In deep learning, it is common to use more network parameters than training points. In such scenarioof over-parameterization, there are usually multiple networks that achieve zero training error so that thetraining algorithm induces an implicit bias on the computed solution. In practice, (stochastic) gradientdescent tends to prefer solutions which generalize well, which provides a possible explanation of thesuccess of deep learning. In this paper we analyze the dynamics of gradient descent in the simplifiedsetting of linear networks and of an estimation problem. Although we are not in an overparameterizedscenario, our analysis nevertheless provides insights into the phenomenon of implicit bias. In fact, wederive a rigorous analysis of the dynamics of vanilla gradient descent, and characterize the dynamicalconvergence of the spectrum. We are able to accurately locate time intervals where the effective rankof the iterates is close to the effective rank of a low-rank projection of the ground-truth matrix. Inpractice, those intervals can be used as criteria for early stopping if a certain regularity is desired. Wealso provide empirical evidence for implicit bias in more general scenarios, such as matrix sensing andrandom initialization. This suggests that deep learning prefers trajectories whose complexity (measuredin terms of effective rank) is monotonically increasing, which we believe is a fundamental concept for thetheoretical understanding of deep learning.
University of Applied Sciences and Arts, Northwestern Switzerland（瑞士西北应用科学与艺术大学）；University of Ljubljana（卢布尔雅那大学）
This article utilizes the projected gradient method (PG) for a non-negative matrix factorization problem (NMF), where one or both matrix factors must have orthonormal columns or rows. We penalise the orthonormality constraints and apply the PG method via a block coordinate descent approach. This means that at a certain time one matrix factor is fixed and the other is updated by moving along the steepest descent direction computed from the penalised objective function and projecting onto the space of non-negative matrices. Our method is tested on two sets of synthetic data for various values of…
Université de Mons（蒙斯大学）；Université de Rennes（雷恩大学）；INRIA（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；IRISA（雷恩第一大学计算机科学研究所）
This paper is concerned with improving the empirical convergence speed of block-coordinate descent algorithms for approximate nonnegative tensor factorization (NTF). We propose an extrapolation strategy in-between block updates, referred to as heuristic extrapolation with restarts (HER). HER significantly accelerates the empirical convergence speed of most existing block-coordinate algorithms for dense NTF, in particular for challenging computational scenarios, while requiring a negligible additional computational budget.
Nonnegative matrix factorization (NMF) has attracted much attention in the last decade as a dimension reduction method in many applications. Due to the explosion in the size of data, naturally the samples are collected and stored distributively in local computational nodes. Thus, there is a growing need to develop algorithms in a distributed memory architecture. We propose a novel distributed algorithm, called \textit{distributed incremental block coordinate descent} (DID), to solve the problem. By adapting the block coordinate descent framework, closed-form update rules are obtained in DID. M…
We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.
In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading contender to explain the generalization power of deep models such as neural networks---and a general blueprint to study it. We apply these techniques to settle a conjecture on implicit regularization in matrix factorization.
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）；Paul G. Allen School of Computer Science and Engineering（保罗·G·艾伦计算机科学与工程学院）；University of Washington（华盛顿大学）
We study the asymmetric low-rank factorization problem: \[\min_{\mathbf{U} \in \mathbb{R}^{m \times d}, \mathbf{V} \in \mathbb{R}^{n \times d}} \frac{1}{2}\|\mathbf{U}\mathbf{V}^\top -\mathbfΣ\|_F^2\] where $\mathbfΣ$ is a given matrix of size $m \times n$ and rank $d$. This is a canonical problem that admits two difficulties in optimization: 1) non-convexity and 2) non-smoothness (due to unbalancedness of $\mathbf{U}$ and $\mathbf{V}$). This is also a prototype for more complex problems such as asymmetric matrix sensing and matrix completion. Despite being non-convex and non-smooth, it has be…
School of Industrial and Systems Engineering at Georgia Tech（佐治亚理工学院工业与系统工程学院）
Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that i…
This work explores non-negative low-rank matrix factorization based on regularized Poisson models (PF or "Poisson factorization" for short) for recommender systems with implicit-feedback data. The properties of Poisson likelihood allow a shortcut for very fast computations over zero-valued inputs, and oftentimes results in very sparse factors for both users and items. Compared to HPF (a popular Bayesian formulation of the problem with hierarchical priors), the frequentist optimization-based approach presented here tends to produce better top-N recommendations with significantly shorter fitting…
National Engineering School of Tunis, University of Tunis El Manar（突尼斯埃尔马纳尔大学突尼斯国立工程学院）；Aix Marseille University（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；Ecole Centrale de Marseille（马赛中央理工学院）
We study the implicit regularization effects of deep learning in tensor factorization. While implicit regularization in deep matrix and 'shallow' tensor factorization via linear and certain type of non-linear neural networks promotes low-rank solutions with at most quadratic growth, we show that its effect in deep tensor factorization grows polynomially with the depth of the network. This provides a remarkably faithful description of the observed experimental behaviour. Using numerical experiments, we demonstrate the benefits of this implicit regularization in yielding a more accurate estimati…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Tel Aviv University（特拉维夫大学）
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion and sensing, a model referred to as deep matrix factorization. Our first finding, supported by theory and experiments, is that adding depth to a matrix factorization enhances an implicit tendency towards low-rank solutions, oftentimes leading to more accurate recovery. Secondly, …
In the pursuit of explaining implicit regularization in deep learning, prominent focus was given to matrix and tensor factorizations, which correspond to simplified neural networks. It was shown that these models exhibit an implicit tendency towards low matrix and tensor ranks, respectively. Drawing closer to practical deep learning, the current paper theoretically analyzes the implicit regularization in hierarchical tensor factorization, a model equivalent to certain deep convolutional neural networks. Through a dynamical systems lens, we overcome challenges associated with hierarchy, and est…
Matrix factorization is a simple and natural test-bed to investigate the implicit regularization of gradient descent. Gunasekar et al. (2017) conjectured that Gradient Flow with infinitesimal initialization converges to the solution that minimizes the nuclear norm, but a series of recent papers argued that the language of norm minimization is not sufficient to give a full characterization for the implicit regularization. In this work, we provide theoretical and empirical evidence that for depth-2 matrix factorization, gradient flow with infinitesimal initialization is mathematically equivalent…
We introduce a probabilistic model with implicit norm regularization for learning nonnegative matrix factorization (NMF) that is commonly used for predicting missing values and finding hidden patterns in the data, in which the matrix factors are latent variables associated with each data dimension. The nonnegativity constraint for the latent factors is handled by choosing priors with support on the nonnegative subspace, e.g., exponential density or distribution based on exponential function. Bayesian inference procedure based on Gibbs sampling is employed. We evaluate the model on several real…
Global Convergence of Four-Layer Matrix Factorization under Random Initialization
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）；Paul G. Allen School of Computer Science and Engineering（保罗·G·艾伦计算机科学与工程学院）；University of Washington（华盛顿大学）；Electrical and Computer Engineering（电气与计算机工程）
Recent efforts to unravel the mystery of implicit regularization in deep learning have led to a theoretical focus on matrix factorization -- matrix completion via linear neural network. As a step further towards practical deep learning, we provide the first theoretical analysis of implicit regularization in tensor factorization -- tensor completion via certain type of non-linear neural network. We circumvent the notorious difficulty of tensor problems by adopting a dynamical systems perspective, and characterizing the evolution induced by gradient descent. The characterization suggests a form …
Queensland University of Technology（昆士兰科技大学）；Singapore University of Technology and Design（新加坡科技设计大学）
With the advancements in computing technology and web-based applications, data is increasingly generated in multi-dimensional form. This data is usually sparse due to the presence of a large number of users and fewer user interactions. To deal with this, the Nonnegative Tensor Factorization (NTF) based methods have been widely used. However existing factorization algorithms are not suitable to process in all three conditions of size, density, and rank of the tensor. Consequently, their applicability becomes limited. In this paper, we propose a novel fast and efficient NTF algorithm using the e…
Stability of Low-Rank Implicit Regularization in Perturbed Deep Matrix Factorization
Department of Informatics and Networked Systems, University of Pittsburgh（信息学与网络系统系，匹兹堡大学）；Department of Mathematics, University of Pittsburgh（数学系，匹兹堡大学）
Aix Marseille Univ（艾克斯-马赛大学）；Université de Toulon（土伦大学）；CNRS（法国国家科学研究中心）；LIS（信息与系统实验室）；Ecole Centrale de Marseille（马赛中央理工学院）
Attempts of studying implicit regularization associated to gradient descent (GD) have identified matrix completion as a suitable test-bed. Late findings suggest that this phenomenon cannot be phrased as a minimization-norm problem, implying that a paradigm shift is required and that dynamics has to be taken into account. In the present work we address the more general setup of tensor completion by leveraging two popularized tensor factorization, namely Tucker and TensorTrain (TT). We track relevant quantities such as tensor nuclear norm, effective rank, generalized singular values and we intro…
Albeit the implicit feedback based recommendation problem - when only the user history is available but there are no ratings - is the most typical setting in real-world applications, it is much less researched than the explicit feedback case. State-of-the-art algorithms that are efficient on the explicit case cannot be automatically transformed to the implicit case if scalability should be maintained. There are few implicit feedback benchmark data sets, therefore new ideas are usually experimented on explicit benchmarks. In this paper, we propose a generic context-aware implicit feedback recom…
We study the convergence of a variant of distributed gradient descent (DGD) on a distributed low-rank matrix approximation problem wherein some optimization variables are used for consensus (as in classical DGD) and some optimization variables appear only locally at a single node in the network. We term the resulting algorithm DGD+LOCAL. Using algorithmic connections to gradient descent and geometric connections to the well-behaved landscape of the centralized low-rank matrix approximation problem, we identify sufficient conditions where DGD+LOCAL is guaranteed to converge with exact consensus…
College of Computer Science, Zhejiang University（浙江大学计算机科学学院）；Zhejiang University-LianlianPay Joint Research Center（浙江大学-连连支付联合研究中心）
Recommendation from implicit feedback is a highly challenging task due to the lack of the reliable observed negative data. A popular and effective approach for implicit recommendation is to treat unobserved data as negative but downweight their confidence. Naturally, how to assign confidence weights and how to handle the large number of the unobserved data are two key problems for implicit recommendation models. However, existing methods either pursuit fast learning by manually assigning simple confidence weights, which lacks flexibility and may create empirical bias in evaluating user's prefe…
Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks). It is an open question whether norms can explain the implicit regularization in matrix factorization. The current paper resolves this open question in the negative, by proving that there exist natural matrix factorization problems on w…
The objective function of a matrix factorization model usually aims to minimize the average of a regression error contributed by each element. However, given the existence of stochastic noises, the implicit deviations of sample data from their true values are almost surely diverse, which makes each data point not equally suitable for fitting a model. In this case, simply averaging the cost among data in the objective function is not ideal. Intuitively we would like to emphasize more on the reliable instances (i.e., those contain smaller noise) while training a model. Motivated by such observat…
NTT Communication Science Laboratories（NTT通信科学实验室）
We propose a method that meta-learns a knowledge on matrix factorization from various matrices, and uses the knowledge for factorizing unseen matrices. The proposed method uses a neural network that takes a matrix as input, and generates prior distributions of factorized matrices of the given matrix. The neural network is meta-learned such that the expected imputation error is minimized when the factorized matrices are adapted to each matrix by a maximum a posteriori (MAP) estimation. We use a gradient descent method for the MAP estimation, which enables us to backpropagate the expected imputa…
Advanced Analytics Institute（高级分析研究所）；University of Technology, Sydney（悉尼科技大学）
The essence of the challenges cold start and sparsity in Recommender Systems (RS) is that the extant techniques, such as Collaborative Filtering (CF) and Matrix Factorization (MF), mainly rely on the user-item rating matrix, which sometimes is not informative enough for predicting recommendations. To solve these challenges, the objective item attributes are incorporated as complementary information. However, most of the existing methods for inferring the relationships between items assume that the attributes are "independently and identically distributed (iid)", which does not always hold in r…
University of California, Irvine（加州大学尔湾分校）；Google（谷歌）；Yahoo Labs（雅虎实验室）；University of Amsterdam（阿姆斯特丹大学）
Despite having various attractive qualities such as high prediction accuracy and the ability to quantify uncertainty and avoid over-fitting, Bayesian Matrix Factorization has not been widely adopted because of the prohibitive cost of inference. In this paper, we propose a scalable distributed Bayesian matrix factorization algorithm using stochastic gradient MCMC. Our algorithm, based on Distributed Stochastic Gradient Langevin Dynamics, can not only match the prediction accuracy of standard MCMC methods like Gibbs sampling, but at the same time is as fast and simple as stochastic gradient desc…
Traditional NMF-based signal decomposition relies on the factorization of spectral data, which is typically computed by means of short-time frequency transform. In this paper we propose to relax the choice of a pre-fixed transform and learn a short-time orthogonal transform together with the factorization. To this end, we formulate a regularized optimization problem reminiscent of conventional NMF, yet with the transform as additional unknown parameters, and design a novel block-descent algorithm enabling to find stationary points of this objective function. The proposed joint transform learni…
Boston University（波士顿大学）；Rice University（莱斯大学）；University of Maryland（马里兰大学）
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-$r$ positive semi-definite (PSD) matrices in $\mathbb{R}^{n \times n}$, with a sensing mechanism that satisfies restricted isometry properties (RIP). The algorithm we study is \emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization $UU^\top$, for $U \in \mathbb{R}^{n \times r}$. Surprisingly, recent work argues that the choice of $r \leq n$ is not pivotal: even setting $U …
The success of gradient descent in ML and especially for learning neural networks is remarkable and robust. In the context of how the brain learns, one aspect of gradient descent that appears biologically difficult to realize (if not implausible) is that its updates rely on feedback from later layers to earlier layers through the same connections. Such bidirected links are relatively few in brain networks, and even when reciprocal connections exist, they may not be equi-weighted. Random Feedback Alignment (Lillicrap et al., 2016), where the backward weights are random and fixed, has been propo…
Technische Universität Berlin（柏林工业大学）；Twitter（推特）；Stanford University（斯坦福大学）
We present Factorbird, a prototype of a parameter server approach for factorizing large matrices with Stochastic Gradient Descent-based algorithms. We designed Factorbird to meet the following desiderata: (a) scalability to tall and wide matrices with dozens of billions of non-zeros, (b) extensibility to different kinds of models and loss functions as long as they can be optimized using Stochastic Gradient Descent (SGD), and (c) adaptability to both batch and streaming scenarios. Factorbird uses a parameter server in order to scale to models that exceed the memory of an individual machine, and…
IRIT, Université de Toulouse, CNRS（图卢兹大学 IRIT 研究所、法国国家科学研究中心）；ISAE-SUPAERO, Université de Toulouse（图卢兹大学 ISAE-SUPAERO 高等航空与航天学院）
We introduce a new non-negative matrix factorization (NMF) method for ordinal data, called OrdNMF. Ordinal data are categorical data which exhibit a natural ordering between the categories. In particular, they can be found in recommender systems, either with explicit data (such as ratings) or implicit data (such as quantized play counts). OrdNMF is a probabilistic latent factor model that generalizes Bernoulli-Poisson factorization (BePoF) and Poisson factorization (PF) applied to binarized data. Contrary to these methods, OrdNMF circumvents binarization and can exploit a more informative repr…
Matrix factorization is one of the best approaches for collaborative filtering, because of its high accuracy in presenting users and items latent factors. The main disadvantages of matrix factorization are its complexity, and being very hard to be parallelized, specially with very large matrices. In this paper, we introduce a new method for collaborative filtering based on Matrix Factorization by combining simulated annealing with levy distribution. By using this method, good solutions are achieved in acceptable time with low computations, compared to other methods like stochastic gradient des…
Multi-label network classification is a well-known task that is being used in a wide variety of web-based and non-web-based domains. It can be formalized as a multi-relational learning task for predicting nodes labels based on their relations within the network. In sparse networks, this prediction task can be very challenging when only implicit feedback information is available such as in predicting user interests in social networks. Current approaches rely on learning per-node latent representations by utilizing the network structure, however, implicit feedback relations are naturally sparse …
Explaining the predictions of neural black-box models is an important problem, especially when such models are used in applications where user trust is crucial. Estimating the influence of training examples on a learned neural model's behavior allows us to identify training examples most responsible for a given prediction and, therefore, to faithfully explain the output of a black-box model. The most generally applicable existing method is based on influence functions, which scale poorly for larger sample sizes and models. We propose gradient rollback, a general approach for influence estimati…
We develop a Bayesian Poisson matrix factorization model for forming recommendations from sparse user behavior data. These data are large user/item matrices where each user has provided feedback on only a small subset of items, either explicitly (e.g., through star ratings) or implicitly (e.g., through views or purchases). In contrast to traditional matrix factorization approaches, Poisson factorization implicitly models each user's limited attention to consume items. Moreover, because of the mathematical form of the Poisson likelihood, the model needs only to explicitly consider the observed …
University of California（加利福尼亚大学）；University of Washington（华盛顿大学）
Matrix factorization is a well-studied task in machine learning for compactly representing large, noisy data. In our approach, instead of using the traditional concept of matrix rank, we define a new notion of link-rank based on a non-linear link function used within factorization. In particular, by applying the round function on a factorization to obtain ordinal-valued matrices, we introduce generalized round-rank (GRR). We show that not only are there many full-rank matrices that are low GRR, but further, that these matrices cannot be approximated well by low-rank linear factorization. We pr…
University of California, Santa Cruz（加利福尼亚大学圣克鲁兹分校）；Google Brain（谷歌大脑）
We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using the unprojected gradient. Following these observations, we derive a more \emph{implicit} form of Krasulina's $k$-PCA update, i.e. a version that uses the information of the future gradient as much as possible. Most interestingly, our implicit Krasulina update avoids the costly Q…
We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural networks, among others. Our approach is to directly estimate the low-rank factor by minimizing a nonconvex quadratic loss function via vanilla gradient descent, following a tailored spectral initialization. When the true rank is small, this algorithm is guaranteed to converge to the ground truth (up to global ambiguity) with near-optimal sample complexity and…
Sharif University of Technology（谢里夫理工大学）；Vienna University of Technology（维也纳工业大学）
Recommendation plays an increasingly important role in our daily lives. Recommender systems automatically suggest items to users that might be interesting for them. Recent studies illustrate that incorporating social trust in Matrix Factorization methods demonstrably improves accuracy of rating prediction. Such approaches mainly use the trust scores explicitly expressed by users. However, it is often challenging to have users provide explicit trust scores of each other. There exist quite a few works, which propose Trust Metrics to compute and predict trust scores between users based on their i…
As part of the effort to understand implicit bias of gradient descent in overparametrized models, several results have shown how the training trajectory on the overparametrized model can be understood as mirror descent on a different objective. The main result here is a characterization of this phenomenon under a notion termed commuting parametrization, which encompasses all the previous results in this setting. It is shown that gradient flow with any commuting parametrization is equivalent to continuous mirror descent with a related Legendre function. Conversely, continuous mirror descent wit…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
We present experiments demonstrating that some other form of capacity control, different from network size, plays a central role in learning multilayer feed-forward networks. We argue, partially through analogy to matrix factorization, that this is an inductive bias that can help shed light on deep learning.
It is well known that the finite step-size ($h$) in Gradient Descent (GD) implicitly regularizes solutions to flatter minima. A natural question to ask is "Does the momentum parameter $β$ play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?". To answer this question, first, we show that the discrete H.B momentum update (GD+M) follows a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. Then, we show that this implicit regularizer for (GD+M) is stronger than that of (GD) by factor …
University of Oxford（牛津大学）；Instadeep Ltd（Instadeep有限公司）；University of Southern California（南加州大学）
We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite $p-$variation with $p=2$. Proofs are based on non-asymptotic estimates for the loss function and for norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervis…
Implicit deep learning has recently become popular in the machine learning community since these implicit models can achieve competitive performance with state-of-the-art deep networks while using significantly less memory and computational resources. However, our theoretical understanding of when and how first-order methods such as gradient descent (GD) converge on \textit{nonlinear} implicit networks is limited. Although this type of problem has been studied in standard feed-forward networks, the case of implicit models is still intriguing because implicit networks have \textit{infinitely} m…
McGill University（麦吉尔大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
In this paper, we propose an implicit gradient descent algorithm for the classic $k$-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed stochastic backward Euler and the recent entropy stochastic gradient descent (Entropy-SGD) for improving the training of deep neural networks. Numerical experiments on various synthetic and real datas…
