Generalized gradients and applications

By F. Clarke. 1975

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Citations in the corpus

OptNet: Differentiable Optimization as a Layer in Neural Networks (2017, background): “…( θ ) is subdifferentiable every-where: ∂z (cid:63) ( θ ) (cid:54) = ∅ , where ∂z (cid:63) ( θ ) denotes the Clarke generalized subdifferential (Clarke, 1975) (an extension of the subgradient to non-convex functions), and has a single unique element (the Jacobian) for all but a measure zero…”
Toward Moderate Overparameterization: Global Convergence Guarantees for Training Shallow Neural Networks (2019, methodology): “If the function is not differentiable but only has isolated non-differentiable points we use φ′ to denote a generalized derivative [11].”
Computing Approximate Equilibria in Sequential Adversarial Games by Exploitability Descent (2019, methodology): “More precisely we use [ Clarke, 1975, Theorem 2.1 ] to state our result.”
A nonsmooth critical point theory approach to some nonlinear elliptic equations in {${\Bbb R}^n$} (2000, methodology): “More precisely, we consider a model elliptic problem first studied by Rabinowitz [13] with the C1-theory and we extend his results by means of the nonsmooth critical point theories of Clarke [5, 6] and Degiovanni et al. [8, 9]; one of the purposes of this paper is to emphasize some differences betwe”
Continuous Subdifferential Approximations and Their Applications (2003, background): “We recall that a locally Lipschitz function f is differentiable almost everywhere, and we can define its Clarke subdifferential [13] (see also [14]) by”
A class of gap functions for variational inequalities (1994, background): “1 of Clarke (1975) G̃ is then also Lipschitz continuous.”
Boosted and reward-regularized classification for apprenticeship learning (2014, methodology): “This technique can be extended to non-smooth and non-convex functionals, yet the functional has to be Lipschitz in order to guarantee that the gradient of the functional exists almost everywhere [4].”
On the theory of subdifferentials (2012, background): “We talk very little or do not talk at all about Clarke’s generalized gradients, well studied and presented in the literature (although the definition of the generalized gradient through convexification of smaller subdifferentials or normal cones allows to get some of its basic properties with extrem”
METRIC REGULARITY—A SURVEY PART 1. THEORY (2016, background): “Concerning the sources of the main concepts, Clarke’s subdifferential was first to appear—it was introduced in Clarke’s 1973 thesis [20] and in printed form first appeared in [21].”
A Morse-Sard theorem for the distance function on Riemannian manifolds (2004, methodology): “This tool was introduced by Clarke [2] in 1973.”
A survey of subdifferential calculus with applications (2002, background): “The basic pattern of sensitivity analysis is well illustrated by the following example taken from Clarke 47].”
Exactly Computing the Local Lipschitz Constant of ReLU Networks (2020, background): “Our contributions are as follows: • We present novel analytic results connecting the Lipschitz constant of an arbitrary function to the supremal norm of the Jacobian, over only the points where the Jacobian is deﬁned.”