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Faster Fixed-Point Methods for Multichain MDPs
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Develops faster value-iteration methods for general (multichain) average-reward MDPs by combining a shifted Halpern iteration with a gain-estimation phase and discounted VI with sublinear value-error reduction, achieving rates in terms of a sharper gain-dropping time parameter.
Problem
Solving general (multichain) average-reward MDPs with value iteration nonasymptotically, where the optimal policy must solve a navigation subproblem (reaching the best recurrent class) as well as the recurrent optimization problem; prior bounds degrade with large complexity parameters like B and 1/D
Approach
Introduces the gain-dropping time parameter T_drop as a sharper complexity measure, shows restricted commutativity of the Bellman operator with the optimal gain, and designs an approximately shifted Halpern iteration (gain estimate from Picard steps then Halpern) plus an average-to-discounted reduction with a warm-start discounted VI, yielding O(1/n) rates independent of 1/Delta.
Key results
- Theorem gives policy suboptimality bound (10/3 T_drop + 13 + 35/n + 20/n^2)/n * ||h0-h||, improving the O(1+1/Delta) dependence of prior Halpern-based work.
- Discounted warm-start Halpern-then-Picard achieves ||T_gamma(V)-V|| bounds of order gamma^n / sum gamma^i, matching a lower bound up to constant 8e.
- Final average-reward result: rho* - rho^pi <= (T_drop+1)(71 min{span h, ...}+2)/(n-1) with gamma=1-1/n.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use gain-dropping time \Tdrop in the policy-suboptimality bound instead of the prior bound's transient-time parameter \mathsf{B} | Approximately Shifted Halpern Iteration versus discounted reduction prior work · General multichain average-reward MDPs | policy suboptimality: O(\frac{\Tdrop + 1}{n} \tinfnorm{h_0 - h}) versus O((\mathsf{B} + \tspannorm{h^{\pi^\star}})\frac{\log n}{n}) | Helped |
| Use gain-dropping time \Tdrop rather than the prior dependence on 1/\Delta | Approximately Shifted Halpern Iteration versus prior Halpern iteration · General multichain average-reward MDPs | policy suboptimality: O(\frac{\Tdrop + 1}{n} \tinfnorm{h_0 - h}) versus O(\frac{1 + 1/\Delta}{n}\tinfnorm{h_0 - h}) | Helped |
| Add a Picard phase after approximately 1/(1-\gamma) Halpern iterations | Halpern-Then-Picard · Contractive fixed-point problems in Banach spaces | fixed-point residual: 8e/(1+\gamma) | Helped |
| Warm-start discounted value iteration using undiscounted value iterations and re-normalization | Warm-Start Halpern-Then-Picard · Discounted MDPs | discounted value error: O(1/t) | Helped |
| Set \gamma = 1 - 1/n and apply the warm-start discounted algorithm | Discounted value iteration with warm start · General multichain average-reward MDPs | policy suboptimality: \left(\Tdrop + 1 \right) \frac{71\min\left\{\spannorm{h}, \spannorm{h^{\pi^\star}} + \Tdrop + \spannorm{h^{\pi^\star}} \ | Helped |
| Use the gain-estimation phase and initialize Halpern iteration with its output | Approximately Shifted Halpern Iteration · General multichain average-reward MDPs | policy suboptimality: \frac{\frac{10}{3}\Tdrop + 13 + \frac{35}{n} + \frac{20}{n^2}}{n}\infnorm{h_0 - h} | Helped |
Surprising findings
- General multichain MDPs are no harder than weakly communicating MDPs in terms of fixed-point error, disproving a conjecture in prior work.
- Lower-bound instance for discounted value error can be modified so that sublinear O(1/t) value-error reduction is possible when complexity parameters are restricted.
Limitations
- Bellman operators assumed exactly computable; stochastic evaluations not studied.
- Lower bound for policy suboptimality in general MDPs not established; gap to fixed-point error rate open outside large-n regime.
Experimental setup
Metrics: policy suboptimality (gain) fixed-point error (Bellman residual) discounted value error
Baselines: Halpern iteration for average-reward MDPs (Zurek and Chen) average-to-discounted reduction with bounded transient time B (Wang et al.) anti-optimality operator discounted VI
Builds on: Halpern iteration discounted-to-average reduction (Wang et al.) Halpern iteration for average-reward MDPs (Zurek & Chen)
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