We investigate contextual online learning with nonparametric (Lipschitz) comparison classes under different assumptions on losses and feedback information. For full information feedback and Lipschitz losses, we design the first explicit algorithm achieving the minimax regret rate (up to log factors). In a partial feedback model motivated by second-price auctions, we obtain algorithms for Lipsch...

Let A(D) be the disc algebra of all continuous complex-valued functions on the unit disc D holomorphic in its interior. Functions from A(D) act on the set of all contraction operators (‖A‖ 1) on Hilbert spaces. It is proved that the following classes of functions from A(D) coincide: (1) the class of operator Lipschitz functions on the unit circle T; (2) the class of operator Lipschitz functions...

We show that in two dimensions or higher, the Mordukhovich-loffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Frechet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with ...

We consider a pair of reflected Brownian motions in a Lipschitz planar domain starting from different points but driven by the same Brownian motion. First we construct such a pair of processes in a certain weak sense, since it is not known whether a strong solution to the Skorohod equation in Lipschitz domains exists. Then we prove that the distance between the two processes converges to zero w...

In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α : H → Rm is injective, with (α(x))k = |〈x, fk〉|, where {f1, · · · , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω : Rm → H that is Li...

2.1 Definition: Lipschitz Function A real valued function f : D ⊆ R → R is called Lipschitz continuous or is said to satisfy a Lipschitz condition if there exists a constant K ≥ 0 such that for all x1, x2 in D |f(x1)− f(x2)| ≤ K|x1 − x2| (3) The inequality is (trivially) satisfied if x1 = x2. Otherwise, for x1 6= x2, one can equivalently define a function to be Lipschitz if and only if |f(x1)−f...

There were investigated 22 cases from which the thymic tissue was removed either during surgery for cardiovascular malformations (n = 14), or for myasthenia gravis (n = 8). Histological sections were stained with routine morphologic methods, and immunohistochemistry was performed for cytokeratin, CD20, CD3, and S100 protein. Aspects characteristic for thymus involution were found in 11 cases wi...