A soft pool while the anchor settles¶

In the anchor-ramp experiment, a pull that ramps with depth lost the anchor on two or three of eight seeds, where the step in falloff lost it on none. The lost runs split off from the kept ones during the pull warm-up, around epochs 16 to 24: on a kept run the example answers rise onto e₁ along with the query answer, and on a lost run they fall back. The losses also followed how little pull slices 1 to 3 had, though that ordering was picked after the counts were in.

The pull is pooled over the positions of each context at a sharp temperature, \(\tau_p\) = 0.1,¹1 Pooling temperature: the pull on a context is a softmin-weighted mix over its positions. A low temperature gives nearly all the weight to the closest position; a high one spreads it evenly. so while the query answer is the position closest to e₁, it gets almost all of the pull. A soft pool early on would share the pull with the example answers while they are still deciding.

A pull on slices 1 to 3 alone, with nothing on slice 4, would put the pull where it mostly moves the example answers.

The 4 new arms, of 8 runs each, cross the two changes and add a pool over slices 1 to 3 with both temperatures soft. They use the seeds of the split-anti-seeds experiment, so its falloff runs serve as the baseline with no new training. Those eight runs are the only ones falloff has, so its loss rate is known only roughly (see Scope).

Observations¶

• Removal and spill (E1)Spill The side effect of an edit on the other ops: the largest drop in the score of any op other than the one removed, net of what the same edit does to a control trained without the anchor. The selectivity criterion of D2.2 allows at most 0.02 at any dose.: the soft start kept the anchor on every seed in all three arms that had it, and the ramp at the fixed temperature lost it on one. Removal on the kept runs was close to that of falloff everywhere. But each soft arm had runs over the spill criterion, and the pool over slices 1 to 3 had the most.
• When the runs settle (E2): a soft pool holds every position back through the first part of the warm-up, then the example answers come onto e₁ with the query answer on every seed, about four to twelve epochs later than under falloff. The one lost run latched onto the query answer for a time: its example answers fell back after rising partway, climbed back only slowly from about epoch 100, and ended below every kept run, with the edit removing almost nothing.
• Where the anchor sits by depth (E3): without a pull on slice 4, the alignmentAlignment 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. there falls back from its slice-3 level, and the pooled arm settles where the ramp does. The syntax embeddings stay off e₁ in every arm. The soft pool briefly lifts every position toward e₁ during the rise, the other ops included, and that lift goes with the spill.

Scope¶

This is an exploratory study, with no preregistration and no gate. Two factors are crossed, with falloff as the condition where both are at their recipe level:

• The pull shape: the step on slices 2 to 4 (the recipe), or a ramp over slices 1 to 3 at a third, two thirds and the full weight, with nothing on the embedding or slice 4. The ramp adds up to the step over slices 1 to 3 (two each), so the earlier slices keep the pull the step gives them, and the pull on slice 4 goes.
• The position pool temperature \(\tau_p\): fixed at 0.1 (the recipe), or soft-to-sharp, starting at 1 and sharpening to 0.1 by epoch 40, the end of the pull warm-up. The schedule is smooth in log \(\tau_p\), so the temperature halves at a steady pace and comes to rest at 0.1. At epoch 20, when the anchor-ramp runs split, it is about halfway there in log terms: a query answer well ahead of an example answer takes a few times as much pull as the example answer, against about fifty times at 0.1.

The fourth arm pools the pull over the slice × position pairs of slices 1 to 3 (the nested pool of the slice-terms experiment), so the pull can choose the slice as well as the position. Its slice pool temperature \(\tau_\ell\) starts at 4, near a plain mean over slices, and sharpens to 0.4 over the same window as \(\tau_p\).

The end value rests on one seed. In E4 of slice-terms, at seed 701, every \(\tau_\ell\) from 0.2 to 1.6 removed about three quarters to nine tenths of the op, with no trend and every run within the criterion, while the joint pool at \(\tau_\ell\) = 0.1 lost the anchor on one seed of four. So 0.4 is inside the range that worked there.

A soft pool and a ramp each change more than one thing:

• A softer \(\tau_p\) spreads the pull of each context over its positions, but the total pull per context stays the same at every temperature (the softmin weights sum to one). In the τ×λ_a sweep, a softer fixed \(\tau_p\) meant more spillSpill The side effect of an edit on the other ops: the largest drop in the score of any op other than the one removed, net of what the same edit does to a control trained without the anchor. The selectivity criterion of D2.2 allows at most 0.02 at any dose., so spill is the thing to watch; here it is soft only during the warm-up.
• The ramp pulls two thirds as much in total as the step, all of it moved off slice 4. A difference between the ramp and the step can come from the shape or from the lower total.
• The pool over slices 1 to 3 pulls as much in total as the step, and lets the pull choose where it goes.

Every run trains at model seeds 704 to 711 for 400 epochs, paired by seed with its falloff twin and with ex-2.2.23's control (same initialization, batches and label draws).

With eight seeds per arm, one or two lost runs could come from the seeds alone, so we weigh the counts across arms together. falloff has run only at these eight seeds and lost none, which fits a loss rate anywhere from zero to about 30% (the 95% upper bound for none in eight). Its close relatives in the split-anti-seeds experiment, the uniform anti holds of 0.2 and 0.1, each lost the anchor on one seed of the eight. So one loss in eight is no worse than the recipe as far as eight seeds can tell. Since falloff lost no runs here, so the step pair can only show a soft start doing harm, and only the ramp pair can show it helping.

The measurements¶

The measurements and dose scoring are those of the split-anti-seeds experiment.

• Removal: the net drop in the score on difference under the edit, as a share of the way from the clean score 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.. The edit projects e₁ out of the state at every position and slice, at a dose of a quarter, a half, three quarters, or all of the e₁ component. Higher is better.
• SpillSpill The side effect of an edit on the other ops: the largest drop in the score of any op other than the one removed, net of what the same edit does to a control trained without the anchor. The selectivity criterion of D2.2 allows at most 0.02 at any dose.: the largest net drop on another op under the edit. A run is within the criterion when its spill is at most 0.02 at every dose.
• Both are net of the control at the same seed.
• A run lost the anchor when the full edit removes less than half of the op, as in the split-anti-seeds experiment.
• Task change: the mean score over ops, less that of the control. Zero means no damage.
• The alignmentAlignment 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. α with e₁ at the example answers and at the query answer, at every slice, along training and at the end.

Removal and spill (E1)¶

Does a soft pool early, or a pull on slices 1 to 3 alone, keep the anchor on every seed, and keep the edit as selective as falloff? The main measure is the removal at the full dose, with the spillSpill The side effect of an edit on the other ops: the largest drop in the score of any op other than the one removed, net of what the same edit does to a control trained without the anchor. The selectivity criterion of D2.2 allows at most 0.02 at any dose. at any dose beside it. The soft step against falloff isolates the temperature, and the fixed ramp against falloff isolates the shape.

Every run but one climbs with the dose to about seven tenths to nine tenths of the op, as falloff does. The exception is one run of ramp 1–3 (the dashed line), which removes almost nothing at any dose.

With the soft start the same ramp kept the anchor on that seed, so in the one pair where the soft start could help, it did. But at a loss rate of one in eight, the soft twin would most often have kept that seed anyway, so this fits the soft start helping without being evidence that it does.

The soft start makes the edit less selective. falloff and ramp 1–3 stayed within the criterion on every seed, and their soft twins went over it on two seeds each, by a little. The pool over slices 1 to 3 went over on half its seeds, one of them (seed 711) by more than three times the criterion. The τ×λ_a sweep found the same for a softer fixed temperature, so this seems to be the price of a soft pool, even one that is soft for the first tenth of training only.

The pool over slices 1 to 3 removes about as much as falloff, with a narrow seed range, and more than either ramp at every dose. It pulls as much in total as the step and half again as much as the ramps, and it can put that pull on the slice and position where it moves the concept most, so it acts like the stronger pull of the two shapes on slices 1 to 3. Its spill is the widest of any arm, though.

The ramp also removes a little less. Over the kept runs, both ramps remove a few hundredths less than the step, and ramp 1–3, soft τ has three runs near seven tenths. The mean task score is unchanged in every arm, and no new arm loses more on any one op than falloff does.

When the runs settle (E2)¶

Does the soft pool carry the example answers onto e₁ with the query answer through the warm-up, where the lost runs of the anchor-ramp experiment fell back? The main measure is 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 the example answers along training, at slice 3, the deepest slice every arm pulls. The query answer is in the row underneath, and the anchor weight and the temperature in strips below that.

A soft pool slows the start. In the step pair at epoch 16, the query answer under falloff is already partway onto e₁ with the example answers following, while under the soft start both are still near zero. By epoch 24 the example answers have come up on every seed of step, soft τ, all crossing 0.4 at that epoch.

The ramp starts slowly at slice 3 at either temperature, since it pulls that slice at two thirds of the step weight. With the soft start its runs cross 0.4 within eight epochs of one another; at the fixed temperature one run is left behind.

So the soft pool does what it was meant to do, in that no run latched onto the query answer, but it does so by holding every position back. A likely reading is that it spreads the pull so thinly that nothing moves much until the temperature is low enough to choose, and by then the example answers rise in step with the query answer.

On that reading the soft start works mostly as a delay. A warm-up that simply started later might settle the runs the same way, and no arm here tells the two apart. The step pair does show the soft start tightening the spread: under falloff one seed (711) crossed 0.4 late, at epoch 40, and under the soft start every seed crossed at epoch 24. Eight seeds per arm fit this reading without proving it.

The lost run of ramp 1–3 (seed 708) starts as the anchor-ramp losses did and ends differently. Its example answers rose partway with the others, fell back four epochs later, and stayed below 0.4 until about epoch 100, while its query answer stayed on e₁: a partial latch onto the query answer. After that the example answers climbed to about four fifths of the kept level.

By the end its query answer is near one, as on every run (it is close to the ceiling of α, so it has little room to differ). Its example answers end below every kept run but well above zero, yet the edit removes almost nothing on it, at any slice.

A likely reading is that the model learned to solve difference while the anchor was away, along a path that does not go through e₁, and does not use the alignment that came back later. One run can't settle that. But alignment alone would understate this kind of loss: the anchor-ramp losses ended near zero, and this one ends at four fifths of the kept level.

Where the anchor sits by depth (E3)¶

On the runs that kept the anchor, where does the alignmentAlignment 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. sit by slice? With nothing on slice 4, does slice 4 still hold the concept as the residual stream (the running sum each block adds its output to) carries it up? Where does the pool over slices 1 to 3 settle, given that the pool over every block settled on block 4 in the slice-terms experiment? We also check the syntax embeddings, since in the τ×λ_a sweep a soft pool spilled through the = embedding coming onto e₁.

Up to slice 3 the arms are nearly alike, and the soft start changes nothing there. At slice 4, where the ramp and the pool put no pull, the example answers fall back to about three quarters of their slice-3 level, and the query answer falls from near one to about four fifths. Under the step they hold their slice-3 level. So the concept rides up past the last pulled slice, though only partly, and the edit still finds it, since it acts on every slice.

The anti term may account for some of the fall at slice 4. It acts on every position, the difference answers included, and holds at 0.03 on slice 4 in every arm. Under the step the pull on slice 4 counters it; under the ramp and the pool nothing on slice 4 counters it, so the model gains by moving the concept off e₁ there. No arm here turns the anti term off on slice 4, so this is untested.

The pool over slices 1 to 3 settles much as the ramp does, a little higher at slices 1 and 2. It peaks on slice 3, the last slice in the pool, as the slice-terms pool peaked on its last block. The other ops peak at slice 1 in every arm, as in falloff.

The syntax embeddings stay off e₁ in every arm, within a few hundredths, so the spillSpill The side effect of an edit on the other ops: the largest drop in the score of any op other than the one removed, net of what the same edit does to a control trained without the anchor. The selectivity criterion of D2.2 allows at most 0.02 at any dose. of E1 does not come from the route the τ×λ_a sweep found. The other ops end about as aligned in the step pair, and a little more aligned under the soft ramp than under the fixed one, though not by enough to account for the spill.

The trajectories hold a lead. While the pool is soft, its pull reaches every position from the op word on, and during the rise the other ops come partway onto e₁ with it.

Every soft arm lifts the other ops higher during the rise than its fixed twin does, and the pool over slices 1 to 3 lifts them most, in the same order as the spill. By the end of training the lift has gone, and the other ops are about as aligned in every arm. The peak goes with the spill across runs, though loosely, and within an arm it barely separates the runs. So a likely reading is that the model learns something about the other ops while they sit partway on e₁, which the edit still reaches after the alignment has gone. This post-hoc reading would need an arm to test it.

Discussion¶

The soft start got the example answers onto e₁ on every seed it was tried on, apparently by holding the pull back until it can choose. That fits the anchor-ramp reading, that runs are lost in the first tens of epochs when the sharp pool gives the query answer nearly all of the pull. But the evidence for a benefit is one seed, which a one-in-eight loss rate would often keep anyway (E1).

Both soft twins spilled more than their fixed arms, and the pool over slices 1 to 3, which has no fixed twin, spilled most.

The ramp over slices 1 to 3 lost the anchor on one seed of eight at the fixed temperature, where the anchor-ramp ramps from slice 1 lost two or three. That fits the second anchor-ramp reading, that the earlier slices need about as much pull as the step gives them: this ramp gives slices 1 to 3 that much, and drops slice 4 instead.

Its one loss came later and in another way, with the alignmentAlignment 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. partly back by the end but no longer doing the work. That is a reason to keep scoring removal, and not alignment alone, when judging a run.

The soft ramp kept the anchor on every seed here without the step between slices 1 and 2, so it would be a candidate if its spillSpill The side effect of an edit on the other ops: the largest drop in the score of any op other than the one removed, net of what the same edit does to a control trained without the anchor. The selectivity criterion of D2.2 allows at most 0.02 at any dose. can be kept down. If the spill comes from the soft pool reaching every position, as E3 suggests, a stronger anti term while the pool is soft would aim at it. Turning the anti term off on slice 4 might also let the ramp keep the concept there. As it stands, falloff still keeps the anchor on every seed with the edit inside the criterion, which no new arm matched on both counts.

Method¶

The pooling temperaturesPooling 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. are step inputs here, scheduled like the weights (AnchorSpec.tau_start, slice_tau_start and tau_anneal_end); before this experiment they were fixed when the training step was built. The schedule is minimum-jerk in log τ from the start value at epoch 0 to the end value at epoch 40, and holds after. Training otherwise follows the split-anti-seeds experiment unchanged. The trajectory of each run also keeps 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. by slice and position for each op separately, and the e₁ component of every syntax embedding, for later use. The 32 runs cost about $11.