Analysis of feature learning in weight-tied autoencoders via the mean field lens
This provides theoretical insights into autoencoder feature extraction, addressing a fundamental open problem in unsupervised learning, though it is incremental as it builds on mean field frameworks and focuses on a specific model class.
The paper tackles the longstanding problem of understanding feature learning in nonlinear autoencoders by analyzing weight-tied autoencoders using mean field theory, showing that training dynamics reveal phases corresponding to learning principal subspaces with nonlinear shrinkage dependent on regularization and stopping time, with experiments on real data matching the theory.
Autoencoders are among the earliest introduced nonlinear models for unsupervised learning. Although they are widely adopted beyond research, it has been a longstanding open problem to understand mathematically the feature extraction mechanism that trained nonlinear autoencoders provide. In this work, we make progress in this problem by analyzing a class of two-layer weight-tied nonlinear autoencoders in the mean field framework. Upon a suitable scaling, in the regime of a large number of neurons, the models trained with stochastic gradient descent are shown to admit a mean field limiting dynamics. This limiting description reveals an asymptotically precise picture of feature learning by these models: their training dynamics exhibit different phases that correspond to the learning of different principal subspaces of the data, with varying degrees of nonlinear shrinkage dependent on the $\ell_{2}$-regularization and stopping time. While we prove these results under an idealized assumption of (correlated) Gaussian data, experiments on real-life data demonstrate an interesting match with the theory. The autoencoder setup of interests poses a nontrivial mathematical challenge to proving these results. In this setup, the "Lipschitz" constants of the models grow with the data dimension $d$. Consequently an adaptation of previous analyses requires a number of neurons $N$ that is at least exponential in $d$. Our main technical contribution is a new argument which proves that the required $N$ is only polynomial in $d$. We conjecture that $N\gg d$ is sufficient and that $N$ is necessarily larger than a data-dependent intrinsic dimension, a behavior that is fundamentally different from previously studied setups.