NeurIPS 2016 Papers

The goal of ordinal embedding is to represent items as points in a low-dimensional Euclidean space given a set of constraints like ``item $i$ is closer to item $j$ than item $k$''. Ordinal constraints like this often come from human judgments. The classic approach to solving this problem is known as non-metric multidimensional scaling. To account for errors and variation in judgments, we consider the noisy situation in which the given constraints are independently corrupted by reversing the correct constraint with some probability. The ordinal embedding problem has been studied for decades, but most past work pays little attention to the question of whether accurate embedding is possible, apart from empirical studies. This paper shows that under a generative data model it is possible to learn the correct embedding from noisy distance comparisons. In establishing this fundamental result, the paper makes several new contributions. First, we derive prediction error bounds for embedding from noisy distance comparisons by exploiting the fact that the rank of a distance matrix of points in $\R^d$ is at most $d+2$. These bounds characterize how well a learned embedding predicts new comparative judgments. Second, we show that the underlying embedding can be recovered by solving a simple convex optimization. This result is highly non-trivial since we show that the linear map corresponding to distance comparisons is non-invertible, but there exists a nonlinear map that is invertible. Third, two new algorithms for ordinal embedding are proposed and evaluated in experiments.

In high-dimensional settings, where the number of features p is typically much larger than the number of samples n, methods which can systematically examine arbitrary combinations of features, a huge 2^p-dimensional space, have recently begun to be explored. However, none of the current methods is able to assess the association between feature combinations and a target variable while conditioning on a categorical covariate, in order to correct for potential confounding effects. We propose the Fast Automatic Conditional Search (FACS) algorithm, a significant discriminative itemset mining method which conditions on categorical covariates and only scales as O(k log k), where k is the number of states of the categorical covariate. Based on the Cochran-Mantel-Haenszel Test, FACS demonstrates superior speed and statistical power on simulated and real-world datasets compared to the state of the art, opening the door to numerous applications in biomedicine.

Most existing approaches to distributed sparse regression assume the data is partitioned by samples. However, for high-dimensional data (D >> N), it is more natural to partition the data by features. We propose an algorithm to distributed sparse regression when the data is partitioned by features rather than samples. Our approach allows the user to tailor our general method to various distributed computing platforms by trading-off the total amount of data (in bits) sent over the communication network and the number of rounds of communication. We show that an implementation of our approach is capable of solving L1-regularized L2 regression problems with millions of features in minutes.

Dimensionality reduction is one of the key issues in the design of effective machine learning methods for automatic induction. In this work, we introduce recursive maxima hunting (RMH) for variable selection in classification problems with functional data. In this context, variable selection techniques are especially attractive because they reduce the dimensionality, facilitate the interpretation and can improve the accuracy of the predictive models. The method, which is a recursive extension of maxima hunting (MH), performs variable selection by identifying the maxima of a relevance function, which measures the strength of the correlation of the predictor functional variable with the class label. At each stage, the information associated with the selected variable is removed by subtracting the conditional expectation of the process. The results of an extensive empirical evaluation are used to illustrate that, in the problems investigated, RMH has comparable or higher predictive accuracy than standard simensionality reduction techniques, such as PCA and PLS, and state-of-the-art feature selection methods for functional data, such as maxima hunting.

Minimizing a convex function over the spectrahedron, i.e., the set of all $d\times d$ positive semidefinite matrices with unit trace, is an important optimization task with many applications in optimization, machine learning, and signal processing. It is also notoriously difficult to solve in large-scale since standard techniques require to compute expensive matrix decompositions. An alternative, is the conditional gradient method (aka Frank-Wolfe algorithm) that regained much interest in recent years, mostly due to its application to this specific setting. The key benefit of the CG method is that it avoids expensive matrix decompositions all together, and simply requires a single eigenvector computation per iteration, which is much more efficient. On the downside, the CG method, in general, converges with an inferior rate. The error for minimizing a $\beta$-smooth function after $t$ iterations scales like $\beta/t$. This rate does not improve even if the function is also strongly convex. In this work we present a modification of the CG method tailored for the spectrahedron. The per-iteration complexity of the method is essentially identical to that of the standard CG method: only a single eigenvecor computation is required. For minimizing an $\alpha$-strongly convex and $\beta$-smooth function, the \textit{expected} error of the method after $t$ iterations is: $O\left({\min\{\frac{\beta{}}{t} ,\left({\frac{\beta\sqrt{\rank(\X^*)}}{\alpha^{1/4}t}}\right)^{4/3}, \left({\frac{\beta}{\sqrt{\alpha}\lambda_{\min}(\X^*)t}}\right)^{2}\}}\right)$. Beyond the significant improvement in convergence rate, it also follows that when the optimum is low-rank, our method provides better accuracy-rank tradeoff than the standard CG method. To the best of our knowledge, this is the first result that attains provably faster convergence rates for a CG variant for optimization over the spectrahedron. We also present encouraging preliminary empirical results.

Many statistical models can be simulated forwards but have intractable likelihoods. Approximate Bayesian Computation (ABC) methods are used to infer properties of these models from data. Traditionally these methods approximate the posterior over parameters by conditioning on data being inside an ε-ball around the observed data, which is only correct in the limit ε→0. Monte Carlo methods can then draw samples from the approximate posterior to approximate predictions or error bars on parameters. These algorithms critically slow down as ε→0, and in practice draw samples from a broader distribution than the posterior. We propose a new approach to likelihood-free inference based on Bayesian conditional density estimation. Preliminary inferences based on limited simulation data are used to guide later simulations. In some cases, learning an accurate parametric representation of the entire true posterior distribution requires fewer model simulations than Monte Carlo ABC methods need to produce a single sample from an approximate posterior.

We address the problem of recovering a high-dimensional but structured vector from linear observations in a general setting where the vector can come from an arbitrary union of subspaces. This setup includes well-studied problems such as compressive sensing and low-rank matrix recovery. We show how to design more efficient algorithms for the union-of subspace recovery problem by using

We study fast learning rates when the losses are not necessarily bounded and may have a distribution with heavy tails. To enable such analyses, we introduce two new conditions: (i) the envelope function $\sup_{f \in \mathcal{F}}|\ell \circ f|$, where $\ell$ is the loss function and $\mathcal{F}$ is the hypothesis class, exists and is $L^r$-integrable, and (ii) $\ell$ satisfies the multi-scale Bernstein's condition on $\mathcal{F}$. Under these assumptions, we prove that learning rate faster than $O(n^{-1/2})$ can be obtained and, depending on $r$ and the multi-scale Bernstein's powers, can be arbitrarily close to $O(n^{-1})$. We then verify these assumptions and derive fast learning rates for the problem of vector quantization by $k$-means clustering with heavy-tailed distributions. The analyses enable us to obtain novel learning rates that extend and complement existing results in the literature from both theoretical and practical viewpoints.

New silicon technology is enabling large-scale electrophysiological recordings in vivo from hundreds to thousands of channels. Interpreting these recordings requires scalable and accurate automated methods for spike sorting, which should minimize the time required for manual curation of the results. Here we introduce KiloSort, a new integrated spike sorting framework that uses template matching both during spike detection and during spike clustering. KiloSort models the electrical voltage as a sum of template waveforms triggered on the spike times, which allows overlapping spikes to be identified and resolved. Unlike previous algorithms that compress the data with PCA, KiloSort operates on the raw data which allows it to construct a more accurate model of the waveforms. Processing times are faster than in previous algorithms thanks to batch-based optimization on GPUs. We compare KiloSort to an established algorithm and show favorable performance, at much reduced processing times. A novel post-clustering merging step based on the continuity of the templates further reduced substantially the number of manual operations required on this data, for the neurons with near-zero error rates, paving the way for fully automated spike sorting of multichannel electrode recordings.

For many machine learning problems, there are some inputs that are known to be positively (or negatively) related to the output, and in such cases training the model to respect that monotonic relationship can provide regularization, and makes the model more interpretable. However, flexible monotonic functions are computationally challenging to learn beyond a few features. We break through this barrier by learning ensembles of monotonic calibrated interpolated look-up tables (lattices). A key contribution is an automated algorithm for selecting feature subsets for the ensemble base models. We demonstrate that compared to random forests, these ensembles produce similar or better accuracy, while providing guaranteed monotonicity consistent with prior knowledge, smaller model size and faster evaluation.

We study probability measures induced by set functions with constraints. Such measures arise in a variety of real-world settings, where prior knowledge, resource limitations, or other pragmatic considerations impose constraints. We consider the task of rapidly sampling from such constrained measures, and develop fast Markov chain samplers for them. Our first main result is for MCMC sampling from Strongly Rayleigh (SR) measures, for which we present sharp polynomial bounds on the mixing time. As a corollary, this result yields a fast mixing sampler for Determinantal Point Processes (DPPs), yielding (to our knowledge) the first provably fast MCMC sampler for DPPs since their inception over four decades ago. Beyond SR measures, we develop MCMC samplers for probabilistic models with hard constraints and identify sufficient conditions under which their chains mix rapidly. We illustrate our claims by empirically verifying the dependence of mixing times on the key factors governing our theoretical bounds.

In this paper, we introduce the public-private framework of data summarization motivated by privacy concerns in personalized recommender systems and online social services. Such systems have usually access to massive data generated by a large pool of users. A major fraction of the data is public and is visible to (and can be used for) all users. However, each user can also contribute some private data that should not be shared with other users to ensure her privacy. The goal is to provide a succinct summary of massive dataset, ideally as small as possible, from which customized summaries can be built for each user, i.e. it can contain elements from the public data (for diversity) and users' private data (for personalization). To formalize the above challenge, we assume that the scoring function according to which a user evaluates the utility of her summary satisfies submodularity, a widely used notion in data summarization applications. Thus, we model the data summarization targeted to each user as an instance of a submodular cover problem. However, when the data is massive it is infeasible to use the centralized greedy algorithm to find a customized summary even for a single user. Moreover, for a large pool of users, it is too time consuming to find such summaries separately. Instead, we develop a fast distributed algorithm for submodular cover, FASTCOVER, that provides a succinct summary in one shot and for all users. We show that the solution provided by FASTCOVER is competitive with that of the centralized algorithm with the number of rounds that is exponentially smaller than state of the art results. Moreover, we have implemented FASTCOVER with Spark to demonstrate its practical performance on a number of concrete applications, including personalized location recommendation, personalized movie recommendation, and dominating set on tens of millions of data points and varying number of users.

We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many corruptions can we tolerate) via polynomial-time algorithms is by now understood. This paper presents and analyzes a non-convex optimization approach that greatly reduces the computational complexity of the above problems, compared to the best available algorithms. In particular, in the fully observed case, with $r$ denoting rank and $d$ dimension, we reduce the complexity from $O(r^2d^2\log(1/\epsilon))$ to $O(rd^2\log(1/\epsilon))$ -- a big savings when the rank is big. For the partially observed case, we show the complexity of our algorithm is no more than $O(r^4d\log(d)\log(1/\epsilon))$. Not only is this the best-known run-time for a provable algorithm under partial observation, but in the setting where $r$ is small compared to $d$, it also allows for near-linear-in-$d$ run-time that can be exploited in the fully-observed case as well, by simply running our algorithm on a subset of the observations.

Fluorescent calcium indicators are a popular means for observing the spiking activity of large neuronal populations. Unfortunately, extracting the spike train of each neuron from raw fluorescence calcium imaging data is a nontrivial problem. We present a fast online active set method to solve this sparse nonnegative deconvolution problem. Importantly, the algorithm progresses through each time series sequentially from beginning to end, thus enabling real-time online spike inference during the imaging session. Our algorithm is a generalization of the pool adjacent violators algorithm (PAVA) for isotonic regression and inherits its linear-time computational complexity. We gain remarkable increases in processing speed: more than one order of magnitude compared to currently employed state of the art convex solvers relying on interior point methods. Our method can exploit warm starts; therefore optimizing model hyperparameters only requires a handful of passes through the data. The algorithm enables real-time simultaneous deconvolution of $O(10^5)$ traces of whole-brain zebrafish imaging data on a laptop.

We introduce the study of fairness in multi-armed bandit problems. Our fairness definition demands that, given a pool of applicants, a worse applicant is never favored over a better one, despite a learning algorithm’s uncertainty over the true payoffs. In the classic stochastic bandits problem we provide a provably fair algorithm based on “chained” confidence intervals, and prove a cumulative regret bound with a cubic dependence on the number of arms. We further show that any fair algorithm must have such a dependence, providing a strong separation between fair and unfair learning that extends to the general contextual case. In the general contextual case, we prove a tight connection between fairness and the KWIK (Knows What It Knows) learning model: a KWIK algorithm for a class of functions can be transformed into a provably fair contextual bandit algorithm and vice versa. This tight connection allows us to provide a provably fair algorithm for the linear contextual bandit problem with a polynomial dependence on the dimension, and to show (for a different class of functions) a worst-case exponential gap in regret between fair and non-fair learning algorithms.

Building discriminative representations for 3D data has been an important task in computer graphics and computer vision research. Convolutional Neural Networks (CNNs) have shown to operate on 2D images with great success for a variety of tasks. Lifting convolution operators to 3D (3DCNNs) seems like a plausible and promising next step. Unfortunately, the computational complexity of 3D CNNs grows cubically with respect to voxel resolution. Moreover, since most 3D geometry representations are boundary based, occupied regions do not increase proportionately with the size of the discretization, resulting in wasted computation. In this work, we represent 3D spaces as volumetric fields, and propose a novel design that employs field probing filters to efficiently extract features from them. Each field probing filter is a set of probing points -- sensors that perceive the space. Our learning algorithm optimizes not only the weights associated with the probing points, but also their locations, which deforms the shape of the probing filters and adaptively distributes them in 3D space. The optimized probing points sense the 3D space "intelligently", rather than operating blindly over the entire domain. We show that field probing is significantly more efficient than 3DCNNs, while providing state-of-the-art performance, on classification tasks for 3D object recognition benchmark datasets.

We combine Riemannian geometry with the mean field theory of high dimensional chaos to study the nature of signal propagation in deep neural networks with random weights. Our results reveal a phase transition in the expressivity of random deep networks, with networks in the chaotic phase computing nonlinear functions whose global curvature grows exponentially with depth, but not with width. We prove that this generic class of random functions cannot be efficiently computed by any shallow network, going beyond prior work that restricts their analysis to single functions. Moreover, we formally quantify and demonstrate the long conjectured idea that deep networks can disentangle exponentially curved manifolds in input space into flat manifolds in hidden space. Our theoretical framework for analyzing the expressive power of deep networks is broadly applicable and provides a basis for quantifying previously abstract notions about the geometry of deep functions.

Word embeddings are a powerful approach to capturing semantic similarity among terms in a vocabulary. In this paper, we develop exponential family embeddings, which extends the idea of word embeddings to other types of high-dimensional data. As examples, we studied several types of data: neural data with real-valued observations, count data from a market basket analysis, and ratings data from a movie recommendation system. The main idea is that each observation is modeled conditioned on a set of latent embeddings and other observations, called the context, where the way the context is defined depends on the problem. In language the context is the surrounding words; in neuroscience the context is close-by neurons; in market basket data the context is other items in the shopping cart. Each instance of an embedding defines the context, the exponential family of conditional distributions, and how the embedding vectors are shared across data. We infer the embeddings with stochastic gradient descent, with an algorithm that connects closely to generalized linear models. On all three of our applications—neural activity of zebrafish, users’ shopping behavior, and movie ratings—we found that exponential family embedding models are more effective than other dimension reduction methods. They better reconstruct held-out data and find interesting qualitative structure.

The amount of data available in the world is growing faster than our ability to deal with it. However, if we take advantage of the internal structure, data may become much smaller for machine learning purposes. In this paper we focus on one of the fundamental machine learning tasks, empirical risk minimization (ERM), and provide faster algorithms with the help from the clustering structure of the data. We introduce a simple notion of raw clustering that can be efficiently computed from the data, and propose two algorithms based on clustering information. Our accelerated algorithm ClusterACDM is built on a novel Haar transformation applied to the dual space of the ERM problem, and our variance-reduction based algorithm ClusterSVRG introduces a new gradient estimator using clustering. Our algorithms outperform their classical counterparts ACDM and SVRG respectively.