{\backslash text\; \{d\; e\; b\; i\; a\; s\}\}\; \backslash |\; \_\; \{\backslash mathcal\; \{H\}\; \_\; \{K\; \_\; \{0\}\}\}\; ^\; \{2\}\; \backslash quad\; \backslash text\; \{s\; .\; t\; .\}\; \backslash widehat\; \{L\}\; \_\; \{K\}\; (Q\; +\; Q\; \_\; \{\backslash text\; \{d\; e\; b\; i\; a\; s\}\})\; \backslash leq\; \backslash lambda\_\; \{K\}\; \backslash right\backslash \}.$$

We should distinguish $Q_{\mathrm{debias}}$ with the $Q_{\mathrm{res}}$ in Section 4.3, and $Q_{\mathrm{res}}$ can not be used for the debiasing purpose because it is designed to give an extreme estimation (for providing the bound), rather than a minimum correction.

In the case of random features approximation, the optimization reduces to

$${\theta }^{\star }=\underset{\theta }{\mathrm{argmin}}\{\mathrm{\parallel }\theta {\mathrm{\parallel }}_2^2,\backslash theta^\{\backslash star\}\; =\; \backslash operatorname\; *\{arg\; min\}\_\{\backslash theta\}\backslash left\backslash \{\backslash |\; \backslash theta\; \backslash |\_\{2\}^\{2\},\; \backslash right.$$

s.t. $(Z\theta -\zeta)^{\top}M(Z\theta -\zeta)\leq \lambda_{K}}$ (18)

(b) $1 - \delta$

(a) Number of transitions
(e) Number of transitions
Figure 1. Results on Inverted-Pendulum (a)-(d) and Puck-Mountain (e)-(h). The plots show the confidence bounds on the average discounted reward as we vary the number of transitions $n$ in (a) & (e), failure probability $\delta$ in (b) & (f), the temperature parameter $\tau$ of the behavior policy in (c) & (g), and the number of features in (d) & (h). The default parameters (when not varied) are: discounted factor $\gamma = 0.95$ ; horizon length $T = 50$ for Inverted-Pendulum and $T = 100$ for Puck-Mountain; number of episodes 20; failure probability $\delta = 0.10$ ; temperature of the behavior policy $\tau = 1$ ; and the feature dimension 10.

(f) $1 - \delta$

(c) Temperature
(g) Temperature

(d) Number of features
(h) Number of features

and $Q_{\mathrm{debias}}(x) = \theta^{\star}{}^{\top}\Phi (x)$ . If the existing estimator $\widehat{Q}$ already satisfies $\widehat{L}_K^V (\widehat{Q})\leq \lambda_K$ , this provides a zero correction (i.e., $\theta^{\star} = 0$ ), since the estimator is already sufficiently accurate. This procedure is summarized in Algorithm 2.

5. Experiments

We evaluate the proposed algorithms in Section 4 on two continuous control tasks: Inverted-Pendulum and Puck-Mountain. Details of the tasks are in Appendix D.

For all of our experiments, we use a Gaussian RBF kernel $K(x_{i},x_{j}) = \exp (-||x_{i} - x_{j}||_{2}^{2} / h^{2})$ for evaluating the kernel Bellman statistics, with a bandwidth selected from a separate batch of training data, and $V$ -statistics to calculate Equation (5) given a set of empirical data.

To construct the behavior and target policies, we first train an optimal Q-function using deep Q-learning, and use its softmax functions as policies, and set the temperature to be higher for the behavior policies to encourage exploration. Note that our algorithms do not require to know the behavior policies since they are behavior-agnostic. A description for how to construct policies is in Appendix D.

5.1. Confidence Bounds for Off-Policy Evaluation

We test Algorithm 1 on the two continuous control tasks, Inverted-Pendulum (Figure 1(a)-(d)) and Puck-Mountain (Figure 1(e)-(h)). We solve the convex optimization in Algorithm 1 using CVXPY (Diamond & Boyd, 2016; Agrawal et al., 2018), which gives us the upper and lower bound $\widehat{\eta}^{+}$ and $\widehat{\eta}^{-}$ . The results are reported w.r.t. the average discounted reward defined by $\eta^{\pi} / (1 - \gamma)$ . The results are averaged over 50 random trials.

We use two types of feature maps $\Phi (\cdot)$ to parameterize the state-action value function $Q(x)\coloneqq \theta^{\top}\Phi (x)$ ..

1. Random Fourier features: $\Phi (x)\coloneqq [\cos (\mu_i^\top x + b_i)]{i = 1}^m$ where $\mu{i}\sim \mathcal{N}(0,\frac{1}{h_{0}^{2}}\mathrm{I}),b_{i}\sim \mathrm{Uniform}[0,2\pi ]$ , and $h_0$ is a bandwidth parameter.
2. Neural features. We use a small neural network to parameterize $Q$ function, and learn the $Q$ function by minimizing the kernel loss on the training dataset, and set $\Phi(\cdot)$ by selecting a set of neural features (the neural feature map before the last linear layer) on the validation set.

Figure 1(a)-(d) demonstrate the evaluation results on Inverted-Pendulum. From Figure 1(a) & (b) we can see that, as we increase the number of transitions, or increase the failure probability $\delta$ , the confidence bounds become

Reward by original $\widehat{Q}_1$ - Reward by correction o

• Reward by original $\widehat{Q}_2$
• Reward by correction of $\widehat{Q}_2$

True Reward $\eta^{\pi}$ Confidence Bounds

(d) Diagnosis for $\widehat{Q}_1$ with different $\delta$

(e) Diagnosis for $\widehat{Q}_2$ with different $\delta$

(c) Norm of $Q_{\mathrm{debias}}$
(f) Norm of $Q_{\mathrm{debias}}$
Figure 2. Post-hoc diagnosis on Inverted-Pendulum. In figure (a)-(c), we vary the number of transitions with fixed $\delta = 0.10$ . In figure (d)-(f), we fixed number of transitions $n = 500$ and vary the failure probability $\delta$ . The default parameters are: discounted factor $\gamma = 0.95$ , the horizon length $T = 50$ , number of transitions $n = 500$ , failure probability $\delta = 0.10$ , temperature of the behavior policy $\tau = 1$ , and the feature dimension 10.

tighter since $\lambda_{K}$ becomes smaller. However, it still covers the ground truth $\eta^{\pi}$ , which indicates that our method gives a reasonable confidence interval for off-policy estimation.

In Figure 1(c), we investigate the performance of our method when we vary a temperature parameter of the behavior policy that controls the concentration of the probability mass. The confidence bounds do not change significantly with different temperature of behavior policies. In Figure 1(d), we study the performance of our algorithms with different number of features (including both random Fourier and neural features), which shows that we can get a tight confidence interval when the number of features is small. This is because when decreasing the number of features, the Q function is constrained in a lower dimensional function space and hence gives a tighter confidence interval. However, decreasing the number of features also increases the risk of model misspecification. We also test our method on Puck-Mountain, and report the results in Figure 1 (e)-(h), which show similar results as we observe in Inverted-Pendulum.

5.2. Post-hoc Diagnosis for Existing Estimators

We implement Algorithm 2 to provide post-hoc diagnosis for existing $Q$ estimators, and test it on Inverted-Pendulum (Figure 2) and Puck-Mountain (Figure 3 in the Appendix D). Figure 2 (a)-(f) show the diagnosis results for two different state-action value function estimators $(\widehat{Q}_1$ and $\widehat{Q}_2)$ on Inverted-Pendulum, which we learn with kernel Bellmen statistics using different number of optimization steps. Here $\widehat{Q}_1$ is a relatively accurate estimator while the estimation of $Q_2$ has larger bias.

Figure 2 (a)-(c) show that as we increase the number of transitions, the norm of the debias term $Q_{\mathrm{debias}}$ keeps zero when $\widehat{Q}_1$ and $\widehat{Q}_2$ are inside the confidence interval. Meanwhile, when $\widehat{Q}_1$ and $\widehat{Q}_2$ can not provide an accurate estimation (falling outside of the confidence interval), our algorithm gives a good post-hoc correction to ensure the corrected estimation are inside the confidence interval. As we can see, such post-hoc diagnosis provides both confidence bounds and correction for existing estimators, which can be useful in real-world applications.

Figure 2 (d)-(f) demonstrate the performance of our algo

rithms when we change the failure probability $\delta$ . Again, the empirical results show that our algorithms can provide meaningful post-hoc diagnosis.

We also test our method on Puck-Mountain, following the same procedure as on Inverted-Pendulum, which shows similar results as we find in Inverted-Pendulum.

6. Conclusion

In this paper, we develop a new variational framework for constructing confidence bounds for off-policy evaluation and extend it to perform post-hoc diagnosis on existing estimators. In future work, we will leverage our framework to develop safe and efficient policy optimization and exploration algorithms based on the kernel Bellman statistics.

Acknowledgment

This work is supported in part by NSF CRII 1830161 and NSF CAREER 1846421.

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