In this paper, we analyze discrete gradient descent and $\epsilon$-noisy gradient descent on a special but important class of functions. We find that when used to minimize a function $L: \mathbb{R}^n \rightarrow \mathbb{R}$ in this class we consider, discrete gradient descent can exhibit strikingly different behavior from continuous gradient descent. On long time scales, discrete gradient descent and continuous gradient descent tend toward different global minima of $L$. Discrete gradient descent preferentially finds global minima at which the graph of the function $L$ is shallowest, while gradient flow shows no such preference.

Time series data analytics has been a problem of substantial interests for decades, and Dynamic Time Warping (DTW) has been the most widely adopted technique to measure dissimilarity between time series. A number of global-alignment kernels have since been proposed in the spirit of DTW to extend its use to kernel-based estimation method such as support vector machine. However, those kernels suffer from diagonal dominance of the Gram matrix and a quadratic complexity w.r.t. the sample size. In this work, we study a family of alignment-aware positive definite (p.d.) kernels, with its feature embedding given by a distribution of \emph{Random Warping Series (RWS)}. The proposed kernel does not suffer from the issue of diagonal dominance while naturally enjoys a \emph{Random Features} (RF) approximation, which reduces the computational complexity of existing DTW-based techniques from quadratic to linear in terms of both the number and the length of time-series. We also study the convergence of the RF approximation for the domain of time series of unbounded length. Our extensive experiments on 16 benchmark datasets demonstrate that RWS outperforms or matches state-of-the-art classification and clustering methods in both accuracy and computational time. Our code and data is available at { \url{https://github.com/IBM/RandomWarpingSeries}}.

Kernel method has been developed as one of the standard approaches for nonlinear learning, which however, does not scale to large data set due to its quadratic complexity in the number of samples. A number of kernel approximation methods have thus been proposed in the recent years, among which the random features method gains much popularity due to its simplicity and direct reduction of nonlinear problem to a linear one. The Random Binning (RB) feature, proposed in the first random-feature paper \cite{rahimi2007random}, has drawn much less attention than the Random Fourier (RF) feature. In this work, we observe that the RB features, with right choice of optimization solver, could be orders-of-magnitude more efficient than other random features and kernel approximation methods under the same requirement of accuracy. We thus propose the first analysis of RB from the perspective of optimization, which by interpreting RB as a Randomized Block Coordinate Descent in the infinite-dimensional space, gives a faster convergence rate compared to that of other random features. In particular, we show that by drawing $R$ random grids with at least $\kappa$ number of non-empty bins per grid in expectation, RB method achieves a convergence rate of $O(1/(\kappa R))$, which not only sharpens its $O(1/\sqrt{R})$ rate from Monte Carlo analysis, but also shows a $\kappa$ times speedup over other random features under the same analysis framework. In addition, we demonstrate another advantage of RB in the L1-regularized setting, where unlike other random features, a RB-based Coordinate Descent solver can be parallelized with guaranteed speedup proportional to $\kappa$. Our extensive experiments demonstrate the superior performance of the RB features over other random features and kernel approximation methods. Our code and data is available at { \url{https://github.com/teddylfwu/RandomBinning}}.

Recent years, transfer learning has attracted much attention in the community of machine learning. In this paper, we mainly focus on the tasks of parameter transfer under the framework of extreme learning machine (ELM). Unlike the existing parameter transfer approaches, which incorporate the source model information into the target by regularizing the di erence between the source and target domain parameters, an intuitively appealing projective-model is proposed to bridge the source and target model parameters. Specifically, we formulate the parameter transfer in the ELM networks by the means of parameter projection, and train the model by optimizing the projection matrix and classifier parameters jointly. Further more, the `L2,1-norm structured sparsity penalty is imposed on the source domain parameters, which encourages the joint feature selection and parameter transfer. To evaluate the e ectiveness of the proposed method, comprehensive experiments on several commonly used domain adaptation datasets are presented. The results show that the proposed method significantly outperforms the non-transfer ELM networks and other classical transfer learning methods.

In this dissertation, we focus on several important problems in structured prediction. In structured prediction, the label has a rich intrinsic substructure, and the loss varies with respect to the predicted label and the true label pair. Structured SVM is an extension of binary SVM to adapt to such structured tasks. In the first part of the dissertation, we study the surrogate losses and its efficient methods. To minimize the empirical risk, a surrogate loss which upper bounds the loss, is used as a proxy to minimize the actual loss. Since the objective function is written in terms of the surrogate loss, the choice of the surrogate loss is important, and the performance depends on it. Another issue regarding the surrogate loss is the efficiency of the argmax label inference for the surrogate loss. Efficient inference is necessary for the optimization since it is often the most time-consuming step. We present a new class of surrogate losses named bi-criteria surrogate loss, which is a generalization of the popular surrogate losses. We first investigate an efficient method for a slack rescaling formulation as a starting point utilizing decomposability of the model. Then, we extend the algorithm to the bi-criteria surrogate loss, which is very efficient and also shows performance improvements. In the second part of the dissertation, another important issue of regularization is studied. Specifically, we investigate a problem of regularization in hierarchical classification when a structural imbalance exists in the label structure. We present a method to normalize the structure, as well as a new norm, namely shared Frobenius norm. It is suitable for hierarchical classification that adapts to the data in addition to the label structure.

Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, variable selection, and high-dimensional analysis. We propose a broad framework for sparse relaxed regularized regression, called SR3. The key idea is to solve a relaxation of the regularized problem, which has three advantages over the state-of-the-art: (1) solutions of the relaxed problem are superior with respect to errors, false positives, and conditioning, (2) relaxation allows extremely fast algorithms for both convex and nonconvex formulations, and (3) the methods apply to composite regularizers such as total variation (TV) and its nonconvex variants. We demonstrate the advantages of SR3 (computational efficiency, higher accuracy, faster convergence rates, greater flexibility) across a range of regularized regression problems with synthetic and real data, including applications in compressed sensing, LASSO, matrix completion, TV regularization, and group sparsity. To promote reproducible research, we also provide a companion Matlab package that implements these examples.

We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger class includes functions whose second derivatives may be singular or unbounded at their minima. Our methods are discretizations of conformal Hamiltonian dynamics, which generalize the classical momentum method to model the motion of a particle with non-standard kinetic energy exposed to a dissipative force and the gradient field of the function of interest. They are first-order in the sense that they require only gradient computation. Yet, crucially the kinetic gradient map can be designed to incorporate information about the convex conjugate in a fashion that allows for linear convergence on convex functions that may be non-smooth or non-strongly convex. We study in detail one implicit and two explicit methods. For one explicit method, we provide conditions under which it converges to stationary points of non-convex functions. For all, we provide conditions on the convex function and kinetic energy pair that guarantee linear convergence, and show that these conditions can be satisfied by functions with power growth. In sum, these methods expand the class of convex functions on which linear convergence is possible with first-order computation.

Independent component analysis (ICA) is a statistical method for transforming an observable multi-dimensional random vector into components that are as statistically independent as possible from each other. Usually the ICA framework assumes a model according to which the observations are generated (such as a linear transformation with additive noise). ICA over finite fields is a special case of ICA in which both the observations and the independent components are over a finite alphabet. In this thesis we consider a formulation of the finite-field case in which an observation vector is decomposed to its independent components (as much as possible) with no prior assumption on the way it was generated. This generalization is also known as Barlow's minimal redundancy representation and is considered an open problem. We propose several theorems and show that this hard problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a series of linear problems. Moreover, we show that there exists a simple transformation (namely, order permutation) which provides a greedy yet very effective approximation of the optimal solution. We further show that while not every random vector can be efficiently decomposed into independent components, the vast majority of vectors do decompose very well (that is, within a small constant cost), as the dimension increases. In addition, we show that we may practically achieve this favorable constant cost with a complexity that is asymptotically linear in the alphabet size. Our contribution provides the first efficient set of solutions to Barlow's problem with theoretical and computational guarantees. Finally, we demonstrate our suggested framework in multiple source coding applications.

Ever since the proof of asymptotic normality of maximum likelihood estimator by Cramer (1946), it has been understood that a basic technique of the Taylor series expansion suffices for asymptotics of $M$-estimators with smooth/differentiable loss function. Although the Taylor series expansion is a purely deterministic tool, the realization that the asymptotic normality results can also be made deterministic (and so finite sample) received far less attention. With the advent of big data and high-dimensional statistics, the need for finite sample results has increased. In this paper, we use the (well-known) Banach fixed point theorem to derive various deterministic inequalities that lead to the classical results when studied under randomness. In addition, we provide applications of these deterministic inequalities for crossvalidation/subsampling, marginal screening and uniform-in-submodel results that are very useful for post-selection inference and in the study of post-regularization estimators. Our results apply to many classical estimators, in particular, generalized linear models, non-linear regression and cox proportional hazards model. Extensions to non-smooth and constrained problems are also discussed.

Abstract--The real estate market is exposed to many fluctuations in prices, because of existing correlations with many variables, some of which cannot be controlled or might even be unknown. Housing prices can increase rapidly (or in some cases, also drop very fast), yet the numerous listings available online where houses are sold or rented are not likely to be updated that often. In some cases, individuals interested in selling a house (or apartment) might include it in some online listing, and forget about updating the price. In other cases, some individuals might be interested in deliberately setting a price below the market price in order to sell the home faster, for various reasons. In this paper we aim at developing a machine learning application that identifies opportunities in the real estate market in real time, i.e., houses that are listed with a price substantially below the market price. This program can be useful for investors interested in the housing market. The application is formally implemented as a regression problem, that tries to estimate the market price of a house given features retrieved from public online listings. For building this application, we have performed a feature engineering stage in order to discover relevant features that allows attaining a high predictive performance. Several machine learning algorithms have been tested, including regression trees, k-NN and neural networks, identifying advantages and handicaps of each of them. The real estate market is rapidly evolving. A recent report published by MSCI estimates the size of the professionally managed real estate investment market in $8.5 trillion in 2017, increasing a total of $1.1 trillion since the previous year [1]. Of course, the real market size is expected to be much larger when counting assets which are not professionally managed or that are not object of investment. When looked from a macroeconomic perspective, there are many aspects that significantly drive the behavior of this market, such as demographics, interest rates, government regulation and, for short, global economic health. However, looking at the market evolution from a global perspective turns out to be too simplistic. Although the market at a global scale is very tightly correlated, there are many aspects influencing the behavior of markets at a local scale, such as political instability or the emergence of highly demanded "hot spots" that can shift rapidly.