Optimistic Optimisation of Composite Objective with Exponentiated Update

TL;DR

This paper introduces a new family of online optimization algorithms that combine exponentiated gradient and p-norm methods, achieving adaptive, optimistic, and efficient solutions with optimal regret bounds for sparse and smooth objectives.

Contribution

It presents a novel algorithmic framework that unifies exponentiated gradient and p-norm approaches, incorporating adaptivity and optimism for improved online and stochastic optimization.

Findings

Achieves sequence-dependent regret bounds matching the best-known for sparse targets.

Provides efficient implementations for common composite objectives and constraints.

Can be adapted to stochastic optimization with accelerated rates for smooth functions.

Abstract

This paper proposes a new family of algorithms for the online optimisation of composite objectives. The algorithms can be interpreted as the combination of the exponentiated gradient and $p$-norm algorithm. Combined with algorithmic ideas of adaptivity and optimism, the proposed algorithms achieve a sequence-dependent regret upper bound, matching the best-known bounds for sparse target decision variables. Furthermore, the algorithms have efficient implementations for popular composite objectives and constraints and can be converted to stochastic optimisation algorithms with the optimal accelerated rate for smooth objectives.