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 128

Oct 6, 2026cs.LG

DeltaTTT: Layerwise Optimization for Nonlinear Recurrent Memory

Sequential test-time training adapts a memory network through successive updates, each computing an inner-loop gradient based on the network's previous state. Intuitively, this state dependence should allow each update to account for what the memory has already learned and better incorporate new information. However, we find that this expected advantage does not consistently materialize in nonlinear memories: a fixed-base parallel TTT baseline outperforms its serial counterpart. Our exploratory experiments point to a key underlying difficulty: nonlinear memories can be harder to optimize than linear ones within a single pass over the sequence. To alleviate this optimization difficulty, we introduce DeltaTTT, which replaces joint inner-loop optimization of a two-layer memory network with layerwise learning. Each layer is assigned a local prediction target and updated through a state-dependent delta rule. This formulation retains a nonlinear readout while enabling chunkwise parallel computation. Experiments on DeltaNet and LaCT backbones show improvements in language modeling and retrieval over their recurrent baselines.

Yining Li, Dongchen Han, Jie Fu +1

Qiuzhen College, Tsinghua University · LeapLab, Tsinghua University · IQuest Research

Deep Learning Optimization Neural Network Optimization Test-Time Training Language Modeling

Oct 6, 2026cs.LG

How Bregman Divergences Shape Shampoo

Understanding the principles behind Shampoo has recently guided the development of more effective neural network optimizers. These methods learn a preconditioner by optimizing the Frobenius or Kullback-Leibler (KL) divergence against the gradient second moment. In this work, we investigate how the choice of divergence shapes preconditioning, which remains unclear and blocks further improvements. To do so, we develop a unified Bregman divergence framework that connects all popular divergences, allowing us to study them jointly. Through empirical spectral analysis of gradient second moments, we examine how divergence choice shapes Kronecker approximation and interacts with finite-sample error in preconditioning. We find that some divergences can better compensate for finite-sample underestimation of the empirical second moment, helping explain the differing behavior of their corresponding Shampoo variants. We further validate this explanation through GPT-2 pretraining experiments. By connecting divergence choice to practical training behavior, we believe our framework provides principled guidance for understanding the foundations of, and further improving, Shampoo.

Bing Liu, Wenjie Zhou, Chengcheng Zhao +4

Zhejiang University · University of the Chinese Academy of Sciences · Peking University +2

Deep Learning Optimization Functional Bregman Divergences Neural Network Optimization

Oct 5, 2026cs.AI

Does Muon Need Fine-Grained Spectral Shaping?

Muon combines current and past gradients into matrix momentum. For

M=UΣV^\top

, the idealized polar update

Q=UV^\top

gives every singular direction the same weight. We refer to this as the flat profile. Several recent optimizers replace this flat profile with fine-grained spectral maps that give each direction its own gain. We ask how much of this spectral detail a Muon update needs. Our spectral diagnostics show that approximately

94

--

97\%

of measured singular modes lie below an estimated noise edge, yet collectively align positively with a reference gradient. We introduce BulkBoost, a two-band spectral reweighting framework with fixed-rank and noise-calibrated variants. The latter uses split-minibatch gradient differences to calibrate a Marchenko--Pastur reference edge for Muon's Nesterov input, separating the bulk below the edge from the spikes above it. Both variants increase the bulk's relative weight through one shared gain while preserving the Frobenius norm of each matrix's unreweighted direction. For a fixed partition, our theory gives the first-order condition under which moving weight toward the bulk lowers the loss. It also quantifies the fraction of the maximal first-order improvement rate, over all per-mode reallocations, that two bands can capture. Across 30 continued-pretraining settings spanning Pythia-14M to 410M and six corpora, two-band reweighting is competitive with the fine-grained power-law profile of Freon and outperforms Spectra. Measured against Muon's flat profile, Freon reduces final loss by

0.022\%

of the pre-adaptation loss on average, whereas the two-band variants achieve reductions of

0.073

--

0.147\%

. These observations suggest that useful departures from the flat profile are surprisingly low-dimensional: a single bulk-to-spike gain captures at least as much benefit as the fine-grained spectral profiles.

Meher Chaitanya, Tianyi Zhou, Aristides Gionis

KTH Royal Institute of Technology Stockholm, Sweden

Deep Learning Optimization Muon Optimizer

Oct 5, 2026cs.LG

Learning Pareto Stationary Fronts via Single-Pass Backpropagation

We propose MOSEL (Multi-Objective Stackelberg Efficient Learning), a framework for a posteriori multi-objective optimization (MOO) in deep neural networks that recovers a full front of Pareto stationary solutions at the computational cost of standard single-objective training. MOSEL reformulates the problem as a bilevel optimization problem that leverages network modularity to decouple representation learning from objective-preference alignment. Casting the bilevel optimization problem as a Stackelberg game enables solving the original a posteriori MOO problem in a single forward-backward pass. As a result, MOSEL matches the time and memory efficiency of standard single-objective training while enabling scalable Pareto stationary front learning. Empirically, MOSEL uncovers diverse and optimal Pareto frontiers in strongly conflicting settings (e.g., fairness-accuracy). Remarkably, even in weakly conflicting regimes such as multi-task learning, it consistently converges to solutions closer to the utopia point, outperforming both standard single-objective training and specialized multi-task learning methods. These results highlight the broader potential of a posteriori MOO learning as a pathway to efficiently learn more diverse and robust representations, ultimately improving generalization.

Elina Rojin Celik, Marcos Medeiros Raimundo, Isabel Valera

Department of Computer Science Saarland University, Germany · Instituto de Computação Universidade Estadual de Campinas, Brazil

Deep Learning Optimization Multi-Objective Optimization Pareto Optimization

Oct 5, 2026cs.LG

Mind the Drift: Diagonal Linear Networks Under Large Learning Rates

Large learning rates can qualitatively change the trajectory of neural network training, often pushing optimization into regimes far from classical gradient-flow behavior. The Edge of Stability (EoS) offers a valuable lens on the dynamics such learning rates induce. We study corresponding dynamics in diagonal linear networks, where we uncover a competition between two distinct implicit biases that jointly determine the sparsity of the recovered solution in regression settings. Complementary to the Gain, which captures the average discretization error accumulated by Gradient Descent relative to Gradient Flow, we derive a closely associated but overlooked quantity: the Drift. Under large learning rates, it describes an imbalance between different discretization errors and represents a systematic shift in the optimization trajectory. While the Gain grows monotonically in certain regimes, and can bias towards denser, flatter interpolators, the impact of the Drift depends on its alignment with potential solutions, which can either counteract or reinforce the effect of the Gain. Consequently, its behavior drives model selection, particularly during early training epochs. To validate our theoretical insights, we introduce an intervention that actively steers the Gain to recover sharper, sparser solutions. Thus, our analysis reveals that large learning rates do not universally hinder the recovery of sparse solutions. On the contrary, they can be harnessed to control the implicit bias of training.

Aniket Sanyal, Tom Jacobs, Rebekka Burkholz

CISPA Helmholtz Center for Information Security · Technical University of Munich · Konrad Zuse School of Excellence in Learning and Intelligent Systems (ELIZA)

Edge of Stability Deep Learning Optimization Neural Network Training Dynamics Implicit Regularization

Oct 5, 2026cs.LG

ORCA: The Annealed Spectral Conditioning Optimizer for Faster, Better LLM Training

Modern LLM optimizers such as Muon often produce weight matrices with higher effective rank than Adam, yet further spectral control has delivered only modest gains. We identify a tension behind this result: concentrated spectra can suppress gradient directions in coupled weight matrices and slow optimization, while constraints maintained throughout training can limit task-specific adaptation and raise the attainable loss floor. We introduce ORCA (Orthogonal Regularization, Cooled After), a minimal optimizer intervention that applies strong but temporary soft orthogonality regularization early in training, then removes it. This allows the weights to benefit from a broader spectrum early on and adapt freely afterward. Across LLaMA, Qwen3, and fine-grained mixture-of-experts models ranging from 130M to 8B parameters, ORCA achieves lower final validation loss than Muon. Its loss reduction relative to Muon matches or exceeds Muon's reduction relative to Adam. Ablations support the early-shaping, later-release design. Further, ORCA requires no architectural changes and adds minimal overhead.

Yuanshi Liu, Boyuan Jiang, Liang Hou +4

School of Intelligence Science and Technology, Peking University · Kling Team

Spectral Regularization Deep Learning Optimization LLM Training

Oct 1, 2026cs.LG

SoftServe: A Scalable Quasi-Newton Method for Deep Learning

Quasi-Newton (QN) methods have long been among the most effective methods for large-scale unconstrained convex optimization. Two obstacles have limited their use in deep learning: non-convexity and enormous parameter sizes. We introduce SoftServe, a family of QN methods designed to overcome these obstacles without line searches or ad hoc curvature corrections. SoftServe derives positivedefinite curvature estimates from the variational objective of Berglund et al. (2025), even in the presence of negative curvature. We develop diagonal and Kroneckerfactored variants that preserve positive definiteness by construction and scale to massive neural networks. Finally, SoftServe relies on the stable coupled Newton-Schulz iteration for the required matrix operations, replacing costly matrix decompositions with GPU-friendly matrix multiplications. SoftServe excels on problems that are severely ill-conditioned, including tasks such as recurrent networks, deep autoencoders, physics-informed neural networks, and a 136M-parameter physics-informed diffusion model, often achieving lower losses than established baselines including Adam, Muon, and SOAP.

Joohwan Ko, Tetiana Parshakova, Diana Cai +1

University of Massachusetts Amherst · CCM, Flatiron Institute · Cornell University

Deep Learning Optimization Second-Order Optimization Nonconvex Optimization Quasi-Newton Methods

Sep 30, 2026cs.LG

Increasing Width Allows Greedy Layer-wise Training to Rival End-to-End Backpropagation in Self-Supervised Learning

End-to-end backpropagation has been the dominant mode of training in deep learning, allowing for the coordination of parameter updates across layers of a neural network. Prior studies have explored alternative -- and, in some cases, simpler -- training mechanisms, showing that they can sometimes achieve performance similar to backpropagation. However, the architectural conditions under which locally optimized networks, which avoid end-to-end backpropagation of error, can learn representations comparable to those learned through end-to-end training remain unclear. We aim to answer this question in the context of self-supervised learning, an important framework for large-scale pretraining in artificial intelligence. Here, we investigate how network width and depth affect the efficacy of greedy layer-wise and end-to-end self-supervised training in convolutional networks. We find that in wider networks, the benefits of end-to-end backpropagation over greedy layer-wise training shrink: in relatively shallow and very wide networks, we even observed higher performance in models trained with greedy layer-wise training. Subsequent analysis of the representations formed by these networks shows that very wide greedy-trained networks exhibit more favorable representational geometry than do networks trained end-to-end with backpropagation. This work shows that width can compensate for restricted credit assignment and identifies differences in representational geometry as a potential mechanism for their improved performance.

Syon Mansur, Joel Zylberberg

Jules Stein Eye Institute University of California, Los Angeles Los Angeles, CA 90095, USA

Deep Learning Optimization Convolutional Neural Networks Neural Network Training Dynamics Self-Supervised Visual Representation Learning

Sep 30, 2026cs.LG

From Spectra to Joint Schedules in LLM Pre-training: 3+3(+2) Scaling-Law Regimes

Power-law learning curves are often treated as fixed properties of a model and its data, although learning-rate and batch-size schedules can change the observed loss. We study this dependence in noisy online SGD with linear random features. Conditional on the representation, an exact Volterra equation separates two response components: a forcing term that propagates unresolved target error and a memory kernel that propagates stochastic-error injections. We prove that either component follows a power law if and only if its cumulative weighted spectral mass has the corresponding low-spectrum scaling; individual eigenvalues and target coefficients need not obey coordinatewise power laws. Under a joint schedule, intrinsic time

T_t=\sum_{s<t}η_s

controls optimization progress, while

r_t=B_t/η_t

controls noise injection. Their interaction yields sharp conditions under which a schedule preserves, changes, or destroys the clean power law, together with a memory ceiling on noise reduction. The power-law random-feature model realizes this mechanism in

3+3(+2)

propagation regimes with phase-dependent compute rates. Controlled nanoGPT experiments show that (1) learning-rate and batch-size schedules with matched

B/η

paths are nearly equivalent in intrinsic time, (2) a forcing-memory surrogate accurately predicts loss across schedules, and (3) its fitted exponents across real-world datasets identify the regime of LLMs in

3+3(+2)

map.

Yichen Wang, Fanghui Liu, Yudong Chen

Department of Computer Sciences, University of Wisconsin–Madison, USA. · School of Mathematical Sciences, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, China.

Language Model Pretraining Deep Learning Optimization Learning Rate Scheduling Language Model Scaling Laws Stochastic Gradient Descent

Sep 28, 2026cs.LG

EvE: An Alternate Optimizer to Adam

Adam and its variants dominate neural network training, but a single run only reveals whether a configuration works well after most of its budget is spent, a poor fit for hyperparameter or architecture search, where configurations must be ranked cheaply and pruned early. We introduce EvE (Evolutionary Explorer), a steady-state, population-of-four differential evolution (DE) optimizer with a targeted Adam fallback: each iteration proposes one candidate via DE, running a short burst of gradient descent only if the DE step fails to improve on the incumbent. Selection is greedy, so on a deterministic objective the best-so-far value is provably monotone non-increasing, and since gradients are used only as a targeted rescue, per-iteration cost stays within a constant factor of a single Adam step regardless of dimension. Under a fixed, evaluation-cost-matched budget, EvE wins or ties Adam on 76% of 70 (problem, dimension) cells across seven scalable benchmarks up to one million variables. On three real neural-network tasks (an MLP on MNIST, and LoRA fine-tuning of a 1.5B-parameter language model on two datasets) EvE finishes the same charged budget 1.7-3.9x faster, at a modest cost in final quality (about one accuracy point on MNIST, 9-11% higher relative test loss on the two fine-tuning tasks; on GSM8K, Adam is about 5 accuracy points more accurate, and fine-tuning lowers accuracy below the base model for both). Inside successive halving on UCI Adult, EvE completes hyperparameter and architecture searches 3.1-3.5x faster, ranking configurations about as consistently with Adam as Adam does with itself across seeds (Kendall's tau 0.66-0.69). EvE is not a total replacement for Adam as a final-stage trainer, but a fast, gradient-aware proxy for the search-heavy, budget-constrained regime one level up.

Shashank Raj, Kalyanmoy Deb

Department of Computer Science The University of Alabama Tuscaloosa, AL 35487, USA

Deep Learning Optimization Differential Evolution Evolutionary Optimization

Sep 24, 2026stat.ML

Low-Rank Friction for Memory-Efficient Transformer Pretraining

iKFAD is a recently proposed optimiser that replaces adaptive learning rates with adaptive friction in the momentum dynamics, yet performs as well as Adam. Its limitation is that the full friction tensor

ξ\in\mathbb{R}^{m\times n}

carries the same

\mathcal{O}(mn)

memory overhead per layer as Adam's second-moment buffer. Here we replace iKFAD's friction tensor

ξ

with a rank-1 outer-product factorisation built from row and column momentum statistics, resulting in Rank-1 iKFAD (R-iKFAD). This reduces the friction memory footprint from

\mathcal{O}(mn)

to

\mathcal{O}(m+n)

per layer, which approximately halves iKFAD's total optimiser state. Despite this reduction, R-iKFAD maintains parity in performance with iKFAD: experiments on GPT2-Nano, TinyViT, DistilBERT and GPT2-S confirm that it matches or exceeds iKFAD while nearly halving the memory footprint and remaining comparably robust to hyperparameters. We analyse the continuous-time dynamics in two damping regimes. For linear damping (

γ>0

) we prove exponential convergence under strong convexity. For

γ=0

, the preferred option in our experiments, the friction is generated entirely from past momentum and switches off as the momentum vanishes, so geometric convergence cannot be shown. We nonetheless prove convergence to the minimiser, together with matching upper and lower bounds on the energy: of order

t^{-1}

when the regularisation scale

ε_{\mathrm{stab}}

is zero, and of order

t^{-1/2}

when it is positive. To our knowledge this is the first convergence rate for a rank-1 factored optimiser in continuous time, and the first such result that does not require positive damping.

Rajit Rajpal, Benedict Leimkuhler

School of Mathematics University of Edinburgh Edinburgh, UK EH9 3FD

Deep Learning Optimization Low-Rank Compression Memory-Efficient Training

Sep 10, 2026cs.LG

AdamX: Cosine similarity meets gradient descent

We introduce AdamX, a first-order optimizer that incorporates cosine similarity as an adaptive mechanism for controlling update magnitudes. The proposed method is scalable, model-agnostic, and straightforward to integrate into existing training pipelines. We further introduce a variance rectification scheme that promotes smoother optimization during the early stages of training. Overall, we provide empirical evidence that AdamX achieves competitive convergence rates across a range of benchmark datasets and architectures. Performance is evaluated in terms of the number of epochs required to reach predefined performance thresholds under a fixed hyperparameter budget. Code and Experiments available at: https://github.com/FranciscoCaldas/adamX.

Francisco Caldas, Ruben Belo, Cláudia Soares

NOVA School of Science of Technology Universidade Nova de Lisboa, Caparica, Portugal

Deep Learning Optimization Adaptive Gradient Methods

Sep 8, 2026cs.CV

BrachistoneLR: A Brachistochrone-Inspired Learning-Rate Schedule and a Controlled Benchmark of Scheduling Policies

The learning-rate schedule is a consequential choice in training deep networks, yet the policies in common use are heuristic, and published comparisons are hard to read, because architecture, dataset, and budget tend to vary alongside the schedule. We study BrachistoneLR, a schedule built by mapping the vertical coordinate of the brachistochrone, the curve of fastest descent under gravity, onto the range between a peak and a floor rate. Expanding the definition shows it to be cosine annealing with the half-period set to E - 1 instead of E, the configuration a standard implementation gives when its period argument is one less than the number of epochs. The rate therefore reaches its floor at the last epoch trained rather than one epoch later, and we show this difference decays as E^-2, making it a short-horizon effect. We then benchmark six schedules over 72 runs on three image classification datasets (MNIST, Fashion-MNIST, CIFAR-10) and four architecture families (fully connected, convolutional, recurrent, residual), fixing the optimizer, data pipeline, and evaluation protocol so that only the schedule varies. Schedules that fall smoothly from peak to floor beat the constant rate and calendar-based decay by margins that grow with task difficulty, reaching 2.5 points of dataset mean on CIFAR-10. Within that leading group, BrachistoneLR, cosine annealing, and warmup-cosine lie within 0.06 accuracy points and 0.17 of a mean rank, which one seed per configuration cannot separate. BrachistoneLR is best on both residual networks and has the highest CIFAR-10 mean, and it sets no milestones, decay factor, warmup length, or restart period. We conclude that the shape of a schedule matters more than its parameterization, that the choice of whether to use a smooth schedule matters more than the choice among them, and that the terminal-rate distinction is worth attention only over short horizons.

Md. Sadekur Rahman Roni, Md. Jalal uddin Chowdhury, Moutusi Dash Nimi

Department of Computer Science and Engineering Leading University Sylhet, Bangladesh · DeepNet Research and Development Lab Sylhet 3100, Bangladesh · Department of Electrical and Electronic Engineering Leading UniversitySep Sylhet, Bangladesh

Deep Learning Optimization Learning Rate Scheduling

Aug 23, 2026cs.LG

Beyond Dense Adam States: Adaptive Log-Space Quantization for Memory-Efficient Optimizers

Optimizer-state quantization is commonly designed for Adam's dense, parameter-aligned first- and second-moment arrays. This abstraction breaks for memory-efficient optimizers, whose states may be factored, confidence-modulated, or maintained in a projected space, so similar reconstruction error can produce different update error. We formulate optimizer-state quantization as a joint problem over representation, topology, and update semantics. We then introduce Adaptive Log-Space (AL) quantization for non-negative states. AL fits each block's observed nonzero logarithmic interval and reserves a separate code for exact zero, enforcing

q = 0 \Leftrightarrow x = 0

; signed momentum and state precision remain independently selectable. Controlled probes show that adaptive ranges reduce update error and temporal drift, exact-zero reservation preserves dormant states, and state topology constrains useful block granularity. End-to-end language-model training evaluates the resulting policy across dense, factored, confidence, and projected optimizer states. On TinyLlama-1.1B, AL8 with uniform 8-bit momentum reaches 72.90 perplexity versus 73.54 for bitsandbytes 8-bit AdamW, with comparable optimizer-state storage and higher throughput. CAME matches reference-level final perplexity across three seeds when its non-negative states use AL16, while a semantic grouping-and-protection policy closes most of quantized Adafactor's 100K-step late-loss gap. These results make state topology and update semantics first-class design constraints for optimizer quantization.

Yan Wang

Independent Researcher

Deep Learning Optimization Memory-Efficient Optimization Low-Bit Quantization Neural Network Quantization

Aug 19, 2026cs.LG

Activation-Keyed Momentum: An Anisotropic Momentum Update via the Delta Rule

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a key and an output-side error that acts as a value. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under

μ

P, and its extra compute stays between

22.2\%

and

25.0\%

of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to

46.39 \pm 4.32\%

fewer steps at 67M and

22.12 \pm 0.80\%

at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.

Euijin Hong, Guannan Qu

Electrical and Computer Engineering, Carnegie Mellon University

Deep Learning Optimization Momentum Methods Neural Network Optimization Accelerated Gradient Methods

Aug 13, 2026cs.LG

Momentum as Residual-Driven Multiplier Correction for Deep Learning Optimization

Momentum-based optimizers are widely used in modern deep learning, yet the relations among momentum recursion, update geometry, and acceleration remain only partially understood. We develop an

\textbf{A}

DMM-

\textbf{I}

nspired

\textbf{M}

omentum (AIM) framework based on residual-penalty variable splitting, which interprets momentum as a multiplier-like correction driven by the splitting residual. AIM recovers the exponential moving average of gradients from an ADMM-style multiplier update and separates two mechanisms that are usually intertwined in practical optimizers: the residual penalty determines the update geometry, whereas the approximation of the objective-related subproblem determines the acceleration form. Building on AIM, we propose

\textbf{R}

elativistic

\textbf{A}

daptive gradient

\textbf{D}

escent with

\textbf{A}

ccelerated

\textbf{R}

esidual (RADAR), which combines relativistic adaptive geometry, decoupled residual correction, and second-order momentum filtering to improve the update direction and momentum estimation. We establish stochastic convergence through a variance-perturbed Lyapunov drift analysis. Experiments on supervised vision learning, language modeling, and reinforcement learning show that RADAR achieves consistent improvements over strong adaptive optimizer baselines.

Zhixin Ren, Yau Lyu, Congrong Li +2

Department of Mathmatical Science, Tsinghua University · School of Vehicle and Mobility, Tsinghua University · College of AI, Tsinghua University

Deep Learning Optimization Momentum Methods Adaptive Gradient Methods Accelerated Gradient Methods

Aug 3, 2026cs.LG

AOS: Adaptive Optimizer Switching via Training-State Signals for Faster Convergence and Better Generalization

Single-optimizer training is a poor fit for the distinct phases of deep network optimization: adaptive methods handle noisy early gradients well but overshoot flat minima, while SGD with momentum generalizes better in the late phase but converges slowly early on. We introduce AOS-R (Adaptive Optimizer Switching, Rule-Based), a lightweight controller that monitors six online gradient-space signals -- gradient noise scale (GNS), Hutchinson curvature trace, loss stagnation, update stability ratio, gradient stability index (GSI), and loss improvement ratio (LIR) -- and switches among AdamW, SGD-M, and Lion as the optimization landscape evolves. State-preserving momentum transfer and a 400-step learning-rate bridge prevent accuracy degradation at every transition point. On CIFAR-100/WRN-28x10, AOS-R reaches 78% top-1 in 81 epochs -- 26% fewer than AdamW (109), 43% fewer than SGD-M (143), and 16% fewer than Lion (96). Across eight model-dataset benchmarks, AOS-R achieves best accuracy on 6 of 8 combinations with a mean +0.4 pp gain and 0.80x convergence speedup over AdamW under a single shared hyperparameter configuration.

Alok Kumar Pandey, Umang Chaturvedi, Aatish Rana +1

Deep Learning Optimization Stochastic Optimization Convergence Adaptive Gradient Methods Neural Network Training Dynamics

Jul 30, 2026cs.LG

Towards joint scaling laws with optimal batch size schedules

Modern deep learning typically keeps the batch size static throughout training, thus overlooking the joint effect of learning rate and batch size on the training dynamics. In this paper, we study the deep learning dynamics through the lens of convex optimization and derive a joint characterization of loss in terms of both schedules, applicable to general optimizers and model architectures. This characterization yields a closed-form optimal batch size schedule for any prescribed learning rate schedule, and further leads to joint scaling laws that consistently outperform static batch size baselines, highlighting the significance of dynamic batch size schedule in large language model training.

Jiaxiang Li, Zhiqi Bu, Shiyun Xu

Meta · Independent researcher

Deep Learning Optimization Language Model Scaling Laws Neural Network Training Dynamics LLM Training

Jul 21, 2026cs.LG

Where Should Optimizer State Live? Tiered State Allocation for Memory-Efficient Mixture-of-Experts Training

Optimizer state is the largest single line item in the memory budget of mixture-of-experts (MoE) training. On a 6.78B-parameter MoE language model AdamW keeps 50.6 GB of first and second moments to update 12.6 GB of bfloat16 weights. We study SkewAdam, an optimizer built on the observation that the three parameter populations of an MoE differ enough in size and gradient statistics that they should not receive the same state. Those populations are the dense backbone, the experts and the router. SkewAdam keeps float32 momentum plus a factored second moment for the backbone (5% of parameters), a factored second moment alone for the experts (95%) and an exact second moment for the router (<0.01%). The resulting state occupies 1.29 GB or 2.6% of AdamW's and peak training memory falls from 81.4 GB to 31.3 GB, within the budget of a 40 GB accelerator. In a controlled comparison from identical initializations over 82M tokens, SkewAdam reaches validation perplexity 108.4, ahead of AdamW (126.8), Muon (120.2) and Lion (393.7), and settles router load balance to within 1% of its uniform floor. The allocation is not what earns that perplexity. A tier ablation reaches the same value while carrying twenty times the state, so the tiers buy memory rather than accuracy. Same-platform runs separate what does earn it. Removing momentum costs 31 perplexity points (tuned Adafactor, 139.7) and replacing the factored second moment and its update clipping with a full second moment costs 10 (tuned AdamW, 118.5), so neither tuned baseline reaches the untuned tiered policy. Where optimizer state lives, these results suggest, matters at least as much as how much of it there is.

Nuemaan Malik

Independent Researcher

Deep Learning Optimization Mixture-of-Experts Language Models Memory-Efficient Optimization Efficient Language Model Training Memory-Efficient Training

Jul 17, 2026cs.LG

Learning Faster without Deeper Networks: A*-Inspired Batch Selection for Efficient CNN Training

Common practice when training Convolutional Neural Networks (CNNs) is to use randomly shuffled mini-batches. This creates two limitations: slower convergence, and a diminishing learning signal, since many samples are quickly classified as easy during training. We address these inefficiencies with A*-Inspired Batch Selection (A*-BS), a lightweight, model-agnostic strategy that formulates mini-batch scheduling as a heuristic search problem. Each batch is treated as a node in a search space and ranked using an A*-like score combining a loss-based difficulty measure with a reuse penalty. This encourages informative gradient updates and batch diversity throughout training, without modifying network architectures or optimization algorithms, so it integrates seamlessly into existing pipelines. We evaluate A*-BS on the twelve 2D classification tasks of the MedMNIST-v2 benchmark, using a deliberately simple architecture of approximately 2.25x10^5 parameters, compared against the ResNet-18 and ResNet-50 baselines reported by the benchmark. On half of these tasks, the lightweight model with A*-BS reaches higher accuracy and AUC than both ResNet baselines, with relative gains of up to 15%. An ablation under identical architecture and hyperparameters shows A*-BS outperforms random batch shuffling on all twelve tasks. Wall-clock measurements further show the lightweight CNN with A*-BS trains substantially faster than ResNet-18 and ResNet-50 on identical hardware. These results indicate that intelligent batch ordering can partially compensate for reduced architectural complexity, offering a computationally efficient alternative to deeper models, with reliability reinforced by strong performance even against deeper, more sophisticated architectures.

Anxhelo Shehu, Enes Stastoli, Arben Cela

AI Laboratory, University Metropolitan Tirana, Tirana, Albania. · Laboratory of Images Signals and Intelligent Systems, ESIEE Paris, Paris, France.

Deep Learning Optimization Convolutional Neural Networks

Jul 14, 2026cs.LG

Reassessing Muon for Matrix Factorization

Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training. Its empirical success has motivated a growing body of theoretical work that interprets Muon as steepest descent under the spectral norm. Yet it remains unclear which of Muon's advantages stem from its update rule itself and which are artifacts of the scale, architecture, and data of modern deep networks. In this work, we isolate the optimizer from these confounding factors by studying Muon on a simple, well-understood, and spectrally structured problem: low-rank matrix factorization. Through a controlled comparison against carefully tuned adaptive baselines, we find that Muon does not consistently outperform AdamW in this setting and that several previously reported advantages are sensitive to hyperparameter choices. Our results provide a more nuanced picture of when spectrum-aware orthogonalization is beneficial and argue for evaluating modern optimizers on controlled problems in addition to end-to-end benchmarks.

Ali Parviz, Gal Mishne, Alex Cloninger

†. Halicioğlu Data Science Institute, UC San Diego. · ∗. Department of Mathematics, UC San Diego.

Deep Learning Optimization Muon Optimizer Adaptive Gradient Methods

Jul 10, 2026cs.LG

LionVote: Per-Layer Learning Rate Adaptation for Lion

Per-layer diagnostics reveal that, at the prescribed learning rate, Lion's effective scale is 2.6-2.8x too high for attention and MLP parameters and ~2x too high for normalization layers on ViT-Tiny/CIFAR-100; this 32% cross-layer-type disparity cannot be reproduced by a single global rate. The measurement comes from LionVote, a per-layer learning rate mechanism in which each parameter tensor maintains a compound level, a persistent integer updated every c epochs by two diagnostics (gradient direction stability and momentum health) resolved by a validation loss tiebreaker. Voting thresholds derive from geometric identities, the EMA time constant, and a noise-floor estimate; cadence is bounded structurally and selected by ablation. On ViT-Tiny/CIFAR-100, LionVote achieves 69.7% top-1 accuracy vs. Lion's 69.0% (p < 0.02, Welch's t-test) and AdamW's 68.8%. Per-layer adaptation value depends on both architectural heterogeneity and task; on uniform CNN architectures tuned SGD with cosine annealing remains dominant, and on ViT architectures gains are task-dependent.

Kris Atallah

New York University, New York, USA · New York University New York, NY, USA

Deep Learning Optimization Layer-Wise Learning Rate Adaptation

Jul 9, 2026cs.LG

Beyond Backpropagation: Monte Carlo Method Can Train Deep Neural Networks

Backpropagation (BP) dominates deep learning training, but its reliance on gradients brings inherent troubles -- vanishing and exploding gradients. The pursuit of gradient-free methods has long been a goal in the field of artificial intelligence. This paper shows that indeed the simplest Monte Carlo algorithm implemented on a single GPU -- randomly mutate a parameter, keep it if the loss decreases, otherwise retry -- can practically train deep networks. This gradient-free method does not even need common techniques such as batch normalization or residual connections to directly train sufficiently deep networks. More remarkably, its flexibility extends to several nontrivial scenarios: it enables pure pruning training, supports discrete weights, accommodates unconventional transfer functions such as Gaussian, and reveals the substantial redundancy of deep networks. We have demonstrated its feasibility on deep networks with more than 20 layers, single-hidden-layer wide networks with up to 16,384 hidden neurons, and even a simple Transformer architecture trained on both image classification (MNIST) and character-level language modeling (Tiny Shakespeare). This simple gradient-free method may offer a complementary perspective for understanding the self-organization and learning mechanisms of neural networks, and also provides an alternative route for building physically inspired deep learning systems.

Hong Zhao

Department of Physics, Xiamen University, Xiamen 361005, China · Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China

Deep Learning Optimization

Jul 7, 2026cs.LG

No Subspace to Track: Non-Identifiability and Optimizer State in Low-Rank Training

Memory-efficient optimizers such as GaLore train large language models by projecting gradients onto a rank-r subspace recomputed every T steps, assuming this subspace is a slowly drifting object that can be tracked. We show that beyond a small reproducible core, there is no such object. Two estimates of the top-r subspace computed at the same step from disjoint minibatches disagree as much as estimates computed T steps apart (0.73 vs 0.74 of the maximal chordal distance sqrt(2r), at Pythia-160M with r=128): the apparent rotation at each refresh is dominated by estimator noise. This holds across four model families in three architecture classes from 70M to 6.9B parameters, strengthening with scale, and more weakly in a vision transformer. Only ~39 of 128 directions are reproducible across minibatches, and averaging cannot recover the rest: under N-fold averaging the gradient's spectral tail shrinks as N^(-1/4) rather than the N^(-1/2) of pure noise, so no averaging budget makes the subspace well defined. What helps instead follows from treating each refresh as a change of coordinates for Adam's state. Carrying the second moment blindly is provably about (r-k*)/2 worse than the best rotation-blind estimator, while the first moment transports exactly through the rotation, the optimal linear map under isotropic gradients and the rule LDAdam uses. At 1B over 40k steps (3 seeds), full LDAdam reaches 18.7 perplexity at beta2=0.999, beating untransported GaLore after its best beta2 fix (19.3); shortening the second-moment memory to beta2=0.99 helps the refreshing optimizers, though for canonical GaLore the effect is small and a full-rank control reverses it. One measurable fact, subspace non-identifiability, clarifies why GaLore works, which patches work, and what to check before trusting a low-rank assumption: the reproducible rank k*.

Noel Thomas

Mohamed bin Zayed University of Artificial Intelligence

Deep Learning Optimization Parameter Identifiability Low-Rank Approximation LLM Training

Jul 5, 2026math.OC

Unified convergence analysis for gradient descent optimization methods in the training of deep neural networks

Gradient based optimization methods are nowadays the methods of choice for training deep neural networks (DNNs) in artificial intelligence (AI) systems. In practically relevant DNN training problems, one does usually not apply the standard gradient descent (GD) optimization method but instead one employs suitable sophisticated GD optimization methods, which incorporate adaptivity and/or acceleration techniques, such as the famous Adam optimizer. It is a key contribution of this work to provide a general unified convergence analysis for GD optimization methods in the training of DNNs with analytic activations such as the softplus and the popular Gaussian error linear unit (GeLU) activation. Our general unified convergence result applies to a large class of gradient based optimization methods such as the standard GD, the momentum, the Nesterov accelerated gradient (NAG), the RMSprop, the Adam, the Adamax, the Nadam, the Nadamax, the Adan, the AdaBelief, the AMSGrad, and the Yogi optimizers. Our analysis employs the theory of Kurdyka-Łojasiewicz (KL) inequalities to establish convergence to critical points in the training of DNNs. To the best of our knowledge, the generality of our convergence analysis is also just in the special situation of the Adam optimizer a new contribution to the literature on the analysis of AI optimization algorithms.

Shokhrukh Ibragimov, Arnulf Jentzen

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

Deep Learning Optimization Neural Network Optimization Convergence Analysis Adaptive Gradient Methods Gradient Descent Dynamics

Jul 4, 2026cs.LG

OmniOpt: Taxonomy, Geometry, and Benchmarking of Modern Optimizers

Optimizer selection for large-scale model training has become a system-level design decision constrained jointly by compute, memory, tuning budget, and task diversity, yet the landscape of over one hundred methods remains fragmented. We therefore present OmniOpt, a unified survey and benchmark cookbook of optimizers for the research community. OmniOpt rests on four coupled components. First, we treat every optimizer update as a structured transformation through a five-stage meta-pipeline, and show that most methods engage only one or two of these stages. Second, we use norm-constrained linear minimization oracles (LMOs) to unify different optimizers. Third, these two views ground a dual-dimension taxonomy, one dimension assigning each method to a mechanism family and the other recording the measurable training objectives it aims to improve. Fourth, and at the core of this paper, we instantiate the full taxonomy in a unified cross-domain benchmark spanning representative optimizers, model scales, and training regimes from language model pretraining to image classification, systematically analyzing each method family across multiple effect objectives and laying out their trade-offs. OmniOpt thus supplies the research community with an operational coordinate system for selecting optimizers under explicit mechanism and objective assumptions, and charts a direction for the future development of the optimizer community.

Siyuan Li, Jiabao Pan, Yumou Liu +9

Shanghai Artificial Intelligence Laboratory · Westlake University · Shanghai University +4

Deep Learning Optimization Neural Network Optimization

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

May 26, 2026cs.LG

The Stability of Singular Distribution: A Spectral Perspective on the Two-Phase Dynamics of Language Model Pre-training

Large language model pre-training typically exhibits a two-phase trajectory: a fast initial loss drop followed by a prolonged slow improvement. We identify an underlying spectral phenomenon, Stability of Singular Distribution (SoSD), where the trace-normalized singular value spectrum stabilizes early, even as parameter matrices continue to evolve. We demonstrate that synchronization between SoSD and the slow-descent regime is widely observed across diverse architectures (GPT-2, LLaMA) and settings, including various schedules (Step-wise, WSD, Cosine Decay), weight decays, and optimizers (AdamW, Muon). By analyzing a simplified Transformer, we prove that growing weight norms inevitably precipitate an early SoSD threshold, after which the rate of loss decrease becomes theoretically bounded by the variation in the singular distribution. We further interpret strategies like WSD and Muon through their ability to modulate the SoSD scale, offering a spectral lens for understanding efficient pre-training dynamics.

Hongtao Zhang, Wenjie Zhou, Chenxi Jia +2

State Key Laboratory of AI Safety, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China · School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences, Beijing, China · University of Chinese Academy of Sciences, Beijing, China +1

Language Model Pretraining Deep Learning Optimization Neural Network Training Dynamics

May 25, 2026cs.LG

Reparametrizing Shampoo and SOAP for Subspace Basis Updates and BFloat16 Storage

Shampoo-based methods, such as KL-Shampoo and SOAP, have demonstrated strong performance in training neural networks and rely on QR decomposition. Because existing QR implementations require single-precision (FP32) arithmetic and remain computationally expensive, these methods become time- and memory-intensive when their preconditioning matrices are large. Moreover, using BFloat16 (BFP16) storage to reduce memory usage can degrade the performance of Shampoo-based methods. We propose a reparametrization of the preconditioner that supports BFP16 storage and forms a complete basis by combining updated basis vectors with unchanged ones. By updating only part of the basis through QR decomposition in a subspace, our approach reduces computational overhead while mitigating the performance degradation caused by BFP16 storage. Our approach applies broadly to Shampoo-based methods that employ QR decomposition, including KL-Shampoo, SOAP, and KL-SOAP. In particular, it improves the performance of SOAP and KL-SOAP under BFP16 storage, enabling KL-SOAP to match or exceed KL-Shampoo. Overall, our approach makes Shampoo-based methods more memory- and time-efficient.

Alan Milligan, Zikun Xu, Simon Lacoste-Julien +2

Mila & Université de Montréal · Microsoft · Concordia University & Mila +1

Deep Learning Optimization Memory-Efficient Optimization Matrix Factorization

May 25, 2026cs.LG

BigMac: Breaking the Pareto Frontier of Compute and Memory in Multimodal LLM Training

Training multimodal large language models (MLLMs) is challenged by both model and data heterogeneity. Existing systems redesign the training pipeline to address these challenges, but remain bound by a Pareto frontier between compute and memory efficiency, improving one only at the expense of the other. We present BigMac, a new training pipeline for multimodal LLMs. The core idea of BigMac is to elegantly nest the encoder and generator computation into the original LLM pipeline, forming a dependency-safe nested pipeline structure. With this design, BigMac reduces the activation memory complexity of the encoder and generator to O(1) while keeping the activation memory complexity of the LLM unchanged. At the same time, it achieves the same computational efficiency as the idealized setting with unlimited memory. As a result, BigMac breaks the Pareto frontier between computational efficiency and memory usage, enabling simultaneous optimization of both computation and memory in MLLM training. We evaluate BigMac on multiple MLLMs and training workloads. Experimental results show that BigMac achieves a 1.08

\times

-1.9

\times

training speedup over baseline systems while maintaining stable memory usage as batch size increases.

Zili Zhang, Chengxu Yang, Shenglong Zhang +8

1Peking University · †Work done during internships at Xiaohongshu, Inc. · 2Independent Researcher +2

Deep Learning Optimization Multimodal Large Language Models Efficient Language Model Training Multimodal Foundation Models LLM Training Memory-Efficient Training

May 25, 2026cs.LG

EMA-Nesterov: Stabilizing Nesterov's Lookahead for Accelerated Deep Learning Optimization

Lookahead-based acceleration methods, such as Nesterov's momentum, are widely used in optimization, but they often become unreliable in deep learning training mainly due to stochastic gradient noise and non-convex loss landscapes. In particular, standard lookahead relies on short-horizon update signals (e.g., differences between consecutive iterates), which are inherently noisy and can lead to unstable extrapolation directions. This work revisits Nesterov's acceleration from a trajectory perspective and argues that effective acceleration in deep learning should harness the low-frequency trends of optimization trajectories rather than extrapolating noisy one-step updates. Leveraging this insight, we propose EMA-Nesterov, a simple modification that replaces the standard Nesterov's lookahead direction with an exponential moving average (EMA) of parameter updates. This yields a stabilized lookahead direction that captures and harnesses the evolving trend of the training trajectory through a low-pass filter, while remaining adaptive to progressive changes via the geometric weighting structure of EMA. We show that EMA-Nesterov retains a theoretical accelerated convergence rate in convex problems that is analogous to Nesterov's accelerated gradient method. Furthermore, we provide empirical evidence on language model pre-training to verify that EMA-Nesterov is broadly applicable across a range of fine-tuned base optimizers, including Adam, SOAP, Muon, as well as complex optimizers that achieve state-of-the-art performance on optimization benchmarks (NanoGPT). Compared to prior lookahead methods, EMA-Nesterov achieves better performance by avoiding the instability of short-horizon lookahead and the non-adaptivity of long-horizon lookahead.

Chung-Yiu Yau, Dawei Li, Athanasios Glentis +3

University of Minnesota · 2Amazon AGI · 3The Chinese University of Hong Kong

Deep Learning Optimization Gradient Descent Accelerated Gradient Methods