Neural Networks as Interacting Particle Systems: Asymptotic Convexity of the Loss Landscape and Universal Scaling of the Approximation Error.

Neural networks, a central tool in machine learning, have demonstrated remarkable, high fidelity performance on image recognition and classification tasks. These successes evince an ability to accurately represent high dimensional functions, potentially of great use in computational and applied mathematics. That said, there are few rigorous results about the representation error and trainability of neural networks. Here we characterize both the error and the scaling of the error with the size of the network by reinterpreting the standard optimization algorithm used in machine learning applications, stochastic gradient descent, as the evolution of a particle system with interactions governed by a potential related to the objective or loss function used to train the network. We show that, when the number $n$ of parameters is large, the empirical distribution of the particles descends on a convex landscape towards a minimizer at a rate independent of $n$. We establish a Law of Large Numbers and a Central Limit Theorem for the empirical distribution, which together show that the approximation error of the network universally scales as $O(n^{-1})$. Remarkably, these properties do not depend on the dimensionality of the domain of the function that we seek to represent. Our analysis also quantifies the scale and nature of the noise introduced by stochastic gradient descent and provides guidelines for the step size and batch size to use when training a neural network. We illustrate our findings on examples in which we train neural network to learn the energy function of the continuous 3-spin model on the sphere. The approximation error scales as our analysis predicts in as high a dimension as $d=25$.