Evolution of Optimization Methods: Algorithms, Scenarios, and Evaluations

Tong Zhang, Jiangning Zhang†, Zhucun Xue, Juntao Jiang, Yicheng Xu, Chengming Xu, Teng Hu, Xingyu Xie, Xiaobin Hu†, Yabiao Wang, Yong Liu†, Shuicheng Yan

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📖 Overview

Foundational optimization algorithms are the core driving force behind deep learning, evolving from early stochastic gradient descent (SGD) to the widely adopted Adam family. However, as the scale of modern foundation models grows massively, this optimization paradigm is forced to expand, encountering new physical and systemic bottlenecks during large-scale training. In particular, stringent differential privacy requirements and distributed training paradigms have exposed critical limitations of conventional approaches regarding privacy protection and memory efficiency.

However, existing reviews on optimization algorithms often focus on narrow technical fields, e.g., first-order and second-order, lacking a comprehensive perspective on the field's evolution, especially regarding Zeroth-order and Scenario-oriented paradigms.

To address these gaps, this survey provides a systematic review of the development of optimization algorithmss, tracing its evolution through four major paradigms:

First-order methods → Second-order methods → Zeroth-order methods → Scenario-oriented paradigms

We conduct comprehensive theoretical analysis and standardized empirical evaluations, objectively pointing out the pros, cons, and fundamental design trade-offs of various methods across different architectures. By synthesizing theoretical insights with extensive empirical evidence, we distill key developmental trends and provides actionable guidance and future research directions for designing next-generation efficient, robust, and trustworthy optimization algorithms.

🎯Contributions

1️⃣ Unified Taxonomy: Establishing a rigorous mathematical taxonomy that unifies disparate conceptual definitions across fundamental optimization primitives.

• 📊 Evolutionary Trajectory tracing the development of foundational algorithms from First-Order to Second-Order and Zeroth-Order methods.
• 🔬 Intrinsic Connections clarifying the complex evolutionary logic and structural relationships between different optimization approaches to provide a coherent framework for the field.

2️⃣ Scenario-Oriented Analysis: Demonstrating how foundational algorithms are fundamentally re-architected into scenario-oriented paradigms to address severe physical bottlenecks.

• 📈 Systems-Aware Engineering highlighting the critical shift from pure algorithmic design to practical solutions that balance theoretical guarantees with strict engineering constraints.
• 🔍 Overcoming Systemic Barriers detailing how these paradigms tackle specific, real-world challenges such as distributed communication barriers and strict differential privacy constraints.

3️⃣ Standardized Evaluation: Introducing a rigorously controlled evaluation framework that strictly separates pure algorithmic performance from large-scale engineering optimizations.

• 🚀 Extensive Benchmarking developing a standardized testbed to evaluate 23 distinct optimizers across diverse architectural proxies, including CNN and Transformer-based models.
• 🔮 Strategic Insights systematically isolating and examining learning rate sensitivity, long-term training scalability, and cross-architecture generalization to guide the design of next-generation optimizers.

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📈Evolution

📈 A Comprehensive Analysis of Optimization Methods: This figure systematically summarizes the development trends and core characteristics of optimization methodologies across different orders.

• Key Insights:Attention to optimization algorithms experienced a sharp increase since 2024. This explosive growth is closely tied to the rapid development of massive models, with first-order methods maintaining a dominant position.

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🚩Timeline

Timeline of prominent optimization algorithms. The evolution highlights key algorithmic milestones, associated research institutions, and publication venues over time.

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🏗️ Architecture Overview

🗂️ A Comprehensive Taxonomy of Optimization Algorithms

📐 Taxonomy Overview: This framework categorizes existing works based on three dominant paradigms, First-Order Methods, Second-Order Methods, and Zeroth-Order Methods, and further structures them according to their fundamental mathematical principles and evolutionary development. Key branches include:

• 🚀 First-Order Methods: Gradient-Driven (e.g., SGD) $\rightarrow$ Adaptive Learning Rate (e.g., Adam) $\rightarrow$ Acceleration to Automation (e.g., Adan, Nadam) $\rightarrow$ Scalar to Preconditioner (e.g., Shampoo) $\rightarrow$ Stability to Temporal (e.g., SPAM) $\rightarrow$ Temporal to Geometry (e.g., SAM).
• ⚙️ Second-Order Methods: Deterministic Curvature to Geometry (e.g., K-FAC, AdaFisher) $\rightarrow$ Approximation to Iterative Update (e.g., ADAHESSIAN).
• 📍 Zeroth-Order Methods: Perturbation Optimization (e.g., FZOO, LeZO) $\rightarrow$ Adaptive to Resource-Aware (e.g., MeZO, ZO-AdaMM) $\rightarrow$ Variance Reduction to Adaptive (e.g., MeZO-SVRG).

📊 Benchmark Evaluation Results on Vision Tasks

📈 Benchmark Evaluation: This comprehensive assessment evaluates 23 representative optimization algorithms across continuous vision architectures (ViT-S and ResNet-50) and varying training horizons:

• ⏱️ Short-Term Convergence (100 Epochs): Evaluates the rapid descent capability and initial exploration efficiency of optimizers within a constrained computational budget.
• 🏃 Long-Term Scalability (300 Epochs): Assesses the algorithm's resilience against late-stage gradient noise and its capacity to continually extract representational power over extended cycles.
• 📊 Ranking Dynamics: Tracks relative performance shifts across epochs, highlighting how algorithms dynamically navigate the trade-off between early acceleration and long-term stability.

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🔥Add Your Paper in our Survey!!!!!

• You are welcome to give us an issue or PR for your optimizer work !!!!!

Note that: Due to the huge paper in arXiv, we are sorry to cover all in our survey. You can directly present a PR into this repo and we will record it for next version update of our survey.

🔥New

• [2026.06.12] We update GitHub to record the available paper by the end of 2026/6/12.

🌟Welcome everyone to follow and join the Scaling Opt community: Scaling Opt Community

• Algorithm Visualizations: Includes visualization scripts for the Rosenbrock Function and Rastrigin Function, allowing users to freely explore optimization behaviors.

• Performance Benchmarks: We primarily recommend benchmarks based on Algoperf, along with other benchmark suites and analysis articles for validating and comparing state-of-the-art optimizers.

• Papers & Blogs Recommendations: The platform curates high-quality papers and blog posts from recent years, continuously updated with the latest daily arXiv publications. Currently, the collection contains nearly one hundred resources.

• Tutorials Sharing: The platform gathers high-quality community resources and is actively developing a tutorial series titled From Classical to Modern Optimizers.

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🔨Installation

To reproduce our benchmarks, you need to clone this repository and install the required dependencies. We strongly recommend using a virtual environment (e.g., Conda).

# 1. Clone the repository
git clone https://github.com/JZhangTon/awesome-optimizer.git
cd awesome-optimizer

# 2. Install required packages
pip install -r requirements.txt

⚙️Usage & 📈Benchmarking

• To start a training run and reproduce our benchmark results, you can execute the provided training scripts. We provide a script for easy benchmarking. See examples/benchmark to see how to use it.

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🗂️Taxonomy of Optimization Algorithms

• 🚀 First-Order Algorithms
• ⚙️ Second-Order Algorithms
• 📍 Zeroth-Order Algorithms
• 🌐 Distributed Optimization
• 🛡️ Privacy-Preserving Optimization
• ⚡ Memory-Efficient Optimization
• 🧩 Tailored Optimization Approaches

🚀 First-Order Algorithms

Abbreviation	Venue & Year	Paper Title	Project	Sub-methods	Fine-grained Methods
Pion		Rethinking Muon Beyond Pretraining: Spectral Failures and High-Pass Remedies for VLA and RLVR	Link	Spectral High Pass	Matrix Orthogonalization
C-Adam		A Theoretical and Experimental Study of a Novel Adaptive Learning Algorithm	Link	Adaptive Learning Rate Methods	Line-of-Sight Adam Variant
SparseOpt		SparseOpt: Addressing Normalization-induced Gradient Skew in Sparse Training	Link	Gradient Normalization & Clipping	Sparse Training Gradient Rebalancing
Anon		Anon: Extrapolating Optimizer Adaptivity Across the Real Spectrum	Link	Adaptive Learning Rate Methods
PS-Clip-SGD		Robust and Fast Training via Per-Sample Clipping	Link	Enhancing Training Stability
Nora		Nora: Normalized Orthogonal Row Alignment for Scalable Matrix Optimizer	Link	Adaptive Step-Size Control	Matrix Orthogonalization
Muon^2		Muon^2: Boosting Muon via Adaptive Second-Moment Preconditioning	Link	Adaptive Step-Size Control	Matrix Orthogonalization
HomeAdam		HomeAdam: Adam and AdamW Algorithms Sometimes Go Home to Obtain Better Provable Generalization	Link	Adaptive Step-Size Control	Second-order moment adaptation
FlashOptim		FlashOptim: Optimizers for Memory-Efficient Training	Link	Memory-Efficient Optimization	Low-Memory Optimizer Design
FANoS		FANoS: Friction-Adaptive Nos´e–Hoover Symplectic Momentum for Stiff Objectives	Link	Accelerating Convergence Rate	Momentum Damping Mechanism
NOVAK		NOVAK: Unified adaptive optimizer for deep neural networks	Link	Hybrid Methods	Gradient Smoothing Hybrid
AdamNX		AdamNX: An Adam improvement algorithm based on a novel exponential decay mechanism for the second-order moment estimate	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
ROOT		ROOT: Robust Orthogonalized Optimizer for Neural Network Training	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
AuON		AuON: A Linear-time Alternative to Orthogonal Momentum Updates	Link	Gradient Normalization & Clipping	Layer-Wise Gradient Normalization
ZetA		ZETA: A HYBRID OPTIMIZER COMBINING RIEMANN ZETA SCALING WITH ADAM FOR ROBUST DEEP LEARNING	Link	Hybrid Methods	Multi-Objective Hybrid
NIRMAL		COMPARATIVE ANALYSIS OF NOVEL NIRMAL OPTIMIZER AGAINST ADAM AND SGD WITH MOMENTUM	Link	Hybrid Methods	Multi-Objective Hybrid
SCSAdamW		Beyond First-Order: Training LLMs with Stochastic Conjugate Subgradients and AdamW	Link	Loss Landscape Optimization	Momentum Landscape Adaptation
adaNPAG		Boosting Accelerated Proximal Gradient Method with Adaptive Sampling for Stochastic Composite Optimization *	Link	Momentum-Enhanced SGD	Accelerated Momentum
SoftSignSGD		SoftSignSGD(S3): An Enhanced Optimizer for Practical DNN Training and Loss Spikes Minimization Beyond Adam	Link	Adaptive Learning Rate Methods	Hybrid Adaptive Strategy
AdaMuon		ADAMUON: ADAPTIVE MUON OPTIMIZER	Link	Adaptive Learning Rate Methods	Hybrid Adaptive Strategy
Accelerated GRAAL		NESTEROV FINDS GRAAL: OPTIMAL AND ADAPTIVE GRADIENT METHOD FOR CONVEX OPTIMIZATION	Link	Hybrid Methods	Multi-Objective Hybrid
DEO		Dimer-Enhanced Optimization: A First-Order Approach to Escaping Saddle Points in Neural Network Training	Link	Loss Landscape Optimization	Curvature-Guided Landscape Exploration
LyAm		LyAm: Robust Non-Convex Optimization for Stable Learning in Noisy Environments	Link	Learning Rate Scheduling	Stability-Aware Adaptive Scheduling
Splus		A Stable Whitening Optimizer for Efficient Neural Network Training	Link	Preconditioned Gradient Methods	Two Metrics' Preconditioner
HGM		Hindsight-Guided Momentum (HGM) Optimizer: An Approach to Adaptive Learning Rates	Link	Learning Rate Scheduling	Gradient Angle Scheduling
AutoSGD		AutoSGD: Automatic Learning Rate Selection for Stochastic Gradient Descent	Link	Learning Rate Scheduling	Scheduler-Free Adaptation
AdamS		AdamS: Momentum Itself Can Be A Normalizer for LLM Pretraining and Post-training	Link	Adaptive Learning Rate Methods	Stateless Adaptation
LightSAM		LightSAM: Parameter-Agnostic Sharpness-Aware Minimization	Link	Loss Landscape Optimization	Sharpness-Aware Minimization (SAM)
ADAGB2		Fast Stochastic Second-Order Adagrad for Nonconvex Bound-Constrained Optimization	Link	Hybrid Methods	Projection Gradient Hybrid
VRAdam		A Physics-Inspired Optimizer: Velocity Regularized Adam	Link	Momentum-Enhanced SGD	Momentum Damping Mechanism
SKA-SGD		STREAMING KRYLOV-ACCELERATED STOCHASTIC GRADIENT DESCENT	Link	Loss Landscape Optimization	Curvature-Guided Landscape Exploration
Adam-Power		GradPower: Powering Gradients for Faster Language Model Pre-Training	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
AlphaGrad		AlphaGrad: Non-Linear Gradient Normalization Optimizer	Link	Adaptive Learning Rate Methods； Low-Memory Optimizer Design； Stateless Optimization Methods	Stateless Adaptation； Structural Redesign； Parameter Characteristic-Driven Updates
AsyncSAM		ASYNCHRONOUS SHARPNESS-AWARE MINIMIZATION FOR FAST AND ACCURATE DEEP LEARNING	Link	Loss Landscape Optimization	Sharpness-Aware Minimization (SAM)
ASGO		ASGO: Adaptive Structured Gradient Optimization	Link	Preconditioned Gradient Methods	Single Metric's Preconditioner
AdaGC		AdaGC: Improving Training Stability for Large Language Model Pretraining	Link	Gradient Normalization & Clipping； Robust Optimization	Noise-Robust Normalization； Dynamic Gradient Clipping； Noise-Robust Gradients
Adadiag		Improving Adaptive Moment Optimization viaPreconditioner Diagonalization	Link	Hybrid Methods	Projection Gradient Hybrid
eagle		EAGLE: EARLY APPROXIMATED-GRADIENT-BASED LEARNING RATE ESTIMATOR	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
Hessian-aware Scaling		First-ish Order Methods: Hessian-aware Scalings of Gradient Descent	Link	Preconditioned Gradient Methods	Single Metric's Preconditioner
GCSAM		GCSAM: Gradient Centralized Sharpness Aware Minimization	Link	Gradient Normalization & Clipping	Mean-Removal Normalization
SGDO		Overshoot: Taking advantage of future gradients in momentum-based stochastic optimization	Link	Momentum-Enhanced SGD	Accelerated Momentum
μ²-SGD		DO STOCHASTIC, FEEL NOISELESS: STABLE STOCHASTIC OPTIMIZATION VIA A DOUBLE MOMENTUM MECHANISM	Link	Momentum-Enhanced SGD	Double-momentum mechanism
Stable-SPAM		Stable-SPAM: How to Train in 4-Bit More Stably than 16-Bit Adam	Link	Gradient Normalization & Clipping； Privacy-Aware Gradient Clipping	Layer-Wise Gradient Normalization； Dynamic Gradient Clipping； Adaptive Clipping
apollo		APOLLO:SGD-LIKE MEMORY, ADAMW-LEVEL PERFORMANCE	Link	Adaptive Learning Rate Methods； Towards LLM Traning	Hybrid Adaptive Strategy； Gradient Projection Mechanism
SPAM		SPAM: SPIKE-AWARE ADAM WITH MOMENTUM RESET FOR STABLE LLM TRAINING	Link	Gradient Normalization & Clipping； Optimizer State Compression	Element-Wise Gradient Scaling； Spike-Aware Gradient Clipping； Sparse State Compression
Coupled Adam		Better Embeddings with Coupled Adam	Link	Adaptive Learning Rate Methods	Layer-Wise Adaptation
SWAN		SWAN: SGD WITH NORMALIZATION AND WHITENING ENABLES STATELESS LLM TRAINING	Link	Towards LLM Traning	Gradient Preconditioning Mechanism
LDAdam		LDADAM: ADAPTIVE OPTIMIZATION FROM LOWDIMENSIONAL GRADIENT STATISTICS	Link	Hybrid Methods	Projection Gradient Hybrid
SSAM		Stabilizing Sharpness-aware Minimization Through A Simple Renormalization Strategy	Link	Loss Landscape Optimization	Renormalized Gradient Norm SAM
MARS		MARS: Unleashing the Power of Variance Reduction for Training Large Models	Link	Momentum-Enhanced SGD； Adaptive Learning Rate Methods	Double-momentum mechanism；Momentum-based Adaptive
VSGD		Variational Stochastic Gradient Descent for Deep Neural Networks	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
MIAdam		A Method for Enhancing Generalization of Adam by Multiple Integrations	Link	Loss Landscape Optimization	Curvature-Guided Landscape Exploration
KOALA++		KOALA++: Efficient Kalman-Based Optimization with Gradient-Covariance Products	Link	Adaptive Learning Rate Methods	Kalman filtering based
PAdamP		ADAPTIVE MOMENT ESTIMATION OPTIMIZATION ALGORITHM USING PROJECTION GRADIENT FOR DEEP LEARNING	Link	Hybrid Methods	Projection Gradient Hybrid
DecGD		A New Adaptive Gradient Method with Gradient Decomposition	Link	Learning Rate Scheduling	Loss-Sensitive Scheduling
Grams		Grams: Gradient Descent with Adaptive Momentum Scaling	Link	Hybrid Methods	Multi-Objective Hybrid
FSGDM		ON THE PERFORMANCE ANALYSIS OF MOMENTUM METHOD: A FREQUENCY DOMAIN PERSPECTIVE	Link	Momentum-Enhanced SGD	Frequency Domain Momentum Analysis
AdEMAMix		THE ADEMAMIX OPTIMIZER:BETTER, FASTER, OLDER	Link	Momentum-Enhanced SGD	Double-momentum mechanism
HVAdam		HVAdam: A Full-Dimension Adaptive Optimizer	Link	Hybrid Methods	Projection Gradient Hybrid
SGD-SaI		No More Adam: Learning Rate Scaling at Initialization is All You Need	Link	Learning Rate Scheduling； Stateless Optimization Methods	Initial Learning Rate Scaling； Parameter Characteristic-Driven Updates
Adam++		Towards Simple and Provable Parameter-Free Adaptive Gradient Methods	Link	Learning Rate Scheduling	Scheduler-Free Adaptation
EXADAM		EXADAM: THE POWER OF ADAPTIVE CROSS-MOMENTS	Link	Adaptive Learning Rate Methods	Hybrid Adaptive Strategy
Cautious Optimizers		Cautious Optimizers: Improving Training with One Line of Code	Link	Momentum-Enhanced SGD	Momentum-Gradient Alignment
AGS-GD		Anisotropic Gaussian Smoothing for Gradient-based Optimization	Link	Hybrid Methods； Auto-Designed Optimizers	Gradient Smoothing Hybrid； Automated Discovery&Theoretical Derivation
CAdam		CAdam: Confidence-Based Optimization for Online Learning	Link	Hybrid Methods	Multi-Objective Hybrid
INNAprop		A SECOND-ORDER-LIKE OPTIMIZER WITH ADAPTIVE GRADIENT SCALING FOR DEEP LEARNING	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
CaAdam		CaAdam: Improving Adam optimizer using connection aware methods	Link	Adaptive Learning Rate Methods	Layer-Wise Adaptation
BADM		BADM: Batch ADMM for Deep Learning	Link	Hybrid Methods	Multi-Objective Hybrid
FAdam		FAdam: Adam is a natural gradient optimizer using diagonal empirical Fisher information	Link	Adaptive Learning Rate Methods； Hybrid Methods	Dynamic Epsilon Adjustment； Multi-Objective Hybrid
MSAM		Momentum-SAM: Sharpness Aware Minimization without Computational Overhead	Link	Loss Landscape Optimization	Momentum Landscape Adaptation
LOMO		Full Parameter Fine-tuning for Large Language Models with Limited Resources	Link	Towards LLM Traning； Memory-Efficient Fine-Tuning for Large Models	Real-Time Computation； Staless Fine-Tuning
BAdam		BAdam: A Memory Efficient Full Parameter Optimization Method for Large Language Models	Link	Hybrid Methods； Towards LLM Traning	Multi-Objective Hybrid； Real-Time Computation； Block-Wise Computation
DP-AdamBC		DP-AdamBC: Your DP-Adam Is Actually DP-SGD (Unless You Apply Bias Correction)	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
Dice-SGD		DIFFERENTIALLY PRIVATE SGD WITHOUT CLIPPING BIAS: AN ERROR-FEEDBACK APPROACH	Link	Gradient Normalization & Clipping	DP-enhanced Gradient Clipping
FESS-GDA		Stochastic Smoothed Gradient Descent Ascent for Federated Minimax Optimization	Link	Hybrid Methods	Gradient Filtering Hybrid
AdaSAM		AdaSAM: Boosting Sharpness-Aware Minimization with Adaptive Learning Rate and Momentum for Training Deep Neural Networks	Link	Hybrid Methods	Multi-Objective Hybrid
SAMPa		SAMPa: Sharpness-aware Minimization Parallelized	Link	Loss Landscape Optimization	Sharpness-Aware Minimization (SAM)
		Lookbehind-SAM: k steps back, 1 step forward	Link	Loss Landscape Optimization	Multi-Step Ascent SAM
F-SAM		Friendly Sharpness-Aware Minimization	Link	Loss Landscape Optimization	Noise Injection Enhancement
FGSAM		Fast Graph Sharpness-Aware Minimization for Enhancing and Accelerating Few-Shot Node Classification	Link	Loss Landscape Optimization	Noise Injection Enhancement
Adan		Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
4-bit shampoo		4-bit Shampoo for Memory-Efficient Network Training	Link	Preconditioned Gradient Methods； Low-Memory Optimizer Design	Two Metrics' Preconditioner； Compression&Approximation of States
Muon	Blog'24	Muon: An optimizer for hidden layers in neural networks	Link	Adaptive Learning Rate Methods	Layer-Wise Adaptation
ADOPT		ADOPT: Modified Adam Can Converge with Any β2 with the Optimal Rate	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
SET-adam		On Suppressing Range of Adaptive Stepsizes of Adam to Improve Generalisation Performance	Link	Adaptive Learning Rate Methods； Hybrid Methods	Second-Order Moment Adaptation； Multi-Objective Hybrid
Adam-Real		Adam on Local Time: Addressing Nonstationarity in RL with Relative Adam Timesteps	Link	Momentum-Enhanced SGD	Scheduled Momentum Reset
SNGM		Stochastic Normalized Gradient Descent with Momentum for Large-Batch Training	Link	Momentum-Enhanced SGD	Momentum Damping Mechanism
Schedule-Free		The Road Less Scheduled	Link	Learning Rate Scheduling	Scheduler-Free Adaptation
AUTODROP		AUTODROP: TRAINING DEEP LEARNING MODELS WITH AUTOMATIC LEARNING RATE DROP	Link	Adaptive Learning Rate Methods； Learning Rate Scheduling	Stateless Adaptation； Scheduler-Free Adaptation
ADAACT		AN ADAPTIVE METHOD STABILIZING ACTIVATIONS FOR ENHANCED GENERALIZATION	Link	Adaptive Learning Rate Methods； Optimizer State Compression	Neuron-Level Adaptation； State Sharing
MoMo		MoMo: Momentum Models for Adaptive Learning Rates	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
RSGDM		Reducing Bias in Deep Learning Optimization: The RSGDM Approach	Link	Momentum-Enhanced SGD	Accelerated Momentum
NYSACT		NYSACT: A SCALABLE PRECONDITIONED GRADIENT DESCENT USING NYSTRÖM APPROXIMATION	Link	Preconditioned Gradient Methods	Single Metric's Preconditioner
SGDF		Signal Processing Meets SGD: From Momentum to Filter	Link	Momentum-Enhanced SGD	Dynamic Momentum Weight
AdaLOMO		AdaLomo: Low-memory Optimization with Adaptive Learning Rate	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
		SGD with Large Step Sizes Learns Sparse Features	Link	Learning Rate Scheduling	Stability-Aware Adaptive Scheduling
look around		Lookaround Optimizer: k steps around, 1 step average	Link	Loss Landscape Optimization	Weight Averaging
GAM		Gradient Norm Aware Minimization Seeks First-Order Flatness and ImprovesGeneralization	Link	Loss Landscape Optimization	Curvature-Guided Landscape Exploration
AE-SAM		AN ADAPTIVE POLICY TO EMPLOY SHARPNESS-AWARE MINIMIZATION	Link	Loss Landscape Optimization	Sharpness-Aware Minimization (SAM)
Aida		A DNN Optimizer that Improves over AdaBelief by Suppression of the Adaptive Stepsize Range	Link	Adaptive Learning Rate Methods	Prediction Deviation Adaptation
Lion		Symbolic Discovery of Optimization Algorithms	Link	Adaptive Learning Rate Methods； Auto-Designed Optimizers	Momentum-based Adaptive; Automated Discovery&Theoretical Derivation
AdamMC		Moment Centralization based Gradient Descent Optimizers for Convolutional Neural Networks	Link	Gradient Normalization & Clipping	Mean-Removal Normalization
MultiAdam		MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks	Link	Gradient Normalization & Clipping	Layer-Wise Gradient Normalization
AdaNorm		AdaNorm: Adaptive Gradient Norm Correction based Optimizer for CNNs	Link	Gradient Normalization & Clipping； Robust Optimization	Element-Wise Gradient Scaling； Noise-Robust Normalization； Noise-Robust Gradients
AGD		AGD: an Auto-switchable Optimizer using Stepwise Gradient Difference for Preconditioning Matrix	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
RLEKF		RLEKF: An Optimizer for Deep Potential with Ab Initio Accuracy	Link	Adaptive Learning Rate Methods	Kalman filtering based
Amos		Amos: AN ADAM-STYLE OPTIMIZER WITH ADAPTIVE WEIGHT DECAY TOWARDS MODEL-ORIENTED SCALE	Link	Learning Rate Scheduling	Scheduler-Free Adaptation
AdaBFE		BFE and AdaBFE: A New Approach in Learning Rate Automation for Stochastic Optimization	Link	Learning Rate Scheduling； Stateless Optimization Methods	Gradient Angle Scheduling； Parameter Characteristic-Driven Updates
DP-SGD		Normalized/Clipped SGD with Perturbation for Differentially Private Non-Convex Optimization	Link	Gradient Normalization & Clipping	Basic Fixed Gradient Clipping
AdamFamily		AdaFamily: A family of Adam-like adaptive gradient methods	Link	Adaptive Learning Rate Methods	Hybrid Adaptive Strategy
SRSGD		Scheduled Restart Momentum for Accelerated Stochastic Gradient Descent	Link	Momentum-Enhanced SGD	Scheduled Momentum Reset
Step-Tuned SGD		Second-order step-size tuning of SGD for non-convex optimization	Link	Learning Rate Scheduling	Scheduler-Free Adaptation
AEGDM		AN ADAPTIVE GRADIENT METHOD WITH ENERGY AND MOMENTUM	Link	Adaptive Learning Rate Methods	Stateless Adaptation
AdaInject		AdaInject: Injection Based Adaptive Gradient Descent Optimizers for Convolutional Neural Networks	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
KOALA		KOALA: A Kalman Optimization Algorithm with Loss Adaptivity	Link	Adaptive Learning Rate Methods	Kalman filtering based
ESAM		EFFICIENT SHARPNESS-AWARE MINIMIZATION FOR IMPROVED TRAINING OF NEURAL NETWORKS	Link	Loss Landscape Optimization	Sharpness-Aware Minimization (SAM)
MADGRAD		Adaptivity without Compromise: A Momentumized, Adaptive, Dual Averaged Gradient Method for Stochastic Optimization	Link	Momentum-Enhanced SGD； Hybrid Methods	Accelerated Momentum； Multi-Objective Hybrid
GDA-AM		GDA-AM: On the effectiveness of solving minimax optimization via Anderson Acceleration	Link	Hybrid Methods	Multi-Objective Hybrid
AdamD		AdamD: Improved bias-correction in Adam	Link	Adaptive Learning Rate Methods	Bias Correction Rules Adaptaion
AdaL		AdaL: Adaptive Gradient Transformation Contributes to Convergences and Generalizations	Link	Adaptive Learning Rate Methods	Hybrid Adaptive Strategy
AngularGrad		AngularGrad: A New Optimization Technique for Angular Convergence of Neural Networks	Link	Momentum-Enhanced SGD	Momentum-Gradient Alignment
SGD-G2		Stochastic Runge-Kutta methods and adaptive SGD-G2 stochastic gradient descent	Link	Learning Rate Scheduling	Scheduler-Free Adaptation
SQuARM-SGD		SQuARM-SGD: Communication-Efficient Momentum SGD for Decentralized Optimization	Link	Momentum-Enhanced SGD； Local Update Strategies； Distributed Hybrid Optimization	Accelerated Momentum； Local SGD； Local Momentum Updates； Compression & Local Updates
SAM		Sharpness-Aware Minimization for Efficiently Improving Generalization	Link	Loss Landscape Optimization	Sharpness-Aware Minimization (SAM)
AvaGrad		Domain-independent Dominance of Adaptive Methods	Link	Adaptive Learning Rate Methods	Decoupled Learning Rate and Adaptability
Madam		MaxVA: Fast Adaptation of Step Sizes by Maximizing Observed Variance of Gradients	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
ACMo		ACMO: ANGLE-CALIBRATED MOMENT METHODS FOR STOCHASTIC OPTIMIZATION	Link	Learning Rate Scheduling	Gradient Angle Scheduling
AdamP / SGDP		AdamP: Slowing Down the Slowdown for Momentum Optimizers on Scale-invariant Weights	Link	Momentum-Enhanced SGD； Hybrid Methods	Momentum Damping Mechanism； Projection Gradient Hybrid
ACProp		Momentum Centering and Asynchronous Update for Adaptive Gradient Methods	Link	Adaptive Learning Rate Methods	Hybrid Adaptive Strategy
Adam+		Adam+: A Stochastic Method with Adaptive Variance Reduction	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
EAdam		EAdam Optimizer: How ∈ Impact Adam	Link	Adaptive Learning Rate Methods	Dynamic Epsilon Adjustment
AdaSGD		AdaSGD: Bridging the gap between SGD and Adam	Link	Adaptive Learning Rate Methods； Hybrid Methods	Hybrid Adaptive Strategy； SGD-Adam Hybrid
ADAS		ADAS: ADAPTIVE SCHEDULING OF STOCHASTIC GRADIENTS	Link	Learning Rate Scheduling	Scheduler-Free Adaptation
LaProp		LaProp: Separating Momentum and Adaptivity in Adam	Link	Adaptive Learning Rate Methods
Multistage SGDM		An Improved Analysis of Stochastic Gradient Descent with Momentum	Link	Learning Rate Scheduling	Stability-Aware Adaptive Scheduling
pbSGD		pbSGD: Powered Stochastic Gradient Descent Methods for Accelerated Non-Convex Optimization	Link	Gradient Normalization & Clipping	Element-Wise Gradient Scaling
clipped-SGD		Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping	Link	Gradient Normalization & Clipping	Basic Fixed Gradient Clipping
Cayley SGD		EFFICIENT RIEMANNIAN OPTIMIZATION ON THE STIEFEL MANIFOLD VIA THE CAYLEY TRANSFORM	Link	Hybrid Methods	Projection Gradient Hybrid
NIGT		Momentum Improves Normalized SGD	Link	Gradient Normalization & Clipping	Noise-Robust Normalization
AdaBelief		AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation； Prediction Deviation Adaptation
RAdam		On the Variance of the Adaptive Learning Rate and Beyond	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
AdamBS		Adam with Bandit Sampling for Deep Learning	Link	Hybrid Methods	Multi-Objective Hybrid
DEAM		DEAM: Adaptive Momentum with Discriminative Weight for Stochastic Optimization	Link	Momentum-Enhanced SGD	Dynamic Momentum Weight
LAMB		Large Batch Optimization for Deep Learning: Training BERT in 76 minutes	Link	Adaptive Learning Rate Methods； Learning Rate Scheduling	Layer-Wise Adaptation； Batch-Aware Scheduling
ADASS		ADASS: Adaptive Sample Selection for Training Acceleration	Link	Hybrid Methods	Gradient Filtering Hybrid
NovoGrad		Stochastic Gradient Methods with Layer-wise Adaptive Moments for Training of Deep Networks	Link	Adaptive Learning Rate Methods	Layer-Wise Adaptation
AdamW / SGDW		Decoupled Weight Decay Regularization	Link	Adaptive Learning Rate Methods
QHadam		QUASI-HYPERBOLIC MOMENTUM AND ADAM FORDEEP FOR DEEP LEARNING	Link	Adaptive Learning Rate Methods
HAdam		On Higher-order Moments in Adam	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
diffGrad		diffGrad: An Optimization Method for Convolutional Neural Networks	Link	Adaptive Learning Rate Methods	Hybrid Adaptive Strategy
NosAdam		Nostalgic Adam: Weighting more of the past gradients when designing the adaptive learning rate	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
Lookahead		Lookahead Optimizer: k steps forward, 1 step back	Link	Hybrid Methods	Multi-Objective Hybrid
AdaBound		Adaptive Gradient Methods with Dynamic Bound of Learning Rate	Link	Learning Rate Scheduling	Element-Wise Learning Rate Scheduling
LazyOptimizer	Blog'19		Link	Momentum-Enhanced SGD	Scheduled Momentum Reset
YOGI		Adaptive Methods for Nonconvex Optimization	Link	Momentum-Enhanced SGD	Double-momentum mechanism
VR-SGD		VR-SGD: A Simple Stochastic Variance Reduction Method for Machine Learning	Link	Hybrid Methods	Gradient Filtering Hybrid
Shampoo		Shampoo: Preconditioned Stochastic Tensor Optimization	Link	Preconditioned Gradient Methods	Two Metrics' Preconditioner
MSVAG		DissectingAdam:TheSign,MagnitudeandVarianceofStochasticGradients	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
PIDOptimizer		A PID Controller Approach for Stochastic Optimization of Deep Networks	Link	Momentum-Enhanced SGD	Momentum Damping Mechanism
LARS		Large batch training of Convolutional Network	Link	Adaptive Learning Rate Methods	Layer-Wise Adaptation
NAdam		Incorporating Nesterov Momentum into Adam	Link	Adaptive Learning Rate Methods	Momentum-based Adaptive
Adam		ADAM: A METHOD FOR STOCHASTIC OPTIMIZATION	Link	Adaptive Learning Rate Methods
SGDM		On the importance of initialization and momentum in deep learning	Link	Momentum-Enhanced SGD	Accelerated Momentum
AdaDelta		ADADELTA:ANADAPTIVELEARNINGRATEMETHOD	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation
AdaGrad		Adaptive Subgradient Methods for Online Learning and Stochastic Optimization	Link	Adaptive Learning Rate Methods	Second-Order Moment Adaptation

⚙️ Second-Order Algorithms

Abbreviation	Venue & Year	Paper Title	Project	Sub-methods	Fine-grained Methods
C-ALADIN		A Global Convergence Analysis of Consensus ALADIN for Convex Optimization	Link	Hessian Approximation & Estimation	Distributed Newton-Type Approximation
S-BFGS		EFFICIENT STOCHASTIC BFGS METHODS INSPIRED BY BAYESIAN PRINCIPLES	Link	Quasi-Newton Methods	Stochastic BFGS
MAC		MAC: AN EFFICIENT GRADIENT PRECONDITIONING USING MEAN ACTIVATION APPROXIMATED CURVATURE	Link	Fisher Information Matrix Application	Curvature-Aware Approximation
RACS		Towards Efficient Optimizer Design for LLM via Structured Fisher Approximation with a Low-Rank Extension	Link	Fisher Information Matrix Application	Diagonal Fisher Approximation
OCAR		Online Curvature-Aware Replay: Leveraging 2nd Order Information for Online Continual Learning	Link	Fisher Information Matrix Application	Diagonal Fisher Approximation
FUSE-PV		FUSE: First-Order and Second-Order Unified SynthEsis in Stochastic Optimization	Link	Quasi-Newton Methods	Stochastic BFGS
SASSHA		SASSHA: Sharpness-aware Adaptive Second-order Optimization with Stable Hessian Approximation	Link	Hessian Approximation & Estimation	Diagonal Hessian Approximation
AdaFisher		ADAFISHER: ADAPTIVE SECOND ORDER OPTIMIZATION VIA FISHER INFORMATION	Link	Fisher Information Matrix Application	Diagonal Fisher Approximation； Block-Diagonal Kronecker Approximation
OptiQ		Second-Order Optimization via Quiescence	Link	Curvature-Guided Preconditioning	Hessian Diagonal Preconditioning
SOAA		EFFICIENT SECOND-ORDER NEURAL NETWORK OPTIMIZATION VIA ADAPTIVE TRUST REGION METHODS	Link	Fisher Information Matrix Application	Diagonal Fisher Approximation
CRNAS		Novel Optimization Techniques for Parameter Estimation	Link	Hessian Approximation & Estimation	Diagonal Hessian Approximation
Athena		Athena: Efficient Block-Wise Post-Training Quantization for Large Language Models Using Second-Order Matrix Derivative Information	Link	Hessian Approximation & Estimation	Block Hessian Approximation
Q-Newton		Q-Newton: Hybrid Quantum-Classical Scheduling for Accelerating Neural Network Training with Newton’s Gradient Descent	Link	Hessian Approximation & Estimation	Block Hessian Approximation
SkechySGD		SketchySGD: Reliable Stochastic Optimization via Randomized Curvature Estimates	Link	Hessian Approximation & Estimation	Stochastic Hessian Sampling
sophia		Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training	Link	Hessian Approximation & Estimation； Curvature-Guided Preconditioning； Second-Order Moment Fusion； Privacy-Aware Gradient Clipping； Stateless Optimization Methods	Diagonal Hessian Approximation； Hessian Diagonal Preconditioning； Noise-Robust Second-Order Momentum； Real-Time Curvature Estimation
Fed-Sophia		Fed-Sophia: A Communication-Efficient Second-Order Federated Learning Algorithm	Link	Hessian Approximation & Estimation； Federated Learning Optimization	Diagonal Hessian Approximation； Federated Second-Order Optimization
HesScale		Revisiting Scalable Hessian Diagonal Approximations for Applications in Reinforcement Learning	Link	Hessian Approximation & Estimation	Diagonal Hessian Approximation
mL-BFGS		mL-BFGS: A Momentum-based L-BFGS for Distributed Large-Scale Neural Network Optimization	Link	Quasi-Newton Methods	Stochastic BFGS； Low-Memory Quasi-Newton
SGDHess		Better SGD using Second-order Momentum	Link	Hessian Approximation & Estimation	Gradient Difference Estimation
AdaHessian		AdaHessian: An Adaptive Second Order Optimizer for Machine Learning	Link	Hessian Approximation & Estimation； Second-Order Moment Fusion	Diagonal Hessian Approximation； Noise-Robust Second-Order Momentum
TKFAC		A Trace-restricted Kronecker-Factored Approximation to Natural Gradient	Link	Fisher Information Matrix Application	Trace-Preserving Fisher Approximation
SGN		On the Promise of the Stochastic Generalized Gauss-Newton Method for Training DNNs	Link	Hessian Approximation & Estimation	Stochastic Hessian Sampling
SpiderSQN		A FAST QUASI-NEWTON-TYPE METHOD FOR LARGESCALE STOCHASTIC OPTIMISATION	Link	Quasi-Newton Methods	Stochastic BFGS
K-BFGS and K-BFGS(L),		Practical Quasi-Newton Methods for Training Deep Neural Networks	Link	Quasi-Newton Methods	Low-Memory Quasi-Newton
K-FAC		Optimizing Neural Networks with Kronecker-factored Approximate Curvature	Link	Fisher Information Matrix Application	Block-Diagonal Kronecker Approximation
Natural Gradient		Natural gradient works efficiently in learning		Fisher Information Matrix Application
BFGS		A limited memory algorithm for bound constrained optimization		Quasi-Newton Methods
Newton's Method	ANL'1982	Newton's method		Hessian Approximation & Estimation
L-BFGS		Updating quasi-newton matrices with limited storage		Quasi-Newton Methods	Stochastic BFGS
Gauss-Newton Method		Quasi-Likelihood Functions, Generalized Linear Models, and the Gauss-Newton Method		Hessian Approximation & Estimation

📍 Zeroth-Order Algorithms

Abbreviation	Venue & Year	Paper Title	Project	Sub-methods	Fine-grained Methods
AdaMeZO		AdaMeZO: Adam-style Zeroth-Order Optimizer for LLM Fine-tuning Without Maintaining the Moments	Link	Adaptive Methods
MEAZO		On Adaptivity in Zeroth-Order Optimization	Link	Memory-Efficient Methods
ZO-SAH		Subspace-based Approximate Hessian Method for Zeroth-Order Optimization	Link	Adaptive Methods	Projection-based Adaptive
FZOO		FZOO: Fast Zeroth-Order Optimizer for Fine-Tuning Large Language Models towards Adam-Scale Speed	Link	Variance Reduction	Structured Variance Control
VR-SZD		A Structured Proximal Stochastic Variance Reduced Zeroth-order Algorithm	Link	Variance Reduction	Snapshot Variance Reduction
KerZOO		KerZOO: Kernel Function Informed Zeroth-Order Optimization for Accurate and Accelerated LLM Fine-Tuning	Link	Memory-efficient Methods	Inference-Level Memory Zeroth-Order
QZO		Fine-tuning Quantized Neural Networks with Zeroth-order Optimization	Link	Memory-efficient Methods	Quantized Zeroth-Order Finetuning
VAMO		VAMO: Efficient Large-Scale Nonconvex Optimization via Adaptive Zeroth Order Variance Reduction	Link	Zeroth-First Order Hybrid； Variance Reduction	Variance Reduction Hybrid； Snapshot Variance Reduction
ZO2		ZO2: Scalable Zeroth-Order Fine-Tuning for Extremely Large Language Models with Limited GPU Memory	Link	Memory-efficient Methods	Inference-Level Memory Zeroth-Order
LORENZA		LORENZA: Enhancing Generalization in Low-Rank Gradient LLM Training and Fine-Tuning via Efficient Zeroth-Order Adaptive SAM Optimization	Link	Adaptive Methods	Momentum-based Adaptive； Low-Rank Adaptive
QuZO		QuZO: Quantized Zeroth-Order Fine-Tuning for Large Language Models	Link	Memory-efficient Methods； Low-Rank Methods	Quantized Zeroth-Order Finetuning； Low-Rank & Quantization
MaZO		MaZO: Masked Zeroth-Order Optimization for Multi-Task Fine-Tuning of Large Language Models	Link	Memory-efficient Methods	Sparse Parameter Zeroth-Order
DiZO		Harmony in Divergence: Towards Fast, Accurate, and Memory-efficient Zeroth-order LLM Fine-tuning	Link	Adaptive Methods	Projection-based Adaptive
TeZO		TeZO: Empowering the Low-Rankness on the Temporal Dimension in the Zeroth-Order Optimization for Fine-tuning LLMs	Link	Memory-efficient Methods	Low-Rank Zeroth-Order Finetuning
ELASTICZO		ELASTICZO: A MEMORY-EFFICIENT ON-DEVICE LEARNING WITH COMBINED ZEROTH- AND FIRST-ORDER OPTIMIZATION	Link	Zeroth-First Order Hybrid	Layer-Wise Hybrid
LOZO		Enhancing zeroth-order fine-tuning for language models with low-rank structures	Link	Memory-efficient Methods	Low-Rank Zeroth-Order Finetuning
Addax		Addax: Utilizing Zeroth-Order Gradients to Improve Memory Efficiency and Performance of SGD for Fine-Tuning Language Models	Link	Zeroth-First Order Hybrid	Weighted Hybrid
ZOQO		ZOQO: Zero-Order Quantized Optimization	Link	Memory-efficient Methods	Quantized Zeroth-Order Finetuning
R-AdaZO		Refining Adaptive Zeroth-Order Optimization at Ease	Link	Adaptive Method	Momentum-based Adaptive
ZO-AdaMM		Zeroth-Order Adaptive Momentum Method for Black-Box Optimization	Link	Adaptive Methods	Momentum-based Adaptive
LeZO		SIMULTANEOUS COMPUTATION AND MEMORY EFFICIENT ZEROTH-ORDER OPTIMIZER FOR FINE-TUNING LARGE LANGUAGE MODELS	Link	Perturbation Optimization； Memory-efficient Methods； Memory-Efficient Fine-Tuning for Large Models	Sparse Perturbation； Sparse Parameter Zeroth-Order； Selective Parameter Fine-Tuning
SuZero		Zeroth-Order Fine-Tuning of LLMs in Random Subspaces	Link	Memory-efficient Methods	Low-Rank Zeroth-Order Finetuning
Sparse MeZO		Sparse MeZO: Less Parameters for Better Performance in Zeroth-Order LLM Fine-Tuning	Link	Perturbation Optimization； Memory-efficient Methods； Memory-Efficient Fine-Tuning for Large Models	Sparse Perturbation； Sparse Parameter Zeroth-Order； Selective Parameter Fine-Tuning
MeZO-SVRG		Variance-reduced Zeroth-Order Methods for Fine-Tuning Language Models	Link	Variance Reduction	Snapshot Variance Reduction
ZO-AdaMU		ZO-AdaMU Optimizer: Adapting Perturbation by the Momentum and Uncertainty in Zeroth-order Optimization	Link	Perturbation Optimization； Memory-efficient Methods	Paired Perturbation Sampling； Inference-Level Memory Zeroth-Order
ZoPro		A Zeroth-Order Proximal Algorithm for Consensus Optimization	Link	Distributed Zero-Order Optimization	Distributed Perturbation Sampling
MeZO		Fine-Tuning Language Models with Just Forward Passes	Link	Memory-efficient Methods	Inference-Level Memory Zeroth-Order
TOP-DP		Topology-aware Differential Privacy for Decentralized Image Classification	Link	Distributed Zero-Order Optimization； Differential Privacy Optimization	Privacy-Preserving Zeroth-Order； DP-SGD Variants； Dynamic Noise Scheduling； Privacy-Utility Balance
SPSA		Global random optimization by simultaneous perturbation stochastic approximation		Perturbation Optimization	Paired Perturbation Sampling

🌐 Distributed Optimization

Abbreviation	Venue & Year	Paper Title	Project	Sub-methods	Fine-grained Methods
AlignFed		AlignFed: Alignment-Aware Asynchronous Federated Fine-Tuning for Large Language Models in Heterogeneous Edge Environments	Link	Federated Learning Optimization	Asynchronous Federated Aggregation
DECA		DECA: Decentralizing Block-Wise Adam for Efficient LLM Full-Parameter Fine-Tuning on Non-IID Data	Link	Federated Learning Optimization	Block-Wise Decentralized Adam
FedSIR		FedSIR: Spectral Client Identification and Relabeling for Federated Learning with Noisy Labels	Link	Federated Learning Optimization
Ringleader ASGD		First Provably Optimal Asynchronous SGD for Homogeneous and Heterogeneous Data	Link	Local Update Strategies	Local-global hybrid updates
FedMuon		FedMuon: Accelerating Federated Learning with Matrix Orthogonalization	Link	Federated Learning Optimization	Federated Momentum Fusion
DLAS-R-FTC		Distributed Optimization and Learning for Automated Stepsize Selection with Finite Time Coordination	Link	Decentralized Communication	Distributed Consensus Optimization
DOME		Communication Efficient, Differentially Private Distributed Optimization using Correlation-Aware Sketching	Link	Gradient Compression & Quantization	Low-Rank Gradient Compression
Deco-SGD		DeCo-SGD: Joint Optimization of Delay Staleness and Gradient Compression Ratio for Distributed SGD	Link	Gradient Compression & Quantization； Local Update Strategies	Adaptive Compression Level； Local-Global Hybrid Updates
TAH-QUANT		TAH-QUANT: Effective Activation Quantization in Pipeline Parallelism over Slow Network	Link	Gradient Compression & Quantization	Quantization Compression
LQ-SGD		Trustworthy Efficient Communication for Distributed Learning using LQ-SGD Algorithm	Link	Gradient Compression & Quantization； Low-Rank Methods	Quantization Compression； Low-Rank & Quantization
FedCurv		Blockchain-Enabled Privacy-Preserving Second-Order Federated Edge Learning in Personalized Healthcare	Link	Federated Learning Optimization	Federated Second-Order Optimization
pFedSOP		pFedSOP : Accelerating Training Of Personalized Federated Learning Using Second-Order Optimization	Link	Federated Learning Optimization	Federated Second-Order Optimization
FedOne		FedOne: Query-Efficient Federated Learning for Black-box Discrete Prompt Learning	Link	Federated Learning Optimization	Client Sampling Optimization
DEC-LOC		DES-LOC: Desynced Low Communication Adaptive Optimizers for Training Foundation Models	Link	Decentralized Communication	Distributed Consensus Optimization
Kuramoto-FedAvg		Kuramoto-FedAvg: Using Synchronization Dynamics to Improve Federated Learning Optimization under Statistical Heterogeneity	Link	Federated Learning Optimization	Federated Momentum Fusion
AbsSADMM		Stochastic ADMM with batch size adaptation for nonconvex nonsmooth optimization	Link	Local Update Strategies	Adaptive Local Steps
ADEF		Accelerated Distributed Optimization with Compression and Error Feedback	Link	Gradient Compression & Quantization	Compression Error Compensation
FedCET		Communication Efficient Federated Learning with Linear Convergence on Heterogeneous Data	Link	Local Update Strategies	Adaptive Local Steps
Interleaved-ShuffleG		The Cost of Shuffling in Private Gradient Based Optimization	Link	Decentralized Communication； Differential Privacy Optimization	Privacy-Preserving Decentralization； Privacy-Utility Balance
FAdamGC		Gradient Correction in Federated Learning with Adaptive Optimization	Link	Federated Learning Optimization	Client Sampling Optimization
LT-ADMM		Communication-Efficient Stochastic Distributed Learning	Link	Decentralized Communication	Distributed Consensus Optimization
HybridSGD		Communication-Efficient, 2D Parallel Stochastic Gradient Descent for Distributed-Memory Optimization	Link	Local Update Strategies	Local-Global Hybrid Updates
DAT-SGD		Enhancing Parallelism in Decentralized Stochastic Convex Optimization	Link	Decentralized Communication	Neighbor Communication Topology
FedSTaS		FedSTaS: Client Stratification and Client Level Sampling for Efficient Federated Learning	Link	Federated Learning Optimization	Client Sampling Optimization
FedIvon		Federated Learning with Uncertainty and Personalization via Efficient Second-order Optimization	Link	Federated Learning Optimization	Personalized Federated Optimization
FAGH		FAGH: Accelerating Federated Learning with Approximated Global Hessian	Link	Federated Learning Optimization	Federated Second-Order Optimization
AdaFedAdam		ACCELERATING FAIR FEDERATED LEARNING: ADAPTIVE FEDERATED ADAM	Link	Federated Learning Optimization	Federated Momentum Fusion
MM-PSGD		Distributed Optimization over Block-Cyclic Data	Link	Federated Learning Optimization	Personalized Federated Optimization
MC-PSGD		Distributed Optimization over Block-Cyclic Data	Link	Federated Learning Optimization	Personalized Federated Optimization
FedLion		FEDLION: FASTER ADAPTIVE FEDERATED OPTIMIZATION WITH FEWER COMMUNICATION	Link	Federated Learning Optimization	Federated Momentum Fusion
FADAS		FADAS: Towards Federated Adaptive Asynchronous Optimization	Link	Federated Learning Optimization	Federated Momentum Fusion
FLeNS		FLeNS: Federated Learning with Enhanced Nesterov-Newton Sketch	Link	Federated Learning Optimization	Federated Momentum Fusion
FedRepOpt		FedRepOpt: Gradient Re-parametrized Optimizers in Federated Learning	Link	Federated Learning Optimization	Federated Momentum Fusion
Fed-Sophia		Fed-Sophia: A Communication-Efficient Second-Order Federated Learning Algorithm	Link	Hessian Approximation & Estimation； Federated Learning Optimization	Diagonal Hessian Approximation； Federated Second-Order Optimization
FedLAP-DP		FedLAP-DP: Federated Learning by Sharing Differentially Private Loss Approximations	Link	Decentralized Communication	Privacy-Preserving Decentralization
AdaCGD		Adaptive Compression for Communication-Efficient Distributed Training	Link	Gradient Compression & Quantization	Adaptive Compression Level
0/1 Adam		Maximizing Communication Efficiency for Large-scale Training via 0/1 Adam	Link	Gradient Compression & Quantization	Quantization Compression
SketchedAMSGrad		Communication-Efficient Adam-Type Algorithms for Distributed Data Mining	Link	Gradient Compression & Quantization	Low-Rank Gradient Compression
SPARQ-SGD		SPARQ-SGD: Event-Triggered and Compressed Communication in Decentralized Stochastic Optimization	Link	Gradient Compression & Quantizatio	Sparsification Compression
1-bit Adam		1-bit Adam: Communication Efficient Large-Scale Training with Adam’s Convergence Speed	Link	Gradient Compression & Quantization	Quantization Compression
BVR-L-SGD		Bias-Variance Reduced Local SGD for Less Heterogeneous Federated Learning	Link	Local Update Strategies	Local SGD
A(DP)^2SGD		A(DP)^2SGD: Asynchronous Decentralized Parallel Stochastic Gradient Descent with Differential Privacy	Link	Decentralized Communication	Neighbor Communication Topology
SQuARM-SGD		SQuARM-SGD: Communication-Efficient Momentum SGD for Decentralized Optimization	Link	Momentum-Enhanced SGD； Local Update Strategies； Distributed Hybrid Optimization	Accelerated Momentum； Local SGD； Local Momentum Updates； Compression & Local Updates
DLCP		Domain-specific Communication Optimization for Distributed DNN Training	Link	Local Update Strategies	Local SGD
APMSqueeze		APMSqueeze: A Communication Efficient Adam-Preconditioned Momentum SGD Algorithm	Link	Gradient Compression & Quantization	Compression Error Compensation
DEED-GD		DEED: A General Quantization Scheme for Communication Efficiency in Bits	Link	Gradient Compression & Quantization	Quantization Compression
DP-PASGD		Differentially Private Federated Learning for Resource-Constrained Internet of Things	Link	Local Update Strategies	Adaptive Local Steps
LAGS-SGD		Layer-wise Adaptive Gradient Sparsification for Distributed Deep Learning with Convergence Guarantees	Link	Gradient Compression & Quantization	Adaptive Compression Level
FedAC		Federated Accelerated Stochastic Gradient Descent	Link	Federated Learning Optimization	Federated Momentum Fusion
rTop-k		rTop-k: A Statistical Estimation Approach to Distributed SGD	Link	Gradient Compression & Quantization	Sparsification Compression
Qsparse-local-SGD		Qsparse-local-SGD: Distributed SGD with Quantization, Sparsification, and Local Computations	Link	Distributed Hybrid Optimization	Compression & Local Updates
SLOWMO		SLOWMO: IMPROVING COMMUNICATION-EFFICIENT DISTRIBUTED SGD WITH SLOW MOMENTUM	Link	Local Update Strategies	Local SGD
SCAFFOLD		SCAFFOLD: Stochastic Controlled Averaging for Federated Learning	Link	Local Update Strategies	Local SGD
LD-SGD		Communication-Efficient Local Decentralized SGD Methods	Link	Decentralized Communication	Neighbor Communication Topology
PowerSGD		PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization	Link	Gradient Compression & Quantization	Low-Rank Gradient Compression
signProx		signProx: One-Bit Proximal Algorithm for Nonconvex Stochastic Optimization	Link	Gradient Compression & Quantization	Quantization Compression
signSGD		signSGD: Compressed Optimisation for Non-Convex Problems	Link	Gradient Compression & Quantization	Quantization Compression

🛡️ Privacy-Preserving Optimization

Abbreviation	Venue & Year	Paper Title	Project	Sub-methods	Fine-grained Methods
PrivCode++		PrivCode++: Latent-Conditioned Differentially Private Code Generation for Comprehensive Guarantees	Link	Differential Privacy Optimization	DP Fine-Tuning for Code Generation
DP-MacAdam		DP-MacAdam: Differentially Private Mechanism with Adaptive Clipping and Adaptive Momentum	Link	Differential Privacy Optimization	Adaptive Clipping and Momentum
DPSR-CG		Revisiting Privacy Amplification by Subsampling in Selective Release DPSGD	Link	Differential Privacy Optimization	Selective Release DP-SGD
PINA		Differentially Private Clustered Federated Learning with Privacy-Preserving Initialization and Normality-Driven Aggregation	Link	Differential Privacy Optimization
DP-aware AdaLN-Zero		DP-aware AdaLN-Zero: Taming Conditioning-Induced Heavy-Tailed Gradients in Differentially Private Diffusion	Link	Differential Privacy Optimization	Dynamic noise scheduling
DP-λCGD		DP-λCGD: Efficient Noise Correlation for Differentially Private Model Training	Link	Differential Privacy Optimization	DP-SGD variants
RaCO-DP		Private Rate-Constrained Optimization with Applications to Fair Learning	Link	Differential Privacy Optimization	DP-SGD Variants
Interleaved-ShuffleG		The Cost of Shuffling in Private Gradient Based Optimization	Link	Decentralized Communication； Differential Privacy Optimization	Privacy-Preserving Decentralization； Privacy-Utility Balance
DPZV		DPZV: Elevating the Tradeoff between Privacy and Utility in Zeroth-Order Vertical Federated Learning	Link	Gradient Noise Injection	Noise-Robust Optimization
Stable-SPAM		Stable-SPAM: How to Train in 4-Bit More Stably than 16-Bit Adam	Link	Gradient Normalization & Clipping； Privacy-Aware Gradient Clipping	Layer-Wise Gradient Normalization； Dynamic Gradient Clipping； Adaptive Clipping
GeoDP-SGD		Analyzing and Optimizing Perturbation of DP-SGD Geometrically	Link	Gradient Noise Injection	Noise-Robust Optimization
Logit-DP		DIFFERENTIALLY PRIVATE OPTIMIZATION FOR NONDECOMPOSABLE OBJECTIVE FUNCTIONS	Link	Privacy-Aware Gradient Clipping	Global Gradient Clipping
DOPPLER		DOPPLER: Differentially Private Optimizers with Low-pass Filter for Privacy Noise Reduction	Link	Differential Privacy Optimization	DP-SGD Variants
DC-SGD		DC-SGD: Differentially Private SGD with Dynamic Clipping through Gradient Norm Distribution Estimation	Link	Differential Privacy Optimization； Gradient Noise Injection； Privacy-Utility Tradeoff； Privacy-Aware Gradient Clipping	Dynamic Noise Scheduling； Dynamic Clipping Threshold； Adaptive Clipping
SPARTA		SPARTA: An Optimization Framework for Differentially Private Sparse Fine-Tuning	Link	Differential Privacy Optimization	DP-SGD Variants
DP-AdamW-BC		DP-AdamW: Investigating Decoupled Weight Decay and Bias Correction in Private Deep Learning	Link	Differential Privacy Optimization	DP-SGD Variants
DP-MicroAdam		DP-MicroAdam: Private and Frugal Algorithm for Training and Fine-tuning	Link	Differential Privacy Optimization	DP-SGD Variants
AClipped-dpSGD		Efficient Private SCO for Heavy-Tailed Data via Averaged Clipping	Link	Privacy-Aware Gradient Clipping	Global Gradient Clipping
sophia		Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training	Link	Hessian Approximation & Estimation； Curvature-Guided Preconditioning； Second-Order Moment Fusion； Privacy-Aware Gradient Clipping； Stateless Optimization Methods	Diagonal Hessian Approximation； Hessian Diagonal Preconditioning； Noise-Robust Second-Order Momentum； Real-Time Curvature Estimation
ANSGD		Learning across Data Owners with Joint Differential Privacy	Link	Differential Privacy Optimization	DP-SGD Variants
DP-Adam		DP-ADAM: CORRECTING DP BIAS IN ADAM’S SECOND MOMENT ESTIMATION	Link	Differential Privacy Optimization； Gradient Noise Injection	DP-SGD Variants； Noise-Robust Optimization
DP-FedSAM		Make Landscape Flatter in Differentially Private Federated Learning	Link	Gradient Noise Injection； Federated Privacy Enhancement	Noise-Robust Optimization； Federated Noise Aggregation
DPIS		DPIS: An Enhanced Mechanism for Differentially Private SGD with Importance Sampling	Link	Differential Privacy Optimization	DP-SGD Variants
DP-SGD-JL		Fast and Memory Efficient Differentially Private-SGD via JL Projections	Link	Differential Privacy Optimization	DP-SGD Variants
TOP-DP		Topology-aware Differential Privacy for Decentralized Image Classification	Link	Distributed Zero-Order Optimization； Differential Privacy Optimization	Privacy-Preserving Zeroth-Order； DP-SGD Variants； Dynamic Noise Scheduling； Privacy-Utility Balance
DP-LSSGD		DP-LSSGD: A Stochastic Optimization Method to Lift the Utility in Privacy-Preserving ERM	Link	Privacy-Utility Tradeoff	Post-Processing Optimization

⚡ Memory-Efficient Optimization

Abbreviation	Venue & Year	Paper Title	Project	Sub-methods	Fine-grained Methods
DL-ZO		Dominant-Layer ZO: A Single Layer Dominates Zeroth-Order Fine-Tuning of LLMs	Link	Memory-efficient Methods	Layer-Selective Fine-Tuning
LQ-SGD		Trustworthy Efficient Communication for Distributed Learning using LQ-SGD Algorithm	Link	Gradient Compression & Quantization； Low-Rank Methods	Quantization Compression； Low-Rank & Quantization
SUMO		SUMO: Subspace-Aware Moment-Orthogonalization for Accelerating Memory-Efficient LLM Training	Link	Low-Rank Gradient Storage	Gradient Low-Rank Projection
AlphaGrad		AlphaGrad: Non-Linear Gradient Normalization Optimizer	Link	Adaptive Learning Rate Methods； Low-Memory Optimizer Design； Stateless Optimization Methods	Stateless Adaptation； Structural Redesign； Parameter Characteristic-Driven Updates
QuZO		QuZO: Quantized Zeroth-Order Fine-Tuning for Large Language Models	Link	Memory-efficient Methods； Low-Rank Methods	Quantized Zeroth-Order Finetuning； Low-Rank & Quantization
GWT		Wavelet Meets Adam: Compressing Gradients for Memory-Efficient Training	Link	Low-Rank Gradient Storage	Gradient Low-Rank Projection
AdaRankGrad		Adarankgrad: Adaptive gradient-rank and moments for memory-efficient llms training and fine-tuning	Link	Low-Rank Methods	Projection&Adjustment
Adam-mini		ADAM-MINI: USE FEWER LEARNING RATES TO GAIN MORE	Link	Low-Memory Optimizer Design	Structural Redesign
SPAM		SPAM: SPIKE-AWARE ADAM WITH MOMENTUM RESET FOR STABLE LLM TRAINING	Link	Gradient Normalization & Clipping； Optimizer State Compression	Element-Wise Gradient Scaling； Spike-Aware Gradient Clipping； Sparse State Compression
SGD-SaI		No More Adam: Learning Rate Scaling at Initialization is All You Need	Link	Learning Rate Scheduling； Stateless Optimization Methods	Initial Learning Rate Scaling； Parameter Characteristic-Driven Updates
A-GNB		HELENE: HESSIAN LAYER-WISE CLIPPING AND GRADIENT ANNEALING FOR ACCELERATING FINETUNING LLM WITH ZEROTH-ORDER OPTIMIZATION	Link	Low-Rank Gradient Storage	Dynamic Gradient Rank
LeZO		SIMULTANEOUS COMPUTATION AND MEMORY EFFICIENT ZEROTH-ORDER OPTIMIZER FOR FINE-TUNING LARGE LANGUAGE MODELS	Link	Perturbation Optimization； Memory-efficient Methods； Memory-Efficient Fine-Tuning for Large Models	Sparse Perturbation； Sparse Parameter Zeroth-Order； Selective Parameter Fine-Tuning
Adapprox		Adapprox: Adaptive Approximation in Adam Optimization via Randomized Low-Rank Matrices	Link	Low-Rank Methods	Projection&Adjustment
Sparse MeZO		Sparse MeZO: Less Parameters for Better Performance in Zeroth-Order LLM Fine-Tuning	Link	Perturbation Optimization； Memory-efficient Methods； Memory-Efficient Fine-Tuning for Large Models	Sparse Perturbation； Sparse Parameter Zeroth-Order； Selective Parameter Fine-Tuning
LOMO		Full Parameter Fine-tuning for Large Language Models with Limited Resources	Link	Towards LLM Traning； Memory-Efficient Fine-Tuning for Large Models	Real-Time Computation； Staless Fine-Tuning
MICROADAM		MICROADAM: Accurate Adaptive Optimization with Low Space Overhead and Provable Convergence	Link	Optimizer State Compression	Sparse State Compression
4-bit shampoo		4-bit Shampoo for Memory-Efficient Network Training	Link	Preconditioned Gradient Methods； Low-Memory Optimizer Design	Two Metrics' Preconditioner； Compression&Approximation of States
ADAACT		AN ADAPTIVE METHOD STABILIZING ACTIVATIONS FOR ENHANCED GENERALIZATION	Link	Adaptive Learning Rate Methods； Optimizer State Compression	Neuron-Level Adaptation； State Sharing
sophia		Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training	Link	Hessian Approximation & Estimation； Curvature-Guided Preconditioning； Second-Order Moment Fusion； Privacy-Aware Gradient Clipping； Stateless Optimization Methods	Diagonal Hessian Approximation； Hessian Diagonal Preconditioning； Noise-Robust Second-Order Momentum； Real-Time Curvature Estimation
tpSGD		Learning with Local Gradients at the Edge	Link	Low-Memory Optimizer Design	Structural Redesign
AdaBFE		BFE and AdaBFE: A New Approach in Learning Rate Automation for Stochastic Optimization	Link	Learning Rate Scheduling； Stateless Optimization Methods	Gradient Angle Scheduling； Parameter Characteristic-Driven Updates
Adafactor		Adafactor: Adaptive Learning Rates with Sublinear Memory Cost	Link	Low-Memory Optimizer Design； Optimizer State Compression	Compression&Approximation of States； State Sharing

🧩 Tailored Optimization Approaches

Abbreviation	Venue & Year	Paper Title	Project	Sub-methods	Fine-grained Methods
MAdam		MAdam: Metric-Aware Multi-Objective Adam	Link	Hybrid Methods	Multi-Objective Adaptive Strategy
KO		KO: Kinetics-inspired Neural Optimizer with PDE Simulation Approaches	Link	Auto-Designed Optimizers	Automated Discovery&Theoretical Derivation
BC-ADMM		BC-ADMM: An Efficient Non-convex Constrained Optimizer with Robotic Applications	Link	Robust Optimization	Structure-Aware Optimization
AdaGC		AdaGC: Improving Training Stability for Large Language Model Pretraining	Link	Gradient Normalization & Clipping； Robust Optimization	Noise-Robust Normalization； Dynamic Gradient Clipping； Noise-Robust Gradients
AGS-GD		Anisotropic Gaussian Smoothing for Gradient-based Optimization	Link	Hybrid Methods； Auto-Designed Optimizers	Gradient Smoothing Hybrid； Automated Discovery&Theoretical Derivation
WarpAdam		WarpAdam: A new Adam optimizer based on Meta-Learning approach	Link	Auto-Designed Optimizers	Evolutionary Strategies&Meta-Adaptive Learning
MADA		MADA: Meta-Adaptive Optimizers through hyper-gradient Descent	Link	Auto-Designed Optimizers	Evolutionary Strategies&Meta-Adaptive Learning
Lion		Symbolic Discovery of Optimization Algorithms	Link	Adaptive Learning Rate Methods； Auto-Designed Optimizers	Momentum-based Adaptive; Automated Discovery&Theoretical Derivation
AdaNorm		AdaNorm: Adaptive Gradient Norm Correction based Optimizer for CNNs	Link	Gradient Normalization & Clipping； Robust Optimization	Element-Wise Gradient Scaling； Noise-Robust Normalization； Noise-Robust Gradients
BGADAM		BGADAM: Boosting based Genetic-Evolutionary ADAM for Neural Network Optimization	Link	Auto-Designed Optimizers	Evolutionary Strategies&Meta-Adaptive Learning
HyperAdam		HyperAdam: A Learnable Task-Adaptive Adam for Network Training	Link	Auto-Designed Optimizers	Evolutionary Strategies&Meta-Adaptive Learning
GADAM		GADAM: Genetic-Evolutionary ADAM for Deep Neural Network Optimization	Link	Auto-Designed Optimizers	Evolutionary Strategies&Meta-Adaptive Learning

🔬Future Prospect

Challenges

• ⚖️ Multi-objective Trade-off Bottlenecks: Optimizing large models often forces a choice between convergence speed, memory efficiency, and distributed scaling, which can sacrifice generalization, introduce latency, or exacerbate instability. The core challenge is breaking these interconnected bottlenecks under a unified framework.
• 💾 Memory and Computational Overheads: Dense optimizer states create severe memory bottlenecks. Structural approximations, such as matrix inversions, significantly decrease global efficiency, and per-step computational latency often negates theoretical advantages in iteration count.
• 🔊 Noise Amplification and Estimation Variance: Anisotropic loss landscapes heavily amplify stochastic mini-batch noise. Random perturbations used for directional gradients suffer from approximation variance that scales poorly with dimensionality, and privacy-preserving noise degrades gradient fidelity.

Trends

• 🤖 Automated Symbolic Discovery: Shifting from fragile heuristic tuning to the automated generation of architecture-specific optimizers, enabling models to inherently navigate complex loss landscapes without manual intervention.
• 🧮 Preconditioning and Orthogonalization: Leveraging structural gradient statistics for preconditioning or matrix orthogonalization (e.g., Kron and Muon) to overcome the representational bottlenecks of simple diagonal scaling and open novel parameter space pathways.

Opptunities

• 🧩 Deep Integration of Multi-Order Algorithms: Moving beyond isolated algorithmic improvements by deeply integrating FO, SO, and ZO algorithms. The fundamental focus will shift from minimizing iteration complexity to improving global wall-clock efficiency.
• 🧭 Dimensionality-Robust Subspace Projection: Discovering advanced subspace projection methods that safely constrain massive search spaces without prematurely restricting algorithmic access to high-quality global solutions.
• ⚖️ State Fidelity Preservation: Designing future memory-efficient architectures to smoothly average out extreme gradient shocks without exceeding memory limits.

🔗Citation

If you find our survey and repository useful for your research project, please consider citing our paper:

@article{zhang2026evolution,
  title={Evolution of Optimization Methods: Algorithms, Scenarios, and Evaluations},
  author={Zhang, Tong and Zhang, Jiangning and Xue, Zhucun and Jiang, Juntao and Xu, Yicheng and Xu, Chengming and Hu, Teng and Xie, Xingyu and Hu, Xiaobin and Wang, Yabiao and others},
  journal={arXiv preprint arXiv:2604.12968},
  year={2026}
}

📫Contact

186368@zju.edu.cn