When developing deep learning models, we usually decide what task we want to solve then search for a model that generalizes well on the task. An intriguing question would be: what if, instead of fixing the task and searching in the model space, we fix the model and search in the task space? Can we find tasks that the model generalizes on? How do they look, or do they indicate anything? These are the questions we address in this paper.

When developing deep learning models, we usually decide what task we want to solve then search for a model that generalizes well on the task. An intriguing question would be: what if, instead of fixing the task and searching in the model space, we fix the model and search in the task space? Can we find tasks that the model generalizes on? How do they look, or do they indicate anything? These are the questions we address in this paper.

We show that feedforward neural networks with ReLU activation generalize on low complexity data, suitably defined. Given i.i.d. data generated from a simple programming language, the minimum description length (MDL) feedforward neural network which interpolates the data generalizes with high probability. We define this simple programming language, along with a notion of description length of such networks. We provide several examples on basic computational tasks, such as checking primality of a natural number, and more. For primality testing, our theorem shows the following. Suppose that we draw an i.i.d. sample of $\Theta(N^{\delta}\ln N)$ numbers uniformly at random from $1$ to $N$, where $\delta\in (0,1)$. For each number $x_i$, let $y_i = 1$ if $x_i$ is a prime and $0$ if it is not. Then with high probability, the MDL network fitted to this data accurately answers whether a newly drawn number between $1$ and $N$ is a prime or not, with test error $\leq O(N^{-\delta})$. Note that the network is not designed to detect primes; minimum description learning discovers a network which does so.