Following the results in Rüschendorf and Uckelmann (2002), we introduce the completely mixable distributions on R and prove that distributions with monotone density and moderate mean are completely mixable. Using this method we solve the minimization problem minXi∼P Ef(X1 + · · ·+ Xn) for convex functions f and marginal distributions P with monotone density. Our results also provide valuable im...

In this paper we study the differentially private Empirical Risk Minimization (ERM) problem in different settings. For smooth (strongly) convex loss function with or without (non)-smooth regularization, we give algorithms that achieve either optimal or near optimal utility bounds with less gradient complexity compared with previous work. For ERM with smooth convex loss function in high-dimensio...

In this work we develop a new algorithm for regularized empirical risk minimization. Our method extends recent techniques of Shalev-Shwartz [02/2015], which enable a dual-free analysis of SDCA, to arbitrary mini-batching schemes. Moreover, our method is able to better utilize the information in the data defining the ERM problem. For convex loss functions, our complexity results match those of Q...

In this paper, we investigate the problem of binary classification with a reject option in which one can withhold the decision of classifying an observation at a cost lower than that of misclassification. Since the natural loss function is non-convex so that empirical risk minimization easily becomes infeasible, the paper proposes minimizing convex risks based on surrogate convex loss functions...

• New method to calculate transition time in the two-state model. Coupling potential shows dual nature, opaque and transparent. Explicit dependency of on second state’s Greens function. The coupling strength incident energy. We propose a simple scattering problem, where two constant potentials are coupled by delta function V 12 = 21 k 0 ? ( x ) . exact analytical expression for ? is derived. no...