# Zeroth-order non-convex learning via hierarchical dual averaging

arXiv: https://arxiv.org/abs/2109.05829 · HTML: https://mlsanity.com/paper/2109.05829/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 58 references that are in the map.
- First posted on arXiv 13 Sep 2021.

## Builds on (most cited first, 16 of 16)

- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [More Adaptive Algorithms for Adversarial Bandits](https://mlsanity.com/paper/1801.03265/) (2018)
- [Bandits and Experts in Metric Spaces](https://mlsanity.com/paper/1312.1277/) (2013)
- [Kernel-based methods for bandit convex optimization](https://mlsanity.com/paper/1607.03084/) (2016)
- [Online Non-Convex Learning: Following the Perturbed Leader is Optimal](https://mlsanity.com/paper/1903.08110/) (2019)
- [Adaptive extra-gradient methods for min-max optimization and games](https://mlsanity.com/paper/2010.12100/) (2020)
- [Efficient Regret Minimization in Non-Convex Games](https://mlsanity.com/paper/1708.00075/) (2017)
- [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018)
- [Learning in Non-convex Games with an Optimization Oracle](https://mlsanity.com/paper/1810.07362/) (2018)
- [Multi-scale exploration of convex functions and bandit convex optimization](https://mlsanity.com/paper/1507.06580/) (2015)
- [Riemannian game dynamics](https://mlsanity.com/paper/1603.09173/) (2016)
- [Online non-convex optimization with imperfect feedback](https://mlsanity.com/paper/2010.08496/) (2020)
- [Hessian barrier algorithms for linearly constrained optimization problems](https://mlsanity.com/paper/1809.09449/) (2018)
- [On the convergence of gradient-like flows with noisy gradient input](https://mlsanity.com/paper/1611.06730/) (2016)
- [On First-Order Bounds, Variance and Gap-Dependent Bounds for Adversarial Bandits](https://mlsanity.com/paper/1903.07890/) (2019)

## Built on by (most cited first, 6 of 6)

- [Online Bilevel Optimization: Regret Analysis of Online Alternating Gradient Methods](https://mlsanity.com/paper/2207.02829/) (2022)
- [Non-Convex Bilevel Optimization with Time-Varying Objective Functions](https://mlsanity.com/paper/2308.03811/) (2023)
- [Nested bandits](https://mlsanity.com/paper/2206.09348/) (2022)
- [Gradient and Projection Free Distributed Online Min-Max Resource Optimization](https://mlsanity.com/paper/2112.03896/) (2021)
- [How to Boost Any Loss Function](https://mlsanity.com/paper/2407.02279/) (2024)
- [Leveraging Similarities in Multi-Armed Bandits](https://mlsanity.com/paper/2606.23414/) (2026)

## Abstract (arXiv)

We propose a hierarchical version of dual averaging for zeroth-order online non-convex optimization - i.e., learning processes where, at each stage, the optimizer is facing an unknown non-convex loss function and only receives the incurred loss as feedback. The proposed class of policies relies on the construction of an online model that aggregates loss information as it arrives, and it consists of two principal components: (a) a regularizer adapted to the Fisher information metric (as opposed to the metric norm of the ambient space); and (b) a principled exploration of the problem's state space based on an adapted hierarchical schedule. This construction enables sharper control of the model's bias and variance, and allows us to derive tight bounds for both the learner's static and dynamic regret - i.e., the regret incurred against the best dynamic policy in hindsight over the horizon of play.