A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval
Institute of Data Science（数据科学研究院）；National University of Singapore（新加坡国立大学）；Department of Electrical and Computer Engineering（电气与计算机工程系）；Department of Mathematics（数学系）
In this work, we study the robust phase retrieval problem where the task is to recover an unknown signal $θ^* \in \mathbb{R}^d$ in the presence of potentially arbitrarily corrupted magnitude-only linear measurements. We propose an alternating minimization approach that incorporates an oracle solver for a non-convex optimization problem as a subroutine. Our algorithm guarantees convergence to $θ^*$ and provides an explicit polynomial dependence of the convergence rate on the fraction of corrupted measurements. We then provide an efficient construction of the aforementioned oracle under a sparse arbitrary outliers model and offer valuable insights into the geometric properties of the loss landscape in phase retrieval with corrupted measurements. Our proposed oracle avoids the need for computationally intensive spectral initialization, using a simple gradient descent algorithm with a constant step size and random initialization instead. Additionally, our overall algorithm achieves nearly linear sample complexity, $\mathcal{O}(d \, \mathrm{polylog}(d))$.
In recent years, the Deep Learning Alternating Minimization (DLAM), which is actually the alternating minimization applied to the penalty form of the deep neutral networks training, has been developed as an alternative algorithm to overcome several drawbacks of Stochastic Gradient Descent (SGD) algorithms. This work develops an improved DLAM by the well-known inertial technique, namely iPDLAM, which predicts a point by linearization of current and last iterates. To obtain further training speed, we apply a warm-up technique to the penalty parameter, that is, starting with a small initial one a…
We present theoretical guarantees for an alternating minimization algorithm for the dictionary learning/sparse coding problem. The dictionary learning problem is to factorize vector samples $y^{1},y^{2},\ldots, y^{n}$ into an appropriate basis (dictionary) $A^*$ and sparse vectors $x^{1*},\ldots,x^{n*}$. Our algorithm is a simple alternating minimization procedure that switches between $\ell_1$ minimization and gradient descent in alternate steps. Dictionary learning and specifically alternating minimization algorithms for dictionary learning are well studied both theoretically and empirically…
We study alternating minimization for matrix completion in the simplest possible setting: completing a rank-one matrix from a revealed subset of the entries. We bound the asymptotic convergence rate by the variational characterization of eigenvalues of a reversible consensus problem. This leads to a polynomial upper bound on the asymptotic rate in terms of number of nodes as well as the largest degree of the graph of revealed entries.
Mean-Field is an efficient way to approximate a posterior distribution in complex graphical models and constitutes the most popular class of Bayesian variational approximation methods. In most applications, the mean field distribution parameters are computed using an alternate coordinate minimization. However, the convergence properties of this algorithm remain unclear. In this paper, we show how, by adding an appropriate penalization term, we can guarantee convergence to a critical point, while keeping a closed form update at each step. A convergence rate estimate can also be derived based on…
Many applications require recovering a ground truth low-rank matrix from noisy observations of the entries, which in practice is typically formulated as a weighted low-rank approximation problem and solved by non-convex optimization heuristics such as alternating minimization. In this paper, we provide provable recovery guarantee of weighted low-rank via a simple alternating minimization algorithm. In particular, for a natural class of matrices and weights and without any assumption on the noise, we bound the spectral norm of the difference between the recovered matrix and the ground truth, by…
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机研究所）
Alternating minimization (AM) procedures are practically efficient in many applications for solving convex and non-convex optimization problems. On the other hand, Nesterov's accelerated gradient is theoretically optimal first-order method for convex optimization. In this paper we combine AM and Nesterov's acceleration to propose an accelerated alternating minimization algorithm. We prove $1/k^2$ convergence rate in terms of the objective for convex problems and $1/k$ in terms of the squared gradient norm for non-convex problems, where $k$ is the iteration counter. Our method does not require …
This paper deals with unsupervised clustering with feature selection. The problem is to estimate both labels and a sparse projection matrix of weights. To address this combinatorial non-convex problem maintaining a strict control on the sparsity of the matrix of weights, we propose an alternating minimization of the Frobenius norm criterion. We provide a new efficient algorithm named K-sparse which alternates k-means with projection-gradient minimization. The projection-gradient step is a method of splitting type, with exact projection on the $\ell^1$ ball to promote sparsity. The convergence …
Convergence Rates of Training Deep Neural Networks via Alternating Minimization Methods
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels (which sample corresponds to which model) are not observed. We give a tractable algorithm for the mixed linear equation problem, and show that under some technical conditions, our algorithm is guaranteed to solve the problem exactly with sample complexity linear in the dimension, and polynomial in $k$, the number of components. Previous approaches have requir…
Microsoft Research India（微软印度研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. More than four decades after it was first proposed, the seminal error reduction algorithm of (Gerchberg and Saxton 1972) and (Fienup 1982) is still the popular choice for solving many variants of this problem. The algorithm is based on alternating minimization; i.e. it alternates between estimating the missing phase information, and the candidate solution. Despite its wide usage in practice, no global convergence guarantees for this algorithm are known. In this paper, w…
National University of Defense Technology（国防科技大学）；Beihang University（北京航空航天大学）；Imperial College London（帝国理工学院）；University of Modena and Reggio Emilia (UNIMORE)（摩德纳-雷焦艾米利亚大学（UNIMORE））
In this paper, we propose a novel solution for non-convex problems of multiple variables, especially for those typically solved by an alternating minimization (AM) strategy that splits the original optimization problem into a set of sub-problems corresponding to each variable, and then iteratively optimize each sub-problem using a fixed updating rule. However, due to the intrinsic non-convexity of the original optimization problem, the optimization can usually be trapped into spurious local minimum even when each sub-problem can be optimally solved at each iteration. Meanwhile, learning-based …
Alternating minimization methods have recently been proposed as alternatives to the gradient descent for deep neural network optimization. Alternating minimization methods can typically decompose a deep neural network into layerwise subproblems, which can then be optimized in parallel. Despite the significant parallelism, alternating minimization methods are rarely explored in training deep neural networks because of the severe accuracy degradation. In this paper, we analyze the reason and propose to achieve a compelling trade-off between parallelism and accuracy by a reformulation called Tuna…
Alternating minimization algorithms for graph regularized tensor completion
Huawei European Research Institute（华为欧洲研究院）；LISN, INRIA, Université Paris-Saclay（法国国家信息与自动化研究所，巴黎萨克雷大学）；LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院科学与工程计算国家重点实验室）；Department of INMA, Université catholique de Louvain（鲁汶天主教大学）
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
CP decomposition (CPD) is prevalent in chemometrics, signal processing, data mining and many more fields. While many algorithms have been proposed to compute the CPD, alternating least squares (ALS) remains one of the most widely used algorithm for computing the decomposition. Recent works have introduced the notion of eigenvalues and singular values of a tensor and explored applications of eigenvectors and singular vectors in areas like signal processing, data analytics and in various other fields. We introduce a new formulation for deriving singular values and vectors of a tensor by consider…
University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）；Umeå University（于默奥大学）
We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second…
University of Michigan（密歇根大学）；Claremont Graduate University（克莱蒙特研究生大学）；University of California Los Angeles（加利福尼亚大学洛杉矶分校）
Methods exploiting sparsity have been popular in imaging and signal processing applications including compression, denoising, and imaging inverse problems. Data-driven approaches such as dictionary learning and transform learning enable one to discover complex image features from datasets and provide promising performance over analytical models. Alternating minimization algorithms have been particularly popular in dictionary or transform learning. In this work, we study the properties of alternating minimization for structured (unitary) sparsifying operator learning. While the algorithm conver…
Microsoft Research India（微软印度研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Alternating minimization represents a widely applicable and empirically successful approach for finding low-rank matrices that best fit the given data. For example, for the problem of low-rank matrix completion, this method is believed to be one of the most accurate and efficient, and formed a major component of the winning entry in the Netflix Challenge. In the alternating minimization approach, the low-rank target matrix is written in a bi-linear form, i.e. $X = UV^†$; the algorithm then alternates between finding the best $U$ and the best $V$. Typically, each alternating step in isolation i…
Nowadays, the availability of large-scale data in disparate application domains urges the deployment of sophisticated tools for extracting valuable knowledge out of this huge bulk of information. In that vein, low-rank representations (LRRs) which seek low-dimensional embeddings of data have naturally appeared. In an effort to reduce computational complexity and improve estimation performance, LRR has been viewed via a matrix factorization (MF) perspective. Recently, low-rank MF (LRMF) approaches have been proposed for tackling the inherent weakness of MF i.e., the unawareness of the dimension…
Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation
Simons Institute for the Theory of Computing, UC Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；University of Illinois, Chicago（伊利诺伊大学芝加哥分校）；Rice University（莱斯大学）；Massachusetts Institute of Technology（麻省理工学院）
University of Hong Kong（香港大学）；University of New South Wales（新南威尔士大学）；Southern University of Science and Technology（南方科技大学）
Wireless federated learning (FL) is an emerging machine learning paradigm that trains a global parametric model from distributed datasets via wireless communications. This paper proposes a unit-modulus wireless FL (UMWFL) framework, which simultaneously uploads local model parameters and computes global model parameters via optimized phase shifting. The proposed framework avoids sophisticated baseband signal processing, leading to both low communication delays and implementation costs. A training loss bound is derived and a penalty alternating minimization (PAM) algorithm is proposed to minimi…
Microsoft Research（微软研究院）；University of California, Irvine（加州大学欧文分校）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）
We consider the problem of sparse coding, where each sample consists of a sparse linear combination of a set of dictionary atoms, and the task is to learn both the dictionary elements and the mixing coefficients. Alternating minimization is a popular heuristic for sparse coding, where the dictionary and the coefficients are estimated in alternate steps, keeping the other fixed. Typically, the coefficients are estimated via $\ell_1$ minimization, keeping the dictionary fixed, and the dictionary is estimated through least squares, keeping the coefficients fixed. In this paper, we establish local…
Softening labels of training datasets with respect to data representations has been frequently used to improve the training of deep neural networks (DNNs). While such a practice has been studied as a way to leverage privileged information about the distribution of the data, a well-trained learner with soft classification outputs should be first obtained as a prior to generate such privileged information. To solve such chicken-egg problem, we propose COLAM framework that Co-Learns DNNs and soft labels through Alternating Minimization of two objectives - (a) the training loss subject to soft lab…
The paper considers a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be partitioned into groups of similar networks, and networks in each group are equipped with a distinct Stochastic Block Model. The goal is to partition the multilayer network into clusters of similar layers, and to identify communities in those layers. Jing et al. (2020) introduced the MMLSBM and developed a clustering methodology, TWIST, based on regularized tensor decomposition. The present paper proposes a different technique, an alternating minimization algorithm (ALMA), that aims at simultaneous re…
We propose and analyze a new family of algorithms for training neural networks with ReLU activations. Our algorithms are based on the technique of alternating minimization: estimating the activation patterns of each ReLU for all given samples, interleaved with weight updates via a least-squares step. The main focus of our paper are 1-hidden layer networks with $k$ hidden neurons and ReLU activation. We show that under standard distributional assumptions on the $d-$dimensional input data, our algorithm provably recovers the true `ground truth' parameters in a linearly convergent fashion. This h…
NYU Tandon（纽约大学坦登工程学院）；IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）；MIT（麻省理工学院）
Despite significant recent advances in deep neural networks, training them remains a challenge due to the highly non-convex nature of the objective function. State-of-the-art methods rely on error backpropagation, which suffers from several well-known issues, such as vanishing and exploding gradients, inability to handle non-differentiable nonlinearities and to parallelize weight-updates across layers, and biological implausibility. These limitations continue to motivate exploration of alternative training algorithms, including several recently proposed auxiliary-variable methods which break t…
Mining and exploring databases should provide users with knowledge and new insights. Tiles of data strive to unveil true underlying structure and distinguish valuable information from various kinds of noise. We propose a novel Boolean matrix factorization algorithm to solve the tiling problem, based on recent results from optimization theory. In contrast to existing work, the new algorithm minimizes the description length of the resulting factorization. This approach is well known for model selection and data compression, but not for finding suitable factorizations via numerical optimization. …
We address the problem of solving mixed random linear equations. We have unlabeled observations coming from multiple linear regressions, and each observation corresponds to exactly one of the regression models. The goal is to learn the linear regressors from the observations. Classically, Alternating Minimization (AM) (which is a variant of Expectation Maximization (EM)) is used to solve this problem. AM iteratively alternates between the estimation of labels and solving the regression problems with the estimated labels. Empirically, it is observed that, for a large variety of non-convex probl…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Google Research（谷歌研究院）；University of Washington（华盛顿大学）
Meta-learning synthesizes and leverages the knowledge from a given set of tasks to rapidly learn new tasks using very little data. Meta-learning of linear regression tasks, where the regressors lie in a low-dimensional subspace, is an extensively-studied fundamental problem in this domain. However, existing results either guarantee highly suboptimal estimation errors, or require $Ω(d)$ samples per task (where $d$ is the data dimensionality) thus providing little gain over separately learning each task. In this work, we study a simple alternating minimization method (MLLAM), which alternately l…
The Hong Kong University of Science and Technology（香港科技大学）；Zhongnan University of Economics and Law（中南财经政法大学）；Wuhan University（武汉大学）
Sparse phase retrieval plays an important role in many fields of applied science and thus attracts lots of attention. In this paper, we propose a \underline{sto}chastic alte\underline{r}nating \underline{m}inimizing method for \underline{sp}arse ph\underline{a}se \underline{r}etrieval (\textit{StormSpar}) algorithm which {emprically} is able to recover $n$-dimensional $s$-sparse signals from only $O(s\,\mathrm{log}\, n)$ number of measurements without a desired initial value required by many existing methods. In \textit{StormSpar}, the hard-thresholding pursuit (HTP) algorithm is employed to s…
Deeply unrolled alternating projections for phase retrieval
University of Washington（华盛顿大学）；NSF-Simons National Institute for Theory and Mathematics in Biology（NSF-Simons生物理论与数学国家研究所）；Department of Computer Science, University of Chicago（芝加哥大学计算机科学系）；Committee on Computational and Applied Mathematics, University of Chicago（芝加哥大学计算与应用数学委员会）；Data Science Institute, University of Chicago（芝加哥大学数据科学研究所）；NSF-Simons AI-Institute for the Sky (SkAI)（NSF-Simons天空人工智能研究所）
The low-tubal-rank tensor model has been recently proposed for real-world multidimensional data. In this paper, we study the low-tubal-rank tensor completion problem, i.e., to recover a third-order tensor by observing a subset of its elements selected uniformly at random. We propose a fast iterative algorithm, called {\em Tubal-Alt-Min}, that is inspired by a similar approach for low-rank matrix completion. The unknown low-tubal-rank tensor is represented as the product of two much smaller tensors with the low-tubal-rank property being automatically incorporated, and Tubal-Alt-Min alternates b…
Colorado School of Mines（科罗拉多矿业学院）；Johns Hopkins University（约翰斯·霍普金斯大学）
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothness condition, which is satisfied by any finite-order polynomial objective function. Further, this work develops new Bregman-divergence based algorithms that are guaranteed to converge to a second-order stationary point for any relatively smooth problem. In addition, the proposed…
We propose Noise-Augmented Privacy-Preserving Empirical Risk Minimization (NAPP-ERM) that solves ERM with differential privacy guarantees. Existing privacy-preserving ERM approaches may be subject to over-regularization with the employment of an l2 term to achieve strong convexity on top of the target regularization. NAPP-ERM improves over the current approaches and mitigates over-regularization by iteratively realizing target regularization through appropriately designed augmented data and delivering strong convexity via a single adaptively weighted dual-purpose l2 regularizer. When the targe…
