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On the Sample Complexity Bounds of Bilevel Reinforcement Learning
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Derives the first sample complexity bound (Õ(ε^-3)) for bilevel reinforcement learning in continuous state-action spaces via a penalty-based first-order, Hessian-free algorithm, and extends the bound to general non-convex-lower-level bilevel optimization.
Problem
Sample complexity of bilevel reinforcement learning (BRL) in continuous/parameterized settings with non-convex lower-level objectives and biased MDP gradient estimates, where existing analyses are limited to tabular settings or require second-order terms.
Approach
Uses a penalty-based proxy objective with closed-form MDP gradients for the reward parameter; treats the inner gradient-descent steps as SGD on a PL-satisfying loss, and derives a recursive optimality-gap lemma for biased SGD to obtain a first-order, Hessian-free sample complexity bound.
Key results
- Establishes Õ(ε^-3) sample complexity for the proposed first-order BRL algorithm in continuous state-action spaces.
- For standard bilevel optimization with non-convex lower level and unbiased gradients, achieves Õ(ε^-3) versus prior Õ(ε^-6) bound.
- Proof-of-concept experiments on MuJoCo tasks are reported to show the first-order algorithm works in practice.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed first-order BRL algorithm in continuous state-action spaces | BRL · Continuous-state/action BRL | sample complexity: $\tilde{\mathcal{O}}(\epsilon^{-3.5})$ -> $\tilde{\mathcal{O}}(\epsilon^{-3})$ | Helped |
| Use the proposed approach with unbiased gradients for both levels | Standard bilevel optimization with non-convex lower level · Bilevel optimization with non-convex lower level | sample complexity: $\tilde{\mathcal{O}}(\epsilon^{-6})$ -> $\tilde{\mathcal{O}}(\epsilon^{-3})$ | Helped |
Limitations
- Relies on strong assumptions including PL condition on penalized objective and bounded/unbiased gradient variance for the upper level.
- Sample complexity bound includes an approximation error term from neural Q-function approximation (epsilon_approx) that does not vanish.
- Empirical evaluation is only a proof-of-concept on MuJoCo tasks.
Experimental setup
Benchmarks: MuJoCo tasks
Metrics: sample complexity bound (theoretical)
Baselines: PARL penalty-based bilevel methods with O(ε^-6) sample complexity
Builds on: penalty-based bilevel optimization policy gradient theorem PARL
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