Weak error analysis for stochastic gradient descent optimization algorithms
This provides a theoretical foundation for SGD applications in areas like natural language processing and fraud detection, but it is incremental as it extends existing error analysis.
The paper tackles the problem of analyzing the error of stochastic gradient descent (SGD) optimization algorithms with respect to test functions that differ from the objective function, proving that under suitable assumptions, this error decays at the same speed as when the test function matches the objective function.
Stochastic gradient descent (SGD) type optimization schemes are fundamental ingredients in a large number of machine learning based algorithms. In particular, SGD type optimization schemes are frequently employed in applications involving natural language processing, object and face recognition, fraud detection, computational advertisement, and numerical approximations of partial differential equations. In mathematical convergence results for SGD type optimization schemes there are usually two types of error criteria studied in the scientific literature, that is, the error in the strong sense and the error with respect to the objective function. In applications one is often not only interested in the size of the error with respect to the objective function but also in the size of the error with respect to a test function which is possibly different from the objective function. The analysis of the size of this error is the subject of this article. In particular, the main result of this article proves under suitable assumptions that the size of this error decays at the same speed as in the special case where the test function coincides with the objective function.