# Adaptive Sequential Machine Learning

## TL;DR

This paper extends a framework for adaptive sequential optimization to machine learning tasks, proposing methods to select sample sizes dynamically to control excess risk, validated through experiments on synthetic and real data.

## Contribution

It introduces an adaptive sampling method based on minimizer change estimates for machine learning, enhancing efficiency in stochastic optimization.

## Key findings

- The proposed method effectively controls excess risk.
- Adaptive sampling reduces computational costs.
- Experimental results validate the approach.

## Abstract

A framework previously introduced in [3] for solving a sequence of stochastic optimization problems with bounded changes in the minimizers is extended and applied to machine learning problems such as regression and classification. The stochastic optimization problems arising in these machine learning problems is solved using algorithms such as stochastic gradient descent (SGD). A method based on estimates of the change in the minimizers and properties of the optimization algorithm is introduced for adaptively selecting the number of samples at each time step to ensure that the excess risk, i.e., the expected gap between the loss achieved by the approximate minimizer produced by the optimization algorithm and the exact minimizer, does not exceed a target level. A bound is developed to show that the estimate of the change in the minimizers is non-trivial provided that the excess risk is small enough. Extensions relevant to the machine learning setting are considered, including a cost-based approach to select the number of samples with a cost budget over a fixed horizon, and an approach to applying cross-validation for model selection. Finally, experiments with synthetic and real data are used to validate the algorithms.