Uspekhi Matematicheskikh Nauk

The problem of enhancing the interpretability and consistency of Baysesian classifier solutions in approximating the empirical data by means of a multilayer perceptron is under consideration. Histogram regression preserves transparency and statistical interpretation but is limited by memory requirements ($O(n)$) and weak scalability, while a multilayer perceptron provides a memory efficient representation ($O(1)$)and high computational efficiency in combination with limited interpretability. The focus is on a unary learning scheme, when the training sample consists of examples in the same target class and additional background points which are uniformly distributed over a compact subset of the feature space. This approach enables one to treat each class separately and implement the failure mechanism outside the data support, which enhances the model reliability. It is proposed to consider the perceptron output as a consistent analogue of the histogram class interval induced by the linearity cells of the perceptron. It is proved that under the natural assumptions of regularity and controlled growth of architecture the output function of a multilayer perseptron is consistent and equivalent to a histogram estimator. Theoretical consistency is rigorously ðroved in the case of a fixed first layer, while numerical experiments confirm the applicability of the results to models all of whose layers are trained. Thus histogram interpretation ensures the statistical verification of the consistency of perceptron approximation and addscredibility to classification solutions in the framework of a unary model.

We propose some accelerated methods for solving optimization problems under the condition of relatively smooth and relatively Lipschitz continuous functions with inexact oracle. We consider the problem of minimizing a convex, differentiable, and relatively smooth function relative to a reference convex function. The first proposed method is based on a similar triangles method with inexact oracle, which uses a special triangular scaling property of the Bregman divergence used. The other proposed methods are non-adaptive and adaptive (tuning to the relative smoothness parameter) accelerated Bregman proximal gradient methods with inexact oracle. These methods are universal in the sense that they apply not only to relatively smooth but also to relatively Lipschitz continuous optimization problems. We also introduce an adaptive intermediate Bregman method, which interpolates between slower but more robust non-accelerated algorithms and faster but less robust accelerated algorithms. We conclude the paper with the results of numerical experiments demonstrating the advantages of the proposed algorithms for the Poisson inverse problem.