Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of $m$ measurements, $y_k = |\mathbf a_k^* \mathbf x|$ for $k = 1, \dots, m$, is it possible to recover $\mathbf x \in \mathbb{C}^n$ (i.e., length-$n$ complex vector)? This **generalized phase retrieval** (GPR) problem is a fundamental task in various disciplines, and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretical explanations. In this paper, we take a step towards bridging this gap. We prove that when the measurement vectors $\mathbf a_k$'s are generic (i.i.d. complex Gaussian) and the number of measurements is large enough ($m \ge C n \log^3 n$), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) there are no spurious local minimizers, and all global minimizers are equal to the target signal $\mathbf x$, up to a global phase; and (2) the objective function has a negative curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.

This article is Part VI in a series looking at data science and machine learning by walking through a Kaggle competition. If you have not done so already, you are strongly encouraged to go back and read the earlier parts – (Part I, Part II, Part III, Part IV and Part V). Continuing on the walkthrough, in this part we build the model that will predict the first booking destination country for each user based on the dataset created in the earlier parts. The first step to building a model is to decide what type of algorithm to use. Below we look at some of the options.

Markov Chain Monte Carlo (MCMC) and Belief Propagation (BP) are the most popular algorithms for computational inference in Graphical Models (GM). In principle, MCMC is an exact probabilistic method which, however, often suffers from exponentially slow mixing. In contrast, BP is a deterministic method, which is typically fast, empirically very successful, however in general lacking control of accuracy over loopy graphs. In this paper, we introduce MCMC algorithms correcting the approximation error of BP, i.e., we provide a way to compensate for BP errors via a consecutive BP-aware MCMC. Our framework is based on the Loop Calculus (LC) approach which allows to express the BP error as a sum of weighted generalized loops. Although the full series is computationally intractable, it is known that a truncated series, summing up all 2-regular loops, is computable in polynomial-time for planar pair-wise binary GMs and it also provides a highly accurate approximation empirically. Motivated by this, we, first, propose a polynomial-time approximation MCMC scheme for the truncated series of general (non-planar) pair-wise binary models. Our main idea here is to use the Worm algorithm, known to provide fast mixing in other (related) problems, and then design an appropriate rejection scheme to sample 2-regular loops. Furthermore, we also design an efficient rejection-free MCMC scheme for approximating the full series. The main novelty underlying our design is in utilizing the concept of cycle basis, which provides an efficient decomposition of the generalized loops. In essence, the proposed MCMC schemes run on transformed GM built upon the non-trivial BP solution, and our experiments show that this synthesis of BP and MCMC outperforms both direct MCMC and bare BP schemes.

In this paper, we address the problems of identifying linear structural equation models and discovering the constraints they imply. We first extend the half-trek criterion to cover a broader class of models and apply our extension to finding testable constraints implied by the model. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification and constraint discovery power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification and constraint discovery algorithms for non-parametric models.

One of the major issues in stochastic gradient descent (SGD) methods is how to choose an appropriate step size while running the algorithm. Since the traditional line search technique does not apply for stochastic optimization methods, the common practice in SGD is either to use a diminishing step size, or to tune a step size by hand, which can be time consuming in practice. In this paper, we propose to use the Barzilai-Borwein (BB) method to automatically compute step sizes for SGD and its variant: stochastic variance reduced gradient (SVRG) method, which leads to two algorithms: SGD-BB and SVRG-BB. We prove that SVRG-BB converges linearly for strongly convex objective functions. As a by-product, we prove the linear convergence result of SVRG with Option I proposed in [10], whose convergence result has been missing in the literature. Numerical experiments on standard data sets show that the performance of SGD-BB and SVRG-BB is comparable to and sometimes even better than SGD and SVRG with best-tuned step sizes, and is superior to some advanced SGD variants.

Tensor networks are approximations of high-order tensors which are efficient to work with and have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing tensor networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize non-linear kernel learning models. For the MNIST data set we obtain less than 1% test set classification error. We discuss an interpretation of the additional structure imparted by the tensor network to the learned model.

We develop a scalable, computationally efficient method for the task of energy disaggregation for home appliance monitoring. In this problem the goal is to estimate the energy consumption of each appliance based on the total energy-consumption signal of a household. The current state of the art models the problem as inference in factorial HMMs, and finds an approximate solution to the resulting quadratic integer program via quadratic programming. Here we take a more principled approach, better suited to integer programming problems, and find an approximate optimum by combining convex semidefinite relaxations with randomized rounding, as well as with a scalable ADMM method that exploits the special structure of the resulting semidefinite program. Simulation results demonstrate the superiority of our methods both in synthetic and real-world datasets.

In recent years, a rapidly increasing number of applications in practice requires solving non-convex objectives, like training neural networks, learning graphical models, maximum likelihood estimation etc. Though simple heuristics such as gradient descent with very few modifications tend to work well, theoretical understanding is very weak. We consider possibly the most natural class of non-convex functions where one could hope to obtain provable guarantees: functions that are ``approximately convex'', i.e. functions $\tf: \Real^d \to \Real$ for which there exists a \emph{convex function} $f$ such that for all $x$, $|\tf(x) - f(x)| \le \errnoise$ for a fixed value $\errnoise$. We then want to minimize $\tf$, i.e. output a point $\tx$ such that $\tf(\tx) \le \min_{x} \tf(x) + \err$. It is quite natural to conjecture that for fixed $\err$, the problem gets harder for larger $\errnoise$, however, the exact dependency of $\err$ and $\errnoise$ is not known. In this paper, we strengthen the known \emph{information theoretic} lower bounds on the trade-off between $\err$ and $\errnoise$ substantially, and exhibit an algorithm that matches these lower bounds for a large class of convex bodies.

The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family has been very popular in machine learning due to its “Occam’s razor” interpretation. Unfortunately, calculating the potentials in the maximum entropy distribution is intractable [BGS14]. We provide computationally efficient versions of this principle when the mean parameters are pairwise moments: we design distributions that approximately match given pairwise moments, while having entropy which is comparable to the maximum entropy distribution matching those moments. We additionally provide surprising applications of the approximate maximum entropy principle to designing provable variational methods for partition function calculations for Ising models without any assumptions on the potentials of the model. More precisely, we show that we can get approximation guarantees for the log-partition function comparable to those in the low-temperature limit, which is the setting of optimization of quadratic forms over the hypercube. ([AN06])

One crucial goal in kernel online learning is to bound the model size. Common approaches employ budget maintenance procedures to restrict the model sizes using removal, projection, or merging strategies. Although projection and merging, in the literature, are known to be the most effective strategies, they demand extensive computation whilst removal strategy fails to retain information of the removed vectors. An alternative way to address the model size problem is to apply random features to approximate the kernel function. This allows the model to be maintained directly in the random feature space, hence effectively resolve the curse of kernelization. However, this approach still suffers from a serious shortcoming as it needs to use a high dimensional random feature space to achieve a sufficiently accurate kernel approximation. Consequently, it leads to a significant increase in the computational cost. To address all of these aforementioned challenges, we present in this paper the Dual Space Gradient Descent (DualSGD), a novel framework that utilizes random features as an auxiliary space to maintain information from data points removed during budget maintenance. Consequently, our approach permits the budget to be maintained in a simple, direct and elegant way while simultaneously mitigating the impact of the dimensionality issue on learning performance. We further provide convergence analysis and extensively conduct experiments on five real-world datasets to demonstrate the predictive performance and scalability of our proposed method in comparison with the state-of-the-art baselines.