Dmitrii Andriianov, Andrey Veprikov, Aleksandr Beznosikovcs.LG
Low-rank adaptation (LoRA) is the standard way to fine-tune large models, yet when its two factors are trained independently, the update ignores the geometry of the low-rank weight change it induces. We introduce LoRA-TSD, an optimizer that treats every LoRA step as a tangent vector of the fixed-rank matrix manifold and takes the spectral-norm steepest-descent step of Muon inside that tangent space, mapping the result back to the factors through a retraction native to the LoRA parametrization. The step avoids expensive operations on full weight matrices, and its retraction is up to $2.8\times$ cheaper than the truncated-SVD retraction used by prior manifold methods. We prove that the Frobenius-norm version of our surrogate recovers LoRA-Pro, and we identify the tangent-projected gradient, the Riemannian gradient of the manifold, as the stationarity measure natural to LoRA training and computable from the factor gradients alone. Under this measure we give the first global convergence guarantees for both LoRA-Pro and LoRA-TSD, with rates that drive the factor-gradient norms to zero. Across six commonsense and natural-language-inference benchmarks with Llama-3.2-1B, Llama-3.1-8B and Qwen3-32B, LoRA-TSD outperforms every competing LoRA optimizer and stays robust to the adapter rank. Code is available at https://github.com/brain-lab-research/LoRA-TSD.
Split federated learning (SFL) has emerged as a powerful paradigm for model training at the edge. However, SFL inherently involves discrete decision variables for model splitting and resource allocation, resulting in a challenging mixed-integer problem. Consequently, prior optimization schemes for SFL are either \textit{heuristic} or \textit{computationally inefficient}, which cannot handle large-scale user populations. To address this limitation, this work establishes an efficient optimization framework for SFL under resource-constrained networks. Our framework jointly optimizes model splitting and resource allocation to minimize training cost, which is defined as the weighted sum of latency and energy costs. We first study the model splitting problem and develop a polynomial-time algorithm that achieves the global optimum. Then, we extend the approach to the joint model splitting and resource allocation problem. In this case, we formulate it as a two-dimensional master problem and develop an efficient approximation method with a $(1+ε)$-approximation guarantee. Extensive experiments show that the proposed approach provides efficient solutions to strike the optimal energy--latency tradeoff.
Optimization over the Stiefel manifold plays a significant role in various machine learning tasks. Existing methods either use the retraction operators, requiring costly orthonormalization for large-scale matrices, or employ landing methods that rely on careful step size selection and penalty parameter tuning. To address these challenges, we propose a retraction-free and penalty parameter-free algorithm that directly lands on the manifold. By leveraging the strongly-convex-like property of the quadratic penalty function and the proximal smoothness of the Stiefel manifold, we establish global convergence guarantees with the best-known iteration complexities under both constant and diminishing step sizes. Then, we reformulate the low-rank adaptation (LoRA) fine-tuning problem for large language models as a manifold optimization problem, introducing Manifold-LoRA for geometry-accelerated adaptation. This approach employs the proposed landing technique and a carefully designed step size strategy to accelerate the training process. Numerical experiments on benchmark datasets demonstrate the efficiency and strong downstream performance of the proposed method.
The discovery of scaling laws has motivated training neural networks on ever increasing quantities of data. This is typically done with a constant decoupled weight decay which causes the network weights to shrink steadily over the course of training. Taking inspiration from the Robbins--Monro conditions, we propose to scale weight decay by the fraction of the peak learning rate $η/η_{\max}$. We prove that this scaled weight decay preserves the asymptotic stationarity guarantees of the corresponding unregularized methods for both stochastic gradient descent and the non-Euclidean spectral optimizer Muon, thereby avoiding the additional asymptotic bias introduced by constant decoupled weight decay. This retains the stability benefits of weight decay without changing the asymptotic optimization target. Using a steady-state analysis, we explain why under standard weight decay the weight norm shrinks steadily as training proceeds, whereas under scaled weight decay it settles to a roughly constant value. When applied to the training of mixture-of-experts models, Muon with scaled weight decay (Muon-SW) consistently outpaces Muon with identical hyperparameters, reaching the same validation loss $\mathbf{30\%}$ faster at our largest scale across models from $72 - 930$ million parameters trained at $\sim 600$ tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.
Mikail Khona, Aditya Vavre, Boxiang Wang +11cs.LG cs.AI
Higher-order optimizers such as Muon and SOAP offer faster convergence than AdamW, but their computational cost and numerical stability challenges have limited adoption at scale. In this work, we adapt and enhance preconditioned gradient methods to overcome the practical challenges of large-scale LLM pretraining. We first identify instabilities in SOAP at large batch sizes and propose algorithmic modifications including per-step QR orthogonalization and improved preconditioning strategies that eliminate loss spikes and enable stable training in these regimes. We then present a unified empirical study of SOAP, Muon, and AdamW using update-RMS matching to ensure fair learning rate transfer across optimizers. As part of this analysis, we empirically evaluate the orthogonalization quality of Muon. Our experiments on multi-billion-parameter models trained on trillions of tokens reveal that SOAP and Muon consistently outperform AdamW at the scales we tested. Notably, at batch sizes of up to 100M tokens for next-token prediction, these optimizers maintain training stability and quality while AdamW degrades. To enable efficient training at large scale, we introduce a layer-wise distributed optimizer compatible with Megatron-LM. Our implementation balances memory and hides communication while avoiding approximations to the optimizer computations, thus retaining their convergence benefits. Additionally, we identify and build specific system-level improvements to further accelerate our layer-wise implementation. To support the research community, we release a codebase that contains emerging algorithms for optimization: https://github.com/NVIDIA-NeMo/Emerging-Optimizers
Xiaolong Huang, Benjamin Thérien, James Harrison +1cs.LG
Learned optimization aims to improve upon hand-designed optimizers (e.g., Adam and Muon) by meta-learning small neural network optimizers over a distribution of tasks. While recent work has greatly advanced the architectural design and inductive biases of learned optimizers (LOs), their meta-training remains biased toward short-unroll learning on particular tasks, resulting in redundant computation and leaving LOs often unable to compete with hand-designed optimizers. We introduce Efficient Long-hOrizon (ELO) learning, an efficient meta-training algorithm that (1) reallocates wasted meta-training compute to longer failure regimes, achieving efficient long-horizon learning, and (2) enforces decoupled progressive expert supervision, providing stable meta-learning signals that additionally improve the generalization of LOs. Our empirical study evaluates ELO for meta-training both element-wise and matrix-based LOs. Across downstream language modeling (GPT-2-124M/350M on FineWeb) and image classification (ViT-B/16, ResNet-50 on ImageNet-1K) tasks, ELO substantially improves the long-unroll performance and out-of-distribution generalization of the base LOs. In particular, ELO-Celo2 consistently outperforms well-tuned AdamW across all evaluated tasks, while remaining competitive with Muon on language modeling. \textit{Notably, all ELO baselines require less than 7 H100 GPU-hours for meta-training.}
Federated Transformer training increasingly relies on local AdamW, whose adaptive updates can provide much stronger local progress than SGD-based training. However, under heterogeneous client data, even globally corrected AdamW updates may remain highly uneven in coordinate-wise reliability. We refer to this phenomenon as coordinate trust mismatch. Existing federated adaptive optimizers mainly address mismatch at the client-update or communication-round level, but still apply the corrected adaptive direction densely and uniformly across coordinates. In this paper, we propose FedACT, a global-aware coordinate trust modulation method for federated AdamW training. FedACT first forms a globally corrected adaptive direction and then reallocates update magnitudes according to a coordinate-wise trust score, assigning larger steps to coordinates jointly supported by local gradients and global correction, while preserving smaller non-zero updates on the remaining coordinates. Extensive experiments on federated vision Transformers, CNNs, LLM pre-training, and LLM fine-tuning show that FedACT consistently improves over strong federated adaptive baselines, with the largest gains on Transformer models under stronger data heterogeneity. Mechanism analyses further show that FedACT improves cross-client direction consistency, suggesting that coordinate-level trust allocation effectively complements round-level global-local correction. Code will be released.
Efficient optimization is essential for training large language models. Although intra-layer selective updates have been explored, a general mechanism that enables fine-grained control while ensuring convergence guarantees is still lacking. To bridge this gap, we propose \textbf{MGUP}, a novel mechanism for selective updates. \textbf{MGUP} augments standard momentum-based optimizers by applying larger step-sizes to a selected fixed proportion of parameters in each iteration, while applying smaller, non-zero step-sizes to the rest. As a nearly {plug-and-play} module, \textbf{MGUP} seamlessly integrates with optimizers such as AdamW, Lion, and Muon. This yields powerful variants such as \textbf{MGUP-AdamW}, \textbf{MGUP-Lion}, and \textbf{MGUP-Muon}. Under standard assumptions, we provide theoretical convergence guarantees for \textbf{MGUP-AdamW} (without weight decay) in stochastic optimization. Extensive experiments across diverse tasks, including MAE pretraining, LLM pretraining, and downstream fine-tuning, demonstrate that our \textbf{MGUP}-enhanced optimizers achieve superior or more stable performance compared to their original base optimizers. We offer a principled, versatile, and theoretically grounded strategy for efficient intra-layer selective updates, accelerating and stabilizing the training of large-scale models. The code is publicly available at https://github.com/MaeChd/MGUP.
Zhihan Zhang, Alexander Le Metzger, Jiuyang Lyu +10cs.AR cs.AI
Embedded devices from wildlife monitoring stations to clinical wearables require local AI inference due to latency, communication, or privacy constraints. Optimizing models for heterogeneous microcontrollers (MCUs) requires simultaneously satisfying hard physical constraints on memory, power, and temperature while preserving accuracy, a multidimensional optimization that is today performed manually by experts. We ask whether an LLM agent can autonomously navigate this complex, multi-turn pipeline guided by real hardware feedback, and introduce a hardware-in-the-loop agent arena in which the agent iteratively refines both model and firmware -- compiling, flashing, and measuring on real hardware -- to enable closed-loop optimization. Frontier models, including Claude Opus 4.7 and Gemini 3.1 Pro, fail entirely without hardware feedback (0% deployment success), whereas our hardware-in-the-loop formulation achieves the first successful deployment within three iterations and can surpass human expert results within seven. This agentic co-optimization achieves 250x compression for vision models with <3.3% accuracy loss and 400x for audio with <6% Feature Error Rate loss, enabling battery-free operation on a commercial MCU via solar harvesting. We demonstrate practical impact in two real-world systems: an elk-detection camera trap (96.7% accuracy) and a phonetic-transcription wearable (8.44% FER) for child development research.
Kannanthodath Induchoodan Ajay Menon, Christian Prehofer, Yunfei Xu +1cs.LG
Personalized federated learning (PFL) is one of the important approaches in federated learning for addressing statistical heterogeneity while enabling client-specific adaptation. Many PFL methods split the model into shared and personalized parameters, which are jointly trained on each client. However, this creates an optimization issue: shared parameters are updated by clients optimizing different local objectives, which can lead to inconsistent shared updates and weaken the shared representation. To address this problem, we propose Federated Shared Parameter Correction (FedSPC), a modular correction method for PFL. FedSPC applies control-variate correction only to the shared parameters of a given PFL method, while leaving personalized parameters unchanged. It can be integrated into three common PFL settings: shared feature extractors, shared classifiers, and fully shared models with local regularization. Experiments on CIFAR-100 and Tiny-ImageNet with ViT, ResNet-34, and VGG-11 show that FedSPC improves performance across representative PFL methods, including FedPer, FedRep, FedBABU, LG-FedAvg, and Ditto.
Franz Louis Cesista, Katherine Crowson, Cédric Simal +1cs.LG cs.AI
Low-Rank Adaptation (LoRA) significantly reduces compute and memory costs for finetuning Deep Learning models but is often harder to tune than dense training: when using factor-wise optimizers such as AdamW, it is sensitive to initialization choices, its optimal learning rates transfer poorly across ranks, and it often fails to beat dense baselines. We derive LoRA-Muon by applying the Muon optimizer's spectral steepest-descent rule to the low-rank setting. Along with our split weight-decay rule, our main claim is that LoRA-Muon is a good low-rank proxy for full-rank Muon and Shampoo-family optimizers. Its optimal learning rates transfer across rank, width, depth, and factor-rescaling. In our compute-matched TinyShakespeare study, a rank-$2$ proxy recovers the dense best tested learning rate, and a rank-$32$ LoRA-Muon run attains lower mean validation loss than the dense baseline in the seed-averaged sweep. We further show that the Spectron optimizer depends on arbitrary factor scaling, so it would likely be a poor fit when finetuning starts from badly imbalanced factors, and that LoRA-RITE's simplified QR-coordinate core implements the same spectral update. LoRA-Muon computes that update without QR-decomposition and avoids storing second moments, making it more accelerator-friendly and memory-efficient.
This paper introduces range regularization for federated learning with linear systematic components to enhance statistical accuracy and induce cross-client regularity conducive to quantization, coding, and resource efficiency. Our approach identifies features with shared weights across different clients and adaptively clusters the weights of personalized features at extreme values, a process we refer to as polar clustering. Theoretical analysis of the associated estimators poses significant challenges due to the seminorm nature and non-decomposability of the regularizer. We develop new proof techniques for the nonasymptotic analysis of statistical accuracy and faithful pattern recovery. Moreover, a fast optimization algorithm that leverages varying degrees of local strong convexity is proposed to reduce iteration complexity. Experiments support the efficacy and efficiency of the proposed approach.
The heterogeneity of client data and systems makes it difficult to achieve satisfactory convergence speed and robustness in federated learning with random client selection. To address this issue, this paper proposes a surrogate-assisted client evolutionary selection framework for federated learning. In this framework, some typical client selection strategies are first used to generate candidate sets, and a metric function that integrates model performance, communication latency, and energy consumption is developed to formulate the client selection problem as a combinatorial optimization one. Subsequently, a surrogate model is constructed using the candidate selections and metric to efficiently approximate the performance of selected client subsets. An evolutionary algorithm is employed to search the combinatorial space of client selections, guided by the surrogate model to accelerate convergence. Experiments on MNIST, CIFAR10, CINIC10, and TinyImageNet demonstrate that the proposed algorithm achieves faster convergence, lower energy consumption, and improved robustness compared to existing methods.
We propose a preconditioning (PC) layer, a weight parameterization via polynomial preconditioner that ensures stable weight conditioning throughout LLM training. The PC module reshapes the singular-value spectrum of weight matrices via low-degree polynomial preconditioning. After training, the preconditioned weights can be merged back into the original architecture, incurring no inference overhead. We demonstrate the advantage of the proposed PC layer over standard transformers in Llama-1B pre-training, for both the AdamW and Muon optimizers. Theoretically, we justify this spectrum-control principle by proving that uniformly bounding each layer's singular values ensures geometric convergence of gradient descent to global minima, for certain deep linear networks. Our code is available at https://github.com/Empath-aln/PC-layer.