## Abstract

We consider the setting of online convex optimization (OCO) with exp-concave losses. The best regret bound known for this setting is O(n log T), where n is the dimension and T is the number of prediction rounds (treating all other quantities as constants and assuming T is sufficiently large), and is attainable via the well-known Online Newton Step algorithm (ONS). However, ONS requires on each iteration to compute a projection (according to some matrix-induced norm) onto the feasible convex set, which is often computationally prohibitive in high-dimensional settings and when the feasible set admits a non-trivial structure. In this work we consider projection-free online algorithms for exp-concave and smooth losses, where by projection-free we refer to algorithms that rely only on the availability of a linear optimization oracle (LOO) for the feasible set, which in many applications of interest admits much more efficient implementations than a projection oracle. We present an LOO-based ONS-style algorithm, which using overall O(T) calls to a LOO, guarantees in worst case regret bounded by O^{e}(n^{2}/^{3}T^{2}/^{3}) (ignoring all quantities except for n, T). However, our algorithm is most interesting in an important and plausible low-dimensional data scenario: if the gradients (approximately) span a subspace of dimension at most ρ, ρ << n, the regret bound improves to O^{e}(ρ^{2}/^{3}T^{2}/^{3}), and by applying standard deterministic sketching techniques, both the space and average additional per-iteration runtime requirements are only O(ρn) (instead of O(n^{2})). This improves upon recently proposed LOO-based algorithms for OCO which, while having the same state-of-the-art dependence on the horizon T, suffer from regret/oracle complexity that scales with √n or worse.

Original language | English |
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Pages (from-to) | 1259-1284 |

Number of pages | 26 |

Journal | Proceedings of Machine Learning Research |

Volume | 195 |

State | Published - 2023 |

Event | 36th Annual Conference on Learning Theory, COLT 2023 - Bangalore, India Duration: 12 Jul 2023 → 15 Jul 2023 |

## Keywords

- exp-concave
- linear optimization oracle
- online convex optimization
- online learning
- projection-free

## All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability