Graph Contrastive Learning (GCL) has emerged as the foremost approach for self-supervised learning on graph-structured data. GCL reduces reliance on labeled data by learning robust representations from various augmented views. However, existing GCL methods typically depend on consistent stochastic augmentations, which overlook their impact on the intrinsic structure of the spectral domain, thereby limiting the model's ability to generalize effectively. To address these limitations, we propose a novel paradigm called AS-GCL that incorporates asymmetric spectral augmentation for graph contrastive learning. A typical GCL framework consists of three key components: graph data augmentation, view encoding, and contrastive loss. Our method introduces significant enhancements to each of these components. Specifically, for data augmentation, we apply spectral-based augmentation to minimize spectral variations, strengthen structural invariance, and reduce noise. With respect to encoding, we employ parameter-sharing encoders with distinct diffusion operators to generate diverse, noise-resistant graph views. For contrastive loss, we introduce an upper-bound loss function that promotes generalization by maintaining a balanced distribution of intra- and inter-class distance. To our knowledge, we are the first to encode augmentation views of the spectral domain using asymmetric encoders. Extensive experiments on eight benchmark datasets across various node-level tasks demonstrate the advantages of the proposed method.

We solve a regularized weighted low-rank approximation problem by a stochastic gradient descent on a manifold. To guarantee the convergence of our stochastic gradient descent, we establish a convergence theorem on manifolds for retraction-based stochastic gradient descents admitting confinements. On sample data from the Netflix Prize training dataset, our algorithm outperforms the existing stochastic gradient descent on Euclidean spaces. We also compare the accelerated line search on this manifold to the existing accelerated line search on Euclidean spaces.

Exponential moving average (EMA) has recently gained significant popularity in training modern deep learning models, especially diffusion-based generative models. However, there have been few theoretical results explaining the effectiveness of EMA. In this paper, to better understand EMA, we establish the risk bound of online SGD with EMA for high-dimensional linear regression, one of the simplest overparameterized learning tasks that shares similarities with neural networks. Our results indicate that (i) the variance error of SGD with EMA is always smaller than that of SGD without averaging, and (ii) unlike SGD with iterate averaging from the beginning, the bias error of SGD with EMA decays exponentially in every eigen-subspace of the data covariance matrix. Additionally, we develop proof techniques applicable to the analysis of a broad class of averaging schemes.

We study fundamental limits of first-order stochastic optimization in a range of nonconvex settings, including L-smooth functions satisfying Quasar-Convexity (QC), Quadratic Growth (QG), and Restricted Secant Inequalities (RSI). While the convergence properties of standard algorithms are well-understood in deterministic regimes, significantly fewer results address the stochastic case, where only unbiased and noisy gradients are available. We establish new lower bounds on the number of noisy gradient queries to minimize these classes of functions, also showing that they are tight (up to a logarithmic factor) in all the relevant quantities characterizing each class. Our approach reformulates the optimization task as a function identification problem, leveraging divergence composition arguments to construct a challenging subclass that leads to sharp lower bounds. Furthermore, we present a specialized algorithm in the one-dimensional setting that achieves faster rates, suggesting that certain dimensional thresholds are intrinsic to the complexity of non-convex stochastic optimization.

It has recently been argued that AI models' representations are becoming aligned as their scale and performance increase. Empirical analyses have been designed to support this idea and conjecture the possible alignment of different representations toward a shared statistical model of reality. In this paper, we propose a learning-theoretic perspective to representation alignment. First, we review and connect different notions of alignment based on metric, probabilistic, and spectral ideas. Then, we focus on stitching, a particular approach to understanding the interplay between different representations in the context of a task. Our main contribution here is relating properties of stitching to the kernel alignment of the underlying representation. Our results can be seen as a first step toward casting representation alignment as a learning-theoretic problem.

Understanding the advantages of deep neural networks trained by gradient descent (GD) compared to shallow models remains an open theoretical challenge. While the study of multi-index models with Gaussian data in high dimensions has provided analytical insights into the benefits of GD-trained neural networks over kernels, the role of depth in improving sample complexity and generalization in GD-trained networks remains poorly understood. In this paper, we introduce a class of target functions (single and multi-index Gaussian hierarchical targets) that incorporate a hierarchy of latent subspace dimensionalities. This framework enables us to analytically study the learning dynamics and generalization performance of deep networks compared to shallow ones in the high-dimensional limit. Specifically, our main theorem shows that feature learning with GD reduces the effective dimensionality, transforming a high-dimensional problem into a sequence of lower-dimensional ones. This enables learning the target function with drastically less samples than with shallow networks. While the results are proven in a controlled training setting, we also discuss more common training procedures and argue that they learn through the same mechanisms. These findings open the way to further quantitative studies of the crucial role of depth in learning hierarchical structures with deep networks.

In recent years, mixup regularization has gained popularity as an effective way to improve the generalization performance of deep learning models by training on convex combinations of training data. While many mixup variants have been explored, the proper adoption of the technique to conditional density estimation and probabilistic machine learning remains relatively unexplored. This work introduces a novel framework for mixup regularization based on probabilistic fusion that is better suited for conditional density estimation tasks. For data distributed according to a member of the exponential family, we show that likelihood functions can be analytically fused using log-linear pooling. We further propose an extension of probabilistic mixup, which allows for fusion of inputs at an arbitrary intermediate layer of the neural network. We provide a theoretical analysis comparing our approach to standard mixup variants. Empirical results on synthetic and real datasets demonstrate the benefits of our proposed framework compared to existing mixup variants.

A substantial body of work in machine learning (ML) and randomized numerical linear algebra (RandNLA) has exploited various sorts of random sketching methodologies, including random sampling and random projection, with much of the analysis using Johnson--Lindenstrauss and subspace embedding techniques. Recent studies have identified the issue of inversion bias -- the phenomenon that inverses of random sketches are not unbiased, despite the unbiasedness of the sketches themselves. This bias presents challenges for the use of random sketches in various ML pipelines, such as fast stochastic optimization, scalable statistical estimators, and distributed optimization. In the context of random projection, the inversion bias can be easily corrected for dense Gaussian projections (which are, however, too expensive for many applications). Recent work has shown how the inversion bias can be corrected for sparse sub-gaussian projections. In this paper, we show how the inversion bias can be corrected for random sampling methods, both uniform and non-uniform leverage-based, as well as for structured random projections, including those based on the Hadamard transform. Using these results, we establish problem-independent local convergence rates for sub-sampled Newton methods.

This paper introduces OGBoost, a scikit-learn-compatible Python package for ordinal regression using gradient boosting. Ordinal variables (e.g., rating scales, quality assessments) lie between nominal and continuous data, necessitating specialized methods that reflect their inherent ordering. Built on a coordinate-descent approach for optimization and the latent-variable framework for ordinal regression, OGBoost performs joint optimization of a latent continuous regression function (functional gradient descent) and a threshold vector that converts the latent continuous value into discrete class probabilities (classical gradient descent). In addition to the stanadard methods for scikit-learn classifiers, the GradientBoostingOrdinal class implements a "decision_function" that returns the (scalar) value of the latent function for each observation, which can be used as a high-resolution alternative to class labels for comparing and ranking observations. The class has the option to use cross-validation for early stopping rather than a single holdout validation set, a more robust approach for small and/or imbalanced datasets. Furthermore, users can select base learners with different underlying algorithms and/or hyperparameters for use throughout the boosting iterations, resulting in a `heterogeneous' ensemble approach that can be used as a more efficient alternative to hyperparameter tuning (e.g. via grid search). We illustrate the capabilities of OGBoost through examples, using the wine quality dataset from the UCI respository. The package is available on PyPI and can be installed via "pip install ogboost".

We present an agentic, autonomous graph expansion framework that iteratively structures and refines knowledge in situ. Unlike conventional knowledge graph construction methods relying on static extraction or single-pass learning, our approach couples a reasoning-native large language model with a continually updated graph representation. At each step, the system actively generates new concepts and relationships, merges them into a global graph, and formulates subsequent prompts based on its evolving structure. Through this feedback-driven loop, the model organizes information into a scale-free network characterized by hub formation, stable modularity, and bridging nodes that link disparate knowledge clusters. Over hundreds of iterations, new nodes and edges continue to appear without saturating, while centrality measures and shortest path distributions evolve to yield increasingly distributed connectivity. Our analysis reveals emergent patterns, such as the rise of highly connected 'hub' concepts and the shifting influence of 'bridge' nodes, indicating that agentic, self-reinforcing graph construction can yield open-ended, coherent knowledge structures. Applied to materials design problems, we present compositional reasoning experiments by extracting node-specific and synergy-level principles to foster genuinely novel knowledge synthesis, yielding cross-domain ideas that transcend rote summarization and strengthen the framework's potential for open-ended scientific discovery. We discuss other applications in scientific discovery and outline future directions for enhancing scalability and interpretability.