Deep Learning Optimization

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7 papers in the last four weeks, up 133% on the four weeks before. 0.1% of all new papers.

Jul 13Week of Sep 28

Latest papers 132

Jul 1, 2026cs.LG

Class-Grouped Normalized Momentum and Faster Hyperparameter Exploration to Tackle Class Imbalance in Federated Learning

Class imbalance poses a critical challenge in federated learning (FL), where underrepresented classes suffer from poor predictive performance yet cannot be addressed by standard centralized techniques due to privacy and heterogeneity constraints. We propose FedCGNM (Federated Class-Grouped Normalized Momentum), a client-side optimizer in FL that partitions classes into a small number of groups based on minimum within-group variance, maintains a momentum per group, normalizes each group momentum to unit length, and uses the summation of the normalized group momentums as an update direction. This design both equalizes gradient magnitude across majority and minority groups and mitigates the noise inherent in rare-class gradients. We further provide a theoretical convergence analysis explicitly accounting for time-varying resampling-rates. Additionally, to efficiently optimize these rates in small-client regimes, we introduce FedHOO, an X-armed-bandit (XAB) based algorithm that exploits federated parallelism that evaluates many combinations of two candidate rates per client at linear cost. Empirical evaluation on four public long-tailed benchmarks and a proprietary chip-defect dataset demonstrates that FedCGNM consistently outperforms baselines, with FedHOO yielding further gains in small-scale federations.

Haemin Park, Diego Klabjan, Martin W. Braun +2

Department of Industrial Engineering & Management Sciences, Northwestern University, Evanston, IL, USA · Intel Corporation, Chandler, AZ, USA

Class-Imbalanced Learning Deep Learning Optimization Hyperparameter Optimization Federated Learning

Jul 1, 2026cs.LG

From Approximation to Emergence: A Theory of Deep Learning

Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the object it controls, the assumptions that make it valid, and the phenomena it leaves unexplained. Written for researchers, graduate students, and mathematically trained practitioners, this monograph offers a rigorous map of deep learning theory as it stands today: powerful, incomplete, and increasingly centered on the question of how learned mechanisms arise from scale, data, architecture, and training.

Zhilin Zhao

Sun Yat-sen University

Neural Network Generalization Deep Learning Optimization In-Context Learning Neural Network Approximation Theory Neural Scaling Laws

Jun 29, 2026cs.LG

Gradient Smoothing: Coupling Layer-wise Updates for Improved Optimization

Deep neural networks with repeated architectural blocks, such as transformers, often exhibit structured relationships across layers that emerge during training. Motivated by this observation, we introduce \emph{Depth-wise Gradient Augmentation}, a general optimization paradigm in which the update applied to each layer is obtained by transforming the collection of block-wise optimizer updates along the depth dimension. Within this framework, we study \emph{Gradient Smoothing}, a family of depth-wise smoothing methods, and instantiate it with a simple local \emph{Window Smoothing} operator. The resulting method operates directly on block-wise updates produced by arbitrary base optimizers (e.g., SGD, Adam, Muon), incurs minimal computational overhead, and is compatible with existing optimization pipelines. We evaluate Gradient Smoothing across a diverse set of architectures and training regimes, including language model pretraining, RL post-training of LLMs for reasoning, diffusion modeling, and image classification with Vision Transformers. Across these settings, Gradient Smoothing consistently improves optimization and generalization performance without modifying model architectures or training objectives. We further show that it promotes more structured representation evolution across depth, consistent with its interpretation as a structured depth-wise preconditioning method. Together, these results establish Depth-wise Gradient Augmentation as a promising framework for exploiting cross-depth structure in optimization and demonstrate Gradient Smoothing as a simple and broadly applicable instantiation.

Haoming Meng, Anton Sugolov, Vardan Papyan

Department of Mathematics, University of Toronto, Toronto, Canada · Vector Institute, Toronto, Canada

Deep Learning Optimization Neural Network Optimization

Jun 28, 2026cs.LG

Dead-Direction Conditioners: Gauge-Equivariant Preconditioning for Deep Networks

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a

G

-equivariant one: it conditions the optimizer's state in the orbit decomposition of a

G

-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient

\barΘ= Θ/G

. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head

O(d_{\rm head})

attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.

Tejas Pradeep Shirodkar

IIIT, Hyderabad

Deep Learning Optimization Neural Network Optimization Adaptive Gradient Methods

Jun 25, 2026cs.DC

DMuon: Efficient Distributed Muon Training with Near-Adam Overhead

Matrix-orthogonalization-based optimizers, exemplified by Muon, have demonstrated strong convergence behavior across a wide range of modern deep learning workloads. The matrix-aware updates offer a compelling alternative to conventional element-wise optimization, particularly as model architectures continue to grow in scale and heterogeneity. Yet contemporary distributed training infrastructure built around the assumption of element-wise optimizers is poorly matched to matrix-level optimizers such as Muon, whose updates couple entire weight matrices and require costly Newton-Schulz iterations. Vanilla Muon implementations incur more than 2x the cost of forward and backward passes. To close this gap, we present DMuon, an open-source distributed Muon implementation that integrates into existing training pipelines as a drop-in module, with no framework-level modifications. Across both embodied foundation model and large language model (LLM) training workloads, DMuon achieves a 1.48x-3.01x speedup in end-to-end step time and a 6.85x-163.00x speedup in optimizer-step time, bringing per-step latency to near-AdamW levels and enabling efficient scaling in our model training.

Vincent Chen, Starrick Liu, Regis Cheng +8

X SQUARE ROBOT TEAM

Deep Learning Optimization Distributed Optimization Muon Optimizer Distributed Training

Jun 24, 2026cs.LG

Tensorion: A Tensor-Aware Generalization of the Muon Optimizer

Common first-order optimizers, such as Adam, implicitly treat each parameter block as an unstructured vector, which disregards the multilinear weight structure present in many modern machine learning models. Recent work has shown that exploiting matrix structure can improve optimization dynamics. A notable example is Muon, which performs steepest descent under the spectral norm constraint. We take the next step and introduce Tensorion, a tensor-aware optimizer that extends Muon's constrained optimization perspective from matrices to higher-order tensors. Tensorion is built around a linear minimization oracle (LMO) over a tensor norm ball. The norm is carefully chosen to balance two objectives: tightly bounding the tensor spectral norm, while still keeping the LMO tractable. This LMO becomes computable because it reduces to operations on adaptively selected unfolding matrices. Notably, when restricted to order-2 tensors (i.e., matrices), Tensorion recovers Muon exactly. Experiments on tensor-based computer vision problems suggest that Tensorion can offer improved convergence behavior and more stable gradient updates compared with Adam-based and existing tensor-aware baselines in the evaluated settings.

Vladimir Bogachev, Vladimir Aletov, Alexander Molozhavenko +2

HSE University · Basic Research of Artificial Intelligence Laboratory (BRAIn Lab)

Deep Learning Optimization Muon Optimizer

Jun 24, 2026cs.LG

An Analysis of Posterior Collapse, Parameterization and Initialization in Variational Deep Gaussian Processes

DGPs are probabilistic models with remarkable prediction performance that concatenate GPs across several layers. Exact inference in DGPs is intractable, and variational inference is often used to approximate the posterior with a parametric distribution tuned by minimizing the Kullback-Leibler divergence. Moreover, finding a good VI approximation is challenging. In particular, a problem of VI is posterior collapse, where VI converges to a variational posterior that matches the prior. In variational DGPs, this implies explaining the data as noise. This work studies posterior collapse in DGPs and identifies its connection to the DSVI algorithm and the widely used linear prior mean function employed in all but the last layer. We show that the benefit of the linear prior mean does not arise from avoiding the non-injective pathology in very deep DGPs, as previously believed, but from improving the conditioning of the optimization problem at initialization. Thus, we propose an alternative initialization of a zero prior mean DGP that mimics a DGP with a linear prior mean at initialization. This enables successful training of DGPs without imposing optimization-driven constraints on the prior, allowing to choose the prior based on modeling assumptions rather than optimization convenience. Our analysis considers three common parameterizations of DGPs and shows that not all of them benefit from a linear prior mean. We also explain why a whitened parameterization of the \DGP provides more stable convergence, something often assumed from experience, but lacking a rigorous analysis. Furthermore, we show that this stability is also beneficial to avoid the posterior collapse problem. Extensive experiments validate our findings: the proposed initialization prevents posterior collapse, improves stability, and achieves performance comparable to (and sometimes better than) DGPs with a linear prior mean.

Francisco Javier Sáez-Maldonado, Juan Maroñas, Daniel Hernández-Lobato

Department of Computer Science and Artificial Intelligence Universidad de Granada Granada, Spain · Department of Quantitative Methods CUNEF Universidad Madrid, Spain · Machine Learning Group at Universidad Autónoma de Madrid +1

Deep Learning Optimization Posterior Collapse Gaussian Processes Variational Inference

Jun 23, 2026cs.LG

Training for the Model You Return: Improving Optimization for Iterate-Averaged Language Models

Many modern Language Model (LM) pipelines return an averaged model, such as an exponential moving average of the training iterates, rather than the final iterate itself. This raises a fundamental question: given that we will return an iterate average, how should we change training to improve the performance of this average? We study this question by formulating optimizer design for the iterate-average estimator as an optimal-control problem. In a continuous-time stochastic quadratic model, we solve for the control strategy that minimizes the error of the returned average subject to a penalty on the size of the intervention. A practical approximation to this controller yields PACE, a lightweight wrapper around AdamW that pulls the live weights toward their exponential moving average with a clipped, per-coordinate control strength. We prove that a stylized version of PACE converges at the standard stochastic convex optimization rate, up to a factor depending on the averaging rule, while in the quadratic setting it can strictly improve the limiting squared error of the iterate-average estimator and can do so by an arbitrarily large factor on some instances. Empirically, our results suggest that PACE improves over AdamW and EMA-evaluated AdamW in supervised fine-tuning of 1-2B parameter LMs and in GPT-2 pretraining on FineWeb for a wide range of learning rates, decay schedules, and other hyperparameters.

Kwok Chun Au, Adam Block

Department of Computer Science, Columbia University · Department of Electrical Engineering, Columbia University

Deep Learning Optimization LLM Training

Jun 22, 2026cs.LG

Fast and Slow Variational Continual Learning

Continual learning remains a major challenge for modern deep networks, partly because commonly used optimizers lack inherent mechanisms for continual adaptation. One such natural mechanism is fast and slow adaptation to balance stability and plasticity. This mechanism has deep roots in neuroscience and biology, but there is no consensus on how to best incorporate it in commonly used optimizers. Here, we show that this can be easily done via the VCL framework, where past posteriors are used as priors in the future. Our key idea is to incorporate slow adaptation via merging of past posteriors to slow down the drift in the knowledge as learning progresses. The merged posterior is then used as the prior in the VCL update to implement the fast-weight updates. These steps can be seamlessly implemented in the IVON optimizer, whose form and costs are nearly identical to that of Adam. We call this new optimizer the Continual IVON (CoVON) optimizer and show that it not only consistently improves over existing VCL optimizers, but also performs better than other weight-regularization strategies across domain-incremental learning, continual pre-training, and fine-tuning of large language models.

Subarnaduti Paul, Yohan Jung, Mohammad Emtiyaz Khan +3

Faculty of Mathematics and Computer Science, University of Bremen, Bremen, Germany · Department of Computer Science & AI, Jeonbuk National University, Republic of Korea · Department of Computer Science, TU Darmstadt, Darmstadt, Germany +3

Continual Learning for LLMs Deep Learning Optimization Continual Learning Domain-Incremental Learning

Jun 19, 2026cs.LG

Towards Understanding the Power and Limits of the Muon Optimizer: A River-Valley Perspective

Recently, Muon has gained substantial attention as an appealing alternative to Adam-like optimizers, with many works highlighting its advantages through spectral normalization and improved conditioning. Yet this positive theoretical narrative contrasts with its empirical performance in large language model (LLM) training, where Muon's gains over Adam/AdamW are often mixed, schedule-sensitive, and not uniformly superior. To address this gap, we develop a trajectory-level theory characterizing both the strengths and limitations of Muon. We introduce a mixed-spiked matrix sensing model whose sensing operator decomposes into signal, spike, and bulk components, capturing a mixture of anisotropic structure and long-tail information reminiscent of LLM training. On top of it, we adopted a river-valley perspective in which we view the landscape as composed of a river direction flowing to the desired solution and hill directions encoding nuisance or task-irrelevant information. In the momentum-free setting, we show that Muon moves faster along the information-bearing river direction during early optimization, but can converge much more slowly near the river bottom than gradient descent. We then extend the river-valley perspective to general nonconvex objectives with momentum by studying points on the spectral river. There, while Muon converges faster early on, its orthogonalized update removes residual scale information, making it prone to overshooting and oscillation near the target solution. Together, these results suggest that our characterizations extend beyond spiked matrix sensing and motivate switching to GD-like refinement optimizers in the final phase, rather than relying only on a fixed learning-rate schedule for Muon. We also provide preliminary evidence supporting this two-stage approach in language model training experiments.

Tianqi Shen, Jinji Yang, Runze Shi +3

Department of Computer Science City University of Hong Kong (CityUHK) · Department of Statistics and Data Science University of Pennsylvania · School of Statistics and Data Science Shanghai University of Finance and Economics

Deep Learning Optimization Muon Optimizer LLM Training

Jun 19, 2026cs.LG

Breaking chains with trees: Deep learning with $\mathcal{O}(\log N)$ parallel time complexity

Modern deep neural network architectures are trained via backpropagation, which requires errors to be sequentially propagated through all layers before parameters can be updated. This introduces two limitations: locking, where layer-wise updates are strictly interdependent and cannot proceed in parallel, and the weight transport problem, which requires symmetric forward and backward pathways for exact gradient computation. These constraints restrict parallelism, increase memory and communication overhead, and pose challenges for scalable learning. In this work, we propose Hierarchical Block-Local Learning (HBLL), a framework that decomposes deep neural networks into hierarchically linked blocks trained using local learning objectives derived from variational principles, eliminating the need for full end-to-end backpropagation while maintaining effective information propagation across the network. HBLL is the first algorithm that is able to train deep neural networks in

\mathcal{O}(\log N)

parallel time complexity, where

N

is the number of network layers. We show that HBLL implicitly defines a family of subnetworks corresponding to different hierarchical paths, enabling flexible inference with different effective numbers of layers. We evaluate HBLL on a set of challenging vision and language modeling tasks, achieving competitive performance. We also extend HBLL to recurrent sequence architectures, applying to settings that otherwise rely on backpropagation through time.

Neeraj Mohan Sushma, Aditya Nagarsekar, Cabrel Teguemne Fokam +4

Center for Cognitive Interaction Technology, Bielefeld University, Germany · Ruhr Universität Bochum, Germany · Birla Institute of Technology and Science, Pilani, Goa Campus +1

Deep Learning Optimization Neural Network Optimization Local Learning

Jun 18, 2026cs.LG

Towards Robust Training in NNGPT AutoML Pipeline: A Loss-Optimizer Pairing Selection Study

The choice of loss function and optimizer is an important decision, that shapes further model training. Yet automated architecture search pipelines (AutoML) benefits significantly more from the optimal pairing selection and vice versa. This paper investigates whether a single recipe is sufficient for heterogeneous architecture pools, or whether the optimal pairing varies across structurally diverse models. We conduct a systematic empirical study of all

3 \times 6 = 18

combinations of six optimizers (SGD+Momentum, Adam, AdamW, RMSprop, Adagrad, Adadelta), paired with three loss functions: Cross-Entropy (CEL), Negative Log-Likelihood (NLL), and the recently introduced genetically evolved NGL loss across the base models presented in LEMUR heterogeneous architecture pool on six image classification datasets (CelebA-Gender, CIFAR-10, CIFAR-100, ImageNette, MNIST, SVHN). The 18 loss-optimizer configurations are applied to each of the 33 compatible base architectures taken from the LEMUR pool, resulting in 594 variants that were generated fully automatically by a source-level injection pipeline and evaluated under fixed hyperparameters, ensuring that observed accuracy differences are attributable solely to the loss-optimizer pairing. Our results confirm that no single pairing is universally optimal. Cross-Entropy with Adam or AdamW is the most robust choice across architecture families and datasets. NGL is a competitive alternative to CEL on standard convolutional classifiers, but only when paired with adaptive optimizers; it degrades substantially with SGD or accumulation-based methods. Adagrad and Adadelta consistently underperform under fixed hyperparameters regardless of loss function, highlighting their sensitivity to learning rate tuning. These findings provide actionable guidance for loss-optimizer selection within NNGPT Framework.

Anton Abramochkin, Radu Timofte, Dmitry Ignatov

Computer Vision Lab, CAIDAS & IFI, University of Würzburg, Germany

Deep Learning Optimization Neural Network Optimization Hyperparameter Optimization

Jun 18, 2026cs.AI

eCNNTO: A Highly Generalizable ConvNet for Accelerating Topology Optimization

This work proposes an element-based Convolutional Neural Network (CNN) to accelerate density-based Topology Optimization (TO), termed eCNNTO. TO generally undergoes a large number of iterations, where finite element analysis is performed in every iteration, leading to the efficiency bottleneck especially when dense meshes are used to achieve high-resolution designs. To address this limitation, eCNNTO is proposed to build upon Kallioras et al. (2020), where a Deep Belief Network (DBN) was trained for every element to predict its near-optimal density from its early history, thereby skipping the great majority of iterations and significantly accelerating the TO procedure. However, the method lacks spatial correlations among neighboring elements and may lead to disconnected features in the final structure. The proposed method employs CNN with residual connections to address this issue. On top of it, a novel training strategy is introduced to further enhance the optimization efficiency, where the training dataset consists of the final stage density histories rather than early ones. This change can also help reduce the required training data size. eCNNTO requires only a small dataset to train and yet it can be generalized to problems with largely different boundary conditions, loading cases, design domain geometries, mesh resolutions, as well as non-design domains. In the end, the generalization capabilities and efficiency of eCNNTO are demonstrated through a variety of examples in two and three dimensions, achieving up to 90% and 97% reduction of iterations, respectively.

Shengbiao Lu, Xiaodong Wei

Global college, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, Shanghai, China

Topology Optimization Deep Learning Optimization Convolutional Neural Networks

Jun 15, 2026cs.AR

NeuronFabric: A Software Reference Architecture for On-Chip Transformer Training with Local Adam

Publicly documented accelerator architectures generally separate training computation from optimizer-state updates or rely on external memory and host orchestration. This paper presents NeuronFabric, a software reference architecture intended for future FPGA and ASIC implementations of transformer training with local Adam updates. A complete C# prototype implements forward pass, backpropagation, and Adam optimization without external machine-learning frameworks. The goal is to validate numerical correctness and memory requirements before hardware implementation. The evaluated model is a 334K-parameter autoregressive transformer (d=88, H=4, f=264, L=4, vocab=256) trained on the Shakespeare corpus. The BF16W configuration achieves evaluation loss 1.5426 after 80K samples, compared with 1.5224 for an FP32 GPU reference, while producing coherent character-level text. The paper introduces BF16W, which stores weights in BF16 while retaining Adam optimizer moments in FP32. This reduces memory requirements for on-chip training. A 334K-parameter FP32 model with Adam moments requires approximately 4.0 MB, matching the BRAM capacity of a Xilinx ZCU102 device. The BF16W variant requires approximately 3.34 MB, leaving memory available for activation storage. We describe the vocabulary-budget constraint observed during earlier experiments, quantify BF16W memory savings, and outline FPGA training as the next stage of development. No FPGA measurements are included in this paper. This publication serves as a public architectural disclosure and software reference implementation for future FPGA and ASIC exploration of the NeuronFabric architecture.

Evgeny Ukladchikov

Independent Researcher

Deep Learning Optimization Transformer FPGA-Based Neural Network Acceleration

Jun 14, 2026math.OC

Schattor: Schatten-family methods for deep learning optimization

Modern deep learning optimization features heterogeneous parameter structures, noisy gradients, and highly nonconvex landscapes, posing significant challenges for both algorithm design and theoretical analysis. Motivated by the limitations of SGD and the success of adaptive optimizers, we propose {\it Schattor}, a family of adaptive first-order methods based on Schatten norms. Schattor unifies SGD and the recently proposed matrix-variate adaptive optimizer Muon within a single Schatten-norm-based framework. We establish dimension-free stationarity guarantees for methods in the Schattor family for stochastic matrix optimization problems via a novel matrix martingale moment bound. We also develop multi-block extensions that adaptively balance block-wise optimization progress and prove dimension-free stationarity guarantees in this more general setting.

Bohao Ma, Junyu Zhang, Chuan He

School of Data Science, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong, China · Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore · Department of Mathematics, Linköping University, Sweden

Deep Learning Optimization Stochastic Optimization Convergence Matrix Optimization Adaptive Gradient Methods

Jun 13, 2026cs.LG

When to use what Schatten- $p$ norm in deep learning?

Schatten-

\infty

based optimizers such as Muon have shown promising empirical performance, but there remains seemingly conflicting observations regarding whether they are beneficial. We resolve this conflict by showing that the conclusion is regime dependent. Even when the objective is smooth in the Schatten-

\infty

geometry, smaller Schatten-

p

geometries can be optimal, specifically in the low-dimensional regime, which we show includes Chinchilla scaling. This conclusion follows from a new noise-robust acceleration result for the SODA framework for

p>2

. The same analysis explains why Muon-like methods do not require warmup, why they naturally favor large batches, and yields a batch size scaling rule for arbitrary

p

.

Thomas Pethick

Deep Learning Optimization Muon Optimizer Nonconvex Optimization Stochastic Optimization Convergence Accelerated Gradient Methods

Jun 12, 2026cs.LG

Beyond a Single Explanation of the Adam--SGD Gap

Prior work has identified several factors that can contribute to the performance gap between Adam and SGD, spanning data aspects, architecture design, and optimization properties. Yet these explanations are often studied in isolation, leaving their relative importance unclear. In this work, we revisit these hypotheses through a controlled empirical study across vision, language, genomics, and graph tasks, spanning modern and classical architectures, and carefully designed training setups. Our results suggest that no single factor consistently explains the Adam--SGD gap. For instance, the Adam advantage can (1) persist under a uniform vocabulary distribution yet nearly disappear under a heavy-tailed one; (2) reverse in favor of SGD in softmax-attention models; and (3) become larger under soft architectural modifications, e.g., when ReLU is replaced by a GeLU nonlinearity. This suggests that the gap arises from nontrivial data and architecture interactions, rather than from a single common factor. Yet, we observe a pattern across our settings: a \emph{crossover batch size} at which the relative advantage shifts from SGD to Adam as the batch size scales. These empirical results are captured by our theoretical gap model, which predicts this batch-size-dependent crossover. Our perspective helps reconcile several existing hypotheses while offering practical insights across domains.

Chenxiang Zhang, Rustem Islamov, Enea Monzio Compagnoni +3

University of Luxembourg · MPI for Intelligent Systems · University of Basel +2

Deep Learning Optimization Adaptive Gradient Methods Stochastic Gradient Descent

Jun 11, 2026cs.LG

Gefen: Optimized Stochastic Optimizer

AdamW is a default optimizer for deep learning, but its moment states add two parameter-sized buffers to training memory, increasing the cost of large-scale pretraining. We propose Gefen, a memory-efficient optimizer that automatically shares second-moment estimates across parameter blocks and quantizes the first moment using a learned codebook. Gefen reduces AdamW's optimizer memory footprint by up to 8x while maintaining performance, saving 6.5 GiB per billion parameters. Prior work shares second moments across parameters grouped along the Hessian's block-diagonal structure, but relies on hand-specified architectural rules and leaves unexplained why such grouping works. We prove that large mixed Hessian entries constrain the ratio of squared gradients toward one, explaining why shared second moments are accurate when the squared gradients they pool are similar. The Hessian need not be computed: its block structure is inherited by squared gradients, allowing blocks to be found directly. Gefen therefore infers block structure from initial squared gradients, requiring no architecture-specific metadata or user-tuned hyperparameters beyond AdamW defaults. Gefen learns an exact histogram-based dynamic-programming quantization codebook and reuses the blocks for first-moment scaling. Across diverse pretraining experiments, Gefen achieves the lowest peak optimizer memory among compared methods that maintain AdamW-level performance. In single-machine and distributed training, the reduced footprint enables larger microbatches and substantially improves throughput over AdamW, making Gefen a drop-in replacement that can train larger models or use larger global batch sizes. We provide the complete Python implementation, including fused CUDA kernels at https://github.com/ndvbd/Gefen

Nadav Benedek, Tomer Koren, Ohad Fried

Reichman University · Tel Aviv University, Google Research

Deep Learning Optimization Memory-Efficient Optimization Adaptive Gradient Methods Memory-Efficient Training

Jun 11, 2026cs.LG

LoRA-Muon: Spectral Steepest Descent on the Low-Rank Manifold

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.

Franz Louis Cesista, Katherine Crowson, Cédric Simal +1

Ateneo de Manila University · EleutherAI · NaXys +1

Deep Learning Optimization Riemannian Optimization Muon Optimizer Low-Rank Adaptation Parameter-Efficient Fine-Tuning

Jun 10, 2026cs.LG

Simplicity Suffices for Parameter Noise Injection in Stochastic Gradient Descent

Injecting noise into the optimization process is a well-established technique for improving the training and generalization of deep neural networks. Yet, despite the breadth of existing approaches, it remains unclear which design choices truly matter in practice. In this work, we investigate parameter noise injection for stochastic gradient descent, focusing on two key questions: how to efficiently pair each training example with its own perturbation in mini-batch training, and whether sophisticated noise parameterizations or multi-sample gradient averaging yield meaningful gains over simpler alternatives. To address the first question, we leverage a distributional identity for linear layers that allows per-example noise injection without breaking batched computation. To address the second, we systematically compare several diagonal Gaussian parameterizations against an isotropic baseline across varying noise levels on CIFAR100. Our results consistently show that simple, lightweight strategies, isotropic noise with a single perturbed forward pass per update step, recover most of the benefit of more complex schemes. These findings suggest that simplicity suffices for parameter noise injection, and that practitioners need not resort to elaborate perturbation designs to reap the optimization and generalization benefits of noisy SGD.

Benjamin Leblanc, Louis-Jacob Lebel, Teddy Kana +1

Universit´e Laval

Neural Network Generalization Deep Learning Optimization Stochastic Gradient Descent

Jun 9, 2026cs.CV

5% > 100%: Flatness Preference is All You Need for Multimodal Parameter-Efficient Fine-Tuning

Parameter-Efficient Fine-Tuning (PEFT) methods provide a streamlined and efficient tool for adapting large models to domain-specific multimodal downstream tasks. Although these methods proved their tangible effects in practice, their principal aspects remain under-explored. Therefore we remain curious about the underlying generalization mechanisms in various PEFT methods and how they can be further enhanced. In this paper, we reveal the flatness preference widely present in various PEFTs, where a small fraction of sharp dimensions dominates the generalization of PEFT. This finding suggests an appealing possibility: we may be satisfied with a better generalization by merely attending to this small fraction of sharp dimensions instead of all of them. Furthermore, we propose Flatness Preference Optimization (FlatPO) to flatten these key sharpness dimensions, leading various PEFTs toward better generalization. Extensive experiments demonstrate the effectiveness of our findings and the proposed method. Code is available at https://github.com/Can-Lin/FlatPO.

Yifan Zhu, Can Lin, Hangjie Yuan +4

School of Computer Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, China · College of Computer Science and Technology, Zhejiang University, Zhejiang 310027, China · ETH Zürich, Switzerland

Deep Learning Optimization Flat Minima Parameter-Efficient Fine-Tuning

Jun 9, 2026cs.LG

FOGO: Forgetting-aware Orthogonalization Optimizer

We argue that forgetting is not confined to continual learning but is a general optimization phenomenon: during standard training, dominant mini-batch gradients suppress rare but useful update directions, causing short-term forgetting at every step. When such knowledge is never revisited, these losses compound into long-term forgetting-the classical failure mode of continual learning. We introduce FOGO, a scalable optimizer that continuously detects and resolves gradient interference across both regimes. FOGO spectrally orthogonalizes momentum updates to prevent dominant directions from monopolizing optimization, then stores representative past directions in a compact codebook memory built on random projection, where pairwise distances are provably preserved in low-dimensional space. At each step, conflicts between the current update and stored directions are resolved via lightweight orthogonal correction and lifted back through a proximal step, with minimal overhead and no data storage. Across class-imbalanced classification, continual visual learning under domain and class shifts, continual fine-tuning of LLaVA-7B, and GPT-2 pretraining, FOGO consistently improves convergence and knowledge retention, outperforming Adam and Muon.

Toan Nguyen, Yang Liu, Trung Le +2

School of Computer Science and Engineering University of New South Wales Sydney, Australia · Department of Data Science & AI Monash University Melbourne, Australia · DEVCOM Army Research Laboratory USA

Deep Learning Optimization Continual Learning Catastrophic Forgetting

Jun 4, 2026cs.LG

Principles and Practice of Deep Representation Learning: or a Mathematical Theory of Memory

In the current era of deep learning and especially generative models, there is significant investment in training very large deep neural networks. Thus far, such models have been "black boxes" that are difficult to understand in the sense that they have opaque internal mechanisms, leading to difficulties in interpretability, reliability, and control. Naturally, this lack of understanding has led to both hype and fear. This book is an attempt to "open the black box" and understand the mechanisms of large deep networks, through the perspective of representation learning, which is a major factor - arguably the single most important one - in the empirical power of deep learning models. A brief outline of this book is as follows. Chapter 1 will summarize the threads that underlie the whole text. Chapters 2, 3, 4, 5, and 6 will explain the design principles of modern neural network architectures through optimization and information theory, reducing the process of architecture development (long having been described as a sort of "alchemy") to undergraduate-level linear algebra and calculus exercises once the underlying principles are introduced. Chapters 7 and 8 will discuss applications of these principles to solve problems in more paradigmatic ways, obtaining new methods and models which are efficient, interpretable, and controllable by design, and yet no less - sometimes even more - powerful than the black-box models they resemble. Chapter 9 will discuss potential future directions for deep learning, the role of representation learning, as well as some open problems.

Sam Buchanan, Druv Pai, Peng Wang +1

UC Berkeley · TTIC · University of Macau +2

Deep Learning Optimization Representation Learning Neural Network Interpretability

Jun 4, 2026cs.LG

Double Preconditioning (DoPr): Optimization for Test-Time Performance, not Validation Loss

Many modern applications of deep learning involve training a neural network via a one-step prediction loss (e.g.,

L^2

regression, cross-entropy), but deploy the network by rolling out along its own predictions. Key examples include autoregressive language modeling, flow-based generative modeling, and robot policy learning. It is well-documented that these settings induce a phenomenon we call test-time feedback (TTF): the mismatch between the training/validation loss and downstream metrics of interest, such as task success rate and generation quality, which grows with task length. While data curation, architecture, and objective design have been proposed to combat train-test shift in TTF settings, this paper proposes optimization as a new design axis to mitigate error accumulation. Specifically, we introduce a new optimization paradigm called double-preconditioning (DoPr) uniquely tailored to the challenges of TTF. DoPr combines gradient-wise preconditioning, as in Adam and Muon, with activation-wise preconditioning (AP), such as in KFAC. We show that the addition of AP yields a drop-in intervention for increasing downstream model performance across a range of TTF settings. Interestingly, these gains in test-time performance do not consistently accompany improvements in validation loss, opening new questions about how to properly evaluate models trained with one-step supervised objectives.

Thomas T. Zhang, Alok Shah, Yifei Zhang +3

University of Pennsylvania · Carnegie Mellon University · Amazon FAR

Deep Learning Optimization

Jun 2, 2026cs.LG

Denoise First, Orthogonalize Later: Understanding Momentum in Muon via Spectral Filtering

Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear. Existing analyses of Muon either remove momentum to study spectral updates in isolation, or retain momentum without explaining why it improves empirical performance. Our work bridges this gap by showing momentum in Muon acts as a spectral filter. Under a structured signal-plus-perturbation gradient model, we prove that momentum suppresses perturbations while preserving the dominant signal, thereby enlarging the spectral gap between them. This enlarged gap stabilizes the singular subspaces of the matrix passed to Muon's orthogonalization step, making the resulting update more reliable. We further show that applying momentum before orthogonalization achieves provably stronger alignment with the signal component of the gradient than either reversing this order or simply removing momentum. Experiments across diverse tasks, including LLM pretraining, support our theoretical analysis. More broadly, our theory offers a starting point for understanding the benefits of momentum in other matrix-based optimizers.

Xianliang Li, Zihan Zhang, Weiyang Liu +1

1The Institute of Statistical Mathematics · 2The Graduate Institute for Advanced Studies, SOKENDAI · 3National Institute of Informatics +3

Deep Learning Optimization Momentum Methods Muon Optimizer Spectral Methods Gradient Descent Dynamics

Jun 1, 2026cs.LG

FOAM: Frequency and Operator Error-Based Adaptive Damping Method for Reducing Staleness-Oriented Error for Shampoo

Shampoo is attracting considerable attention for its superior performance on large-scale optimization benchmarks; yet it faces a significant practical bottleneck: the prohibitive computational overhead of matrix inversion. To mitigate this, practitioners typically rely on stale preconditioner updates, creating a fundamental trade-off between computational efficiency and optimization fidelity. In this work, we provide a theoretical study of staleness through the complementary lenses of convergence and stability. While staleness improves computational efficiency, it inherently degrades performance and introduces numerical instability. Crucially, we identify that damping, acting as a numerical stabilizer, can effectively suppress these negative effects. Guided by this analysis, we propose FOAM, an adaptive algorithm that stabilizes training by dynamically controlling both the damping factor and the eigendecomposition frequency based on an approximation of the staleness-oriented error. Experimental results demonstrate that FOAM reduces wall-clock time compared to standard Shampoo while maintaining robust convergence.

Kyunghun Nam, Sumyeong Ahn

KENTECH, Naju, Republic of Korea.

Deep Learning Optimization Optimization Convergence Analysis Algorithmic Stability Second-Order Optimization

May 29, 2026cs.LG

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks

We study optimal learning-rate selection in two-layer and three-layer linear neural networks trained to learn linear target functions. In particular, we derive the exact closed-form expressions for the gradients and test loss after one and two steps of gradient descent, enabling a precise characterization of early training dynamics. We characterize how learning rates should scale under the gradient approximation in the first two steps, and prove that performing updates with this approximation yields a tractable surrogate loss with a tight, small approximation error. This formulation enables the theoretical analysis of layer-wise learning rates and reveals a distinct early-training regime: test loss can be minimized by unequal learning rates at the initial step, while equal learning rates become optimal in subsequent steps. Our numerical experiments validate the theory and demonstrate the importance of balancing layer-wise learning rates early during training. The code is available at: https://github.com/TDCSZ327/Layer-Balancing.

Tianyu Pang, Vignesh Kothapalli, Shenyang Deng +3

Dartmouth College · Stanford University · Virginia Tech

Deep Learning Optimization Neural Network Training Dynamics Gradient Descent Dynamics Layer-Wise Learning Rate Adaptation

May 29, 2026cs.LG

Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling

Sign-based and LMO-inspired optimizers have recently attracted substantial attention in deep learning due to their strong performance and low memory footprint. However, their fixed-magnitude updates can hurt terminal convergence: they decouple update mechanisms from gradient magnitudes and fail to account for parameter heterogeneity, often leading to oscillation rather than convergence. We propose SoftSignum, a smooth relaxation of sign-based optimization that replaces the hard sign map with a temperature-controlled soft-sign transformation, enabling a parameter-wise transition from sign-like updates to magnitude-sensitive SGD-like steps. We complement it with an adaptive quantile-based temperature schedule and extend the same principle to matrix-valued optimizers, obtaining SoftMuon. We also develop a generalized geometry-relaxation framework based on strongly convex regularizers and Fenchel conjugates, proving convergence in stochastic non-convex setting. Experiments on diverse deep learning tasks, including LLM pretraining, show that SoftSignum and SoftMuon consistently improve over their hard sign-based counterparts and standard AdamW.

Dmitrii Feoktistov, Timofey Belinsky, Andrey Veprikov +2

HSE University, Moscow, Russia · BRAIn Lab, Moscow, Russia · Yandex Research, Moscow, Russia +3

Stochastic Optimization Sign-Based Optimization Deep Learning Optimization Nonconvex Stochastic Optimization Muon Optimizer

May 29, 2026cs.LG

Inconsistency-Aware Minimization: Improving Generalization with Unlabeled Data

Estimating the generalization gap and developing optimization methods that improve generalization are crucial for deep learning models, for both theoretical understanding and practical applications. Leveraging unlabeled data for these purposes offers significant advantages in real-world scenarios. This paper introduces a novel generalization measure, local inconsistency, derived from an information-geometric perspective on the parameter space of neural networks. A key feature of local inconsistency is that it can be computed without explicit labels. We establish theoretical underpinnings by connecting local inconsistency to the Fisher information matrix and the loss Hessian. Empirically, we demonstrate that local inconsistency correlates with the generalization gap. Based on these findings, we propose Inconsistency-Aware Minimization (IAM), which incorporates local inconsistency into the training objective. We demonstrate that in standard supervised learning settings, IAM enhances generalization, achieving performance comparable to that of existing methods such as Sharpness-Aware Minimization. Furthermore, IAM exhibits efficacy in semi- and self-supervised learning scenarios, where the local inconsistency is computed from unlabeled data.

Hee-Sung Kim, Hyeonseong Kim, Sungyoon Lee

Department of Computer Science, Hanyang University, Seoul, Korea · Agency for Defense Development, Daejeon, Korea

Neural Network Generalization Deep Learning Optimization

May 27, 2026stat.ML

Is Backpropagation Optimal? When Synthetic Gradients Improve Sample Efficiency

Backpropagation is the default learning rule for artificial neural networks and is often treated as the settled approach whenever differentiability is available. In this work, we revisit this convention through a theoretical lens of sample efficiency. We introduce a unified vectorized feedback framework for loss-based and reward-based learning on computational graphs, in which synthetic gradients emerge as a natural alternative to backpropagation. We characterize the conditions under which synthetic gradients can achieve a lower gradient-estimation mean squared error than backpropagation. We construct examples illustrating that this sample efficiency advantage can be arbitrarily large. Experiments on contextual bandits and reinforcement learning tasks demonstrate the potential of our theoretical findings.

Yibo Jacky Zhang, Zeyu Tang, Sanmi Koyejo

Stanford University

Deep Learning Optimization Backpropagation Data-Efficient Learning