Chapter 16 — Latent Euclidean Distance: Convenient, but Not Necessarily Reliable

The big picture

A tiny story

Two children stand very close on opposite sides of a wall. A ruler says, “Tiny gap!” Their feet say, “Find the door.”

Original LeWorldModel v3 uses a similarly simple planning ruler. It compares the final imagined latent (z_H) with the separately encoded goal latent (z_g):

[ J = \lVert zH-z_g\rVert_2^2 = \sum_i (z{H,i}-z_{g,i})^2 . ]

The ruler is fast, needs no reward labels, and gives one score per candidate plan. Those are excellent baseline properties. They do not make the score shortest-path length, travel time, success probability, safety, or task progress.

The real rule

• The goal is used by the terminal cost; it is not a secret future observation fed into the dynamics predictor.
• Squaring a nonnegative Euclidean gap preserves its ordering within one fixed representation, but the source-faithful LeWM criterion is the summed squared gap.
• Euclidean distance is symmetric: (d(A,B)=d(B,A)). Reachability may be directed or depend on an action budget.
• Raw cost scale depends on latent dimension and coordinate scale. Compare rankings within a named checkpoint before comparing naked magnitudes across models.
• SIGReg can make a population broadly Gaussian without making nearby latent points connected by valid actions.

The planner needs an ordering: candidate A should cost less than candidate B when A is the better decision. A high average correlation on easy pairs can hide a few decisive inversions near a wall or contact boundary.

The trick that can fool us

Make four start–goal pairs with the same straight-line gap:

1. both points in one open room;
2. points close across a wall;
3. points aligned with the doorway;
4. points in opposite rooms, far from the doorway.

Use simulator geometry only as an evaluation oracle. Compare latent ranking, shortest feasible route, budgeted reachability, the first action selected under a fixed CEM budget, and the executed result.

First score real encoded endpoints. Then score model-generated endpoints. If the ranking is already wrong with oracle dynamics, inspect representation or cost geometry. If it becomes wrong only after rollout, inspect dynamics drift. If ranking is good but the plan fails, search, support, action conversion, or execution may be responsible.

Experiment receipt

The baseline claim comes from LeWorldModel v3 and frozen author commit 8edfeb3. It supports this narrow wording: LeWM ranks terminal latents with a simple symmetric squared distance whose agreement with route length or reachability must be tested.

The later Objective Bottleneck study reports weak latent-L2/spatial-distance alignment and behavior changes after a cost substitution in one independently implemented TwoRoom system. It is a same-system follow-up, not a second independent replication and not a universal reachability metric.

Record checkpoint, preprocessing, latent width, pair generator, oracle definition, action budget, CEM budget, and seeds. The wall picture is a tutorial counterexample; it is not evidence that every LeWM checkpoint fails it.

Quick check

1. Why is squared latent distance a useful baseline?
2. What can a symmetric endpoint ruler never express by itself?
3. Which oracle swap separates a bad cost ranking from a bad rollout?