Optimized Weight Initialization on the Stiefel Manifold for Deep ReLU Neural Networks

Despite recent algorithmic advances, we still lack principled ways to leverage the well-documented rescaling symmetries in ReLU neural network parameters. While two properly rescaled weights implement the same function, the training dynamics can be dramatically different. To offer a fresh perspective on exploiting this phenomenon, we build on the recent path-lifting framework, which provides a compact factorization of ReLU networks. We introduce a geometrically motivated criterion to rescale neural network parameters which minimization leads to a conditioning strategy that aligns a kernel in the path-lifting space with a chosen reference. We derive an efficient algorithm to perform this alignment. In the context of random network initialization, we analyze how the architecture and the initialization scale jointly impact the output of the proposed method. Numerical experiments illustrate its potential to speed up training.

Stochastic gradient descent (SGD), one of the most fundamental optimization algorithms in machine learning (ML), can be recast through a continuous-time approximation as a Fokker-Planck equation for Langevin dynamics, a viewpoint that has motivated many theoretical studies. Within this framework, we study the relationship between the quasi-stationary distribution derived from this equation and the initial distribution through the Kullback-Leibler (KL) divergence. As the quasi-steady-state distribution depends on the expected cost function, the KL divergence eventually reveals the connection between the expected cost function and the initialization distribution. By applying this to deep neural network models (DNNs), we can express the bounds of the expected loss function explicitly in terms of the initialization parameters. Then, by minimizing this bound, we obtain an optimal condition of the initialization variance in the Gaussian case. This result provides a concrete mathematical criterion, rather than a heuristic approach, to select the scale of weight initialization in DNNs. In addition, we experimentally confirm our theoretical results by using the classical SGD to train fully connected neural networks on the MNIST and Fashion-MNIST datasets. The result shows that if the variance of the initialization distribution satisfies our theoretical optimal condition, then the corresponding DNN model always achieves lower final training loss and higher test accuracy than the conventional He-normal initialization. Our work thus supplies a mathematically grounded indicator that guides the choice of initialization variance and clarifies its physical meaning of the dynamics of parameters in DNNs.

The optimization of neural networks under weight decay remains poorly understood from a theoretical standpoint. While weight decay is standard practice in modern training procedures, most theoretical analyses focus on unregularized settings. In this work, we investigate the loss landscape of the $\ell_2$-regularized training loss for two-layer ReLU networks. We show that the landscape becomes benign -- i.e., free of spurious local minima -- under large overparametrization, specifically when the network width $m$ satisfies $m \gtrsim \min(n^d, 2^n)$, where $n$ is the number of data points and $d$ the input dimension. More precisely in this regime, almost all constant activation regions contain a global minimum and no spurious local minima. We further show that this level of overparametrization is not only sufficient but also necessary via the example of orthogonal data. Finally, we demonstrate that such loss landscape results primarily hold relevance in the large initialization regime. In contrast, for small initializations -- corresponding to the feature learning regime -- optimization can still converge to spurious local minima, despite the global benignity of the landscape.