Deep Relaxation: partial differential equations for optimizing deep neural networks
This provides a theoretical framework for improving optimization in deep learning, though it appears incremental as it builds on existing relaxation techniques from statistical physics.
The paper connects non-convex optimization for training deep neural networks to nonlinear partial differential equations (PDEs), showing that a modified algorithm based on viscous Hamilton-Jacobi PDEs performs better in expectation than stochastic gradient descent.
In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already been used successfully in this context are reinterpreted as solutions of a viscous Hamilton-Jacobi PDE. Using a stochastic control interpretation allows we prove that the modified algorithm performs better in expectation that stochastic gradient descent. Well-known PDE regularity results allow us to analyze the geometry of the relaxed energy landscape, confirming empirical evidence. The PDE is derived from a stochastic homogenization problem, which arises in the implementation of the algorithm. The algorithms scale well in practice and can effectively tackle the high dimensionality of modern neural networks.