Yeah, I didn’t mean to imply that you guys said ‘simple --> large volume’ anywhere. I just think it’s a point worth emphasizing, especially around here where I think people will imagine “Solomonoff Induction-like” when they hear about a “bias towards simple functions”
Also, very briefly on your comment on feature learning—the GP limit is used to calculate the volume of functions locally to the initialization. The fact that kernel methods do not learn features should not be relevant given this interpretation
But in the infinite-width setting, Bayesian inference in general is given by a GP limit, right? Initialization doesn’t matter. This means that the arguments for lack of feature learning still go through. It’s technically possible that there could be feature learning in finite-width randomly-sampled networks, but it seems strange that finiteness would help here(and any such learning would be experimentally inaccessible). This is a major reason that I’m skeptical of the “SGD as a random sampler” picture.
[First thank you for your comments and observations—it’s always interesting to read pushback]
First, I think my point about using the GP to measure the volume occupied functions locally to where SGD trained networks are initialised is important. We are not really comparing NNs to NNGPs (well, technically we are, but we are interpreting what the NNGP does differently). We are trying to argue that SGD acts as a random sampler—it will find functions with probability proportional to the volume of those functions local to where the optimiser is in parameter-space. We argue that this quantity is well approximated by the NNGP.
This is relevant to the comments on features: if you look at the definition of $P_B(f|S)$ it’s fairly clear that (assuming training by random sampling) initialising and freezing all but the last layer and then random sampling over that will, in expectation, give precisely the same posterior distribution as if you were to random sample over the whole network. This property holds for finite and infinite width networks. This may seem counterintuitive, but the term P(S|f) in the definition of $P_B(f|S)$ ensures that if the random initialisation of the frozen layers does not allow for 100% training accuracy, that random initialisation is ignored. Therefore, if an optimiser samples functions proportional to their volume, you won’t get any difference in performance if you learn features (optimise the whole network) or do not learn features (randomly initialise and freeze all but the last layer and then train just the last).
Given therefore that the posteriors are the same, it implies that feature learning is not aiding inductive bias—rather, feature learning is important for expressivity reasons. The reason why you can’t just use frozen initial layers and obtain the same inductive bias on SOTA architectures is most likely because you can’t make the layers wide enough, to ensure that the network is expressive enough with high probability. Imagenet for example has input dimension of ~200000 so you would need some very wide layers to approach the wide-layer limit.
Furthermore (and on a slightly different note), it is known that infintesimal GD converges to the Boltzmann distribution for any DNN (very similar to random sampling) https://arxiv.org/abs/2004.01190. This means that the coloured noise in SGD is the only possible source for drastically improved inductive bias (which would have to emerge only on large scales, as we do not observe it at smaller scales). I have also not heard as good a theoretical justification for why this noise would dramatically aid generalisation.
Given this, I think it a sensible null hypothesis that optimisers are approximately performing random sampling from a well-biased parameter-space (with some subtleties, see my other comment about tempered posteriors), at substantially larger scales. This to me makes more sense than “optimisers perform random sampling at small/medium scales, but as you move to bigger scales coloured noise in SGD is the dominant source of inductive bias”.
Finally, I would like to point out that this is my impression from the literature, and my work. I am aware that there’s a lot I don’t know, and if anyone can point out why this line of argument is not correct, or can steelman a case for SGD inductive bias appearing at larger scales, I would be very interested to hear it.
First thank you for your comments and observations—it’s always interesting to read pushback
And thanks for engaging with my random blog comments! TBC, I think you guys are definitely on the right track in trying to relate SGD to function simplicity, and the empirical work you’ve done fleshing out that picture is great. I just think it could be even better if it was based around a better SGD scaling limit ;)
Therefore, if an optimiser samples functions proportional to their volume, you won’t get any difference in performance if you learn features (optimise the whole network) or do not learn features (randomly initialise and freeze all but the last layer and then train just the last).
Right, this is an even better argument that NNGPs/random-sampled nets don’t learn features.
Given therefore that the posteriors are the same, it implies that feature learning is not aiding inductive bias—rather, feature learning is important for expressivity reasons
I think this only applies to NNGP/random-sampled nets, not SGD-trained nets. To apply to SGD-trained nets, you’d need to show that the new features learned by SGD have the same distribution as the features found in an infinitely-wide random net, but I don’t think this is the case. By illustration, some SGD-trained nets can develop expressive neurons like ‘car detector’, enabling them to fit the data with a relatively small number of such neurons. If you used an NNGP to learn the same thing, you wouldn’t get a single ‘car detector’ neuron, but rather some huge linear combination of high-frequency features that can approximate the cars seen in the dataset. I think this would probably generalize worse than the network with an actual ‘car detector’(this isn’t empirical evidence of course, but I think what we know about SGD-trained nets and the NNGP strongly suggests a picture like this)
Furthermore (and on a slightly different note), it is known that infintesimal GD converges to the Boltzmann distribution for any DNN (very similar to random sampling)
Interesting, haven’t seen this before. Just skimming the paper, it sounds like the very small learning rate + added white noise might result in different limiting behavior from usual SGD. Generally it seems that there are a lot of different possible limits one can take; empirically SGD-trained nets do seem to have ‘feature learning’ so I’m skeptical of limits that don’t have that(I assume they don’t have them for theoretical reasons, anyway. Would be interesting to actually examine the features found in networks trained like this, and to see if they can do transfer learning at all) re:‘colored noise’, not sure to what extent this matters. I think a more likely source of discrepancy is the lack of white noise in normal training(I guess this counts as ‘colored noise’ in a sense) and the larger learning rate.
if anyone can point out why this line of argument is not correct, or can steelman a case for SGD inductive bias appearing at larger scales, I would be very interested to hear it.
Not to be a broken record, but I strongly recommend checking out Greg Yang’s work. He clearly shows that there exist infinite-width limits of SGD that can do feature/transfer learning.
[Advance apologies if I haven’t explained stuff well enough here. I think the important theme here is that we should maintain a way of thinking about the random sampling picture that is distinct from NNGPs.]
Right, this is an even better argument that NNGPs/random-sampled nets don’t learn features.
Ah I see I need to explain myself further—the following is very counterintuitive but I think it’s right. Learning features involves the movement of weights in the early layers, by definition. The claim I am making is that the reason why feature learning is good is not because it improves inductive bias—it is because it allows the network to be compressed. That is probably at the core of our disagreement.
Imagine taking a network and making it so thin that it is only just able to represent the function it needs to. Now try the training with last layer only after randomly initialising the others. You can’t—because the randomly initialised first layers will drastically decrease its expressivity, so you can’t express the true function. Now do the same, but with very wide layers—by the lottery ticket hypothesis (and in the limit of infinite width) this will work well because of the (near) unlimited expressivity. Hence, for narrow networks you have to “learn features” to make sure you are expressive enough, but for wide ones you do not.
Consider ResNet18 on Imagenet. Imagenet has an input dimension of $3\times256\times256\approx200000$. The widths of the layers within the resnet are at least two orders of magnitude smaller (so you are nowhere near the limit of infinite width). This is the case of the thin network I talked about earlier—you have to learn features precisely because the network isn’t expressive enough for you to get away without. I’m pretty sure this is the motivation for making networks deep in the first place—for expressivity reasons.
So my claim is features are important to keep the number of parameters small, but do not in themselves aid inductive bias. I know that the first pushback to this will be “but transfer learning improves inductive bias.” Of course—you are basically taking a network that has just been trained on millions of images, and then using this on a set of new images—there will be some information in common across images that has been encoded in the earlier layers. The hierarchical nature of neural networks allows this to happen, but fundamentally not in a way that could not be explained by the random sampling picture.
So, in conclusion, I don’t think SGD needs to be doing any “feature learning” beyond what can be achieved in the random sampling fashion. Note that the random sampling arguments apply not only in the limit of infinite width.
[However, it is worth noting that this is conjecture, although I think it is the most natural conclusion from what we know about DNNs. That said, I will only be happy to accept it when we have found a good way of rigorously comparing the posteriors of a random-sample trained finite width neural network and its corresponding SGD trained version]
I think this would probably generalize worse than the network with an actual ‘car detector’(this isn’t empirical evidence of course, but I think what we know about SGD-trained nets and the NNGP strongly suggests a picture like this)
What do we know about SGD-trained nets that suggests this?
Not to be a broken record, but I strongly recommend checking out Greg Yang’s work. He clearly shows that there exist infinite-width limits of SGD that can do feature/transfer learning.
I’ve read the new feature learning paper! We’re big fans of his work, although again I don’t think it contradicts anything I’ve just said.
The claim I am making is that the reason why feature learning is good is not because it improves inductive bias—it is because it allows the network to be compressed. That is probably at the core of our disagreement.
Yes, I think so. Let’s go over the ‘thin network’ example—we want to learn some function which can be represented by a thin network. But let’s say a randomly-initialized thin network’s intermediate functions won’t be able to fit the function—that is (with high probability over the random initialization) we won’t be able to fit the function just by changing the parameters of the last layer. It seems there are a few ways we can alter the network to make fitting possible:
(A) Expand the network’s width until (with high probability) it’s possible to fit the function by only altering the last layer
(B) Keeping the width the same, re-sample the parameters in all layers until we find a setting that can fit the function
(C) Keeping the width the same, train the network with SGD
By hypothesis, all three methods will let us fit the target function. You seem to be saying[I think, correct me if I’m wrong] that all three methods should have the same inductive bias as well. I just don’t see any reason this should be the case—on the face of it, I would guess that all three have different inductive biases(though A and B might be similar). They’re clearly different in some respects -- (C) can do transfer learning but (A) cannot(B is unclear).
What do we know about SGD-trained nets that suggests this?
My intuition here is that SGD-trained nets can learn functions non-linearly while NTK/GP can only do so linearly. So in the car detector example, SGD is able to develop a neuron detecting cars through some as-yet unclear ‘feature learning’ mechanism. The NTK/GP can do so as well, sort of, since they’re universal function approximators. However, the way they do this is by taking a giant linear combination of random functions which is able to function identically to a car detector on the data points given. It seems like this might be more fragile/generalize worse than the neurons produced by SGD. Though that is admittedly somewhat conjectural at this stage, since we don’t really have a great understanding of how feature learning in SGD works.
I’ve read the new feature learning paper! We’re big fans of his work, although again I don’t think it contradicts anything I’ve just said.
ETA: Let me elaborate upon what I see as the significance of the ‘feature learning in infinite nets’ paper. We know that NNGP/NTK models can’t learn features, but SGD can: I think this provides strong evidence that they are learning using different mechanisms, and likely have substantially different inductive biases. The question is whether randomly sampled finite nets can learn features as well. Since they are equivalent to NNGP/NTK at infinite width, any feature learning they do can only come from finiteness. In contrast, in the case of SGD, it’s possible to do feature learning even in the infinite-width limit. This suggests that even if randomly-sampled finite nets can do feature learning, the mechanism by which they do so is different from SGD, and hence their inductive bias is likely to be different as well.
I’d like to add some points to this interesting discussion:
As far as I understand, feature learning is not necessary for some standard types of transfer learning. E.g.: one can train an NNGP on a large dataset, and then use the learned posterior as prior for “fine-tuning” on some new dataset. This is hard to scale using actual GP techniques, but if wide neural nets (with random sampling or SGD) do approximate NNGPs, this could be a way they achieve transfer learning without feature learning.
You say
In contrast, in the case of SGD, it’s possible to do feature learning even in the infinite-width limit
That is true, but one of the points in Greg Yang’s paper, as far as I remember, was also to say that people weren’t using the scaling limit that would lead to that. That has made me wonder whether feature learning may be happening in our biggests models or not. The work on multimodal neurons in CLIP suggests there is feature learning. But what about GPT-3? In any case, I don’t think it’ll be happening by the mechanism Yang proposes as people aren’t using his initialization scheme. Perhaps, then the mechanism by which finite randomly-sampled NNs could conceivably feature-learn, could be the same as the one SGD is using. I am not sure either way. For me to evaluate the empirical evidence better, I’d need a sense about whether the evidence we have is in sufficiently large models or not (as I do think that randomly-sampled NNs for infinite width won’t do feature learning—though I’m not sure how to prove that, without a better definition of feature learning).
Another point is in answer to your comment that NNGP often underpeforms NTK. I think there’s actually more evidence on the contrary (see https://arxiv.org/abs/2007.15801 ), even if there’re instancs of both ways.
Overall, I think the work in Jascha Sohl-Dickstein’s groun (e.g. the paper linked above) has been great for disentangling these issues, and they seem to point at a complex/nuanced picture, which really leads me to believe we don’t have a clear answer about whether NNGPs will be a good model of SGD in practice (as of today; practice may also change). However, my general observation is that I’m not aware of any evidence that shows that SGD-trained nets beat architecture-equivalent NNGPs by a significant margin, consistently over a wide range of tasks in practice. Chris’ work on Bayesian picture of SGD tried to do this, but the problems are indeed, not quite large enough to be confident. In here https://arxiv.org/abs/2012.04115 we also explore NNGPs (but through a different lens), over SOTA architectures, but still small tasks. So I think the question still remains open as to how would NNGPs perform for more complex datasets.
By hypothesis, all three methods will let us fit the target function. You seem to be saying[I think, correct me if I’m wrong] that all three methods should have the same inductive bias as well.
Not exactly the same—it is known that there is a width dependence on inductive biases. I believe that typically wide networks are better, although I know of some counterexamples.
They’re clearly different in some respects -- (C) can do transfer learning but (A) cannot
I think this is the main source of our disagreement. First of all, while the posterior of an NNGP is equivalent to that of a trained-by-random-sampling infinitely wide NN, it does not contain all the same information. It is a collapsed version of an infinitely wide neural network that does not contain any information about the weights in each layer. This was one of Greg Yang’s points—by definition, a kernel method cannot learn features as you are ignoring the effects of the initial layers, as from a function perspective they are irrelevant—in other words, you have just thrown that information away.
This is not the same as saying that an extremely wide trained-by-random-sampling neural network would not learn features—there is a possibility that the first time you reach 100% training accuracy corresponds to effectively randomly initialised initial layers + trained last layer, but in expectation all the layers should be distinct from an entirely random intialisation.
(B is unclear).
Assuming that the network is so compressed that it can barely represent the true function without substantial fine-tuning of weights in all layers, weights in early layers would absolutely have to be very different from random initialisation.
However, the way they do this is by taking a giant linear combination of random functions which is able to function identically to a car detector on the data points given. It seems like this might be more fragile/generalize worse than the neurons produced by SGD. Though that is admittedly somewhat conjectural at this stage, since we don’t really have a great understanding of how feature learning in SGD works.
You can make arguments that this is what would happen for very wide networks—but then SGD is probably doing the same thing, unless you’re assuming that it learns a few (e.g.) car detector neurons and then the rest are completely redundant. I would expect the car detector neurons to show up in narrower networks, but by my point immediately above, I don’t see why this has to be an SGD-only property.
My intuition here is that SGD-trained nets can learn functions non-linearly while NTK/GP can only do so linearly.
Yes but again an NNGP has thrown away all information about the weights. The NTK limit effectively passes all the gradient to the last layer, so again, by definition, it is a linear model.
Since they are equivalent to NNGP/NTK at infinite width, any feature learning they do can only come from finiteness. In contrast, in the case of SGD, it’s possible to do feature learning even in the infinite-width limit.
Same point as above. The Greg Yang paper shows you need to do the clever reparameterisation to make sure not all the gradient gets passed to the last layer (as it does in NTK). The NNGP flattens the neural network so again there can be no feature learning by that representation. So I think the conclusion “can only come from finiteness” is wrong. The second point is correct, but only because you haven’t collapsed the network into a kernel. If you were to take an extremely wide neural network and train the whole thing by random sampling with some extra steps (e.g. encouraging orthogonality of intermediate outputs between different classes), I don’t see why you wouldn’t have some degree of ‘feature learning’ here.
Perhaps this is a physicist vs mathematician type of thinking though. I think I see where you are coming from, but I don’t think the no feature learning arguments are valid, as I think I outlined.
Perhaps this is a physicist vs mathematician type of thinking though
Good guess ;)
This is not the same as saying that an extremely wide trained-by-random-sampling neural network would not learn features—there is a possibility that the first time you reach 100% training accuracy corresponds to effectively randomly initialised initial layers + trained last layer, but in expectation all the layers should be distinct from an entirely random intialisation.
I see—so you’re saying that even though the distribution of output functions learned by an infinitely-wide randomly-sampled net is unchanged if you freeze everything but the last layer, the distribution of intermediate functions might change. If true, this would mean that feature learning and inductive bias are ‘uncoupled’ for infinite-width randomly-sampled nets. I think this is false, however—that is, I think it’s provable that the distribution of intermediate functions does not change in the infinite-width limit when you condition on the training data, even when conditioning over all layers. I can’t find a reference offhand though, I’ll report back if I find anything resolving this one way or another.
Haha some things are pretty obvious—it’s always really nice to get a very different perspective on an idea, thank you for continuing the conversation!
I see—so you’re saying that even though the distribution of output functions learned by an infinitely-wide randomly-sampled net is unchanged if you freeze everything but the last layer, the distribution of intermediate functions might change. If true, this would mean that feature learning and inductive bias are ‘uncoupled’ for infinite randomly-sampled nets
That is exactly what I’m saying. I don’t know if it is testable in practice, but it is in theory … I would be very interested to see anything about this—let me know if you find anything!
If it turns out that, in the limit of infinite width, feature learning does not work, what are your thoughts about my case for feature learning for the narrow (but trained-by-random-sampling) case? I would guess you find this case significantly more compelling than the infinite width case?
I just came across this paper which derives an expression for the posterior distribution of the weights in each layer in the infinite-width limit. The result: the distribution is unchanged from the prior in every layer but the last. So it indeed seems that there is no feature learning in this limit.
Yeah, I didn’t mean to imply that you guys said ‘simple --> large volume’ anywhere. I just think it’s a point worth emphasizing, especially around here where I think people will imagine “Solomonoff Induction-like” when they hear about a “bias towards simple functions”
But in the infinite-width setting, Bayesian inference in general is given by a GP limit, right? Initialization doesn’t matter. This means that the arguments for lack of feature learning still go through. It’s technically possible that there could be feature learning in finite-width randomly-sampled networks, but it seems strange that finiteness would help here(and any such learning would be experimentally inaccessible). This is a major reason that I’m skeptical of the “SGD as a random sampler” picture.
[First thank you for your comments and observations—it’s always interesting to read pushback]
First, I think my point about using the GP to measure the volume occupied functions locally to where SGD trained networks are initialised is important. We are not really comparing NNs to NNGPs (well, technically we are, but we are interpreting what the NNGP does differently). We are trying to argue that SGD acts as a random sampler—it will find functions with probability proportional to the volume of those functions local to where the optimiser is in parameter-space. We argue that this quantity is well approximated by the NNGP.
This is relevant to the comments on features: if you look at the definition of $P_B(f|S)$ it’s fairly clear that (assuming training by random sampling) initialising and freezing all but the last layer and then random sampling over that will, in expectation, give precisely the same posterior distribution as if you were to random sample over the whole network. This property holds for finite and infinite width networks. This may seem counterintuitive, but the term P(S|f) in the definition of $P_B(f|S)$ ensures that if the random initialisation of the frozen layers does not allow for 100% training accuracy, that random initialisation is ignored. Therefore, if an optimiser samples functions proportional to their volume, you won’t get any difference in performance if you learn features (optimise the whole network) or do not learn features (randomly initialise and freeze all but the last layer and then train just the last).
Given therefore that the posteriors are the same, it implies that feature learning is not aiding inductive bias—rather, feature learning is important for expressivity reasons. The reason why you can’t just use frozen initial layers and obtain the same inductive bias on SOTA architectures is most likely because you can’t make the layers wide enough, to ensure that the network is expressive enough with high probability. Imagenet for example has input dimension of ~200000 so you would need some very wide layers to approach the wide-layer limit.
Furthermore (and on a slightly different note), it is known that infintesimal GD converges to the Boltzmann distribution for any DNN (very similar to random sampling) https://arxiv.org/abs/2004.01190. This means that the coloured noise in SGD is the only possible source for drastically improved inductive bias (which would have to emerge only on large scales, as we do not observe it at smaller scales). I have also not heard as good a theoretical justification for why this noise would dramatically aid generalisation.
Given this, I think it a sensible null hypothesis that optimisers are approximately performing random sampling from a well-biased parameter-space (with some subtleties, see my other comment about tempered posteriors), at substantially larger scales. This to me makes more sense than “optimisers perform random sampling at small/medium scales, but as you move to bigger scales coloured noise in SGD is the dominant source of inductive bias”.
Finally, I would like to point out that this is my impression from the literature, and my work. I am aware that there’s a lot I don’t know, and if anyone can point out why this line of argument is not correct, or can steelman a case for SGD inductive bias appearing at larger scales, I would be very interested to hear it.
And thanks for engaging with my random blog comments! TBC, I think you guys are definitely on the right track in trying to relate SGD to function simplicity, and the empirical work you’ve done fleshing out that picture is great. I just think it could be even better if it was based around a better SGD scaling limit ;)
Right, this is an even better argument that NNGPs/random-sampled nets don’t learn features.
I think this only applies to NNGP/random-sampled nets, not SGD-trained nets. To apply to SGD-trained nets, you’d need to show that the new features learned by SGD have the same distribution as the features found in an infinitely-wide random net, but I don’t think this is the case. By illustration, some SGD-trained nets can develop expressive neurons like ‘car detector’, enabling them to fit the data with a relatively small number of such neurons. If you used an NNGP to learn the same thing, you wouldn’t get a single ‘car detector’ neuron, but rather some huge linear combination of high-frequency features that can approximate the cars seen in the dataset. I think this would probably generalize worse than the network with an actual ‘car detector’(this isn’t empirical evidence of course, but I think what we know about SGD-trained nets and the NNGP strongly suggests a picture like this)
Interesting, haven’t seen this before. Just skimming the paper, it sounds like the very small learning rate + added white noise might result in different limiting behavior from usual SGD. Generally it seems that there are a lot of different possible limits one can take; empirically SGD-trained nets do seem to have ‘feature learning’ so I’m skeptical of limits that don’t have that(I assume they don’t have them for theoretical reasons, anyway. Would be interesting to actually examine the features found in networks trained like this, and to see if they can do transfer learning at all) re:‘colored noise’, not sure to what extent this matters. I think a more likely source of discrepancy is the lack of white noise in normal training(I guess this counts as ‘colored noise’ in a sense) and the larger learning rate.
Not to be a broken record, but I strongly recommend checking out Greg Yang’s work. He clearly shows that there exist infinite-width limits of SGD that can do feature/transfer learning.
[Advance apologies if I haven’t explained stuff well enough here. I think the important theme here is that we should maintain a way of thinking about the random sampling picture that is distinct from NNGPs.]
Ah I see I need to explain myself further—the following is very counterintuitive but I think it’s right. Learning features involves the movement of weights in the early layers, by definition. The claim I am making is that the reason why feature learning is good is not because it improves inductive bias—it is because it allows the network to be compressed. That is probably at the core of our disagreement.
Imagine taking a network and making it so thin that it is only just able to represent the function it needs to. Now try the training with last layer only after randomly initialising the others. You can’t—because the randomly initialised first layers will drastically decrease its expressivity, so you can’t express the true function. Now do the same, but with very wide layers—by the lottery ticket hypothesis (and in the limit of infinite width) this will work well because of the (near) unlimited expressivity. Hence, for narrow networks you have to “learn features” to make sure you are expressive enough, but for wide ones you do not.
Consider ResNet18 on Imagenet. Imagenet has an input dimension of $3\times256\times256\approx200000$. The widths of the layers within the resnet are at least two orders of magnitude smaller (so you are nowhere near the limit of infinite width). This is the case of the thin network I talked about earlier—you have to learn features precisely because the network isn’t expressive enough for you to get away without. I’m pretty sure this is the motivation for making networks deep in the first place—for expressivity reasons.
So my claim is features are important to keep the number of parameters small, but do not in themselves aid inductive bias. I know that the first pushback to this will be “but transfer learning improves inductive bias.” Of course—you are basically taking a network that has just been trained on millions of images, and then using this on a set of new images—there will be some information in common across images that has been encoded in the earlier layers. The hierarchical nature of neural networks allows this to happen, but fundamentally not in a way that could not be explained by the random sampling picture.
So, in conclusion, I don’t think SGD needs to be doing any “feature learning” beyond what can be achieved in the random sampling fashion. Note that the random sampling arguments apply not only in the limit of infinite width.
[However, it is worth noting that this is conjecture, although I think it is the most natural conclusion from what we know about DNNs. That said, I will only be happy to accept it when we have found a good way of rigorously comparing the posteriors of a random-sample trained finite width neural network and its corresponding SGD trained version]
What do we know about SGD-trained nets that suggests this?
I’ve read the new feature learning paper! We’re big fans of his work, although again I don’t think it contradicts anything I’ve just said.
Yes, I think so. Let’s go over the ‘thin network’ example—we want to learn some function which can be represented by a thin network. But let’s say a randomly-initialized thin network’s intermediate functions won’t be able to fit the function—that is (with high probability over the random initialization) we won’t be able to fit the function just by changing the parameters of the last layer. It seems there are a few ways we can alter the network to make fitting possible:
(A) Expand the network’s width until (with high probability) it’s possible to fit the function by only altering the last layer
(B) Keeping the width the same, re-sample the parameters in all layers until we find a setting that can fit the function
(C) Keeping the width the same, train the network with SGD
By hypothesis, all three methods will let us fit the target function. You seem to be saying[I think, correct me if I’m wrong] that all three methods should have the same inductive bias as well. I just don’t see any reason this should be the case—on the face of it, I would guess that all three have different inductive biases(though A and B might be similar). They’re clearly different in some respects -- (C) can do transfer learning but (A) cannot(B is unclear).
My intuition here is that SGD-trained nets can learn functions non-linearly while NTK/GP can only do so linearly. So in the car detector example, SGD is able to develop a neuron detecting cars through some as-yet unclear ‘feature learning’ mechanism. The NTK/GP can do so as well, sort of, since they’re universal function approximators. However, the way they do this is by taking a giant linear combination of random functions which is able to function identically to a car detector on the data points given. It seems like this might be more fragile/generalize worse than the neurons produced by SGD. Though that is admittedly somewhat conjectural at this stage, since we don’t really have a great understanding of how feature learning in SGD works.
ETA: Let me elaborate upon what I see as the significance of the ‘feature learning in infinite nets’ paper. We know that NNGP/NTK models can’t learn features, but SGD can: I think this provides strong evidence that they are learning using different mechanisms, and likely have substantially different inductive biases. The question is whether randomly sampled finite nets can learn features as well. Since they are equivalent to NNGP/NTK at infinite width, any feature learning they do can only come from finiteness. In contrast, in the case of SGD, it’s possible to do feature learning even in the infinite-width limit. This suggests that even if randomly-sampled finite nets can do feature learning, the mechanism by which they do so is different from SGD, and hence their inductive bias is likely to be different as well.
I’d like to add some points to this interesting discussion:
As far as I understand, feature learning is not necessary for some standard types of transfer learning. E.g.: one can train an NNGP on a large dataset, and then use the learned posterior as prior for “fine-tuning” on some new dataset. This is hard to scale using actual GP techniques, but if wide neural nets (with random sampling or SGD) do approximate NNGPs, this could be a way they achieve transfer learning without feature learning.
You say
That is true, but one of the points in Greg Yang’s paper, as far as I remember, was also to say that people weren’t using the scaling limit that would lead to that. That has made me wonder whether feature learning may be happening in our biggests models or not. The work on multimodal neurons in CLIP suggests there is feature learning. But what about GPT-3? In any case, I don’t think it’ll be happening by the mechanism Yang proposes as people aren’t using his initialization scheme. Perhaps, then the mechanism by which finite randomly-sampled NNs could conceivably feature-learn, could be the same as the one SGD is using. I am not sure either way. For me to evaluate the empirical evidence better, I’d need a sense about whether the evidence we have is in sufficiently large models or not (as I do think that randomly-sampled NNs for infinite width won’t do feature learning—though I’m not sure how to prove that, without a better definition of feature learning).
Another point is in answer to your comment that NNGP often underpeforms NTK. I think there’s actually more evidence on the contrary (see https://arxiv.org/abs/2007.15801 ), even if there’re instancs of both ways.
Overall, I think the work in Jascha Sohl-Dickstein’s groun (e.g. the paper linked above) has been great for disentangling these issues, and they seem to point at a complex/nuanced picture, which really leads me to believe we don’t have a clear answer about whether NNGPs will be a good model of SGD in practice (as of today; practice may also change). However, my general observation is that I’m not aware of any evidence that shows that SGD-trained nets beat architecture-equivalent NNGPs by a significant margin, consistently over a wide range of tasks in practice. Chris’ work on Bayesian picture of SGD tried to do this, but the problems are indeed, not quite large enough to be confident. In here https://arxiv.org/abs/2012.04115 we also explore NNGPs (but through a different lens), over SOTA architectures, but still small tasks. So I think the question still remains open as to how would NNGPs perform for more complex datasets.
Not exactly the same—it is known that there is a width dependence on inductive biases. I believe that typically wide networks are better, although I know of some counterexamples.
I think this is the main source of our disagreement. First of all, while the posterior of an NNGP is equivalent to that of a trained-by-random-sampling infinitely wide NN, it does not contain all the same information. It is a collapsed version of an infinitely wide neural network that does not contain any information about the weights in each layer. This was one of Greg Yang’s points—by definition, a kernel method cannot learn features as you are ignoring the effects of the initial layers, as from a function perspective they are irrelevant—in other words, you have just thrown that information away.
This is not the same as saying that an extremely wide trained-by-random-sampling neural network would not learn features—there is a possibility that the first time you reach 100% training accuracy corresponds to effectively randomly initialised initial layers + trained last layer, but in expectation all the layers should be distinct from an entirely random intialisation.
Assuming that the network is so compressed that it can barely represent the true function without substantial fine-tuning of weights in all layers, weights in early layers would absolutely have to be very different from random initialisation.
You can make arguments that this is what would happen for very wide networks—but then SGD is probably doing the same thing, unless you’re assuming that it learns a few (e.g.) car detector neurons and then the rest are completely redundant. I would expect the car detector neurons to show up in narrower networks, but by my point immediately above, I don’t see why this has to be an SGD-only property.
Yes but again an NNGP has thrown away all information about the weights. The NTK limit effectively passes all the gradient to the last layer, so again, by definition, it is a linear model.
Same point as above. The Greg Yang paper shows you need to do the clever reparameterisation to make sure not all the gradient gets passed to the last layer (as it does in NTK). The NNGP flattens the neural network so again there can be no feature learning by that representation. So I think the conclusion “can only come from finiteness” is wrong. The second point is correct, but only because you haven’t collapsed the network into a kernel. If you were to take an extremely wide neural network and train the whole thing by random sampling with some extra steps (e.g. encouraging orthogonality of intermediate outputs between different classes), I don’t see why you wouldn’t have some degree of ‘feature learning’ here.
Perhaps this is a physicist vs mathematician type of thinking though. I think I see where you are coming from, but I don’t think the no feature learning arguments are valid, as I think I outlined.
Good guess ;)
I see—so you’re saying that even though the distribution of output functions learned by an infinitely-wide randomly-sampled net is unchanged if you freeze everything but the last layer, the distribution of intermediate functions might change. If true, this would mean that feature learning and inductive bias are ‘uncoupled’ for infinite-width randomly-sampled nets. I think this is false, however—that is, I think it’s provable that the distribution of intermediate functions does not change in the infinite-width limit when you condition on the training data, even when conditioning over all layers. I can’t find a reference offhand though, I’ll report back if I find anything resolving this one way or another.
Haha some things are pretty obvious—it’s always really nice to get a very different perspective on an idea, thank you for continuing the conversation!
That is exactly what I’m saying. I don’t know if it is testable in practice, but it is in theory … I would be very interested to see anything about this—let me know if you find anything!
If it turns out that, in the limit of infinite width, feature learning does not work, what are your thoughts about my case for feature learning for the narrow (but trained-by-random-sampling) case? I would guess you find this case significantly more compelling than the infinite width case?
I just came across this paper which derives an expression for the posterior distribution of the weights in each layer in the infinite-width limit. The result: the distribution is unchanged from the prior in every layer but the last. So it indeed seems that there is no feature learning in this limit.