I think it can be a problem if you recommend a book and expect the other person to have a social obligation to read it (and needs to make an effortful excuse or pay social capital if it’s not read). It might be hard to fully get rid of this, but I think the utility comparison that should be made is “social friction from someone not following a book recommendation” vs. “utility to the other person from you recommending a book based on knowledge of the book and the person’s preferences/interests”. I suspect that in most contexts this is both an EV-positive exchange and the person correctly decides not to read/finish the book. Maybe a good social norm would be to not get upset if someone doesn’t read your book rec, and also to not feel pressured to read a book that was recommended if you started it/ read a summary and decided it’s not for you
Dmitry Vaintrob
Very cool and well-presented—thanks for taking the time to write this down. I thought about this question at some point and ended up deciding that the compressed sensing picture isn’t very well shaped for this, but didn’t have a complete argument for this—it’s nice to have confirmation
On the friendship fallacy and Owen Barfield
I just finished reading the book “The Fellowship: The Literary Lives of the Inklings”, by Philip and Carol Zaleski. It’s a book about an intellectually appealing and socially cohesive group of writers in Oxford who met weekly and critiqued each other’s work, which included JRR Tolkien and CS Lewis. The book is very centered on Christianity (the writers also write Christian apologetics), but this works well, as understanding either Lewis or Tolkien or the Inklings in general without the lens of their deeply held thoughtful Christianity is about as silly as trying to analyze the Lion King without reading Hamlet.
But there is a core character in the book who is treated sympathetically and who I really hate: Owen Barfield, the “founding” Inkling. From his youth, he is a follower of Rudolf Steiner and a devoted Anthroposophist (a particularly benign group of Christian Occultists). Barfield was Lewis’s friend, existing always in his shadow (Lewis was very famous in his lifetime as a philosopher and Christian apologist, a kind of Jordan Peterson of his time if you imagine Jordan Peterson had brains and real literary/academic credentials). He worked in a law firm and consistently saw himself as a thwarted philosopher/writer/poet, and he found recognition late in life after he wrote a Lewis biography and after his woo-adjacent ideas became more popular in the 60s.
Throughout his life, Barfield created a personal philosophy of “all the things I like/ think are interesting are kind of the same thing”, and he was very sad when people he liked disapproved of, or failed to identify as “sort of the same thing” the different things that he mixed into his philosophy. While he generally is a bit of an “intellectual klutz”, his fundamental failure is the “Friendship Fallacy”: the idea of treating ideas as friends, as something deserving of loyalty. When he encounters different ideas he likes, he “wants them to get along,” and when ideas fail to convince skeptics or produce results or interface with reality (or indeed, with faith), he simply fails to impose any kind of falsifiability requirement and treats this as a loyalty test he must pass. He totally lacks the kind of internal courage needed to kill one’s darlings (whether philosophical or literary) and to treat his own ideas with skepticism and view towards falsification—perhaps the core trait of a good thinker (Feynman’s “You must not fool yourself—and you are the easiest person to fool”).
Interestingly, I don’t extend this antipathy to the Christianity of the group’s other famous members. Unlike Barfield, Tolkien and Chesterton largely succeed (imo) in separating the domains of the literary, the psychological, and the religious. They don’t pretend to be scientific authorities or predict things “in the world”. Tolkien in particular is very anti-progress and a bit of a luddite, but in my understanding his work as a linguist is very good for his time. In fact, it’s funny that his deeply Christian mentality created one of the most “atheist nerd”-like behaviors of creating thoroughly crafted fictional languages of fantasy cultures. I’ve been surprised to learn from reading a couple of his biographies that his linguistic worldbuilding in fact preceded his fantasy work: he designed Elvish before writing any work in his canon, and wrote the work to flesh out the mythology behind expressions and poems. He famously said about his work “The making of language and mythology are related functions”. In fact, he viewed the work of producing plausible cultures and languages—in my view an admirable (though non-academic) kind of secular scholarship analogous to studying alternative physical systems, etc. -- as an explicitly Christian task of “subcreation”, a sort of worship-by-imitation of God.
It’s a bit hard to exactly formulate a razor between the kind of “lazy scientism” of Barfield and various other forms of “pseudoscientific woo” and the serious and purely mystical/ inspirational deep religiosity of people like Tolkien (and to a lesser extent Lewis—another interesting thing I learned was that he started out as a devoted atheist in a world where this was actually socially fraught, and was converted through a philosophical struggle involving Barfield and Tolkien in particular). But maybe the idea of a “philosophy without struggle”: a tendency towards confirmation and a total lack of earnest self-questioning goes a part of the way towards explaining this distinction. Another part is the difference between a purely metaphysical personal religion and a more woo idea of a religion that “makes predictions about the world”. I think the thing that really took me aback a bit was the level of academic embrace of Barfield late in his life, not just as a Lewis biographer but as a respected academic philosopher with honorary professorships and the works—a confirmation (if ever more are needed) that lazy pseudointellectualism and confirmation bias are very much not incompatible with academic success. Another theme that I think is interesting is the fact that Lewis and Tolkien were at times genuinely interested and even somewhat inspired by his ideas (though they had no time for occultism or 60-esque woo). The extent to which this happened is hard to gauge (he outlived them and wrote a lot about how he influenced them in his biographies/reminiscences, and this was then picked up by scholars). But unquestioningly, this did occur to some extent. And whether or not you class Tolkien/Lewis as “valuable thinkers”, the history of science and philosophy does seem to abound with examples of clear and robust thinkers whose good ideas were to some extent inspired by charismatic charlatans and woo.
Below are my personal notes on Barfield that I wrote after reading the book.
I despise Barfield. Not in the visceral sense that the first syllable of his name may (Anthroposophically) evoke. Indeed I identify with the underdog/late-bloomer shape of his biography, with his striving towards a higher calling. I readily adopt the book’s sympathy towards him as a literary character with fortunes tied to an idea deeply espoused, a thwarted writer with some modicum of undiscovered talent. My antipathy isn’t even in the specifics of what he espouses: a mild but virulently wrong view of science and philosophy adjacent to all the stupid of my parents’ generation of `anthroposophy’ (Atlantis, Consciousness and Quantum Mechanics, anti-Evolutionism, Vibes). But I despise him as one of a Fundamental Mistake. That of confusing science and personality. Being loyal to a scientific or philosophical discipline isn’t like being loyal to a person: if it’s consistently fucking up and you need to make excuses for its behavior to all your reasonable smart friends, you’re not being a good friend but rather a bad scientist. Barfield is almost an archetype of Bad Science if you project out the crazy/dogmatic/ political/ evil-Nazi component. He really is a nice man. But within his mild-mannered Christian friendliness which I respect, he is inflexible and unscientific. He doesn’t update. He glows when people endorse his preferred view (Anthroposophy and Steiner) and sadly laments when they disagree with him—because he can’t help but feel like ``there’s something there″. He wants to seamlessly draw parallels between all the nice things he and other nice people believe. He draws lines of identification back and forth between all the things he likes (Coleridge <> Himself <> Quantum Mechanics <> Anthroposophy <> Steiner <> Religion <> Consciousness <> Complementary dualism/”polarity”). He has “nothing but symbols” in his brain, and the symbols in his brain aren’t strong enough to notice that they fail to signify. A person without significance, with a philosophy without significance, possessed of a brain without the capacity to grasp the concept of what it means to signify. The first of these is a tragedy (people should matter) and his late-found fame mediated through famous friends is a sweet story, maybe one he even deserves as the first-mover of the Inklings, the reason for the Lewis-Tolkien friendship, etc. The second is a neutral: theories that fail to achieve significance “in their lifetime” may be bunk but may have value: Greek Atomism, various prescient ideas about physics and computers (Babbage/ Lovelace), etc. But the third is a profound personal failing, and it’s only through luck and through (mostly well-placed) trust in much smarter and more rigorous friends that he avoided attaching this vapid form of mentation to something truly vile: Nazism (which he very briefly flirted with, charmed by its interest in magic and the occult), various fundamentalisms (including an anti-evolutionary fundamentalism: his friends believed in evolution but he didn’t really buy it “on vibes”; he was never a fundamentalist), Communism, etc.
Thanks for this post. I would argue that part of an explanation here could also be economic: modernity brings specialization and a move from the artisan economy of objects as uncommon, expensive, multipurpose, and with a narrow user base (illuminated manuscripts, decorative furniture) to a more utilitarian and targeted economy. Early artisans need to compete for a small number of rich clients by being the most impressive, artistic, etc., whereas more modern suppliers follow more traditional laws of supply and demand and track more costs (cost-effectiveness, readability and reader’s time vs. beauty and remarkableness). And consumers similarly can decouple their needs: art as separate from furniture and architecture, poetry and drama as separate from information and literature. I think another aspect of this shift, that I’m sad we’ve lost, is the old multipurpose scientific/philosophical treatises with illustrations or poems (my favorite being de Rerum Natura, though you could argue that Nietzsche and Wagner tried to revive this with their attempts at Gesamtkunstwerke).
I’m managing to get verve and probity, but having issues with wiles
I really liked the post—I was confused by the meaning and purpose no-coincidence principle when I was a ARC, and this post clarifies it well. I like that this is asking for something that is weaker than a proof (or a probabilistic weakening of proof), as [related to the example of using the Riemann hypothesis], in general you expect from incompleteness for there to be true results that lead to “surprising” families of circuits which are not provable by logic. I can also see Paul’s point of how this statement is sort of like P vs. BPP but not quite.
More specifically, this feels like a sort of 2nd-order boolean/polynomial hierarchy statement whose first-order version is P vs. BPP. Are there analogues of this for other orders?
Looks like a conspiracy of pigeons posing as lw commenters have downvoted your post
Thanks!
I haven’t grokked your loss scales explanation (the “interpretability insights” section) without reading your other post though.
Not saying anything deep here. The point is just that you might have two cartoon pictures:
every correctly classified input is either the result of a memorizing circuit or of a single coherent generalizing circuit behavior. If you remove a single generalizing circuit, your accuracy will degrade additively.
a correctly classified input is the result of a “combined” circuit consisting of multiple parallel generalizing “subprocesses” giving independent predictions, and if you remove any of these subprocesses, your accuracy will degrade multiplicatively.
A lot of ML work only thinks about picture #1 (which is the natural picture to look at if you only have one generalizing circuit and every other circuit is a memorization). But the thing I’m saying is that picture #2 also occurs, and in some sense is “the info-theoretic default” (though both occur simultaneously—this is also related to the ideas in this post)
Thanks for the questions!
You first introduce the SLT argument that tells us which loss scale to choose (the “Watanabe scale”, derived from the Watanabe critical temperature).
Sorry, I think the context of the Watanabe scale is a bit confusing. I’m saying that in fact it’s the wrong scale to use as a “natural scale”. The Watanabe scale depends only on the number of training datapoints, and doesn’t notice any other properties of your NN or your phenomenon of interest.
Roughly, the Watanabe scale is the scale on which loss improves if you memorize a single datapoint (so memorizing improves accuracy by 1/n with n = #(training set) and in a suitable operationalization, improves loss by , and this is the Watanabe scale).
It’s used in SLT roughly because it’s the minimal temperature scale where “memorization doesn’t count as relevant”, and so relevant measurements become independent of the n-point sample. However in most interp experiments, the realistic loss reconstruction loss reconstruction is much rougher (i.e., further from optimal loss) than the 1/n scale where memorization becomes an issue (even if you conceptualize #(training set) as some small synthetic training set that you were running the experiment on).
For your second question: again, what I wrote is confusing and I really want to rewrite it more clearly later. I tried to clarify what I think you’re asking about in this shortform. Roughly, the point here is that to avoid having your results messed up by spurious behaviors, you might want to degrade as much as possible while still observing the effect of your experiment. The idea is that if you found any degradation that wasn’t explicitly designed with your experiment in mind (i.e., is natural), but where you see your experimental results hold, then you have “found a phenomenon”. The hope is that if you look at the roughest such scale, you might kill enough confounders and interactions to make your result be “clean” (or at least cleaner): so for example optimistically you might hope to explain all the loss of the degraded model at the degradation scale you chose (whereas at other scales, there are a bunch of other effects improving the loss on the dataset you’re looking at that you’re not capturing in the explanation).
The question now is when degrading, what order you want to “kill confounders” in to optimally purify the effect you’re considering. The “natural degradation” idea seems like a good place to look since it kills the “small but annoying” confounders: things like memorization, weird specific connotations of the test sentences you used for your experiment, etc. Another reasonable place to look is training checkpoints, as these correspond to killing “hard to learn” effects. Ideally you’d perform several kinds of degradation to “maximally purify” your effect. Here the “natural scales” (loss on the level Claude 1 e.g., or Bert) are much too fine for most modern experiments, and I’m envisioning something much rougher.
The intuition here comes from physics. Like if you want to study properties of a hydrogen atom that you don’t see either in water or in hydrogen gas, a natural thing to do is to heat up hydrogen gas to extreme temperatures where the molecules degrade but the atoms are still present, now in “pure” form. Of course not all phenomena can be purified in this way (some are confounded by effects both at higher and at lower temperature, etc.).
Thanks! Yes the temperature picture is the direction I’m going in. I had heard the term “rate distortion”, but didn’t realize the connection with this picture. Might have to change the language for my next post
This seems overstated
In some sense this is the definition of the complexity of an ML algorithm; more precisely, the direct analog of complexity in information theory, which is the “entropy” or “Solomonoff complexity” measurement, is the free energy (I’m writing a distillation on this but it is a standard result). The relevant question then becomes whether the “SGLD” sampling techniques used in SLT for measuring the free energy (or technically its derivative) actually converge to reasonable values in polynomial time. This is checked pretty extensively in this paper for example.
A possibly more interesting question is whether notions of complexity in interpretations of programs agree with the inherent complexity as measured by free energy. The place I’m aware of where this is operationalized and checked is our project with Nina on modular addition: here we do have a clear understanding of the platonic complexity, and the local learning coefficient does a very good job of asymptotically capturing it with very good precision (both for memorizing and generalizing algorithms, where the complexity difference is very significant).
Citation? [for Apollo]
Look at this paper (note I haven’t read it yet). I think their LIB work is also promising (at least it separates circuits of small algorithms)
Thanks for the reference, and thanks for providing an informed point of view here. I would love to have more of a debate here, and would quite like being wrong as I like tropical geometry.
First, about your concrete question:
As I understand it, here the notion of “density of polygons’ is used as a kind of proxy for the derivative of a PL function?
Density is a proxy for the second derivative: indeed, the closer a function is to linear, the easier it is to approximate it by a linear function. I think a similar idea occurs in 3D graphics, in mesh optimization, where you can improve performance by reducing the number of cells in flatter domains (I don’t understand this field, but this is done in this paper according to some energy curvature-related energy functional). The question of “derivative change when crossing walls” seems similar. In general, glancing at the paper you sent, it looks like polyhedral currents are a locally polynomial PL generalization of currents of ordinary functions (and it seems that there is some interesting connection made to intersection theory/analogues of Chow theory, though I don’t have nearly enough background to read this part carefully). Since the purpose of PL functions in ML is to approximate some (approximately smooth, but fractally messy and stochastic) “true classification”, I don’t see why one wouldn’t just use ordinary currents here (currents on a PL manifold can be made sense of after smoothing, or in a distribution-valued sense, etc.).
In general, I think the central crux between us is whether or not this is true:
tropical geometry might be relevant ML, for the simple reason that the functions coming up in ML with ReLU activation are PL
I’m not sure I agree with this argument. The use of PL functions is by no means central to ML theory, and is an incidental aspect of early algorithms. The most efficient activation functions for most problems tend to not be ReLUs, though the question of activation functions is often somewhat moot due to the universal approximation theorem (and the fact that, in practice, at least for shallow NNs anything implementable by one reasonable activation tends to be easily implementable, with similar macroscopic properties, by any other). So the reason that PL functions come up is that they’re “good enough to approximate any function” (and also “asymptotic linearity” seems genuinely useful to avoid some explosion behaviors). But by the same token, you might expect people who think deeply about polynomial functions to be good at doing analysis because of the Stone-Weierstrass theorem.
More concretely, I think there are two core “type mismatches” between tropical geometry and the kinds of questions that appear in ML:
Algebraic geometry in general (including tropical geometry) isn’t good at dealing with deep compositions of functions, and especially approximate compositions.
(More specific to TG): the polytopes that appear in neural nets are as I explained inherently random (the typical interpretation we have of even combinatorial algorithms like modular addition is that the PL functions produce some random sharding of some polynomial function). This is a very strange thing to consider from the point of view of a tropical geometer: like as an algebraic geometer, it’s hard for me to imagine a case where “this polynomial has degree approximately 5… it might be 4 or 6, but the difference between them is small”. I simply can’t think of any behavior that is at all meaningful from an AG-like perspective where the questions of fan combinatorics and degrees of polynomials are replaced by questions of approximate equality.
I can see myself changing my view if I see some nontrivial concrete prediction or idea that tropical geometry can provide in this context. I think a “relaxed” form of this question (where I genuinely haven’t looked at the literature) is whether tropical geometry has ever been useful (either in proving something or at least in reconceptualizing something in an interesting way) in linear programming. I think if I see a convincing affirmative answer to this relaxed question, I would be a little more sympathetic here. However, the type signature here really does seem off to me.
If I understand correctly, you want a way of thinking about a reference class of programs that has some specific, perhaps interpretability-relevant or compression-related properties in common with the deterministic program you’re studying?
I think in this case I’d actually say the tempered Bayesian posterior by itself isn’t enough, since even if you work locally in a basin, it might not preserve the specific features you want. In this case I’d probably still start with the tempered Bayesian posterior, but then also condition on the specific properties/explicit features/ etc. that you want to preserve. (I might be misunderstanding your comment though)
Statistical localization in disordered systems, and dreaming of more realistic interpretability endpoints
[epistemic status: half fever dream, half something I think is an important point to get across. Note that the physics I discuss is not my field though close to my interests. I have not carefully engaged with it or read the relevant papers—I am likely to be wrong about the statements made and the language used.]
A frequent discussion I get into in the context of AI is “what is an endpoint for interpretability”. I get into this argument from two sides:
arguing with interpretability purists, who say that the only way to get robust safety from interpretability is to mathematically prove that behaviors are safe and/or no deception is going on.
arguing with interpretability skeptics, who say that the only way to get robust safety from interpretability is to prove that behaviors are safe and/or no deception is going on.
My typical response to this is that no, you’re being silly: imagine discussing any other phenomenon in this way: “the only way to show that the sun will rise tomorrow is to completely model the sun on the level of subatomic particles and prove that they will not spontaneously explode”. Or asking a bridge safety expert to model every single particle and provably lower-bound the probability of them losing structural coherence in a way not observed by bulk models.
But there’s a more fundamental intuition here, that I started developing when I started trying to learn statistical physics. There are a few lossy ways of expressing it. One is to talk about renormalization, how assumptions about renormalizability of systems is a “theorem” in statistical mechanics, but is not (and probably never will be) proven mathematically, (in some sense, it feels much more like a “truly new flavor of axiom” than even complexity-theoretic things like P vs. NP). But that’s still not it. There is a more general intuition, that’s hard to get across (in particular for someone who, like me, is only a dabbler in the subject) -- that some genuinely incredibly complex and information-laden systems have some “strong locality” properties, which are (insofar as the physical meaning of the word holds meaning) both provable and very robust to changing and expanding the context.
For a while, I thought that this is just a vibe—a way to guide thinking, but not something that can be operationalized in a way that may significantly convince people without a similar intuition.
However, recently I’ve become more hopeful that an “explicitly formalizable” notion of robust interpretability may fall out of this language in a somewhat natural way.
This is closely related to recent discussions and writeups we’ve been doing with Lauren Greenspan on scale and renormalization in (statistical) QFT and connections to ML.
One direction to operationalize this is through the notion of “localization” in statistical physics, and in particular “Anderson localization”. The idea (if I understand it correctly) is that in certain disordered systems (think of a semiconductor, which is an “ordered” metal with a disordered system of “impurity atoms” sprinkled inside), you can prove a kind of screening property: that from the point of view of the localized dynamics near a particular spin, you can provably ignore spins far away from the point you’re studying (or rather, replace them by an “ordered” field that modifies the local dynamics in a fully controllable way). This idea of of local interactions being “screened” from far-away details is ubiquitous. In a very large and very robust class of systems, interactions are purely local, except for mediation by a small number of hierarchical “smooth” couplings that see only high-level summary statistics of the “non-local” spins and treat them as a background—and moreover, these “locality” properties are provable (insofar as we assume the extra “axioms” of thermodynamics), assuming some (once again, hierarchical and robustly adjustable) assumptions of independence. There are a number of related principles here that (if I understand correctly) get used in similar contexts, sometimes interchangeably: one I liked is “local perturbations perturb locally” (“LPPL”) from this paper.
Note that in the above paragraph I did something I generally disapprove of: I am trying to extract and verbalize “vibes” from science that I don’t understand on a concrete level, and I am almost certainly getting a bunch of things wrong. But I don’t know of another way of gesturing in a “look, there’s something here and it’s worth looking into” way without doing this to some extent.
Now AI systems, just like semiconductors, are statistical systems with a lot of disorder. In particular in a standard operationalization (as e.g. in PDLT), we can conceptualize of neural nets as a field theory. There is a “vacuum theory” that depends only on the architecture, and then adding new datapoints corresponds to adding particles. PDLT only studies a certain perturbative picture here, but it seems plausible that an extension of these techniques may extend to non-perturbative scales (and hope for this is a big part of the reason that Lauren and I have been thinking and writing about renormalization). In a “dream” version of such an extension, the datapoints would form a kind of disordered system, with both ordered components, hierarchical relationships, and some assumption of inherent randomness outside of the relationships. A great aspect of “numerical” QFT, such as gets applied in condensed matter models, is that you don’t need a really great model of the hierarchical relationships: sometimes you can just play around and turn on a handful of extra parameters until you find something that works. (Again, at the moment this is an imprecise interpretation of things I have not deeply engaged with.)
Of course doing this makes some assumptions—but the assumptions are on the level of the data (i.e. particles), not the weights/ model internals (i.e., fields—the place where we are worried about misalignment, etc.). And if you assume these assumptions and write down a “localization theorem” result, then plausibly the kind of statement you will get is something along the lines of the following:
“the way this LLM is completing this sentence is a combination of a sophisticated collection of hierarchical relationships, but I know that the behavior here is equivalent to behaviors on other similar sentences up to small (provably) low-complexity perturbations”.
More generally, the kinds of information this kind of picture would give is a kind of “local provably robust interpretability”—where the text completion behavior of a model is provably (under suitable “disordered system” assumptions) reducible to a collection of several local circuits that depend on understandable phenomena at a few different scales. A guiding “complexity intuition” for me here is provided by the notrivial but tractable grammar task diagrams in the paper Marks et al. (See pages 25-27, and note the shape of these diagrams is more or less straightup typical of the shape of a nonrenormalized interaction diagram you see before you start applying renormalization to simplify a statistical system).
An important caveat here is that in physical models of this type (and in pictures that include renormalization more generally), one does not make—or assume—any “fundamentality” assumptions. In many cases a number of alternative (but equivalent, once the “screening” is factored in) pictures exist, with various levels of granularity, elegance, etc. (this already can be seen in the 2D Ising model—a simple magnet model—where the same behaviors can either be understood in a combinatorial “spin-to-spin interaction” way, which mirrors the “fundamental interpretability” desires of mechinterp, and through this “recursive screening out” model that is more renormalization-flavored; the results are the same (to a very high level of precision), even when looking at very localized effects involving collections of a few spins. So the question of whether an interpretation is “fundamental” or uses the “right latents” is to a large extent obviated here; the world of thermodynamics is much more anarchical and democratic than the world of mathematical formalism and “elegant proof”, at least in this context.
Having handwavily described a putative model, I want to quickly say that I don’t actually believe in this model. There are a bunch of things I probably got wrong, there are a bunch of other, better tools to use, and so on. But the point is not the model: it’s that this kind of stuff exists. There exist languages that show that arbitrarily complex, arbitrarily expressive behaviors are provably reducible to local interactions, where behaviors can be understood as clusters of hierarchical interactions that treat all but a few parts of the system at every point as “screened out noise”.
I think that if models like this are possible, then a solution to “the interpretability component to safety” is possible in this framework. If you have provably localized behaviors then for example you have a good idea where to look for deception: e.g., deception cannot occur on the level “very low-level” local interactions, as they are too simple to express the necessary reasoning, and perhaps it can be carefully operationalized and tracked in the higher-level interactions.
As you’ve no doubt noticed, this whole picture is splotchy and vague. It may be completely wrong. But there also may be something in this direction that works. I’m hoping to think more about this, and very interested in hearing people’s criticisms and thoughts.
What application do you have in mind? If you’re trying to reason about formal models without trying to completely rigorously prove things about them, then I think thinking of neural networks as stochastic systems is the way to go. Namely, you view the weights as a random variable solving a stochastic optimization problem to produce a weight-valued random variable, then conditioning it on whatever knowledge about the weights/activations you assume is available. This can be done both in the Bayesian “thermostatic” sense as a model of idealized networks, and in the sense of modeling the NN as SGD-like systems. Both methods are explored explicitly (and give different results) in suitable high width limits by the PDLT and tensor networks paradigms (the latter also looks at “true SGD” with nonnegligible step size).
Here you should be careful about what you condition on, as conditioning on exact knowledge of too much input-output behavior of course blows stuff up, and you should think of a way of coarse-graining, i.e. “choose a precision scale” :). Here my first goto would be to assume the tempered Boltzmann distribution on the loss at an appropriate choice of temperature for what you’re studying.
If you’re trying to do experiments, then I would suspect that a lot of the time you can just blindly throw whatever ML-ish tools you’d use in an underdetermined, “true inference” context and they’ll just work (with suitable choices of hyperparameters)
This is where this question of “scale” comes in. I want to add that (at least morally/intuitively) we are also thinking about discrete systems like lattices, and then instead of a regulator you have a coarsegraining or a “blocking transformation”, which you have a lot of freedom to choose. For example in PDLT, the object that plays the role of coarsegraining is the operation that takes a probability distribution on neurons and applies a single-layer NN to it.
Thanks for the reference—I’ll check out the paper (though there are no pointer variables in this picture inherently).
I think there is a miscommunication in my messaging. Possibly through overcommitting to the “matrix” analogy, I may have given the impression that I’m doing something I’m not. In particular, the view here isn’t a controversial one—it has nothing to do with Everett or einselection or decoherence. Crucially, I am saying nothing at all about quantum branches.
I’m now realizing that when you say map or territory, you’re probably talking about a different picture where quantum interpretation (decoherence and branches) is foregrounded. I’m doing nothing of the sort, and as far as I can tell never making any “interpretive” claims.
All the statements in the post are essentially mathematically rigorous claims which say what happens when you
start with the usual QM picture, and posit that
your universe divides into at least two subsystems, one of which you’re studying
one of the subsystems your system is coupled to is a minimally informative infinite-dimensional environment (i.e., a bath).
Both of these are mathematically formalizable and aren’t saying anything about how to interpret quantum branches etc. And the Lindbladian is simply a useful formalism for tracking the evolution of a system that has these properties (subdivisions and baths). Note that (maybe this is the confusion?) subsystem does not mean quantum branch, or decoherence result. “Subsystem” means that we’re looking at these particles over here, but there are also those particles over there (i.e. in terms of math, your Hilbert space is a tensor product
Also, I want to be clear that we can and should run this whole story without ever using the term “probability distribution” in any of the quantum-thermodynamics concepts. The language to describe a quantum system as above (system coupled with a bath) is from the start a language that only involves density matrices, and never uses the term “X is a probability distribution of Y”. Instead you can get classical probability distributions to map into this picture as a certain limit of these dynamics.
As to measurement, I think you’re once again talking about interpretation. I agree that in general, this may be tricky. But what is once again true mathematically is that if you model your system as coupled to a bath then you can set up behaviors that behave exactly as you would expect from an experiment from the point of view of studying the system (without asking questions about decoherence).
This is fascinating! If there’s nothing else going on with your prompting, this looks like an incredibly hacky mid-inference intervention. My guess would be that openai applied some hasty patch against a sycophancy steering vector and this vector caught both actual sycophantic behaviors and descriptions of sycophantic behaviors in LLMs (I’d guess “sycophancy” as a word isn’t so much the issue as the LLM behavior connotation). Presumably the patch they used activates at a later token in the word “sycophancy” in an AI context. This is incredibly low-tech and unsophisticated—like much worse than the stories of repairing Apollo missions with duct tape. Even a really basic finetuning would not exhibit this behavior (otoh, I suppose stuff like this works for humans, where people will sometimes redirect mid-sentence).
FWIW, I wasn’t able to reconstruct this exact behavior (working in an incognito window with a fresh chatgpt instance), but it did suspiciously avoid talking about sycophancy and when I asked about sycophancy specifically, it got stuck in inference and returned an error