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On the Sample Complexity of Reinforcement Learning with a Generative Model
Mohammad Gheshlaghi Azar, Remi Munos, Bert Kappen
TL;DR
The paper studies how many generative-model samples are needed to estimate optimal action-value functions in discounted MDPs. It analyzes model-based QVI using variance-sensitive concentration and proves matching upper and lower sample-complexity bounds. The resulting bounds close the RL gap and improve dependence on the discount factor.
Problem
The paper asks whether the sample-complexity bound for estimating optimal action-value functions in discounted MDPs can be made tight, resolving a gap between existing upper and lower bounds.
Method
The paper analyzes model-based QVI with variance-sensitive concentration and constructs a structured MDP class for a matching lower bound.
Results
QVI attains a new upper bound, and every RL algorithm has a matching lower bound, closing the sample-complexity gap and improving dependence on 1/(1−γ).
Takeaways & Limitations
QVI is optimal for estimating the optimal action-value function in discounted MDPs with a generative model under the paper's stated dependencies.
Takeaways & Limitations
The paper studies optimal action-value estimation rather than exploration and does not compare its results with PAC-MDP or upper-confidence-bound algorithms.
Abstract
from arXiv · showhide
We consider the problem of learning the optimal action-value function in the discounted-reward Markov decision processes (MDPs). We prove a new PAC bound on the sample-complexity of model-based value iteration algorithm in the presence of the generative model, which indicates that for an MDP with N state-action pairs and the discount factor γ\in[0,1) only O(N\log(N/δ)/((1-γ)^3ε^2)) samples are required to find an ε-optimal estimation of the action-value function with the probability 1-δ. We also prove a matching lower bound of Θ(N\log(N/δ)/((1-γ)^3ε^2)) on the sample complexity of estimating the optimal action-value function by every RL algorithm. To the best of our knowledge, this is the first matching result on the sample complexity of estimating the optimal (action-) value function in which the upper bound matches the lower bound of RL in terms of N, ε, δand 1/(1-γ). Also, both our lower bound and our upper bound significantly improve on the state-of-the-art in terms of 1/(1-γ).
1. Introduction
The paper addresses whether the existing sample-complexity bound for model-based Q-value iteration is tight. It develops improved upper and lower bounds that close the gap for estimating optimal action-value functions in discounted MDPs.
- Model-based Q-value iteration estimates transition dynamics and uses Bellman recursion to estimate the optimal action-value function.
- An existing QVI bound had an unresolved gap of order 1/(1−γ)^2 from the best lower bound, leaving its tightness unclear.
- The paper proves a new QVI upper bound and a matching lower bound, with each improving the corresponding prior result by an order of 1/(1−γ).
- The paper focuses on finite state-action discounted infinite-horizon MDPs and states that analogous upper-bound orders can be obtained for policy iteration and linear programming.
- The upper-bound analysis replaces the previous maximum-value argument with variance-sensitive concentration and a Bellman-like variance equation.
- The lower-bound construction adds high-probability self-transitions at intermediate states, making small estimation errors propagate through the MDP recursion.
2. Background
The paper works with finite discounted MDPs and defines value functions through expected discounted rewards. Model-based QVI estimates transition probabilities from generative-model samples and iterates an empirical Bellman optimality operator.
- A discounted MDP is represented by state and action sets, transition distribution, reward function, and discount factor γ.
- The state-action space Z=X×A is finite, with N state-action pairs, and immediate rewards lie in [0,1].
- A policy assigns an action to each state, while value and action-value functions equal expected sums of discounted rewards under that policy.
- 2.1. Model-based Q-value Iteration (QVI): QVI samples each state-action pair n times from the generative model and estimates transition probabilities using empirical state-transition counts.
- 2.1. Model-based Q-value Iteration (QVI): The algorithm iterates action-value functions from an initial Q0 through an empirical Bellman optimality operator to estimate Q*.
3. Main Results
The paper gives PAC guarantees for QVI and a matching lower bound for every RL algorithm estimating the optimal action-value function. Together, these results characterize the required transition samples under the stated correctness and MDP assumptions.
- PAC bounds measure the uniform error between the optimal action-value function Q∗ and its sample estimate.
- A total sampling budget suffices for QVI to achieve uniform error ∥Q∗−Qk∥≤ε with probability at least 1−δ after k iterations.The theorem assumes ε, δ ∈ (0,1); constants can be chosen as c = 68 and c0 = 12.
- The lower-bound result applies to every (ε,δ,T)-correct RL algorithm on a specified class of MDPs.The algorithm may allocate transition samples arbitrarily rather than using the same number for every state-action pair.
4. Analysis
The analysis proves matching sample-complexity bounds for QVI and general RL algorithms by combining variance-sensitive concentration with a likelihood-ratio lower-bound construction.
- QVI upper bound: QVI analysis improves the crude ˜O(β^2/√n) error to the sharper ˜O(β^1.5/√n) bound.The improvement uses Bernstein’s inequality, variance recursion, and an O(β^1.5) bound on the propagated variance term.
- Variance analysis: The variance of discounted rewards satisfies a Bellman-like recursion, enabling bounds on the accumulated variance through the transition operator.This recursion replaces the immediate reward with an immediate-variance term and supports the subsequent operator bound.
- QVI upper bound: The proofs establish a high-probability bound on QVI’s estimation error by controlling the distance between Q* and the empirical optimum.Lemma 7 bounds ∥Q*−bQ*∥, and the proof of Theorem 1 combines this with the iteration error to obtain the PAC result.
- Lower-bound construction: The lower-bound construction uses MDPs with N = 3KL state-action pairs, three state sets, and transitions from Y1 to itself or corresponding absorbing states in Y2.The construction assigns reward 1 in Y1 and reward 0 elsewhere, with each state in Y2 connected to one state in Y1.
- Lower-bound proof: Any algorithm using fewer than β^3N/(c1ε^2) log(N/(c2δ)) transitions cannot be (ε, δ, T)-correct over the constructed MDP class.The proof identifies many under-sampled state-action pairs and uses two MDPs whose distinguishability limits accurate estimation.
5. Conclusion and Future Works
The paper concludes that QVI attains a minimax sample-complexity bound for optimal action-value estimation with a generative model, while leaving exploration as a separate problem.
- Conclusion: QVI is presented as an optimal learning algorithm whose dependencies on 1/ε, N, δ, and β match the lower bound.The conclusion describes this as the first minimax bound for estimating the optimal action-value function in discounted-reward MDPs.
- Future work: The analysis improves β-dependence by replacing Hoeffding-based concentration with Bernstein-based analysis in model-based value-iteration methods.The authors suggest that related PAC-MDP results could also benefit from this improved analysis.
- Future work: The paper studies estimation of the optimal action-value function rather than exploration, so it does not compare against exploration-focused PAC-MDP and upper-confidence-bound algorithms.The authors identify improving exploration results using their analysis as a possible direction.