Generalized Momentum-Based Methods: A Hamiltonian Perspective
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We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method to a broad class of momentum methods in the setting of (possibly) constrained minimization in Banach spaces. Our perspective leads to a generic and unifying non-asymptotic analysis of convergence of these methods in both the function value (in the setting of convex optimization) and in the norm of the gradient (in the setting of unconstrained, possibly nonconvex, optimization). The convergence analysis is intuitive and based on the conserved quantities of the time-dependent Hamiltonian that we introduce and that produces generalized momentum methods as its equations of motion.
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