What is the library, if any, used for the equation/formula rendering? Looks tastefully done.

> The mathematics on this page is being rendered by MathJax, which generates both the text spoken by screen readers, as well as the visual layout for sighted users.

Lovely. Thank you.

also very curious, visualization is great!

> reasoning as pathfinding through token space

Sounds like it's hillclimbing all the way down!

Cool framing. I'm excited for the parallel construction of ___ as pathfinding through weight space.

Seems like thinking should both expose solutions to head towards but also expose problems to flag/navigate around. It would be interesting get a measure of both. In my experience, preventing absolutely silly logic flaws, that even a child would avoid, seems to be the biggest benefit for enabling thinking. And, looking at the thinking output (for models that expose it), this seems to follow the output patterns of "but wait!" that fills the thinking output.

Wang's argument contradicts the view that intelligence necessarily improves through more reasoning and undermines any assumption that recursive self-improvement (RSI) will automatically converge toward greater intelligence.

Content aside, that's a beautiful webiste

Normalizes user voice to goal so the action has more chances to success you can see this on model that dont hide traces sending very badly spelled or incomplete sentences

I'm pretty sure that the current approach of baking in chain-of-thought reasoning into weights is not the end of the line. It just seems to accidental and ad-hoc.

P. S. The animated bird is truly magical!

My understanding of the math is weak (to say the least), but intuitively what I'm getting from this is:

1) Pretraining creates a bias _towards_ "correct answers" with each inference step

2) A longer walk is more likely to land in the correct region, given it's easier to verify you have the correct answer than to create it

3) RL selects the subset of walks that lands in the correct region and tilts the distribution in favor of such walks

4) By 2, that trains towards longer walks

...Is that correct?

Yes. One point worth making though is that pre-training trains the model to produce high probability completions - in practice, most of what it's trained on is a good completion as a human would judge it, i.e. it's not factually false data, so you get "creates a bias _towards_ "correct answers" with each inference step", but the model isn't really learning what correct or incorrect means in any way a human would consciously think about it.

Same reason the human brain has a train of thought: there's a lot of stuff you "know" as episodic memories of words you've heard/read, but haven't yet internalized via practice into reflex reactions. It's a lot faster and cheaper to acquire a memory of some words, than it is to acquire an entire new skill, so you end up with far more memories-of-words than concrete skills.

Reasoning (i.e. bubbling those remembered words up into your mental context, and then using a special "acting as constrained by words" bridging reflex on them, a.k.a. "system-2 thinking") is how you leverage memories-of-words into useful behavior.

An especially useful version of this is planning, where you chain through "mere words" until you end up at words that declare to yourself what you're about to do; and then you rely on your bridging reflex to react in a way that's constrained by the plan you generated.

That is how I have been thinking about LLM reasoning pretty much since it arrived.

I am not sure I support his statement about RL “searching for successful paths and then training the model to reproduce them”.

Don’t we train the model to use a certain method to explore the space in search for an answer?

As soon as you have tool calling and a way to verify your result, such as a theorem prover for example you are essentially running an optimization algorithm.

> Don’t we train the model to use a certain method to explore the space in search for an answer?

Not really - you just sample random high-probability completions and then the verifier tells you which ones to nudge the model towards and which to nudge away from. There's lots of clever ways you could improve the search process, but standard practice right now is just random sampling.

> Consider the problem: “what’s 1 +2 +3 +⋯ +10”? One solution is to add the numbers one at a time: 1, 3, 6, 10, and so on up to 55. Another way is to pair them from the outside in: 1 +10, 2 +9, 3 +8, 4 +7 and 5 +6 each make 11, and five pairs of 11 make 55. Both of these paths land on the correct answer, 55.

Watching my son work through difficult math problems has forced me into this realization. In my head there is one "correct" path, and I often have to stop myself from interrupting his chain of thought. He often is just getting there a different way.