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Nonlocal models have recently had a major impact in nonlinear continuummechanics and are used to describe physical systems/processes which cannot beaccurately described by classical, calculus based "local" approaches. In part,this is due to their multiscale nature that enables aggregation of micro-levelbehavior to obtain a macro-level description of singular/irregular phenomenasuch as peridynamics, crack propagation, anomalous diffusion and transportphenomena. At the core of these models are nonlocal differential operators,including nonlocal analogs of the gradient/Hessian. This paper initiates theuse of such nonlocal operators in the context of optimization and learning. Wedefine and analyze the convergence properties of nonlocal analogs of(stochastic) gradient descent and Newton's method on Euclidean spaces. Ourresults indicate that as the nonlocal interactions become less noticeable, theoptima corresponding to nonlocal optimization converge to the "usual" optima.At the same time, we argue that nonlocal learning is possible in situationswhere standard calculus fails. As a stylized numerical example of this, weconsider the problem of non-differentiable parameter estimation on a non-smoothtranslation manifold and show that our nonlocal gradient descent recovers theunknown translation parameter from a non-differentiable objective function.

Optimization and Learning With Nonlocal Calculus