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Model-Based Reinforcement Learning with a Generative Model is Minimax Optimal
Alekh Agarwal, Sham Kakade, Lin F. Yang
TL;DR
The paper asks whether maximum-likelihood plug-in planning is minimax optimal for obtaining ε-optimal policies from a generative model in the non-asymptotic regime. It analyzes planning in the empirical MDP, using direct uniform-convergence arguments and an absorbing MDP construction to address statistical dependence. The result establishes minimax-optimal policy guarantees while allowing any sufficiently accurate planning algorithm in the empirical MDP.
Problem
The paper asks whether the natural maximum-likelihood plug-in approach is non-asymptotically minimax optimal for policy quality with generative-model samples.
Method
The paper runs a black-box optimization oracle on the sample-based MDP and directly establishes uniform convergence despite the returned policy depending on the empirical model.
Results
The model-based approach is non-asymptotically minimax optimal, and any near-optimal policy in the empirical MDP can support the corresponding true-MDP guarantee under the theorem’s conditions.
Takeaways & Limitations
The result permits any efficient planning algorithm in the empirical MDP, simplifying algorithm design by avoiding dependence between the planning algorithm and sampling procedure.
Abstract
from arXiv · showhide
This work considers the sample and computational complexity of obtaining an $ε$-optimal policy in a discounted Markov Decision Process (MDP), given only access to a generative model. In this work, we study the effectiveness of the most natural plug-in approach to model-based planning: we build the maximum likelihood estimate of the transition model in the MDP from observations and then find an optimal policy in this empirical MDP. We ask arguably the most basic and unresolved question in model based planning: is the naive "plug-in" approach, non-asymptotically, minimax optimal in the quality of the policy it finds, given a fixed sample size? Here, the non-asymptotic regime refers to when the sample size is sublinear in the model size. With access to a generative model, we resolve this question in the strongest possible sense: our main result shows that \emph{any} high accuracy solution in the plug-in model constructed with $N$ samples, provides an $ε$-optimal policy in the true underlying MDP (where $ε$ is the minimax accuracy with $N$ samples at every state, action pair). In comparison, all prior (non-asymptotically) minimax optimal results use model free approaches, such as the Variance Reduced Q-value iteration algorithm (Sidford et al 2018), while the best known model-based results (e.g. Azar et al 2013) require larger sample sizes in their dependence on the planning horizon or the state space. Notably, we show that the model-based approach allows the use of \emph{any} efficient planning algorithm in the empirical MDP, which simplifies algorithm design as this approach does not tie the algorithm to the sampling procedure. The core of our analysis is avnovel "absorbing MDP" construction to address the statistical dependency issues that arise in the analysis of model-based planning approaches, a construction which may be helpful more generally.
1 Introduction
The paper studies whether the natural model-based plug-in approach is minimax optimal with sublinear samples and answers affirmatively. It also shows that this result permits flexible planning algorithms and uses an absorbing MDP construction to address dependence issues.
- 1 Introduction: The generative-model setting asks how many samples are needed to obtain a near-optimal policy from next-state queries for any state-action pair.This question is framed as analogous to sample complexity in supervised learning.
- 1 Introduction: The plug-in approach estimates transitions by maximum likelihood, then computes an optimal policy in the resulting empirical MDP.The paper studies this approach in the non-asymptotic regime, where sample size is sublinear in model size.
- 1 Introduction: The model-based planning approach is shown to be non-asymptotically minimax optimal for finding a near-optimal policy.This addresses the unresolved question of whether the naïve plug-in method achieves minimax policy quality at a fixed sample size.
- 1 Introduction: Unlike prior minimax-optimal model-free methods, the result allows any efficient planning algorithm to operate on the empirical MDP.This decouples planning from the sampling procedure and simplifies algorithm design.
- 1 Introduction: Obtaining an ε-optimal policy is subtler than estimating an ε-optimal value function because naïve greedy extraction can amplify error by a factor of 1 −γ.Earlier work avoided this amplification only when sample size was at least O(|S|2|A|), linear in model size.
- 1 Introduction: Prior model-based sub-optimality is attributed to analysis rather than an inherent limitation of the approach.The paper develops a novel absorbing MDP construction to handle dependence issues arising in the analysis of earlier model-based results.
2 Setting
The setting is a discounted finite MDP with known deterministic rewards and generative access to sampled transitions. The planner builds an empirical transition model from N samples per state-action pair and evaluates an approximately optimized policy in that model against the true MDP.
- An MDP consists of finite states and actions, a transition kernel, rewards in [0, 1], and a discount factor γ ∈ (0, 1).
- A stationary policy maps each state to an action, and an ε-optimal policy is within ε of the optimal value at every state.
- The setting focuses on the non-asymptotic regime where the sample size is sublinear in model size, while the cited improvement to the ε range is left for future work.
- The generative model returns a next-state sample for any queried state-action pair, with N samples collected at every pair.
- The empirical transition model uses observed transition counts, while the empirical MDP retains the original reward function and replaces only the transition kernel.
- An optimization oracle returns a near-optimal policy for the empirical MDP, allowing statistical analysis to be separated from the choice of planning algorithm.
3 Main results
The paper analyzes a black-box planning oracle on the sample-based MDP and establishes that sufficiently accurate empirical planning yields an O(ε)-sub-optimal policy in the original MDP with minimax-optimal samples. The approach also supports several efficient planning algorithms, including methods with nearly linear running time in the total sample size.
- Main results: The analysis directly establishes uniform convergence for value estimates while using a black-box optimization oracle on the sample-based MDP.This avoids tying the planning algorithm to the sampling procedure.
- Main results: With ǫopt roughly (1 −γ)ǫ, the returned policy is O(ǫ)-sub-optimal in the original MDP.The guarantee applies to any policy satisfying the empirical optimality condition in Theorem 1.
- Main results: The sample requirement is known to be unimprovable up to a logarithmic factor for ǫ ∈(0, 1].This optimality follows from existing lower bounds.
- Computational implications: Value iteration provides a natural implementation, while policy iteration can achieve the same iteration complexity but requires solving linear systems of size |S|2.Value iteration uses O[(1 −γ)−1 · log ǫ−1opt] iterations, each taking O(|S||A|N) time.
- Computational implications: A randomized planning algorithm can achieve an ǫopt-optimal policy in time nearly linear in the total sample size, up to additional logarithmic factors.The scheme first reads O(|S||A|N) data points.
4 Analysis
The analysis develops uniform-convergence guarantees for policies optimized in the empirical MDP, addressing dependence between the empirical model and its value estimates through absorbing MDPs.
- Analytical ingredients: The analysis represents policy values through Bellman equations, transition operators, variance terms, and resolvent bounds.These ingredients include Qπ = r + γPπQπ, componentwise transition variance, and bounds on (I −γPπ)^−1.
- Bounding errors: Crude empirical-value bounds lose an additional factor of 1/(1 −γ), motivating the more careful variance-aware analysis.The paper states that removing this factor requires significantly more careful analysis.
- Absorbing MDP construction: The proof introduces an auxiliary absorbing MDP construction to decouple empirical-model dependence from value estimation.For a chosen state s, the construction makes s absorbing and assigns reward (1 −γ)u, while leaving the rest of the MDP unchanged.
- Uniform convergence: The absorbing construction supports uniform convergence over a finite set of terminal values u using concentration and union bounds.The analysis establishes simultaneous bounds for all u in a finite set Us, with probability guarantees stated in Lemma 6 and Proposition 1.
5 Conclusion
The conclusion places the work within the broader problem of planning with approximate models and points to possible reuse of its technical ideas in exploration and related settings.
- 5 Conclusion: The paper addresses the longstanding reinforcement-learning question of how to plan effectively with an approximate model of the world.The conclusion frames planning with approximate models as a fundamental question in reinforcement learning.
- 5 Conclusion: The paper’s technical components may be reusable in broader contexts, as earlier variance-structure results supported later minimax-optimal exploration algorithms.The conclusion specifically connects prior variance results to exploration work and expresses hope that this paper’s techniques will similarly transfer.