Acquisition

Which duel to ask next. The answer turned out to be "it does not matter much".

The rules

Acquisition is selectable, because the choice is an empirical claim and both alternatives are kept so the comparison stays runnable.

The measurement

cargo run -p auracle-session --example learn_synthetic --release -- --compare 20

20 seeds, 72 duels, refit every 12, against the synthetic user.

The methodology matters more than usual here, because the effect sizes are small:

• Common random numbers. Pool fill, the user's coin flip at duel $t$, the MCMC seed at round $r$, and refinement seeds are all shared across arms, so only the acquisition draw differs. Without this the between-seed variance would swamp everything.
• One fixed held-out exam under a single reference scale, so arms that built different pools are still answering the same questions.
• $\pm$ is two standard errors of the paired difference.

Three metrics: cosine similarity to the true $\theta^*$ (↑), rank correlation on the exam (↑), and excess nats against the true model (↓).

Static pool — i.i.d. prior draws, refine_steps: 0

Thompson is the one clear loser ($t = 2.2 / 6.1 / -8.0$). It is a best-arm rule: it converges on identifying the top patch, which is not what a duel is for here. Finding the single best patch in a pool and learning the shape of a taste are different objectives, and optimizing the first does not deliver the second.

BALD and uniform pairing are within two standard errors on every metric.

Evolving pool — refine_steps: 12, refinement between rounds

A static i.i.d. pool is a weak regime to conclude from on its own: prior draws are spread over feature space by construction, which is exactly where uniform pairs already achieve near-optimal $\lVert \varphi_a - \varphi_b \rVert$ coverage and an information-seeking rule has no redundancy to prune. The shipped pool is not that pool (refinement injects children near the current best, and insertion evicts the worst) so the comparison runs an evolving regime too.

Same answer. Thompson loses; BALD and uniform pairing tie on every metric.

Why Random is the default

Measured in both the regime the product starts in and the regime it evolves into, uniform pairing is indistinguishable from BALD. A rule with four tuning constants that ties a rule with none should not ship on a tie.

Two supporting reasons survived checking, one did not:

• The info_gain BALD reports had zero consumers in the frontend.
• BALD's repeat avoidance is real but barely needed over a pool this size that uniform pairing already samples without repeating (gated by duels_spread_over_candidates_not_just_pairs).
• Random makes every duel an unbiased calibration sample rather than one in ten, a virtue that holds regardless of which rule learns $\theta$ faster.

A retraction worth recording

One earlier justification was withdrawn for a bad reason, and the record should say so.

The "pool grows and concentrates" argument was dismissed on the grounds that insertion caps the pool, but a capped size is not an unchanging spread, and evicting the worst member could in principle concentrate a pool. Dismissing the concentration argument because it was unmeasured, while treating a measurement from the other regime as decisive, had the burden of proof backwards.

The evolving run above is that measurement. It happens to show the concentration never materializes, but the default rests on the measured tie, not on the dismissal.

What Bald is still for

It decisively beats the best-arm rule, so it is the right thing to reach for if acquisition ever needs to do something uniform pairing cannot:

All measured, none currently worth the tie.

A correction worth recording

An earlier version of the BALD rule scored its enjoyment term on unnormalized utility and used an absolute softmax temperature of 0.05 nats.

Both are scale bets, and both lost. The enjoyment term grew without bound as the posterior sharpened, and $\exp(\Delta J / T)$ ran to $e^{10}$ — so the "softmax" was an argmax. That version was measurably worse than random, and it is the version an independent replication measured.

It also produced the duel repetition observed in the running app: the same defect, seen from two directions. Fixed, BALD ties random, and the tables above are the fixed rule.

The general lesson: a temperature with units of nats is a bet about the scale of the quantity it divides, and a quantity that grows as a model sharpens will eventually break that bet.

Where uncertainty would earn its keep

The design's argument for acquisition is that $\theta$'s posterior uncertainty lets early sessions ask informative questions (duels the model cannot rank) while a confident model mostly serves things you will like. That remains the right frame, and it is also the frame in which the measurement says the informative-question machinery is not currently paying for itself.

At session horizon — tens of duels, a 48-patch pool kept spread by its own dynamics — there is not enough redundancy in the question set for an information-seeking rule to exploit. A much larger pool, or a much longer session, is where the tie would be expected to break.