Adaptive Gradient Methods

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13 papers in the last four weeks, up 160% on the four weeks before. 0.1% of all new papers.

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Latest papers 113

May 7, 2026cs.LG

A Rod Flow Model for Adam at the Edge of Stability

Cohen et al. (arXiv:2207.14484) observed that adaptive gradient methods such as Adam operate at the edge of stability. While there has been significant work on continuous-time modeling of gradient descent at the edge of stability, extending these models to momentum methods remains underdeveloped. In the gradient descent setting, Regis et al. (arXiv:2602.01480) introduced rod flow, which models consecutive iterates as an extended one-dimensional object -- a "rod." Here we extend rod flow to Adam by working in the joint phase space of parameters and first moment

(w, m)

and treating the second moment

ν

as a smooth auxiliary variable. We also develop rod flows for heavy ball momentum, Nesterov momentum, and scalar and per-component versions of RMSProp, Adam, and NAdam. For all eight optimizers, we empirically evaluate rod flow on representative machine learning architectures, where it tracks the discrete iterates through the edge-of-stability regime significantly more accurately than the corresponding stable flow.

Eric Regis, Sinho Chewi

Yale University

Edge of Stability Momentum Methods Adaptive Gradient Methods Gradient Descent Dynamics

May 7, 2026cs.LG

Revisiting Adam for Streaming Reinforcement Learning

Learning from a sequence of interactions, as soon as observations are perceived and acted upon, without explicitly storing them, holds the promise of simpler, more efficient and adaptive algorithms. For over a decade, however, deep reinforcement learning walked the contrary path, augmenting agents with replay buffers or parallel sampling routines, in an effort to tame learning instability. Recently, this topic has been revisited by Elsayed et al. (2024), focusing on update computation through eligibility traces and modifications to the optimisation routine, resulting in the StreamQ algorithm. In this work we take a step back, investigating the efficacy of established updates, such as those implemented by DQN and C51 within this online setting. Not only do we find that they perform well, but through analysing how the optimisation algorithm generally, and Adam in particular, interacts with these updates, we contend that two properties are essential for robust performance: i) the derivative of the objective is to be bounded and ii) weight updates are variance-adjusted. Rigorous and exhaustive experimentation demonstrates that C51, which exhibits both characteristics, is competitive with StreamQ across a subset of 55 Atari games. Using these insights, we derive a variance-adjusted algorithm based on eligibility traces, termed Adaptive Q

(λ)

, which approaches double the human baseline on the same subset, surpassing existing methods by all performance metrics.

Florin Gogianu, Adrian Catalin Lutu, Razvan Pascanu

Bitdefender, Bucharest, Romania · Mila, Montreal, Canada

Distributional RL Reinforcement Learning Deep Q-Learning Adaptive Gradient Methods

May 7, 2026cs.LG

Directional Consistency as a Complementary Optimization Signal: The GONO Framework

We identify and formalize an underexplored phenomenon in deep learning optimization: directional alignment and loss convergence can be decoupled. An optimizer can exhibit near-perfect directional consistency (cc_t -> 1, measured via consecutive gradient cosine similarity) while the loss remains high or decreases slowly. This observation reveals that existing optimizers such as Adam, SGD, and RMSprop lack explicit mechanisms to exploit temporal consistency in gradient directions, relying instead on magnitude-based signals that fail to distinguish plateaus, saddle points, and genuine convergence. Motivated by this, we introduce GONO (Gradient-Oriented Norm-Adaptive Optimizer), which adapts Adam's momentum coefficient beta_1 based on cc_t: amplifying momentum under directional consistency and suppressing it during oscillation. We prove GONO matches Adam's O(1/sqrt(T)) convergence rate and reduces exactly to Adam when the signal is uninformative. Empirically, cc_t achieves oscillation detection with F1=1.00 (vs. 0.45 for gradient norm), and GONO remains competitive with AdamW on MNIST (98.15%), CIFAR-10 (43.14%), and ResNet-18 (75.44%), establishing directional alignment as a theoretically grounded, practically actionable optimization signal. Code: https://github.com/victordaniel/gono-optimizer

Victor Daniel Gera

Department of Artificial Intelligence Anurag University, Hyderabad, India

Deep Learning Optimization Stochastic Optimization Convergence Adaptive Gradient Methods Gradient Alignment

May 7, 2026cs.LG

Revealing Modular Gradient Noise Imbalance in LLMs: Calibrating Adam via Signal-to-Noise Ratio

The impressive performance of large language models (LLMs) arises from their massive scale and heterogeneous module composition. However, this structural heterogeneity introduces additional optimization challenges. While adaptive optimizers such as Adam(W) provide per-parameter adaptivity, they do not explicitly account for module-level gradient heterogeneity, resulting in slower convergence, suboptimal performance, or training instability. Existing approaches typically rely on manually tuned module-specific learning rates or specific optimization strategies, which are computationally costly and difficult to generalize across tasks or models. To establish a more principled approach, we first analyze the noise-damping behavior of Adam in high-noise modules and introduce \textbf{Module-wise Learning Rate Scaling via SNR (MoLS)}. MoLS estimates module-level SNRs to scale Adam updates, allowing automated module-wise learning rate allocation without manual tuning. Empirical results through multiple LLM training benchmarks demonstrate that MoLS improves convergence speed and generalization, achieving performance comparable to carefully tuned module-specific learning rates, while remaining compatible with memory-efficient training algorithms.

Ziqing Wen, Zhouyang Liu, Jiahuan Wang +4

College of Computer Science and Technology, National University of Defense Technology, Changsha, Hunan, China

Deep Learning Optimization Learning Rate Scheduling Adaptive Gradient Methods

May 5, 2026cs.LG

On Adaptivity in Zeroth-Order Optimization

We investigate the effectiveness of adaptive zeroth-order (ZO) optimization for memory-constrained fine-tuning of large language models (LLMs). Contrary to prior claims, we show that adaptive ZO methods such as ZO-Adam offer no convergence advantage over well-tuned ZO-SGD, while incurring significant memory overhead. Our analysis reveals that in high dimensions, ZO gradients lack coordinate-wise heterogeneity, rendering adaptive mechanisms memory inefficient. Leveraging this insight, we propose MEAZO, a memory-efficient adaptive ZO optimizer that tracks only a single scalar for global step size adaptation. We support our method with theoretical convergence guarantees under standard assumptions. Experiments across multiple LLM families and tasks demonstrate that MEAZO matches ZO-Adam's performance with the memory footprint of ZO-SGD. Additional experiments on synthetic quadratic problems and LLM fine-tuning further demonstrate MEAZO's enhanced robustness to step size choices, particularly in grouped or block-structured optimization settings.

Hassan Dbouk, Nidham Gazagnadou, Matthias Reisser +1

Qualcomm AI Research

Zeroth-Order Optimization LLM Fine-Tuning Memory-Efficient Optimization Stochastic Optimization Convergence Adaptive Gradient Methods

May 5, 2026cs.LG

FIBER: A Differentially Private Optimizer with Filter-Aware Innovation Bias Correction

Differentially private (DP) training protects individual examples by adding noise to gradients, but the injected noise interacts nontrivially with adaptive optimizers. Recent DP methods temporally filter privatized gradients to reduce variance; however, filtering also changes the DP noise statistics seen by AdamW's second-moment accumulator. As a result, bias corrections derived for unfiltered DP noise, such as subtracting sigma_w squared, can become miscalibrated when filtering is present. We propose FiBeR, a DP optimizer designed for temporally filtered privatized gradients. FiBeR (i) performs denoising in innovation space by filtering the residual stream and integrating it to form the filtered gradient estimate, (ii) decouples the two-point observation geometry from the innovation gain to enable independent tuning, and (iii) introduces a filter-aware second-moment calibration that subtracts the attenuated DP noise contribution A(omega) sigma_w squared, where A(omega) is derived in closed form for the innovation filter and can be computed for general stable linear filters. Across vision and language benchmarks, FiBeR consistently demonstrates substantial improvements in the performance of DP optimizers, surpassing state-of-the-art results under equivalent privacy constraints on multiple tasks.

Duc Dm, Thao Do, Minh Son Hoang +3

School of Computing, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea · Center for Environmental Intelligence (CEI), VinUniversity, Hanoi, Vietnam · Khoury College of Computer Sciences, Northeastern University, USA

Differentially Private Stochastic Gradient Descent Privacy-Preserving ML Differential Privacy Adaptive Gradient Methods

May 4, 2026cs.AI

Anon: Extrapolating Adaptivity Beyond SGD and Adam

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\geq 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.

Yiheng Zhang, Kaiyan Zhao, Shaowu Wu +5

Wuhan University · University of Macau · University of Calgary

Nonconvex Stochastic Optimization Adaptive Gradient Methods

May 1, 2026cs.LG

AdaMeZO: Adam-style Zeroth-Order Optimizer for LLM Fine-tuning Without Maintaining the Moments

Fine-tuning LLMs is necessary for various dedicated downstream tasks, but classic backpropagation-based fine-tuning methods require substantial GPU memory. To this end, a recent work, MeZO, which relies solely on forward passes to fine-tune LLMs, significantly reduces GPU requirements at the cost of slower convergence due to its indifference to loss landscapes. Standard solutions, such as Adam, explore loss landscapes by estimating the first- and second-order moments and storing them in memory to guide the model's movement through dimensions with lower curvature and vice versa. However, directly applying Adam negates MeZO's advantage as it will triple the memory requirement. In light of this, we propose AdaMeZO, a zeroth-order optimizer that leverages Adam-style first- and second-moment estimates without maintaining them in memory. We present a theoretical analysis of AdaMeZO, corroborated by extensive experiments demonstrating AdaMeZO's performance, showing that AdaMeZO can outperform MeZO while requiring up to

70\%

fewer forward passes. Trajectory visualizations affirm AdaMeZO's ability to adapt to diverse loss landscapes.

Zhijie Cai, Haolong Chen, Guangxu Zhu

Shenzhen Research Institute of Big Data · The Chinese University of Hong Kong-Shenzhen · Shenzhen Loop Area Institute

Memory-Efficient Fine-Tuning Zeroth-Order Optimization LLM Fine-Tuning Memory-Efficient Optimization Adaptive Gradient Methods

Apr 30, 2026cs.AI

Learning Rate Engineering: From Coarse Single Parameter to Layered Evolution

Learning rate scheduling has evolved from the single global fixed rate of early SGD to sophisticated layer-wise adaptive strategies. We systematize this evolution into five generations: (Gen1) global fixed learning rates, (Gen2) global scheduling, (Gen3) parameter-level adaptation, (Gen4) layer-level differentiation, and (Gen5) joint layer-time scheduling. We trace the fundamental motivation behind each transition, showing how the shift from one-size-fits-all to tailoring by layer and time addresses the impossible trinity of transfer learning: lower layers require small updates to preserve general knowledge while higher layers need large updates to adapt to new tasks. Building on this taxonomy, we propose Discriminative Adaptive Layer Scaling (DALS), a unified framework that integrates phase-adaptive cosine scheduling, depth-aware Grokfast gradient filtering, and LARS-style trust ratios into a single coherent optimizer. We benchmark 18 strategies including three DALS variants across all five generations on five datasets: synthetic, CIFAR-10 (from scratch), RTE, TREC-6, and IMDb (fine-tuning). On synthetic, DALS achieves the best accuracy at 98.0%, while DALS-Fast reaches 90% in just 3 epochs. The cross-dataset analysis reveals striking regime-dependent patterns -- no single strategy wins across all regimes. Critically, STLR+Discriminative, the ULMFiT champion, catastrophically fails on from-scratch tasks (43.6% on TREC-6 from scratch vs. 96.8% with RAdam), confirming that directional decay biases are harmful without pretrained features. DALS avoids either extreme, achieving the best synthetic result while maintaining competitive fine-tuning performance.

Ming-Hong Yao, Di Wang, Jian Cui +5

School of AI, Jilin International Sciences University, Changchun, China.

Deep Learning Optimization Fine-Tuning Learning Rate Scheduling Adaptive Gradient Methods Layer-Wise Learning Rate Adaptation

Apr 29, 2026cs.LG

NeuroPlastic: A Plasticity-Modulated Optimizer for Biologically Inspired Learning Dynamics

Optimization algorithms are fundamental to modern deep learning, yet most widely used methods rely on update rules based primarily on local gradient statistics. We introduce NeuroPlastic, a plasticity-modulated optimizer that augments gradient-based updates with an adaptive multi-signal modulation mechanism inspired by multi-factor synaptic plasticity, a concept from neurobiology. NeuroPlastic dynamically scales gradient updates using interacting components that capture gradient, activity-like, and memory-like statistics, forming a lightweight modulation layer compatible with standard deep learning training pipelines. Across image classification benchmarks, NeuroPlastic consistently improves over a controlled gradient-only ablation, with more pronounced gains on the Fashion-MNIST benchmark and in reduced-data regimes. In transfer experiments on CIFAR-10 with ResNet-18, the method remains stable and competitive without retuning. These results suggest that multi-signal plasticity-inspired modulation can provide a useful extension to conventional gradient-driven optimization, particularly when learning signals are limited or noisy, and offer a promising direction for gradient-based methods in deep learning.

Douglas Jiang, Yuechen Wang, Jiayi Wang +3

Department of Neurology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA 02115, USA · Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, MA 02115, USA · Equal contributions +2

Neural Network Optimization Adaptive Gradient Methods

Apr 24, 2026cs.LG

Hidden Failure Modes of Gradient Modification under Adam in Continual Learning, and Adaptive Decoupled Moment Routing as a Repair

Many continual-learning methods modify gradients upstream (e.g., projection, penalty rescaling, replay mixing) while treating Adam as a neutral backend. We show this composition has a hidden failure mode. In a high-overlap, non-adaptive 8-domain continual LM, all shared-routing projection baselines collapse close to vanilla forgetting (12.5--12.8 vs. 13.2). A 0.5% replay buffer is the strongest shared alternative but still reaches 11.6, while fixed-strength decoupling falls below vanilla at 14.1. Only adaptive decoupled routing remains stable at 9.4, improving over vanilla by 3.8 units. On a 16-domain stream, its gain over the strongest shared-routing projection baseline grows to 4.5--4.8 units. The failure is largely invisible on clean benchmarks. We explain this effect through Adam's second-moment pathway: in the tested regime, projection induces a 1/(1-alpha) inflation of the old-direction effective learning rate, matching measurements within 8% across eight alpha values. The same conflict appears with penalty methods, replay mixing, and at 7B scale under LoRA. Our fix routes the modified gradient only to the first moment while preserving magnitude-faithful second-moment statistics, with overlap-aware adaptive strength. This simple change is the only tested configuration that consistently avoids collapse across methods, optimizers, and scale.

Yuelin Hu, Zhenbo Yu, Zhengxue Cheng +2

Shanghai Jiao Tong University · Shanghai Maritime University

Continual Learning for LLMs Adaptive Gradient Methods Catastrophic Forgetting

Apr 19, 2026cs.LG

A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad, Shampoo and Muo

A unified framework for first-order optimization algorithms fornonconvex unconstrained optimization is proposed that uses adaptivelypreconditioned gradients and includes popular methods such as full anddiagonal AdaGrad, AdaNorm, as well as adpative variants of Shampoo andMuon. This framework also allows combining heterogeneous geometriesacross different groups of variables while preserving a unifiedconvergence analysis. A fully stochastic global rate-of-convergenceanalysis is conducted for all methods in the framework, with andwithout two types of momentum, using reasonable assumptions on thevariance of the gradient oracle and without assuming boundedstochastic gradients or small enough stepsize.

S. Gratton, Ph. L. Toint

Université de Toulouse, INP, IRIT, Toulouse, France. · Université de Toulouse, INP, IRIT, Toulouse, France, and NAXYS, University of Namur, Namur, Belgium.

Momentum Methods Nonconvex Stochastic Optimization Stochastic Optimization Convergence Adaptive Gradient Methods

Mar 19, 2026cs.LG

Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method

The adaptive moment estimation (Adam) optimizer proposed by Kingma & Ba (2014) is presumably the most popular stochastic gradient descent (SGD) optimization method for the training of deep neural networks (DNNs) in artificial intelligence (AI) systems. Despite its groundbreaking success in the training of AI systems, it still remains an open research problem to provide a complete error analysis of Adam, not only for optimizing DNNs but even when applied to strongly convex stochastic optimization problems (SOPs). Previous error analysis results for strongly convex SOPs in the literature provide conditional convergence analyses that rely on the assumption that Adam does not diverge to infinity but remains uniformly bounded. It is the key contribution of this work to establish uniform a priori bounds for Adam and, thereby, to provide -- for the first time -- an unconditional error analysis for Adam for a large class of strongly convex SOPs.

Steffen Dereich, Thang Do, Arnulf Jentzen

Institute for Mathematical Stochastics, Faculty of Mathematics and Computer Science, University of Münster, Germany · School of Data Science, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen), China · Department of Probability and Statistic, Institute of Mathematics, Vietnam Academy of Science and Technology, Vietnam +2

Stochastic Optimization Stochastic Optimization Convergence Adaptive Gradient Methods Stochastic Gradient Descent

Jan 30, 2026cs.LG

Why GRPO Needs Normalization: A Local-Curvature Perspective on Adaptive Gradients

Reinforcement learning (RL) has become a key driver of language model reasoning. Among RL algorithms, Group Relative Policy Optimization (GRPO) is the de facto standard, avoiding the need for a critic by using per-prompt baselines and variance normalization. Yet why and when this normalization helps remains unclear. In this work, we answer both questions through the lens of local curvature of the sequence-level policy gradient: standard deviation normalization implements an adaptive gradient. Theoretically, we prove that under mild conditions GRPO attains an improved convergence rate over unnormalized REINFORCE, with gains characterized by the average within-prompt reward standard deviation across prompts and iterations. We further introduce IS-GRPO, an importance-sampling variant of GRPO whose expected update remains aligned with the full gradient, and prove a convergence guarantee of the same form as REINFORCE's, which is tighter in practice during real training runs by a factor we measure. Empirically, on GSM8K and MATH we validate the curvature--variance link and measure the quantities that govern our bounds along real training runs. At 1.5B scale, per-prompt normalization outperforms both unnormalized and globally normalized baselines, with gains that emerge in a middle phase of training where per-prompt variances become heterogeneous. At 7B scale, the normalization schemes become statistically indistinguishable while IS-GRPO retains a small lead, in line with the theory's prediction that the gains require heterogeneity.

Cheng Ge, Caitlyn Heqi Yin, Hao Liang +1

Department of Aeronautics and Astronautics, MIT · Department of Statistics, University of Wisconsin–Madison · Department of Informatics, King’s College London +1

RL for Language Model Reasoning Group Relative Policy Optimization Adaptive Gradient Methods

Jan 29, 2026cs.LG

Why $β_1 = β_2$ Is Dynamically Special in Adam

Adam has been at the core of large-scale training for almost a decade, yet the role of its two momentum parameters remains poorly understood. Recent work shows that tying

β_{1}=β_{2}

can preserve Adam's strong performance despite collapsing two memory scales into one, raising a basic question: what becomes dynamically special when the memories are tied? We identify a concrete mechanism. In the continuous-time limit, each normalized-update coordinate decomposes into a sign component, an explicit magnitude-lag term proportional to the difference between the two memory times, and additional transition, curvature, and nonlinear ratio terms. This lag channel vanishes exactly when

β_{1}=β_{2}

, making the diagonal the unique regime in which this mismatch-induced response is structurally absent. A full-history discrete decomposition on real training gradients recovers this change in composition: tied updates are sign-dominated, whereas the lag term becomes substantial off the diagonal and leaves a comparatively small residual. Across six vision and language tasks, tied configurations also typically exhibit smoother update-norm trajectories. Overall, our results identify memory-scale mismatch as a concrete source of magnitude sensitivity in Adam and provide a mechanistic account of why tied momentum is dynamically distinctive.

Alberto Fernández-Hernández, Cristian Pérez-Corral, Jose I. Mestre +2

Universitat Politècnica de València Valencia, Spain · Universitat Jaume I Castelló de la Plana, Spain

Momentum Methods Adaptive Gradient Methods Neural Network Training Dynamics Gradient Descent Dynamics

Jan 21, 2026cs.LG

ZENITH: Automated Gradient Norm Informed Stochastic Optimization

Training deep computer vision models requires manual oversight or hyperparameter tuning of the learning rate (LR) schedule. While existing adaptive optimizers schedule the LR automatically, they suffer from computational and memory overhead, incompatibility with regularization, and suboptimal LR choices. In this work, we introduce the ZENITH (Zero-overhead Evolution using Norm-Informed Training History) optimizer, which adapts the LR using the temporal evolution of the gradient norm. Image classification experiments spanning 6 CNN architectures and 6 benchmarks demonstrate that ZENITH achieves higher test accuracy in lower wall-clock time than baselines. It also yielded superior mAP in object detection, keypoint detection, and instance segmentation on MS COCO using the R-CNN family of models. Furthermore, its compatibility with regularization enables even better generalization.

Dhrubo Saha

Georgia Institute of Technology

Deep Learning Optimization Learning Rate Scheduling Adaptive Gradient Methods Stochastic Gradient Descent

Nov 10, 2025math.OC

Adam symmetry theorem: characterization of the convergence of the stochastic Adam optimizer

Beside the standard stochastic gradient descent (SGD) method, the Adam optimizer due to Kingma & Ba (2014) is currently probably the best-known optimization method for the training of deep neural networks in artificial intelligence (AI) systems. Despite the popularity and the success of Adam it remains an \emph{open research problem} to provide a rigorous convergence analysis for Adam even for the class of strongly convex SOPs. In one of the main results of this work we establish convergence rates for Adam in terms of the number of gradient steps (convergence rate \nicefrac{1}{2} w.r.t. the size of the learning rate), the size of the mini-batches (convergence rate 1 w.r.t. the size of the mini-batches), and the size of the second moment parameter of Adam (convergence rate 1 w.r.t. the distance of the second moment parameter to 1) for the class of strongly convex SOPs. In a further main result of this work, which we refer to as \emph{Adam symmetry theorem}, we illustrate the optimality of the established convergence rates by proving for a special class of simple quadratic strongly convex SOPs that Adam converges as the number of gradient steps increases to infinity to the solution of the SOP (the unique minimizer of the strongly convex objective function) if and \emph{only} if the random variables in the SOP (the data in the SOP) are \emph{symmetrically distributed}. In particular, in the standard case where the random variables in the SOP are not symmetrically distributed we \emph{disprove} that Adam converges to the minimizer of the SOP as the number of Adam steps increases to infinity. We also complement the conclusions of our convergence analysis and the Adam symmetry theorem by several numerical simulations that indicate the sharpness of the established convergence rates and that illustrate the practical appearance of the phenomena revealed in the \emph{Adam symmetry theorem}.

Steffen Dereich, Thang Do, Arnulf Jentzen +1

Institute for Mathematical Stochastics, Faculty of Mathematics and Computer Science, University of Münster, Germany · School of Data Science, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen), China · School of Data Science and School of Artificial Intelligence, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen), China +1

Stochastic Optimization Stochastic Optimization Convergence Adaptive Gradient Methods

Sep 3, 2025stat.ML

Understanding and Improving Shampoo and SOAP via Kullback-Leibler Minimization

Shampoo and its efficient variant, SOAP, employ structured second-moment estimations and have shown strong performance for training neural networks (NNs). In practice, however, Shampoo typically requires step-size grafting with Adam to be competitive, and SOAP mitigates this by applying Adam in Shampoo's eigenbasis -- at the cost of additional memory overhead from Adam in both methods. Prior analyses have largely relied on the Frobenius norm to motivate these estimation schemes. We instead recast their estimation procedures as covariance estimation under Kullback-Leibler (KL) divergence minimization, revealing a previously overlooked theoretical limitation and motivating principled redesigns. Building on this perspective, we develop

\textbf{KL-Shampoo}

and

\textbf{KL-SOAP}

, practical schemes that match or exceed the performance of Shampoo and SOAP in NN pre-training while achieving SOAP-level per-iteration runtime. Notably, KL-Shampoo does not rely on Adam to attain competitive performance, eliminating the memory overhead introduced by Adam. Across our experiments, KL-Shampoo consistently outperforms SOAP, Shampoo, and even KL-SOAP, establishing the KL-based approach as a promising foundation for designing structured methods in NN optimization. An implementation of KL-Shampoo/KL-SOAP is available at https://github.com/yorkerlin/KL-Methods

Wu Lin, Scott C. Lowe, Felix Dangel +3

1Vector Institute, Toronto, Canada · University of Cambridge, Cambridge, United Kingdom · 3Microsoft, United States +1

Deep Learning Optimization Neural Network Optimization Second-Order Optimization Adaptive Gradient Methods

May 17, 2025cs.LG

On the $O(\frac{\sqrt{d}}{K^{1/4}})$ Convergence Rate of AdamW Measured by $\ell_1$ Norm

As the default optimizer for training large language models, AdamW has achieved remarkable success in deep learning. However, its convergence behavior is not theoretically well-understood. This paper establishes the convergence rate

\frac{1}{K}\sum_{k=1}^KE\left[||\nabla f(x^k)||_1\right]\leq O(\frac{\sqrt{d}C}{K^{1/4}})

for AdamW measured by

\ell_1

norm, where

K

represents the iteration number,

d

denotes the model dimension, and

C

matches the constant in the optimal convergence rate of SGD. Theoretically, we have

||\nabla f(x)||_2\ll ||\nabla f(x)||_1\leq \sqrt{d}||\nabla f(x)||_2

for any high-dimensional vector

x

and

E\left[||\nabla f(x)||_1\right]\geq\sqrt{\frac{2d}π}E\left[||\nabla f(x)||_2\right]

when each element of

\nabla f(x)

is generated from Gaussian distribution

\mathcal N(0,1)

. Empirically, our experimental results on real-world deep learning tasks reveal

||\nabla f(x)||_1=\varTheta(\sqrt{d})||\nabla f(x)||_2

. Both support that our convergence rate can be considered to be analogous to the optimal

\frac{1}{K}\sum_{k=1}^KE\left[||\nabla f(x^k)||_2\right]\leq O(\frac{C}{K^{1/4}})

convergence rate of SGD in the ideal case. We also extend our result to NAdamW, an AdamW variant that employs a double-momentum mechanism, and demonstrate that it maintains the same convergence rate.

Huan Li, Yiming Dong, Zhouchen Lin

Institute of Robotics and Automatic Information Systems, College of Artificial Intelligence, Nankai University, Tianjin, China. · National Key Lab of General AI, School of Intelligence Science and Technology, Peking University, Beijing, China.

Optimization Convergence Analysis Stochastic Optimization Convergence Adaptive Gradient Methods

Jun 20, 2024math.OC

Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses

The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (particularly, in the situation of constant learning rates). In practice, human-tuned deterministic learning rate schedules or small constant learning rates are often used, and implementations in machine learning frameworks like Tensorflow and Pytorch typically employ constant learning rates. We propose a learning-rate-adaptive approach for SGD methods, adjusting the learning rate based on empirical estimates for the objective function values. Specifically, we propose a learning-rate-adaptive variant of the Adam optimizer and implement it for several machine learning problems, including deep learning methods for partial differential equations such as deep Kolmogorov methods, physics-informed neural networks, and deep Ritz methods. We refer to https://github.com/deeplearningmethods/adaptive-learning-rate for the Python source codes for the numerical simulations in this work. Our results show that the proposed adaptive Adam variant achieves faster reductions of the objective function value compared to Adam with default learning rates. For certain quadratic minimization problems, we rigorously prove that an adaptive SGD variant converges to the global minimizer. This proof uses properties of invariant measures of the SGD dynamics and a generalized convergence analysis for SGD with random predictable learning rates which we develop in this work.

Steffen Dereich, Arnulf Jentzen, Adrian Riekert

Institute for Mathematical Stochastics, University of Münster, Germany · School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen), China · Applied Mathematics: Institute for Analysis and Numerics, University of Münster, Germany

Stochastic Optimization Convergence Adaptive Gradient Methods Stochastic Gradient Descent

May 26, 2023cs.LG

Generalizing Adam to Manifolds for Efficiently Training Transformers

One of the primary reasons behind the success of neural networks has been the emergence of an array of new, highly-successful optimizers, perhaps most importantly the Adam optimizer. It is widely used for training neural networks, yet notoriously hard to interpret. Lacking a clear physical intuition, Adam is difficult to generalize to manifolds. Some attempts have been made to directly apply parts of the Adam algorithm to manifolds or to find an underlying structure, but a full generalization has remained elusive. In this work a new approach is presented that leverages the special structure of the manifolds which are relevant for optimization of neural networks, such as the Stiefel manifold, the symplectic Stiefel manifold and the Grassmann manifold: all of these are homogeneous spaces and as such admit a global tangent space representation. This is a common vector space, often called the Lie subspace, that makes the generalization of all steps in the Adam optimizer (as well as other optimizers) possible. It is thus possible to extend the Adam optimizer to manifolds without a projection step, something that was not possible before. The resulting algorithm is then applied to train transformers and a symplectic autoencoder for which orthogonality constraints are enforced up to machine precision and we conclusively demonstrate the advantage of the proposed optimizer over existing methods.

Benedikt Brantner

Max-Planck-Institut für Plasmaphysik Boltzmannstraße 2, 85748 Garching, Deutschland · Technische Universität München, Zentrum Mathematik Boltzmannstraße 3, 85748 Garching, Deutschland

Stiefel Manifold Optimization Riemannian Optimization Adaptive Gradient Methods

Jan 26, 2022cs.CR

Privacy-Preserving Logistic Regression Training with A Faster Gradient Variant

Training logistic regression over encrypted data has emerged as a prominent approach to addressing security concerns in recent years. In this paper, we introduce an efficient gradient variant, termed the \textit{quadratic gradient}, which is specifically designed for privacy-preserving logistic regression while remaining equally effective in plaintext optimization. By incorporating this quadratic gradient, we enhance Nesterov's Accelerated Gradient (NAG), Adaptive Gradient (AdaGrad), and Adam algorithms. We evaluate these enhanced algorithms across various datasets, with experimental results demonstrating state-of-the-art convergence rates that significantly outperform traditional first-order gradient methods. Furthermore, we apply the enhanced NAG method to implement homomorphic logistic regression training, achieving comparable performance within only four iterations. The proposed quadratic-gradient approach offers a unified framework that synergizes the advantages of first-order gradient methods and second-order Newton-type methods, suggesting broad applicability to diverse numerical optimization tasks.

John Chiang

Logistic Regression Gradient Descent Privacy-Preserving ML Adaptive Gradient Methods Accelerated Gradient Methods

Date pendingstat.ML

Adapt or Forget: Provable Tradeoffs Between Adam and SGD in Nonstationary Optimization

We provide a theoretical analysis of Adam under non-stationary stochastic objectives, separating two regimes: Euclidean tracking under adaptive strong monotonicity of the Adam-preconditioned mean-gradient operator, and high-probability projected stationarity guarantees under general

L

-smooth objectives. In the tracking regime, we derive finite-time expected and high-probability bounds that decompose sharply into four components: initialization, objective drift, a first-moment tracking error governed by

\beta_1

, and a preconditioner perturbation governed by

\beta_2

. We characterize the burn-in time required for the transient terms to decay to the asymptotic tracking bound under constant and step-decay schedules. We also prove a high-probability bound on the average projected stationarity gap for Adam under distribution shift. Across both analyses, our bounds reveal a noise--drift tradeoff: in noise-dominated regimes, first-moment averaging and adaptive preconditioning can yield favorable upper guarantees, whereas in drift-dominated regimes, stale first-moment information and preconditioner perturbations can enlarge Adam's tracking guarantee, potentially allowing vanilla SGD to attain a smaller tracking error. Our explicit

(\beta_1,\beta_2,\epsilon)

-dependent bounds identify mechanisms through which adaptive step-sizing can help or hurt under nonstationarity and provide theoretical explanations consistent with Adam's empirical instability and stabilization under distribution shift.

Sharan Sahu, Abir Sarkar, Cameron J. Hogan +1

Department of Statistics and Data Science Cornell University

Stochastic Optimization Stochastic Optimization Convergence Adaptive Gradient Methods Stochastic Gradient Descent