Adaptive Optimizers

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Sep 29, 2026cs.AI

OptiCom : A Unified Framework for State-Conditioned Composition in LLM-Driven Optimization

Large language models (LLMs) are increasingly deployed to solve complex scientific and practical problems via iterative optimization. However, dynamically coordinating diverse search mechanisms as candidate quality, failure modes, and resource budgets evolve remains a critical open challenge. Targeted empirical diagnostics reveal that mechanism effectiveness is highly state-dependent. Motivated by this, we analyze how individual decisions drive final outcomes, decomposing the expected terminal improvement under a shared budget into cumulative decision opportunities minus cumulative selection losses. Guided by this opportunity-loss theoretical foundation, we propose OptiCom, a unified framework that represents LLM-driven optimizers within a shared configuration space: C=(A,Q,O,E,M,S), corresponding to artifact, query, operator, evaluation, memory, and strategy. Operating within this space, a fast LLM-based Optimization Controller dynamically composes immediate mechanisms through structured Action Packages, while a slower Strategy Adapter refines long-term selection preferences, operator weights, and templates based on accumulated trajectory feedback. Comprehensive evaluations across 32 benchmark groups demonstrate the superiority of framework: OptiCom achieves an average Max-score rank of 1.72 among 14 evaluated configurations, securing the top score in 23 groups. Ultimately, these results highlight the broad applicability and high extensibility of OptiCom as a general-purpose paradigm for robust LLM test-time scaling.

Chenxing Wei, Sichen Liu, Lizhao Liu +6

School of Computing and Data Science, The University of Hong Kong · Shenzhen Loop Area Institute · ByteDance +3

Optimization Modeling Adaptive Optimizers Unified Framework

Sep 29, 2026math.OC

Second-Moment Stochastic Approximation Methods

Classical stochastic approximation methods rely on estimators of the first moment (mean) of a random regression function. We study methods that employ estimators of both the first and the second moments, which include modern deep-learning optimizers such as Adam and Muon as special cases. We derive second-moment stochastic approximation methods through the lens of optimal preconditioning for solving matrix equations, and develop a two-stage framework for their convergence analysis. The first stage focuses on the analysis of conceptual (impractical) methods that rely on the exact first and second moments. In the second stage, we replace the exact moments with their respective estimators, and invoke Dvoretzky's theorem to show that the resulting practical methods converge almost surely to a neighborhood of the target solution. The size of the neighborhood depends on the biases and variances of the first- and second-moment estimators. We derive concrete bounds for Muon and a spectral variant of Adam that determine the radius of their neighborhood of convergence.

Tao Jiang, Lin Xiao

Fundamental AI Research (FAIR), Meta Superintelligence Labs

Stochastic Approximation Adaptive Optimizers Muon Spectral Preconditioning Random

Sep 28, 2026cs.LG

Rotated Manifold Optimization for Low-Rank Adaptation

We propose a novel optimizer for low-rank adaptation (LoRA) that explicitly incorporates the gauge symmetry of low-rank factorization. Our optimizer extends recent matrix optimizers for full-parameter training to the manifold of fixed-rank matrices by interpreting them as normalization under a rotated basis. We show how rotation and normalization can be integrated with the fixed-rank manifold efficiently. Our optimizer converges faster to lower held-out loss and achieves better or comparable downstream performance on both supervised finetuning and reinforcement learning tasks.

Yuhui Ding, Javier Zazo, James Hensman

ETH Zurich · Microsoft Research Cambridge

Low-Rank Structure Adaptive Optimizers Manifolds Symmetry

Sep 22, 2026quant-ph

Bridge of $Ψ$ 's: Quantum Circuit Optimization with Schrödinger Bridges

Quantum circuit optimization replaces a circuit with an equivalent one of fewer gates and lower depth, reducing execution cost and error rate. We ask whether a generative model can learn this transformation directly from examples, rather than selecting from a fixed rewrite library or rigid algebraic routines. We present Bridge of

Ψ

's (BOPS), a generative model based on Schrödinger bridges, using a custom denoiser architecture, that learns a transformation from a source circuit into an equivalent optimized circuit. We train it on data constructed to be hard for existing optimizers, by applying rewrite rules backwards so that each input has a known lower-cost target. On held-out 8 qubits

\times

64 depth Clifford+

T

circuits, BOPS reduces gate count by

2.46\times

and depth by

2.45\times

in geometric mean, outperforming all nine baseline optimizers. This constitutes the first generative model bridging quantum circuits and frontier machine learning methods, opening up the quantum compilation stack to learned optimization along multiple axes.

Lino S. Hofstetter, Lia Yeh, Prakash Murali

ETH Zürich Zürich, Switzerland · University of Cambridge Cambridge, United Kingdom

Quantum Circuits Schrödinger Bridges Circuits Bridge Adaptive Optimizers

Sep 14, 2026cs.LG

Rank-Efficient LoRA via Joint Tangent-Space Optimization under Isotropic Curvature

Low-Rank Adaptation (LoRA) is an effective approach for adapting large pretrained models by learning low-rank weight updates. In practice, the LoRA rank is used to control an adapter's parameter budget and representational capacity. We show that this view is incomplete: while the nominal rank determines the representational capacity, the optimizer shapes how much of that capacity is used in the induced weight-space updates. In a case study of GPT-2 adaptation with LoRA, we observe a strong rank-dependent optimizer effect. Despite using the same nominal rank, AdamW often produces per-step updates with concentrated singular spectra and low effective rank, whereas Muon uses a richer set of directions and benefits more consistently from increasing LoRA rank. These observations motivate ISO-LoRA, an optimizer that couples the LoRA factor updates through spectral descent on the induced tangent perturbation in weight space. ISO-LoRA promotes updates that distribute energy more evenly across singular directions, improving rank utilization while preserving compatibility with the LoRA parameterization. We complement this design with theoretical guarantees showing that ISO-LoRA can achieve higher effective rank than standard factor-wise optimizers through a one-step analysis under a stylized spiked-gradient model. We validate this design on language-model adaptation across 0.1B-7B-parameter models, where ISO-LoRA improves effective rank and downstream performance, with the strongest gains at moderate-to-large LoRA ranks. Our results highlight rank utilization as a key factor in LoRA optimization and suggest that optimizer design offers an important path toward stronger parameter-efficient adaptation.

Zihan Zhu, Zhehang Du, Xuyang Chen +5

University of Pennsylvania · DRW Associates LLC · University of Chicago

Low-Rank Structure Adaptive Optimizers Large Language Model Adaptation Local Curvature Representational Capacity

Sep 2, 2026cs.LG

Frontier LLMs are effective batch optimizers: Assessing reasoning models in continuous and discrete settings

Frontier large language models (LLMs) have become attractive priors for optimization due to their large-scale pretraining that enables them to navigate a variety of optimization settings. However, the effectiveness of modern reasoning LLMs in batch optimization settings remains underexplored. Here we investigate the performance of the current generation of frontier LLMs as batch optimizers in both continuous and discrete settings. We find that while LLMs are competitive zero-shot batch optimizers for numerical test functions, their performance is brittle compared to classical non-LLM optimization approaches. However, LLM priors are significantly better in semantically rich settings, indicating that their batch optimization behavior is highly effective when navigating and reasoning over the discrete spaces most similar in structure to their pretraining data.

Frank Hu, Shriram Chennakesavalu, David Graff

Prescient Design, Genentech South San Francisco, CA, USA

Adaptive Optimizers Optimization Modeling Frontiers Large Reasoning Models Prior Knowledge Integration Discrete

Aug 11, 2026cs.AI

RLMOpt: Adaptive Prompt Optimization via Recursive Language Models

Prompt optimizers automate the search for prompts that improve language-model performance, but existing methods rely on a predefined optimization procedure: the algorithm determines which candidates to explore and how the search progresses, while the language model generates or refines prompt proposals. We introduce RLMOpt, a prompt optimizer that makes the search policy itself language-model-driven through a recursive language model (RLM). The RLM agent operates over a tool-based environment, inspecting task information, analyzing failures, generating candidates, allocating evaluation budget, and deciding when to stop. A deterministic harness complements the agent by enforcing objective scoring, Pareto-based selection, and regression constraints. We evaluate RLMOpt across four benchmarks spanning structured clinical information extraction (Chia), multi-hop question answering (HotpotQA), verifiable instruction following (IFBench-2025), and multi-turn tool-calling agents (BFCL). In a matched comparison at a single seed, RLMOpt obtains the best held-out score on all four benchmarks and leads the four-task mean (0.610 against 0.589 for GEPA). Repeating each benchmark across seeds yields 11 matched benchmark-seed comparisons, in which RLMOpt outperforms GEPA in 9 cases. Across all 11 runs, it never produced a prompt that underperformed its seed, whereas GEPA fell below its starting point twice. It is also more efficient, achieving these results with fewer search rollouts while producing prompts that are 27-79% the size of those produced by GEPA. Our results further show that optimization gains are determined primarily by the headroom available in the seed prompt, rather than by the search budget. Efficient optimization therefore depends on reaching the available headroom reliably and with minimal search

Subhash Bangalore Satheesha, Nirvik Pande, Deepthi Duddempudi +1

Autonomize AI · Carnegie Mellon University

Automatic Prompt Optimization Prompt Engineering Adaptive Optimizers

Aug 5, 2026cs.LG

The Loss Does Not See the Basis, but Adam Does

Gradient descent on a factored model

W = UV^\top

is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under

(U, V) \mapsto (UQ, VQ)

. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum, "shared-scalar" Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A "spectral schedule" reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants

W_Q^\top W_K

56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.

Devender Singh

Department of Computer Science Memorial University of Newfoundland

Low-Rank Structure Adam Adaptive Optimizers Gradient

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

Adaptive Optimizers Neural Network Optimization Batch Convergence

Jul 28, 2026cs.LG

Guiding Posterior Exploration with Optimizer-Derived Geometry

Sampling-based methods offer a principled approach to uncertainty quantification in Bayesian neural networks. Their practical use, however, is often challenged by the computational cost of exploring high-dimensional and multimodal posterior distributions. To overcome these difficulties, Bayesian Deep Ensembles, i.e., warmstarting the sampling from several optimized solutions, have proven to be an effective strategy. In this paper, we demonstrate that curvature estimates computed during the warmstart as a byproduct in adaptive optimizers such as AdamW can inform the sampling phase at negligible additional cost. Specifically, our proposed preconditioned sampling strategy based on optimizer-derived geometries can substantially reduce or even eliminate the need for a lengthy sampling burn-in phase and leads to greater numerical stability. This approach consistently maintains or improves predictive performance and uncertainty quantification without any additional computational costs. We confirm the consistency of our findings across various datasets and network architectures.

Moritz Schlager, Emanuel Sommer, Thomas Möllenhoff +1

TUM, MCML · LMU Munich, MCML · RIKEN AIP

Adaptive Sampling Bayesian Neural Networks Shape Optimization Adaptive Optimizers Uncertainty Quantification Stochastic Exploration+3

Jul 27, 2026cs.AI

Are Prompt Optimizers Blind? Cross-Modal Visual Feedback for Automatic Prompt Optimization

Automatic prompt optimization (APO) has been widely adopted to adapt vision-language models (VLMs) to downstream tasks without weight updates, yielding promising results. However, on multimodal tasks, the effectiveness of APO is fundamentally bottlenecked by a blind feedback channel: the optimizer reads the question, the prediction, and the gold answer, but never the input image on which the model failed, and therefore cannot diagnose visually grounded errors. As a remedy, we introduce Cross-Modal Visual Feedback (CMVF). CMVF incorporates (1) a failure-conditioned visual diagnosis stage, in which a stronger optimizer VLM inspects each failed image without access to predictions or labels, and (2) an error-aware aggregation stage that compresses these observations into reusable, task-level visual blind-spot patterns that drive the prompt rewrite. Crucially, the image is consumed only during optimization; the deployed artifact is an ordinary text prompt that runs at the same inference cost as any text-only baseline. Extensive results across 12 VQA datasets and 4 target VLMs demonstrate that CMVF consistently ranks first, improving over the strongest baseline on every target by 2.4 points on average, with gains of up to 6.5 points on individual benchmarks. Moreover, the optimizer self-organizes into expert-style visual checklists that transfer across models without re-optimization.

Haoyue Liu, Xiaoyu Ma, Ye Chen +2

The Chinese University of Hong Kong, Shenzhen, China · Xi’an Jiaotong University, China · Peking University, China +1

Automatic Prompt Optimization Prompt Engineering Adaptive Optimizers Feedback

Jul 24, 2026cs.LG

Hyperball May Not Be a Free Lunch

For scale-invariant deep networks, Hyperball-style optimizers have shown strong performance in large-scale training by fixing the norms of matrix-valued parameters and normalizing updates. However, the source of their advantage remains unclear. Starting from the angular displacement between consecutive parameter states, we derive an angular effective learning rate that accounts for the parameter-update angle, parameter norm, and update norm. We also show that the conventional norm-based measure is a special case under parameter-update orthogonality. We then decompose optimizer updates into radial and tangential components and analyze how radial updates affect one-step angular displacement. Under the training configurations considered, numerical results show that the radial component has only a limited direct effect on the angular effective learning rate. It therefore cannot explain why MuonH converges more slowly than MuonWD early in training but overtakes it later. To further isolate the underlying mechanism, we devise a heuristic experiment that modifies only the learning-rate schedule so that the dynamics of each optimizer reproduce those of the other. The results suggest that their main difference stems from the evolution of the effective step size rather than an intrinsically superior update direction induced by Hyperball. Our pretraining experiments further show that more aggressive learning-rate decay can accelerate MuonH early in training but may impair its later performance. Thus, maintaining a constant angular velocity does not eliminate the learning-rate-scheduling problem; careful scheduling remains essential to realizing the potential of Hyperball-style optimizers. Our code is publicly available at https://github.com/mangocrazz/hyperball-may-not-be-a-free-lunch.

Yihao Xiao, Jialong Sun, Zitian Gao +5

1IQuest Research · 4Shenzhen University of Advanced Technology · 2Peking University +2

Batch Adaptive Optimizers Orthogonality Recipe

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

Adaptive Optimizers Mixture-Of-Experts Adam Load Balancing

Jul 15, 2026cs.AI

Do Agent Optimizers Compound? A Continual-Learning Evaluation on Terminal-Bench 2.0

Most reported gains from agent-optimization methods are one-shot: an agent is optimized against a fixed benchmark and the resulting improvement is reported as if it were a stable property of the method. This does not test the setting that matters for deployed agents, where optimization is applied recursively as new failures and new tasks appear over time. The central question this raises is whether optimizer-driven gains compound: after an agent has been optimized once, can it be optimized again on newly arrived tasks without eroding the gains the first round produced? We study this question with a two-phase continual-learning evaluation built from hard tasks in Terminal-Bench 2.0, comparing three approaches to agent-harness optimization (GEPA, Meta Harness, and RELAI's Verifiable Continual Learning, RELAI-VCL) under identical optimization budgets. All three methods improve over the baseline agent in the conventional, static, single-phase setting. However, once new tasks are introduced, the methods diverge sharply: GEPA's optimized agent transfers below the unoptimized baseline, Meta Harness transfers well but fails to improve further once given a second optimization budget, and RELAI-VCL is the only method that both transfers positively to unseen tasks and continues improving after those tasks are folded into the optimization objective, reaching the highest pass rate at every evaluated stage and the highest lifelong average pass rate overall (76.4% vs. 66.0% for GEPA, 64.6% for Meta Harness, and 58.7% for the baseline). Our key observation was that optimization gains compounded only when regression control was built into the optimization loop, providing an inductive bias against shortcut solutions that fail to generalize.

Wenxiao Wang, Priyatham Kattakinda, Soheil Feizi

Agentic Optimization Replay-Based Continual Learning Adaptive Optimizers Optimization Modeling

Jul 8, 2026cs.LG

Causal Optimizer Interaction Calculus: Hidden Geometric Relaxation and Identifiable Interventions

Optimizer experiments observe responses to algorithmic configurations without uniquely revealing hidden mechanisms. We develop a causal optimizer interaction calculus that separates pathwise realization, Mobius decomposition, and experimental identification. Under a fixed innovation coupling, every finite-horizon innovation-driven optimizer admits a behaviorally minimal pathwise realization. For any finite effect support and intervention design, an incidence operator gives the complete observational gauge, exact identifiability, sharp quotient stability, held-out predictions, and exact noiseless configuration complexity. Smooth hidden relaxation generates interactions through inverse hidden-state stiffness. Building on this structural law, we prove an observable-readout transfer theorem: arbitrary smooth update or trace readouts inherit an explicit five-term interaction through first and second hidden responses. Unlike the reduced optimal value, a general readout has no universal interaction sign. Its Boolean effects remain exact integrals of continuous interaction curvature and can therefore be identified by factorial interventions. We also derive Gaussian quotient minimax risk, exact confidence sets and tests, misspecification decomposition, certified downstream decisions, and optimal replication. A controlled real-data experiment on a 65-dimensional strongly convex logistic model validates the complete reduced-value chain. Boolean effects and independently integrated curvature agree within 4.21e-11, while nine held-out continuous intensities agree within 8.88e-13. Gaussian campaigns attain the predicted coverage and power, and 4,500 real-minibatch observations reject an order-two interaction model. Neural trace audits provide complementary evidence that the declared response classes remain informative in nonconvex training.

Zavier Li

Xidian University Xi’an, China

Shape Optimization Adaptive Optimizers Gradient Trajectory-Level Credit Modular Framework

Jul 7, 2026cs.LG

Efficient Long-Horizon Learning for Learned Optimization

Learned optimization aims to improve upon hand-designed optimizers (e.g., Adam and Muon) by meta-learning small neural network optimizers over a distribution of tasks. While recent work has greatly advanced the architectural design and inductive biases of learned optimizers (LOs), their meta-training remains biased toward short-unroll learning on particular tasks, resulting in redundant computation and leaving LOs often unable to compete with hand-designed optimizers. We introduce Efficient Long-hOrizon (ELO) learning, an efficient meta-training algorithm that (1) reallocates wasted meta-training compute to longer failure regimes, achieving efficient long-horizon learning, and (2) enforces decoupled progressive expert supervision, providing stable meta-learning signals that additionally improve the generalization of LOs. Our empirical study evaluates ELO for meta-training both element-wise and matrix-based LOs. Across downstream language modeling (GPT-2-124M/350M on FineWeb) and image classification (ViT-B/16, ResNet-50 on ImageNet-1K) tasks, ELO substantially improves the long-unroll performance and out-of-distribution generalization of the base LOs. In particular, ELO-Celo2 consistently outperforms well-tuned AdamW across all evaluated tasks, while remaining competitive with Muon on language modeling. \textit{Notably, all ELO baselines require less than 7 H100 GPU-hours for meta-training.}

Xiaolong Huang, Benjamin Thérien, James Harrison +1

Mila - Quebec AI Institute · Concordia University · Université de Montréal +1

Meta-Learning Adaptive Optimizers Muon

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

Low-Rank Structure Adaptive Optimizers Subspace Identifiability Vision Transformer

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

Adaptive Optimizers Taxonomy Omnimem

Jun 25, 2026cs.LG

Designing Reward Signals for Portable Query Generation: A Case Study in Industrial Semantic Job Search

Job-search platforms rely on low-bandwidth query interfaces that often fail to capture the high-dimensional complexity of candidate profiles. We present an end-to-end RLAIF (Reinforcement Learning from AI Feedback) framework to generate \emph{portable} job search queries, terms that abstract away seeker-specific identifiers while preserving generalizable qualifications. This task introduces a highly adversarial reward surface where policy optimization frequently exploits flaws in LLM-as-judge rubrics, resulting in degenerate verbatim-copying behaviors. We conducted comprehensive empirical experiments to isolate the impact of optimization mechanics against structured reward engineering. Our results demonstrate that for critic-free optimizers, performance is overwhelmingly dictated by robust reward shaping, rendering the specific choice of algorithm largely immaterial. While critic-free per-rollout baseline methods (RLOO and REINFORCE++) natively resist reward-hacking, the group-relative advantage normalization in GRPO appears uniquely sensitive to spurious reward signals, making it disproportionately susceptible to exploitation. We show that introducing a deterministic, rule-based reward floor to correct for rewards assigned to verbatim copying mitigates this failure mode, resulting in a substantial

+0.147

quality improvement on a cross-family evaluation judge. Ultimately, we show that the training-time reward model inflates performance gains by

2.4\times

, confirming that the training success is fundamentally dependent on enforcing reward-shaping disciplines rather than selecting alternative optimizers.

Ping Liu, Qianqi Shen, Jianqiang Shen +11

LinkedIn Corporation, Mountain View, CA, USA

Jobs Reinforcement Learning From Human Feedback Queries Adaptive Optimizers Case Study

Jun 24, 2026cs.LG

Improving Neural Network Training by Decoupling the Magnitude and Direction of Weight Vectors

Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object. Yet every weight matrix carries two distinct quantities -- a \emph{magnitude} and a \emph{direction} -- and all optimizers stepping in the matrix as a whole couple their dynamics: the directional change from an update depends on the current magnitude, while the magnitude drifts as a byproduct of learning the direction, so neither is governed directly by the learning rate. Typical training therefore leans on surrounding recipes such as weight decay and warmup to keep learning stable at scale, though these regulate the coupling only indirectly; other recent methods instead constrain the weight to a fixed-norm sphere, but add no learnable magnitude, leaving scale control to normalization layers alone. We propose \emph{Magnitude--Direction (MD) Decoupling}, an optimizer modification that factorizes each weight into a fixed-norm direction on a hypersphere and learnable per-row and per-column magnitude gains, updated at separate learning rates, all while the model still sees a single fused weight tensor. The method is agnostic to the base optimizer and removes the need for weight decay and warmup. Across both Adam and Muon, MD Decoupling improves on well-tuned baselines, transfers the optimal LR across model width without retuning, and continues to help at scale on large Mixture-of-Experts (MoE) models. Treating magnitude and direction as separately controlled quantities thus yields more predictable training dynamics and a simple, broadly applicable improvement to modern optimizers.

Alexander Hägele, Alejandro Hernández-Cano, Atli Kosson +1

Machine Learning and Optimization Lab, EPFL

Adaptive Optimizers Decoupling Spectral Norm

Jun 23, 2026cs.LG

ASAP: Agent-System Co-Design for Wall-Clock-Centered Auto HPO Research for ML Experiments

Hyperparameter Optimization (HPO) is essential for maximizing machine learning model performance, and its core challenge is sample efficiency: finding strong configurations within a limited budget. Because every HPO tool relies on a surrogate prior that imparts its own inductive bias, individual tools struggle once problems become sufficiently diverse and drift from these priors. Motivated by the reasoning and generalization capabilities of LLMs, recent work has explored using LLMs for HPO and reports improved per-iteration performance. Yet these methods share two limitations with a common origin: they use the LLM as a single-tool replacement evaluated by iteration count. (i) Deployed in place of prior tools, the LLM is itself constrained by its pretraining objective to one family of inductive-biased proposals; this single-source setup still fails to handle the full diversity of problems. (ii) Per-iteration evaluation ignores that, in real runs, LLM inference or tool execution is paid serially on top of model evaluation every round, so iteration-count gains do not translate into end-to-end wall-clock gains. We present ASAP, an agent-system co-design that addresses both limitations. On the agent side, ASAP uses the LLM to integrate a diverse pool of inductive-biased optimizers and to select among their proposals each round. On the system side, ASAP re-architects the loop to reduce end-to-end wall-clock while preserving regret quality: a prefix-stable prompt maximizes KV-cache reuse across rounds; speculation parallelism hides the remaining LLM and tool latency under model evaluation via a relative-error accept test; and a Self-Tuner adapts the speculation threshold from execution logs off the critical path. Extensive experiments on diverse modern HPO tasks show that ASAP consistently outperforms baselines, underscoring the value of tool integration and agent-system co-design.

Taicheng Guo, Haomin Zhuang, Kehan Guo +4

University of Notre Dame United States

General Grid Search Framework Agentic Optimization Human-In-The-Loop LLM Inference Optimization Adaptive Optimizers Co-Design

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

Replay-Based Continual Learning Adaptive Optimizers Continual Learning Parameter-Efficient Adaptation Modern Deep Networks Fast-Slow+1

Jun 22, 2026cs.LG

Muown Implicitly Performs Angular Step-size Decay

Matrix-aware optimizers such as Muon and Muown have recently shown strong empirical performance for pre-training Transformers. In particular, Muown separates each weight matrix into row magnitudes and an un-normalized direction variable, updating the former with Adam and the latter with Muon. We show that the directional update of Muown is equivalent to a Riemannian step on the normalized directions, while the magnitude of the un-normalized parameterization only modulates the angular step size. This explains the step-size stability of Muown and suggests making the angular step size explicit. The resulting method, AngularMuown, optimizes directly over the normalized directions and uses a schedulable angular multiplier decoupled from the radial magnitude update. AngularMuown improves over Muown and, at the time of writing, a preliminary version is leading the per-optimizer category of the modded nanoGPT speedrunning competition. Further experiments on Qwen2-0.5B, and 1.1B parameter mixture-of-experts models confirm the algorithm scales beyond small models. An implementation of the algorithm is available at https://github.com/fhueb/angular-muown

Florian Hübler, Kai Lion, Antonio Orvieto +1

Department of Computer Science · ETH Zurich, Switzerland · ELLIS Institute Tübingen, MPI-IS +1

Muon Adaptive Optimizers Generative Pretrained Transformers Step Accuracy

Jun 16, 2026cs.LG

MGUP: A Momentum-Gradient Alignment Update Policy for Stochastic Optimization

Efficient optimization is essential for training large language models. Although intra-layer selective updates have been explored, a general mechanism that enables fine-grained control while ensuring convergence guarantees is still lacking. To bridge this gap, we propose \textbf{MGUP}, a novel mechanism for selective updates. \textbf{MGUP} augments standard momentum-based optimizers by applying larger step-sizes to a selected fixed proportion of parameters in each iteration, while applying smaller, non-zero step-sizes to the rest. As a nearly {plug-and-play} module, \textbf{MGUP} seamlessly integrates with optimizers such as AdamW, Lion, and Muon. This yields powerful variants such as \textbf{MGUP-AdamW}, \textbf{MGUP-Lion}, and \textbf{MGUP-Muon}. Under standard assumptions, we provide theoretical convergence guarantees for \textbf{MGUP-AdamW} (without weight decay) in stochastic optimization. Extensive experiments across diverse tasks, including MAE pretraining, LLM pretraining, and downstream fine-tuning, demonstrate that our \textbf{MGUP}-enhanced optimizers achieve superior or more stable performance compared to their original base optimizers. We offer a principled, versatile, and theoretically grounded strategy for efficient intra-layer selective updates, accelerating and stabilizing the training of large-scale models. The code is publicly available at https://github.com/MaeChd/MGUP.

Da Chang, Ganzhao Yuan

Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences · Pengcheng Laboratory · University of Chinese Academy of Sciences +1

Adaptive Optimizers Stochastic Optimization Downstream Fine-Tuning Convergence

Jun 16, 2026cs.CV

Theoretical Grounding of Out-Of-Distribution Detection With Reinforcement Learning Optimizer

Out-of-distribution (OOD) detection in dynamic open-world environments requires a model to continually adapt to evolving data distributions while generalizing to covariate-shifted inputs and rejecting semantic-shifted OOD examples. Most existing OOD detection methods optimize only the current-step objective and do not explicitly account for how post-deployment environment changes affect future OOD behavior. In this paper, we establish a theoretical grounding for dynamic OOD detection using a reinforcement learning (RL)-guided optimizer that explicitly favors updates that reduce the semantic OOD false positive rate over time. We develop a novel augmented optimizer that uses an RL-guided correction term on top of standard gradient descent (GD) and show its improvement over both future-domain generalization and semantic-OOD rejection. We analyze temporal error decomposition in terms of model-change and environment-change generalization errors and develop a new theoretical framework for comparing the generalization errors under both GD and RL-guided optimizers.

Salimeh Sekeh, Xin Zhang

San Diego State University

Out-Of-Distribution Detection Off-Policy Learning Adaptive Optimizers Distributions Theory

Jun 15, 2026cs.LG

Fantastic Pretraining Optimizers and Where to Find Them II: Hyperball Optimization

Matrix based optimizers such as Muon can substantially speed up language model pretraining, but their gains over AdamW are observed to shrink as model size and data scale grow when using standard constant decoupled weight decay. We propose Hyperball, a simple optimizer wrapper that addresses this issue. Given a base optimizer such as Adam or Muon, Hyperball sets the Frobenius norms of weight matrices and their corresponding optimizer updates to fixed constants. On Qwen3 style models up to 1.2B parameters, Muon Hyperball achieves 20--30% token equivalent speedup over weight decay baselines. Hyperball also improves learning rate transfer across widths and depths compared to decoupled weight decay. This method is motivated by prior theory showing that training with weight decay leads to an equilibrium weight norm that only depends on the training hyperparameters. Through this mechanism, the weight decay then decides the angular learning rate, i.e. how fast the direction of the weight matrix changes.

Kaiyue Wen, Xingyu Dang, Kaifeng Lyu +2

†Stanford University. · ‡Princeton University. · §Tsinghua University.

Adaptive Optimizers Weight Decay Large Language Model Pretraining Batch Muon Time-To-First-Token

Jun 12, 2026cs.LG

Zero-order Parameter-free Optimization for LMO-based Methods: Novel Approach for Efficient Fine-tuning

Fine-tuning large language models (LLMs) has become a central application of modern optimization, enabling pretrained models to adapt to diverse downstream tasks and domain-specific data. A major obstacle in large-scale fine-tuning is the memory overhead of backpropagation, which requires storing activations, gradients, and optimizer states. Zeroth-order (ZO) optimization offers a memory-efficient alternative, but its performance is highly sensitive to the stepsize and smoothing parameter, often requiring costly task-specific tuning. Parameter-free (PF) optimization addresses this issue by adapting algorithmic parameters without prior knowledge of problem-dependent constants. Moreover, large-scale fine-tuning can benefit from geometry-aware updates that account for the heterogeneous structure of parameter blocks, which can be modeled through methods that exploit linear minimization oracle (LMO). In this work, we study PF adaptation for LMO-based ZO optimization and introduce

\texttt{AdaNAGED}

, a method that unifies gradient-free training, adaptive tuning, and non-Euclidean update geometry. We establish convergence guarantees and validate the method on large-scale LLM fine-tuning task with

\texttt{OPT}-1.3\mathrm{B}

model.

Dmitriy Bystrov, Daniil Medyakov, Dmitry Bylinkin +1

1Moscow Independent Research Institute of Artificial Intelligence · 2Basic Research of Artificial Intelligence Laboratory (BRAIn Lab) · 3Innopolis University

Parameter-Efficient Fine-Tuning Methods Zeroth-Order Optimization Large Language Model Fine-Tuning Model Fine-Tuning Adaptive Optimizers Zeroth-Order+2

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

Adaptive Optimizers Continual Learning Forgetting Prompt Optimization

Jun 5, 2026stat.CO

Large-scale empirical tuning and comparison of default optimizers for variational inference

Black-box variational inference (BBVI) is a methodology for posterior approximation that relies on stochastic optimization. In practice, the stochastic optimizers underpinning BBVI generally require extensive problem-specific tuning, which undermines its promise as a truly "black box" inference algorithm. However, over the past decade, many new adaptive stochastic optimization algorithms have been developed that reduce or remove entirely the need for tuning. In this work, we investigate this new collection of adaptive methods in the context of BBVI, with the goal of establishing the current state of the art in tuning-free optimization-based inference. In particular, we present a large-scale empirical evaluation of 56 stochastic gradient-based optimization algorithms applied to 1092 Bayesian inference optimization problems, involving over 550,000 individual optimization runs and 15 core-years of compute. The optimization algorithms we evaluate are chosen to represent a wide spectrum of recent approaches and the benchmark problems are chosen to span a range of difficulty, with posterior target dimension 1-10^4, condition number 1-10^8, and a range of variational families. Our results show that no single method dominates, but running a selection of 5 algorithms suffices to reliably get close to the best-possible observed performance. We thus provide a strong baseline for applications where expert tuning is not possible and for comparison when developing new stochastic optimization algorithms.

Trevor Campbell, Jonathan H. Huggins, Kyurae Kim +1

Department of Statistics, UBC · Department of Mathematics & Statistics, Boston University · Faculty of Computing & Statistics, Boston University +1

Variational Inference Variational Posterior Stochastic Optimization Adaptive Optimizers Comparison

Jun 5, 2026cs.LG

Reversible Foundations: Training a 120B Sparse MoE through State-Preserving Scaling

This paper reports on training a hundred-billion-parameter sparse mixture of experts on a single eight-GPU node, end to end. LightningLM 0.1V is a recurrence-backbone language model family grown in four stages from a small dense seed, through a 5B and a 9B mixture of experts, to a 120B model with 460 routed experts under top-12 routing. Each larger model is grown from the trained weights of the smaller one; active parameters rise monotonically from 1.78B at the dense seed to 5.93B at 120B (about 5% of the 118.67B stored). The full lineage runs on single nodes, the larger stages at 8K context, reaching a released training loss of 1.78 at 120B scale. This is a systems and experience report. It is organized around three disciplines. Reversibility: a reversible recurrence stack reconstructs activations in the backward pass instead of storing them, holding activation memory flat as the model grows. State-preserving growth: each expansion (dense to MoE, shallow to deep, few experts to many) is given as a reproducible principle paired with the failure that results from getting it wrong; several failures are silent. Single-node economics: the 120B trains through TQP, a strategy of quantized base expert weights and trained low-rank adapters that carries optimizer state on 2.26B adapter parameters rather than 100B+ resident in routed experts, cutting expert-path optimizer state by a factor of ~45. What is new is the integration of known primitives, not any primitive in isolation: one grown lineage running end to end on a single node, documented at practitioner level, with per-domain held-out loss as evidence that targeted capabilities (multilingual Indic competence, code) were learned by construction. Model family, tokenizer, and training code are released.

Rohan Shravan

The School of AI Bengaluru, India

Mixture-Of-Experts Architectures Large Language Model Training Mixture-Of-Experts Adaptive Optimizers Cpu-Gpu Hybrid Designs Reverse

Jun 4, 2026cs.LG

Towards Serverless Semi-Decentralized Federated Learning with Heterogeneous Optimizers

We investigate cluster formation, involving the number and composition of clusters, in decentralized federated learning (FL) with heterogeneous machine learning (ML) optimizers. While clustering in centralized FL has enabled scalability and resource savings, its value and development in fully decentralized environments have yet to be explored. Optimizing cluster formation in such environments is challenging, especially due to the complex coupling between network graph structures, local data heterogeneity, and different local ML model optimizers. To address these challenges, we propose serverless semi-decentralized FL (SSD-FL), a methodology requiring no persistent server infrastructure. In SSD-FL, cluster formation occurs via a lightweight, one-time device-to-device (D2D) initialization phase, after which actual ML model training (alongside consensus and convergence processes) is fully serverless. Functionally, SSD-FL segments global rounds into intra-cluster and inter-cluster regimes, ensuring global convergence and consensus through novel "effective loss functions" that integrate device-specific ML optimizers with network graph-based regularization. Next, SSD-FL leverages the consensus gap via the Cheeger inequality to develop an iterative clustering algorithm evaluated against our derived convergence and consensus bounds, which incorporate a unique scoring metric to quantify data and optimizer heterogeneity across devices. Finally, experimental evaluation against three categories of decentralized FL methodologies validate that SSD-FL improves both convergence speeds and communication efficiency across various network graphs, datasets, and local optimizer regimes.

Su Wang, Mung Chiang, H. Vincent Poor

Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ, USA · Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA

Federated Learning Heterogeneity Cluster Adaptive Optimizers Cloud Convergence

Jun 1, 2026cs.LG

Beyond $\ell_2$ -norm and $\ell_\infty$ -norm: A Curvature-Inspired $\ell_p$ -Norm Scheme for Deep Neural Networks

The existing optimizers for deep neural networks (DNNs) typically rely on either the

\ell_2

norm or the

\ell_\infty

norm, resulting in optimizers that do not adapt well to substantial changes in curvature across parameter dimensions. Generally, the training process of DNNs often exhibits strong curvature anisotropy in the early period, whereas in the later period, the training process of DNNs tends to move toward flatter regions with weaker anisotropy. Particularly, optimizers based on the

\ell_2

-norm are usually dominated by high-curvature directions, restricting updates of optimizers along with lower curvature direction and thus leading to a slower convergence rate. While optimizers based on the

\ell_\infty

-norm are prone to oscillations in flatter regions, due to the coordinate-wise updates of the same magnitude. To address these two extreme cases generated by

\ell_2

and

\ell_\infty

norms, we propose a novel

\ell_p

-norm scheme with a dynamical value of

p

and incorporate it into stochastic gradient descent (SGD) and SGD with momentum (SGDM), leading to two novel optimizers with better generalization performance:

{\ell_p}

-SGD (LPSGD) and

{\ell_p}

-SGDM (LPSGDM). Particularly, the resulting optimizers suppress the dominance of high-curvature directions in the early period by utilizing a large

p

(

p>2

), followed by a gradual decrease of

p

toward 2 to enable more stable and refined updates, where the latter process is motivated by the cosine annealing strategy. We establish theoretical guarantees of the resulting algorithms and analyze that both LPSGD and LPSGDM achieve an

O(T^{-1/2})

convergence rate for the nonconvex setting. Extensive experiments are conducted on benchmark datasets, including CIFAR-10, CIFAR-100, and ImageNet-1K, with multiple DNNs such as VGG-11, ResNet-18, and ResNet-50.

Jianhao Xu, Zhuang Yang

School of Computer Science and Technology, Soochow University, Suzhou, 215006, China

Neural Network Optimization Adaptive Optimizers Spectral Norm Local Curvature Nonconvex

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

Adaptive Optimizers Stochastic Gradient Descent Sign Heterogeneity

May 28, 2026cs.LG

A Theoretical and Experimental Study of a Novel Adaptive Learning Algorithm

A crucial component of machine learning algorithms is minimizing loss functions with less computational cost and less oscillations. While adaptive learning rate-based optimizers have been widely used for real-world tasks, they do not guarantee convergence, which is why AMSGrad was later introduced to investigate the non-convergence behaviour of Adam. In this paper, popular adaptive optimization methods like Adam and AMSGrad are critically reviewed with an emphasis on their fundamental design concepts. To address limitations of the above mentioned optimizers, a new optimizer variant, C-Adam, is proposed based on the line of sight approach. A theoretical proof for convergence is also provided and the optimizer is validated through a number of real-life based numerical experiments.

Sakshi Kumari, Shyam Kumar M, Sushmitha P

#Department of Mathematics Indian Institute of Technology Patna Bihta 801 106 India · *Department of Mechanical Engineering Indian Institute of Technology Kharagpur2026 Kharagpur IndiaMay 28

Adam Adaptive Optimizers Convergence Loss Function

May 20, 2026cs.LG

Correcting Stochastic Update Bias in Preconditioned Language Model Optimizers

Preconditioned optimizers are central to language model training, but their stochastic update rules are usually treated as direct approximations to population preconditioned descent. We show that this view misses two finite-sample biases. First, the gradient and preconditioner are typically estimated from the same minibatch, introducing gradient--preconditioner coupling bias. Second, even when the preconditioner estimate is unbiased, its inverse or inverse-root is generally biased because inversion is nonlinear. We propose a single-batch bias-correction framework that addresses both effects: cross-fitted preconditioning estimates the numerator and preconditioner from independent microbatch groups, while variance-corrected inversion uses microbatch variability to subtract the leading delta-method bias term. The framework applies to diagonal moment, diagonal curvature, and matrix preconditioning methods, instantiated in AdamW, Sophia, and Shampoo. Bias correction reduces held-out pretraining loss on Qwen2.5-0.5B by

0.15

,

0.07

, and

0.11

nats, respectively; the effects on mixed-quality pretraining and downstream instruction tuning are consistently neutral-to-positive. Together, these results establish bias correction as a practical mechanism for reducing finite-sample update bias and improving the performance of preconditioned optimizers.

Nikhil Nayak, Julia White, Urchade Zaratiana +7

Fastino Labs

Large Language Model Pretraining Spectral Preconditioning Adaptive Optimizers Pretraining

May 18, 2026math.OC

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible update rules consistently improve final validation loss, reduce expert load imbalance in sparse MoE models, and in several cases control final vocabulary-logit growth, improve router stability, and overall training stability over the corresponding AdamW updates.

Tim Tsz-Kit Lau, Weijie Su

University of Pennsylvania, Philadelphia, PA 19104, USA.

Adaptive Optimizers Optimizer Design Symmetry Equivariance Model Weights Multilayer Perceptrons+1

May 15, 2026cs.CV

Learn2Splat: Extending the Horizon of Learned 3DGS Optimization

3D Gaussian Splatting (3DGS) optimization is most commonly performed using general-purpose first-order optimizers such as Adam or SGD. Although robust across scenes, they update each parameter independently without exploiting the structural and spatial relationships among Gaussians, which slows convergence. Recent works introduced learned optimizers that predict correlated updates informed by inter-parameter and inter-Gaussian dependencies. However, those are trained for a fixed number of optimization iterations and rely on manually scheduled learning rates to avoid degradation. In this paper, we introduce Learn2Splat, a learned optimizer for 3DGS that avoids degradation over extended optimization horizons without auxiliary mechanisms. To enable this, we propose a meta-learning scheme that extends the optimization horizon via a checkpoint buffer and an optimizer rollout strategy, combined with an architecture that encodes gradient scale information in its latent states. Results show higher novel view synthesis quality at equal wall-clock time, while remaining stable over long horizons, with zero-shot generalization to unseen datasets and settings. To support our findings, we build a unified framework to train and evaluate learned and standard optimizers across sparse and dense view settings. Code and models will be released publicly. Our project page is available at https://autonomousvision.github.io/learn2splat .

Naama Pearl, Stefano Esposito, Haofei Xu +6

University of Tübingen, Tübingen AI Center · ETH Zurich · Meta Reality Labs

Recent 3D Gaussian Splatting Novel View Synthesis Adaptive Optimizers Horizon

May 12, 2026cs.LG

Pion: A Spectrum-Preserving Optimizer via Orthogonal Equivalence Transformation

We introduce Pion, a spectrum-preserving optimizer for large language model (LLM) training based on orthogonal equivalence transformation. Unlike additive optimizers such as Adam and Muon, Pion updates each weight matrix through left and right orthogonal transformations, preserving its singular values throughout training. This yields an optimization mechanism that modulates the geometry of weight matrices while keeping their spectral norm fixed. We derive the Pion update rule, systematically examine its design choices, and analyze its convergence behavior along with several key properties. Empirical results show that Pion offers a stable and competitive alternative to standard optimizers for both LLM pretraining and finetuning.

Kexuan Shi, Hanxuan Li, Zeju Qiu +3

1The Chinese University of Hong Kong · 2Max Planck Institute for Intelligent Systems · 3Westlake University

Adaptive Optimizers Spectral Preconditioning Parameter-Efficient Fine-Tuning Methods Large Language Model Pretraining Orthogonality Muon

May 11, 2026cs.LG

Optimistic Dual Averaging Unifies Modern Optimizers

We introduce SODA, a generalization of Optimistic Dual Averaging, which provides a common perspective on state-of-the-art optimizers like Muon, Lion, AdEMAMix and NAdam, showing that they can all be viewed as optimistic instances of this framework. Based on this framing, we propose a practical SODA wrapper for any base optimizer that eliminates weight decay tuning through a theoretically-grounded

1/k

decay schedule. Empirical results across various scales and training horizons show that SODA consistently improves performance without any additional hyperparameter tuning.

Thomas Pethick, Wanyun Xie, Roman Machacek +1

Independent Researcher · EPFL (LIONS) · University of Bern

Adaptive Optimizers Federated Averaging Taco

May 9, 2026cs.LG

AdaPreLoRA: Adafactor Preconditioned Low-Rank Adaptation

Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian

J_{G}

of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner

J_{G}^* {F}_t J_{G}

induced by any

{W}

-space preconditioner

{F}_t

is singular, and consequently the standard chain rule cannot be uniquely inverted to map a preconditioned

{W}

-space direction back to a factor-space update. We cast existing LoRA optimizers in a unified framework parameterized by two choices: (i) which invertible surrogate for

J_{G}^* {F}_t J_{G}

to use, and (ii) which

{F}_t

on

{W}

to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for

J_{G}^* J_{G}

, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware

{F}_t

paired with a closed-form factor-space solve at

{O}((m+n)r)

memory remains underexplored. We propose \textbf{AdaPreLoRA}, which fills this gap by adopting the Adafactor diagonal Kronecker preconditioner

{H}_t

on

{W}

and selecting from the resulting factor-space solution family the element minimizing an

{H}_t

-weighted imbalance between the two factor contributions; by construction, the resulting factor update is the closest LoRA approximation to the preconditioned

{W}

-space direction under the

{H}_t

-weighted norm. Across GPT-2 (E2E), Mistral-7B and Qwen2-7B (GLUE, ARC, GSM8K), and diffusion-model personalization, AdaPreLoRA is competitive with or improves over a representative set of LoRA optimizers while keeping peak GPU memory at the LoRA optimizer level.

Ziyun Liu, Fengmiao Bian, Jian-Feng Cai

Department of Mathematics The Hong Kong University of Science and Technology

Spectral Preconditioning Low-Rank Structure Adaptive Optimizers Jacobian

May 7, 2026cs.LG

Accelerating LMO-Based Optimization via Implicit Gradient Transport

Recent optimizers such as Lion and Muon have demonstrated strong empirical performance by normalizing gradient momentum via linear minimization oracles (LMOs). While variance reduction has been explored to accelerate LMO-based methods, it typically incurs substantial computational overhead due to additional gradient evaluations. At the same time, the theoretical understanding of LMO-based methods remains fragmented across unconstrained and constrained formulations. Motivated by these limitations, we propose \emph{LMO-IGT}, a new class of stochastic LMO-based methods leveraging implicit gradient transport (IGT). We further introduce a unified framework for stochastic LMO-based optimization together with a new stationarity measure, the \emph{regularized support function} (RSF), which bridges gradient-norm and Frank--Wolfe-gap notions within a common framework. By evaluating stochastic gradients at transported points, LMO-IGT accelerates convergence while retaining the single-gradient-per-iteration structure of standard stochastic LMO. Our analysis establishes that stochastic LMO achieves an iteration complexity of

\mathcal{O}(\varepsilon^{-4})

, variance-reduced LMO achieves

\mathcal{O}(\varepsilon^{-3})

at the cost of additional gradient evaluations, and LMO-IGT achieves

\mathcal{O}(\varepsilon^{-3.5})

using only a single stochastic gradient per iteration. Empirically, LMO-IGT consistently improves over stochastic LMO counterparts with negligible overhead. Among its instantiations, Muon-IGT achieves the strongest overall performance across evaluated settings, demonstrating that IGT provides an effective and practical acceleration mechanism for modern LMO-based optimization.

Won-Jun Jang, Si-Hyeon Lee

School of Electrical Engineering · Korea Advanced Institute of Science and Technology (KAIST) · Daejeon, South Korea

Adaptive Optimizers Gradient Variance Reduction Wasserstein Gradient Flows

May 6, 2026cs.NE

Direct From Darwin: Deriving Advanced Optimizers From Evolutionary First Principles

Evolutionary computation has long promised to deliver both high-performance optimization tools as well as rigorous scientific simulations of Darwinian evolution. However, modern algorithms frequently abandon evolutionary fidelity for physics-inspired heuristics or superficial biological metaphors. This paper derives a suite of advanced gradient-based optimization algorithms directly from evolutionary first principles. We introduce Darwinian Lineage Simulations (DLS) to prove that, in an asexual context, Fisher's and Wright's historically opposed views of evolution are actually formally equivalent; One can partition Fisher's deterministically-evolving total population into Wright's randomly-drifting sub-populations. We prove that proper bookkeeping requires introducing a specific kind of structured noise (the DLS noise relation). Crucially, any bookkeeping choices which satisfy this relation will yield a faithful simulation of evolution. Using this vast representational freedom, we prove that a broad family of battle-tested optimization algorithms are already perfectly compatible with evolutionary dynamics. These include: Stochastic Gradient Descent as well as many regularizations/approximations of Newton's method and Natural Gradient Descent. By simply adding DLS noise (i.e., evolutionarily faithful genetic drift), these algorithms become scientifically valid in silico simulations of Darwinian evolution. Finally, we demonstrate that even the state-of-the-art Adam optimizer can be brought into evolutionary compliance through a minor mathematical surgery.

Daniel Grimmer

Philosophy, Yale University, New Haven, 06511, USA

Evolutionary Optimization Evolutionary Optimization Methods Adaptive Optimizers Adam Population Dynamics Robotwin

May 4, 2026cs.AI

Anon: Extrapolating Adaptivity Beyond SGD and Adam

Adaptive optimizers such as Adam and non-adaptive methods like SGD exhibit distinct generalization capabilities across different architectures. Prior tunable optimizers attempt to bridge this gap by strictly interpolating between SGD and Adam, effectively confining adaptivity within the 0-to-1 bound. However, this restricted interpolation is fundamentally insufficient: we reveal that optimal adaptivity often requires extrapolation, such as negative adaptivity for classical CNNs and adaptivity of at least one (

γ\geq 1

) for Transformers. Extrapolating adaptivity theoretically violates the strict non-decreasing pre-conditioner assumption, often leading to divergence in existing methods. To break this barrier, we propose Anon, an optimizer that achieves fully continuous adaptivity extrapolation across the entire real-number spectrum. To guarantee provable stability in these out-of-bound regimes, we introduce Incremental Delay Update (IDU), a novel mechanism that bypasses hard max-tracking strategies. We theoretically establish Anon's convergence in both convex and non-convex settings. Empirically, by exploring previously unreachable adaptivity landscapes, Anon demonstrates highly competitive and scalable performance among state-of-the-art element-wise optimizers on representative image classification, diffusion, and large language modeling tasks.

Yiheng Zhang, Kaiyan Zhao, Shaowu Wu +5

Wuhan University · University of Macau · University of Calgary

Adaptive Optimizers Adam Stochastic Gradient Descent Convergence Convolutional Neural Networks Stance

Apr 24, 2026cs.LG

Shape of Memory: a Geometric Analysis of Machine Unlearning in Second-Order Optimizers

We argue that current definitions of machine unlearning are underspecified for second-order optimizers. We compare first-order and second-order learners for their ability to handle the data deletion task with varying degrees of eigendecomposition to mimic the loss model memory. While both first and second-order methods realign with the ideal counterfactul in terms of performance and gradient, the second-order optimizer shows significant volatility in the optimizer state. This indicates residual information, supposedly deleted, that isn't detectable by first-order analysis. Various eigendecay treatments show that stability and information loss is regained only under controlled state pertubation where geometric information (or memory) is erased.

Kennon Stewart

Second Street Labs, Detroit, MI, USA · Department of Statistics, University of Michigan, Ann Arbor, MI, USA

Machine Unlearning Zeroth-Order Optimization Adaptive Optimizers

Jan 31, 2026cs.LG

Spectral Imbalance Causes Forgetting in Low-Rank Continual Adaptation

Parameter-efficient continual learning aims to adapt pre-trained models to sequential tasks without forgetting previously acquired knowledge. Most existing approaches treat continual learning as avoiding interference with past updates, rather than considering what properties make the current task-specific update naturally preserve previously acquired knowledge. From a knowledge-decomposition perspective, we observe that low-rank adaptations exhibit highly imbalanced singular value spectra: a few dominant components absorb most of the adaptation energy, thereby (i) more likely to disrupt previously acquired knowledge and (ii) making the update more vulnerable to interference from subsequent tasks. To enable explicit balance among components, we decouple the magnitude of the task update from its directional structure and formulate it as a constrained optimization problem on a restricted Stiefel manifold. We address this problem using a projected first-order method compatible with standard deep-learning optimizers used in vision-language models. Our method mitigates both backward and forward forgetting, consistently outperforming continual learning baselines. The implementation code is available at https://github.com/haodotgu/EBLoRA.

Hao Gu, Mao-Lin Luo, Zi-Hao Zhou +3

School of Computer Science and Engineering, Southeast University, Nanjing 210096, China · Key Laboratory of Computer Network and Information Integration (Southeast University), Ministry of Education, China

Replay-Based Continual Learning Forgetting Singular Value Decomposition Adaptive Optimizers Decoupling Class Imbalance

Aug 17, 2025cs.LG

L-SR1: Learned Symmetric-Rank-One Preconditioning

End-to-end deep learning has achieved impressive results but often relies on large labeled datasets, exhibits limited generalization to unseen scenarios, and incurs substantial computational cost. Classical optimization methods, in contrast, are more data-efficient and lightweight but frequently suffer from slow convergence. Learned optimizers aim to bridge this gap, yet existing approaches have focused primarily on first-order methods, while learned second-order optimization has received much less attention. We introduce L-SR1, a learned second-order optimizer inspired by the classical Symmetric Rank-One (SR1) method. At its core, L-SR1 employs a Projection-Guided Secant Mechanism (PGSM) that generates positive semi-definite preconditioners and biases meta-training toward the quasi-Newton secant relation. Through controlled analytic benchmarks, we study stability, generalization across problem dimensions, and search direction quality, and further evaluate L-SR1 on Monocular Human Mesh Recovery (HMR), where it outperforms both classical and learned optimization-based baselines. With a compact model and no reliance on task-specific fine-tuning or annotated data, L-SR1 demonstrates strong generalization and can be integrated into a broad range of iterative optimization problems to accelerate convergence and reduce the required number of iterations.

Gal Lifshitz, Shahar Zuler, Ori Fouks +1

Tel Aviv University, Tel Aviv, Israel.

Spectral Preconditioning Adaptive Optimizers Low-Rank Structure Deep Learning