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Dynamic Policy Programming
Mohammad Gheshlaghi Azar, Vicenc Gomez, Hilbert J. Kappen
TL;DR
Exact dynamic programming can be intractable for high-dimensional or continuous systems, while Monte Carlo estimation can produce large per-iteration errors. The paper proposes DPP, which accumulates approximation errors and bounds performance loss using their average. DPP-RL outperformed QL and VI on the three reported benchmarks, and the paper identifies limits on the comparisons it makes.
Problem
Dynamic programming is intractable for high-dimensional discrete or continuous systems, and Monte Carlo estimation can make per-iteration approximation errors large and difficult to control.
Source: §1 · pp. 1–2Method
DPP uses an incremental policy update that accumulates prior approximation errors, with approximate performance bounds expressed in terms of average accumulated error.
Source: §1 · p. 2, §4.1 · p. 9Results
Across three benchmarks, DPP-RL outperformed QL and VI; its advantage over QL was about two orders of magnitude on linear MDP and Combination lock, and more than fourfold on grid world.
Source: §5.1.1 · pp. 16–18Takeaways & Limitations
The average-error bound suggests DPP may perform better than AVI and API when estimation variance is large, because accumulated errors can average out during learning.
Source: §1 · p. 2The study focuses on optimal-policy estimation rather than exploration, and does not compare against PAC-MDP methods or actor-critic methods that use different sampling strategies.
Source: §7 · p. 21
Abstract
from arXiv · showhide
In this paper, we propose a novel policy iteration method, called dynamic policy programming (DPP), to estimate the optimal policy in the infinite-horizon Markov decision processes. We prove the finite-iteration and asymptotic l\infty-norm performance-loss bounds for DPP in the presence of approximation/estimation error. The bounds are expressed in terms of the l\infty-norm of the average accumulated error as opposed to the l\infty-norm of the error in the case of the standard approximate value iteration (AVI) and the approximate policy iteration (API). This suggests that DPP can achieve a better performance than AVI and API since it averages out the simulation noise caused by Monte-Carlo sampling throughout the learning process. We examine this theoretical results numerically by com- paring the performance of the approximate variants of DPP with existing reinforcement learning (RL) methods on different problem domains. Our results show that, in all cases, DPP-based algorithms outperform other RL methods by a wide margin.
1. Introduction
Exact dynamic programming is intractable for high-dimensional discrete and continuous systems, so approximate methods use function approximation and Monte Carlo sampling. The paper introduces DPP, whose guarantees depend on accumulated average error rather than the per-iteration error used for AVI and API.
1. Introduction
- Dynamic programming is intractable for high-dimensional discrete or continuous systems, motivating function approximation and Monte Carlo sampling in approximate dynamic programming.API and AVI are cited examples of these approximate methods.Source: §1 · p. 1
- AVI and API performance-loss bounds use the supremum of per-iteration approximation errors, which can be large and hard to control under Monte Carlo estimation variance.The introduction motivates bounds based instead on average accumulated error.Source: §1 · p. 2
- DPP's approximate performance-loss bound depends on average accumulated error, because its incremental policy update accumulates errors from prior iterations.Unlike AVI and API, DPP does not only minimize the current iteration's approximation error.Source: §1 · p. 2
- The paper develops DPP convergence and approximation results, extends the method to sample-based reinforcement learning, and evaluates it on several problem domains.The stated examples include an optimal replacement problem and a stochastic grid world.Source: §1 · p. 2
2. Preliminaries
The preliminaries establish norm notation and define discounted Markov decision processes, policies, value functions, and Bellman operators. These definitions frame the goal of finding a policy that attains optimal value across states.
2. Preliminaries
- The preliminaries introduce Euclidean, weighted Euclidean, and supremum norms for real-valued functions on a finite set.The weighted norm uses a probability measure over that set.Source: §2 · pp. 2–3
2.1 Markov Decision Processes
- A discounted MDP is specified by state and action spaces, a transition distribution, a reward kernel, and a discount factor; the setup assumes finite spaces and bounded rewards.The discount factor lies in [0, 1).Source: §2.1 · p. 3
- A policy determines action distributions, while Qπ gives expected discounted reward after taking an action and then following π; the goal is a policy attaining optimal value at every state.The optimal action-value function is defined as the supremum of Qπ over policies.Source: §2.1 · pp. 3–4
- The Bellman and Bellman optimality operators use policy transitions and action maximization, and both are contractions in the supremum norm with factor γ.The paper also defines a state-action transition kernel induced by a policy.Source: §2.1 · p. 4
3. Dynamic Policy Programming
DPP is derived by moving from a baseline-policy optimization toward a single-loop update of action preferences and policy. Its recursion is incremental, with a soft-max form that preserves this update structure.
3.1 From Bellman Equation to DPP Recursion
- The derivation modifies reward optimization with a relative-entropy penalty for deviating from a baseline policy, yielding a closed-form policy update.The optimal policy depends on the baseline policy, value function, and transition probabilities.Source: §3.1 · p. 5
- Replacing repeated policy and value updates with simultaneous optimization produces a smoother single-loop iteration based on action preferences.The preceding double-loop scheme alternates a policy update and a value-function update.Source: §3.1 · pp. 5–6
- DPP updates action preferences using a soft-max operator and remains incremental in policy even as η approaches infinity.The paper attributes the incremental update to the extra smoothness introduced by replacing the double-loop update with a single loop.Source: §3.1 · pp. 6–7
11 end
DPP has a finite-iteration performance-loss guarantee under stated assumptions and bounded initial preferences. In the greedy limit, its action values converge to optimal values, and its induced policy converges asymptotically to the optimal policy.
3.2 Performance Guarantee
- Under Assumption 1 and a uniform upper bound on initial preferences, Theorem 1 gives DPP a finite-iteration performance-loss guarantee.The supplied passage does not include the inequality's explicit bound.Source: §3.2 · pp. 7–8
- Choosing η = ∞ replaces the soft-max policy and operator with the greedy policy and max-operator.Source: §3.2 · p. 8
- As k tends to infinity, DPP's action values converge to Q*(x, a), and its induced policy asymptotically converges to the optimal policy.Source: §3.2 · p. 8
- Under a mild assumption, the action preferences have a unique limit at infinity.Source: §3.2 · p. 8
4. Dynamic Policy Programming with Approximation
Approximate DPP is analyzed through finite-iteration and asymptotic performance-loss bounds, with the asymptotic bound based on average accumulated error. The section also introduces DPP-RL, a sample-based policy update.
4. Dynamic Policy Programming with Approximation
- Because exact DPP applies only to small problems and requires model knowledge, the section extends it with approximation bounds and sample-based algorithms.The section introduces DPP-RL and SADPP for Monte-Carlo sampling and linear function approximation, respectively.Source: §4 · p. 8
4.1 The ℓ∞-norm performance-loss bounds for approximate DPP
- Approximate DPP has finite-iteration and asymptotic performance-loss bounds; asymptotically, its bound uses average accumulated error rather than AVI/API’s per-iteration error supremum.The finite-iteration bound is stated for the policy induced at round k.Source: §4.1 · pp. 9–10
- With bounded i.i.d. zero-mean errors, approximate DPP asymptotically cancels noise and converges to the optimal policy, while AVI and API have no such guarantee.The paper connects this result to averaging simulation noise from Monte-Carlo sampling.Source: §4.1 · p. 10
4.2 Reinforcement Learning with Dynamic Policy Programming
- DPP-RL replaces the Bellman-operator update with a sample estimate and assumes access to a generative model for state-action transitions.The next state is sampled from P(·|x, a).Source: §4.2 · p. 10
10 end
The section establishes almost-sure convergence of DPP-RL under stated conditions, discusses its learning-step choice, and describes SADPP’s sample-based linear approximation.
10 end
- Despite using an approximate DPP update, DPP-RL’s induced action-values converge almost surely to Q* under Assumption 1 and bounded initial preferences.The paper says the approximation errors are asymptotically averaged out by DPP-RL.Source: §10 · p. 11
- Unlike Q-learning and SARSA, DPP-RL uses no decaying learning step, which the authors say may avoid sensitivity to learning-step choice.The authors expect its convergence rate to remain close to DPP’s fast rate.Source: §10 · p. 11
4.3 Approximate Dynamic Policy Programming with Linear Function Approximation
- SADPP approximates each DPP-operator step by regularized least-squares fitting of linear combinations of bounded basis functions to sampled operator estimates.The quadratic regularization is added to avoid overfitting with a small dataset.Source: §4.3 · pp. 12–13
5. Numerical Results
The experiments compare DPP-RL with QL and VI on three discrete benchmarks, and SADPP with RFQI on an optimal-replacement task. DPP-based methods outperform their comparators in the reported tests, with lower estimation variance also reported.
5.1 DPP-RL
- The linear MDP is a 2,500-state chain with two absorbing ends; its optimal policy reaches the closest absorbing state as soon as possible.Transitions from interior states are action-directed and inversely weighted by distance.Source: §5.1 · pp. 13–14
- In the combination lock, only the all-+1 sequence reaches the absorbing goal; a -1 action resets to a randomly selected earlier state.The goal is harder to reach than in the linear MDP because only x2499 leads to it.Source: §5.1 · pp. 14–15
- The 50 × 50 grid world places absorbing firewalls around the grid and at its center, with the optimal policy seeking to survive while avoiding them.Transition probability includes a 0.6 move in the chosen direction and random moves weighted by inverse Euclidean distance.Source: §5.1 · pp. 15–16
5.1.1 Experimental Setup and Results
- The DPP-RL comparison uses synchronous QL and batch model-based VI, with equal computational budgets and 1 × 10^5 samples per state-action pair.The run-time limits are 30 seconds for the linear and combination-lock problems and 60 seconds for grid world.Source: §5.1.1 · p. 16
- DPP-RL converges near-optimally within seconds and beats QL and VI on all three benchmarks, by about two orders versus QL on linear/lock and over 4 times on grid.The comparisons average performance loss over 50 runs.Source: §5.1.1 · pp. 16–18, Figure 3 · p. 17
5.2 SADPP
- SADPP is evaluated with function approximation and limited samples per iteration to examine sample complexity for near-optimal performance with low variance.It is compared with a regularized fitted Q-iteration variant using a fixed number of basis functions.Source: §5.2 · p. 18
5.2.1 Optimal replacement problem
- The optimal-replacement task is an infinite-horizon discounted MDP in which the product is kept or replaced, with the optimal choice switching at a unique use threshold.The threshold is obtained from the Bellman equation.Source: §5.2.1 · p. 18
5.2.2 Experimental setup and results
- SADPP asymptotically outperforms RFQI on average; SADPP's N = 50 variance is comparable to RFQI's N = 500 variance.The figure reports averages and standard deviations of error for the optimal-replacement problem.Source: §5.2.2 · p. 19
6. Related Work
The paper situates DPP among incremental RL and stochastic-control methods, contrasting their algorithmic forms and available guarantees. It presents convergence analysis for DPP and performance-loss bounds for an incremental RL algorithm as distinctions from these related approaches.
6. Related Work
- Actor-critic and policy-gradient actor-critic use value-function guidance and gradient policy updates; the paper says it is the first to prove finite-iteration or asymptotic performance-loss bounds for an incremental RL algorithm.The cited actor-critic gradient guarantee assumes an unbiased gradient estimate.Source: §6 · pp. 19–20
- Unlike the restricted stochastic-control formulations of Kappen and Todorov, DPP is more general but requires Bellman-like iterations.PoWER and SAEM also use EM to maximize a lower bound expressible as a KL-divergence.Source: §6 · pp. 20–21
- Compared with REPS, DPP is policy-iteration-like rather than actor-critic, converges for any inverse temperature η, and has a convergence analysis provided in this paper.Source: §6 · p. 21
7. Discussion and Future Works
The discussion reports DPP-RL’s theoretical guarantees and numerical convergence-rate advantage, then identifies bounds, comparisons, and problem scales left for future work. Appendix A derives a policy solution through Lagrangian optimization.
7. Discussion and Future Works
- DPP-RL is reported to outperform other RL methods in convergence rate across the tested MDPs; the authors suggest avoiding a decaying learning step may explain this.The discussion notes that DPP-RL does not rely on stochastic approximation for estimating the optimal policy.Source: §7 · p. 21
- The study focuses on optimal-policy estimation rather than exploration, and excludes PAC-MDP and actor-critic comparisons because their exploration or sampling settings differ.Both DPP-RL and SADPP use a generative model, whereas actor-critic methods use trajectories of state-action pairs.Source: §7 · p. 21
- Approximate DPP is analyzed with ℓ∞-norm performance-loss bounds; establishing bounds in the ℓp norms used by many regression methods remains future work.The discussion proposes bounding each value-function component’s performance loss, following Munos (2005).Source: §7 · p. 21
- Finite-sample PAC bounds for SADPP and DPP-RL remain future work; the SADPP route would require ℓ2 error-propagation analysis and standard regression bounds.The discussion frames these bounds in the spirit of results for fitted value iteration and fitted Q-iteration.Source: §7 · pp. 21–22
- Applying DPP to large-scale action problems remains an extension because exact summations in the update become expensive.The authors propose Monte Carlo estimation of MηΨk(x), its expectation under the soft-max policy πk.Source: §7 · p. 22
Appendix A. Proof of Proposition 1
- Appendix A obtains a policy expression by maximizing a Lagrangian and solving its multipliers under the stated constraints.The derivation substitutes the resulting expression into the optimization in equation (4).Source: Appendix A · p. 22
Appendix B. Proof of Theorem 1
Appendix B derives a finite-iteration performance-loss bound by relating the induced policy’s action values to an auxiliary sequence. Appendix C uses convergence of that sequence to establish convergence of the policy.
Appendix B. Proof of Theorem 1
- The finite-iteration result bounds the performance loss of the policy induced by DPP at step k.The proof proceeds by relating Qπk to Q* through an auxiliary action-value function Qk.Source: Appendix B · p. 23
- The proof bounds the auxiliary action-value error and uses that bound, together with its relation to Ψk, to derive the policy performance bound.Lemma 8 supplies an upper bound on ∥Qk − Q*∥; Lemma 7 relates Ψk to Qk.Source: Appendix B · p. 23
- The auxiliary sequence starts at Q0 = Ψ0 and is iterated by a recursion, with the induced policy πk used in the analysis.The excerpt states the role of the recursion but does not provide its full expression.Source: Appendix B · p. 23
Appendix C. Proof of Theorem 3
- Because Qk converges to Q* and πk consequently converges to π*, Ψk has a limit, which the proof then computes.The stated convergence of Qk is linear; the argument also uses Ψk’s expression in terms of Qk, Q0, and πk−1.Source: Appendix C · p. 24
Appendix D. Proof of Theorem 4
Appendix D establishes an approximate-DPP performance-loss bound through an auxiliary action-value sequence. Appendix E proves DPP-RL convergence by showing its sample errors average to zero under boundedness conditions.
Appendix D. Proof of Theorem 4
- The main approximate-DPP result is a finite-iteration ℓ∞-norm performance-loss bound for the policy induced by the preference iterates.The proof relates Q* to Qπk through an auxiliary action-value function Qk, first bounding Q*−Qk.Source: Appendix D · pp. 24, 28
- The auxiliary sequence is initialized with Q0 = Ψ0, and the proof relates Ψk to Qk before bounding their distance from Q*.The text describes Qk as obtained by iterating a recursion and identifies (35) as an approximate version of (31).Source: Appendix D · pp. 25–26
Appendix E. Proof of Theorem 6
- For DPP-RL, the update is expressed as an approximate-DPP update with sample-estimation error, whose conditional mean given prior samples is zero.The proof treats these errors as martingale differences and establishes their uniform boundedness under its assumptions.Source: Appendix E · pp. 28–29
- Under bounded second moments, the average accumulated DPP-RL estimation error converges almost surely to zero, supporting the stated asymptotic policy convergence.The proof invokes the strong law for martingale differences and combines the result with Corollary 5.Source: Appendix E · pp. 29–30