Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved generalization performance, yet the underlying mechanism remains poorly understood. In this work, we represent stochastic optimizers as random dynamical systems, which often converge to a fractal attractor set (rather than a point) with a smaller intrinsic dimension. Building on this connection and inspired by Lyapunov dimension theory, we introduce a novel notion of dimension, coined the `sharpness dimension', and prove a generalization bound based on this dimension. Our results show that generalization in the chaotic regime depends on the complete Hessian spectrum and the structure of its partial determinants, highlighting a complexity that cannot be captured by the trace or spectral norm considered in prior work. Experiments across various MLPs and transformers validate our theory while also providing new insights into the recently observed phenomenon of grokking.

Deep learning (DL) has become a cornerstone of modern machine learning (ML) praxis. We introduce the R package mlr3torch, which is an extensible DL framework for the mlr3 ecosystem. It is built upon the torch package, and simplifies the definition, training, and evaluation of neural networks for both tabular data and generic tensors (e.g., images) for classification and regression. The package implements predefined architectures, and torch models can easily be converted to mlr3 learners. It also allows users to define neural networks as graphs. This representation is based on the graph language defined in mlr3pipelines and allows users to define the entire modeling workflow, including preprocessing, data augmentation, and network architecture, in a single graph. Through its integration into the mlr3 ecosystem, the package allows for convenient resampling, benchmarking, preprocessing, and more. We explain the package's design and features and show how to customize and extend it to new problems. Furthermore, we demonstrate the package's capabilities using three use cases, namely hyperparameter tuning, fine-tuning, and defining architectures for multimodal data. Finally, we present some runtime benchmarks.

Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature-aware optimization framework that augments existing first-order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step-normalized secant curvature indicator to control the correction strength. The framework is plug-and-play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second-order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high-dimensional heat equation, Gray--Scott system, Belousov--Zhabotinsky system, and 2D Kuramoto--Sivashinsky system.

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

Lion optimizer is a popular learning-based optimization algorithm in machine learning, which shows impressive performance in training many deep learning models. Although convergence property of the Lion optimizer has been studied, its generalization analysis is still missing. To fill this gap, we study generalization property of the Lion via algorithmic stability based on the mathematical induction. Specifically, we prove that the Lion has a generalization error of $O(\frac{1}{Nτ^T})$, where $N$ is training sample size, and $τ>0$ denotes the smallest absolute value of non-zero element in gradient estimator, and $T$ is the total iteration number. In addition, we obtain an interesting byproduct that the SignSGD algorithm has the same generalization error as the Lion. To enhance generalization of the Lion, we design a novel efficient Cautious Lion (i.e., CLion) optimizer by cautiously using sign function. Moreover, we prove that our CLion has a lower generalization error of $O(\frac{1}{N})$ than $O(\frac{1}{Nτ^T})$ of the Lion, since the parameter $τ$ generally is very small. Meanwhile, we study convergence property of our CLion optimizer, and prove that our CLion has a fast convergence rate of $O(\frac{\sqrt{d}}{T^{1/4}})$ under $\ell_1$-norm of gradient for nonconvex stochastic optimization, where $d$ denotes the model dimension. Extensive numerical experiments demonstrate effectiveness of our CLion optimizer.

In optimization problems, some variable subsets may have a joint non-linear or non-monotonical influence on the function value. Therefore, knowledge of variable dependencies may be crucial for effective optimization, and many state-of-the-art optimizers leverage it to improve performance. However, some real-world problem instances may be the subject of noise of various origins. In such a case, variable dependencies relevant to optimization may be hard or impossible to tell using dependency checks sufficient for problems without noise, making highly effective operators, e.g., Partition Crossover (PX), useless. Therefore, we use Statistical Linkage Learning (SLL) to decompose problems with noise and propose a new SLL-dedicated mask construction algorithm. We prove that if the quality of the SLL-based decomposition is sufficiently high, the proposed clustering algorithm yields masks equivalent to PX masks for the noise-free instances. The experiments show that the optimizer using the proposed mechanisms remains equally effective despite the noise level and outperforms state-of-the-art optimizers for the problems with high noise.

We present mlr3mbo, a comprehensive and modular toolbox for Bayesian optimization in R. mlr3mbo supports single- and multi-objective optimization, multi-point proposals, batch and asynchronous parallelization, input and output transformations, and robust error handling. While it can be used for many standard Bayesian optimization variants in applied settings, researchers can also construct custom BO algorithms from its flexible building blocks. In addition to an introduction to the software, its design principles, and its building blocks, the paper presents two extensive empirical evaluations of the software on the surrogate-based benchmark suite YAHPO Gym. To identify robust default configurations for both numeric and mixed-hierarchical optimization regimes, and to gain further insights into the respective impacts of individual settings, we run a coordinate descent search over the mlr3mbo configuration space and analyze its results. Furthermore, we demonstrate that mlr3mbo achieves state-of-the-art performance by benchmarking it against a wide range of optimizers, including HEBO, SMAC3, Ax, and Optuna.

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions mostly act either after orthogonalization by rescaling updates or before it with heavier whitening-based preconditioners. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon in three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). These variants rebalance the momentum matrix before finite-step Newton--Schulz using row/column squared-norm statistics and only $\mathcal{O}(m+n)$ auxiliary state. We show that finite-step orthogonalization is governed by input spectral properties, especially stable rank and condition number, and that row/column normalization is a zeroth-order whitening surrogate that removes marginal scale mismatch. For the hidden matrix weights targeted by {\method}, the row-normalized variant R is the natural default and preserves the $\widetilde{\mathcal{O}}(T^{-1/4})$ stationarity guarantee of Muon-type methods. In LLaMA2 pretraining on C4, the default R variant consistently outperforms Muon on 130M and 350M models, yielding faster convergence and lower validation perplexity.

Spectral optimizers such as Muon have recently shown strong empirical performance in large-scale language model training, but the source and extent of their advantage remain poorly understood. We study this question through the linear associative memory problem, a tractable model for factual recall in transformer-based models. In particular, we go beyond orthogonal embeddings and consider Gaussian inputs and outputs, which allows the number of stored associations to greatly exceed the embedding dimension. Our main result sharply characterizes the recovery rates of one step of Muon and SGD on the logistic regression loss under a power law frequency distribution. We show that the storage capacity of Muon significantly exceeds that of SGD, and moreover Muon saturates at a larger critical batch size. We further analyze the multi-step dynamics under a thresholded gradient approximation and show that Muon achieves a substantially faster initial recovery rate than SGD, while both methods eventually converge to the information-theoretic limit at comparable speeds. Experiments on synthetic tasks validate the predicted scaling laws. Our analysis provides a quantitative understanding of the signal amplification of Muon and lays the groundwork for establishing scaling laws across more practical language modeling tasks and optimizers.

In a controlled experiment on modular arithmetic ($p = 9973$), varying only example ordering while holding all else constant, two fixed-ordering strategies achieve 99.5\% test accuracy by epochs 487 and 659 respectively from a training set comprising 0.3\% of the input space, well below established sample complexity lower bounds for this task under IID ordering. The IID baseline achieves 0.30\% after 5{,}000 epochs from identical data. An adversarially structured ordering suppresses learning entirely. The generalizing model reliably constructs a Fourier representation whose fundamental frequency is the Fourier dual of the ordering structure, encoding information present in no individual training example, with the same fundamental emerging across all seeds tested regardless of initialization or training set composition. We discuss implications for training efficiency, the reinterpretation of grokking, and the safety risks of a channel that evades all content-level auditing.