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Tilted Quantile Gradient Updates for Quantile-Constrained Reinforcement Learning
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Proposes Tilted Quantile Policy Optimization (TQPO), a PPO-based quantile-constrained safe RL method that estimates quantile gradients by sampling and uses an asymmetric (tilted) update rate for the Lagrange multiplier.
Problem
Safe RL with probabilistic safety requires bounding the cost quantile (chance constraint) rather than the expected cost, but quantile gradients are hard to estimate and the Lagrange multiplier overshoots, causing over-conservatism and low return.
Approach
Estimate quantile gradients directly via a likelihood-ratio-style sampling estimator (no expectation-form approximation or distribution assumption), and replace the fixed Lagrange multiplier step size with a tilted, pinball-loss-like step size (eta+ and eta- from the CDF of the quantile at the threshold) that speeds up the decrease of lambda after overshoot.
Key results
- TQPO achieves higher return than QCPO on all tasks, e.g. SimpleEnv 95%: 14.2±0.3 vs 2.1±0.2 return with safety probability 95% vs 97%.
- On DynamicEnv 95%, TQPO reaches return 12.5±0.2 vs QCPO 3.5±0.1 with safety probability 94%.
- PPO-Lag's 90% cost quantile significantly exceeds the threshold and its safety probability is below 70%, while quantile-constrained methods satisfy the constraint.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use quantile constraints and sampling-based quantile gradient estimation instead of QCPO's approximation | TQPO · SimpleEnv 95% and DynamicEnv 95% | Return: 2.1±0.2 -> 14.2±0.3; 3.5±0.1 -> 12.5±0.2 | Helped |
| use quantile constraints instead of expectation constraints | TQPO vs PPO-Lag · SimpleEnv, DynamicEnv, and GremlinEnv | Safety probability | Helped |
| add tilted update to QCPO (ablation) | QCPO · SimpleEnv 95% and DynamicEnv 95% | Return: 2.1±0.2 -> 4.7±0.2; 3.5±0.1 -> 8.0±0.3 | Helped |
| add tilted update to QCPO (ablation) | QCPO · SimpleEnv 95% and DynamicEnv 95% | Safety probability: 97%±1% -> 95%±1%; 97%±0% -> 95%±1% | Mixed |
| add tilted update (ablation) | TQPO · SimpleEnv 95% and DynamicEnv 95% | Return: 10.3±0.3 -> 14.2±0.3; 5.26±0.2 -> 12.5±0.2 | Helped |
| add tilted update (ablation) | TQPO · SimpleEnv 95% and DynamicEnv 95% | Safety probability: 96%±1% -> 95%±1%; 95%±1% -> 94%±1% | Mixed |
| replace adaptive tilt with fixed rates η+=0.2, η-=0.8 (ablation) | TQPO · SimpleEnv 95% and DynamicEnv 95% | Return: 13.9±0.8 -> 14.2±0.3; 11.9±0.2 -> 12.5±0.2 | Hurt |
| replace adaptive tilt with fixed rates η+=0.2, η-=0.8 (ablation) | TQPO · SimpleEnv 95% and DynamicEnv 95% | Safety probability: 92%±2% -> 95%±1%; 92%±1% -> 94%±1% | Hurt |
Surprising findings
- The expectation-constrained PPO-Lag satisfies its expected-cost constraint yet has safety probability below 70%, showing expectation constraints fail for chance constraints.
- A naive fixed tilt (eta+=0.2, eta-=0.8) leads to undershooting and a biased safety probability of 92% in ablation.
Limitations
- Convergence proof relies on strong assumptions (Lipschitz gradients, bounded log-gradients, strictly concave objective, Slater's condition).
- Evaluated only on three Safety Gym-style navigation tasks with five seeds.
- Quantile CDF F_q(d) estimated per epoch by sampling; the tilt is heuristic.
Experimental setup
Datasets: SimpleEnv DynamicEnv GremlinEnv
Benchmarks: SimpleEnv DynamicEnv GremlinEnv
Metrics: average episode return safety probability average episode cost cost quantile training time
Baselines: QCPO PPO-Lag
Builds on: PPO QCPO PPO-Lag Safety Gym
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