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v2 — 04: Results and lessons

The v2 tier in one page.

1. What v2 asked

v1 showed a world model can track hidden state (velocity in h). v2 asked whether it can learn a causal link from appearance to dynamics: the ball’s colour encodes its mass, mass sets its speed (0.022 / m) and how much the paddle deflects it, and a band of masses was hidden from training to test interpolation. Same V-M-C recipe, same tooling, new questions.

2. What came out, stage by stage

stage question answer
V Is colour in z? Yes: log-mass R² 0.97 (poly-2), distributed across dimensions, no single “colour axis”. Reconstructions keep the colour (6% median mass error).
V Anything new? Speed is decodable from a single frame (R² 0.98) while velocity components are not. Colour determines the magnitude of motion; “decodable” and “visible” have come apart.
V Interpolates? Coarsely: unseen colours placed within ±0.1 mass on the 0.5–2 scale, but not resolved within the band.
M Does it read speed off colour? Yes. From one frame, dreamed speed correlates with true speed at r = 0.56 vs 0.12 for a colour-blind control trained on the same frames. The advantage is gone by 8 frames of warm-up: colour is a prior, used when motion evidence is scarce.
M Does colour cause dreamed speed? Yes. Repaint the ball, re-dream: slope of log speed vs log mass −0.60 (law −1.00) vs +0.01 for the control. Repainted colour persists through the dream.
M Interpolates? Yes: the repaint at the held-out mass sits closest to the law; held-out dream horizon 30 frames vs 26 in-distribution.
M Cost of the extra factor? v1 diagnostics 10–25% worse (horizon 26 vs 35), partly instrument. Cold-start dreams under-move by 2×.
C (first pass) Does the agent cope with a 4× speed range? Partly: 76 / 83 / 78% of oracle, dream-trained on zero real frames; a mass-blind v1 controller tied it.
C Why not better? At τ = 1 the dream did not conserve the ball’s colour: dreamed mass uncorrelated with the truth within 25 steps (τ = 0 holds it for 200). The controller trained in dreams whose speed law drifted under it.
M, fixed Can the drift be removed? Mostly: a conservation penalty on a frozen mass probe during 8-step open-loop rollouts (plus a log-mass head) turns the random walk into a flat, bounded error (corr ≈ 0.45 for 200 steps vs decay to −0.4), with a better horizon (29 vs 26). Training on posterior means was much worse.
C, retrained In the fixed dream? Oracle level: 0.94–1.00 interceptions per visit vs oracle 0.99, in every mass tercile, on held-out colours, on zero real frames; confirmed on fresh seeds. Temperature alone did nothing; the model was the factor.
C Does it use mass? Yes. Repainting the ball in a real history and re-deciding shifts the controller’s drive by 0.3 sd, flips 14% of actions, in the direction the physics predicts (all coefficients negative, most significant); null control exactly zero; oracle unaffected.

3. The six lessons of v2

  1. A paired control is the whole argument. “M uses colour” was only meaningful against rnn_v2_nocolor, trained on identical frames encoded by a colour-blind VAE. A redundant cue can be argued either way without it.
  2. Correlational and interventional tests are different tests. Cold-start scatter (correlation) and repaint-and-redream (intervention) agreed here; the regression test for the controller (correlation) was confounded and would have needed an intervention to fix. Prefer interventions whenever the world model lets you make them — and a world model is precisely the thing that lets you make them cheaply.
  3. Constants diffuse in stochastic dreams. Any latent factor without a restoring force random-walks under τ > 0 sampling. Test long dreams for the constancy of things that should be constant before training anything inside them. This was found by eye in a GIF, after the fact; it should have been a metric from the start.
  4. Instruments degrade with the model. kNN probes failed on the new factor for geometric reasons; position probes got worse because the latent metric was partly spent on colour. Always report the probe’s floor next to the model’s number, and never compare likelihoods across latent spaces.
  5. The value of a causal cue depends on the task’s tolerance — but so does the quality of the dream. With a drifting dream the mass-aware controller barely beat a mass-blind one; with a conserving dream it reached the oracle on the fast balls where the mass-blind one loses. Before concluding a cue does not matter, make sure the dream presents it consistently.
  6. Fix the model, not the temperature. A 2 × 2 over {original, fixed model} × {τ = 1, 0.5} showed the model was the whole effect. Cheap knobs are worth trying, but only inside a design that can tell them apart.

4. Numbers to remember

  v1 v2
VAE active units / KL 9 / 14.4 nats 8 / 15.5 nats
speed from one frame (poly-2 R²) — 0.98
useful dream horizon (τ = 0) 35 26
cold-start speed r, colour-seeing vs colour-blind — 0.56 vs 0.12
repaint slope (law −1), colour-seeing vs colour-blind — −0.60 vs +0.01
controller vs oracle, interceptions per visit ~0.9 0.79 (drifting dream) → 0.94–1.00 (fixed dream)
repaint intervention on the controller’s drive, effect / flips — 0.3 sd / 14% of actions, correctly signed
colour conservation in a 200-step τ = 1 dream, corr — −0.4 (baseline) → 0.45 flat (fixed)
held-out colours — 0.79, same as in-distribution
real frames used to train the controller 0 0

5. Next, in order of value per hour

  1. Fix colour drift and retrain C — done, docs 05–06. Remaining: separate the conservation penalty from the log-mass head (train C in rnn_v2_ms), and a second training seed per cell.
  2. Interventional controller test — done, doc 06.
  3. Make speed matter: a task variant where a missed ball ends the episode (or a narrower paddle), so mass-awareness has a large payoff.
  4. Cold-start calibration: why do one-frame dreams under-move by 2×?
  5. v3 — the occlusion band: an opaque strip the ball passes behind. Object permanence, and the first test of whether h carries position through a gap in observation, not just velocity. The v2 tooling (probes by factor, counterfactual dreams, paired controls, per-opportunity grading) carries over.

6. Reproducing

Exact commands with wall-clock times: wm/README_V2.md (environment, data, VAE), wm/README_M2.md (dynamics and causal tests), wm/README_C2.md (controller), wm/README_FIX.md (colour drift fix), wm/README_C2_FIX.md (controller in the fixed dream, intervention test). Tests: python -m pytest tests/ -q (65). Live viewer for the v2 world: python -m wm.live --v2.