I think this is a very nice way to present the key ideas. However, in practice I think the discretisation is actually harder to reason about than the continuous version. There are deeper problems, but I'd start by wondering how you would ever compute c(f) defined this way, since it seems to depend in an intricate way on the details of e.g. the floating point implementation.
I'll note that the volume codimension definition of the RLCT is essentially what you have written down here, and you don't need any mathematics beyond calculus to write that down. You only need things like resolutions of singularities if you actually want to compute that value, and the discretisation doesn't seem to offer any advantage there.
I think this is a very nice way to present the key ideas. However, in practice I think the discretisation is actually harder to reason about than the continuous version. There are deeper problems, but I'd start by wondering how you would ever compute c(f) defined this way, since it seems to depend in an intricate way on the details of e.g. the floating point implementation.
I would say that the discretization is going to be easier for people with a computer science background to grasp, even though formally I agree it's going to be less pleasant to reason about or to do computations with. Still, if properties of NNs that only appeared when they are continuous functions on were essential for their generalization, we might be in trouble as people keep lowering the precision of their floating point numbers. This explanation makes it clear that while assuming NNs are continuous (or even analytic!) might be useful for theoretical purposes, the claims about generalization hold just as well in a more realistic discrete setting.
I'll note that the volume codimension definition of the RLCT is essentially what you have written down here, and you don't need any mathematics beyond calculus to write that down. You only need things like resolutions of singularities if you actually want to compute that value, and the discretisation doesn't seem to offer any advantage there.
Yes, my definition is inspired by the volume codimension definition, though here we don't need to take a limit as some because the counting measure makes our life easy. The problem you have in a smooth setting is that descending the Lebesgue measure in a dumb way to subspaces with positive codimension gives trivial results, so more care is necessary to recover and reason about the appropriate notions of volume.
None of this is specific to singular learning theory. The basic idea that the parameter-function map might be degenerate and biased towards simple functions predates SLT(at least this most recent wave of interest in its application to neural nets anyway) and indeed goes back to the 90s, no algebraic geometry required. As far as I can tell, the non-trivial content of SLT is that the averaging over parameters with a given loss is dominated by singular points in the limit because volume clusters there as you take an ever-narrower interval around the minimum set. That's interesting, but I don't have a strong expectation it will end up being applicable to real neural nets since I don't see a mechanism by which SGD is supposed to be attracted to such points(I can see why SGD would be attracted to broad basins generally, but that's not SLT-specific -- the SLT-specific part is attraction to weird points where many broad basins intersect)
None of this is specific to singular learning theory. The basic idea that the parameter-function map might be degenerate and biased towards simple functions predates SLT(at least this most recent wave of interest in its application to neural nets anyway) and indeed goes back to the 90s, no algebraic geometry required.
Sure, I'm aware that people have expressed these ideas before, but I have trouble understanding what is added by singular theory on top of this description. To me, much of singular learning theory looks like trying to do these kinds of calculations in an analytic setting where things become quite a bit more complicated, for example because you no longer have the basic counting function to measure the effective dimensionality of a singularity, forcing you to reach for concepts like "real log canonical threshold" instead.
As far as I can tell, the non-trivial content of SLT is that the averaging over parameters with a given loss is dominated by singular points in the limit because volume clusters there as you take an ever-narrower interval around the minimum set.
I'm not sure why we should expect that beyond the argument I already give in the post. The geometry of the loss landscape is already fully accounted for by the Boltzmann factor; what else does singular learning theory add here?
Maybe this is also what you're confused about when you say "I don't see a mechanism by which SGD is supposed to be attracted to such points".
I’m not sure why we should expect that beyond the argument I already give in the post. The geometry of the loss landscape is already fully accounted for by the Boltzmann factor; what else does singular learning theory add here?
So I believe the point of SLT is that the Boltzmann-weighted integral over the state-space simplifies in certain settings as the number of data points approaches infinity. That integral is going to be dominated by a narrow 'band' around the minimum set, and to evaluate it generally you have to consider the entire minimum set. But when there are singularities, places where there are cusps or intersections of the minimum set, the narrow band's effective dimensionality can go up(this is illustrated in the tweet I linked). This means that as you can just consider the behavior near the 'cuspiest' singularity(I think this is what the RLCT measures) to understand the whole integral.
(...uh, I think. I actually haven't looked into the details enough to write with confidence, but the above is my impression from what reading I have done and jesse's tweet)
To me that just sounds like you're saying the integral is dominated by the contribution of the simplest functions that are of minimum loss, and the contribution factor scales like where is the effective dimensionality near the singularity representing this function, equivalently the complexity of said function. That's exactly what I'm saying in my post - where is the added content here?
Here's a concrete toy example where SLT and this post give different answers (SLT is more specific). Let . And let . Then the minimal loss is achieved at the set of parameters where or (note that this looks like two intersecting lines, with the singularity being the intersection). Note that all in that set also give the same exact . The theory in your post here doesn't say much beyond the standard point that gradient descent will (likely) select a minimal or near-minimal , but it can't distinguish between the different values of within that minimal set.
SLT on the other hand says that gradient descent will be more likely to choose the specific singular value .
Now I'm not sure this example is sufficiently realistic to demonstrate why you would care about SLT's extra specificity, since in this case I'm perfectly happy with any value of in the minimal set - they all give the exact same . If I were to try to generalize this into a useful example, I would try to find a case where has a minimal set that contains multiple different . For example, only evaluates on a subset of points (the 'training data') but different choices of minimal give different values outside of that subset of training data. Then we can consider which has the best generalization to out-of-training data - do the parameters predicted by SLT yield that are best at generalizing?
Disclaimer: I have a very rudimentary understanding of SLT and may be misrepresenting it.
I don't think this representation of the theory in my post is correct. The effective dimension of the singularity near the origin is much higher, e.g. because near every other minimal point of this loss function the Hessian doesn't vanish, while for the singularity at the origin it does vanish. If you discretized this setup by looking at it with a lattice of mesh , say, you would notice that the origin is surrounded by many parameters that give nearly identical loss, while near other parts of the space the number of such parameters is far fewer.
The reason you have to do some kind of "translation" between the two theories is that SLT can see not just exactly optimal points but also nearly optimal points, and bad singularities are surrounded by many more nearly optimal points than better-behaved singularities. You can interpret the discretized picture above as the SLT picture seen at some "resolution" or "scale" , i.e. if you discretized the loss function by evaluating it on a lattice with mesh you get my picture. Of course, this loses the information of what happens as and in some thermodynamic limit, which is what you recover when you do SLT.
I just don't see what this thermodynamic limit tells you about the learning behavior of NNs that we didn't know before. We already know NNs approximate Solomonoff induction if the -complexity is a good approximation to Kolmogorov complexity and so forth. What additional information is gained by knowing what looks like as a smooth function as opposed to a discrete function?
In addition, the strong dependence of SLT on being analytic is bad, because analytic functions are rigid: their value in a small open subset determines their value globally. I can see why you need this assumption because quantifying what happens near a singularity becomes incredibly difficult for general smooth functions, but because of the rigidity of analytic functions the approximation that "we can just pretend NNs are analytic" is more pernicious than e.g. "we can just pretend NNs are smooth". Typical approximation theorems like Stone-Weierstrass also fail to save you because they only work in the sup-norm and that's completely useless for determining behavior at singularities. So I'm yet to be convinced that the additional details in SLT provide a more useful account of NN learning than my simple description above.
The effective dimension of the singularity near the origin is much higher, e.g. because near every other minimal point of this loss function the Hessian doesn't vanish, while for the singularity at the origin it does vanish. If you discretized this setup by looking at it with a lattice of mesh , say, you would notice that the origin is surrounded by many parameters that give nearly identical loss, while near other parts of the space the number of such parameters is far fewer.
As I read it, the arguments you make in the original post depend only on the macrostate , which is the same for both the singular and non-singular points of the minimal loss set (in my example), so they can't distinguish these points at all. I see that you're also applying the logic to points near the minimal set and arguing that the nearly-optimal points are more abundant near the singularities than near the non-singularities. I think that's a significant point not made at all in your original point that brings it closer to SLT, so I'd encourage you to add it to the post.
I think there's also terminology mismatch between your post and SLT. You refer to singularities of (i.e. its derivative is degenerate) while SLT refers to singularities of the set of minimal loss parameters. The point in my example is not singular at all in SLT but is singular. This terminology collision makes it sound like you've recreated SLT more than you actually have.
I'm not too sure how to respond to this comment because it seems like you're not understanding what I'm trying to say.
I agree there's some terminology mismatch, but this is inevitable because SLT is a continuous model and my model is discrete. If you want to translate between them, you need to imagine discretizing SLT, which means you discretize both the codomain of the neural network and the space of functions you're trying to represent in some suitable way. If you do this, then you'll notice that the worse a singularity is, the lower the -complexity of the corresponding discrete function will turn out to be, because many of the neighbors map to the same function after discretization.
The content that SLT adds on top of this is what happens in the limit where your discretization becomes infinitely fine and your dataset becomes infinitely large, but your model doesn't become infinitely large. In this case, SLT claims that the worst singularities dominate the equilibrium behavior of SGD, which I agree is an accurate claim. However, I'm not sure what this claim is supposed to tell us about how NNs learn. I can't make any novel predictions about NNs with this knowledge that I couldn't before.
In this case, SLT claims that the worst singularities dominate the equilibrium behavior of SGD, which I agree is an accurate claim. However, I'm not sure what this claim is supposed to tell us about how NNs learn
I think the implied claim is something like "analyzing the singularities of the model will also be helpful for understanding SGD in more realistic settings" or maybe just "investigating this area further will lead to insights which are applicable in more realistic settings". I mostly don't buy it myself.
the worse a singularity is, the lower the -complexity of the corresponding discrete function will turn out to be
This is where we diverge. Please let me know where you think my error is in the following. Returning to my explicit example (though I wrote originally but will instead use in this post since that matches your definitions).
1. Let be the constant zero function and
Disclaimer: I'm by no means an expert on singular learning theory and what I present below is a simplification that experts might not endorse. Still, I think it might be more comprehensible for a general audience than going into digressions about blowing up singularities and birational invariants.
Here is my current understanding of what singular learning theory is about in a simplified (though perhaps more realistic?) discrete setting.
Suppose you represent a neural network architecture as a map A:2N→F where 2={0,1}, 2N is the set of all possible parameters of A (seen as floating point numbers, say) and F is the set of all possible computable functions from the input and output space you're considering. In thermodynamic terms, we could identify elements of 2N as "microstates" and the corresponding functions that the NN architecture A maps them to as "macrostates".
Furthermore, suppose that F comes together with a loss function L:F→R evaluating how good or bad a particular function is. Assume you optimize L using something like stochastic gradient descent on the function L with a particular learning rate.
Then, in general, we have the following results:
The intuition for why we would expect (3) to be true in practice has to do with the nature of the function approximator A. When c(f) is small, it probably means that we only need a small number of bits of information on top of the definition of A itself to define f, because "many" of the possible parameter values for A are implementing the function f. So f is probably a simple function.
On the other hand, if f is a simple function and A is sufficiently flexible as a function approximator, we can probably implement the functionality of f using only a small number of the N bits in the codomain of A, which leaves us the rest of the bits to vary as we wish. This makes |A−1(f)| quite large, and by extension the complexity c(f) quite small.
The vague concept of "flexibility" mentioned in the paragraph above requires A to have singularities of many effective dimensions, as this is just another way of saying that the image of A has to contain functions with a wide range of A-complexities. If A is a one-to-one function, this clean version of the theory no longer works, though if A is still "close" to being singular (for instance, because many of the functions in its image are very similar) then we can still recover results like the one I mentioned above. The basic insights remain the same in this setting.
I'm wondering what singular learning theory experts have to say about this simplification of their theory. Is this explanation missing some important details that are visible in the full theory? Does the full theory make some predictions that this simplified story does not make?