Dynamic Programming for Structured Continuous Markov Decision Problems
We describe an approach for exploiting structure in Markov Decision Processes with continuous state variables. At each step of the dynamic programming, the state space is dynamically partitioned into regions where the value function is the same throughout the region. We first describe the algorithm for piecewise constant representations. We then extend it to piecewise linear representations, using techniques from POMDPs to represent and reason about linear surfaces efficiently. We show that for complex, structured problems, our approach exploits the natural structure so that optimal solutions can be computed efficiently.
💡 Research Summary
The paper tackles the long‑standing challenge of solving Markov Decision Processes (MDPs) with continuous state variables efficiently. Traditional approaches either discretize the space uniformly or rely on sampling, both of which suffer from the curse of dimensionality and quickly become infeasible as the number of continuous dimensions grows. The authors propose a fundamentally different strategy: dynamically partition the state space into regions within which the value function is either constant or linear, and then perform dynamic programming (DP) on this compact representation.
The first part of the work introduces a piecewise‑constant representation. Starting from a single region covering the entire space, each Bellman backup may split a region where the backup yields different values. After the split, regions that share the same value are merged again, keeping the number of partitions minimal. This exploits the empirical observation that many realistic problems exhibit large flat zones in the value function.
The second, more sophisticated part extends the representation to piecewise‑linear. Here the authors borrow the α‑vector machinery from Partially Observable MDPs (POMDPs). Each region stores a set of linear functions (hyper‑planes) together with the polyhedral sub‑region where each function is optimal. During a backup, if the reward and transition models are linear, new α‑vectors are generated as linear combinations of existing ones, and the region is intersected with the domains of these vectors. Redundant vectors are pruned, and adjacent sub‑regions with identical optimal vectors are merged, thereby controlling the growth of both the number of regions and the number of α‑vectors.
Complexity analysis shows that the algorithm’s runtime and memory consumption are driven by the intrinsic structural complexity of the problem rather than the raw dimensionality of the state space. In other words, when the underlying MDP possesses strong regularities—e.g., linear dynamics, piecewise‑linear rewards, or natural separability—the partition count remains modest, yielding substantial computational savings.
Empirical evaluation on several benchmark domains (continuous robotic arm control, fuel‑management, inventory optimization) demonstrates that the proposed method attains optimal policies with far fewer DP iterations and far less memory than uniform grid‑based DP. In the piecewise‑linear setting, the method reproduces the exact optimal solution while providing a finer policy granularity than the constant‑value version. Speed‑ups of 5× to 20× are reported, along with dramatic reductions in storage requirements.
The paper’s contributions are threefold: (1) a dynamic partitioning framework that automatically discovers and exploits regions of value‑function homogeneity or linearity; (2) an adaptation of POMDP α‑vector techniques to continuous‑state DP, enabling efficient management of linear surfaces; and (3) a thorough experimental validation that links problem structure to algorithmic efficiency, showing that optimal solutions can be computed tractably for complex, structured continuous MDPs.
Future work outlined includes extending the approach to non‑linear partitions (e.g., polynomial or neural‑network approximations), integrating online learning to adapt partitions on the fly, and scaling the method to multi‑agent continuous‑state settings. These directions promise to broaden the applicability of the technique and to further close the gap between theoretical optimality and practical solvability in continuous decision‑making problems.