In this note we propose a new variant of the hybrid variance-reduced proximal gradient method in [7] to solve a common stochastic composite nonconvex optimization problem under standard assumptions. We simply replace the independent unbiased estimator in our hybrid- SARAH estimator introduced in [7] by the stochastic gradient evaluated at the same sample, leading to the identical momentum-SARAH estimator introduced in [2]. This allows us to save one stochastic gradient per iteration compared to [7], and only requires two samples per iteration. Our algorithm is very simple and achieves optimal stochastic oracle complexity bound in terms of stochastic gradient evaluations (up to a constant factor). Our analysis is essentially inspired by [7], but we do not use two different step-sizes.

Neural Networks typically learn by adjusting weights via nonlinear optimization in a training phase. Often, variants of gradient descent are used. These techniques require some differentiability. Therefore, non-smooth but piecewise linear activation functions like ReLU or the Heaviside function raise the question if techniques of linear and mixed integer linear programming are also suited for network training. Learning to near optimality can be performed with Linear Programs (LP) of exponential size for certain network architectures, see [2].

The chi-square statistic is a useful tool for understanding the relationship between two categorical variables. For the sake of example, let's say you work for a tech company that has rolled out a new product and you want to assess the relationship between this product and customer churn. In the age of data, tech or otherwise, many companies undergo to risk of taking evidence that is either anecdotal or perhaps a high level visualization to indicate certainty of a given relationship. The chi-square statistic gives us a way to quantify and assess the strength of a given pair of categorical variables. Let's explore chi-square from this lens of customer churn.

We propose a novel stochastic network model, called Fractal Gaussian Network (FGN), that embodies well-defined and analytically tractable fractal structures. Such fractal structures have been empirically observed in diverse applications. FGNs interpolate continuously between the popular purely random geometric graphs (a.k.a. the Poisson Boolean network), and random graphs with increasingly fractal behavior. In fact, they form a parametric family of sparse random geometric graphs that are parametrized by a fractality parameter $\nu$ which governs the strength of the fractal structure. FGNs are driven by the latent spatial geometry of Gaussian Multiplicative Chaos (GMC), a canonical model of fractality in its own right. We asymptotically characterize the expected number of edges and triangle in FGNs. We then examine the natural question of detecting the presence of fractality and the problem of parameter estimation based on observed network data, in addition to fundamental properties of the FGN as a random graph model. We also explore fractality in community structures by unveiling a natural stochastic block model in the setting of FGNs.

We give curvature-dependant convergence rates for the optimization of weakly convex functions defined on a manifold of 1-bounded geometry via Riemannian gradient descent and via the dynamic trivialization algorithm. In order to do this, we give a tighter bound on the norm of the Hessian of the Riemannian exponential than the previously known. We compute these bounds explicitly for some manifolds commonly used in the optimization literature such as the special orthogonal group and the real Grassmannian. Along the way, we present self-contained proofs of fully general bounds on the norm of the differential of the exponential map and certain cosine inequalities on manifolds, which are commonly used in optimization on manifolds.

Linear algebra is to machine learning as flour to bakery: every machine learning model is based in linear algebra, as every cake is based in flour. It is not the only ingredient, of course. Machine learning models need vector calculus, probability, and optimization, as cakes need sugar, eggs, and butter. Applied machine learning, like bakery, is essentially about combining these mathematical ingredients in clever ways to create useful (tasty?) models. This document contains introductory level linear algebra notes for applied machine learning. It is meant as a reference rather than a comprehensive review. If you ever get confused by matrix multiplication, don't remember what was the $L_2$ norm, or the conditions for linear independence, this can serve as a quick reference. It also a good introduction for people that don't need a deep understanding of linear algebra, but still want to learn about the fundamentals to read about machine learning or to use pre-packaged machine learning ...

Incoming students must have a strong background in computer science, including a solid understanding of complexity theory and good programming skills, as well as a good background in mathematics. Specifically, the first-year courses assume at least one year of college-level probability and statistics, as well as matrix algebra and multivariate calculus. For our introductory ML course, there's a self-assessment test [PDF] which will give you some idea about the background we expect students to have (for the MS you're looking at the "modest requirements"). Generally, you need to have some reasonable programming skills, with experience in Matlab/R/scipy-numpy especially helpful, and Java and Python being more useful than C, and a solid math background, especially in probability/statistics, linear algebra, and matrix and tensor calculus. There was significant variation in all of these scores, and they are only a small portion of applicants' qualifications.

The exogeneity bias and instrument validation have always been critical topics in statistics, machine learning and biostatistics. In the era of big data, such issues typically come with dimensionality issue and, hence, require even more attention than ever. In this paper we ensemble two well-known tools from machine learning and biostatistics -- stable variable selection and random graph -- and apply them to estimating the house pricing mechanics and the follow-up socio-economic effect on the 2010 Sydney house data. The estimation is conducted on an over-200-gigabyte ultrahigh dimensional database consisting of local education data, GIS information, census data, house transaction and other socio-economic records. The technique ensemble carefully improves the variable selection sparisty, stability and robustness to high dimensionality, complicated causal structures and the consequent multicollinearity, which is ultimately helpful on the data-driven recovery of a sparse and intuitive causal structure. The new ensemble also reveals its efficiency and effectiveness on endogeneity detection, instrument validation, weak instruments pruning and selection of proper instruments. From the perspective of machine learning, the estimation result both aligns with and confirms the facts of Sydney house market, the classical economic theories and the previous findings of simultaneous equations modeling. Moreover, the estimation result is totally consistent with and supported by the classical econometric tool like two-stage least square regression and different instrument tests (the code can be found at https://github.com/isaac2math/solar_graph_learning).

The aim of improvements in data driven exercises in insurance has led to the desire to gather additional data than traditionally available. In addition to underwriting characteristics such as age, gender and address, technology now allows the collection of many more variables. Examples include dynamic data from sensors for driving behaviour in vehicles, appliance and electrical usage in homes and static data from external databases on traffic violations, crime scores or credit scores. High dimensional models arise if modelling sensor data at multiple time points and the individual variables that comprise summary scores. Reasoning with a large number of variables can become unnecessarily complex without any actuarial judgment. For example, it may not be necessary to include hundreds of rating factors as predictors if many of them are known to be related or unnecessary. This discussion proposes the use of graph theory as a means of translating intuitive reasoning to mathematical properties. This is done via graphical models, which involve the use of graph theory to formulate probabilistic models (Lauritzen, 1996). The approach has been used in applications such as medical expert systems (Franklin et al., 1989), natural language processing (Blei et al., 2003), image processing, bioinformatics and others (Wainwright and Jordan, 2008).

This post will be quite an interesting one. We will show how a 2D plane can be transformed into another one. Understanding these concepts is a crucial step for some more advanced linear algebra/machine learning methods (e.g. So, let's proceed and we will learn how to connect a matrix-vector multiplication with a linear transformation. In this post we will introduce a linear transformation. A linear transformation can also be seen as a simple function.