Orthogonal gradient updates have emerged as a promising direction in optimization for machine learning. However, traditional approaches such as SVD/QR decomposition incur prohibitive computational costs of O(n^3) and underperform compared to well-tuned SGD with momentum, since momentum is applied only after strict orthogonalization. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and producing semi-orthogonal matrices via Newton-Schulz iterations, reducing complexity to O(n^2). Nevertheless, quadratic costs remain a bottleneck. In this work, we study the semi-orthogonal properties of momentum-based updates and develop a method to bound momentum updates under a spectral-norm trust region, preserving directional information without requiring explicit semi-orthogonalization. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without constructing semi-orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. Our approach combines hyperbolic-cosine RMS scaling transformations with normalization, demonstrating both effectiveness and computational efficiency compared to Newton-Schulz methods. We further introduce a hybrid variant (Hybrid-AuON) that applies a single Newton-Schulz iteration. Experiments across vision and language benchmarks show that AuON and its hybrid variant achieve performance comparable to strong baselines such as AdamW and Muon. Code is available at: https://github.com/ryyzn9/AuON

Structured sparsity regularization offers a principled way to compact neural networks, but its non-differentiability breaks compatibility with conventional stochastic gradient descent and requires either specialized optimizers or additional post-hoc pruning without formal guarantees. In this work, we propose $D$-Gating, a fully differentiable structured overparameterization that splits each group of weights into a primary weight vector and multiple scalar gating factors. We prove that any local minimum under $D$-Gating is also a local minimum using non-smooth structured $L_{2,2/D}$ penalization, and further show that the $D$-Gating objective converges at least exponentially fast to the $L_{2,2/D}$-regularized loss in the gradient flow limit. Together, our results show that $D$-Gating is theoretically equivalent to solving the original group sparsity problem, yet induces distinct learning dynamics that evolve from a non-sparse regime into sparse optimization. We validate our theory across vision, language, and tabular tasks, where $D$-Gating consistently delivers strong performance-sparsity tradeoffs and outperforms both direct optimization of structured penalties and conventional pruning baselines.

Asynchronous stochastic gradient methods are central to scalable distributed optimization, particularly when devices differ in computational capabilities. Such settings arise naturally in federated learning, where training takes place on smartphones and other heterogeneous edge devices. In addition to varying computation speeds, these devices often hold data from different distributions. However, existing asynchronous SGD methods struggle in such heterogeneous settings and face two key limitations. First, many rely on unrealistic assumptions of similarity across workers' data distributions. Second, methods that relax this assumption still fail to achieve theoretically optimal performance under heterogeneous computation times. We introduce Ringleader ASGD, the first asynchronous SGD algorithm that attains the theoretical lower bounds for parallel first-order stochastic methods in the smooth nonconvex regime, thereby achieving optimal time complexity under data heterogeneity and without restrictive similarity assumptions. Our analysis further establishes that Ringleader ASGD remains optimal under arbitrary and even time-varying worker computation speeds, closing a fundamental gap in the theory of asynchronous optimization.

When different objectives conflict with each other in multi-task learning, gradients begin to interfere and slow convergence, thereby potentially reducing the final model's performance. To address this, we introduce SON-GOKU, a scheduler that computes gradient interference, constructs an interference graph, and then applies greedy graph-coloring to partition tasks into groups that align well with each other. At each training step, only one group (color class) of tasks are activated, and the grouping partition is constantly recomputed as task relationships evolve throughout training. By ensuring that each mini-batch contains only tasks that pull the model in the same direction, our method improves the effectiveness of any underlying multi-task learning optimizer without additional tuning. Since tasks within these groups will update in compatible directions, multi-task learning will improve model performance rather than impede it. Empirical results on six different datasets show that this interference-aware graph-coloring approach consistently outperforms baselines and state-of-the-art multi-task optimizers. We provide extensive theory showing why grouping and sequential updates improve multi-task learning, with guarantees on descent, convergence, and accurately identifying what tasks conflict or align.

This paper addresses the statistical estimation of Gaussian Mixture Models (GMMs) with unknown diagonal covariances from independent and identically distributed samples. We employ the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures, to simultaneously estimate the number of components and their parameters. Our main contribution extends the BLASSO methodology to multivariate GMMs with component-specific unknown diagonal covariance matrices-a significantly more flexible setting than previous approaches requiring known and identical covariances. We establish non-asymptotic recovery guarantees with nearly parametric convergence rates for component means, diagonal covariances, and weights, as well as for density prediction. A key theoretical contribution is the identification of an explicit separation condition on mixture components that enables the construction of non-degenerate dual certificates-essential tools for establishing statistical guarantees for the BLASSO. Our analysis leverages the Fisher-Rao geometry of the statistical model and introduces a novel semi-distance adapted to our framework, providing new insights into the interplay between component separation, parameter space geometry, and achievable statistical recovery.

Classical influence functions face significant challenges when applied to deep neural networks, primarily due to non-invertible Hessians and high-dimensional parameter spaces. We propose the local Bayesian influence function (BIF), an extension of classical influence functions that replaces Hessian inversion with loss landscape statistics that can be estimated via stochastic-gradient MCMC sampling. This Hessian-free approach captures higher-order interactions among parameters and scales efficiently to neural networks with billions of parameters. We demonstrate state-of-the-art results on predicting retraining experiments.

A central goal in AI interpretability and data attribution is interpreting and mapping the global structure of the data distribution as seen by a trained neural network (Carter et al., 2019; Pepin Lehalleur et al., 2025; Olah, 2015). One approach is to start local, by quantifying a suitable measure of similarity between pairs of individual samples--that is, by defining a kernel. "Interpreting" the global structure of the data distribution then becomes a problem of analyzing the geometric structure in this kernel (e.g., via clustering techniques), and "mapping" becomes a problem of visualizing points in this kernel space (e.g., via dimensionality reduction techniques). This kernel-based approach has been used successfully with similarity measures derived from activations or representations. For example, it is possible to define a kernel via cosine similarity between the hidden vectors of sparse autoencoders (SAEs).

In certain real-world optimization scenarios, practitioners are not interested in solving multiple problems but rather in finding the best solution to a single, specific problem. When the computational budget is large relative to the cost of evaluating a candidate solution, multiple heuristic alternatives can be tried to solve the same given problem, each possibly with a different algorithm, parameter configuration, initialization, or stopping criterion. The sequential selection of which alternative to try next is crucial for efficiently identifying the one that provides the best possible solution across multiple attempts. Despite the relevance of this problem in practice, it has not yet been the exclusive focus of any existing review. Several sequential alternative selection strategies have been proposed in different research topics, but they have not been comprehensively and systematically unified under a common perspective. This work presents a focused review of single-problem multi-attempt heuristic optimization. It brings together suitable strategies to this problem that have been studied separately through algorithm selection, parameter tuning, multi-start and resource allocation. These strategies are explained using a unified terminology within a common framework, which supports the development of a taxonomy for systematically organizing and classifying them.

Autonomous Raman instruments on Mars rovers, deep-sea landers, and field robots must interpret raw spectra distorted by fluorescence baselines, peak shifts, and limited ground-truth labels. Using curated subsets of the RRUFF database, we evaluate one-dimensional convolutional neural networks (CNNs) and report four advances: (i) Baseline-independent classification: compact CNNs surpass $k$-nearest-neighbors and support-vector machines on handcrafted features, removing background-correction and peak-picking stages while ensuring reproducibility through released data splits and scripts. (ii) Pooling-controlled robustness: tuning a single pooling parameter accommodates Raman shifts up to $30 \,\mathrm{cm}^{-1}$, balancing translational invariance with spectral resolution. (iii) Label-efficient learning: semi-supervised generative adversarial networks and contrastive pretraining raise accuracy by up to $11\%$ with only $10\%$ labels, valuable for autonomous deployments with scarce annotation. (iv) Constant-time adaptation: freezing the CNN backbone and retraining only the softmax layer transfers models to unseen minerals at $\mathcal{O}(1)$ cost, outperforming Siamese networks on resource-limited processors. This workflow, which involves training on raw spectra, tuning pooling, adding semi-supervision when labels are scarce, and fine-tuning lightly for new targets, provides a practical path toward robust, low-footprint Raman classification in autonomous exploration.

Feature selection is crucial for fuzzy decision systems (FDSs), as it identifies informative features and eliminates rule redundancy, thereby enhancing predictive performance and interpretability. Most existing methods either fail to directly align evaluation criteria with learning performance or rely solely on non-directional Euclidean distances to capture relationships among decision classes, which limits their ability to clarify decision boundaries. However, the spatial distribution of instances has a potential impact on the clarity of such boundaries. Motivated by this, we propose Spatially-aware Separability-driven Feature Selection (S$^2$FS), a novel framework for FDSs guided by a spatially-aware separability criterion. This criterion jointly considers within-class compactness and between-class separation by integrating scalar-distances with spatial directional information, providing a more comprehensive characterization of class structures. S$^2$FS employs a forward greedy strategy to iteratively select the most discriminative features. Extensive experiments on ten real-world datasets demonstrate that S$^2$FS consistently outperforms eight state-of-the-art feature selection algorithms in both classification accuracy and clustering performance, while feature visualizations further confirm the interpretability of the selected features.