# Near-Linear Time Algorithm with Near-Logarithmic Regret Per Switch for Mixable/Exp-Concave Losses

arXiv: https://arxiv.org/abs/2109.13786 · HTML: https://mlsanity.com/paper/2109.13786/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 49 references that are in the map.
- First posted on arXiv 28 Sep 2021.

## Builds on (most cited first, 6 of 6)

- [Strongly Adaptive Online Learning](https://mlsanity.com/paper/1502.07073/) (2015)
- [Minimizing Dynamic Regret and Adaptive Regret Simultaneously](https://mlsanity.com/paper/2002.02085/) (2020)
- [Adaptive Regret of Convex and Smooth Functions](https://mlsanity.com/paper/1904.11681/) (2019)
- [Recursive Experts: An Efficient Optimal Mixture of Learning Systems in Dynamic Environments](https://mlsanity.com/paper/2009.09249/) (2020)
- [Minimax Optimal Algorithms for Adversarial Bandit Problem with Multiple Plays](https://mlsanity.com/paper/1911.11122/) (2019)
- [Optimal and Efficient Algorithms for General Mixable Losses against Switching Oracles](https://mlsanity.com/paper/2108.06411/) (2021)

## Built on by (most cited first, 1 of 1)

- [Dimension-Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2610.20187/) (2026)

## Abstract (arXiv)

We investigate the problem of online learning, which has gained significant attention in recent years due to its applicability in a wide range of fields from machine learning to game theory. Specifically, we study the online optimization of mixable loss functions with logarithmic static regret in a dynamic environment. The best dynamic estimation sequence that we compete against is selected in hindsight with full observation of the loss functions and is allowed to select different optimal estimations in different time intervals (segments). We propose an online mixture framework that uses these static solvers as the base algorithm. We show that with the suitable selection of hyper-expert creations and weighting strategies, we can achieve logarithmic and squared logarithmic regret per switch in quadratic and linearithmic computational complexity, respectively. For the first time in literature, we show that it is also possible to achieve near-logarithmic regret per switch with sub-polynomial complexity per time. Our results are guaranteed to hold in a strong deterministic sense in an individual sequence manner.