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Efficiently Breaking the Curse of Horizon in Off-Policy Evaluation with Double Reinforcement Learning

Nathan Kallus, Masatoshi Uehara

arXiv:1909.05850v6stat.MLcs.LGmath.OC

TL;DR

Long-horizon off-policy evaluation is difficult because overlap between behavior and target policies diminishes. The paper derives efficiency bounds under different process structures and proposes a Double Reinforcement Learning estimator for time-invariant MDPs. Its bounds distinguish near-on-policy feasibility in time-varying models from consistent evaluation under MDP structure, including from one trajectory.

  • Problem

    As horizons grow, overlap between behavior and target policies diminishes exponentially, making off-policy evaluation increasingly difficult.

    Source: §1 · pp. 1–2

  • Method

    The paper derives efficiency bounds for non-Markov, time-varying Markov, and time-invariant Markov models, then proposes a Double Reinforcement Learning estimator using stationary density ratios and q-functions.

    Source: §1 · p. 3

  • Results

    The bounds show near-on-policy consistency is feasible in non-Markov and time-varying models, while time-invariant MDPs permit consistent evaluation from one trajectory regardless of density-ratio growth.

    Source: §1 · p. 3, §1.4 · p. 12

  • Takeaways & Limitations

    The proposed estimator achieves the MDP efficiency bound while retaining double robustness, providing an efficient way to leverage time-invariant structure.

    Source: §9 · p. 42

    Without stationarity and mixing, dependence within a trajectory makes the MSE rate difficult to characterize theoretically.

    Source: Theorem 11 (Efficiency of ˆρDRL(M3)… · p. 29

Abstract

from arXiv · show

Off-policy evaluation (OPE) in reinforcement learning is notoriously difficult in long- and infinite-horizon settings due to diminishing overlap between behavior and target policies. In this paper, we study the role of Markovian and time-invariant structure in efficient OPE. We first derive the efficiency bounds for OPE when one assumes each of these structures. This precisely characterizes the curse of horizon: in time-variant processes, OPE is only feasible in the near-on-policy setting, where behavior and target policies are sufficiently similar. But, in time-invariant Markov decision processes, our bounds show that truly-off-policy evaluation is feasible, even with only just one dependent trajectory, and provide the limits of how well we could hope to do. We develop a new estimator based on Double Reinforcement Learning (DRL) that leverages this structure for OPE using the efficient influence function we derive. Our DRL estimator simultaneously uses estimated stationary density ratios and $q$-functions and remains efficient when both are estimated at slow, nonparametric rates and remains consistent when either is estimated consistently. We investigate these properties and the performance benefits of leveraging the problem structure for more efficient OPE.

1. Introduction

The introduction and literature review frame long-horizon OPE as a challenge shaped by diminishing target–behavior overlap and examine how Markovian and time-invariant structure bears on efficiency. They motivate establishing efficiency limits and developing an estimator that exploits this structure, while contrasting it with approaches that do not and discussing evaluation under M3.

1. Introduction

  • With long horizons, overlap can decay exponentially; the bounds distinguish near-on-policy feasibility in NMDP/TMDP from truly off-policy feasibility in time-invariant MDPs.The paper proposes double reinforcement learning that jointly learns average visitation distributions and q-functions, achieving the MDP efficiency bound under certain mixing conditions.Source: §1 · pp. 2–3

1.1. Organization

  • The paper’s roadmap proceeds from model definitions and efficiency bounds to estimator analyses, single-trajectory estimation methods, numerical experiments, and conclusions.It notes that standard DR and DRL extensions are efficient in NMDP/TMDP but not under the MDP model, motivating a separate MDP estimator.Source: §1.1 · p. 4

1.2. Problem Setup and Notation

  • The setup defines the target-policy discounted cumulative reward from data generated by a behavior policy, distinguishing NMDP, time-varying MDP, and time-invariant MDP structures.It considers transition-sampling and trajectory-sampling settings, with dependent transitions permitted in the latter.Source: §1.2 · pp. 5–6

1.3. Efficiency Bounds

  • The efficiency bounds characterize the best-possible asymptotic MSE and provide limits for evaluating estimators across the three process models.The paper derives bounds in identifying iid settings, then shows they can be achieved in those settings and more complex settings such as a single growing trajectory.Source: §1.3 · p. 8

1.4. Summary of Literature on OPE

  • Although RL commonly assumes time-invariant MDPs, prior OPE methods largely had not exploited that model’s structure to improve estimation efficiency.OPE also corresponds to estimating a dynamic policy’s total treatment effect under sequential ignorability.Source: §1.4 · pp. 9–10
  • Even with full trajectories, importance sampling can have infinite variance in infinite-horizon problems because of diminishing overlap, the curse of horizon (Liu et al. 2018a).The paper says its results in M1 and M2 characterize more precisely when this curse applies.Source: §1.4 · p. 11
  • Doubly robust estimation is efficient under M1 in finite horizons, while infinite-horizon efficiency requires additional conditions (Kallus and Uehara 2020a).The paper derives those additional conditions for M1 in Section 3.Source: §1.4 · p. 11
  • Under M2, IS and DR estimators fail to be efficient, while DRL is efficient with nuisance estimates converging as slowly as N^-1/4; infinite-horizon efficiency requires additional conditions.The cited rate allows black-box machine-learning methods; the paper derives the infinite-horizon conditions in Section 3.Source: §1.4 · pp. 11–12
  • In M3, consistent evaluation is possible from a single trajectory with known initial distribution, regardless of growth in μ_t; M2 estimators can fail when μ_t explodes.The review also notes that IS, DR, and DRL estimators do not exploit time-invariance and fail to be efficient under M3.Source: §1.4 · p. 12

2. Efficiency Bounds in Infinite Horizons

The section derives infinite-horizon efficiency bounds for non-Markov, time-varying Markov, and time-invariant Markov processes. The bounds distinguish settings where regular estimation is feasible from those where Markovian, time-invariant structure yields a faster transition-based rate.

2. Efficiency Bounds in Infinite Horizons

  • The analysis extends infinite-horizon efficiency bounds to NMDP and TMDP, then studies the bound under time-invariant MDP structure.The bounds are computed in identifying ideal sampling settings, with achievability also considered for more complex sampling such as a single growing trajectory.Source: §2 · p. 12

2.1. Efficiency Bounds in Non-Markov and Time-Variant Markov Decision Processes

  • When the NMDP or TMDP bound is finite, the best-achievable leading MSE term is the corresponding bound divided by N; an infinite bound precludes regular square-root-consistent estimation.Sufficient finiteness conditions use discounted growth of the cumulative density ratios for NMDP and marginal density ratios for TMDP.Source: §2.1 · p. 14
  • In NMDP, non-degenerate rewards and transitions plus E[log(η_k)] ≥ −log(γ) imply an infinite bound, while TMDP’s bound depends on marginal rather than cumulative density-ratio growth.The TMDP curse is milder; with stationary or ergodic behavior-policy distributions and overlapping stationary distributions, the marginal ratios are uniformly bounded.Source: §2.1 · p. 15

2.2. Efficiency Bounds in Time-Invariant Markov Decision Processes

The efficiency bound in either model is therefore

  • Under MDP, MSE scales inversely with observed transitions, yielding a factor-T faster rate than under NMDP or TMDP when mixing assumptions hold.For dependent trajectory sampling, the same bound is achievable with n = N(T + 1) under certain mixing assumptions.Source: The efficiency bound in either model… · p. 16

3. Efficient Estimators for Infinite Horizons under NMDP and TMDP

The section extends DRL estimators to infinite-horizon non-Markov and time-varying Markov settings, then shows that standard DR-style estimators remain inefficient under the time-invariant MDP model.

3.1. Non-Markov Decision Process

  • Under NMDP, the infinite-horizon DRL extension uses q-estimates for a truncation horizon ωN, so implementation requires trajectories only through that truncation.The estimator does not require estimating q-functions that depend on all future rewards.Source: §3.1 · p. 18
  • Under NMDP, cross-fitting permits black-box nuisance estimates without Donsker or metric-entropy assumptions, provided the stated rate conditions hold.The required truncation cannot be infinite, even with infinitely long trajectories; the text gives ζν + ζq > 1 as a rate condition in one setting.Source: §3.1 · p. 19

3.2. Time-Variant Markov Decision Process

  • Under TMDP, the infinite-horizon extension is feasible with trajectories of length ω(log N), and cross-fitting requires only nuisance-rate conditions.The section extends the finite-horizon DRL estimator to the infinite-horizon case.Source: §3.2 · p. 20

3.3. Inefficiency under MDP

  • Standard DR-type estimators remain inefficient under MDP structure: even oracle nuisances yield variance vanishing only as O(1/N), despite O(NT) transitions.Using time-invariant q-functions alone does not improve the asymptotic variance.Source: §3.3 · p. 21

4. Efficient Estimator for Markov Decision Process

The paper constructs an MDP estimator from the efficient influence function and establishes its efficiency under transition sampling and, with mixing conditions, dependent trajectory sampling. Its robustness and cross-fitting properties allow nuisance estimates to converge at subparametric rates.

4. Efficient Estimator for Markov Decision Process

  • The proposed MDP estimator achieves the efficiency bound under appropriate conditions for transition and trajectory sampling, including dependent observations.Trajectory-sampling results require sufficient mixing and a specialized sample-splitting procedure.Source: §4 · p. 21

4.1. Efficient Estimation Under Transition Sampling

  • With cross-fitting under transition sampling, the estimator remains efficient with subparametric nuisance-estimation rates, without strong metric-entropy restrictions.This allows use of black-box machine-learning methods without requiring good metric-entropy conditions.Source: §4.1 · pp. 23–24
  • The transition-sampling estimator is consistent if either the density-ratio estimate or the q-function estimate is consistent.This double-robustness result does not itself provide a rate or asymptotic distribution.Source: §4.1 · p. 24
  • Under model double robustness, the estimator has an Op(n^-1/2) error when one nuisance is consistent and the other converges at a root-n rate.The same result applies to the adaptive version under its additional metric-entropy condition.Source: §4.1 · p. 24

4.2. Efficient Estimation Under Trajectory Sampling

  • Trajectory sampling can use a single long trajectory, with efficiency characterized when observations are sufficiently mixing.The analysis considers T → ∞ with N ≥ 1, including N = 1.Source: §4.2 · pp. 25–26

Theorem 10 (Efficiency structure under mixing). Suppose P∞

  • With the oracle q-function, the estimator's time-averaged second-term variables form a martingale difference sequence, eliminating cross-time covariances.This structure supports the efficiency result under mixing; feasible estimation must approximate the oracle average to op((NT)^-1/2).Source: Theorem 10 (Efficiency structure under… · p. 26
  • For one trajectory, cross-time-fitting uses four folds and estimates nuisances from a nonadjacent fold, which mixing makes asymptotically near-independent.With N ≥ 2 trajectories, cross-trajectory-fitting instead splits across independent trajectories.Source: Theorem 10 (Efficiency structure under… · pp. 27–28

Theorem 11 (Efficiency of ˆρDRL(M3) with cross-trajectory-fitting). Define κw

  • Cross-fitted trajectory estimators avoid strong plug-in conditions beyond a slow, subparametric rate, but require slightly stronger mixing than the oracle result.Cross-trajectory-fitting is feasible for N ≥ 2; cross-time-fitting is used when N = 1.Source: Theorem 11 (Efficiency of ˆρDRL(M3)… · pp. 28–29

5. Asymptotically valid confidence intervals

The section gives a route to asymptotically valid confidence intervals from the estimators' asymptotic normality and consistent variance estimates. For efficient estimators, the resulting intervals cannot be improved asymptotically.

5. Asymptotically valid confidence intervals

  • A consistent estimate of the asymptotic variance, combined with asymptotic normality, yields confidence intervals with asymptotic coverage 1 − α.For the MDP estimator under transition sampling, the proposed variance estimate is the sample variance of the estimating function.Source: §5 · p. 30
  • Estimating the efficiency bounds EB(M1), EB(M2), and EB(M3) suffices to construct asymptotically valid confidence intervals for the corresponding estimators.The paper states that the sample-variance approach also gives consistent variance estimates for the M1 and M2 estimators.Source: §5 · p. 30
  • Because the estimators are efficient, their confidence intervals cannot be improved asymptotically.Tests based on an efficient estimator are locally uniformly powerful.Source: §5 · pp. 30–31

6. Modeling the Ratio of Average Visitation Distributions

The section studies stationary density-ratio estimation for the MDP DRL estimator and contrasts it with marginalized importance sampling. It characterizes the density ratio through a fixed-point relation and gives a parametric estimation result.

6.1. Importance Sampling Using Stationary Density Ratios

  • Setting the DRL q-estimate to zero produces an importance-sampling-type estimator, but marginalized importance sampling lacks the DRL estimator's martingale-difference and doubly robust structures.The missing structures can leave time covariance and plug-in variance inflation, complicating asymptotic MSE analysis and suggesting the estimator is not efficient.Source: §6.1 · pp. 31–32

6.2. Efficient Semiparametric Estimation

  • The paper's conditional moment characterization identifies w when the corresponding equation holds for all integrable test functions and its integral equation has a unique solution.The passage notes that this is not a standard moment equation because it depends on the unknown p(0)πb(s).Source: §6.2 · pp. 32–33
  • The stationary density ratio w is characterized by a fixed-point equation, unlike the time-varying relation that obtains each µk from µk−1.The passage describes this as analogous to the difference between q-equations in time-variant finite-horizon and time-invariant infinite-horizon problems.Source: §6.2 · p. 33

Theorem 15 (Efficient estimation of w(s;β∗) under transition-sampling). Define

  • For a correctly specified parametric model w(s;β*), the proposed estimator converges at Op(n^-1/2) regardless of the choice of estimating function fw.The variance-minimizing choice of fw is unclear because p(0)πb(s) is unknown.Source: Theorem 15 (Efficient estimation of… · p. 34

7. Modeling the q-function

The section develops q-function estimation for the paper’s DRL estimator, extending LSTDQ-style estimating equations to parametric and nonparametric models. Under correct parametric specification, the q-function estimate has a root-n rate regardless of the choice of estimating function, while efficient estimation requires a particular choice.

7. Modeling the q-function

  • The q-function can be estimated from a parametric model by solving recursive estimating equations, a framework that includes LSTDQ.With a linear q-model and features as the estimating function, the equations yield LSTDQ.Source: §7 · p. 35

Theorem 17 (Efficient estimation of q(s,a;β) under transition sampling). Suppose

  • The efficient estimating function achieves the bound, but it is not known in practice and may need to be estimated parametrically or with sieve GMM.Sieve GMM uses a basis expansion for the estimating function.Source: Theorem 17 (Efficient estimation of… · p. 36
  • Regardless of the choice of fq, the q-function estimate q(·,·; β̂_fq) has rate ∥q(·,·; β̂_fq) − q∥₂ = Op(n−1/2).Source: Theorem 17 (Efficient estimation of… · p. 36
  • A basis expansion for q supports nonparametric estimation without correct parametric specification, subject to regularity and smoothness conditions.The resulting estimate can be used in the DRL estimator.Source: Theorem 17 (Efficient estimation of… · p. 36
  • The parametric DM estimator has a bound no greater than the M3 efficiency bound, but it assumes a correctly specified q-model and is not regular for M3.Unlike DRL, DM requires parametric-rate q-estimation to attain 1/n MSE scaling.Source: Theorem 17 (Efficient estimation of… · pp. 36–37

8. Experimental Results

The experiments compare OPE estimators in Taxi and CartPole, including settings with misspecified nuisance models, varying overlap, and flexible function approximation. DRL performs strongly across the reported settings, with its advantage especially apparent when overlap is weak or nuisance models are misspecified.

8.1. Taxi Environment

  • The Taxi experiments vary overlap and nuisance-model correctness across three settings, including cases where only the w-model or only the q-model is correct.The target policy is fixed within each setting, while the behavior policy mixes it with another policy using α = 0.2 or 0.6.Source: §8.1 · p. 38
  • DRL outperforms standard IS and DR estimators in every Taxi setting, and has the lowest MSE across settings in the large-sample regime.In that regime, DRL is similar to DM when q is well specified and to MIS when w is well specified with good overlap, but performs better when those conditions fail.Source: §8.1 · pp. 38–39
  • In Taxi’s small-to-medium sample regime, DRL performs best except when overlap is good and w is well specified, where MIS performs better.When overlap is bad or w is misspecified, MIS performs much worse; in setting (2) with α = 0.2, DRL performs much better than all other estimators across sample sizes.Source: Figure 8 · p. 39, §8.1 · p. 40

8.2. CartPole Environment

  • The CartPole environment has continuous four-dimensional states, binary actions, and an absorbing state that makes trajectories highly non-stationary.The study uses flexible models for w and q, whose precise convergence may not be guaranteed.Source: §8.2 · p. 40
  • For CartPole, the study varies evaluation-policy temperature and trajectory count, using 40 replications and relative MSE normalized by an estimated policy-value difference.The behavior policy uses temperature 1.0; Fig. 12 fixes τ = 1.3 while varying N.Source: §8.2 · pp. 40–41
  • In CartPole at τ = 1.3, DRL clearly outperforms the other estimators as the number of trajectories varies.The reported comparison is in Fig. 12, with relative MSE on a logarithmic scale and 90% confidence intervals.Source: Figure 12 · p. 41, §8.2 · p. 42
  • The CartPole comparison includes MIS, DM, DRL3, and DualDICE, with the w-estimators and q-estimators fitted using flexible neural-network models.DualDICE performs consistently badly across settings; the authors attribute this to instability in its two-network minimax optimization.Source: §8.2 · pp. 41–42
  • In CartPole, the authors attribute DRL’s advantage to its relative insensitivity to flexible w- and q-estimators, whose high variance affects MIS and DM.They report that MIS is always worse than DR and suggest w-estimation is harder than q-estimation in this environment.Source: §8.2 · p. 42

9. Conclusions

The conclusion establishes an efficiency bound for OPE in time-invariant MDPs when N may be finite and T tends to infinity. It presents an estimator that attains the bound while retaining double robustness.

9. Conclusions

  • The paper establishes an OPE efficiency bound for time-invariant MDPs when N may be finite and T tends to infinity.The bound quantifies the best asymptotic estimation rate in this setting.Source: §9 · p. 42
  • Many IS and DR estimators do not fully leverage MDP structure and have suboptimal MSE scaling in rate, not only in leading coefficient.Source: §9 · p. 42
  • The proposed estimator is the first described as efficient for this bound while also enjoying double robustness.The conclusion suggests future work could combine this approach with balancing, stability, or blending to improve finite-sample performance.Source: §9 · p. 42

Endnotes

The endnotes clarify the paper’s estimand and modeling scope. They distinguish mean-reward evaluation from estimating a whole value or quality function, and explain restrictions used in the semiparametric analysis.

Endnotes

  • Here, OPE means estimating a policy’s mean reward, rather than its whole value or quality function.Source: Endnotes · p. 43
  • The evaluation setup does not require standard Borel measurable spaces, a restriction often used in control settings because of measurability issues in optimization.Source: Endnotes · p. 43
  • The analysis restricts state, action, reward, and policy distributions to share base measures so semiparametric perturbations can be considered easily.Source: Endnotes · p. 43

A. Notation

This section collects notation for the policy value, state-action quantities, density ratios, and empirical averages used in the paper.

A. Notation

  • Table 2 gathers notation for policies, value and q-functions, density ratios, and expectations.It also defines sample splits and empirical-process notation used with cross-fitting.Source: Table 2 · p. 49, A. Notation · p. 49
  • The notation includes the policy value as a long-run discounted quantity and the average visitation distribution for a policy.The excerpted definitions identify these quantities but do not show their complete formulas.Source: A. Notation · p. 49
  • The notation distinguishes the target and behavior policies and defines the instantaneous density ratio as their action-probability ratio.The table also distinguishes cumulative and marginal density ratios.Source: A. Notation · p. 49

B.1. Some Definitions

This section introduces P-Donsker classes and dependence coefficients used to describe regularity and dependence in stationary processes.

B.1. Some Definitions

  • A function class is P-Donsker when its empirical process, scaled by √n, converges in distribution to a tight limit in l∞(F).A square-integrable envelope and finite uniform entropy integral are sufficient for this property.Source: §B.1 · p. 50
  • The mixing-coefficient definitions are set in a stationary process whose variables are identically distributed but may be dependent.The excerpt introduces the definitions but does not include their formulas.Source: §B.1 · p. 50
  • Mixing coefficients quantify dependence between subsequences separated by m steps, and faster decay as m grows corresponds to greater ergodicity.For an independent process, all these coefficients are zero.Source: §B.1 · p. 50

B.2. Semiparametric Theory

This appendix reviews semiparametric tools for scalar estimands, including influence functions, tangent spaces, regularity, and efficiency bounds.

B.2. Semiparametric Theory

  • The review focuses on scalar estimands and treats the observed data as a sample from a model of distributions.It sets notation for the estimand and its estimator.Source: §B.2 · p. 50

B.2.1. Definitions

  • The tangent space is the L2-closure of score functions, and the efficient influence function is the unique gradient that belongs to it.Regular estimators are defined by limiting behavior that is insensitive to local changes in the data-generating process.Source: §B.2.1 · p. 51

B.2.2. Characterizations

  • For an asymptotically linear estimator, regularity is equivalent to its influence function being a gradient of the estimand.The efficient influence function is the projection of any gradient onto the tangent space.Source: §B.2.2 · p. 52

B.2.3. Strategy to calculate the EIF

  • The EIF construction strategy is to find a gradient, characterize the tangent space, and either verify membership or project onto that space.Source: §B.2.3 · p. 52

B.2.4. Optimalites

  • The efficiency bound is the EIF variance and provides a local asymptotic minimax lower bound for regular estimation.An estimator is efficient when it is regular and has the EIF-based limiting normal distribution; asymptotic linearity with the EIF is necessary and sufficient.Source: §B.2.4 · pp. 52–53

C. Proofs

The proofs establish the efficient influence function and efficiency bound, verify the DRL estimator’s efficiency and consistency under stated conditions, and derive asymptotic results using mixing arguments for dependent cross-fitting. They also derive an asymptotic variance expression for the EIS estimator and compare the direct method’s variance with the nonparametric efficient bound.

Proofs of Theorems 1 and 2

  • Theorem 1 and 2 proofs adapt Kallus and Uehara (2020a); unknown initial density removes the k = 0 term, while Theorem 1 also changes conditioning sets.The proofs are presented as corollaries of that work, with the stated differences in the efficiency-bound setup.Source: C. Proofs · p. 53

Proof of Theorem 3

  • Theorem 3 derives the efficient influence function and efficiency bound, and the same influence function remains valid under M3b.The proof establishes that the gradient belongs to the relevant tangent spaces.Source: C. Proofs · p. 57

Proof of Theorem 4

  • The DRL efficiency proof bounds its component terms and, under assumptions (4a), (4b), and (4d), uses a central limit theorem to establish efficiency and regularity.The proof identifies terms (24) and (26) as O_p(1) before combining the results.Source: C. Proofs · pp. 57–59

Proofs of Theorems 6 and 7

  • The cross-fitting and adaptive proofs show a remainder term is o_p(1), then invoke a central limit theorem to conclude the final statement.For the adaptive version, the proof uses a finite uniform entropy integral to establish the Donsker condition.Source: C. Proofs · p. 60, Proving efficiency · pp. 60–61

Proof of Theorem 8

  • Theorem 8’s proof establishes DRL consistency when either the q-function model is consistent or the density-ratio model is correct.The proof treats the q-model and ratio-model cases separately.Source: Proving efficiency · p. 62

Proof of Theorem 12

  • For Theorem 12, cross-fitting dependence is handled using time separation and mixing inequalities to control conditional means and variances, then Chebyshev’s inequality concludes the proof.The proof uses folds separated by T/4 time steps rather than treating them as independent.Source: Proving efficiency · pp. 65–66

Proof of Theorem 13

  • Theorem 13’s proof controls the empirical-process term through tight Gaussian-process convergence and uses the assumed L2 convergence of the estimated influence function.Source: Proving efficiency · p. 68

Proof of Theorem 14

  • For Theorem 14’s cross-fitting version, both analyzed terms are Op(1/√n), with the first having conditional mean zero and conditional variance Op(1/n).Source: Proving efficiency · p. 68

Calculation of asymptotic variance

  • The EIS estimator has the displayed asymptotic linear expansion, whose influence term includes an adjustment involving derivatives of the density ratio and moment condition.The expansion is stated at β* with an op(1) remainder.Source: Proving efficiency · p. 70

Proof of Lemma 2

  • By Cauchy–Schwarz, the direct method’s variance in (61) is smaller than the efficient bound under the nonparametric model.Source: Proving efficiency · p. 71

D. Additional Experimental Results

This section provides additional results from the Section 8 experiment for α = 0.4 and α = 0.8, shown in Figures 14–19.

D. Additional Experimental Results

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