The Nonlinear Library AF - My Criticism of Singular Learning Theory by Joar Skalse
Nov 19, 2023
20:01
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: My Criticism of Singular Learning Theory, published by Joar Skalse on November 19, 2023 on The AI Alignment Forum.
In this post, I will briefly give my criticism of Singular Learning Theory (SLT), and explain why I am skeptical of its significance. I will especially focus on the question of generalisation --- I do not believe that SLT offers any explanation of generalisation in neural networks. I will also briefly mention some of my other criticisms of SLT, describe some alternative solutions to the problems that SLT aims to tackle, and describe some related research problems which I would be more excited about.
(I have been meaning to write this for almost 6 months now, since I attended the SLT workshop last June, but things have kept coming in the way.)
For an overview of SLT, see this sequence. This post will also refer to the results described in this post, and will also occasionally touch on VC theory. However, I have tried to make it mostly self-contained.
The Mystery of Generalisation
First of all, what is the mystery of generalisation? The issue is this; neural networks are highly expressive, and typically overparameterised. In particular, when a real-world neural network is trained on a real-world dataset, it is typically the case that this network is able to express many functions which would fit the training data well, but which would generalise poorly.
Moreover, among all functions which do fit the training data, there are more functions (by number) that generalise poorly, than functions that generalise well. And yet neural networks will typically find functions that generalise well.
To make this point more intuitive, suppose we have a 500,000-degree polynomial, and that we fit this to 50,000 data points. In this case, we have 450,000 degrees of freedom, and we should by default expect to end up with a function which generalises very poorly. But when we train a neural network with 500,000 parameters on 50,000 MNIST images, we end up with a neural network that generalises well. Moreover, adding more parameters to the neural network will typically make generalisation better, whereas adding more parameters to the polynomial is likely to make generalisation worse.
A simple hypothesis might be that some of the parameters in a neural network are redundant, so that even if it has 500,000 parameters, the dimensionality of the space of all functions which it can express is still less than 500,000. This is true. However, the magnitude of this effect is too small to solve the puzzle. If you get the MNIST training set, and assign random labels to the test data, and then try to fit the network to this function, you will find that this often can be done.
This means that while neural networks have redundant parameters, they are still able to express more functions which generalise poorly, than functions which generalise well. Hence the puzzle.
The anwer to this puzzle must be that neural networks have an inductive bias towards low-complexity functions. That is, among all functions which fit a given training set, neural networks are more likely to find a low-complexity function (and such functions are more likely to generalise well, as per Occam's Razor). The next question is where this inductive bias comes from, and how it works. Understanding this would let us better understand and predict the behaviour of neural networks, which would be very useful for AI alignment.
I should also mention that generalisation only is mysterious when we have an amount of training data that is small relative to the overall expressivity of the learning machine. Classical statistical learning theory already tells us that any sufficiently well-behaved learning machine will generalise well in the limit of infinite training data. For an overview of these results, see this post. Thus, the quest...
In this post, I will briefly give my criticism of Singular Learning Theory (SLT), and explain why I am skeptical of its significance. I will especially focus on the question of generalisation --- I do not believe that SLT offers any explanation of generalisation in neural networks. I will also briefly mention some of my other criticisms of SLT, describe some alternative solutions to the problems that SLT aims to tackle, and describe some related research problems which I would be more excited about.
(I have been meaning to write this for almost 6 months now, since I attended the SLT workshop last June, but things have kept coming in the way.)
For an overview of SLT, see this sequence. This post will also refer to the results described in this post, and will also occasionally touch on VC theory. However, I have tried to make it mostly self-contained.
The Mystery of Generalisation
First of all, what is the mystery of generalisation? The issue is this; neural networks are highly expressive, and typically overparameterised. In particular, when a real-world neural network is trained on a real-world dataset, it is typically the case that this network is able to express many functions which would fit the training data well, but which would generalise poorly.
Moreover, among all functions which do fit the training data, there are more functions (by number) that generalise poorly, than functions that generalise well. And yet neural networks will typically find functions that generalise well.
To make this point more intuitive, suppose we have a 500,000-degree polynomial, and that we fit this to 50,000 data points. In this case, we have 450,000 degrees of freedom, and we should by default expect to end up with a function which generalises very poorly. But when we train a neural network with 500,000 parameters on 50,000 MNIST images, we end up with a neural network that generalises well. Moreover, adding more parameters to the neural network will typically make generalisation better, whereas adding more parameters to the polynomial is likely to make generalisation worse.
A simple hypothesis might be that some of the parameters in a neural network are redundant, so that even if it has 500,000 parameters, the dimensionality of the space of all functions which it can express is still less than 500,000. This is true. However, the magnitude of this effect is too small to solve the puzzle. If you get the MNIST training set, and assign random labels to the test data, and then try to fit the network to this function, you will find that this often can be done.
This means that while neural networks have redundant parameters, they are still able to express more functions which generalise poorly, than functions which generalise well. Hence the puzzle.
The anwer to this puzzle must be that neural networks have an inductive bias towards low-complexity functions. That is, among all functions which fit a given training set, neural networks are more likely to find a low-complexity function (and such functions are more likely to generalise well, as per Occam's Razor). The next question is where this inductive bias comes from, and how it works. Understanding this would let us better understand and predict the behaviour of neural networks, which would be very useful for AI alignment.
I should also mention that generalisation only is mysterious when we have an amount of training data that is small relative to the overall expressivity of the learning machine. Classical statistical learning theory already tells us that any sufficiently well-behaved learning machine will generalise well in the limit of infinite training data. For an overview of these results, see this post. Thus, the quest...