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The paper proposes an extended principle of empirical risk minimization to solve the regression problem. It is based on using aggregate functions instead of arithmetic mean to calculate risk. This can be justified if the loss distribution of emissions is significant or distorted, causing a shift in the risk assessment of the average loss from the very beginning. Therefore, in such cases, when optimizing characteristics in the regression problem the robust estimate of average value-at-risk should be initially used. Such intermediate risk assessment can be constructed using avg functions, which are the solution to the problem of penalty function minimization in case of mean deviation. This approach allows, on one hand, to determine a much broader class of secondary functions, and, on the other hand, to determine the average differentiable functions that approximate the average non-differentiable functions, such as a median or quintile. As a result, it is possible to construct gradient methods for solving the regression problem that, in a sense, can approximate robust techniques such as Least Median and Least Quantile. This paper proposes a new gradient scheme for solving the minimization problem of the intermediate risk. It is an analog of the used in the SAG algorithm circuit when the risk is calculated by arithmetic mean. An illustrative example presents the construction of robust procedures for characteristics assessment in a linear regression based on the use of the avg function, which approximates the median.

Принцип минимизации эмпирического риска на основе агрегирующих функций средних потерь для решения задач регрессии