Optimal Initialization Scale for Neural Networks With Locally Quadratic Loss Landscapes: An SGD Dynamics Perspective

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. Under the assumptions of a locally quadratic neural-network loss landscape and a quasi-stationary SGD regime, we derive an explicit upper bound for the expected loss in terms of the initialization variance. 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 within the locally quadratic SGD regime. In addition, we experimentally confirm our theoretical results by using classical SGD to train shallow fully connected neural networks on the MNIST dataset and a two-layer CNN model on the CIFAR-10 dataset. The results show that, for the simple network model, if the variance of the initialization distribution satisfies our theoretical optimal condition, then the corresponding network achieves lower final training loss and higher test accuracy than the conventional He-normal initialization. For the two-layer CNN model, He-normal initialization gives the best performance among the tested initializations. This observation is consistent with the proposed optimal condition, since the variance ratio of He-normal initialization remains close to the optimal value not only at the overall level but also approximately at the layer-wise level. This suggests that satisfying the proposed condition at both global and layer-wise scales may be important for achieving lower loss in this architecture. We also discuss the limitation of our optimal condition on complex networks. Our work thus supplies a mathematically grounded indicator for choosing the initialization variance and clarifies its physical meaning in terms of the parameter dynamics in neural network models.