Causal Reinforcement Learning and Off-Policy Evaluation

Reinforcement Learning (RL) provides a powerful framework for sequential decision-making under uncertainty, where an agent learns to optimize its actions based on interactions with an environment. Such sequential decision-making is central to dynamic treatment regimes and temporal systems. However, standard RL approaches often rely heavily on correlational patterns learned from interaction data. Integrating causal inference principles offers a path towards more reliable policy evaluation, understanding agent behavior, and developing agents that generalize better across changing conditions. Here, the intersection of causal inference and RL is examined, with a specific focus on the challenges and causal solutions for Off-Policy Evaluation (OPE).

Causal Perspective on Reinforcement Learning

In a typical Markov Decision Process (MDP) setting, an agent interacts with an environment over discrete time steps $t$. At each step, the agent observes a state $s_t \in \mathcal{S}$, takes an action $a_t \in \mathcal{A}$ according to its policy $\pi(a_t | s_t)$, receives a reward $r_t = R(s_t, a_t)$, and transitions to a new state $s_{t+1} \sim P(s_{t+1} | s_t, a_t)$. The goal is often to find a policy $\pi$ that maximizes the expected cumulative discounted reward, or return $G_t = \sum_{k=0}^{\infty} \gamma^k r_{t+k}$, where $\gamma \in [0, 1)$ is the discount factor.

Causal inference introduces several valuable perspectives:

1. Environment Dynamics as Causal Mechanisms: The transition function $P(s_{t+1} | s_t, a_t)$ and reward function $R(s_t, a_t)$ can be viewed as structural causal mechanisms. Modeling them using causal frameworks (like SCMs adapted for time) rather than purely predictive models can improve understanding and extrapolation to environments where some mechanisms might change.
2. Actions as Interventions: Choosing an action $a_t$ is an intervention on the system. The agent's policy $\pi$ dictates a sequence of interventions based on observed states.
3. Off-Policy Evaluation as a Causal Question: OPE aims to estimate the expected return of an evaluation policy $\pi_e$ using data generated by a different behavior policy $\pi_b$. This is fundamentally a causal question: "What would the expected return $E[G_0^{\pi_e}]$ be if we intervened to deploy policy $\pi_e$?" We are asking for the expected outcome under an intervention, $E[Y^{do(\pi=\pi_e)}]$, where $Y$ represents the return.

The Challenge of Confounding in Off-Policy Evaluation

A significant challenge in OPE arises from confounding. The actions $a_t$ chosen by the behavior policy $\pi_b$ are based on the state $s_t$. This state $s_t$ not only influences the action $a_t$ but also potentially influences future states $s_{t+1}, s_{t+2}, \dots$ and rewards $r_t, r_{t+1}, \dots$ through the environment dynamics, independent of the action's direct effect. If we naively evaluate $\pi_e$ using data from $\pi_b$, we might incorrectly attribute outcomes caused by the states encountered under $\pi_b$ to the actions $\pi_e$ would have taken in those states.

Consider the following simplified causal graph representing one step of an MDP:

A simplified causal graph for one time step in an MDP. The state $S_t$ influences the action $A_t$ (via the policy $\pi_b$) and also directly influences the reward $R_t$ and next state $S_{t+1}$ (via environment dynamics). This makes $S_t$ a confounder for the effect of $A_t$ on future outcomes when evaluating a different policy $\pi_e$.

Standard Importance Sampling (IS) attempts to correct for the distribution shift between policies by re-weighting trajectories:

$$\hat{V}_{IS}^{\pi_e} = \frac{1}{N} \sum_{i=1}^{N} \left( \prod_{t=0}^{T-1} \frac{\pi_e(a_{i,t} | s_{i,t})}{\pi_b(a_{i,t} | s_{i,t})} \right) G_{i,0}$$

where $i$ indexes trajectories and $G_{i,0}$ is the return of trajectory $i$. While theoretically unbiased under certain assumptions (including positivity: $\pi_b(a|s) > 0$ whenever $\pi_e(a|s) > 0$), IS often suffers from extremely high variance, especially for long trajectories, as the product of importance ratios can explode or vanish.

Causal Methods for Off-Policy Evaluation

Causal inference provides techniques to develop OPE estimators, often by using models of the environment or value functions alongside policy models.

Estimation

Drawing parallels with methods in static treatment effect estimation (related to Double Machine Learning discussed in Chapter 3), OPE estimators can be constructed that combine importance weighting with a model of expected future returns (e.g., a Q-function $Q^{\pi_e}(s, a)$ or value function $V^{\pi_e}(s)$ estimated under the evaluation policy). A common form of the Doubly Robust (DR) estimator for the expected return at the initial state $s_0$ is:

$$\hat{V}_{DR}^{\pi_e} = \frac{1}{N} \sum_{i=1}^{N} \sum_{t=0}^{T-1} \gamma^t \left( \rho_{i, 0:t-1} r_{i,t} - (\rho_{i, 0:t} \hat{Q}^{\pi_e}(s_{i,t}, a_{i,t}) - \rho_{i, 0:t-1} \hat{V}^{\pi_e}(s_{i,t})) \right)$$

(Simplified forms exist). Here, $\rho_{i, 0:t} = \prod_{k=0}^{t} \frac{\pi_e(a_{i,k} | s_{i,k})}{\pi_b(a_{i,k} | s_{i,k})}$ is the cumulative importance ratio up to time $t$, and $\hat{Q}^{\pi_e}, \hat{V}^{\pi_e}$ are estimated value functions for the evaluation policy.

The important property is double robustness: the estimator is consistent (converges to the true value $E[G_0^{\pi_e}]$) if either the policy ratio model (implicit in $\rho$) or the value function model ($\hat{Q}^{\pi_e}, \hat{V}^{\pi_e}$) is correctly specified, not necessarily both. This often leads to lower variance than pure IS, especially if the value function model captures significant aspects of the environment dynamics.

Connections to Marginal Structural Models (MSMs)

The techniques used in OPE, particularly DR estimators, bear resemblance to methods like MSMs used in epidemiology and econometrics to estimate the causal effects of time-varying treatments in the presence of time-varying confounders. In the RL context, actions $a_t$ are the time-varying treatments, states $s_t$ include time-varying confounders (influenced by past actions and influencing future actions and outcomes), and the return $G_t$ is the outcome. Applying MSM principles involves using inverse probability weighting (similar to IS) or G-computation (similar to model-based value estimation) or combinations (like DR) to adjust for the confounding effects of the state history. This requires assumptions analogous to sequential ignorability: $Y^{a_0, ..., a_T} \perp A_t | H_t$ for all $t$, where $H_t = (S_0, A_0, R_0, ..., S_t)$ is the history up to time $t$ and $Y$ is the potential outcome (return) under a sequence of actions.

Handling Unobserved Confounding

Standard OPE methods assume all relevant state information $s_t$ needed to achieve conditional independence (sequential ignorability) is observed. If critical state components are unobserved (latent state variables), these methods can yield biased estimates. This scenario mirrors the unobserved confounding problem discussed in Chapter 4.

Advanced techniques are being explored:

• Proximal Causal Inference for RL: Adapting proximal methods, which use proxy variables related to the unobserved confounders (as covered in Chapter 4), for the RL setting. This involves finding variables that satisfy specific conditional independence relationships, allowing identification and estimation even with hidden state confounding.
• Sensitivity Analysis: Quantifying how OPE estimates might change if unobserved confounding were present.

Causal Inference for Policy Learning and Generalization

Evaluating existing policies, causal inference can inform the learning process itself:

• Causal Model-Based RL: Learning an explicit causal model of the environment's transitions $P(s_{t+1}|s_t, do(a_t))$ and rewards $R(s_t, do(a_t))$ can offer advantages over purely predictive models. A causal model aims to capture the underlying mechanisms, potentially allowing for:
  • More effective planning.
  • Better generalization to unseen states or slightly different environments where some causal modules remain invariant.
  • Reasoning about interventions on the environment itself.
• Causal Discovery in RL: Applying causal discovery algorithms (from Chapter 2 and the section on Time Series Causal Discovery) to the stream of $(s_t, a_t, r_t, s_{t+1})$ data generated during interaction. Discovering the causal graph governing the environment can guide model building and potentially identify instrumental variables or other structures useful for dealing with confounding.
• Transfer Learning: Causal models facilitate transfer learning. If we know which causal mechanisms change between a source task and a target task, we can reuse the invariant parts of the learned model, potentially speeding up learning in the target environment.

Practical Notes

Integrating causal inference into RL involves several practical considerations:

• Assumption Validity: Causal OPE methods rely on assumptions like sequential ignorability (or alternatives if using methods like proximal inference) and positivity. Validating these assumptions is difficult and often requires domain knowledge. Sensitivity analysis is important.
• Model Specification: Doubly robust methods require estimating either policy ratios or value functions. The performance depends heavily on the quality of these models, especially in high-dimensional state/action spaces (linking back to Chapter 3).
• Computational Cost: Causal methods, particularly those involving complex models or discovery algorithms, can be computationally intensive.
• Software Implementation: While standard RL libraries are mature, libraries focused on causal RL and robust OPE are less common, though research implementations exist (e.g., within frameworks like RLlib or independently).

In summary, applying a causal lens to RL, especially for OPE, moves from correlational pattern matching towards understanding the effects of interventions (actions) in dynamic systems. While standard IS provides a basic correction, methods like Doubly Robust estimation offer significant variance reduction by incorporating environment or value models. Addressing unobserved state confounding and using causal models for policy learning represent active and important areas of research for building more reliable and adaptable agents.