By Vincent Granville Ph.D. Published in September 2022. This book covers the foundations of machine learning, with modern approaches to solving complex problems. Emphasis is on scalability, automation, testing, optimizing, and interpretability (explainable AI). For instance, regression techniques -- including logistic and Lasso -- are presented as a single method, without using advanced linear algebra. There is no need to learn 50 versions when one does it all and more.

The Stochastic Gradient Langevin Dynamics (SGLD) [49], first proposed by Welling and Teh, has drawn great attention of researchers when dealing with optimization or sampling tasks[2, 33, 40]. As a samplingalgorithm, SGLD canbe viewed asa"randombatch"version of the Unadjusted Langevin Algorithm (ULA), which is the Euler-Maruyama discretization of the Langevin diffusion, a stochastic process converging to a target Gibbs' distribution under suitable settings. As an optimization algorithm, SGLD can be viewed as a variant of the classical Stochastic Gradient Descent (SGD) [44], by adding independent Gaussian noise in each iteration of SGD. At recent decades, SGD and its variants [44, 25, 11, 37] have received a great deal of attention when solving high-dimensional tasks, ranging from computer vision, natural language processing, to high dimensional sampling, statistical optimization, etc. Also much theoretical analysis for SGD has been done by former researchers, including loss landscape of SGD iteration [46, 47], its dynamical stability [50] and diffusion approximation [32, 21, 17]. The combination of the SGD algorithm and the Langevin diffusion, can improve the behavior of both methods: for SGD, by taking another independent diffusion term into consideration, though not converging to a fixed point, the algorithm may be able to admit better ergodic properties and obtain better performance near saddle points [26, 52]. Besides, the application of the methodology of random mini-batch to Langevin diffusion could result in some efficient methods that could reduce computational cost while preserving the dynamical and statistical properties. Examples include the SGLD algorithm we study in the paper and the random batch methods for interacting particle systems [22, 23].

In the machine learning literature stochastic gradient descent has recently been widely discussed for its purported implicit regularization properties. Much of the theory, that attempts to clarify the role of noise in stochastic gradient algorithms, has widely approximated stochastic gradient descent by a stochastic differential equation with Gaussian noise. We provide a novel rigorous theoretical justification for this practice that showcases how the Gaussianity of the noise arises naturally.

Abstract: This paper attempts to solve the famous Vehicle Routing Problem by considering multiple constraints including capacitated vehicles, single depot, and distance using two approaches namely, cluster first and route the second algorithm and using integer linear programming. A set of nodes are provided as input to the system and a feasible route is generated as output, giving clusters of nodes and the route to be traveled within the cluster. For clustering the nodes, we have adopted the DBSCAN algorithm, and the routing is done using the approximation algorithm, Christofide's algorithm. Abstract: Recently, the applications of the methodologies of Reinforcement Learning (RL) to NP-Hard Combinatorial optimization problems have become a popular topic. This is essentially due to the nature of the traditional combinatorial algorithms, often based on a trial-and-error process.

Sliced mutual information (SMI) is defined as an average of mutual information (MI) terms between one-dimensional random projections of the random variables. It serves as a surrogate measure of dependence to classic MI that preserves many of its properties but is more scalable to high dimensions. However, a quantitative characterization of how SMI itself and estimation rates thereof depend on the ambient dimension, which is crucial to the understanding of scalability, remain obscure. This work provides a multifaceted account of the dependence of SMI on dimension, under a broader framework termed $k$-SMI, which considers projections to $k$-dimensional subspaces. Using a new result on the continuity of differential entropy in the 2-Wasserstein metric, we derive sharp bounds on the error of Monte Carlo (MC)-based estimates of $k$-SMI, with explicit dependence on $k$ and the ambient dimension, revealing their interplay with the number of samples. We then combine the MC integrator with the neural estimation framework to provide an end-to-end $k$-SMI estimator, for which optimal convergence rates are established. We also explore asymptotics of the population $k$-SMI as dimension grows, providing Gaussian approximation results with a residual that decays under appropriate moment bounds. All our results trivially apply to SMI by setting $k=1$. Our theory is validated with numerical experiments and is applied to sliced InfoGAN, which altogether provide a comprehensive quantitative account of the scalability question of $k$-SMI, including SMI as a special case when $k=1$.

Nonlinear dynamics are ubiquitous in science and engineering applications, but the physics of most complex systems is far from being fully understood. Discovering interpretable governing equations from measurement data can help us understand and predict the behavior of complex dynamic systems. Although extensive work has recently been done in this field, robustly distilling explicit model forms from very sparse data with considerable noise remains intractable. Moreover, quantifying and propagating the uncertainty of the identified system from noisy data is challenging, and relevant literature is still limited. To bridge this gap, we develop a novel Bayesian spline learning framework to identify parsimonious governing equations of nonlinear (spatio)temporal dynamics from sparse, noisy data with quantified uncertainty. The proposed method utilizes spline basis to handle the data scarcity and measurement noise, upon which a group of derivatives can be accurately computed to form a library of candidate model terms. The equation residuals are used to inform the spline learning in a Bayesian manner, where approximate Bayesian uncertainty calibration techniques are employed to approximate posterior distributions of the trainable parameters. To promote the sparsity, an iterative sequential-threshold Bayesian learning approach is developed, using the alternative direction optimization strategy to systematically approximate L0 sparsity constraints. The proposed algorithm is evaluated on multiple nonlinear dynamical systems governed by canonical ordinary and partial differential equations, and the merit/superiority of the proposed method is demonstrated by comparison with state-of-the-art methods.

Data Science is the buzzword, and a lot of enthusiasts are interested in learning its fundamentals to make a lucrative career in this field. Linear Algebra is one of the important concepts to learn how to perform data transformation techniques like pre-processing, dimensionality reduction, etc. There are many courses available at your fingertip, but it is difficult to choose the right course suited for your requirement. That's precisely the intent of this post - it makes your course search easy by listing down the five free courses to learn linear algebra foundations for data science. Before I go straight into listing down the courses for you, let me first explain the commonly asked questions – why do we need to learn linear algebra in the first place?

Stochastic Programming is a powerful modeling framework for decision-making under uncertainty. In this work, we tackle two-stage stochastic programs (2SPs), the most widely used class of stochastic programming models. Solving 2SPs exactly requires optimizing over an expected value function that is computationally intractable. Having a mixed-integer linear program (MIP) or a nonlinear program (NLP) in the second stage further aggravates the intractability, even when specialized algorithms that exploit problem structure are employed. Finding high-quality (first-stage) solutions -- without leveraging problem structure -- can be crucial in such settings. We develop Neur2SP, a new method that approximates the expected value function via a neural network to obtain a surrogate model that can be solved more efficiently than the traditional extensive formulation approach. Neur2SP makes no assumptions about the problem structure, in particular about the second-stage problem, and can be implemented using an off-the-shelf MIP solver. Our extensive computational experiments on four benchmark 2SP problem classes with different structures (containing MIP and NLP second-stage problems) demonstrate the efficiency (time) and efficacy (solution quality) of Neur2SP. In under 1.66 seconds, Neur2SP finds high-quality solutions across all problems even as the number of scenarios increases, an ideal property that is difficult to have for traditional 2SP solution techniques. Namely, the most generic baseline method typically requires minutes to hours to find solutions of comparable quality.

Treatment effect estimation is a fundamental problem in causal inference. We focus on designing efficient randomized controlled trials, to accurately estimate the effect of some treatment on a population of $n$ individuals. In particular, we study sample-constrained treatment effect estimation, where we must select a subset of $s \ll n$ individuals from the population to experiment on. This subset must be further partitioned into treatment and control groups. Algorithms for partitioning the entire population into treatment and control groups, or for choosing a single representative subset, have been well-studied. The key challenge in our setting is jointly choosing a representative subset and a partition for that set. We focus on both individual and average treatment effect estimation, under a linear effects model. We give provably efficient experimental designs and corresponding estimators, by identifying connections to discrepancy minimization and leverage-score-based sampling used in randomized numerical linear algebra. Our theoretical results obtain a smooth transition to known guarantees when $s$ equals the population size. We also empirically demonstrate the performance of our algorithms.

Kernel matrix-vector multiplication (KMVM) is a foundational operation in machine learning and scientific computing. However, as KMVM tends to scale quadratically in both memory and time, applications are often limited by these computational constraints. In this paper, we propose a novel approximation procedure coined \textit{Faster-Fast and Free Memory Method} ($\fthreem$) to address these scaling issues of KMVM for tall~($10^8\sim 10^9$) and skinny~($D\leq7$) data. Extensive experiments demonstrate that $\fthreem$ has empirical \emph{linear time and memory} complexity with a relative error of order $10^{-3}$ and can compute a full KMVM for a billion points \emph{in under a minute} on a high-end GPU, leading to a significant speed-up in comparison to existing CPU methods. We demonstrate the utility of our procedure by applying it as a drop-in for the state-of-the-art GPU-based linear solver FALKON, \emph{improving speed 1.5-5.5 times} at the cost of $<1\%$ drop in accuracy. We further demonstrate competitive results on \emph{Gaussian Process regression} coupled with significant speedups on a variety of real-world datasets.