Pairwise learning refers to learning tasks with loss functions depending on a pair of training examples, which includes ranking and metric learning as specific examples. Recently, there has been an increasing amount of attention on the generalization analysis of pairwise learning to understand its practical behavior. However, the existing stability analysis provides suboptimal high-probability generalization bounds. In this paper, we provide a refined stability analysis by developing generalization bounds which can be \sqrt{n} -times faster than the existing results, where n is the sample size. This implies excess risk bounds of the order O(n {-1/2}) (up to a logarithmic factor) for both regularized risk minimization and stochastic gradient descent. We also introduce a new on-average stability measure to develop optimistic bounds in a low noise setting.

We focus on the problem of efficient sampling and learning of probability densities by incorporating symmetries in probabilistic models. We first introduce Equivariant Stein Variational Gradient Descent algorithm -- an equivariant sampling method based on Stein's identity for sampling from densities with symmetries. Equivariant SVGD explicitly incorporates symmetry information in a density through equivariant kernels which makes the resultant sampler efficient both in terms of sample complexity and the quality of generated samples. Subsequently, we define equivariant energy based models to model invariant densities that are learned using contrastive divergence. By utilizing our equivariant SVGD for training equivariant EBMs, we propose new ways of improving and scaling up training of energy based models.

The softmax is the standard transformation used in machine learning to map real-valued vectors to categorical distributions. Unfortunately, this transform poses serious drawbacks for gradient descent (ascent) optimization. We reveal this difficulty by establishing two negative results: (1) optimizing any expectation with respect to the softmax must exhibit sensitivity to parameter initialization (softmax gravity well''), and (2) optimizing log-probabilities under the softmax must exhibit slow convergence (softmax damping''). Both findings are based on an analysis of convergence rates using the Non-uniform \L{}ojasiewicz (N\L{}) inequalities. To circumvent these shortcomings we investigate an alternative transformation, the \emph{escort} mapping, that demonstrates better optimization properties.

Meta learning has demonstrated tremendous success in few-shot learning with limited supervised data. In those settings, the meta model is usually overparameterized. While the conventional statistical learning theory suggests that overparameterized models tend to overfit, empirical evidence reveals that overparameterized meta learning methods still work well -- a phenomenon often called benign overfitting.'' To understand this phenomenon, we focus on the meta learning settings with a challenging bilevel structure that we term the gradient-based meta learning, and analyze its generalization performance under an overparameterized meta linear regression model. While our analysis uses the relatively tractable linear models, our theory contributes to understanding the delicate interplay among data heterogeneity, model adaptation and benign overfitting in gradient-based meta learning tasks.

The alternating direction method of multipliers (ADMM) is an important tool for solving complex optimization problems, but it involves minimization sub-steps that are often difficult to solve efficiently. The Primal-Dual Hybrid Gradient (PDHG) method is a powerful alternative that often has simpler substeps than ADMM, thus producing lower complexity solvers. Despite the flexibility of this method, PDHG is often impractical because it requires the careful choice of multiple stepsize parameters. There is often no intuitive way to choose these parameters to maximize efficiency, or even achieve convergence. We propose self-adaptive stepsize rules that automatically tune PDHG parameters for optimal convergence.

As machine learning has become more prevalent, researchers have begun to recognize the necessity of ensuring machine learning systems are fair. Recently, there has been an interest in defining a notion of fairness that mitigates over-representation in traditional clustering. In this paper we extend this notion to hierarchical clustering, where the goal is to recursively partition the data to optimize a specific objective. For various natural objectives, we obtain simple, efficient algorithms to find a provably good fair hierarchical clustering. Empirically, we show that our algorithms can find a fair hierarchical clustering, with only a negligible loss in the objective.

We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the noise inherent to the stochastic approximation scheme dominates any noise from asynchrony. We also give empirical evidence demonstrating the strong performance of asynchronous, parallel stochastic optimization schemes, demonstrating that the robustness inherent to stochastic approximation problems allows substantially faster parallel and asynchronous solution methods. In short, we show that for many stochastic approximation problems, as Freddie Mercury sings in Queen's \emph{Bohemian Rhapsody}, Nothing really matters.''

Additive manufacturing (AM) is a rapidly evolving technology that has attracted applications across a wide range of fields due to its ability to fabricate complex geometries. However, one of the key challenges in AM is achieving consistent print quality. This inconsistency is often attributed to uncontrolled melt pool dynamics, partly caused by spatter which can lead to defects. Therefore, capturing and controlling the evolution of the melt pool is crucial for enhancing process stability and part quality. In this study, we developed a framework to support decision-making in AM operations, facilitating quality control and minimizing defects via machine learning (ML) and polynomial symbolic regression models. We implemented experimentally validated computational tools as a cost-effective approach to collect large datasets from laser powder bed fusion (LPBF) processes. For a dataset consisting of 281 process conditions, parameters such as melt pool dimensions (length, width, depth), melt pool geometry (area, volume), and volume indicated as spatter were extracted. Using machine learning (ML) and polynomial symbolic regression models, a high R2 of over 95 % was achieved in predicting the melt pool dimensions and geometry features for both the training and testing datasets, with either process conditions (power and velocity) or melt pool dimensions as the model inputs. In the case of volume indicated as spatter, R2 improved after logarithmic transforming the model inputs, which was either the process conditions or the melt pool dimensions. Among the investigated ML models, the ExtraTree model achieved the highest R2 values of 96.7 % and 87.5 %.

Artificial intelligence (AI) has transformed the geoscience community with deep learning models (DLMs) that are trained to complete specific tasks within workflows. This success has led to the development of geoscience foundation models (FMs), which promise to accomplish multiple tasks within a workflow or replace the workflow altogether. However, lack of robust evaluation frameworks, even for traditional DLMs, leaves the geoscience community ill prepared for the inevitable adoption of FMs. We address this gap by designing an evaluation framework that jointly incorporates three crucial aspects to current DLMs and future FMs: performance uncertainty, learning efficiency, and overlapping training-test data splits. To target the three aspects, we meticulously construct the training, validation, and test splits using clustering methods tailored to geoscience data and enact an expansive training design to segregate performance uncertainty arising from stochastic training processes and random data sampling. The framework's ability to guard against misleading declarations of model superiority is demonstrated through evaluation of PhaseNet, a popular seismic phase picking DLM, under 3 training approaches. Furthermore, we show how the performance gains due to overlapping training-test data can lead to biased FM evaluation. Our framework helps practitioners choose the best model for their problem and set performance expectations by explicitly analyzing model performance at varying budgets of training data.

Compared with standard learning, adversarially robust learning is widely recognized to demand significantly more training examples. Recent works propose the use of self-supervised adversarial training (SSAT) with external or synthetically generated unlabeled data to enhance model robustness. However, SSAT requires a substantial amount of extra unlabeled data, significantly increasing memory usage and model training times. To address these challenges, we propose novel methods to strategically select a small subset of unlabeled data essential for SSAT and robustness improvement. Our selection prioritizes data points near the model's decision boundary based on latent clustering-based techniques, efficiently identifying a critical subset of unlabeled data with a higher concentration of boundary-adjacent points. While focusing on near-boundary data, our methods are designed to maintain a balanced ratio between boundary and non-boundary data points to avoid overfitting. Our experiments on image benchmarks show that integrating our selection strategies into self-supervised adversarial training can largely reduce memory and computational requirements while achieving high model robustness. In particular, our latent clustering-based selection method with k-means is the most effective, achieving nearly identical test-time robust accuracies with 5 to 10 times less external or generated unlabeled data when applied to image benchmarks. Additionally, we validate the generalizability of our approach across various application scenarios, including a real-world medical dataset for COVID-19 chest X-ray classification.