Empirical Risk Minimization for Time Series: Nonparametric Performance Bounds for Prediction
This provides theoretical guarantees for time series forecasting, but it is incremental as it extends empirical risk minimization principles to a specific domain.
The paper tackles the problem of 1-step-ahead prediction for univariate time series using empirical risk minimization, showing that the selected algorithm asymptotically achieves optimal predictive performance within a class of recursive algorithms.
Empirical risk minimization is a standard principle for choosing algorithms in learning theory. In this paper we study the properties of empirical risk minimization for time series. The analysis is carried out in a general framework that covers different types of forecasting applications encountered in the literature. We are concerned with 1-step-ahead prediction of a univariate time series generated by a parameter-driven process. A class of recursive algorithms is available to forecast the time series. The algorithms are recursive in the sense that the forecast produced in a given period is a function of the lagged values of the forecast and of the time series. The relationship between the generating mechanism of the time series and the class of algorithms is unspecified. Our main result establishes that the algorithm chosen by empirical risk minimization achieves asymptotically the optimal predictive performance that is attainable within the class of algorithms.