Study 03: Why higher rank is not always better

Intervention Analysis

We perturb the same Spot-flat policies by doubling PPO epochs per update from 5 to 10, producing the intentionally collapsed regime described in the paper. We then measure effective rank, the swing–stance phase gap, and reward in the same runs.

Jacobian effective rank reaches a uniform ceiling across swing and stance, and feature rank rises sharply. At the same time, the swing-dominant phase gap falls to approximately zero. Higher rank therefore coincides with a loss of phase-dependent structure, rather than a healthier representation. The disappearance of this pattern after changing one training setting supports its interpretation as learned structure.

Effective rank increases and the phase gap approaches zero under over-training, while gait reward changes little — Representation collapse under over-training (healthy baseline desaturated, over-trained saturated). (a) Feature-space erank and functional Jacobian erank both increase. (b) The phase gap Δφ approaches zero for SimBa, while MLP remains near zero. (c) Standard gait reward is insensitive to the loss of phase-dependent structure; the large SimBa configuration even receives a higher reward.

Gram matrix under healthy and collapsed training — Gram matrix visualization. (a) Healthy training produces well-defined geometric structure. (b) Collapsed training yields smooth, non-linear patterns, reflecting a loss of structure.

Results

Quantity	Effect of over-training
PPO learning epochs per iteration	5 → 10
SimBa feature effective rank	+66%
Jacobian effective rank	saturates to a uniform ceiling
Phase gap Δφ, SimBa	→ ≈ 0
Phase gap Δφ, MLP	flat near zero throughout
Gait reward	largely unchanged, or increasing despite collapse

In-distribution tracking and reward remain largely unchanged. Phase-conditioned rank therefore identifies a change in the over-trained policies that reward alone would not reveal.