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Safe Policy Improvement by Minimizing Robust Baseline Regret

Marek Petrik, Yinlam Chow, Mohammad Ghavamzadeh

arXiv:1607.03842v1stat.ML

TL;DR

With limited data, decision-makers need policies guaranteed to perform at least as well as a baseline. The paper minimizes robust regret using an inaccurate dynamics model with known error bounds, enabling statewise improvement or fallback; its approximate algorithm outperforms the standard model-based approach when the model is uncertain.

  • Problem

    Limited data make it important to compute policies guaranteed to perform at least as well as a baseline before deploying an untested strategy.

    Source: §1 · p. 1

  • Method

    The paper uses an inaccurate model with known error bounds to minimize robust regret against the baseline, allowing statewise use of a learned policy or fallback to the baseline.

    Source: §1 · p. 2, §6 · p. 10

  • Results

    The formulation is NP-hard, and its simple approximate algorithm significantly outperforms the standard model-based approach when the model is uncertain.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The method interleaves an improved policy with the baseline according to statewise model error, rather than requiring a single all-or-nothing policy choice.

    Source: §1 · p. 2, §6 · p. 10

    Even rectangular uncertainty sets do not guarantee tractability for robust regret minimization; identifying tractable uncertainty-set structures is left for future work.

    Source: §2 · p. 3

Abstract

from arXiv · show

An important problem in sequential decision-making under uncertainty is to use limited data to compute a safe policy, i.e., a policy that is guaranteed to perform at least as well as a given baseline strategy. In this paper, we develop and analyze a new model-based approach to compute a safe policy when we have access to an inaccurate dynamics model of the system with known accuracy guarantees. Our proposed robust method uses this (inaccurate) model to directly minimize the (negative) regret w.r.t. the baseline policy. Contrary to the existing approaches, minimizing the regret allows one to improve the baseline policy in states with accurate dynamics and seamlessly fall back to the baseline policy, otherwise. We show that our formulation is NP-hard and propose an approximate algorithm. Our empirical results on several domains show that even this relatively simple approximate algorithm can significantly outperform standard approaches.

1 Introduction

With limited data, decision-makers need safe policies that are guaranteed to perform at least as well as the baseline before deploying an untested strategy. This paper proposes minimizing robust regret to improve the baseline where model uncertainty permits and fall back to it elsewhere.

1 Introduction

  • Only a few methods for learning from batch data guarantee good deployed performance, although decision-makers seek guarantees against the current baseline before adopting an untested policy.The paper frames this as a practical barrier to using learned strategies in sequential decision-making.Source: §1 · p. 1
  • The paper studies model-based safe policy computation for infinite-horizon discounted MDPs using a simulator and statewise model-error function.Safety means guaranteeing performance at least as good as the baseline.Source: §1 · p. 1
  • Existing methods either adopt a learned policy with provable improvement or return the baseline, limiting improvement when model uncertainty varies across states.The desired alternative combines the learned and baseline policies state by state.Source: §1 · pp. 1–2
  • Minimizing negative robust regret offers a way to use the learned policy where improvement is significant or uncertainty is small, and otherwise use the baseline.This unifies uncertainty in the learned and baseline policies rather than comparing pessimistic learned-policy and optimistic baseline estimates.Source: §1 · p. 2
  • The paper formulates robust baseline-regret minimization, shows it is NP-hard, and proposes a simple approximate algorithm that outperforms the standard model-based approach when the model is uncertain.The paper also reports that solutions may require randomized policies and derives a performance-loss bound.Source: §1 · p. 2

2 Preliminaries

The paper models an infinite-horizon discounted MDP with known rewards and estimated transitions whose errors are bounded. It uses a robust uncertainty set containing the true transition model to define safety relative to a deterministic baseline.

2 Preliminaries

  • An infinite-horizon γ-discounted MDP comprises state and action spaces, bounded rewards, transition probabilities, an initial-state distribution, and a discount factor.The paper distinguishes randomized stationary policies from deterministic stationary policies.Source: §2 · p. 2
  • The true reward is assumed known, while historical data provide an estimated transition model that may differ from the unknown true transitions.The paper says extending the setup to reward estimation is straightforward but omits it for brevity.Source: §2 · p. 2
  • The error function bounds the ℓ1 difference between estimated and true transition probabilities and can be derived from samples or domain properties.The bound is specified for each state-action pair.Source: §2 · pp. 2–3
  • The robust MDP uncertainty set contains transition models within the state-action error bounds around the estimated model, including the true transition model.Nature adversarially selects transitions from the set.Source: §2 · p. 3
  • A policy is safe when its true return is guaranteed to be no lower than the deterministic baseline policy’s true return.The paper’s robust-regret problem remains potentially intractable even with rectangular uncertainty sets.Source: §2 · p. 3

3 Robust Policy Improvement Model

The section formulates safe improvement as maximizing a guaranteed return margin over the baseline across all models in the uncertainty set. It analyzes the needed policy class, performance loss, computational complexity, and an approximation that treats baseline transitions as known.

3 Robust Policy Improvement Model

  • The robust improvement problem seeks the largest guaranteed return margin over the baseline for every model in the uncertainty set, making any solution safe under the true transition model.The true transition probability is assumed to belong to that uncertainty set.Source: §3 · p. 3

3.1 Policy Class

  • Randomized policies may be essential: Example 4 achieves ζ = 1/2, while deterministic policies guarantee no positive improvement, and their loss can be unbounded.The theorem states that a deterministic policy need not attain the solution and that the loss from restricting to deterministic policies has no constant bound.Source: §3.1 · pp. 4–5

3.2 Performance Bound

  • A solution is safe, and its performance loss relative to the optimal policy is bounded by the theorem's inequality, which is tight.The bound uses state-occupancy distributions of the optimal and baseline policies in the true MDP.Source: §3.2 · p. 5

3.3 Computational Complexity

  • Both optimization problems are NP-hard, though the problem is polynomial-time solvable when the baseline-induced Markov chain is known precisely.Under that condition, the problem reduces to an ℓ1-constraint robust MDP.Source: §3.3 · pp. 5–6

3.4 Approximate Algorithm

  • The approximate algorithm assumes zero error in baseline transition probabilities, enabling polynomial-time robust regret minimization.The authors motivate this assumption by baseline-generated data and by transition probabilities often affecting baseline and improved returns similarly, leaving little effect on their difference.Source: §3.4 · p. 6

4 Standard Policy Improvement Methods

This section presents simulator-based and robust-MDP alternatives as standard ways to approximate robust baseline-regret minimization. Their safety and performance guarantees differ, and the two RMDP algorithms use different uncertainty sets.

4 Standard Policy Improvement Methods

  • The section compares two standard safe-policy methods and places reward-adjusted MDP as an additional method described in Appendix G.The methods are interpreted as approximations of the proposed baseline-regret minimization problem.Source: §4 · p. 7

4.1 Solving the Simulator

  • Solving the estimated simulator for reward maximization can incur performance loss and has no guarantee of safety under model uncertainty.Its performance guarantee is weaker than the proposed optimization problem's guarantee.Source: §4.1 · p. 7

4.2 Solving Robust MDP

  • The RMDP algorithm returns its robust solution only when its worst-case return exceeds the baseline's robust return; otherwise it returns the baseline.The algorithm is safe and has a tighter performance-loss bound than the simulator-based policy.Source: §4.2 · p. 7, §4.1 · p. 7
  • Algorithms 1 and 2 approximate the same problem with different uncertainty sets: Algorithm 1 treats baseline-suggested actions as error-free, while Algorithm 2 uses the simulator's error function.Although πR has the same performance bound as πS, its performance loss may be worse by an arbitrarily large margin.Source: §4.2 · p. 8

5 Experimental Evaluation

The experiments compare robust baseline-regret minimization with simpler methods in an illustrative example, a grid domain, and energy arbitrage. They examine performance as model uncertainty varies with the available samples.

5.1 An Illustrative Example

  • In the illustrative example, plain robust optimization rejects even the optimal policy as unsafe, while baseline-regret minimization returns it.The robust return of π⋆ is −9 and the optimistic return of πB is +10; under the regret comparison, the difference is 1.Source: §5.1 · p. 8

5.2 Example Grid Problem; 5.3 Energy Arbitrage

  • Across the grid and energy-arbitrage experiments, RBC safely improves on the baseline under large model error and has the best performance in most energy-arbitrage cases.The grid results show significant improvement when samples are few; the energy-arbitrage comparison averages results over five runs.Source: §5.2 · p. 9, Figure 2 · p. 9, §5.3 · p. 10

5.2 Example Grid Problem

  • With few grid transition samples, RBC outperforms other methods by using the baseline in high-error states and improving only where model error is small.The comparison reports improvement percentage in total return over the baseline, averaged over 40 runs; EXP can perform worse than the baseline, especially with few samples.Source: §5.2 · p. 9

5.3 Energy Arbitrage

  • Energy arbitrage concentrates uncertainty in price transitions: battery charging and discharging dynamics are modeled with high confidence, reducing the uncertain transitions to the ten price states.The domain has three state components—charge, capacity, and price—and the state of charge and actions are discretized into ten levels.Source: §5.3 · p. 9

6 Conclusion

The paper concludes that robust baseline-regret minimization can interleave an improved policy with the baseline according to statewise model error, while the standard robust formulation may be overly conservative. The optimization can require randomized policies and is NP-hard, motivating further work on approximation guarantees and tractable uncertainty structures.

6 Conclusion

  • The standard robust formulation may be overly conservative; the proposed approach interleaves an improved policy with the baseline according to error at each state.The paper considers a policy safe when it is guaranteed to improve on the baseline.Source: §6 · p. 10
  • The optimization problem may require randomized solutions, has a derived performance bound, and is NP-hard to solve.Source: §6 · p. 10
  • Future work includes tighter performance guarantees for approximate algorithms and identifying uncertainty-set structures that make the problem tractable.Source: §6 · p. 10

A Error Bound

This appendix outlines a sample-based construction of transition-model error bounds and gives a supporting comparison argument for transitions in the uncertainty set. The proof uses a triangle-inequality bound to relate those transitions to the true model.

A Error Bound

  • The error function is constructed so the true transition model belongs to the uncertainty set with probability at least 1 − δ, using samples at each state-action pair.Source: A Error Bound · p. 12

B Proof of Lemma 11

  • Lemma 11 concerns bounding the return difference for a policy across MDPs parameterized by the true transition model and an uncertainty realization.Source: B Proof of Lemma 11 · p. 13
  • For any transition model ξ in the uncertainty set, the proof bounds its ℓ1 distance from the true transition distribution by 2e(x, a).The argument applies the triangle inequality to the true-to-simulator and simulator-to-ξ distances.Source: B Proof of Lemma 11 · p. 13

C Proof of Theorem 5

The proof establishes safety through the baseline’s feasibility in the regret objective and bounds the returned policy’s performance loss. A constructed example shows the bound can be tight.

C Proof of Theorem 5

  • The baseline policy is feasible for (2), making its objective non-negative and enabling the safety condition to be derived algebraically.The proof then establishes that Φ(πS) ≤ Φ(πB).Source: C Proof of Theorem 5 · p. 14
  • The proof derives the performance-loss bound by bounding the difference between optimal and solution returns relative to the baseline.It introduces Φ(πS) by adding and subtracting the baseline return.Source: C Proof of Theorem 5 · p. 14
  • The bound is tight in an example where the optimal policy returns 1 + 2ε and the solution to (2) returns 1, giving loss 2ε.The baseline takes action a1, while the optimal policy takes a2.Source: C Proof of Theorem 5 · p. 14, Figure 4 · p. 15

D Proof of Theorem 6

The proof reduces Boolean satisfiability to the optimization over uncertainty realizations, then constructs a robust MDP to show the outer policy maximization does not make the problem easier.

D Proof of Theorem 6

  • For fixed π, the returns ρ(π, ξ) and ρ(πB, ξ) are represented by MDPs M1 and M2, where ξ acts as the policy selecting uncertainty realizations.The two MDPs share state and action sets.Source: D Proof of Theorem 6 · p. 15
  • Variable states encode truth assignments with actions T and F, while paired literal states encode literal values and the negation operation.Actions 0 and 1 at literal states represent the referenced variable’s value.Source: D Proof of Theorem 6 · p. 16
  • The inner minimization in (2) is NP-hard, and deciding whether its optimal value exceeds 3k −2 is as hard as the corresponding SAT problem.The reduction establishes the threshold decision’s hardness through a SAT construction.Source: D Proof of Theorem 6 · p. 17
  • The reduction extends to the outer maximization by constructing robust MDP R so that maximal improvement induces M1 and the baseline induces M2.The construction makes a threshold condition hold if and only if the SAT problem is satisfiable.Source: D Proof of Theorem 6 · p. 17

E Proof of Theorem 8

The proof bounds the optimal-policy gap relative to a policy optimized under the estimated dynamics, using Lemma 11 and the stated model-error term.

E Proof of Theorem 8

  • The gap Δ between the optimal policy and πsim under the true MDP is bounded above by the optimal-policy gap against πsim’s return under bP, plus γRmax.The displayed inequality is Δ ≤ ρ(π⋆, P⋆) − ρ(πsim, bP) + γRmax.Source: E Proof of Theorem 8 · p. 18
  • The proof uses πsim’s optimality in the MDP with transition probabilities bP to justify a step in the bound.This is identified as justification (a).Source: E Proof of Theorem 8 · p. 18
  • A separate step applies Lemma 11 with π = π⋆ and ξ = bP.This is identified as justification (b) for the bound.Source: E Proof of Theorem 8 · p. 18

F Performance Bound on the Solution of the Robust Algorithm

Theorem 12 states that Algorithm 2 returns a safe policy and bounds its performance loss. Its policy choice falls back to the baseline when the condition for using the candidate policy is violated.

F Performance Bound on the Solution of the Robust Algorithm

G Solving the Reward-Adjusted MDP

The reward-adjusted MDP (RaMDP) gives a lower-bound performance estimate under the stated assumption, and Algorithm 3 uses it to choose between its optimized policy and the baseline. The section also describes grid and energy-arbitrage problem setups.

G Solving the Reward-Adjusted MDP

  • Algorithm 3 approximates robust-regret minimization by optimizing in the RaMDP when its value exceeds the robust baseline estimate, and otherwise returns the baseline.The RaMDP's performance for any policy is a lower bound on its performance in the true MDP.Source: G Solving the Reward-Adjusted MDP · p. 21
  • Under Assumption 1, Algorithm 3 returns a policy at least as good as the baseline, with a tighter performance-loss bound than the simulator solution.The safety guarantee is stated for the true MDP.Source: G Solving the Reward-Adjusted MDP · pp. 21, 23
  • Algorithm 3 is more efficient but may be overly conservative; Algorithm 2 returns a less conservative safe policy at extra computational cost.The reward adjustment assumes a maximum-value state is accessible from every state with maximum reward.Source: G Solving the Reward-Adjusted MDP · p. 23

H.1 Grid Problem

  • The 12-by-3 website/customer-state grid uses column-only rewards and left/right transitions that fail to random columns with row-specific probabilities 0.9, 0.2, and 0.3.Rows represent more complex customer states, such as overall satisfaction.Source: §H.1 · p. 24

H.2 Energy Arbitrage

  • Energy arbitrage models a degrading battery with charge/discharge actions, while prices evolve as a Markov process and transaction rewards subtract a battery-degradation penalty.The model uses finite known price levels, and the degradation function represents capacity loss from charging or discharging.Source: §H.2 · p. 25
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