Learning Discretized Neural Networks under Ricci Flow

In this paper, we consider Discretized Neural Networks (DNNs) consisting of low-precision weights and activations, which suffer from either infinite or zero gradients caused by the non-differentiable discrete function in the training process. In this case, most training-based DNNs use the standard Straight-Through Estimator (STE) to approximate the gradient w.r.t. discrete value. However, the standard STE will cause the gradient mismatch problem, i.e., the approximated gradient direction may deviate from the steepest descent direction. In other words, the gradient mismatch implies the approximated gradient with perturbations. To address this problem, we introduce the duality theory to regard the perturbation of the approximated gradient as the perturbation of the metric in Linearly Nearly Euclidean (LNE) manifolds. Simultaneously, under the Ricci-DeTurck flow, we prove the dynamical stability and convergence of the LNE metric with the $L^2$-norm perturbation, which can provide a theoretical solution for the gradient mismatch problem. In practice, we also present the steepest descent gradient flow for DNNs on LNE manifolds from the viewpoints of the information geometry and mirror descent. The experimental results on various datasets demonstrate that our method achieves better and more stable performance for DNNs than other representative training-based methods.