This paper develops a general causal inference method for treatment effects models under selection on unobservables. A large set of covariates that admits an unknown, possibly nonlinear factor structure is exploited to control for the latent confounders. The key building block is a local principal subspace approximation procedure that combines $K$-nearest neighbors matching and principal component analysis. Estimators of many causal parameters, including average treatment effects and counterfactual distributions, are constructed based on doubly-robust score functions. Large-sample properties of these estimators are established, which only require relatively mild conditions on the principal subspace approximation. The results are illustrated with an empirical application studying the effect of political connections on stock returns of financial firms, and a Monte Carlo experiment. The main technical and methodological results regarding the general local principal subspace approximation method may be of independent interest.

Proximities are at the heart of almost all machine learning methods. If the input data are given as numerical vectors of equal lengths, euclidean distance, or a Hilbertian inner product is frequently used in modeling algorithms. In a more generic view, objects are compared by a (symmetric) similarity or dissimilarity measure, which may not obey particular mathematical properties. This renders many machine learning methods invalid, leading to convergence problems and the loss of guarantees, like generalization bounds. In many cases, the preferred dissimilarity measure is not metric, like the earth mover distance, or the similarity measure may not be a simple inner product in a Hilbert space but in its generalization a Krein space. If the input data are non-vectorial, like text sequences, proximity-based learning is used or ngram embedding techniques can be applied. Standard embeddings lead to the desired fixed-length vector encoding, but are costly and have substantial limitations in preserving the original data's full information. As an information preserving alternative, we propose a complex-valued vector embedding of proximity data. This allows suitable machine learning algorithms to use these fixed-length, complex-valued vectors for further processing. The complex-valued data can serve as an input to complex-valued machine learning algorithms. In particular, we address supervised learning and use extensions of prototype-based learning. The proposed approach is evaluated on a variety of standard benchmarks and shows strong performance compared to traditional techniques in processing non-metric or non-psd proximity data.

We construct an experimental setup in which changing the scale of initialization strongly impacts the implicit regularization induced by SGD, interpolating from good generalization performance to completely memorizing the training set while making little progress on the test set. Moreover, we find that the extent and manner in which generalization ability is affected depends on the activation and loss function used, with $\sin$ activation being the most extreme. In the case of the homogeneous ReLU activation, we show that this behavior can be attributed to the loss function. Our empirical investigation reveals that increasing the scale of initialization could cause the representations and gradients to be increasingly misaligned across examples in the same class. We further demonstrate that a similar misalignment phenomenon occurs in other scenarios affecting generalization performance, such as changes to the architecture or data distribution.

In this paper, we give explicit descriptions of versions of (Local-) Backtracking Gradient Descent and New Q-Newton's method to the Riemannian setting.Here are some easy to state consequences of results in this paper, where X is a general Riemannian manifold of finite dimension and $f:X\rightarrow \mathbb{R}$ a $C^2$ function which is Morse (that is, all its critical points are non-degenerate). {\bf Theorem.} For random choices of the hyperparameters in the Riemanian Local Backtracking Gradient Descent algorithm and for random choices of the initial point $x_0$, the sequence $\{x_n\}$ constructed by the algorithm either (i) converges to a local minimum of $f$ or (ii) eventually leaves every compact subsets of $X$ (in other words, diverges to infinity on $X$). If $f$ has compact sublevels, then only the former alternative happens. The convergence rate is the same as in the classical paper by Armijo. {\bf Theorem.} Assume that $f$ is $C^3$. For random choices of the hyperparametes in the Riemannian New Q-Newton's method, if the sequence constructed by the algorithm converges, then the limit is a critical point of $f$. We have a local Stable-Center manifold theorem, near saddle points of $f$, for the dynamical system associated to the algorithm. If the limit point is a non-degenerate minimum point, then the rate of convergence is quadratic. If moreover $X$ is an open subset of a Lie group and the initial point $x_0$ is chosen randomly, then we can globally avoid saddle points. As an application, we propose a general method using Riemannian Backtracking GD to find minimum of a function on a bounded ball in a Euclidean space, and do explicit calculations for calculating the smallest eigenvalue of a symmetric square matrix.

We consider the problem of learning the best-fitting single neuron as measured by the expected square loss $\mathbb{E}_{(x,y)\sim \mathcal{D}}[(\sigma(w^\top x)-y)^2]$ over some unknown joint distribution $\mathcal{D}$ by using gradient descent to minimize the empirical risk induced by a set of i.i.d. samples $S\sim \mathcal{D}^n$. The activation function $\sigma$ is an arbitrary Lipschitz and non-decreasing function, making the optimization problem nonconvex and nonsmooth in general, and covers typical neural network activation functions and inverse link functions in the generalized linear model setting. In the agnostic PAC learning setting, where no assumption on the relationship between the labels $y$ and the input $x$ is made, if the optimal population risk is $\mathsf{OPT}$, we show that gradient descent achieves population risk $O(\mathsf{OPT})+\epsilon$ in polynomial time and sample complexity when $\sigma$ is strictly increasing. For the ReLU activation, our population risk guarantee is $O(\mathsf{OPT}^{1/2})+\epsilon$. When labels take the form $y = \sigma(v^\top x) + \xi$ for zero-mean sub-Gaussian noise $\xi$, we show that the population risk guarantees for gradient descent improve to $\mathsf{OPT} + \epsilon$. Our sample complexity and runtime guarantees are (almost) dimension independent, and when $\sigma$ is strictly increasing, require no distributional assumptions beyond boundedness. For ReLU, we show the same results under a nondegeneracy assumption for the marginal distribution of the input.

A machine learning algorithm (such as classification, clustering or regression) uses a training dataset to determine weight factors that can be applied to unseen data for predictive purposes. Behind every machine learning model is an optimization algorithm that relies heavily on calculus. In this article, we discuss one such optimization algorithm, namely, the Gradient Descent Approximation (GDA) and we'll show how it can be used to build a simple regression estimator. In one-dimension, we can find the maximum and minimum of a function using derivatives. Let us consider a simple quadratic function f(x) as shown below.

K-Means Clustering Algorithm K-Means Clustering With Python will help you to comprehensively learn all the concepts of the k-means algorithm in machine learning. K-means Clustering is one of the most common data analysis technique used to get an intuition about the structure of the data. It has various applications such as, Identifying Fake news, Filtering spam mails & Customer Segmentation. This "K-means clustering" tutorial will help you to comprehensively learn all the concepts of the k-means algorithm in machine learning. K-means Clustering is one of the most common data analysis technique used to get an intuition about the structure of the data.

Most of the work positions in Deep Learning, Machine Learning, NLP, Computer Vision, or basically any of the Artificial Intelligence (AI) work require you to have at least a Bachelor's degree in Computer Science or some related area.But if you're from the United States or some other country where most people can't afford to go to the best universities, you need to find other ways to get yourself educated.Fortunately, nowadays you don't have to get a formal degree with the short supply of qualified professionals from these fields - demonstrating your expertise in other forms, such as the courses you've completed, is enough to get you a position.But with that comes a lot of people all trying to sell their own Artificial Intelligence course.In this article, I will discuss some of the best free Artificial Intelligence Courses that come from MIT, Stanford, Amazon, Harvard, and others that you can take, regardless of where you live and how much money you have - I personally took these courses on my own, or I got them recommended by close friends who took them so I can be sure they're good. (BTW I'm not sponsored by any of these. 🙄)You may also be interested in reading about the 5 Best Artificial Intelligence Books in 2020 and the top 5 Interesting FREE AI Books for absolute Beginners by Springer.    THE best free online Artificial Intelligence courses 1. Machine Learning (Andrew Ng)This Machine Learning course by Andrew Ng is probably the most popular course offered by an independent teacher.Andrew Ng co-founded Google Brain and was Chief Scientist in Baidu's A.I research division and can express information in a simplified way that you will be able to easily understand.This course is so awesome because it doesn't have a steep learning curve - which is extremely important for people who have never heard of Machine Learning - it doesn't assume that you have any previous knowledge and gradually guides you through complicated subjects to make your learning experience challenging but enjoyable.Furthermore, it avoids complex math which is probably the biggest fear for people that want to get into Machine Learning and AI.    2. CS50's Introduction to Artificial Intelligence with Python (Harvard)This 7-week Harvard course will teach you how to use machine learning in Python and explore the concepts and algorithms used in modern artificial intelligence - you will immerse yourself in ideas that give rise to technologies such as machine translation and handwriting recognition.It includes hands-on projects where you can learn about algorithms for graph searching, adversarial search, classification, optimization, logical inference, and probability theory and how to incorporate them into your own Python code.   3.

In machine learning, we need lots of data to build an efficient model, but dealing with a larger dataset is not an easy task we need to work hard in preprocessing the data and as a data scientist we will come across a situation dealing with a large number of variables here PCA (principal component analysis) is dimension reduction technique helps in dealing with those problems. In this article, we will demonstrate how to work on larger data and images using a famous dimension reduction technique PCA( principal component analysis). PCA is a dimensionality reduction that is often used to reduce the dimension of the variables of a larger dataset that is compressed to the smaller one which contains most of the information to build an efficient model. In a real-time scenario when you are working reducing the number of variables in the dataset you need compromise on model accuracy but using PCA will give good accuracy. The idea of PCA is to reduce the variables in the dataset and preserve data as much as possible.

Millions of mosques around the world are suffering some problems such as ventilation and difficulty getting rid of bacteria, especially in rush hours where congestion in mosques leads to air pollution and spread of bacteria, in addition to unpleasant odors and to a state of discomfort during the pray times, where in most mosques there are no enough windows to ventilate the mosque well. This paper aims to solve these problems by building a model of smart mosques domes using weather features and outside temperatures. Machine learning algorithms such as k Nearest Neighbors and Decision Tree were applied to predict the state of the domes open or close. The experiments of this paper were applied on Prophet mosque in Saudi Arabia, which basically contains twenty seven manually moving domes. Both machine learning algorithms were tested and evaluated using different evaluation methods. After comparing the results for both algorithms, DT algorithm was achieved higher accuracy 98% comparing with 95% accuracy for kNN algorithm. Finally, the results of this study were promising and will be helpful for all mosques to use our proposed model for controlling domes automatically.