Currently, we do not have a good theoretical understanding of how or why neural networks actually work. For example, we know that large neural networks are sufficiently expressive to compute almost any kind of function. Moreover, most functions that fit a given set of training data will not generalise well to new data. And yet, if we train a neural network we will usually obtain a function that gives good generalisation. What is the mechanism behind this phenomenon?
There has been some recent research which (I believe) sheds some light on this issue. I would like to call attention to this blog post:
(You may notice that I had some involvement with this research, but the main credit should go to Chris Mingard and Guillermo Valle-Perez!)
I believe that research of this type is very relevant for AI alignment. It seems quite plausible that neural networks, or something similar to them, will be used as a component of AGI. If that is the case, then we want to be able to reliably predict and reason about how neural networks behave in new situations, and how they interact with other systems, and it is hard to imagine how that would be possible without a deep understanding of the dynamics at play when neural networks learn from data. Understanding their inductive bias seems particularly important, since this is the key to understanding everything from why they work in the first place, to phenomena such as adversarial examples, to the risk of mesa-optimisation. I hence believe that it makes sense for alignment researchers to keep an eye on what is happening in this space.
If you want some more stuff to read in this genre, I can also recommend these two posts:
We do have a lot of empirical evidence that simple functions indeed have bigger volumes in parameter-space (exponentially bigger volumes, in fact). See especially the 2nd and 3rd paper linked in the OP.
It's true that we don't have a rigorous mathematical proof for "why" simple functions should have larger volumes in parameter-space, but I can give an intuitive argument for why I think it would make sense to expect this to be the case.
First of all, we can look at the claim "backwards"; suppose a function has a large volume in parameter-space. This means that a lot of the parameters in a sense are redundant, so it should be possible to compress the representation of this function. Conversely, if a function has a small volume in parameter-space, then all the parameters of the network are necessary to pinpoint that function, and so writing out the entire structure of the network might be one of the shortest ways to represent that function. Therefore, we should expect functions with large volumes in parameter-space to be simple, and functions with small volumes to be complex.
There is a mathematical theorem, called the Levin bound, that roughly formalises this intuition. This bound says essentially that if we have a space of functions F, and a space of parameters P, and a parameter-function map m : P -> F, where m itself has low complexity relative to the most complex functions in F (and F is overparameterised by P), then m will be exponentially biased towards simple functions (ie, if an element f of F has low complexity, then there is an exponentially larger number of elements of P that map to f via m).
The Levin bound doesn't apply directly to neural networks, because it assumes that P is finite and discrete, but it gives some extra backing to the intuition above.
Currently, we do not have a good theoretical understanding of how or why neural networks actually work. For example, we know that large neural networks are sufficiently expressive to compute almost any kind of function. Moreover, most functions that fit a given set of training data will not generalise well to new data. And yet, if we train a neural network we will usually obtain a function that gives good generalisation. What is the mechanism behind this phenomenon?
There has been some recent research which (I believe) sheds some light on this issue. I would like to call attention to this blog post:
Neural Networks Are Fundamentally Bayesian
This post provides a summary of the research in these three papers, which provide a candidate for a theory of generalisation:
https://arxiv.org/abs/2006.15191
https://arxiv.org/abs/1909.11522
https://arxiv.org/abs/1805.08522
(You may notice that I had some involvement with this research, but the main credit should go to Chris Mingard and Guillermo Valle-Perez!)
I believe that research of this type is very relevant for AI alignment. It seems quite plausible that neural networks, or something similar to them, will be used as a component of AGI. If that is the case, then we want to be able to reliably predict and reason about how neural networks behave in new situations, and how they interact with other systems, and it is hard to imagine how that would be possible without a deep understanding of the dynamics at play when neural networks learn from data. Understanding their inductive bias seems particularly important, since this is the key to understanding everything from why they work in the first place, to phenomena such as adversarial examples, to the risk of mesa-optimisation. I hence believe that it makes sense for alignment researchers to keep an eye on what is happening in this space.
If you want some more stuff to read in this genre, I can also recommend these two posts:
Recent Progress in the Theory of Neural Networks
Understanding "Deep Double Descent"
EDIT: Here is a second post, which talks more about the "prior" of neural networks:
Deep Neural Networks are biased, at initialisation, towards simple functions