Representative problems include AUC maximization [14, 25, 42, 63, 66], metric learning [8, 31], ranking [1, 13] and learning with minimum error entropy loss functions [29]. For example, in supervised metric learning we wish to find a distance function between pairs of examples so that examples within the same class are relatively close while examples from different classes are far apartfromeachother.

InInternational Journalof Parallel, Emergentand Distributed Systems, volume 27, pages 387-408, 2012.

InthecaseofSEkernel, the regularity assumption implies the existence ofall weak derivativesoff. The third equation follows from the definition of Zn(x). The first inequality holds by Assumption 2. We utilize Proposition 1 to conclude thatkZn(x)k2 σ Thesecond inequality holds bydefinition oflight-tailed distributions. Notice thatthecareful choice ofτ andθ ensures θζi(x) h0, which will be validated next. The seventh line is obtained by replacing the valueof θ.

Our sequential optimization setting is as follows. Consider an objective functionf defined over a domainX Rd, whered N is the dimension of the input.