We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

This paper investigates the asymptotic distribution of the maximum-likelihood estimate (MLE) in multinomial logistic models in the high-dimensional regime where dimension and sample size are of the same order. While classical large-sample theory provides asymptotic normality of the MLE under certain conditions, such classical results are expected to fail in high-dimensions as documented for the binary logistic case in the seminal work of Sur and Cand\`es [2019]. We address this issue in classification problems with 3 or more classes, by developing asymptotic normality and asymptotic chi-square results for the multinomial logistic MLE (also known as cross-entropy minimizer) on null covariates. Our theory leads to a new methodology to test the significance of a given feature. Extensive simulation studies on synthetic data corroborate these asymptotic results and confirm the validity of proposed p-values for testing the significance of a given feature.

A heart murmur is an atypical sound produced by the flow of blood through the heart. It can be a sign of a serious heart condition, so detecting heart murmurs is critical for identifying and managing cardiovascular diseases. However, current methods for identifying murmurous heart sounds do not fully utilize the valuable insights that can be gained by exploring intrinsic properties of heart sound signals. To address this issue, this study proposes a new discriminatory set of multiscale features based on the self-similarity and complexity properties of heart sounds, as derived in the wavelet domain. Self-similarity is characterized by assessing fractal behaviors, while complexity is explored by calculating wavelet entropy. We evaluated the diagnostic performance of these proposed features for detecting murmurs using a set of standard classifiers. When applied to a publicly available heart sound dataset, our proposed wavelet-based multiscale features achieved comparable performance to existing methods with fewer features. This suggests that self-similarity and complexity properties in heart sounds could be potential biomarkers for improving the accuracy of murmur detection.

Understanding black-box Machine Learning methods on multidimensional data is a key challenge in Machine Learning. While many powerful Machine Learning methods already exist, these methods are often unexplainable or perform poorly on complex data. This paper proposes visual knowledge discovery approaches based on several forms of lossless General Line Coordinates. These are an expansion of the previously introduced General Line Coordinates Linear and Dynamic Scaffolding Coordinates to produce, explain, and visualize non-linear classifiers with explanation rules. To ensure these non-linear models and rules are accurate, General Line Coordinates Linear also developed new interactive visual knowledge discovery algorithms for finding worst-case validation splits. These expansions are General Line Coordinates non-linear, interactive rules linear, hyperblock rules linear, and worst-case linear. Experiments across multiple benchmark datasets show that this visual knowledge discovery method can compete with other visual and computational Machine Learning algorithms while improving both interpretability and accuracy in linear and non-linear classifications. Major benefits from these expansions consist of the ability to build accurate and highly interpretable models and rules from hyperblocks, the ability to analyze interpretability weaknesses in a model, and the input of expert knowledge through interactive and human-guided visual knowledge discovery methods.

To be able to produce accurate and reliable predictions of visibility has crucial importance in aviation meteorology, as well as in water- and road transportation. Nowadays, several meteorological services provide ensemble forecasts of visibility; however, the skill, and reliability of visibility predictions are far reduced compared to other variables, such as temperature or wind speed. Hence, some form of calibration is strongly advised, which usually means estimation of the predictive distribution of the weather quantity at hand either by parametric or non-parametric approaches, including also machine learning-based techniques. As visibility observations - according to the suggestion of the World Meteorological Organization - are usually reported in discrete values, the predictive distribution for this particular variable is a discrete probability law, hence calibration can be reduced to a classification problem. Based on visibility ensemble forecasts of the European Centre for Medium-Range Weather Forecasts covering two slightly overlapping domains in Central and Western Europe and two different time periods, we investigate the predictive performance of locally, semi-locally and regionally trained proportional odds logistic regression (POLR) and multilayer perceptron (MLP) neural network classifiers. We show that while climatological forecasts outperform the raw ensemble by a wide margin, post-processing results in further substantial improvement in forecast skill and in general, POLR models are superior to their MLP counterparts.

Many convex optimization problems with important applications in machine learning are formulated as empirical risk minimization (ERM). There are several examples: linear and logistic regression, LASSO, kernel regression, quantile regression, $p$-norm regression, support vector machines (SVM), and mean-field variational inference. To improve data privacy, federated learning is proposed in machine learning as a framework for training deep learning models on the network edge without sharing data between participating nodes. In this work, we present an interior point method (IPM) to solve a general ERM problem under the federated learning setting. We show that the communication complexity of each iteration of our IPM is $\tilde{O}(d^{3/2})$, where $d$ is the dimension (i.e., number of features) of the dataset.

This paper develops computationally efficient data-driven model predictive control (MPC) for Agile quadrotor flight. Agile quadrotors in high-speed flights can experience high levels of aerodynamic effects. Modeling these turbulent aerodynamic effects is a cumbersome task and the resulting model may be overly complex and computationally infeasible. Combining Gaussian Process (GP) regression models with a simple dynamic model of the system has demonstrated significant improvements in control performance. However, direct integration of the GP models to the MPC pipeline poses a significant computational burden to the optimization process. Therefore, we present an approach to separate the GP models to the MPC pipeline by computing the model corrections using reference trajectory and the current state measurements prior to the online MPC optimization. This method has been validated in the Gazebo simulation environment and has demonstrated of up to $50\%$ reduction in trajectory tracking error, matching the performance of the direct GP integration method with improved computational efficiency.

Sparse linear regression is a central problem in high-dimensional statistics. We study the correlated random design setting, where the covariates are drawn from a multivariate Gaussian $N(0,\Sigma)$, and we seek an estimator with small excess risk. If the true signal is $t$-sparse, information-theoretically, it is possible to achieve strong recovery guarantees with only $O(t\log n)$ samples. However, computationally efficient algorithms have sample complexity linear in (some variant of) the condition number of $\Sigma$. Classical algorithms such as the Lasso can require significantly more samples than necessary even if there is only a single sparse approximate dependency among the covariates. We provide a polynomial-time algorithm that, given $\Sigma$, automatically adapts the Lasso to tolerate a small number of approximate dependencies. In particular, we achieve near-optimal sample complexity for constant sparsity and if $\Sigma$ has few ``outlier'' eigenvalues. Our algorithm fits into a broader framework of feature adaptation for sparse linear regression with ill-conditioned covariates. With this framework, we additionally provide the first polynomial-factor improvement over brute-force search for constant sparsity $t$ and arbitrary covariance $\Sigma$.

A multivariate regression model of affine and diffeomorphic transformation sequences - FineMorphs - is presented. Leveraging concepts from shape analysis, model states are optimally "reshaped" by diffeomorphisms generated by smooth vector fields during learning. Affine transformations and vector fields are optimized within an optimal control setting, and the model can naturally reduce (or increase) dimensionality and adapt to large datasets via suboptimal vector fields. An existence proof of solution and necessary conditions for optimality for the model are derived. Experimental results on real datasets from the UCI repository are presented, with favorable results in comparison with state-of-the-art in the literature and densely-connected neural networks in TensorFlow.

In deep learning, often the training process finds an interpolator (a solution with 0 training loss), but the test loss is still low. This phenomenon, known as benign overfitting, is a major mystery that received a lot of recent attention. One common mechanism for benign overfitting is implicit regularization, where the training process leads to additional properties for the interpolator, often characterized by minimizing certain norms. However, even for a simple sparse linear regression problem $y = \beta^{*\top} x +\xi$ with sparse $\beta^*$, neither minimum $\ell_1$ or $\ell_2$ norm interpolator gives the optimal test loss. In this work, we give a different parametrization of the model which leads to a new implicit regularization effect that combines the benefit of $\ell_1$ and $\ell_2$ interpolators. We show that training our new model via gradient descent leads to an interpolator with near-optimal test loss. Our result is based on careful analysis of the training dynamics and provides another example of implicit regularization effect that goes beyond norm minimization.