Spectral clipping of leading singular values in gradient matrices stabilizes SGD for non-convex problems with heavy-tailed noise and achieves the optimal convergence rate O(K^{(2-2α)/(3α-2)}).
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GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection
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abstract
Training Large Language Models (LLMs) presents significant memory challenges, predominantly due to the growing size of weights and optimizer states. Common memory-reduction approaches, such as low-rank adaptation (LoRA), add a trainable low-rank matrix to the frozen pre-trained weight in each layer, reducing trainable parameters and optimizer states. However, such approaches typically underperform training with full-rank weights in both pre-training and fine-tuning stages since they limit the parameter search to a low-rank subspace and alter the training dynamics, and further, may require full-rank warm start. In this work, we propose Gradient Low-Rank Projection (GaLore), a training strategy that allows full-parameter learning but is more memory-efficient than common low-rank adaptation methods such as LoRA. Our approach reduces memory usage by up to 65.5% in optimizer states while maintaining both efficiency and performance for pre-training on LLaMA 1B and 7B architectures with C4 dataset with up to 19.7B tokens, and on fine-tuning RoBERTa on GLUE tasks. Our 8-bit GaLore further reduces optimizer memory by up to 82.5% and total training memory by 63.3%, compared to a BF16 baseline. Notably, we demonstrate, for the first time, the feasibility of pre-training a 7B model on consumer GPUs with 24GB memory (e.g., NVIDIA RTX 4090) without model parallel, checkpointing, or offloading strategies.
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Intrinsic Muon provides closed-form linear maximization oracles on multiple Riemannian matrix manifolds for unitarily invariant norms, with convergence rates depending only on manifold dimension or rank.
Muon with Nesterov momentum and inexact polar decomposition achieves optimal convergence rates of O(ε^(-(3α-2)/(α-1))) under heavy-tailed noise for ε-stationary points in non-convex settings.
BROS achieves memory-efficient single-loop stochastic bilevel optimization with O(ε^{-2}) sample complexity by performing updates in randomized subspaces and using Rademacher bi-probe correction for unbiased estimation.
Pro-KLShampoo projects KL-Shampoo preconditioners to a spike-and-flat parametric form on an r-dimensional subspace and recovers the full algebraic preconditioner via orthogonalization, outperforming KL-Shampoo on GPT-2 and LLaMA pre-training scales.
AdamO modifies Adam with an orthogonality correction to ensure the spectral radius of the TD update operator stays below one, providing a theoretical stability guarantee for offline RL.
Muon² adds adaptive second-moment preconditioning to Muon, improving spectrum conditioning for faster orthogonalization, outperforming Muon on GPT and LLaMA pre-training from 60M to 1.3B parameters while cutting Newton-Schulz iterations by 40%.
STQuant dynamically allocates quantization bits for optimizer states in multimodal model training, reducing memory by 84.4% to an average 5.1 bits while preserving quality on GPT-2 and ViT.
PoLAR-VBLL combines orthogonalized low-rank adapters with variational Bayesian last-layer inference to enable scalable, well-calibrated uncertainty quantification in fine-tuned LLMs.
SCT pre-trains LLMs by keeping weights as compact SVD factors with Stiefel QR retraction, delivering up to 199x memory reduction per layer and allowing 70B-parameter training on a Steam Deck.
BOOST delivers 1.46-2.27x end-to-end speedups for low-rank bottleneck LLMs by redesigning tensor parallelism around the bottleneck structure plus supporting optimizations.
CR-Net uses cross-layer low-rank residuals in a dual-path network plus specialized recomputation to outperform prior low-rank methods on 60M-7B model pre-training while using less compute and memory.
A framework combining stochastic zeroth-order optimization and dynamic low-rank surrogate modeling with an implicit projector-splitting integrator enables end-to-end training of hybrid neural networks containing black-box physical layers and reaches near-digital accuracy on vision, audio, and text任务
DP-GRAPE reduces memory in differentially private neural network training by using random Gaussian projections on gradients instead of SVD, achieving comparable privacy-utility tradeoffs to DP-SGD and scaling to 6.7B parameter models.
PrefixMemory-Tuning decouples the prefix from attention to overcome performance limits of traditional prefix-tuning and reaches competitive results with modern PEFT methods on LLM adaptation benchmarks.
GWT projects gradients into wavelet subspaces to compress optimizer states for memory-efficient LLM training while claiming performance parity with full-rank updates.
The paper introduces an incentive-secure Proof-of-Learning protocol for blockchain consensus that claims provable security against two attacks, reduced computational overhead, and guarantees even with untrusted problem providers and verifiers.
Pion is an optimizer that preserves the singular values of weight matrices in LLM training by applying orthogonal equivalence transformations.
LoPT achieves competitive task performance in LLM post-training by limiting task gradients to the upper model half and training the lower half with local feature reconstruction.
ELAS pre-trains low-rank LLMs by applying 2:4 activation sparsity after squared ReLU to cut memory and accelerate training with minimal performance loss.
HTMuon modifies Muon to produce heavier-tailed updates and weight spectra via HT-SR theory, yielding up to 0.98 lower perplexity on LLaMA pretraining and serving as a plug-in for other Muon variants.
Convergence analysis shows Muon outperforms gradient descent by exploiting low-rank structure in neural network Hessians.
A Lingua Franca reactor-based method is proposed to address nondeterminism in agentic AI for human-in-the-loop cyber-physical systems such as driving coaches.
Muon+ adds one normalization step after polar orthogonalization in the Muon optimizer, yielding lower training and validation perplexity and faster pre-training across 60M-7B models.
citing papers explorer
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Gradient Clipping Beyond Vector Norms: A Spectral Approach for Matrix-Valued Parameters
Spectral clipping of leading singular values in gradient matrices stabilizes SGD for non-convex problems with heavy-tailed noise and achieves the optimal convergence rate O(K^{(2-2α)/(3α-2)}).
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Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds
Intrinsic Muon provides closed-form linear maximization oracles on multiple Riemannian matrix manifolds for unitarily invariant norms, with convergence rates depending only on manifold dimension or rank.
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Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition
Muon with Nesterov momentum and inexact polar decomposition achieves optimal convergence rates of O(ε^(-(3α-2)/(α-1))) under heavy-tailed noise for ε-stationary points in non-convex settings.
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BROS: Bias-Corrected Randomized Subspaces for Memory-Efficient Single-Loop Bilevel Optimization
BROS achieves memory-efficient single-loop stochastic bilevel optimization with O(ε^{-2}) sample complexity by performing updates in randomized subspaces and using Rademacher bi-probe correction for unbiased estimation.
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Pro-KLShampoo: Projected KL-Shampoo with Whitening Recovered by Orthogonalization
Pro-KLShampoo projects KL-Shampoo preconditioners to a spike-and-flat parametric form on an r-dimensional subspace and recovers the full algebraic preconditioner via orthogonalization, outperforming KL-Shampoo on GPT-2 and LLaMA pre-training scales.
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AdamO: A Collapse-Suppressed Optimizer for Offline RL
AdamO modifies Adam with an orthogonality correction to ensure the spectral radius of the TD update operator stays below one, providing a theoretical stability guarantee for offline RL.
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Muon$^2$: Boosting Muon via Adaptive Second-Moment Preconditioning
Muon² adds adaptive second-moment preconditioning to Muon, improving spectrum conditioning for faster orthogonalization, outperforming Muon on GPT and LLaMA pre-training from 60M to 1.3B parameters while cutting Newton-Schulz iterations by 40%.
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STQuant: Spatio-Temporal Adaptive Framework for Optimizer Quantization in Large Multimodal Model Training
STQuant dynamically allocates quantization bits for optimizer states in multimodal model training, reducing memory by 84.4% to an average 5.1 bits while preserving quality on GPT-2 and ViT.
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Scalable Variational Bayesian Fine-Tuning of LLMs via Orthogonalized Low-Rank Adapters
PoLAR-VBLL combines orthogonalized low-rank adapters with variational Bayesian last-layer inference to enable scalable, well-calibrated uncertainty quantification in fine-tuned LLMs.
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Spectral Compact Training: Pre-Training Large Language Models via Permanent Truncated SVD and Stiefel QR Retraction
SCT pre-trains LLMs by keeping weights as compact SVD factors with Stiefel QR retraction, delivering up to 199x memory reduction per layer and allowing 70B-parameter training on a Steam Deck.
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BOOST: BOttleneck-Optimized Scalable Training Framework for Low-Rank Large Language Models
BOOST delivers 1.46-2.27x end-to-end speedups for low-rank bottleneck LLMs by redesigning tensor parallelism around the bottleneck structure plus supporting optimizations.
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CR-Net: Scaling Parameter-Efficient Training with Cross-Layer Low-Rank Structure
CR-Net uses cross-layer low-rank residuals in a dual-path network plus specialized recomputation to outperform prior low-rank methods on 60M-7B model pre-training while using less compute and memory.
-
Low-rank surrogate modeling and stochastic zero-order optimization for training of neural networks with black-box layers
A framework combining stochastic zeroth-order optimization and dynamic low-rank surrogate modeling with an implicit projector-splitting integrator enables end-to-end training of hybrid neural networks containing black-box physical layers and reaches near-digital accuracy on vision, audio, and text任务
-
Memory-Efficient Differentially Private Training with Gradient Random Projection
DP-GRAPE reduces memory in differentially private neural network training by using random Gaussian projections on gradients instead of SVD, achieving comparable privacy-utility tradeoffs to DP-SGD and scaling to 6.7B parameter models.
-
PrefixMemory-Tuning: Modernizing Prefix-Tuning by Decoupling the Prefix from Attention
PrefixMemory-Tuning decouples the prefix from attention to overcome performance limits of traditional prefix-tuning and reaches competitive results with modern PEFT methods on LLM adaptation benchmarks.
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GWT: Scalable Optimizer State Compression for Large Language Model Training
GWT projects gradients into wavelet subspaces to compress optimizer states for memory-efficient LLM training while claiming performance parity with full-rank updates.
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Proof-of-Learning with Incentive Security
The paper introduces an incentive-secure Proof-of-Learning protocol for blockchain consensus that claims provable security against two attacks, reduced computational overhead, and guarantees even with untrusted problem providers and verifiers.
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Pion: A Spectrum-Preserving Optimizer via Orthogonal Equivalence Transformation
Pion is an optimizer that preserves the singular values of weight matrices in LLM training by applying orthogonal equivalence transformations.
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Rethinking Local Learning: A Cheaper and Faster Recipe for LLM Post-Training
LoPT achieves competitive task performance in LLM post-training by limiting task gradients to the upper model half and training the lower half with local feature reconstruction.
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ELAS: Efficient Pre-Training of Low-Rank Large Language Models via 2:4 Activation Sparsity
ELAS pre-trains low-rank LLMs by applying 2:4 activation sparsity after squared ReLU to cut memory and accelerate training with minimal performance loss.
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HTMuon: Improving Muon via Heavy-Tailed Spectral Correction
HTMuon modifies Muon to produce heavier-tailed updates and weight spectra via HT-SR theory, yielding up to 0.98 lower perplexity on LLaMA pretraining and serving as a plug-in for other Muon variants.
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On the Convergence Analysis of Muon
Convergence analysis shows Muon outperforms gradient descent by exploiting low-rank structure in neural network Hessians.
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Agentic Driving Coach: Robustness and Determinism of Agentic AI-Powered Human-in-the-Loop Cyber-Physical Systems
A Lingua Franca reactor-based method is proposed to address nondeterminism in agentic AI for human-in-the-loop cyber-physical systems such as driving coaches.
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MUON+: Towards More Effective Muon via One Additional Normalization Step for LLM Pre-training
Muon+ adds one normalization step after polar orthogonalization in the Muon optimizer, yielding lower training and validation perplexity and faster pre-training across 60M-7B models.
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AdaFRUGAL: Adaptive Memory-Efficient Training with Dynamic Control
AdaFRUGAL automates FRUGAL's static hyperparameters with linear decay on subspace ratio and loss-aware update frequency, delivering competitive accuracy with lower memory and faster training on C4, VietVault, and GLUE.
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Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey
A comprehensive survey of PEFT algorithms for large models, covering their performance, overhead, applications, and real-world system implementations.
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Small Language Models (SLMs) Can Still Pack a Punch: A survey (updated 2026)
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