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Optimal Strong Regret and Violation in Constrained MDPs via Policy Optimization
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Proposes CPD-PO, an efficient primal-dual policy optimization algorithm for episodic CMDPs that achieves optimal O~(sqrt(T)) strong regret and strong violation, using PO-DB as primal learner and a UCB-like binary dual update.
Problem
Online learning in episodic constrained MDPs with bandit feedback, aiming for sublinear strong regret and strong cumulative constraint violation using efficient policy optimization rather than occupancy-measure linear programming.
Approach
A primal-dual scheme pairing an adversarial-MDP policy optimization learner (PO-DB) with a UCB-like dual update that chooses each Lagrange multiplier from {0, (L+1)/rho}, without regularization or occupancy-measure projection, which yields optimal sqrt(T) strong regret and violation.
Key results
- Algorithm CPD-PO attains R_T = O~(L^3 m |X| sqrt(|A| T) + L^5 m / rho) strong regret with high probability, improving on the O~(T^0.93) bound of prior policy-optimization work.
- Strong constraint violation V_T = O~(L^3 m |X| sqrt(|A| T) + L^5 m), with no dependence on Slater parameter rho.
- Matches the optimal O~(sqrt(T)) order for strong regret and violation while avoiding occupancy-measure projections.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CPD-PO, combining PO-DB as the primal algorithm with a UCB-like dual update, instead of the prior primal-dual policy optimization method | Online learning in episodic CMDPs · Episodic CMDPs | strong regret: \widetilde{\mathcal{O}}(T^{0.93}) -> \widetilde{\mathcal{O}}(\sqrt{T}) | Helped |
| Use CPD-PO, combining PO-DB as the primal algorithm with a UCB-like dual update, instead of the prior primal-dual policy optimization method | Online learning in episodic CMDPs · Episodic CMDPs | strong constraint violation: \widetilde{\mathcal{O}}(T^{0.93}) -> \widetilde{\mathcal{O}}(\sqrt{T}) | Helped |
Limitations
- Bounds depend on |X| and L^3 m factors, which are not optimal in all parameters.
- Theoretical results only; no empirical experiments reported in the provided text.
- Requires Slater's condition with known-order margin parameter rho.
Experimental setup
Metrics: strong regret strong constraint violation
Baselines: LP-based occupancy-measure algorithm of Efroni et al. (2020) Stradi et al. regularized primal-dual (T^0.93) OptDual-CMDP OptPrimalDual-CMDP
Builds on: PO-DB (policy optimization with dilated bonus) Efroni et al. 2020 CMDP primal-dual Stradi et al. 2024 (T^0.93 strong regret) Slater's condition
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