This work investigates training Conditional Random Fields (CRF) by Stochastic Dual Coordinate Ascent (SDCA). SDCA enjoys a linear convergence rate and a strong empirical performance for independent classification problems. However, it has never been used to train CRF. Yet it benefits from an exact line search with a single marginalization oracle call, unlike previous approaches. In this paper, we adapt SDCA to train CRF and we enhance it with an adaptive non-uniform sampling strategy. Our preliminary experiments suggest that this method matches state-of-the-art CRF optimization techniques.

We study black-box attacks on machine learning classifiers where each query to the model incurs some cost or risk of detection to the adversary. We focus explicitly on minimizing the number of queries as a major objective. Specifically, we consider the problem of attacking machine learning classifiers subject to a budget of feature modification cost while minimizing the number of queries, where each query returns only a class and confidence score. We describe an approach that uses Bayesian optimization to minimize the number of queries, and find that the number of queries can be reduced to approximately one tenth of the number needed through a random strategy for scenarios where the feature modification cost budget is low.

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

Modern statistical inference tasks often require iterative optimization methods to approximate the solution. Convergence analysis from optimization only tells us how well we are approximating the solution deterministically, but overlooks the sampling nature of the data. However, due to the randomness in the data, statisticians are keen to provide uncertainty quantification, or confidence, for the answer obtained after certain steps of optimization. Therefore, it is important yet challenging to understand the sampling distribution of the iterative optimization methods. This paper makes some progress along this direction by introducing a new stochastic optimization method for statistical inference, the moment adjusted stochastic gradient descent. We establish non-asymptotic theory that characterizes the statistical distribution of the iterative methods, with good optimization guarantee. On the statistical front, the theory allows for model misspecification, with very mild conditions on the data. For optimization, the theory is flexible for both the convex and non-convex cases. Remarkably, the moment adjusting idea motivated from "error standardization" in statistics achieves similar effect as Nesterov's acceleration in optimization, for certain convex problems as in fitting generalized linear models. We also demonstrate this acceleration effect in the non-convex setting through experiments.

Many predicted structured objects (e.g., sequences, matchings, trees) are evaluated using the F-score, alignment error rate (AER), or other multivariate performance measures. Since inductively optimizing these measures using training data is typically computationally difficult, empirical risk minimization of surrogate losses is employed, using, e.g., the hinge loss for (structured) support vector machines. These approximations often introduce a mismatch between the learner's objective and the desired application performance, leading to inconsistency. We take a different approach: adversarially approximate training data while optimizing the exact F-score or AER. Structured predictions under this formulation result from solving zero-sum games between a predictor seeking the best performance and an adversary seeking the worst while required to (approximately) match certain structured properties of the training data. We explore this approach for word alignment (AER evaluation) and named entity recognition (F-score evaluation) with linear-chain constraints.

In this work, we introduce the {\em average top-$k$} (\atk) loss as a new aggregate loss for supervised learning, which is the average over the $k$ largest individual losses over a training dataset. We show that the \atk loss is a natural generalization of the two widely used aggregate losses, namely the average loss and the maximum loss, but can combine their advantages and mitigate their drawbacks to better adapt to different data distributions. Furthermore, it remains a convex function over all individual losses, which can lead to convex optimization problems that can be solved effectively with conventional gradient-based methods. We provide an intuitive interpretation of the \atk loss based on its equivalent effect on the continuous individual loss functions, suggesting that it can reduce the penalty on correctly classified data. We further give a learning theory analysis of \matk learning on the classification calibration of the \atk loss and the error bounds of \atk-SVM. We demonstrate the applicability of minimum average top-$k$ learning for binary classification and regression using synthetic and real datasets.

We derive computationally tractable methods to select a small subset of experiment settings from a large pool of given design points. The primary focus is on linear regression models, while the technique extends to generalized linear models and Delta's method (estimating functions of linear regression models) as well. The algorithms are based on a continuous relaxation of an otherwise intractable combinatorial optimization problem, with sampling or greedy procedures as post-processing steps. Formal approximation guarantees are established for both algorithms, and numerical results on both synthetic and real-world data confirm the effectiveness of the proposed methods.

Two-dimensional embeddings remain the dominant approach to visualize high dimensional data. The choice of embeddings ranges from highly non-linear ones, which can capture complex relationships but are difficult to interpret quantitatively, to axis-aligned projections, which are easy to interpret but are limited to bivariate relationships. Linear project can be considered as a compromise between complexity and interpretability, as they allow explicit axes labels, yet provide significantly more degrees of freedom compared to axis-aligned projections. Nevertheless, interpreting the axes directions, which are linear combinations often with many non-trivial components, remains difficult. To address this problem we introduce a structure aware decomposition of (multiple) linear projections into sparse sets of axis aligned projections, which jointly capture all information of the original linear ones. In particular, we use tools from Dempster-Shafer theory to formally define how relevant a given axis aligned project is to explain the neighborhood relations displayed in some linear projection. Furthermore, we introduce a new approach to discover a diverse set of high quality linear projections and show that in practice the information of $k$ linear projections is often jointly encoded in $\sim k$ axis aligned plots. We have integrated these ideas into an interactive visualization system that allows users to jointly browse both linear projections and their axis aligned representatives. Using a number of case studies we show how the resulting plots lead to more intuitive visualizations and new insight.

A regularized optimization problem over a large unstructured graph is studied, where the regularization term is tied to the graph geometry. Typical regularization examples include the total variation and the Laplacian regularizations over the graph. When applying the proximal gradient algorithm to solve this problem, there exist quite affordable methods to implement the proximity operator (backward step) in the special case where the graph is a simple path without loops. In this paper, an algorithm, referred to as "Snake", is proposed to solve such regularized problems over general graphs, by taking benefit of these fast methods. The algorithm consists in properly selecting random simple paths in the graph and performing the proximal gradient algorithm over these simple paths. This algorithm is an instance of a new general stochastic proximal gradient algorithm, whose convergence is proven. Applications to trend filtering and graph inpainting are provided among others. Numerical experiments are conducted over large graphs.

Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.