Gathering labeled data to train well-performing machine learning models is one of the critical challenges in many applications. Active learning aims at reducing the labeling costs by an efficient and effective allocation of costly labeling resources. In this article, we propose a decision-theoretic selection strategy that (1) directly optimizes the gain in misclassification error, and (2) uses a Bayesian approach by introducing a conjugate prior distribution to determine the class posterior to deal with uncertainties. By reformulating existing selection strategies within our proposed model, we can explain which aspects are not covered in current state-of-the-art and why this leads to the superior performance of our approach. Extensive experiments on a large variety of datasets and different kernels validate our claims.

In the artificial neuron, I replace the dot product with the weighted Lehmer mean, which may emulate different cases of a generalized mean. The single neuron instance is replaced by a multiplet of neurons which have the same averaging weights. A group of outputs feed forward, in lieu of the single scalar. The generalization parameter is typically set to a different value for each neuron in the multiplet. I further extend the concept to a multiplet taken from the Gini mean. Derivatives with respect to the weight parameters and with respect to the two generalization parameters are given. Some properties of the network are investigated, showing the capacity to emulate the classical exclusive-or problem organically in two layers and perform some multiplication and division. The network can instantiate truncated power series and variants, which can be used to approximate different functions, provided that parameters are constrained. Moreover, a mean case slope score is derived that can facilitate a learning-rate novelty based on homogeneity of the selected elements. The multiplet neuron equation provides a way to segment regularization timeframes and approaches.

Stochastic optimization algorithms have become indispensable in modern machine learning. An unresolved foundational question in this area is the difference between with-replacement sampling and without-replacement sampling -- does the latter have superior convergence rate compared to the former? A groundbreaking result of Recht and R\'e reduces the problem to a noncommutative analogue of the arithmetic-geometric mean inequality where $n$ positive numbers are replaced by $n$ positive definite matrices. If this inequality holds for all $n$, then without-replacement sampling indeed outperforms with-replacement sampling. The conjectured Recht-R\'e inequality has so far only been established for $n = 2$ and a special case of $n = 3$. We will show that the Recht-R\'e conjecture is false for general $n$. Our approach relies on the noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values, which we show are false as soon as $n = 5$.

Covariance data as represented by symmetric positive definite (SPD) matrices are ubiquitous throughout technical study as efficient descriptors of interdependent systems. Euclidean analysis of SPD matrices, while computationally fast, can lead to skewed and even unphysical interpretations of data. Riemannian methods preserve the geometric structure of SPD data at the cost of expensive eigenvalue computations. In this paper, we propose a geometric method for unsupervised clustering of SPD data based on the Thompson metric. This technique relies upon a novel "inductive midrange" centroid computation for SPD data, whose properties are examined and numerically confirmed. We demonstrate the incorporation of the Thompson metric and inductive midrange into X-means and K-means++ clustering algorithms.

This article will discuss what I consider to be the three levels of data science competency, namely: level 1 (basic level); level 2 (intermediate level); and level 3 (advanced level). Competency increases from level 1 to 3. We shall use Python as the default language, even though other platforms such as R, SAS, and Matlab could be used as programming languages for data science. The views provided here are my views and are based on my own journey to data science. At level one, a data science aspirant should be able to work with datasets generally presented in comma-separated values (CSV) file format. They should have competency in data basics; data visualization; and linear regression.

We introduce AdaHessian, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the Hessian. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and ADAM. The main disadvantage of traditional second order methods is their heavier per-iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in AdaHessian, including: (i) a new variance reduction estimate of the Hessian diagonal with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that AdaHessian achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of ADAM. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that AdaHessian: (i) achieves 1.80\%/1.45\% higher accuracy on ResNets20/32 on Cifar10, and 5.55\% higher accuracy on ImageNet as compared to ADAM; (ii) outperforms ADAMW for transformers by 0.27/0.33 BLEU score on IWSLT14/WMT14 and 1.8/1.0 PPL on PTB/Wikitext-103; and (iii) achieves 0.032\% better score than AdaGrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of AdaHessian is comparable to first-order methods, and that it exhibits robustness towards its hyperparameters. The code for AdaHessian is open-sourced and publicly available.

To predict an epileptic event means the ability to determine in advance the time of the seizure with the highest possible accuracy. A correct prediction benchmark for epilepsy events in clinical applications is a typical problem in biomedical signal processing that helps to an appropriate diagnosis and treatment of this disease. In this work, we use Pearson's product-moment correlation coefficient from generalized Gaussian distribution parameters coupled with a linear-based classifier to predict between seizure and non-seizure events in epileptic EEG signals. The performance in 36 epileptic events from 9 patients showing good performance with 100% of effectiveness for sensitivity and specificity greater than 83% for seizures events in all brain rhythms. Pearson's test suggests that all brain rhythms are highly correlated in non-seizure events but no during the seizure events. This suggests that our model can be scaled with the Pearson's product-moment correlation coefficient for the detection of epileptic seizures.

We develop a general non-asymptotic analysis of learning rates in kernel ridge regression (KRR), applicable for arbitrary Mercer kernels with multi-dimensional support. Our analysis is based on an operator-theoretic framework, at the core of which lies two error bounds under reproducing kernel Hilbert space norms encompassing a general class of kernels and regression functions, with remarkable extensibility to various inferential goals through augmenting results. When applied to KRR estimators, our analysis leads to error bounds under the stronger supremum norm, in addition to the commonly studied weighted $L_2$ norm; in a concrete example specialized to the Mat\'ern kernel, the established bounds recover the nearly minimax optimal rates. The wide applicability of our analysis is further demonstrated through two new theoretical results: (1) non-asymptotic learning rates for mixed partial derivatives of KRR estimators, and (2) a non-asymptotic characterization of the posterior variances of Gaussian processes, which corresponds to uncertainty quantification in kernel methods and nonparametric Bayes.

In multi-stage processes, decisions happen in an ordered sequence of stages. Many of them have the structure of dual funnel problem: as the sample size decreases from one stage to the other, the information increases. A related example is a selection process, where applicants apply for a position, prize, or grant. In each stage, more applicants are evaluated and filtered out, and from the remaining ones, more information is collected. In the last stage, decision-makers use all available information to make their final decision. To train a classifier for each stage becomes impracticable as they can underfit due to the low dimensionality in early stages or overfit due to the small sample size in the latter stages. In this work, we proposed a \textit{Multi-StaGe Transfer Learning} (MSGTL) approach that uses knowledge from simple classifiers trained in early stages to improve the performance of classifiers in the latter stages. By transferring weights from simpler neural networks trained in larger datasets, we able to fine-tune more complex neural networks in the latter stages without overfitting due to the small sample size. We show that it is possible to control the trade-off between conserving knowledge and fine-tuning using a simple probabilistic map. Experiments using real-world data demonstrate the efficacy of our approach as it outperforms other state-of-the-art methods for transfer learning and regularization.

In this paper, we study the asymptotical properties of least squares regularized regression with indefinite kernels in reproducing kernel Kre\u{\i}n spaces (RKKS). The classical approximation analysis cannot be directly applied to study its asymptotical behavior under the framework of learning theory as this problem is in essence non-convex and outputs stationary points. By introducing a bounded hyper-sphere constraint to such non-convex regularized risk minimization problem, we theoretically demonstrate that this problem has a globally optimal solution with a closed form on the sphere, which makes our approximation analysis feasible in RKKS. Accordingly, we modify traditional error decomposition techniques, prove convergence results for the introduced hypothesis error based on matrix perturbation theory, and derive learning rates of such regularized regression problem in RKKS. Under some conditions, the derived learning rates in RKKS are the same as that in reproducing kernel Hilbert spaces (RKHS), which is actually the first work on approximation analysis of regularized learning algorithms in RKKS.