We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL) networks. In particular, we prove that the globally minimizing weights and biases for the $\mathcal{L}^2$ cost obtained constructively in [Chen-Munoz Ewald 2023] for underparametrized ReLU DL networks can generically not be approximated via the gradient descent flow. We therefore conclude that the method introduced in [Chen-Munoz Ewald 2023] is disjoint from the gradient descent method.

Deep learning is renowned for its theory-practice gap, whereby principled theory typically fails to provide much beneficial guidance for implementation in practice. This has been highlighted recently by the benign overfitting phenomenon: when neural networks become sufficiently large to interpolate the dataset perfectly, model performance appears to improve with increasing model size, in apparent contradiction with the well-known bias-variance tradeoff. While such phenomena have proven challenging to theoretically study for general models, the recently proposed Interpolating Information Criterion (IIC) provides a valuable theoretical framework to examine performance for overparameterized models. Using the IIC, a PAC-Bayes bound is obtained for a general class of models, characterizing factors which influence generalization performance in the interpolating regime. From the provided bound, we quantify how the test error for overparameterized models achieving effectively zero training error depends on the quality of the implicit regularization imposed by e.g. the combination of model, optimizer, and parameter-initialization scheme; the spectrum of the empirical neural tangent kernel; curvature of the loss landscape; and noise present in the data.

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction that is useful for various data science problems. However, many applications involve heterogeneous data that varies in quality due to noise characteristics associated with different sources of the data. Methods that deal with this mixed dataset are known as heteroscedastic methods. Current methods like HePPCAT make Gaussian assumptions of the basis coefficients that may not hold in practice. Other methods such as Weighted PCA (WPCA) assume the noise variances are known, which may be difficult to know in practice. This paper develops a PCA method that can estimate the sample-wise noise variances and use this information in the model to improve the estimate of the subspace basis associated with the low-rank structure of the data. This is done without distributional assumptions of the low-rank component and without assuming the noise variances are known. Simulations show the effectiveness of accounting for such heteroscedasticity in the data, the benefits of using such a method with all of the data versus retaining only good data, and comparisons are made against other PCA methods established in the literature like PCA, Robust PCA (RPCA), and HePPCAT. Code available at https://github.com/javiersc1/ALPCAH

Single-call stochastic extragradient methods, like stochastic past extragradient (SPEG) and stochastic optimistic gradient (SOG), have gained a lot of interest in recent years and are one of the most efficient algorithms for solving large-scale min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. However, despite their undoubted popularity, current convergence analyses of SPEG and SOG require a bounded variance assumption. In addition, several important questions regarding the convergence properties of these methods are still open, including mini-batching, efficient step-size selection, and convergence guarantees under different sampling strategies. In this work, we address these questions and provide convergence guarantees for two large classes of structured non-monotone VIPs: (i) quasi-strongly monotone problems (a generalization of strongly monotone problems) and (ii) weak Minty variational inequalities (a generalization of monotone and Minty VIPs). We introduce the expected residual condition, explain its benefits, and show how it can be used to obtain a strictly weaker bound than previously used growth conditions, expected co-coercivity, or bounded variance assumptions. Equipped with this condition, we provide theoretical guarantees for the convergence of single-call extragradient methods for different step-size selections, including constant, decreasing, and step-size-switching rules. Furthermore, our convergence analysis holds under the arbitrary sampling paradigm, which includes importance sampling and various mini-batching strategies as special cases.

We present in this paper a family of generalized simultaneous perturbation-based gradient search (GSPGS) estimators that use noisy function measurements. The number of function measurements required by each estimator is guided by the desired level of accuracy. We first present in detail unbalanced generalized simultaneous perturbation stochastic approximation (GSPSA) estimators and later present the balanced versions (B-GSPSA) of these. We extend this idea further and present the generalized smoothed functional (GSF) and generalized random directions stochastic approximation (GRDSA) estimators, respectively, as well as their balanced variants. We show that estimators within any specified class requiring more number of function measurements result in lower estimator bias. We present a detailed analysis of both the asymptotic and non-asymptotic convergence of the resulting stochastic approximation schemes. We further present a series of experimental results with the various GSPGS estimators on the Rastrigin and quadratic function objectives. Our experiments are seen to validate our theoretical findings.

Description of the problem and related works Fitting a system of nonparametric ordinary differential equations (ODEs) ẋ = f (t, x) to longitudinal data could lead to scientific breakthroughs in disciplines where ODEs or dynamical systems have been used for a long time, including physics, chemistry, and biology, see [1]. By nonparametric, we mean that there is no need to specify the functional form of the vector-field f using a pre-defined finite dimensional parameter. Instead, this force field belongs to a functional space and the number of parameters that characterize this vector field depends on the amount of data available. This provides a great advantage in situations where the form of the vector field is unknown but data is available for learning. The functional spaces considered are Reproducing Kernel Hilbert Spaces (RKHS) [2], allowing for efficient optimization among other desirable properties. A particular difficulty arises when the data is sparse and noisy. This is often the case for longitudinal healthcare data obtained during hospital visits.

Over the recent past data-driven algorithms for solving stochastic optimal control problems in face of model uncertainty have become an increasingly active area of research. However, for singular controls and underlying diffusion dynamics the analysis has so far been restricted to the scalar case. In this paper we fill this gap by studying a multivariate singular control problem for reversible diffusions with controls of reflection type. Our contributions are threefold. We first explicitly determine the long-run average costs as a domain-dependent functional, showing that the control problem can be equivalently characterized as a shape optimization problem. For given diffusion dynamics, assuming the optimal domain to be strongly star-shaped, we then propose a gradient descent algorithm based on polytope approximations to numerically determine a cost-minimizing domain. Finally, we investigate data-driven solutions when the diffusion dynamics are unknown to the controller. Using techniques from nonparametric statistics for stochastic processes, we construct an optimal domain estimator, whose static regret is bounded by the minimax optimal estimation rate of the unreflected process' invariant density. In the most challenging situation, when the dynamics must be learned simultaneously to controlling the process, we develop an episodic learning algorithm to overcome the emerging exploration-exploitation dilemma and show that given the static regret as a baseline, the loss in its sublinear regret per time unit is of natural order compared to the one-dimensional case.

This study investigates a range of psychological, lexical, semantic, and readability features of book reviews to elucidate the factors underlying their perceived popularity. To this end, we conduct statistical analyses of various features, including the types and frequency of opinion and emotion-conveying terms, connectives, character mentions, word uniqueness, commonness, and sentence structure, among others. Additionally, we utilize two readability tests to explore whether reading ease is positively associated with review popularity. Finally, we employ traditional machine learning classifiers and transformer-based fine-tuned language models with n-gram features to automatically determine review popularity. Our findings indicate that, with the exception of a few features (e.g., review length, emotions, and word uniqueness), most attributes do not exhibit significant differences between popular and non-popular review groups. Furthermore, the poor performance of machine learning classifiers using the word n-gram feature highlights the challenges associated with determining popularity in creative domains. Overall, our study provides insights into the factors underlying review popularity and highlights the need for further research in this area, particularly in the creative realm.

Generalization is at the core of machine learning models. However, the definition of generalization is not entirely clear. We employ set theory to introduce the concepts of algorithms, hypotheses, and dataset generalization. We analyze the properties of dataset generalization and prove a theorem on surrogate generalization procedures. This theorem leads to our generalization method. Through a generalization experiment on the MNIST dataset, we obtain 13,541 sample bases. When we use the entire training set to evaluate the model's performance, the models achieve an accuracy of 99.945%. However, if we shift the sample bases or modify the neural network structure, the performance experiences a significant decline. We also identify consistently mispredicted samples and find that they are all challenging examples. The experiments substantiated the accuracy of the generalization definition and the effectiveness of the proposed methods. Both the set-theoretic deduction and the experiments help us better understand generalization.

Initially, those scholars employ these historical elements to forecast whether the criminal would re-offend. Subsequently, the binary outcome of recidivism serves as a proxy variable for recidivism risk. Some computer scientists also employ the probability (or score) assigned by the model for an offender's likelihood of re-offense as a gauge for their recidivism risk (Etzler et al., 2023; Ma et al., 2022; Wang et al., 2022). While such configurations may seem intuitively compelling, they often embody an oversimplified and deterministic viewpoint, which stands in contradiction to contemporary social science theories. Firstly, historical factors alone are insufficient predictors of human actions.