This paper presents an extension of stochastic gradient descent for the minimization Lipschitz continuous loss functions. Our motivation is use in non-smooth non-convex optimization problems, which are frequently encountered applications such as machine learning. Using Clarke $$\epsilon $$ -subdifferential, we prove non-asymptotic convergence to approximate stationary point expectation proposed...

This work relates to multivariate beta operator which is expressed shortly as Bn. We show that the operator preserves Lipschitz constant of a Lipschitz continuous function and semi-additivity of the relevant operand. Furthermore, we provide an r-th order generalisation, B n , of Bn.

Interpolating discrete volume data into a continuous form adapts implicit surface techniques for rendering volumetric iso-surfaces. One such algorithm uses the Lipschitz condition to create an octree representation that accelerates volume rendering. Furthermore, only one preprocessing step is needed to create the Lipschitz-octree representation that accelerates rendering of iso-surfaces for any...

Given a real valued and time-inhomogeneous martingale diffusion X , we investigate the properties of functions defined by the conditional expectation f(t,Xt) = E [g(XT )|Ft]. We show that whenever g is monotonic or Lipschitz continuous then f(t, x) will also be monotonic or Lipschitz continuous in x. If g is convex then f(t, x) will be convex in x and decreasing in t. We also define the margina...

Sufficient conditions for the existence and Lipschitz continuity of optimal strategies for static team optimization problems are studied. Revised statements and proofs of some results appeared in the literature are presented. Their extensions are discussed. As an example of application, optimal production in a multidivisional firm is considered.

If X is a metric space then CX and LX denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of CX modulo LX is the least cardinality of any set U \LX where U generates CX . For a large class of separable metric spaces X we prove that the relative rank of CX modulo LX is uncountable. When X is the Baire space NN, this rank is א1. A large pa...