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JMLRJournal of Machine Learning Research
Iterated Learning in Dynamic Social Networks
http://jmlr.org/papers/v20/18-539.html
2019
A classic finding by (Kalish et al., 2007) shows that no language can be learned iteratively by rational agents in a self-sustained manner. In other words, if $A$ teaches a foreign language to $B$, who then teaches what she learned to $C$, and so on, the language will quickly get lost and agents will wind up teaching their own common native language. If so, how can linguistic novelty ever be sustained? We address this apparent paradox by considering the case of iterated learning in a social network: we show that by varying the lengths of the learning sessions over time or by keeping the networks dynamic, it is possible for iterated learning to endure forever with arbitrarily small loss.
Pyro: Deep Universal Probabilistic Programming
http://jmlr.org/papers/v20/18-403.html
2019
Pyro is a probabilistic programming language built on Python as a platform for developing advanced probabilistic models in AI research. To scale to large data sets and high-dimensional models, Pyro uses stochastic variational inference algorithms and probability distributions built on top of PyTorch, a modern GPU-accelerated deep learning framework. To accommodate complex or model-specific algorithmic behavior, Pyro leverages Poutine, a library of composable building blocks for modifying the behavior of probabilistic programs.
Monotone Learning with Rectified Wire Networks
http://jmlr.org/papers/v20/18-281.html
2019
We introduce a new neural network model, together with a tractable and monotone online learning algorithm. Our model describes feed-forward networks for classification, with one output node for each class. The only nonlinear operation is rectification using a ReLU function with a bias. However, there is a rectifier on every edge rather than at the nodes of the network. There are also weights, but these are positive, static, and associated with the nodes. Our rectified wire networks are able to represent arbitrary Boolean functions. Only the bias parameters, on the edges of the network, are learned. Another departure in our approach, from standard neural networks, is that the loss function is replaced by a constraint. This constraint is simply that the value of the output node associated with the correct class should be zero. Our model has the property that the exact norm-minimizing parameter update, required to correctly classify a training item, is the solution to a quadratic program that can be computed with a few passes through the network. We demonstrate a training algorithm using this update, called sequential deactivation (SDA), on MNIST and some synthetic datasets. Upon adopting a natural choice for the nodal weights, SDA has no hyperparameters other than those describing the network structure. Our experiments explore behavior with respect to network size and depth in a family of sparse expander networks.
TensorLy: Tensor Learning in Python
http://jmlr.org/papers/v20/18-277.html
2019
Tensors are higher-order extensions of matrices. While matrix methods form the cornerstone of traditional machine learning and data analysis, tensor methods have been gaining increasing traction. However, software support for tensor operations is not on the same footing. In order to bridge this gap, we have developed TensorLy, a Python library that provides a high-level API for tensor methods and deep tensorized neural networks. TensorLy aims to follow the same standards adopted by the main projects of the Python scientific community, and to seamlessly integrate with them. Its BSD license makes it suitable for both academic and commercial applications. TensorLy's backend system allows users to perform computations with several libraries such as NumPy or PyTorch to name but a few. They can be scaled on multiple CPU or GPU machines. In addition, using the deep-learning frameworks as backend allows to easily design and train deep tensorized neural networks. TensorLy is available at https://github.com/tensorly/tensorly
Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations
http://jmlr.org/papers/v20/18-190.html
2019
The success of deep convolutional architectures is often attributed in part to their ability to learn multiscale and invariant representations of natural signals. However, a precise study of these properties and how they affect learning guarantees is still missing. In this paper, we consider deep convolutional representations of signals; we study their invariance to translations and to more general groups of transformations, their stability to the action of diffeomorphisms, and their ability to preserve signal information. This analysis is carried by introducing a multilayer kernel based on convolutional kernel networks and by studying the geometry induced by the kernel mapping. We then characterize the corresponding reproducing kernel Hilbert space (RKHS), showing that it contains a large class of convolutional neural networks with homogeneous activation functions. This analysis allows us to separate data representation from learning, and to provide a canonical measure of model complexity, the RKHS norm, which controls both stability and generalization of any learned model. In addition to models in the constructed RKHS, our stability analysis also applies to convolutional networks with generic activations such as rectified linear units, and we discuss its relationship with recent generalization bounds based on spectral norms.
Joint PLDA for Simultaneous Modeling of Two Factors
http://jmlr.org/papers/v20/18-134.html
2019
Probabilistic linear discriminant analysis (PLDA) is a method used for biometric problems like speaker or face recognition that models the variability of the samples using two latent variables, one that depends on the class of the sample and another one that is assumed independent across samples and models the within-class variability. In this work, we propose a generalization of PLDA that enables joint modeling of two sample-dependent factors: the class of interest and a nuisance condition. The approach does not change the basic form of PLDA but rather modifies the training procedure to consider the dependency across samples of the latent variable that models within-class variability. While the identity of the nuisance condition is needed during training, it is not needed during testing since we propose a scoring procedure that marginalizes over the corresponding latent variable. We show results on a multilingual speaker-verification task, where the language spoken is considered a nuisance condition. The proposed joint PLDA approach leads to significant performance gains in this task for two different data sets, in particular when the training data contains mostly or only monolingual speakers.
Determining the Number of Latent Factors in Statistical Multi-Relational Learning
http://jmlr.org/papers/v20/18-037.html
2019
Statistical relational learning is primarily concerned with learning and inferring relationships between entities in large-scale knowledge graphs. Nickel et al. (2011) proposed a RESCAL tensor factorization model for statistical relational learning, which achieves better or at least comparable results on common benchmark data sets when compared to other state-of-the-art methods. Given a positive integer $s$, RESCAL computes an $s$-dimensional latent vector for each entity. The latent factors can be further used for solving relational learning tasks, such as collective classification, collective entity resolution and link-based clustering. The focus of this paper is to determine the number of latent factors in the RESCAL model. Due to the structure of the RESCAL model, its log-likelihood function is not concave. As a result, the corresponding maximum likelihood estimators (MLEs) may not be consistent. Nonetheless, we design a specific pseudometric, prove the consistency of the MLEs under this pseudometric and establish its rate of convergence. Based on these results, we propose a general class of information criteria and prove their model selection consistencies when the number of relations is either bounded or diverges at a proper rate of the number of entities. Simulations and real data examples show that our proposed information criteria have good finite sample properties.
Random Feature-based Online Multi-kernel Learning in Environments with Unknown Dynamics
http://jmlr.org/papers/v20/18-030.html
2019
Kernel-based methods exhibit well-documented performance in various nonlinear learning tasks. Most of them rely on a preselected kernel, whose prudent choice presumes task-specific prior information. Especially when the latter is not available, multi-kernel learning has gained popularity thanks to its flexibility in choosing kernels from a prescribed kernel dictionary. Leveraging the random feature approximation and its recent orthogonality-promoting variant, the present contribution develops a scalable multi-kernel learning scheme (termed Raker) to obtain the sought nonlinear learning function `on the fly,' first for static environments. To further boost performance in dynamic environments, an adaptive multi-kernel learning scheme (termed AdaRaker) is developed. AdaRaker accounts not only for data-driven learning of kernel combination, but also for the unknown dynamics. Performance is analyzed in terms of both static and dynamic regrets. AdaRaker is uniquely capable of tracking nonlinear learning functions in environments with unknown dynamics, and with with analytic performance guarantees Tests with synthetic and real datasets are carried out to showcase the effectiveness of the novel algorithms.
Spectrum Estimation from a Few Entries
http://jmlr.org/papers/v20/18-027.html
2019
Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis, we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering spectral properties of the underlying matrix from a sampling of its entries. In this paper, we address the problem of directly recovering the spectrum, which is the set of singular values, and also in sample-efficient approaches for recovering a spectral sum function, which is an aggregate sum of a fixed function applied to each of the singular values. Our approach is to first estimate the Schatten $k$-norms of a matrix for a small set of values of $k$, and then apply Chebyshev approximation when estimating a spectral sum function or apply moment matching in Wasserstein distance when estimating the singular values directly. The main technical challenge is in accurately estimating the Schatten norms from a sampling of a matrix. We introduce a novel unbiased estimator based on counting small structures called network motifs in a graph and provide guarantees that match its empirical performance. Our theoretical analysis shows that Schatten norms can be recovered accurately from strictly smaller number of samples compared to what is needed to recover the underlying low-rank matrix. Numerical experiments suggest that we significantly improve upon a competing approach of using matrix completion methods, below the matrix completion threshold, above which matrix completion algorithms recover the underlying low-rank matrix exactly.
Accelerated Alternating Projections for Robust Principal Component Analysis
http://jmlr.org/papers/v20/18-022.html
2019
We study robust PCA for the fully observed setting, which is about separating a low rank matrix $\BL$ and a sparse matrix $\BS$ from their sum $\BD=\BL+\BS$. In this paper, a new algorithm, dubbed accelerated alternating projections, is introduced for robust PCA which significantly improves the computational efficiency of the existing alternating projections proposed in (Netrapalli et al., 2014) when updating the low rank factor. The acceleration is achieved by first projecting a matrix onto some low dimensional subspace before obtaining a new estimate of the low rank matrix via truncated SVD. Exact recovery guarantee has been established which shows linear convergence of the proposed algorithm. Empirical performance evaluations establish the advantage of our algorithm over other state-of-the-art algorithms for robust PCA.
spark-crowd: A Spark Package for Learning from Crowdsourced Big Data
http://jmlr.org/papers/v20/17-743.html
2019
As the data sets increase in size, the process of manually labeling data becomes unfeasible by small groups of experts. Thus, it is common to rely on crowdsourcing platforms which provide inexpensive, but noisy, labels. Although implementations of algorithms to tackle this problem exist, none of them focus on scalability, limiting the area of application to relatively small data sets. In this paper, we present spark-crowd, an Apache Spark package for learning from crowdsourced data with scalability in mind.
Multiplicative local linear hazard estimation and best one-sided cross-validation
http://jmlr.org/papers/v20/17-663.html
2019
This paper develops detailed mathematical statistical theory of a new class of cross-validation techniques of local linear kernel hazards and their multiplicative bias corrections. The new class of cross-validation combines principles of local information and recent advances in indirect cross-validation. A few applications of cross-validating multiplicative kernel hazard estimation do exist in the literature. However, detailed mathematical statistical theory and small sample performance are introduced via this paper and further upgraded to our new class of best one-sided cross-validation. Best one-sided cross-validation turns out to have excellent performance in its practical illustrations, in its small sample performance and in its mathematical statistical theoretical performance.
Delay and Cooperation in Nonstochastic Bandits
http://jmlr.org/papers/v20/17-631.html
2019
We study networks of communicating learning agents that cooperate to solve a common nonstochastic bandit problem. Agents use an underlying communication network to get messages about actions selected by other agents, and drop messages that took more than $d$ hops to arrive, where $d$ is a delay parameter. We introduce Exp3-Coop, a cooperative version of the Exp3 algorithm and prove that with $K$ actions and $N$ agents the average per-agent regret after $T$ rounds is at most of order $\sqrt{\bigl(d+1 + \tfrac{K}{N}\alpha_{\le d}\bigr)(T\ln K)}$, where $\alpha_{\le d}$ is the independence number of the $d$-th power of the communication graph $G$. We then show that for any connected graph, for $d=\sqrt{K}$ the regret bound is $K^{1/4}\sqrt{T}$, strictly better than the minimax regret $\sqrt{KT}$ for noncooperating agents. More informed choices of $d$ lead to bounds which are arbitrarily close to the full information minimax regret $\sqrt{T\ln K}$ when $G$ is dense. When $G$ has sparse components, we show that a variant of Exp3-Coop, allowing agents to choose their parameters according to their centrality in $G$, strictly improves the regret. Finally, as a by-product of our analysis, we provide the first characterization of the minimax regret for bandit learning with delay.
Smooth neighborhood recommender systems
http://jmlr.org/papers/v20/17-629.html
2019
Recommender systems predict users' preferences over a large number of items by pooling similar information from other users and/or items in the presence of sparse observations. One major challenge is how to utilize user-item specific covariates and networks describing user-item interactions in a high-dimensional situation, for accurate personalized prediction. In this article, we propose a smooth neighborhood recommender in the framework of the latent factor models. A similarity kernel is utilized to borrow neighborhood information from continuous covariates over a user-item specific network, such as a user's social network, where the grouping information defined by discrete covariates is also integrated through the network. Consequently, user-item specific information is built into the recommender to battle the `cold-start” issue in the absence of observations in collaborative and content-based filtering. Moreover, we utilize a “divide-and-conquer” version of the alternating least squares algorithm to achieve scalable computation, and establish asymptotic results for the proposed method, demonstrating that it achieves superior prediction accuracy. Finally, we illustrate that the proposed method improves substantially over its competitors in simulated examples and real benchmark data--Last.fm music data.
Automated Scalable Bayesian Inference via Hilbert Coresets
http://jmlr.org/papers/v20/17-613.html
2019
The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scale of massive modern data sets, and modifications to make them so are typically model-specific, require expert tuning, and can break theoretical guarantees on inferential quality. Building on the Bayesian coresets framework, this work instead takes advantage of data redundancy to shrink the data set itself as a preprocessing step, providing fully-automated, scalable Bayesian inference with theoretical guarantees. We begin with an intuitive reformulation of Bayesian coreset construction as sparse vector sum approximation, and demonstrate that its automation and performance-based shortcomings arise from the use of the supremum norm. To address these shortcomings we develop Hilbert coresets, i.e., Bayesian coresets constructed under a norm induced by an inner-product on the log-likelihood function space. We propose two Hilbert coreset construction algorithms---one based on importance sampling, and one based on the Frank-Wolfe algorithm---along with theoretical guarantees on approximation quality as a function of coreset size. Since the exact computation of the proposed inner-products is model-specific, we automate the construction with a random finite-dimensional projection of the log-likelihood functions. The resulting automated coreset construction algorithm is simple to implement, and experiments on a variety of models with real and synthetic data sets show that it provides high-quality posterior approximations and a significant reduction in the computational cost of inference.
Approximations of the Restless Bandit Problem
http://jmlr.org/papers/v20/17-547.html
2019
The multi-armed restless bandit problem is studied in the case where the pay-off distributions are stationary $\varphi$-mixing. This version of the problem provides a more realistic model for most real-world applications, but cannot be optimally solved in practice, since it is known to be PSPACE-hard. The objective of this paper is to characterize a sub-class of the problem where good approximate solutions can be found using tractable approaches. Specifically, it is shown that under some conditions on the $\varphi$-mixing coefficients, a modified version of UCB can prove effective. The main challenge is that, unlike in the i.i.d. setting, the distributions of the sampled pay-offs may not have the same characteristics as those of the original bandit arms. In particular, the $\varphi$-mixing property does not necessarily carry over. This is overcome by carefully controlling the effect of a sampling policy on the pay-off distributions. Some of the proof techniques developed in this paper can be more generally used in the context of online sampling under dependence. Proposed algorithms are accompanied with corresponding regret analysis.
Train and Test Tightness of LP Relaxations in Structured Prediction
http://jmlr.org/papers/v20/17-535.html
2019
Structured prediction is used in areas including computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation for the striking observation that approximations based on linear programming (LP) relaxations are often tight (exact) on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that this training tightness generalizes to test data.
Scalable Kernel K-Means Clustering with Nystr\"om Approximation: Relative-Error Bounds
http://jmlr.org/papers/v20/17-517.html
2019
Kernel $k$-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear $k$-means clustering algorithm. However, kernel $k$-means clustering is computationally expensive when the non-linear feature map is high-dimensional and there are many input points. Kernel approximation, e.g., the Nystr\"om method, has been applied in previous works to approximately solve kernel learning problems when both of the above conditions are present. This work analyzes the application of this paradigm to kernel $k$-means clustering, and shows that applying the linear $k$-means clustering algorithm to $\frac{k}{\epsilon} (1 + o(1))$ features constructed using a so-called rank-restricted Nystr\"om approximation results in cluster assignments that satisfy a $1 + \epsilon$ approximation ratio in terms of the kernel $k$-means cost function, relative to the guarantee provided by the same algorithm without the use of the Nystr\"om method. As part of the analysis, this work establishes a novel $1 + \epsilon$ relative-error trace norm guarantee for low-rank approximation using the rank-restricted Nystr\"om approximation. Empirical evaluations on the $8.1$ million instance MNIST8M dataset demonstrate the scalability and usefulness of kernel $k$-means clustering with Nystr\"om approximation. This work argues that spectral clustering using Nystr\"om approximation---a popular and computationally efficient, but theoretically unsound approach to non-linear clustering---should be replaced with the efficient and theoretically sound combination of kernel $k$-means clustering with Nystr\"om approximation. The superior performance of the latter approach is empirically verified.
An Approach to One-Bit Compressed Sensing Based on Probably Approximately Correct Learning Theory
http://jmlr.org/papers/v20/17-504.html
2019
In this paper, the problem of one-bit compressed sensing (OBCS) is formulated as a problem in probably approximately correct (PAC) learning. It is shown that the Vapnik-Chervonenkis (VC-) dimension of the set of half-spaces in $\R^n$ generated by $k$-sparse vectors is bounded below by $k ( \lfloor\lg (n/k) \rfloor +1 )$ and above by $\lfloor 2k \lg (en) \rfloor $. By coupling this estimate with well-established results in PAC learning theory, we show that a consistent algorithm can recover a $k$-sparse vector with $O(k \lg n)$ measurements, given only the signs of the measurement vector. This result holds for \textit{all} probability measures on $\R^n$. The theory is also applicable to the case of noisy labels, where the signs of the measurements are flipped with some unknown probability.
Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions
http://jmlr.org/papers/v20/17-501.html
2019
Graphical Lasso (GL) is a popular method for learning the structure of an undirected graphical model, which is based on an $l_1$ regularization technique. The objective of this paper is to compare the computationally-heavy GL technique with a numerically-cheap heuristic method that is based on simply thresholding the sample covariance matrix. To this end, two notions of sign-consistent and inverse-consistent matrices are developed, and then it is shown that the thresholding and GL methods are equivalent if: (i) the thresholded sample covariance matrix is both sign-consistent and inverse-consistent, and (ii) the gap between the largest thresholded and the smallest un-thresholded entries of the sample covariance matrix is not too small. By building upon this result, it is proved that the GL method---as a conic optimization problem---has an explicit closed-form solution if the thresholded sample covariance matrix has an acyclic structure. This result is then generalized to arbitrary sparse support graphs, where a formula is found to obtain an approximate solution of GL. Furthermore, it is shown that the approximation error of the derived explicit formula decreases exponentially fast with respect to the length of the minimum-length cycle of the sparsity graph. The developed results are demonstrated on synthetic data, functional MRI data, traffic flows for transportation networks, and massive randomly generated data sets. We show that the proposed method can obtain an accurate approximation of the GL for instances with the sizes as large as $80,000\times 80,000$ (more than 3.2 billion variables) in less than 30 minutes on a standard laptop computer running MATLAB, while other state-of-the-art methods do not converge within 4 hours
Dynamic Pricing in High-dimensions
http://jmlr.org/papers/v20/17-357.html
2019
We study the pricing problem faced by a firm that sells a large number of products, described via a wide range of features, to customers that arrive over time. Customers independently make purchasing decisions according to a general choice model that includes products features and customers' characteristics, encoded as $d$-dimensional numerical vectors, as well as the price offered. The parameters of the choice model are a priori unknown to the firm, but can be learned as the (binary-valued) sales data accrues over time. The firm's objective is to maximize its revenue. We benchmark the performance using the classic regret minimization framework where the regret is defined as the expected revenue loss against a clairvoyant policy that knows the parameters of the choice model in advance, and always offers the revenue-maximizing price. This setting is motivated in part by the prevalence of online marketplaces that allow for real-time pricing. We assume a structured choice model, parameters of which depend on $s_0$ out of the $d$ product features. Assuming that the market noise distribution is known, we propose a dynamic policy, called Regularized Maximum Likelihood Pricing (RMLP) that leverages the (sparsity) structure of the high-dimensional model and obtains a logarithmic regret in $T$. More specifically, the regret of our algorithm is of $O(s_0 \log d \cdot \log T)$. Furthermore, we show that no policy can obtain regret better than $O(s_0 (\log d + \log T))$. {In addition, we propose a generalization of our policy to a setting that the market noise distribution is unknown but belongs to a parametrized family of distributions. This policy obtains regret of $O(\sqrt{(\log d)T})$. We further show that no policy can obtain regret better than $\Omega(\sqrt{T})$ in such environments.}
Forward-Backward Selection with Early Dropping
http://jmlr.org/papers/v20/17-334.html
2019
Forward-backward selection is one of the most basic and commonly-used feature selection algorithms available. It is also general and conceptually applicable to many different types of data. In this paper, we propose a heuristic that significantly improves its running time, while preserving predictive performance. The idea is to temporarily discard the variables that are conditionally independent with the outcome given the selected variable set. Depending on how those variables are reconsidered and reintroduced, this heuristic gives rise to a family of algorithms with increasingly stronger theoretical guarantees. In distributions that can be faithfully represented by Bayesian networks or maximal ancestral graphs, members of this algorithmic family are able to correctly identify the Markov blanket in the sample limit. In experiments we show that the proposed heuristic increases computational efficiency by about 1-2 orders of magnitude, while selecting fewer or the same number of variables and retaining predictive performance. Furthermore, we show that the proposed algorithm and feature selection with LASSO perform similarly when restricted to select the same number of variables, making the proposed algorithm an attractive alternative for problems where no (efficient) algorithm for LASSO exists.
Scalable Approximations for Generalized Linear Problems
http://jmlr.org/papers/v20/17-279.html
2019
In stochastic optimization, the population risk is generally approximated by the empirical risk which is in turn minimized by an iterative algorithm. However, in the large-scale setting, empirical risk minimization may be computationally restrictive. In this paper, we design an efficient algorithm to approximate the population risk minimizer in generalized linear problems such as binary classification with surrogate losses and generalized linear regression models. We focus on large-scale problems where the iterative minimization of the empirical risk is computationally intractable, i.e., the number of observations $n$ is much larger than the dimension of the parameter $p$ ($n \gg p \gg 1$). We show that under random sub-Gaussian design, the true minimizer of the population risk is approximately proportional to the corresponding ordinary least squares (OLS) estimator. Using this relation, we design an algorithm that achieves the same accuracy as the empirical risk minimizer through iterations that attain up to a quadratic convergence rate, and that are computationally cheaper than any batch optimization algorithm by at least a factor of $\mathcal{O}(p)$. We provide theoretical guarantees for our algorithm, and analyze the convergence behavior in terms of data dimensions. Finally, we demonstrate the performance of our algorithm on well-known classification and regression problems, through extensive numerical studies on large-scale datasets, and show that it achieves the highest performance compared to several other widely used optimization algorithms.
scikit-multilearn: A Python library for Multi-Label Classification
http://jmlr.org/papers/v20/17-100.html
2019
The scikit-multilearn is a Python library for performing multi-label classification. It is compatible with the scikit-learn and scipy ecosystems and uses sparse matrices for all internal operations; provides native Python implementations of popular multi-label classification methods alongside a novel framework for label space partitioning and division and includes modern algorithm adaptation methods, network-based label space division approaches, which extracts label dependency information and multi-label embedding classifiers. The library provides Python wrapped access to the extensive multi-label method stack from Java libraries and makes it possible to extend deep learning single-label methods for multi-label tasks. The library allows multi-label stratification and data set management. The implementation is more efficient in problem transformation than other established libraries, has good test coverage and follows PEP8. Source code and documentation can be downloaded from http://scikit.ml and also via pip The project is BSD-licensed.
Non-Convex Projected Gradient Descent for Generalized Low-Rank Tensor Regression
http://jmlr.org/papers/v20/16-607.html
2019
In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polynomial-time algorithms they are often slow in practice. On the other hand, limited theoretical guarantees exist for non-convex methods. In this paper we provide a general framework that provides theoretical guarantees for learning high-dimensional tensor regression models under different low-rank structural assumptions using the projected gradient descent algorithm applied to a potentially non-convex constraint set $\Theta$ in terms of its localized Gaussian width (due to Gaussian design). We juxtapose our theoretical results for non-convex projected gradient descent algorithms with previous results on regularized convex approaches. The two main differences between the convex and non-convex approach are: (i) from a computational perspective whether the non-convex projection operator is computable and whether the projection has desirable contraction properties and (ii) from a statistical error bound perspective, the non-convex approach has a superior rate for a number of examples. We provide three concrete examples of low-dimensional structure which address these issues and explain the pros and cons for the non-convex and convex approaches. We supplement our theoretical results with simulations which show that, under several common settings of generalized low rank tensor regression, the projected gradient descent approach is superior both in terms of statistical error and run-time provided the step-sizes of the projected descent algorithm are suitably chosen.
Convergence Rate of a Simulated Annealing Algorithm with Noisy Observations
http://jmlr.org/papers/v20/16-588.html
2019
In this paper we propose a modified version of the simulated annealing algorithm for solving a stochastic global optimization problem. More precisely, we address the problem of finding a global minimizer of a function with noisy evaluations. We provide a rate of convergence and its optimized parametrization to ensure a minimal number of evaluations for a given accuracy and a confidence level close to 1. This work is completed with a set of numerical experimentations and assesses the practical performance both on benchmark test cases and on real world examples.
Parsimonious Online Learning with Kernels via Sparse Projections in Function Space
http://jmlr.org/papers/v20/16-585.html
2019
Despite their attractiveness, popular perception is that techniques for nonparametric function approximation do not scale to streaming data due to an intractable growth in the amount of storage they require. To solve this problem in a memory-affordable way, we propose an online technique based on functional stochastic gradient descent in tandem with supervised sparsification based on greedy function subspace projections. The method, called parsimonious online learning with kernels (POLK), provides a controllable tradeoff between its solution accuracy and the amount of memory it requires. We derive conditions under which the generated function sequence converges almost surely to the optimal function, and we establish that the memory requirement remains finite. We evaluate POLK for kernel multi-class logistic regression and kernel hinge-loss classification on three canonical data sets: a synthetic Gaussian mixture model, the MNIST hand-written digits, and the Brodatz texture database. On all three tasks, we observe a favorable trade-off of objective function evaluation, classification performance, and complexity of the nonparametric regressor extracted by the proposed method.
Transport Analysis of Infinitely Deep Neural Network
http://jmlr.org/papers/v20/16-243.html
2019
We investigated the feature map inside deep neural networks (DNNs) by tracking the transport map. We are interested in the role of depth---why do DNNs perform better than shallow models?---and the interpretation of DNNs---what do intermediate layers do? Despite the rapid development in their application, DNNs remain analytically unexplained because the hidden layers are nested and the parameters are not faithful. Inspired by the integral representation of shallow NNs, which is the continuum limit of the width, or the hidden unit number, we developed the flow representation and transport analysis of DNNs. The flow representation is the continuum limit of the depth, or the hidden layer number, and it is specified by an ordinary differential equation (ODE) with a vector field. We interpret an ordinary DNN as a transport map or an Euler broken line approximation of the flow. Technically speaking, a dynamical system is a natural model for the nested feature maps. In addition, it opens a new way to the coordinate-free treatment of DNNs by avoiding the redundant parametrization of DNNs. Following Wasserstein geometry, we analyze a flow in three aspects: dynamical system, continuity equation, and Wasserstein gradient flow. A key finding is that we specified a series of transport maps of the denoising autoencoder (DAE), which is a cornerstone for the development of deep learning. Starting from the shallow DAE, this paper develops three topics: the transport map of the deep DAE, the equivalence between the stacked DAE and the composition of DAEs, and the development of the double continuum limit or the integral representation of the flow representation. As partial answers to the research questions, we found that deeper DAEs converge faster and the extracted features are better; in addition, a deep Gaussian DAE transports mass to decrease the Shannon entropy of the data distribution. We expect that further investigations on these questions lead to the development of an interpretable and principled alternatives to DNNs.
Adaptation Based on Generalized Discrepancy
http://jmlr.org/papers/v20/15-192.html
2019
We present a new algorithm for domain adaptation improving upon a discrepancy minimization algorithm, (DM), previously shown to outperform a number of algorithms for this problem. Unlike many previously proposed solutions for domain adaptation, our algorithm does not consist of a fixed reweighting of the losses over the training sample. Instead, the reweighting depends on the hypothesis sought. The algorithm is derived from a less conservative notion of discrepancy than the DM algorithm called <i>generalized discrepancy</i>. We present a detailed description of our algorithm and show that it can be formulated as a convex optimization problem. We also give a detailed theoretical analysis of its learning guarantees which helps us select its parameters. Finally, we report the results of experiments demonstrating that it improves upon discrepancy minimization.