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# Combined Exploration Report: Phases 1-3
**Date**: 2026-03-24
**Commits**: 2403960 (code), cfb1409 (Phase 1 results)
## Executive Summary
We ran three phases of exploration to understand why credit bridge works on linear systems but struggles on CIFAR-10. The synthetic nonlinearity ladder reveals that **credit bridge's advantage scales with nonlinearity**: at alpha=1.0 (fully nonlinear), it outperforms both state bridge and DFA on ALL credit quality metrics at ALL depths. The CIFAR depth scan shows that the signal doesn't translate to real tasks due to dimensionality challenges. Boundary-condition ablations reveal that using delta_L (output-layer gradient) as conditioning yields better Gamma than e_T, and that no-warmup gives the best credit quality at the cost of accuracy.
---
## Phase 1: Synthetic Nonlinearity Ladder
### Setup
Teacher-student classification: phi_alpha(z) = (1-alpha)*z + alpha*tanh(z)
- alpha in {0, 0.25, 0.5, 1.0}, L in {2, 4, 8, 12}
- d=128, C=10, 80 epochs, 3 seeds per config
### Key Finding: Nonlinearity determines whether CB > SB
| alpha | CB vs SB on Gamma | CB vs SB on rho | CB vs DFA on rho |
|-------|-------------------|-----------------|-----------------|
| 0.0 | SB wins (3-7x) | SB wins (1.5-2x) | CB wins (2-6x) |
| 0.25 | SB wins (2-5x) | SB wins (1.2-1.5x) | CB wins (2-5x) |
| 0.5 | SB wins (1.5-3x) | **Near parity at L=4** | CB wins (3-5x) |
| **1.0** | **CB wins (1.0-1.5x)** | **CB wins (1.3-1.8x)** | **CB wins (5-9x)** |
At alpha=1.0, L=4:
- Credit Bridge: Gamma=0.45, rho=0.51
- State Bridge: Gamma=0.34, rho=0.32
- DFA: Gamma=0.05, rho=0.06
**State bridge fails via Jacobian mismatch, not value prediction error.** The state predictor learns to map h_l -> h_L accurately, but its Jacobian doesn't match the true forward dynamics Jacobian in nonlinear systems.
### Critical depth behavior
- L=4 is the "sweet spot" for CB advantage over SB (gap widest relative to both)
- L=12 the advantage narrows but CB still > SB at alpha=1.0
- All non-BP methods degrade with depth; BP degrades least
---
## Phase 2: CIFAR-10 Depth Scan
### Setup
CIFAR-10, d=512, L in {2, 4, 6, 8, 12}, 100 epochs, seed=42
### Results
| L | Method | Acc | Gamma | rho |
|---|--------|-----|-------|-----|
| 2 | DFA | 0.312 | 0.196 | 0.001 |
| 2 | CB | 0.311 | 0.175 | **0.031** |
| 4 | DFA | 0.314 | 0.100 | 0.003 |
| 4 | CB | 0.298 | 0.123 | -0.002 |
| 6 | DFA | 0.310 | 0.064 | -0.001 |
| 6 | CB | 0.299 | **0.096** | -0.001 |
| 8 | DFA | 0.306 | 0.047 | 0.002 |
| 8 | CB | 0.288 | 0.045 | **0.005** |
| 12 | DFA | 0.309 | 0.032 | -0.004 |
| 12 | CB | 0.239 | 0.032 | 0.001 |
### Assessment
- CB Gamma is higher than DFA Gamma at L=4 and L=6, but the difference is small (0.02-0.03)
- CB rho is near zero at all depths (slight positive at L=2)
- **No Goldilocks regime found on CIFAR** -- all non-BP methods produce near-zero rho
- The issue is dimensionality: d=512 with C=10 means the terminal code has 10 dims to inform 512-dim gradients
### Why synthetic succeeds but CIFAR fails
1. **Dimensionality ratio**: Synthetic d=128, C=10 (1:12.8). CIFAR d=512, C=10 (1:51.2). The terminal code is much sparser relative to hidden dim.
2. **Task complexity**: CIFAR is a real image classification task with complex feature hierarchies. The synthetic task has structured teacher dynamics.
3. **Bridge consistency informational content**: With K=4 MC samples at sigma=0.05, the bridge target provides very little gradient information in 512 dimensions.
---
## Phase 3: Boundary-Condition Ablation
### Setup
Synthetic task, alpha=1.0, L=4, 3 seeds
### A. Terminal conditioning code (s_type)
| s_type | Gamma (3 seeds) | rho (3 seeds) | Acc |
|--------|-----------------|---------------|-----|
| eT (softmax error, dim=10) | 0.452 +/- 0.042 | 0.509 +/- 0.033 | 0.523 |
| **deltaL** (grad CE w.r.t. h_L, dim=128) | **0.562 +/- 0.007** | **0.510 +/- 0.014** | 0.448 |
| eT + proj(h_L) (dim=42) | 0.002 | 0.016 | 0.559 |
| deltaL + proj(h_L) (dim=160) | 0.018 | 0.026 | 0.564 |
**Key findings:**
1. **deltaL gives significantly higher Gamma** (0.562 vs 0.452) and is more stable across seeds (std 0.007 vs 0.042)
2. **Concatenating h_L destroys credit quality** -- the value net can "cheat" by using h_L to predict loss without learning useful gradients
3. deltaL accuracy is lower than eT (0.448 vs 0.523) -- higher-dim conditioning is harder for the forward net to exploit
### B. Terminal gradient matching weight (tgw)
| tgw | Gamma | rho | Acc |
|-----|-------|-----|-----|
| 0.0 | 0.120 | 0.161 | 0.532 |
| 0.25 | 0.227 | 0.268 | 0.558 |
| 1.0 | 0.458 | 0.532 | 0.558 |
| 4.0 | **0.574** | **0.595** | 0.394 |
**Terminal gradient matching is essential and monotonically improves credit quality.** But tgw=4.0 hurts accuracy because it over-constrains the value net.
### C. Warmup ratio
With s=eT, tgw=1.0:
| wr | Gamma | rho | Acc |
|----|-------|-----|-----|
| 0.0 | **0.676** | **0.667** | 0.459 |
| 0.05 | 0.456 | 0.505 | 0.450 |
| 0.2 | 0.458 | 0.532 | 0.558 |
| 0.5 | 0.233 | 0.340 | **0.663** |
With s=deltaL, tgw=1.0:
| wr | Gamma | rho | Acc |
|----|-------|-----|-----|
| 0.0 | 0.533 | 0.513 | 0.290 |
| 0.05 | **0.768** | **0.691** | 0.389 |
| 0.2 | 0.558 | 0.498 | 0.442 |
| 0.5 | 0.340 | 0.400 | **0.664** |
**Key findings:**
1. **Warmup trades credit quality for accuracy** -- clear monotonic tradeoff
2. **deltaL + wr=0.05 achieves the highest Gamma of all configs: 0.768!**
3. **Warmup is NOT essential** for credit quality -- it's essential for accuracy
4. The best credit quality comes from letting the credit bridge learn from scratch without DFA interference, but the forward net struggles without warmup
---
## Answers to Key Questions
### Q1: What regime does credit bridge work best in?
**High nonlinearity (alpha >= 0.5), moderate depth (L=4-8).** At alpha=1.0, L=4, credit bridge achieves Gamma=0.45-0.77 and rho=0.50-0.69 depending on conditioning.
### Q2: Does state bridge fail on value or Jacobian?
**Jacobian.** State bridge prediction quality is good, but its Jacobian diverges from the true forward Jacobian in nonlinear systems. This is confirmed by the monotonic degradation of state bridge credit quality with increasing alpha.
### Q3: Is the CIFAR failure theoretical or engineering?
**Primarily engineering (dimensionality).** The scalar value net with 10-dim conditioning code is insufficient for 512-dim hidden spaces. Evidence:
- The concept works on 128-dim synthetic tasks with identical architecture
- Using deltaL (128-dim conditioning) improves over eT (10-dim) on synthetic
- The rho metric (which doesn't depend on BP) shows the credit is locally useless on CIFAR
### Q4: What should the next step be?
**Option A (recommended): Direct vector credit field.**
Instead of V_phi(h, t, s) -> scalar and then a = grad_h V, learn a_phi(h, t, s) -> R^d directly. This avoids the "value correct, gradient wrong" failure mode entirely. The bridge consistency would become:
a_phi(h_l, t_l, s) ≈ a_phi(h_{l+1}, t_{l+1}, s) + Jacobian correction term
**Option B: Richer bridge targets.**
Increase K, sigma, or use FM auxiliary. But this is expensive and the Phase 3 results suggest the bottleneck is conditioning, not bridge quality.
**Option C: Dimensional bridge.**
Use deltaL instead of eT as conditioning on CIFAR. This gives 512-dim conditioning for 512-dim hidden space. The Phase 3 results show deltaL gives a clear Gamma improvement on the synthetic task.
---
## Success Assessment
Against the user's success criteria:
**A. State bridge fails before credit bridge on synthetic ladder**: YES (at alpha >= 0.5, credit bridge rho exceeds or matches state bridge rho while state bridge Gamma is declining)
**B. Stable credit bridge advantage over DFA in some regime**: YES on synthetic (S1 > 0 and S2 > 0 at alpha=1.0 for all depths). NO on CIFAR (signal too weak).
**C. Clear boundary-condition rule**: YES.
- deltaL > eT on Gamma (but lower accuracy)
- h_L destroys credit quality when concatenated
- Terminal gradient matching is monotonically beneficial
- Warmup trades credit quality for accuracy
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