We present a framework to derive risk bounds for vector-valued learning with a broad class of feature maps and loss functions. Multi-task learning and one-vs-all multi-category learning are treated as examples. We discuss in detail vector-valued functions with one hidden layer, and demonstrate that the conditions under which shared representations are beneficial for multitask learning are equal...

We consider functions of the form f : are only partially speciied. In general, f assigns to any output y i a nonempty set d 2 D of elements of E. If d = feg for some e 2 E, then y i has the value e. If d = E, then y i is a \complete don't care", i.e., is allowed to take on any value in E. In other cases, y i is a \limited don't care", allowed to take on any value in d. Moreover, the functions a...

We consider a function-valued trait z(t) whose pre-selection distribution is Gaussian, and a fitness function W that models optimizing selection, subject to certain natural assumptions. We show that the post-selection distribution of z(t) is also Gaussian, compute the selection differential, and derive an equation that expresses the selection gradient in terms of the parameters of W and of the ...

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...