# Online Optimisation for Online Learning and Control -- From No-Regret to Generalised Error Convergence

## TL;DR

This paper develops a new theoretical framework for online learning and control, providing convergence guarantees and applying them to online regression and adaptive control with dynamic uncertainties.

## Contribution

It introduces a generalized convergence notion for online algorithms and applies it to derive new performance guarantees for online regression and adaptive control.

## Key findings

- Theoretical guarantees on asymptotic prediction accuracy.
- Generalized learning guarantees for online regression.
- A model-reference adaptive controller with online performance bounds.

## Abstract

This paper presents early work aiming at the development of a new framework for the design and analysis of algorithms for online learning based prediction and control. Firstly, we consider the task of predicting values of a function or time series based on incrementally arriving sequences of inputs by utilising online programming. Introducing a generalisation of standard notions of convergence, we derive theoretical guarantees on the asymptotic behaviour of the prediction accuracies when prediction models are updated by a no-external-regret algorithm. We prove generalised learning guarantees for online regression and provide an example of how this can be applied to online learning-based control. We devise a model-reference adaptive controller with novel online performance guarantees on tracking success in the presence of a priori dynamic uncertainty. Our theoretical results are accompanied by illustrations on simple regression and control problems.