# SGD with AdaGrad Stepsizes: Full Adaptivity with High Probability to Unknown Parameters, Unbounded Gradients and Affine Variance

arXiv: https://arxiv.org/abs/2302.08783 · HTML: https://mlsanity.com/paper/2302.08783/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 29 references that are in the map.
- First posted on arXiv 17 Feb 2023.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [AdaGrad stepsizes: Sharp convergence over nonconvex landscapes](https://mlsanity.com/paper/1806.01811/) (2018)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [Tight Analyses for Non-Smooth Stochastic Gradient Descent](https://mlsanity.com/paper/1812.05217/) (2018)
- [On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization](https://mlsanity.com/paper/1808.05671/) (2018)
- [Scale-Free Online Learning](https://mlsanity.com/paper/1601.01974/) (2016)
- [High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails](https://mlsanity.com/paper/2106.14343/) (2021)
- [A High Probability Analysis of Adaptive SGD with Momentum](https://mlsanity.com/paper/2007.14294/) (2020)
- [The Power of Adaptivity in SGD: Self-Tuning Step Sizes with Unbounded Gradients and Affine Variance](https://mlsanity.com/paper/2202.05791/) (2022)
- [Lipschitz and Comparator-Norm Adaptivity in Online Learning](https://mlsanity.com/paper/2002.12242/) (2020)
- [Making SGD Parameter-Free](https://mlsanity.com/paper/2205.02160/) (2022)
- [High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize](https://mlsanity.com/paper/2204.02833/) (2022)
- [A new regret analysis for Adam-type algorithms](https://mlsanity.com/paper/2003.09729/) (2020)
- [On the Convergence Proof of AMSGrad and a New Version](https://mlsanity.com/paper/1904.03590/) (2019)

## Built on by (most cited first, 30 of 30)

- [AdaGrad under Anisotropic Smoothness](https://mlsanity.com/paper/2406.15244/) (2024)
- [On Convergence of Adam for Stochastic Optimization under Relaxed Assumptions](https://mlsanity.com/paper/2402.03982/) (2024)
- [Provable Complexity Improvement of AdaGrad over SGD: Upper and Lower Bounds in Stochastic Non-Convex Optimization](https://mlsanity.com/paper/2406.04592/) (2024)
- [Revisiting Convergence of AdaGrad with Relaxed Assumptions](https://mlsanity.com/paper/2402.13794/) (2024)
- [Tuning-Free Stochastic Optimization](https://mlsanity.com/paper/2402.07793/) (2024)
- [How Free is Parameter-Free Stochastic Optimization?](https://mlsanity.com/paper/2402.03126/) (2024)
- [High Probability Convergence of Adam Under Unbounded Gradients and Affine Variance Noise](https://mlsanity.com/paper/2311.02000/) (2023)
- [AdAdaGrad: Adaptive Batch Size Schemes for Adaptive Gradient Methods](https://mlsanity.com/paper/2402.11215/) (2024)
- [On the O(frac√dK1/4) Convergence Rate of AdamW Measured by ℓ1 Norm](https://mlsanity.com/paper/2505.11840/) (2025)
- [Non-Euclidean SGD for Structured Optimization: Unified Analysis and Improved Rates](https://mlsanity.com/paper/2511.11466/) (2025)
- [Dynamic Byzantine-Robust Learning: Adapting to Switching Byzantine Workers](https://mlsanity.com/paper/2402.02951/) (2024)
- [Preconditioning for Accelerated Gradient Descent Optimization and Regularization](https://mlsanity.com/paper/2410.00232/) (2024)
- [Simple Convergence Proof of Adam From a Sign-like Descent Perspective](https://mlsanity.com/paper/2507.05966/) (2025)
- [The High Line: Exact Risk and Learning Rate Curves of Stochastic Adaptive Learning Rate Algorithms](https://mlsanity.com/paper/2405.19585/) (2024)
- [Can Adaptive Gradient Methods Converge under Heavy-Tailed Noise? A Case Study of AdaGrad](https://mlsanity.com/paper/2605.18694/) (2026)
- [Lean and Mean Adaptive Optimization via Subset-Norm and Subspace-Momentum with Convergence Guarantees](https://mlsanity.com/paper/2411.07120/) (2024)
- [Adaptive Algorithms with Sharp Convergence Rates for Stochastic Hierarchical Optimization](https://mlsanity.com/paper/2509.15399/) (2025)
- [Design Criteria for SGD Preconditioners: Local Conditioning, Noise Floors, and Basin Stability](https://mlsanity.com/paper/2511.19716/) (2025)
- [A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon](https://mlsanity.com/paper/2604.17423/) (2026)
- [On Adaptivity in Zeroth-Order Optimization](https://mlsanity.com/paper/2605.03869/) (2026)
- [A General Reduction for High-Probability Analysis with General Light-Tailed Distributions](https://mlsanity.com/paper/2403.02873/) (2024)
- [Faster Stochastic Optimization with Arbitrary Delays via Asynchronous Mini-Batching](https://mlsanity.com/paper/2408.07503/) (2024)
- [Locally Optimal Descent for Dynamic Stepsize Scheduling](https://mlsanity.com/paper/2311.13877/) (2023)
- [On the O(frac√dT1/4) Convergence Rate of RMSProp and Its Momentum Extension Measured by ℓ1 Norm](https://mlsanity.com/paper/2402.00389/) (2024)
- [Learning Rate Annealing Improves Tuning Robustness in Stochastic Optimization](https://mlsanity.com/paper/2503.09411/) (2025)
- [A regret minimization approach to fixed-point iterations](https://mlsanity.com/paper/2509.21653/) (2025)
- [Stochastic convergence of parallel asynchronous adaptive first-order methods](https://mlsanity.com/paper/2606.01787/) (2026)
- [Zero-order Parameter-free Optimization for LMO-based Methods: Novel Approach for Efficient Fine-tuning](https://mlsanity.com/paper/2606.14970/) (2026)
- [Why Clipping Matters in AdaGrad? Toward a High-Probability Theory under Generalized Smoothness](https://mlsanity.com/paper/2609.30276/) (2026)
- [Revisiting AdaGrad in Stochastic Convex Optimization: Last Iterates, High Probability, and Lower Bounds](https://mlsanity.com/paper/2609.32729/) (2026)

## Abstract (arXiv)

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings: they either assume some knowledge of the problem parameters, impose strong global Lipschitz conditions, or fail to give bounds that hold with high probability. We provide a comprehensive analysis of this basic method without any of these limitations, in both the convex and non-convex (smooth) cases, that additionally supports a general ``affine variance'' noise model and provides sharp rates of convergence in both the low-noise and high-noise~regimes.