Stochastic descent methods (of the gradient and mirror varieties) have become increasingly popular in optimization. In fact, it is now widely recognized that the success of deep learning is not only due to the special deep architecture of the models, but also due to the behavior of the stochastic descent methods used, which play a key role in reaching "good" solutions that generalize well to unseen data. In an attempt to shed some light on why this is the case, we revisit some minimax properties of stochastic gradient descent (SGD) on the square loss of linear models---originally developed in the 1990's---and extend them to generic stochastic mirror descent (SMD) algorithms on general loss functions and nonlinear models. In particular, we show that there is a fundamental identity which holds for SMD (and SGD) under very general conditions, and that this identity implies the minimax optimality of SMD (and SGD) with sufficiently small step size, for a general class of loss functions and general nonlinear models. We further show that this identity can be used to naturally establish other properties of SMD (and SGD), such as convergence and "implicit regularization" for over-parameterized linear models, which have been shown in certain cases in the literature. We also show how this identity can be used in the so-called "highly over-parameterized" nonlinear setting to provide insights into why SMD (and SGD) may have similar convergence and implicit regularization properties for deep learning.

The Gaussian mechanism is an essential building block used in multitude of differentially private data analysis algorithms. In this paper we revisit the Gaussian mechanism and show that the original analysis has several important limitations. Our analysis reveals that the variance formula for the original mechanism is far from tight in the high privacy regime ($\varepsilon \to 0$) and it cannot be extended to the low privacy regime ($\varepsilon \to \infty$). We address these limitations by developing an optimal Gaussian mechanism whose variance is calibrated directly using the Gaussian cumulative density function instead of a tail bound approximation. We also propose to equip the Gaussian mechanism with a post-processing step based on adaptive estimation techniques by leveraging that the distribution of the perturbation is known. Our experiments show that analytical calibration removes at least a third of the variance of the noise compared to the classical Gaussian mechanism, and that denoising dramatically improves the accuracy of the Gaussian mechanism in the high-dimensional regime.

Microbes are everywhere, including in and on our bodies, and have been shown to play key roles in a variety of prevalent human diseases. Consequently, there has been intense interest in the design of bacteriotherapies or "bugs as drugs," which are communities of bacteria administered to patients for specific therapeutic applications. Central to the design of such therapeutics is an understanding of the causal microbial interaction network and the population dynamics of the organisms. In this work we present a Bayesian nonparametric model and associated efficient inference algorithm that addresses the key conceptual and practical challenges of learning microbial dynamics from time series microbe abundance data. These challenges include high-dimensional (300+ strains of bacteria in the gut) but temporally sparse and non-uniformly sampled data; high measurement noise; and, nonlinear and physically non-negative dynamics. Our contributions include a new type of dynamical systems model for microbial dynamics based on what we term interaction modules, or learned clusters of latent variables with redundant interaction structure (reducing the expected number of interaction coefficients from $O(n^2)$ to $O((\log n)^2)$); a fully Bayesian formulation of the stochastic dynamical systems model that propagates measurement and latent state uncertainty throughout the model; and introduction of a temporally varying auxiliary variable technique to enable efficient inference by relaxing the hard non-negativity constraint on states. We apply our method to simulated and real data, and demonstrate the utility of our technique for system identification from limited data and gaining new biological insights into bacteriotherapy design.

This article targets whom have a basic understanding for TensorFlow Core API. Learning TensorFlow Core API, which is the lowest level API in TensorFlow, is a very good step for starting learning TensorFlow because it let you understand the kernel of the library. Here is a very simple example of TensorFlow Core API in which we create and train a linear regression model. The loss returned is 53.76. Existence of error, specially for large error, means that the parameters used must be updated.

This project was part of one my recent job interview skill test for a "Machine learning engineer" position. I had to complete the project in 48 hours which includes writing a 10-page report in latex. The dataset has classes and highly imbalanced. The primary objective of this project was to handle data imbalance issue. In the following subsections, I describe three techniques I used to overcome the data imbalance problem.

Since machine learning uses an array of mathematical concepts such as linear algebra and optimisation techniques, among others, it can be said that the numerical analysis holds a prime spot in the study of ML. Algorithms will be rendered void if only mathematical concepts are based on without giving thought to a certain numerical approximation in ML. In fact, numerical methods are at the heart of ML fields like deep learning (DL). Here, we will discuss a particular numerical analysis technique called Nyström method. It has now found its application in ML. Latest research has led to variations of the Nyström method for custom implementations in machine learning.

Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) have been widely used in distributed machine learning, e.g., training large collaborative filtering systems and deep neural networks. Due to current technical limit, however, establishing convergence properties of Async-MSGD for these highly complicated nonoconvex problems is generally infeasible. Therefore, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problem - streaming PCA. This allows us to make progress toward understanding Aync-MSGD and gaining new insights for more general problems. Specifically, by exploiting the diffusion approximation of stochastic optimization, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.

Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network's nodes into sets called "communities" such that there are dense connections within communities but sparse connections between them. A popular and statistically principled method to perform such clustering is to use a family of generative models known as stochastic block models (SBMs). In this paper, we show that maximum likelihood estimation in an SBM is a network analog of a well-known continuum surface-tension problem that arises from an application in metallurgy. To illustrate the utility of this bridge, we implement network analogs of three surface-tension algorithms, with which we successfully recover planted community structure in synthetic networks and which yield fascinating insights on empirical networks from the field of hyperspectral video segmentation.

The 2017 ASSISTments Data Mining competition aims to use data from a longitudinal study for predicting a brand-new outcome of students which had never been studied before by the educational data mining research community. Specifically, it facilitates research in developing predictive models that predict whether the first job of a student out of college belongs to a STEM (the acronym for science, technology, engineering, and mathematics) field. This is based on the student's learning history on the ASSISTments blended learning platform in the form of extensive clickstream data gathered during the middle school years. To tackle this challenge, we first estimate the expected knowledge state of students with respect to different mathematical skills using a deep knowledge tracing (DKT) model and an enhanced DKT (DKT+) model. We then combine the features corresponding to the DKT/DKT+ expected knowledge state with other features extracted directly from the student profile in the dataset to train several machine learning models for the STEM/non-STEM job prediction. Our experiments show that models trained with the combined features generally perform better than the models trained with the student profile alone. Detailed analysis of the student's knowledge state reveals that, when compared with non-STEM students, STEM students generally show a higher mastery level and a higher learning gain in mathematics.

Embeddings of medical concepts such as medication, procedure and diagnosis codes in Electronic Medical Records (EMRs) are central to healthcare analytics. Previous work on medical concept embedding takes medical concepts and EMRs as words and documents respectively. Nevertheless, such models miss out the temporal nature of EMR data. On the one hand, two consecutive medical concepts do not indicate they are temporally close, but the correlations between them can be revealed by the time gap. On the other hand, the temporal scopes of medical concepts often vary greatly (e.g., \textit{common cold} and \textit{diabetes}). In this paper, we propose to incorporate the temporal information to embed medical codes. Based on the Continuous Bag-of-Words model, we employ the attention mechanism to learn a "soft" time-aware context window for each medical concept. Experiments on public and proprietary datasets through clustering and nearest neighbour search tasks demonstrate the effectiveness of our model, showing that it outperforms five state-of-the-art baselines.