This work characterizes the benefits of averaging techniques widely used in conjunction with stochastic gradient descent (SGD). In particular, this work sharply analyzes: (1) mini-batching, a method of averaging many samples of the gradient to both reduce the variance of a stochastic gradient estimate and for parallelizing SGD and (2) tail-averaging, a method involving averaging the final few iterates of SGD in order to decrease the variance in SGD's final iterate. This work presents the first tight non-asymptotic generalization error bounds for these schemes for the stochastic approximation problem of least squares regression. Furthermore, this work establishes a precise problem-dependent extent to which mini-batching can be used to yield provable near-linear parallelization speedups over SGD with batch size one. These results are utilized in providing a highly parallelizable SGD algorithm that obtains the optimal statistical error rate with nearly the same number of serial updates as batch gradient descent, which improves significantly over existing SGD-style methods. Finally, this work sheds light on some fundamental differences in SGD's behavior when dealing with agnostic noise in the (non-realizable) least squares regression problem. In particular, the work shows that the stepsizes that ensure optimal statistical error rates for the agnostic case must be a function of the noise properties. The central analysis tools used by this paper are obtained through generalizing the operator view of averaged SGD, introduced by Defossez and Bach (2015) followed by developing a novel analysis in bounding these operators to characterize the generalization error. These techniques may be of broader interest in analyzing various computational aspects of stochastic approximation.

We propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrices to have equal entries and also encourage zeros in the precision matrix. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze an EEG dataset to demonstrate our method's interpretability and classification accuracy.

We study the problem of treatment effect estimation in randomized experiments with high-dimensional covariate information, and show that essentially any risk-consistent regression adjustment can be used to obtain efficient estimates of the average treatment effect. Our results considerably extend the range of settings where high-dimensional regression adjustments are guaranteed to provide valid inference about the population average treatment effect. We then propose cross-estimation, a simple method for obtaining finite-sample-unbiased treatment effect estimates that leverages high-dimensional regression adjustments. Our method can be used when the regression model is estimated using the lasso, the elastic net, subset selection, etc. Finally, we extend our analysis to allow for adaptive specification search via cross-validation, and flexible non-parametric regression adjustments with machine learning methods such as random forests or neural networks.

We provide tight upper and lower bounds on the complexity of minimizing the average of $m$ convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic and randomized settings respectively, and that a gradient oracle is sufficient for the optimal rate. For non-smooth functions, having access to prox oracles reduces the complexity and we present optimal methods based on smoothing that improve over methods using just gradient accesses.

The text is supported by a strong outline. The authors preserve much of the introductory material, but add the latest techniques and developments in data mining, thus making this a comprehensive resource for both beginners and practitioners. The focus is data-all aspects. The presentation is broad, encyclopedic, and comprehensive, with ample references for interested readers to pursue in-depth research on any technique. "This interesting and comprehensive introduction to data mining emphasizes the interest in multidimensional data mining--the integration of online analytical processing (OLAP) and data mining. Some chapters cover basic methods, and others focus on advanced techniques. The structure, along with the didactic presentation, makes the book suitable for both beginners and specialized readers."

Natural Language Processing (NLP) is a vast area of Computer Science that is concerned with the interaction between Computers and Human Language[1]. Within NLP many tasks are – or can be reformulated as – classification tasks. In classification tasks we are trying to produce a classification function which can give the correlation between a certain'feature' and a class . This Classifier first has to be trained with a training dataset, and then it can be used to actually classify documents. Training means that we have to determine its model parameters.

Neural networks designed for sequence predictions have recently gained renewed interested by achieving state-of-the-art performance across areas such as speech recognition, machine translation or language modeling. However, these models are quite computationally demanding, which in turn can limit their application. In the area of language modeling, recent advances have been made leveraging massively large models that could only be trained on a large GPU cluster for weeks at a time. While impressive, these processing-intensive practices favor exploring on large computational infrastructures that are typically too expensive for academic environments and impractical in a production setting, limiting the speed of research, reproducibility, and usability of the results. Recognizing this computational bottleneck, Facebook AI Research (FAIR) designed a novel softmax function approximation tailored for GPUs to efficiently train neural network based language models over very large vocabularies.

Many articles have been written about the top machine learning algorithms: click here and here for instance. Most of them seem to define top as oldest, and thus most used, ignoring modern, efficient algorithms fit for big data, such as indexation, attribution modeling, collaborative filtering, or recommendation engines used by companies such as Amazon, Google, or Facebook. I received this morning and advertisement for a (self-published) book called Master Machine Learning Algorithms, and I could not resist to post the author's list of top 10 machine learning algorithms:: Some of these techniques such as Naive Bayes (variables are almost never uncorrelated), Linear Discriminant Analysis (clusters are almost never separated by hyperplanes), or Linear Regression (numerous model assumptions - including linearity - are almost always violated in real data) have been so abused that I would hesitate teaching them. This is not a criticism of the book; most textbooks mention pretty much the same algorithms, and in this case, even skipping all graph-related algorithms. Even k Nearest Neighbors have modern, fast implementations not covered in traditional books - we are indeed working on this topic and expect to have an article published shortly about it.

The Higgs Boson is a landmark discovery that will help us to understand the basic nature of the universe. It was discovered first by the ATLAS experiment at the Large Hadron Collider, CERN in 2012. The Higg's Boson decays into two tau particles giving rise to a small signal buried in background noise. The goal of the Higgs Boson Machine Learning Challenge was to classify the characterizing events detected by ATLAS into "tau tau decay of a Higgs boson" versus "background." First step was to analyze the data and look for Missingness in the data. We found that the missing columns have some interesting pattern and they depend on the columns "PRI_jet_column", which is the number of jets having integer values of 0,1,2, or 3 where larger values has been caped at 3. The Jets are the experimental signatures of quarks and gluons produced in high-energy processes such as head-on proton-proton collisions. For PRI_jet_column 0, there were 10 columns having NULL values (-999), these are the columns which describe the Jet when it is equal to 0. For example, "DER_mass_jet_jet", the invariant mass (20) of the two jets (undefined if PRI jet num 1).So, it does not make sense to take into account the attributes of the jet(s), since they don't exist. For "PRI_jet_column" 1, there were 7 columns having NULL values and they describe the jets when their number is 2, So we deleted these 7 columns. For "PRI_jet_column" 2 or 3, we did not delete any columns.

Linear regression is a prediction method that is more than 200 years old. Simple linear regression is a great first machine learning algorithm to implement as it requires you to estimate properties from your training dataset, but is simple enough for beginners to understand. In this tutorial, you will discover how to implement the simple linear regression algorithm from scratch in Python. How To Implement Simple Linear Regression From Scratch With Python Photo by Kamyar Adl, some rights reserved. This section is divided into two parts, a description of the simple linear regression technique and a description of the dataset to which we will later apply it.