Improving Energy Conserving Descent for Machine Learning: Theory and Practice

We develop the theory of Energy Conserving Descent (ECD) and introduce ECDSep, a gradient-based optimization algorithm able to tackle convex and non-convex optimization problems. The method is based on the novel ECD framework of optimization as physical evolution of a suitable chaotic energy-conserving dynamical system, enabling analytic control of the distribution of results - dominated at low loss - even for generic high-dimensional problems with no symmetries. Compared to previous realizations of this idea, we exploit the theoretical control to improve both the dynamics and chaos-inducing elements, enhancing performance while simplifying the hyper-parameter tuning of the optimization algorithm targeted to different classes of problems. We empirically compare with popular optimization methods such as SGD, Adam and AdamW on a wide range of machine learning problems, finding competitive or improved performance compared to the best among them on each task. We identify limitations in our analysis pointing to possibilities for additional improvements.

Born-Infeld for AI: Energy-Conserving Descent for Optimization

We introduce a novel framework for optimization based on energy-conserving Hamiltonian dynamics in a strongly mixing (chaotic) regime and establish its key properties analytically and numerically. The prototype is a discretization of Born-Infeld dynamics, with a squared relativistic speed limit depending on the objective function. This class of frictionless, energy-conserving optimizers proceeds unobstructed until slowing naturally near the minimal loss, which dominates the phase space volume of the system. Building from studies of chaotic systems such as dynamical billiards, we formulate a specific algorithm with good performance on machine learning and PDE-solving tasks, including generalization. It cannot stop at a high local minimum and cannot overshoot the global minimum, yielding an advantage in non-convex loss functions, and proceeds faster than GD+momentum in shallow valleys.