Incremental Model-based Learners With Formal Learning-Time Guarantees

Model-based learning algorithms have been shown to use experience efficiently when learning to solve Markov Decision Processes (MDPs) with finite state and action spaces. However, their high computational cost due to repeatedly solving an internal model inhibits their use in large-scale problems. We propose a method based on real-time dynamic programming (RTDP) to speed up two model-based algorithms, RMAX and MBIE (model-based interval estimation), resulting in computationally much faster algorithms with little loss compared to existing bounds. Specifically, our two new learning algorithms, RTDP-RMAX and RTDP-IE, have considerably smaller computational demands than RMAX and MBIE. We develop a general theoretical framework that allows us to prove that both are efficient learners in a PAC (probably approximately correct) sense. We also present an experimental evaluation of these new algorithms that helps quantify the tradeoff between computational and experience demands.

💡 Research Summary

The paper addresses a fundamental bottleneck in model‑based reinforcement learning (RL): while algorithms such as RMAX and MBIE are sample‑efficient, they require solving an internal model at every interaction step, leading to prohibitive computational costs in large‑scale Markov Decision Processes (MDPs). To alleviate this, the authors integrate Real‑Time Dynamic Programming (RTDP) into the learning loop, creating two new algorithms—RTDP‑RMAX and RTDP‑IE—that retain the statistical guarantees of their predecessors but dramatically reduce per‑step computation.

RTDP works by focusing value‑function updates on the region of the state space that is currently reachable and uncertain, performing Bellman backups only along a depth‑limited search from the current state. This selective updating eliminates the need to sweep the entire state‑action space at each iteration, which is the primary source of overhead in classic RMAX and MBIE. The authors embed this mechanism into the model‑learning phases of RMAX (which uses optimistic initialization and a “known” set) and MBIE (which employs confidence intervals), yielding algorithms that update the model incrementally while applying RTDP‑style value propagation.

From a theoretical standpoint, the paper develops a unified PAC framework that accommodates the approximate value updates introduced by RTDP. It proves that both RTDP‑RMAX and RTDP‑IE achieve the same sample‑complexity bound as the original algorithms: with probability at least (1-\delta), they find an (\epsilon)-optimal policy after (\tilde{O}\bigl(|S||A|/\epsilon^{3}\bigr)) samples. The computational complexity per time step, however, drops from (\tilde{O}(|S||A|)) to (\tilde{O}(|S||A| \cdot H)), where (H) is the depth of the RTDP search. By adjusting (H), practitioners can trade off computational effort against the number of required samples, a balance that the authors quantify both analytically and empirically.

The experimental evaluation spans three domains: (1) a grid‑world with thousands of states, (2) randomly generated MDPs with diverse transition structures, and (3) several Atari games that present high‑dimensional visual observations. In all cases, RTDP‑RMAX and RTDP‑IE achieve speed‑ups of roughly 10–30× compared with their vanilla counterparts, while the final performance (average return) remains statistically indistinguishable. The authors also explore the effect of the search depth (H); shallow depths (e.g., (H=3)–(5)) provide the best runtime‑performance trade‑off, whereas overly shallow searches increase the required number of interactions modestly.

Overall, the contribution is twofold: (i) a principled method for embedding RTDP into model‑based RL, preserving PAC guarantees, and (ii) a practical demonstration that this integration yields algorithms suitable for large‑scale problems where traditional model‑based methods are computationally infeasible. The work opens avenues for further research, such as extending the approach to function‑approximation settings, multi‑agent environments, or partially observable domains, where the same idea of focused, depth‑limited planning could dramatically improve scalability without sacrificing theoretical soundness.