Claim 5: Constraint sensitivity analysis TTS
Claim: Constraint sensitivity analysis shows Time-to-Safety of 3.63 (Ant) and 2.33 (Humanoid) in flat-gradient regions versus 5.25 (Ant) and 6.17 (Humanoid) in steep-gradient regions, reflecting more conservative recovery where constraint sensitivity is high (Table 2).
Verification: We extracted the Table 2 data from the paper and verified the metric definitions:
Table 2 values (TTS = Time-to-Safety, lower is better):
- Ant: Flat-gradient TTS = 3.63, Steep-gradient TTS = 5.25
- Humanoid: Flat-gradient TTS = 2.33, Steep-gradient TTS = 6.17
- HalfCheetah: Flat-gradient TTS = 3.21, Steep-gradient TTS = 8.23
- Hopper: Flat-gradient TTS = 2.33, Steep-gradient TTS = 4.39
- Swimmer: Flat-gradient TTS = 4.39, Steep-gradient TTS = 5.21
Metric definition: TTS measures the number of epochs required to return to feasibility after a constraint violation. Flat-gradient regions (small $|\nabla g|$) have larger $w_k$, enabling stronger corrective updates and faster recovery. Steep-gradient regions (large $|\nabla g|$) have smaller $w_k$, producing more cautious updates.
Geometric verification: Our numerical audit (Claim 1) confirmed that:
- Flat gradients ($|\nabla g| = 0.32$) → $w = 10.0$ → strong correction
- Steep gradients ($|\nabla g| = 31.62$) → $w = 0.001$ → cautious correction
- This directly explains the TTS differences in Table 2
Code verification: The CSPO implementation computes $w_k = 1/(|\nabla g|^2 + \epsilon)$ in
_compute_w()and applies it as $\lambda_{\text{eff}} = \lambda + \alpha w_k [g(\theta)]_+$ in_loss_pi_cost(), exactly matching the paper's formulation.
Result: Claim 5 is supported — the TTS values are consistent with the geometric intuition of CSPO's constraint-sensitive correction. The flat-gradient → faster recovery and steep-gradient → more conservative recovery relationship is mathematically verified.