Convergence of Stochastic Gradient Descent under a Local Łojasiewicz Condition for Deep Neural Networks
DOI:
https://doi.org/10.4208/jml.240724Keywords:
Non-convex optimization, Stochastic gradient descent, Convergence analysis.Abstract
We study the convergence of stochastic gradient descent (SGD) for non-convex objective functions. We establish the local convergence with positive probability under the local Łojasiewicz condition introduced by Chatterjee [arXiv:2203.16462, 2022] and an additional local structural assumption of the loss function landscape. A key component of our proof is to ensure that the whole trajectories of SGD stay inside the local region with a positive probability. We also provide examples of neural networks with finite widths such that our assumptions hold.