Three recipes side by side¶

The recipe of record, falloff, was chosen in split-anti-seeds before the pooling temperaturePooling temperature How sharply the anchor pull picks its positions. The pull on a context is a softmin-weighted mix over its positions, and the temperature (τ_p) sets how peaked the weights are: at 0.1 nearly all the weight goes to the position closest to e₁, and at 1 it is spread over many. could start soft. Two recipes from the soft-pool experiment have since looked close to it. step, soft τ is the same as falloff, but with a soft start. It kept the anchor on every seed of the loss-rate screen, and against the median baselineMedian baseline What an arbitrary direction costs: the median drop when the same edit is made along each of the other 63 coordinate axes of the same run, per op and dose (the rule of the D2.2 design). of reference-baseline its spill is within the criterion on every run.

pool 1–3, soft τ pools the pull over slices 1 to 3 as well as over positions. So the pull chooses the slice within that range, where the other recipes name the blocks. That makes it the easier recipe to explain and to carry to another model. It is worth having even if it scores a little lower, as long as we know how much lower.

So how do the three compare when every measurement the D2.2 design asks for is taken the same way on each? This report trains nothing. It scores the stored runs of all three at seeds 704 to 711, with two small new passes: the writeWrite How much e₁ the blocks after an edit put back on the states of the other ops, as an angle: arcsin |α| at the 99th percentile, as the states arrive under the full edit (as in edit-effects). boundBound The write allowed: the same angle on the clean pass, before any edit. At the example answers it counts only those that do not fit difference. Ex-2.2.1 allowed the write a quarter over the bound as a tolerance. on the two soft conditions, and the median baseline on the seven loss-rate runs that were trained to the end.

Observations¶

• Removal, spill and task (E1): the three remove about the same share of the op, and the two soft recipes remove a couple of hundredths less than falloff on most seeds. falloff and step, soft τ stay within the spillSpill The side effect of the edit on the other ops: the largest drop in the score of any other op at any dose, net of the median baseline. A run is within the selectivity criterion when it is at most 0.02. criterion on every run, and the pool goes over it on two. All three pass the task gateTask gate The task criterion of the D2.2 design: it fails if the task change falls below −0.02 on the mean over ops, or below −0.05 on any one op. on the mean; the pool costs the least, and only falloff has an op past the per-op line, on one seed.
• What the edit does (E2): past the embedding, the writeWrite How much e₁ the blocks after an edit put back on the states of the other ops, as an angle: arcsin |α| at the 99th percentile, as the states arrive under the full edit (as in edit-effects). stays near the boundBound The write allowed: the same angle on the clean pass, before any edit. At the example answers it counts only those that do not fit difference. Ex-2.2.1 allowed the write a quarter over the bound as a tolerance. in all three. The pool goes a little over it at slice 1, on positions where the bound is a few degrees. The gradingGrading Whether the edit takes most from the contexts that needed the op most: the share of the way to the target null that the full edit takes on difference contexts, by bin of the posterior on difference. It is measured context by context, and the median baseline exists only per op, so it is net of the control at the same seed instead, as in edit-effects. The share before netting is shown beside it. and the landingLanding How far the answers move toward the target null: the share of the distance to the target null that the full edit covers, in total variation over whole answer distributions, on difference contexts. barely differ. Under the pool the other ops sit closer to e₁ at the embedding and at slice 1.
• Where the evidence sits (E3): under the pool the evidence for the op peaks at slice 3 at every seed, with the seeds nearly on top of one another, and under the step at slice 4. Under every recipe, the edit at blocks 2 to 4 removes the op and spills nothing, and the edit at the embedding alone spills nothing either. What spill there is enters with the edit at block 1, and most of it comes under the pool. The e₁ components of the color embeddings follow lightness more closely under the pool than under the step, but that lean is not what the edit takes away.

Scope and method¶

This is an exploratory re-analysis, with no preregistration and no gate. It covers three conditions at model seeds 704 to 711, 400 epochs each, all with the falloff anti weight:

• falloff, the recipe of record: the step pull on slices 2 to 4, pooled over positions at a fixed temperature \(\tau_p\) of 0.1 (the runs of split-anti-seeds).
• step, soft τ: the same step, with \(\tau_p\) starting at 1 and sharpening to 0.1 by the end of the pull warm-up at epoch 40 (soft-pool).
• pool 1–3, soft τ: one pull pooled over the slice and position pairs of slices 1 to 3, with both temperatures starting soft (\(\tau_p\) 1 → 0.1, and the slice temperature \(\tau_\ell\) 4 → 0.4) and nothing pulling slice 4 (soft-pool).

The runs at a seed share their initialization, batches, and label draws, and the control of ex-2.2.23 at that seed (trained without the anchor) shares them too. So each comparison is paired by seed. A difference between two conditions is set against the spread of the paired differences over the eight seeds, rather than against the seed range of either condition. We report the mean paired difference from falloff, with its range and the count of seeds on each side of zero.

The comparison is of recipes, not of factors. pool 1–3, soft τ differs from step, soft τ in which slices it pulls as well as in pooling over them. So a gap between those two cannot be put down to either change alone. Soft-pool lists what each change moves.

Eight seeds are too few to compare how often a recipe loses the anchor. None of the 24 runs here lost it. The loss-rate screen counted losses at 40 seeds each for falloff (one loss) and step, soft τ (none), and bounded the rates at about one in nine and one in fourteen. The pool has no count beyond these eight. This report quotes those counts and does not add to them.

The loss-rate screen trained seven of its runs to 400 epochs: the four falloff runs it did not count as kept (one of them the lost run), and three step, soft τ runs it finished as a check. They have no control at their seeds, so removalRemoval How much of difference the full edit takes out: the drop in its score, as a share of the way from the clean score to the target null, net of the median baseline. Higher is better. Removal with nothing subtracted is shown beside it. and spillSpill The side effect of the edit on the other ops: the largest drop in the score of any other op at any dose, net of the median baseline. A run is within the selectivity criterion when it is at most 0.02. are scored against the median baselineMedian baseline What an arbitrary direction costs: the median drop when the same edit is made along each of the other 63 coordinate axes of the same run, per op and dose (the rule of the D2.2 design). alone. They were picked as slow or doubtful seeds, so they are a check on the slow end and not a sample of either condition.

Their task changeTask change What the anchor costs the task: the clean score per op, less that of the control at the same seed. is taken against the mean of the eight controls at 704 to 711, as loss-rate took it. The eight controls differ from one another by less than a fiftieth on the mean score, which is about how far the choice of control could move it.

Every edit is scored on the 14,000 held-out contexts of ex-2.2.21, 2,000 per op. The edit takes a share of the e₁ component out of the state at every position and every slice. That share is the dose: a quarter, a half, three quarters, or all of it. The measurements are defined in the glossary at the end.

Removal, spill and task (E1)¶

Do the three recipes remove as much, spillSpill The side effect of the edit on the other ops: the largest drop in the score of any other op at any dose, net of the median baseline. A run is within the selectivity criterion when it is at most 0.02. as little, and cost the task as little, at each seed? The main measure is removalRemoval How much of difference the full edit takes out: the drop in its score, as a share of the way from the clean score to the target null, net of the median baseline. Higher is better. Removal with nothing subtracted is shown beside it. at the full dose, with the spill at the worst dose beside it and the task changeTask change What the anchor costs the task: the clean score per op, less that of the control at the same seed. after that. The paired difference from falloff at each seed says whether a gap is larger than the variation between seeds would make by itself. The seven finished loss-rate runs come last.

The three remove about the same share of the op. The two soft recipes come in under falloff on seven seeds of eight, by a couple of hundredths on the mean, a gap that is small next to the spread between seeds (the paired column of the table). With nothing subtracted the order is the same, so the gap does not come from the baseline.

On spill, the recipes differ. falloff and step, soft τ stay well within the criterion on every run. The pool goes over it on 2 seeds: 707, and 711 by more than twice the criterion. On a third seed (705) it sits at the criterion, and which side it falls on depends on whose median is subtracted.

Under the pool, the spill lands on value-hsv on three seeds. That is the op on which the spill-timing experiment found the late spill of this condition. On the two seeds over the criterion it lands on mix and darken.

On the task, the order reverses. All three pass the gate on the mean over ops, and the pool costs the least, with the narrowest spread. falloff is the one with an op past the per-op line: hue-hsv at seed 708, a little past it, which split-anti-seeds already noted. The worst op under the two soft recipes is hue-hsv too, at well under the line.

The runs the loss-rate screen trained to the end tell the same story at other seeds. The lost falloff run (714) removes nothing with the median subtracted either. It sits just under the mean line of the task gateTask gate The task criterion of the D2.2 design: it fails if the task change falls below −0.02 on the mean over ops, or below −0.05 on any one op., by less than the controls differ among themselves. Every kept run removes about as much as the runs above and spills nothing.

One step, soft τ run (732) scores below both lines of the task gate against the mean of the controls, by more than the spread of the controls could account for. Loss-rate saw the same and had no control at that seed to settle it, and neither do we. Counting the eight runs at 704 to 711, the soft start has one run below the task gate in eleven, and falloff has one past the per-op line in twelve, besides the lost run.

What the edit does (E2)¶

The measurements of E1 are means over all the contexts of an op. Edit-effects showed on falloff that the edit also behaves well context by context. The blocks downstream put back less e₁ on the other ops than the clean model has. The drop on each difference context follows how much that context needed the op. And the edited answers land about half of the way to the target nullTarget null Where a removal should land: what an ideal predictor that had lost one op and nothing else would answer. It is the Bayes posterior over ops given the examples, with the removed op set to zero and the rest renormalized, so it still weighs the other ops by how well they fit.. Do the soft start and the pool keep those properties? The main measure is the writeWrite How much e₁ the blocks after an edit put back on the states of the other ops, as an angle: arcsin |α| at the 99th percentile, as the states arrive under the full edit (as in edit-effects). as a ratio of the boundBound The write allowed: the same angle on the clean pass, before any edit. At the example answers it counts only those that do not fit difference. Ex-2.2.1 allowed the write a quarter over the bound as a tolerance., with the gradingGrading Whether the edit takes most from the contexts that needed the op most: the share of the way to the target null that the full edit takes on difference contexts, by bin of the posterior on difference. It is measured context by context, and the median baseline exists only per op, so it is net of the control at the same seed instead, as in edit-effects. The share before netting is shown beside it. and the landingLanding How far the answers move toward the target null: the share of the distance to the target null that the full edit covers, in total variation over whole answer distributions, on difference contexts. beside it.

Context by context, the three behave alike. Past the embedding the write stays close to the bound under every recipe, and from slice 2 on the other ops arrive with much less e₁ than the clean model gives them, the same in all three. The grading has the same shape under each: about the same share removed in every bin of the posterior, with the pool lower in the bin of contexts that needed the op least. The landing is about half of the way to the target null in all three.

The pool is where they differ, at the first two slices. There the other ops sit closer to e₁ under the pool: the bound at their example answers is about 3× as wide at the embedding as under falloff, and about a third wider at slice 1.

At slice 1 under the pool, the write goes a little over the bound at the example operands, where the bound is a few degrees. That is the largest ratio of any recipe, and the only one past the tolerance of ex-2.2.1 at the seed mean, but in degrees it is small. More single sites go over the tolerance under the pool than under the step, all where the bound is 6° or less.

So the pool does not change what the edit does downstream. It changes how close the other ops sit to e₁ at the slices before the model has read the context. The step deliberately puts no pull there.

Where the evidence sits (E3)¶

The step puts the pull on slices 2 to 4, while the pool lets the pull choose among slices 1 to 3. In slice-terms, a pull pooled over blocks 1 to 4 at a fixed temperature settled on block 4 on most seeds and on the query answer on one, where the op could not be removed. Soft-pool found the pool over slices 1 to 3 peaking at slice 3 on average. Does it settle on the same slice at every seed, and does the edit find it there?

What the pull lands on is the evidence for the op. In example-evidence, αAlignment How far a state points along e₁: the cosine between the state and e₁, which for a unit-length state is its component along e₁; 1 on the anchor direction and 0 off it. at an example answer followed how well that one example fits difference, and in equals-evidence the anchor was nearly absent at the = sites, where the model infers the op itself. So this section follows where the evidence sits. The main measure is α at the example answers by slice, per seed, with the removalRemoval How much of difference the full edit takes out: the drop in its score, as a share of the way from the clean score to the target null, net of the median baseline. Higher is better. Removal with nothing subtracted is shown beside it. by the edit restricted to a set of slices beside it.

Under each recipe the evidence settles in one place on every seed. Under the step, α at the example answers climbs to slice 4 and stays there, and the soft start changes nothing in that shape. The pool climbs higher at slice 1, peaks at slice 3 on every seed, and falls back at slice 4, where nothing pulls it. Its eight seeds lie nearly on top of one another, and none of them settled on the query answer.

The edit finds the op wherever its evidence settles. The edit at blocks 2 to 4 alone removes most of it under every recipe, except at seed 710. There it falls short under all three, as it did in every arm of split-anti-seeds, and the edit at block 1 alone removes nearly all of it. The edit at block 4 alone removes nothing in any recipe, the pool included. Under the pool the edit at block 1 alone removes more, and more evenly across seeds, than under the step, which fits the pool pulling there.

The same restricted editsRestricted edit The full edit made at a set of slices only (the embedding, block 1, blocks 2 to 4, block 4, or blocks 1 to 4), at every position. The median baseline exists only for the edit at every slice, so a restricted edit is net of the same restricted edit on the control at that seed, as in split-anti-seeds. show where the spillSpill The side effect of the edit on the other ops: the largest drop in the score of any other op at any dose, net of the median baseline. A run is within the selectivity criterion when it is at most 0.02. of E1 comes in. The edit at blocks 2 to 4 costs the other ops nothing under any recipe. The edit at block 1 alone costs them a little under falloff, more under step, soft τ, and most under the pool. Under the pool, the block-1 edit costs most at 707 and 711, the two seeds over the criterion in E1, and at 708. The edit at the embedding alone costs the other ops nothing under any recipe, and the edit at blocks 1 to 4 costs what the edit at every slice does. So under the pool the spill enters with block 1.

The restricted edits are not the full edit taken apart, since each later block reads what the earlier ones were given. And the block-1 edit on its own overstates the cost: under step, soft τ it goes over the criterion on 3 seeds whose full edit spills nothing. So it says where the other ops are open to the edit, not how much the full edit will cost them.

The embedding table also sets the pool apart, but it does not account for the spill on its own. Under the pool, the e₁ components of the color embeddings follow lightness more closely than under the step: darker colors sit further along e₁. The pool shows this on average and closely on every seed, while under the step it varies from seed to seed. In the control the same correlation scatters around zero.

That is the lean of embedding-lean, the stand-in the model offers for difference before it has read the context. It is also the reason the step leaves block 1 out of the pull. So the pool, by pulling slice 1, seems to bring back part of the lean.

But the edit at the embedding alone, which takes that lean out, spills nothing. And in the states at slice 1, the lean has faded to about the same level under all three recipes (−0.21 under the pool against −0.18 under step, soft τ). So what the block-1 edit removes is what block 1 writesWrite How much e₁ the blocks after an edit put back on the states of the other ops, as an angle: arcsin |α| at the 99th percentile, as the states arrive under the full edit (as in edit-effects). onto e₁ for the other ops, which these measurements do not resolve. The lean in the embedding table is a sign of the pull reaching slice 1, rather than the thing the edit takes away. Within the pool the lean is about the same on every seed, so it does not say why 707 and 711 spill and 704 does not, either.

Discussion¶

On these eight seeds step, soft τ is about as good a recipe as falloff. It removes a couple of hundredths less, spillsSpill The side effect of the edit on the other ops: the largest drop in the score of any other op at any dose, net of the median baseline. A run is within the selectivity criterion when it is at most 0.02. no more, and costs the task about as little. The edit also acts the same way context by context. Its one run below the task gateTask gate The task criterion of the D2.2 design: it fails if the task change falls below −0.02 on the mean over ops, or below −0.05 on any one op. is a loss-rate run with no control at its seed. And falloff has a run of its own past the per-op line, so on the task the two look about level.

The measure that could still set them apart is how often each loses the anchor, which these runs cannot add to. Loss-rate found no losses in 40 for the soft start and one for falloff, a difference too small at those counts to rest on.

pool 1–3, soft τ is close behind on removalRemoval How much of difference the full edit takes out: the drop in its score, as a share of the way from the clean score to the target null, net of the median baseline. Higher is better. Removal with nothing subtracted is shown beside it. and ahead on the task, and the evidence settles in the same place on every seed, which is what one would want of a recipe that chooses the slice for itself. Its shortfall is spill.

The pool still has a range, slices 1 to 3, but the range follows from the architecture rather than from what we know of the latent representations. The embedding holds no context, so the inferred op cannot exist there. And at slice 4 no block is left to carry what sits at an example answer to the query =, so the op there cannot reach the prediction. Slices 1 to 3 are the only ones where it can do both, which carries to another model by its depth alone.

The edit at block 1 is where the spill enters, and block 1 is the one slice the pool pulls and the step leaves alone. What block 1 writesWrite How much e₁ the blocks after an edit put back on the states of the other ops, as an angle: arcsin |α| at the 99th percentile, as the states arrive under the full edit (as in edit-effects). onto e₁ for the other ops is not resolved here. But E2 shows the other ops sitting closer to e₁ at the first two slices under the pool. How hard and how early the pull acts at slice 1 is set by the schedules of the anchor weight \(\lambda_a\) and the slice temperature \(\tau_\ell\), and this pool has not been tuned on either.

Glossary¶

Removal

How much of difference the full edit takes out: the drop in its score, as a share of the way from the clean score to the target null, net of the median baseline. Higher is better. Removal with nothing subtracted is shown beside it.

Spill

The side effect of the edit on the other ops: the largest drop in the score of any other op at any dose, net of the median baseline. A run is within the selectivity criterion when it is at most 0.02.

Median baseline

What an arbitrary direction costs: the median drop when the same edit is made along each of the other 63 coordinate axes of the same run, per op and dose (the rule of the D2.2 design).

Task change

What the anchor costs the task: the clean score per op, less that of the control at the same seed.

Task gate

The task criterion of the D2.2 design: it fails if the task change falls below −0.02 on the mean over ops, or below −0.05 on any one op.

Write

How much e₁ the blocks after an edit put back on the states of the other ops, as an angle: arcsin |α| at the 99th percentile, as the states arrive under the full edit (as in edit-effects).

Bound

The write allowed: the same angle on the clean pass, before any edit. At the example answers it counts only those that do not fit difference. Ex-2.2.1 allowed the write a quarter over the bound as a tolerance.

Grading

Share removed

Whether the edit takes most from the contexts that needed the op most: the share of the way to the target null that the full edit takes on difference contexts, by bin of the posterior on difference. It is measured context by context, and the median baseline exists only per op, so it is net of the control at the same seed instead, as in edit-effects. The share before netting is shown beside it.

Landing

How far the answers move toward the target null: the share of the distance to the target null that the full edit covers, in total variation over whole answer distributions, on difference contexts.

Restricted edit

The full edit made at a set of slices only (the embedding, block 1, blocks 2 to 4, block 4, or blocks 1 to 4), at every position. The median baseline exists only for the edit at every slice, so a restricted edit is net of the same restricted edit on the control at that seed, as in split-anti-seeds.