# Refining Adaptive Zeroth-Order Optimization at Ease

arXiv: https://arxiv.org/abs/2502.01014 · HTML: https://mlsanity.com/paper/2502.01014/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 3 Feb 2025.

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

- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Fine-Tuning Language Models with Just Forward Passes](https://mlsanity.com/paper/2305.17333/) (2023)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [Revisiting Zeroth-Order Optimization for Memory-Efficient LLM Fine-Tuning: A Benchmark](https://mlsanity.com/paper/2402.11592/) (2024)
- [Zeroth-Order Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1805.10367/) (2018)
- [A Comprehensive Linear Speedup Analysis for Asynchronous Stochastic Parallel Optimization from Zeroth-Order to First-Order](https://mlsanity.com/paper/1606.00498/) (2016)
- [Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity](https://mlsanity.com/paper/2310.17998/) (2023)
- [ZO-AdaMU Optimizer: Adapting Perturbation by the Momentum and Uncertainty in Zeroth-order Optimization](https://mlsanity.com/paper/2312.15184/) (2023)
- [On the Convergence of Prior-Guided Zeroth-Order Optimization Algorithms](https://mlsanity.com/paper/2107.10110/) (2021)
- [Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance](https://mlsanity.com/paper/2404.01436/) (2024)
- [Federated Zeroth-Order Optimization using Trajectory-Informed Surrogate Gradients](https://mlsanity.com/paper/2308.04077/) (2023)
- [Adaptive First-and Zeroth-order Methods for Weakly Convex Stochastic Optimization Problems](https://mlsanity.com/paper/2005.09261/) (2020)
- [Optimizing Black-box Metrics with Iterative Example Weighting](https://mlsanity.com/paper/2102.09492/) (2021)

## Built on by (most cited first, 12 of 12)

- [Zeroth-Order Optimization is Secretly Single-Step Policy Optimization](https://mlsanity.com/paper/2506.14460/) (2025)
- [Downgrade to Upgrade: Optimizer Simplification Enhances Robustness in LLM Unlearning](https://mlsanity.com/paper/2510.00761/) (2025)
- [On Adaptivity in Zeroth-Order Optimization](https://mlsanity.com/paper/2605.03869/) (2026)
- [Power Homotopy for Zeroth-Order Non-Convex Optimizations](https://mlsanity.com/paper/2511.13592/) (2025)
- [SeedFlood: A Step Toward Scalable Decentralized Fine-Tuning of LLMs](https://mlsanity.com/paper/2602.18181/) (2026)
- [PRAC: Principal-Random Subspace for LLM Activation Compression and Memory-Efficient Training](https://mlsanity.com/paper/2602.23111/) (2026)
- [Position: Zeroth-Order Optimization in Deep Learning Is Underexplored, Not Underpowered](https://mlsanity.com/paper/2605.15622/) (2026)
- [Accelerating Zeroth-Order Spectral Optimization with Partial Orthogonalization from Power Iteration](https://mlsanity.com/paper/2605.09034/) (2026)
- [Revisiting Zeroth-Order Hessian Approximation: A Single-Step Policy Optimization Lens](https://mlsanity.com/paper/2605.30960/) (2026)
- [SubZero+: Memory-Efficient Adaptive Zeroth-Order LLM Fine-Tuning in Random Subspaces](https://mlsanity.com/paper/2608.15665/) (2026)
- [ZoAQ: Adaptive Zeroth-Order Querying via Query-Reuse Coupling](https://mlsanity.com/paper/2609.22115/) (2026)
- [Sample-Adaptive Zeroth-Order Post-Training](https://mlsanity.com/paper/2610.12566/) (2026)

## Abstract (arXiv)

Recently, zeroth-order (ZO) optimization plays an essential role in scenarios where gradient information is inaccessible or unaffordable, such as black-box systems and resource-constrained environments. While existing adaptive methods such as ZO-AdaMM have shown promise, they are fundamentally limited by their underutilization of moment information during optimization, usually resulting in underperforming convergence. To overcome these limitations, this paper introduces Refined Adaptive Zeroth-Order Optimization (R-AdaZO). Specifically, we first show the untapped variance reduction effect of first moment estimate on ZO gradient estimation, which improves the accuracy and stability of ZO updates. We then refine the second moment estimate based on these variance-reduced gradient estimates to better capture the geometry of the optimization landscape, enabling a more effective scaling of ZO updates. We present rigorous theoretical analysis to show (a) the first analysis to the variance reduction of first moment estimate in ZO optimization, (b) the improved second moment estimates with a more accurate approximation of its variance-free ideal, (c) the first variance-aware convergence framework for adaptive ZO methods, which may be of independent interest, and (d) the faster convergence of R-AdaZO than existing baselines like ZO-AdaMM. Our extensive experiments, including synthetic problems, black-box adversarial attack, and memory-efficient fine-tuning of large language models (LLMs), further verify the superior convergence of R-AdaZO, indicating that R-AdaZO offers an improved solution for real-world ZO optimization challenges.