# Anon: Extrapolating Adaptivity Beyond SGD and Adam

arXiv: https://arxiv.org/abs/2605.02317 · HTML: https://mlsanity.com/paper/2605.02317/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 41 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 30 of 30)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [Lookahead Optimizer: k steps forward, 1 step back](https://mlsanity.com/paper/1907.08610/) (2019)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning](https://mlsanity.com/paper/2006.00719/) (2020)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [Descending through a Crowded Valley - Benchmarking Deep Learning Optimizers](https://mlsanity.com/paper/2007.01547/) (2020)
- [Closing the Generalization Gap of Adaptive Gradient Methods in Training Deep Neural Networks](https://mlsanity.com/paper/1806.06763/) (2018)
- [Adaptive Inertia: Disentangling the Effects of Adaptive Learning Rate and Momentum](https://mlsanity.com/paper/2006.15815/) (2020)
- [Escaping Saddle Points with Adaptive Gradient Methods](https://mlsanity.com/paper/1901.09149/) (2019)
- [Domain-independent Dominance of Adaptive Methods](https://mlsanity.com/paper/1912.01823/) (2019)
- [Lookaround Optimizer: k steps around, 1 step average](https://mlsanity.com/paper/2306.07684/) (2023)
- [MADA: Meta-Adaptive Optimizers through hyper-gradient Descent](https://mlsanity.com/paper/2401.08893/) (2024)
- [Momentum Centering and Asynchronous Update for Adaptive Gradient Methods](https://mlsanity.com/paper/2110.05454/) (2021)
- [AGD: an Auto-switchable Optimizer using Stepwise Gradient Difference for Preconditioning Matrix](https://mlsanity.com/paper/2312.01658/) (2023)
- [HVAdam: A Full-Dimension Adaptive Optimizer](https://mlsanity.com/paper/2511.20277/) (2025)

## Abstract (arXiv)

Adaptive optimizers such as Adam and non-adaptive methods like SGD exhibit distinct generalization capabilities across different architectures. Prior tunable optimizers attempt to bridge this gap by strictly interpolating between SGD and Adam, effectively confining adaptivity within the 0-to-1 bound. However, this restricted interpolation is fundamentally insufficient: we reveal that optimal adaptivity often requires extrapolation, such as negative adaptivity for classical CNNs and adaptivity of at least one (γ ≥ 1) for Transformers. Extrapolating adaptivity theoretically violates the strict non-decreasing pre-conditioner assumption, often leading to divergence in existing methods. To break this barrier, we propose Anon, an optimizer that achieves fully continuous adaptivity extrapolation across the entire real-number spectrum. To guarantee provable stability in these out-of-bound regimes, we introduce Incremental Delay Update (IDU), a novel mechanism that bypasses hard max-tracking strategies. We theoretically establish Anon's convergence in both convex and non-convex settings. Empirically, by exploring previously unreachable adaptivity landscapes, Anon demonstrates highly competitive and scalable performance among state-of-the-art element-wise optimizers on representative image classification, diffusion, and large language modeling tasks.