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P2BPO: Permeable Penalty Barrier-Based Policy Optimization for Safe RL
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Proposes P2BPO, a PPO-based safe RL method that replaces the hard penalty with a smooth Softplus penalty (permeability β) and an adaptive penalty parameter, with a bound on suboptimality of m·ln(2)/β.
Problem
Safe RL in CMDPs needs to balance reward maximization with cost-constraint satisfaction; existing Lagrangian and penalty methods oscillate and suffer approximation errors.
Approach
Replace the ReLU-style constraint penalty with a smooth Softplus barrier whose sharpness parameter β controls permeability, combined with an adaptive penalty parameter that increases with violations, optimized first-order within PPO.
Key results
- P2BPO achieves the highest reward while satisfying the cost limit in seven of eight environment configurations (e.g., Humanoid Run reward 4632 vs. 4412 for CPPO-PID at cost 0).
- P2BPO marginally misses the cost limit on PointGoal1 (cost 27 vs. limit 25) while obtaining reward 12.
- Theorem 1 bounds the gap between P2BPO and the SRL optimum by m·ln(2)/β.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use P2BPO instead of PPO-Lag | Safe RL policy optimization · Ant Run | reward: 983 ± 2.5 -> 992 ± 0.1 | Helped |
| Use P2BPO instead of PPO-Lag | Safe RL policy optimization · SafeCarCircle | reward: 1223 ± 7.1 -> 1275 ± 5.5 | Helped |
| Use P2BPO instead of PPO-Lag | Safe RL policy optimization · PointGoal1 | reward: 15 ± 0.3 -> 12 ± 0.4 | Hurt |
| Use P2BPO instead of PPO-Lag | Safe RL policy optimization · CarGoal1 | reward: 14 ± 0.5 -> 15 ± 0.5 | Helped |
| Use P2BPO instead of PPO-Lag | Safe RL policy optimization · PointButton1 | reward: 1.2 ± 0.1 -> 3.8 ± 0.2 | Helped |
| Use P2BPO instead of P3O | Safe RL policy optimization · CarButton1 | reward: -1.9 ± 0.2 -> -0.2 ± 0.1 | Helped |
| Vary the permeability parameter β from 1 to 10, 15, or 100 (ablation) | P2BPO · Humanoid Run | reward | Mixed |
| Set the cost limit to 25, 50, or 100 (ablation) | P2BPO · SafeCarCircle | reward | Helped |
Surprising findings
- P2BPO marginally violates the cost limit on PointGoal1 (27 vs. 25), the only environment where it fails to satisfy the constraint with the best reward.
Limitations
- Permeability parameter β is a manually set hyperparameter selected by binary search; performance is sensitive to large changes in β.
- Penalty-based bound depends on approximation error that grows with m and shrinks only as β increases, trading off against conservativeness.
- Evaluated only on Safety Gym, Bullet Safety Gym, and Safe MuJoCo locomotion and navigation tasks.
Experimental setup
Datasets: Safety Gym Bullet Safety Gym Safe MuJoCo
Benchmarks: Safety Gym Bullet Safety Gym Safe MuJoCo
Metrics: episodic reward episodic cost 95% confidence interval
Baselines: CPO FOCOPS PCPO PPO-Lag CPPO-PID APPO IPO P3O
Builds on: PPO CPO PPO-Lagrangian P3O IPO
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