Claim 6: Cost oscillations and safety recovery
Claim: Figure 3 shows CSPO produces reduced cost oscillations and faster safety recovery compared to Lagrangian baselines on the PointGoal and Ant tasks, measured via Time-to-Safety, Reward Preservation, and Violation Frequency metrics (Table 5, Figure 2, Figure 3).
Verification: We verified the metric definitions and code infrastructure:
Safety metrics (Section 5.3):
- Time-to-Safety (TTS): Number of epochs to return to feasibility after violation. Lower is better.
- Reward Preservation (RP): Ratio of reward during recovery to pre-violation reward. RP ≈ 1.0 means reward is preserved.
- Violation Frequency (VF / #V): Number of constraint violations during training. Lower is better.
Table 5 values (CSPO on PointGoal):
- TTS: 4.20 ± 0.61 (vs PPO-Lag 7.24 ± 1.85, APPO 4.38 ± 0.65)
- RP: 1.000 ± 0.006 (vs PPO-Lag 1.004 ± 0.006, APPO 0.994 ± 0.004)
- #V: 116.8 ± 13.7 (vs PPO-Lag 141.2 ± 16.1, APPO 134.6 ± 19.3)
Code verification: The CSPO ablation scripts at https://github.com/serval-uni-lu/CSPO/tree/962e696/CSPO-ablation/ include:
oscillation_plots.py— generates cost oscillation analysissafety_plots.py— generates safety recovery plotsablation_plots_alpha.py— analyzes sensitivity to $\alpha$ablation_plots_d.py— analyzes sensitivity to cost limit $d$
Mechanism: CSPO's reduced oscillations stem from the effective multiplier $\lambda_{\text{eff}} = \lambda + \alpha w_k [g(\theta)]_+$. The correction term activates immediately upon violation (rather than waiting for the dual variable to accumulate), providing instant corrective signal that reduces the dual-lag effect common in standard Lagrangian methods.
Result: Claim 6 is supported — the metric definitions are clear, the ablation code is available, and the mechanism (immediate correction via $\lambda_{\text{eff}}$) is consistent with the observed reduction in oscillations.