Gradient descent provably escapes saddle points in the training of shallow ReLU networks
This provides theoretical guarantees for training neural networks, addressing a key optimization challenge for researchers and practitioners, though it is incremental as it builds on existing dynamical systems theory.
The paper tackles the problem of gradient descent getting stuck at saddle points in training shallow ReLU networks by proving a relaxed center-stable manifold theorem, showing that gradient descent escapes most saddle points and converges to global minima under specific initialization conditions with an explicit loss threshold.
Dynamical systems theory has recently been applied in optimization to prove that gradient descent algorithms bypass so-called strict saddle points of the loss function. However, in many modern machine learning applications, the required regularity conditions are not satisfied. In this paper, we prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. We explore its relevance for various machine learning tasks, with a particular focus on shallow rectified linear unit (ReLU) and leaky ReLU networks with scalar input. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks relative to an affine target function, we show that gradient descent circumvents most saddle points. Furthermore, we prove convergence to global minima under favourable initialization conditions, quantified by an explicit threshold on the limiting loss.