TL;DR

For neural networks trained to perfect loss, the broadness of optima in parameter space is given by the number and norm of independent orthogonal features the neural network has. The inner product that defines this "feature norm" and independence/orthogonality is the $L_{2}$ product of Hilbert space.

Introduction - why do we care about broadness?

Recently, there's been some discussion of what determines the broadness of optima of neural networks in parameter space.

People care about this because the size of the optimum basin may influence how easy it is for gradient descent (or other local optimisation algorithms which you might use to train a neural network) to find the corresponding solution, since it's probably easier to stumble across a broader optimum's attractor.

People also care about this because there's some hypotheses floating around saying that the broadness of optima is correlated with their generality somehow. The broader the basin of your solution is in parameter space, the more likely it seems to be that the network will successfully generalise to new data in deployment. So understanding what kinds of circuit configurations are broad seems like it might help us grasp better how AIs learn things, and how an AGI might really work internally.

Meaning, understanding what kind of solutions are broad is probably relevant to start developing general predictive theories of AI and AI training dynamics, which we'd like to have so we can answer questions like "What goals is my AGI going to develop, given that it has architecture $A$ , and I'm training it with loss function $L$ on data set $D$ ?"

Measuring broadness

Warning: calculus and linear algebra incoming!

Say we have a loss function $L (f, x, y)$ for a neural network $f (x, Θ)$ , with some data points $x$ and labels $y (x)$ , and we've trained the network to perfect loss, setting the parameters to some set $^Θ$ .

$\begin{matrix} L & = \sum x l (f (x, Θ), y (x)) \frac{d L (^Θ)}{d Θ} & = 0 \end{matrix}$ ^[1]

Now, lets try to look at how much the loss changes when we make small changes to the parameters $^Θ$ . To do this, we're going to perform a polynomial expansion of $L$ up to quadratic order. This isn't going to describe the function behaviour perfectly, but within a small neighbourhood of $^Θ$ , it'll be mostly accurate.

$\begin{matrix} L (Θ) - L (^Θ) & = \frac{d L (^Θ)}{d Θ} (Θ -^Θ) + (Θ -^Θ)^{T} \frac{d^{2} L (^Θ)}{d^{2} Θ} (Θ -^Θ) + O ((Θ -^Θ)^{3}) = (Θ -^Θ)^{T} \frac{d^{2} L (^Θ)}{d^{2} Θ} (Θ -^ \end{matrix}$ ...

Selection Theorems: Modularity