Chapter 26 — Linear Probes: Which Physical Variables Are Encoded in the Latent State?

The big picture

A tiny story

An airport scanner sees a key inside a suitcase. That proves the scanner can recover key information from this view. It does not prove the traveler knows the key is there or will use it.

A probe works the same way. Freeze a trained representation, collect its latent vectors, and train a small readout from those vectors to simulator labels. A successful readout shows statistical accessibility for this representation, readout, split, and distribution. It does not prove the predictor or planner uses the information.

The real rule

Name the attachment point: encoder output, projected state, predictor input, or predictor output are different objects. Freeze checkpoint, evaluation mode, crop, resize, normalization, pooling, projector, and history construction. If probe gradients update the encoder, the experiment has become supervised fine-tuning.

Split complete episodes before extracting embeddings or fitting target normalization. Train on training episodes, choose hyperparameters on validation episodes, and evaluate once on untouched test episodes. Adjacent frames on opposite sides of a random frame split are near duplicates, not independent generalization.

LeWM v3 reports:

• TwoRoom: two-dimensional agent position;
• PushT: agent location, block location, block angle;
• OGBench-Cube: joint position/velocity, end-effector position/yaw, gripper state, block position/quaternion/yaw.

Target distance is a tutorial extension, not a v3 probe target. The paper reports MSE and Pearson correlation for linear and nonlinear probes. Correlation can stay high with scale bias; average MSE can hide a bad dimension or rare contact slice.

A linear probe tests simple affine accessibility. An MLP tests accessibility to a richer hypothesis class and may exploit shortcuts. Report capacity, regularization, optimization and tuning budget for both.

The trick that can fool us

Imagine every TwoRoom episode crosses the door at roughly the same time, and a faint corner mark brightens with frame index. A random-frame split lets an MLP—and perhaps a linear probe—predict position from time.

Now split by episode, randomize policy speed, and remove the mark. If performance falls, the original result still meant exactly this: position was recoverable where time and position were coupled.

Use controls: constant-mean predictor, simple pixels or privileged observable baseline, untrained encoder, shuffled labels, and training-versus-test curves. Slice TwoRoom by open floor, wall, and doorway; slice PushT by free motion, approach, first contact, sliding, and rotation where labels are reliable.

To test use, intervene after probing. Perturb a variable-associated subspace and compare with matched-size random directions, norm-preserving noise, and no intervention on the same episodes and candidate actions. A behavior change can still come from entanglement or an out-of-distribution latent, so even this supports “behaviorally implicated under this intervention,” not automatic causality.

Experiment receipt

Later results make the boundary concrete:

• Objective Bottleneck reports terminal latent L2 versus true endpoint distance at Pearson (r=0.426), while a ridge position probe reaches held-out (R^2=0.9922), in one TwoRoom follow-up system.
• SCALE reports a state-regression control that can match or exceed full-embedding decodability on some tasks without matching distance alignment; it improves 13 of 15 task–solver averages versus 15 of 15 for SCALE.
• Decision-Metric Alignment reports probe (R^2) differences below 0.03 across four non-collapsed PushT variants, while Plan–Real Spearman moves from +0.280 to about +0.410–+0.420 and online success spans 49.3%–92.7%. Each configuration has one training run; three seeds measure evaluation variation.

These are separate, author-reported protocols. They show that decodability can separate from planner-facing geometry; they do not identify one universal mechanism. The original v3 probe implementation details are insufficient for an exact reconstruction of every reported probe.

Quick check

1. What must remain frozen for a probe to describe the original representation?
2. What does MLP success establish beyond linear success—and what does it not?
3. Which intervention moves a claim from recoverability toward behavioral relevance?