Evaluating Factor-Wise Auxiliary Dynamics Supervision for Latent Structure and Robustness in Simulated Humanoid Locomotion

Chayanin Chamachot

cs.ROMar 22, 2026

Evaluating Factor-Wise Auxiliary Dynamics Supervision for Latent Structure and Robustness in Simulated Humanoid Locomotion

Authors: Chayanin Chamachot

Organizations: Department of Computer Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand

Abstract

We evaluate whether factor-wise auxiliary dynamics supervision produces useful latent structure or improved robustness in simulated humanoid locomotion. DynaMITE -- a transformer encoder with a factored 24-d latent trained by per-factor auxiliary losses during proximal policy optimization (PPO) -- is compared against Long Short-Term Memory (LSTM), plain Transformer, and Multilayer Perceptron (MLP) baselines on a Unitree G1 humanoid across four Isaac Lab tasks. The supervised latent shows no evidence of decodable or functionally separable factor structure: probe R^2 ~ 0 for all five dynamics factors, clamping any subspace changes reward by < 0.05, and standard disentanglement metrics (MIG, DCI, SAP) are near zero. An unsupervised LSTM hidden state achieves higher probe R^2 (up to 0.10). A 2x2 factorial ablation (n = 10 seeds) isolates the contributions of the tanh bottleneck and auxiliary losses: the auxiliary losses show no measurable effect on either in-distribution (ID) reward (+0.03, p = 0.732) or severe out-of-distribution (OOD) reward (+0.03, p = 0.669), while the bottleneck shows a small, consistent advantage in both regimes (ID: +0.16, p = 0.207; OOD: +0.10, p = 0.208). The bottleneck advantage persists under severe combined perturbation but does not amplify, indicating a training-time representation benefit rather than a robustness mechanism. LSTM achieves the best nominal reward on all four tasks (p < 0.03); DynaMITE degrades less under combined-shift stress (2.3% vs. 16.7%), but this difference is attributable to the bottleneck compression, not the auxiliary supervision. For locomotion practitioners: auxiliary dynamics supervision does not produce an interpretable estimator and does not measurably improve reward or robustness beyond what the bottleneck alone provides; recurrent baselines remain the stronger choice for nominal performance.

Figures & tables

Figure 1: Overview of the DynaMITE architecture. A two-layer transformer encoder processes an 8-step (160 ms) observation–action history to produce a 24-dimensional factored latent vector $\bm{z}\in\mathbb{R}^{24}$ , decomposed into five factor subspaces (friction, mass, motor strength, contact stiffness, action delay). Each subspace is trained with a dedicated auxiliary dynamics-prediction loss during PPO training. The latent is concatenated with the current observation and fed to the policy $\pi(a\mid s,\bm{z})$ and value $V(s,\bm{z})$ heads. Auxiliary losses are active only during training.

Model	History	Latent	Aux Loss	Params
MLP	None	No	No	266–362k
LSTM	Hidden state	No	No	176–215k
Transformer	8 steps	No	No	330–342k
DynaMITE	8 steps	24-d factored	Yes	342–392k

Table 1: Model architecture summary. Parameter counts vary by task due to different observation dimensions.

Method	Flat	Push	Rand.	Terrain
MLP	$-4.83$	$-5.01$	$-5.32$	$-4.82$
LSTM	$\mathbf{-4.01}$	$\mathbf{-4.30}$	$\mathbf{-4.18}$	$\mathbf{-4.06}$
Transf.	$-5.02$	$-4.83$	$-4.77$	$-4.46$
DynaMITE	$-4.88$	$-4.60$	$-4.48$	$-4.49$

Table 2: In-distribution reward comparison (5 seeds, deterministic eval). Bold indicates best per task. All LSTM vs. DynaMITE paired

t

-tests are significant (

p<0.03

Figure 2: In-distribution reward (5 seeds, deterministic evaluation). LSTM achieves the best reward on all tasks; DynaMITE ranks second on three of four tasks but is significantly worse than LSTM on all.

Task	Mean Diff	$t$	$p$
Flat	$+0.870$	8.49	0.0011
Push	$+0.305$	4.69	0.0094
Randomized	$+0.303$	3.34	0.029
Terrain	$+0.435$	8.55	0.0010

Table 3: Paired

t

-tests: LSTM vs. DynaMITE (matched training seeds,

n=5

Method	L0 (ID)	L1	L2	L4
DynaMITE	$-4.38$	$-4.47$	$-4.50$	$\mathbf{-4.63}$
LSTM	$\mathbf{-3.56}$	$\mathbf{-4.23}$	$\mathbf{-4.40}$	$-5.12$
Transf.	$-4.67$	$-4.73$	$-4.72$	$-4.86$
MLP	$-5.68$	$-5.61$	$-5.55$	$-5.64$

Table 4: Combined-shift reward across severity levels (randomized task, 5 seeds, 50-episode deterministic eval per level). Bold indicates best at each level.

Method	L0	L1	L2	L3	L4
DynaMITE	$2.15$	$2.56$	$3.18$	$4.23$	$6.13$
LSTM	$\mathbf{1.31}$	$\mathbf{2.19}$	$\mathbf{3.01}$	$4.59$	$6.94$
Transf.	$2.08$	$2.58$	$3.17$	$4.35$	$6.48$
MLP	$2.24$	$2.76$	$3.40$	$4.47$	$6.30$

Table 5: Combined-shift tracking error across severity levels (randomized task, 5 seeds, 50-episode deterministic eval per level).

Figure 3: Combined-shift stress test (randomized task, 5 seeds). LSTM achieves the best reward at low severity but degrades steeply; DynaMITE’s reward is lower at baseline but more stable. The crossover occurs at severity level 3. Neither model dominates across all levels.

Model	ID	Severe	Worst	Degrad.
DynaMITE	$-4.48$	$-4.58$	$-4.63$	$\mathbf{2.3\%}$
LSTM	$-4.18$	$-4.88$	$-5.12$	$16.7\%$
Transf.	$-4.77$	$-4.81$	$-4.86$	$0.8\%$
MLP	$-5.32$	$-5.66$	$-5.68$	$6.4\%$

Table 6: Severe OOD degradation (combined-shift sweep, randomized task, 5 seeds, deterministic eval). Degradation

=|\text{Severe}-\text{ID}|

; percentages relative to

|\text{ID}|

Figure 4: Pareto front: in-distribution reward vs. severe OOD reward. No model dominates both axes. LSTM achieves the best ID reward; DynaMITE has the highest mean severe OOD reward.

Model	ID	Severe OOD	Worst
DynaMITE	$-4.61$	$\mathbf{-4.58}$	$\mathbf{-4.78}$
LSTM	$\mathbf{-4.14}$	$-4.67$	$-5.12$
Transf.	$-4.77$	$-4.85$	$-5.08$
MLP	$-5.00$	$-5.76$	$-5.97$

Table 7: Pareto analysis (5 seeds, deterministic eval): ID reward

=

4-task mean; Severe OOD Mean

=

average across three randomized-task sweeps.

Push	DynaMITE	LSTM	Transf.	MLP
1.0	$\mathbf{5.6}$	$9.2$	$6.7$	$6.3$
2.0	$\mathbf{5.9}$	$15.9$	$8.1$	$7.6$
3.0	$\mathbf{6.0}$	$20.6$	$8.3$	$7.8$
4.0	$\mathbf{6.0}$	$19.7$	$8.3$	$8.1$
5.0	$\mathbf{6.1}$	$19.8$	$8.0$	$8.2$
6.0	$\mathbf{6.1}$	$19.5$	$8.1$	$8.7$

Table 8: Steps to recover command tracking after push (flat task, 5 seeds

\times

50 episodes per magnitude). Bold indicates fastest.

Push (m/s)	DynaMITE	LSTM
1.0	$3.96\pm 0.53$	$\mathbf{3.05\pm 0.14}$
2.0	$4.03\pm 0.37$	$\mathbf{3.32\pm 0.08}$
3.0	$4.36\pm 0.39$	$\mathbf{3.78\pm 0.08}$
4.0	$5.01\pm 0.26$	$\mathbf{4.62\pm 0.07}$
5.0	$5.76\pm 0.22$	$\mathbf{5.50\pm 0.11}$
6.0	$6.74\pm 0.17$	$\mathbf{6.41\pm 0.08}$

Table 9: Peak tracking error after push (flat task, 5 seeds

\times

50 episodes). Bold indicates lower peak error.

Method	0	0.5–1	1–2	2–3	3–5	5–8	S.
DynaMITE	$-4.37$	$-4.44$	$-4.50$	$-4.56$	$\mathbf{-4.63}$	$\mathbf{-4.78}$	0.41
LSTM	$\mathbf{-3.58}$	$\mathbf{-4.05}$	$\mathbf{-4.23}$	$\mathbf{-4.45}$	$-4.70$	$-5.09$	1.52
Transf.	$-4.65$	$-4.72$	$-4.76$	$-4.81$	$-4.94$	$-5.08$	0.42
MLP	$-5.65$	$-5.64$	$-5.79$	$-5.75$	$-5.89$	$-5.97$	$\mathbf{0.33}$

Table 10: Push magnitude sweep (randomized task, 5 seeds, 50-ep eval per level). Bold indicates best.

Sweep	Task	DynaMITE	LSTM	Cross.
Combined	rand.	$\mathbf{2.3\%}$	16.7%	L3
Push mag.	rand.	$\mathbf{5.1\%}$	17.1%	L4
Push mag.	push	$\mathbf{4.4\%}$	12.2%	L5
Push mag.	terrain	$\mathbf{6.6\%}$	21.5%	L5
Friction	rand.	$-0.4\%$ *	1.5%	None

Table 11: Severe OOD degradation across all sweep types and tasks (5 seeds). DynaMITE degrades less than LSTM on all push-related sweeps; crossover

=

level where DynaMITE first matches or exceeds LSTM.

Figure 5: OOD sweep comparison across four models and five seeds. LSTM degrades more steeply than DynaMITE under push magnitude perturbation across all three tasks.

Figure 6: Latent correlation heatmap showing the Pearson correlation between each factor subspace dimension and the corresponding ground-truth dynamics parameter.

Factor (dims)	Avg $\|\Delta\text{Reward}\|$	Interpretation
Friction (0–3)	0.007	Negligible
Mass (4–9)	0.012	Negligible
Motor (10–15)	0.021	Negligible
Contact (16–19)	0.020	Negligible
Delay (20–23)	0.020	Negligible

Table 12: Latent intervention analysis: average absolute reward change per factor subspace (3 seeds

\times

5 factors

\times

3 DR levels

=

90 conditions, 50-ep eval).

Factor	DM Lin.	DM MLP	LSTM Lin.	LSTM MLP
Friction	$-0.000$	$-0.001$	$0.026$	$\mathbf{0.101}$
Mass	$0.000$	$-0.002$	$0.012$	$\mathbf{0.018}$
Motor	$0.000$	$-0.002$	$0.010$	$\mathbf{0.015}$
Contact	$0.000$	$-0.000$	$0.006$	$\mathbf{0.045}$
Delay	$0.000$	$-0.000$	$0.011$	$\mathbf{0.041}$
Overall	0.000	$-$ 0.001	0.013	0.044

Table 13: Latent probe

R^{2}

for DynaMITE vs. LSTM (5 seeds, 5-fold CV,

\sim

36k samples). Bold indicates best per factor.

Variant	Reward	$\Delta$	$p$	Worse
Full	$\mathbf{-4.46}$	—	—	—
No Aux	$-4.51$	$-0.05$	0.641	6/10
No Latent	$-4.64$	$-0.18$	0.225	9/10
Single	$-4.69$	$-0.23$	$\mathbf{0.048}$	8/10
Aux Only	$-4.64$	$-0.18$	0.251	7/10

Table 14: Ablation study on the randomized task (10 seeds, paired

t

-test vs. full DynaMITE). Bold

p

-value indicates

p<0.05

	No Aux Loss	Aux Loss
Bottleneck	$-4.51\pm 0.20$	$\mathbf{-4.46\pm 0.25}$
No Bottleneck	$-4.64\pm 0.36$	$-4.64\pm 0.34$

Table 15:

2\times 2

factorial: bottleneck vs. auxiliary loss (10 seeds, randomized task). Each cell averages 10 paired seeds.

	No Aux Loss	Aux Loss
Bottleneck	$-4.59\pm 0.21$	$\mathbf{-4.55\pm 0.22}$
No Bottleneck	$-4.68\pm 0.28$	$-4.66\pm 0.23$

Table 16:

2\times 2

factorial under severe combined-shift (avg. of Levels 3–4, 10 seeds, 50-ep eval per level).

Variant	ID	Severe	$\Delta$	Degrad.
Full (B+A)	$-4.46$	$-4.55$	$-0.09$	$2.1\%$
No Aux (B)	$-4.51$	$-4.59$	$-0.08$	$1.9\%$
No Latent	$-4.64$	$-4.68$	$-0.04$	$0.9\%$
Aux Only (A)	$-4.64$	$-4.66$	$-0.02$	$0.5\%$

Table 17: Factorial degradation analysis (10 seeds, randomized task). Degradation

=|\text{Severe}-\text{ID}|/|\text{ID}|

Claim	Evidence	Status
Latent decodable?	Probe $R^{2}\approx 0$ ; MIG/DCI/SAP $\approx 0$	No
Factor-specific effect?	$\|\Delta r\|<0.05$	No evidence
Aux heads learn targets?	MSE $\approx 3.7$ (48% drop)	Not accurately
LSTM best ID?	$p<0.03$ , all 4 tasks	Yes
DynaMITE lower OOD sens.?	0.25 vs. 1.57	Directional
DynaMITE faster recovery?	${\sim}6$ vs. 9–21 steps	Directional

Table 18: Summary of key claims and evidence status.

Factor	Mean $\cos(\nabla_{\text{PPO}},\nabla_{\text{aux}})$	Std
Friction	$-0.001$	0.005
Mass	$-0.002$	0.010
Motor	$-0.005$	0.016
Contact	$-0.008$	0.026
Delay	$-0.002$	0.009

Table 19: Cosine similarity between PPO and auxiliary gradients (3 seeds, averaged over training).

Figure 7: Cosine similarity between PPO and auxiliary gradients over training (3 seeds). All five factor-specific auxiliary gradients remain orthogonal to the PPO gradient ( $|\cos|<0.01$ ) throughout the entire training run.

Metric	DynaMITE (24-d)	LSTM (128-d)
Effective rank	$\mathbf{4.78\pm 0.72}$	$32.20\pm 3.85$
Participation ratio	$\mathbf{2.27\pm 0.15}$	$4.96\pm 1.52$
Condition number	$315.9\pm 204.1$	$\mathbf{108.4\pm 19.9}$

Table 20: Representation geometry comparison (5 seeds, SVD on

\sim

36k representations per model).

Figure 8: Representation geometry. DynaMITE’s 24-d latent collapses to effective rank ${\sim}5$ ; LSTM’s 128-d hidden state uses ${\sim}32$ effective dimensions.

Factor	DynaMITE (nats)	LSTM (nats)
Overall	$\mathbf{0.233\pm 0.052}$	$0.028\pm 0.033$
Friction	$\mathbf{0.054\pm 0.021}$	$0.024\pm 0.020$
Mass	$\mathbf{0.091\pm 0.010}$	$0.011\pm 0.013$
Motor	$\mathbf{0.037\pm 0.031}$	$0.013\pm 0.015$
Contact	$\mathbf{0.026\pm 0.011}$	$0.010\pm 0.015$
Delay	$0.007\pm 0.004$	$0.005\pm 0.003$

Table 21: Mutual information between representations and dynamics factors (5 seeds, KNN estimation

k=5

\sim

36k samples, nats).

Figure 9: Mutual information between learned representations and ground-truth dynamics factors (KNN estimation). DynaMITE retains more MI with dynamics parameters than LSTM (0.233 vs. 0.028 nats), but absolute values for both models are low.

Metric	DynaMITE	LSTM
MIG	$0.0013$	$0.0013$
DCI Disent.	$0.0195$	$\mathbf{0.0929}$
DCI Complete	$0.0168$	$\mathbf{0.0542}$
DCI Info.	$\mathbf{0.0846}$	$0.1397$
SAP	$0.0001$	$0.0006$

Table 22: Standard disentanglement metrics (5 seeds). Lower DCI-I is better (prediction error).

Scenario	Favored	Evidence
Nominal ID reward	LSTM	Confirmed
Moderate mismatch	LSTM	Directional
Severe mismatch	DynaMITE	Directional
Fast re-adaptation	DynaMITE	Directional
Inspect latent	Neither	Confirmed

Table 23: Observed patterns under our simulation protocol.

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