(2001). Empirical Processes in M-Estimation. Journal of the American Statistical Association: Vol. 96, No. 454, pp. 779-780.

Empirical Processes in M-Estimation

A Gentle Introduction to Deep Learning for Graphs

The problem of multiple kernel learning based on penalized empirical risk minimization is discussed. The complexity penalty is determined jointly by the empirical $L_2$ norms and the reproducing kernel Hilbert space (RKHS) norms induced by the kernels with a data-driven choice of regularization parameters. The main focus is on the case when the total number of kernels is large, but only a relatively small number of them is needed to represent the target function, so that the problem is sparse. The goal is to establish oracle inequalities for the excess risk of the resulting prediction rule showing that the method is adaptive both to the unknown design distribution and to the sparsity of the problem.

Sparsity in multiple kernel learning

We introduce and analyze stochastic optimization methods where the input to each update is perturbed by bounded noise. We show that this framework forms the basis of a unified approach to analyzing asynchronous implementations of stochastic optimization algorithms, by viewing them as serial methods operating on noisy inputs. Using our perturbed iterate framework, we provide new analyses of the Hogwild! algorithm and asynchronous stochastic coordinate descent that are simpler than earlier analyses, remove many assumptions of previous models, and in some cases yield improved upper bounds on the convergence rates. We proceed to apply our framework to develop and analyze KroMagnon: a novel, parallel, sparse stochastic variance-reduced gradient (SVRG) algorithm. We demonstrate experimentally on a 16-core machine that the sparse and parallel version of SVRG is in some cases more than four orders of magnitude faster than the standard SVRG algorithm.

https://amplab.cs.berkeley.edu/wp-content/uploads/2015/07/mania-etal.arxiv2015perturbed1.pdf

Perturbed Iterate Analysis for Asynchronous Stochastic Optimization

Interpretability in machine learning (ML) is crucial for high stakes decisions and troubleshooting. In this work, we provide fundamental principles for interpretable ML, and dispel common misunderstandings that dilute the importance of this crucial topic. We also identify 10 technical challenge areas in interpretable machine learning and provide history and background on each problem. Some of these problems are classically important, and some are recent problems that have arisen in the last few years. These problems are: (1) Optimizing sparse logical models such as decision trees; (2) Optimization of scoring systems; (3) Placing constraints into generalized additive models to encourage sparsity and better interpretability; (4) Modern case-based reasoning, including neural networks and matching for causal inference; (5) Complete supervised disentanglement of neural networks; (6) Complete or even partial unsupervised disentanglement of neural networks; (7) Dimensionality reduction for data visualization; (8) Machine learning models that can incorporate physics and other generative or causal constraints; (9) Characterization of the "Rashomon set" of good models; and (10) Interpretable reinforcement learning. This survey is suitable as a starting point for statisticians and computer scientists interested in working in interpretable machine learning.

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges

Given a statistical estimation problem where regularization is performed according to the structure of a large, dense graph G, we consider fitting the statistical estimate using a sparsified surrogate graph G̃, which shares the vertices of G but has far fewer edges, and is thus more tractable to work with computationally. We examine three types of sparsification: spectral sparsification, which can be seen as the result of sampling edges from the graph with probabilities proportional to their effective resistances, and two simpler sparsifiers, which sample edges uniformly from the graph, either globally or locally. We provide strong theoretical and experimental results, demonstrating that sparsification before estimation can give statistically sensible solutions, with significant computational savings.

/pdf/graph-sparsification-approaches-for-laplacian-smoothing-6uuviv6sdl.pdf

Graph Sparsification Approaches for Laplacian Smoothing

We consider the problem of estimating a function defined over $n$ locations on a $d$-dimensional grid (having all side lengths equal to $n^{1/d}$). When the function is constrained to have discrete total variation bounded by $C_n$, we derive the minimax optimal (squared) $\ell_2$ estimation error rate, parametrized by $n$ and $C_n$. Total variation denoising, also known as the fused lasso, is seen to be rate optimal. Several simpler estimators exist, such as Laplacian smoothing and Laplacian eigenmaps. A natural question is: can these simpler estimators perform just as well? We prove that these estimators, and more broadly all estimators given by linear transformations of the input data, are suboptimal over the class of functions with bounded variation. This extends fundamental findings of Donoho and Johnstone [1998] on 1-dimensional total variation spaces to higher dimensions. The implication is that the computationally simpler methods cannot be used for such sophisticated denoising tasks, without sacrificing statistical accuracy. We also derive minimax rates for discrete Sobolev spaces over $d$-dimensional grids, which are, in some sense, smaller than the total variation function spaces. Indeed, these are small enough spaces that linear estimators can be optimal---and a few well-known ones are, such as Laplacian smoothing and Laplacian eigenmaps, as we show. Lastly, we investigate the problem of adaptivity of the total variation denoiser to these smaller Sobolev function spaces.

Total Variation Classes Beyond 1d: Minimax Rates, and the Limitations of Linear Smoothers

We develop parallel and distributed Frank-Wolfe algorithms; the former on shared memory machines with mini-batching, and the latter in a delayed update framework. Whenever possible, we perform computations asynchronously, which helps attain speedups on multicore machines as well as in distributed environments. Moreover, instead of worst-case bounded delays, our methods only depend (mildly) on \emph{expected} delays, allowing them to be robust to stragglers and faulty worker threads. Our algorithms assume block-separable constraints, and subsume the recent Block-Coordinate Frank-Wolfe (BCFW) method~\citep{lacoste2013block}. Our analysis reveals problem-dependent quantities that govern the speedups of our methods over BCFW. We present experiments on structural SVM and Group Fused Lasso, obtaining significant speedups over competing state-of-the-art (and synchronous) methods.

Parallel and Distributed Block-Coordinate Frank-Wolfe Algorithms

We study additive models built with trend filtering, that is, additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k\geq0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) total variation regularizer that it uses. On the theory side, we derive fast error rates for additive trend filtering estimates, and show these rates are minimax optimal when the underlying function is additive and has component functions whose derivatives are of bounded variation. We also show that these rates are unattainable by additive smoothing splines (and by additive models built from linear smoothers, in general). On the computational side, we use backfitting, to leverage fast univariate trend filtering solvers; we also describe a new backfitting algorithm whose iterations can be run in parallel, which (as far as we can tell) is the first of its kind. Lastly, we present a number of experiments to examine the empirical performance of trend filtering.

Additive models with trend filtering

We study parallel and distributed Frank-Wolfe algorithms; the former on shared memory machines with mini-batching, and the latter in a delayed update framework. In both cases, we perform computations asynchronously whenever possible. We assume block-separable constraints as in Block-Coordinate Frank-Wolfe (BCFW) method (Lacoste-Julien et al., 2013), but our analysis subsumes BCFW and reveals problemdependent quantities that govern the speedups of our methods over BCFW. A notable feature of our algorithms is that they do not depend on worst-case bounded delays, but only (mildly) on expected delays, making them robust to stragglers and faulty worker threads. We present experiments on structural SVM and Group Fused Lasso, and observe significant speedups over competing state-of-the-art (and synchronous) methods.

/pdf/parallel-and-distributed-block-coordinate-frank-wolfe-2idqexnmis.pdf

Veeranjaneyulu Sadhanala

Papers

Graph Sparsification Approaches for Laplacian Smoothing

Total Variation Classes Beyond 1d: Minimax Rates, and the Limitations of Linear Smoothers

Parallel and Distributed Block-Coordinate Frank-Wolfe Algorithms

Additive models with trend filtering

Parallel and distributed block-coordinate frank-wolfe algorithms