First, light apparently slows down in some materials because the photons are constantly interacting with electrons, and these interactions create secondary waves that are slightly out of phase with the original light. The end result is a modified wave that effectively travels more slowly. So light going in a straight line through air travels more quickly than light going through a lens.

Second, getting more into quantum physics, light typically follows the shortest path from one point to another because that path tends to provide the most constructive interference between different possible routes. (The "why" of this is more involved; Feynman's book QED gives a good intro.)

Third, if you imagine a lifeguard running to rescue someone in the ocean, then they will take the fastest path, which is not directly toward the person in the ocean. Rather, they will run a bit more on the beach in order to have to swim less because travel through the water is slower. The end result is piecewise linear = two straight lines of travel, with a bend at the water line.

To summarize, you can ask "_why_ does light bend going into and out of lenses?" and the answer involves seeing light no longer as a particle but as a wave function (the quantum perspective), and then taking advantage of that wave function's tendency to prefer fastest-travel paths, and then noticing that the apparent bend is in fact the path of fastest travel.

There's one easy part of your situation, which is that your immediate next step is clear. I think it's clear because it sounds like the two of you have a fundamental disagreement about how to choose a direction. There's also a power imbalance if you've consistently been going along with pivots that you don't want to execute on. If you think you can resolve things, then maybe I'm wrong. I'm basing my advice on the fact that this is 24 months in — enough time for you to know how easy it'd be to get back to "we like working together and are on a path to success."

Now for the hard parts: * You want to act as professionally as possible in walking away. Communicate clearly, being fair to both yourself and the other people involved. I think it's easy to err to far on either side (either too defensive/passive "nice to yourself" or too deferential "nice to others"). * It can be tempting to think of this is as your cofounder's fault, but that's not constructive, and it's better to learn toward your future actions. For example, how could you have seen this problem earlier? My guess is that you were never "cofounder compatible," and this is a chance to get better at identifying that compatibility with others (useful for both startups and other jobs).

It's not a fun situation, and I hope things go well for you.

* AI could get better at thinking intuitively about math concepts. * AI could get better at looking for solutions people can understand. * AI could get better at teaching people about ideas that at first seem abstruse. * AI could get better at understanding its own thought, so that progress is not only a result, but also a method for future progress.

In math, there's an urban legend that the first Greek who proved sqrt(2) is irrational (sometimes credited to Hippasus of Metapontum) was thrown overboard to drown at sea for his discovery. This is almost certainly false, but it does capture the spirit of a mission in pure math. The unspoken dream is this:

~ "Every beautiful question will one day have a beautiful answer."

At the same time, ever since the pure and abstract nature of Euclid's Elements, mathematics has gradually become a more diverse culture. We've accepted more and more kinds of "numbers:" negative, irrational, transcendental, complex, surreal, hyperreal, and beyond those into group theory and category theory. Math was once focused on measurement of shapes or distances, and went beyond that into things like graph theory and probabilities and algorithms.

In each of these evolutions, people are implicitly asking the question:

"What is math?"

Imagine the work of introducing the sqrt() symbol into ancient mathematics. It's strange because you're defining a symbol as answering a previously hard question (what x has x^2=something?). The same might be said of integration as the opposite of a derivative, or of sine defined in terms of geometric questions. Over and over again, new methods become part of the canon by proving to be both useful, and in having properties beyond their definition.

AI may one day fall into this broader scope of math (or may already be there, depending on your view). If an LLM can give you a verified but unreadable proof of a conjecture, it's still true. If it can give you a crazy counterexample, it's still false. I'm not saying math should change, but that there's already a nature of change and diversity within what math is, and that AI seems likely to feel like a branch of this in the future; or a close cousin the way computer science already is.

It seems to me that the term “fake alignment” implies the model has its own agenda and is ignoring training. But if you look at its scratchpad, it seems to be struggling with the conflict of received agendas (vs having “its own” agenda). I’d argue that the implication of the term “faked alignment” is a bit unfair this way.

At the same time, it is a compelling experimental setup that can help us understand both how LLMs deal with value conflicts, and how they think about values overall.

Speaking more abstractly or philosophically, why could a model never internalize something read between the lines? Humans do, and we're part of the same physical system — we're already our own kinds of computers that take away more from a text than what is explicitly there. It's possible.

This recent work is highly relevant: https://learnandburn.ai/p/how-to-tell-if-an-llm-is-just-gues...

It uses an idea called semantic entropy which is more sophisticated than the standard entropy of the token logits, and is more appropriate as a statistical quantification of when an LLM is guessing or has high certainty. The original paper is in Nature, by authors from Oxford.

Or if not, does anyone know how to reach the author? I may have missed it, but I didn't even see the author's name anywhere on the site.

Context:

A random forest is an ML model that can be trained to predict an output value based on a list of input features: eg, predicting a house's value based on square footage, location, etc. This paper focuses on regression models, meaning the output value is a real number (or a vector thereof). Classical ML theory suggests that models with many learned parameters are more likely to overfit the training data, meaning that when you predict an output for a test (non-training) input, the predicted value is less likely to be correct because the model is not generalizing well (it does well on training data, but not on test data - aka, it has memorized, but not understood).

Historically, a surprise is that random forests can have many parameters yet don't overfit. This paper explores the surprise.

What the paper says:

The perspective of the paper is to see random forests (and related models) as _smoothers_, which is a kind of model that essentially memorizes the training data and then makes predictions by combining training output values that are relevant to the prediction-time (new) input values. For example, k-nearest neighbors is a simple kind of smoother. A single decision tree counts as a smoother because each final/leaf node in the tree predicts a value based on combining training outputs that could possibly reach that node. The same can be said for forests.

So the authors see a random forest as a way to use a subset of training data and a subset of (or set of weights on) training features, to provide an averaged output. While a single decision tree can overfit (become "spikey") because some leaf nodes can be based on single training examples, a forest gives a smoother prediction function since it is averaging across many trees, and often other trees won't be spikey for the same input (their leaf node may be based on many training points, not a single one).

Finally, the authors refer to random forests as _adaptive smoothers_ to point out that random forests become even better at smoothing in locations in the input space that either have high variation (intuitively, that have a higher slope), or that are far from the training data. The word "adaptive" indicates that the predicted function changes behavior based on the nature of the data — eg, with k-NN, an adaptive version might increase the value of k at some places in the input space.

The way random forests act adaptively is that (a) the prediction function is naturally more dense (can change value more quickly) in areas of high variability because those locations will have more leaf nodes, and (b) the prediction function is typically a combination of a wider variety of possible values when the input is far from the training data because in that case the trees are likely to provide a variety of output values. These are both ways to avoid overfitting to training data and to generalize better to new inputs.

Disclaimer: I did not carefully read the paper; this is my quick understanding.

However, I'm having trouble understanding the meaning of the numbers in table 4. Does anyone understand all the columns there?

What I'm particularly interested in is finding the unnormalized coefficients from the color matching experiments, or some way to un-normalize those coefficients. (By "those coefficients," I mean the trichromatic coefficients u{a,b,c}_\lambda listed in table 3.) I don't know if that data is in table 4 so maybe those are two separate questions.

Note that many of the diagrams are interactive 3d graphics (I didn't realize that at first, and it makes the page more interesting.)

Although it's not spelled out in the article, I'm hoping that the feature of agency along with an emotional system would enable constructive social behavior. Agency is helpful because it would empower AI models to meaningfully speak to each other, for example. Human emotions like empathy, social alignment, curiosity, or persistence could all help AI models to get along well with others.

Context for the article: I'm working on an ambitious long-term project to write a book about consciousness from a scientific and analytic (versus, say, a meditation-oriented) perspective. I didn't write this fact in the article, but what I'd love to happen is that I meet people with a similar optimistic perspective, and to learn and improve my communication skills via follow-up conversations.

If anyone is interested in chatting more about the topic of the article, please do email me. My email is in my HN profile. Thanks!

I think the contradiction you see is that the model would have to form a completion to the external input it receives. I'm suggesting that the model would have many inputs: one would be the typical input stream, just as LLMs see, but another would be its own internal recent vectors, akin to a recent stream of thought. A "mode" is not built in to the model; at each token point, it can output whatever vector it wants, and one choice is to output the special "" token, which means it's not talking. So the "mode" idea is a hoped-for emergent behavior.

Some more details on using two input streams:

All of the input vectors (internal + external), taken together, are available to work with. It may help to think in terms of the typical transformer architecture, where tokens mostly become a set of vectors, and the original order of the words are attached as positional information. In other words, transformers don't really see a list of words, but a set of vectors, and the position info of each token becomes a tag attached to each vector.

So it's not so hard to merge together two input streams. They can become one big set of vectors, still tagged with position information, but now also tagged as either "internal" or "external" for the source.

It's a fair question, and I don't have all the answers. But for this question, there might be training data available from everyday human conversations. For example, we could use a speech-to-text model that's able to distinguish speakers, and look for points where one person decided to start speaking (that would be training data for when to switch modes). Ideally, the speech-to-text model would be able to include text even when both people spoke at once (this would provide more realistic and complete training data).

I've noticed that the audio mode in ChatGPT's app is good at noticing when I'm done speaking to it, and it reacts accurately enough that I suspect it's more sophisticated than "wait for silence." If there is a "notice the end of speaking" model - which is not a crazy assumption - then I can imagine a slightly more complicated model that notices a combination of "now is a good time to talk + I have something to say."

If you like deep dives, I suggest two follow-up tools for mathematical typesetting:

1. LuaTeX, which is TeX + Lua scripting support. (luatex.org) An example:

https://tex.stackexchange.com/questions/70/what-is-a-simple-...

This has become my default, and I enjoy it. (I use lualatex.)

2. A friend of mine was inspired to write a modern macOS app for math typesetting. It's called MadHat:

https://madhat.design/

One of the coolest features is "no compilation," which I view as a slight exaggeration, but what's true is that it can work quickly and incrementally so you don't have a compile-and-wait cycle, even on long documents.