# Multi-scale exploration of convex functions and bandit convex optimization

arXiv: https://arxiv.org/abs/1507.06580 · HTML: https://mlsanity.com/paper/1507.06580/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2015.

## Built on by (most cited first, 20 of 20)

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- [Optimal Regret Algorithm for Pseudo-1d Bandit Convex Optimization](https://mlsanity.com/paper/2102.07387/) (2021)
- [A Reduction from Delayed to Immediate Feedback for Online Convex Optimization with Improved Guarantees](https://mlsanity.com/paper/2602.02634/) (2026)
- [Adaptive Bandit Convex Optimization with Heterogeneous Curvature](https://mlsanity.com/paper/2202.06150/) (2022)
- [Bandit Learning to Rank with Position-Based Click Models: Personalized and Equal Treatments](https://mlsanity.com/paper/2311.04528/) (2023)
- [Distilled Thompson Sampling: Practical and Efficient Thompson Sampling via Imitation Learning](https://mlsanity.com/paper/2011.14266/) (2020)
- [A Bayesian Approach to Online Planning](https://mlsanity.com/paper/2406.02103/) (2024)
- [Multi-Agent Stage-wise Conservative Linear Bandits](https://mlsanity.com/paper/2510.00602/) (2025)
- [Capacity-Constrained Online Convex Optimization with Delayed Feedback](https://mlsanity.com/paper/2606.11711/) (2026)
- [Distributed Online Bandit Nonconvex Optimization with One-Point Residual Feedback via Dynamic Regret](https://mlsanity.com/paper/2409.15680/) (2024)

## Abstract (arXiv)

We construct a new map from a convex function to a distribution on its domain, with the property that this distribution is a multi-scale exploration of the function. We use this map to solve a decade-old open problem in adversarial bandit convex optimization by showing that the minimax regret for this problem is tildeO(poly(n) √T), where n is the dimension and T the number of rounds. This bound is obtained by studying the dual Bayesian maximin regret via the information ratio analysis of Russo and Van Roy, and then using the multi-scale exploration to solve the Bayesian problem.