Scaling up model sizes can lead to fundamentally new capabilities in many machine learning (ML) tasks. However, training big models requires strong distributed system expertise to carefully design model-parallel execution strategies that suit the model architectures and cluster setups. In this paper, we develop AMP, a framework that automatically derives such strategies. AMP identifies a valid space of model parallelism strategies and efficiently searches the space for high-performed strategies, by leveraging a cost model designed to capture the heterogeneity of the model and cluster specifications. Unlike existing methods, AMP is specifically tailored to support complex models composed of uneven layers and cluster setups with more heterogeneous accelerators and bandwidth. We evaluate AMP on popular modelsand cluster setups from public clouds and show that AMP returns parallel strategies that match the expert-tuned strategies on typical cluster setups. On heterogeneous clusters or models with heterogeneous architectures, AMP finds strategies with 1.54$\times$ and 1.77$\times$ higher throughput than state-of-the-art model-parallel systems, respectively.

Semi-supervised learning (SSL) is a machine learning methodology that leverages unlabeled data in conjunction with a limited amount of labeled data. Although SSL has been applied in various applications and its effectiveness has been empirically demonstrated, it is still not fully understood when and why SSL performs well. Some existing theoretical studies have attempted to address this issue by modeling classification problems using the so-called Gaussian Mixture Model (GMM). These studies provide notable and insightful interpretations. However, their analyses are focused on specific purposes, and a thorough investigation of the properties of GMM in the context of SSL has been lacking. In this paper, we conduct such a detailed analysis of the properties of the high-dimensional GMM for binary classification in the SSL setting. To this end, we employ the approximate message passing and state evolution methods, which are widely used in high-dimensional settings and originate from statistical mechanics. We deal with two estimation approaches: the Bayesian one and the l2-regularized maximum likelihood estimation (RMLE). We conduct a comprehensive comparison between these two approaches, examining aspects such as the global phase diagram, estimation error for the parameters, and prediction error for the labels. A specific comparison is made between the Bayes-optimal (BO) estimator and RMLE, as the BO setting provides optimal estimation performance and is ideal as a benchmark. Our analysis shows that with appropriate regularizations, RMLE can achieve near-optimal performance in terms of both the estimation error and prediction error, especially when there is a large amount of unlabeled data. These results demonstrate that the l2 regularization term plays an effective role in estimation and prediction in SSL approaches.

The committee machine is a simple and natural model for a 2-layer neural network. The results of the paper also apply to many related models: you are allowed an arbitrary function mapping the K hidden values to the final binary output.) This paper studies the problem of learning the weights W under a natural random model. We are given m random examples (X,Y) where the input X (in R n) is iid Gaussian and Y (in { 1,-1}) is the associated output of the network. The unknown weights W are iid from a known prior.

Factorizing low-rank matrices has many applications in machine learning and statistics. For probabilistic models in the Bayes optimal setting, a general expression for the mutual information has been proposed using heuristic statistical physics computations, and proven in few specific cases. Here, we show how to rigorously prove the conjectured formula for the symmetric rank-one case. This allows to express the minimal mean-square-error and to characterize the detectability phase transitions in a large set of estimation problems ranging from community detection to sparse PCA. We also show that for a large set of parameters, an iterative algorithm called approximate message-passing is Bayes optimal. There exists, however, a gap between what currently known polynomial algorithms can do and what is expected information theoretically. Additionally, the proof technique has an interest of its own and exploits three essential ingredients: the interpolation method introduced in statistical physics by Guerra, the analysis of the approximate message-passing algorithm and the theory of spatial coupling and threshold saturation in coding. Our approach is generic and applicable to other open problems in statistical estimation where heuristic statistical physics predictions are available.

Competition – it motivates all of us to become better at what we do. Marketers are no different, especially when they have to stay on top of the SEO game. They research the latest SEO trends; they use the best SEO tools out there. The problem is, unlike a sports competition, the goal posts for SEO are always changing. This, of course, makes the game more interesting, but also pretty frustrating for those trying to "keep up."

SEO is a constantly changing and growing industry. No longer is search engine optimization seen as internet "black magic," but it is now regarded as an essential part of any serious digital marketing strategy. Last year, it was estimated that businesses invested more that $65 billion on SEO services, and that number is projected to climb to over $70 billion by 2018. We've come a long way as an industry -- and from the looks of it, our best days are still ahead of us. The hardest thing in the world of search is predicting what will come next.

We consider the problem of recovering a vector $\beta_o \in \mathbb{R}^p$ from $n$ random and noisy linear observations $y= X\beta_o + w$, where $X$ is the measurement matrix and $w$ is noise. The LASSO estimate is given by the solution to the optimization problem $\hat{\beta}_{\lambda} = \arg \min_{\beta} \frac{1}{2} \|y-X\beta\|_2^2 + \lambda \| \beta \|_1$. Among the iterative algorithms that have been proposed for solving this optimization problem, approximate message passing (AMP) has attracted attention for its fast convergence. Despite significant progress in the theoretical analysis of the estimates of LASSO and AMP, little is known about their behavior as a function of the regularization parameter $\lambda$, or the thereshold parameters $\tau^t$. For instance the following basic questions have not yet been studied in the literature: (i) How does the size of the active set $\|\hat{\beta}^\lambda\|_0/p$ behave as a function of $\lambda$? (ii) How does the mean square error $\|\hat{\beta}_{\lambda} - \beta_o\|_2^2/p$ behave as a function of $\lambda$? (iii) How does $\|\beta^t - \beta_o \|_2^2/p$ behave as a function of $\tau^1, \ldots, \tau^{t-1}$? Answering these questions will help in addressing practical challenges regarding the optimal tuning of $\lambda$ or $\tau^1, \tau^2, \ldots$. This paper answers these questions in the asymptotic setting and shows how these results can be employed in deriving simple and theoretically optimal approaches for tuning the parameters $\tau^1, \ldots, \tau^t$ for AMP or $\lambda$ for LASSO. It also explores the connection between the optimal tuning of the parameters of AMP and the optimal tuning of LASSO.

We study optimal estimation for sparse principal component analysis when the number of non-zero elements is small but on the same order as the dimension of the data. We employ approximate message passing (AMP) algorithm and its state evolution to analyze what is the information theoretically minimal mean-squared error and the one achieved by AMP in the limit of large sizes. For a special case of rank one and large enough density of non-zeros Deshpande and Montanari [1] proved that AMP is asymptotically optimal. We show that both for low density and for large rank the problem undergoes a series of phase transitions suggesting existence of a region of parameters where estimation is information theoretically possible, but AMP (and presumably every other polynomial algorithm) fails. The analysis of the large rank limit is particularly instructive.

We consider the Principal Component Analysis problem for large tensors of arbitrary orderk under a single-spike (or rank-one plus noise) model. On the one hand, we use information theory, and recent results in probability theory, to establish necessaryand sufficient conditions under which the principal component can be estimated using unbounded computational resources. It turns out that this is possible as soon as the signal-to-noise ratio β becomes larger than C k log k (and in particular β can remain bounded as the problem dimensions increase). On the other hand, we analyze several polynomial-time estimation algorithms, based on tensor unfolding, power iteration and message passing ideas from graphical models.We show that, unless the signal-to-noise ratio diverges in the system dimensions, none of these approaches succeeds. This is possibly related to a fundamental limitationof computationally tractable estimators for this problem. We discuss various initializations for tensor power iteration, and show that a tractable initialization based on the spectrum of the unfolded tensor outperforms significantly baseline methods, statistically and computationally. Finally, we consider thecase in which additional side information is available about the unknown signal. We characterize the amount of side information that allows the iterative algorithms to converge to a good estimate.

We propose an artificial immune model for intrusion detection in distributed systems based on a relatively recent theory in immunology called Danger theory. Based on Danger theory, immune response in natural systems is a result of sensing corruption as well as sensing unknown substances. In contrast, traditional self-nonself discrimination theory states that immune response is only initiated by sensing nonself (unknown) patterns. Danger theory solves many problems that could only be partially explained by the traditional model. Although the traditional model is simpler, such problems result in high false positive rates in immune-inspired intrusion detection systems. We believe using danger theory in a multi-agent environment that computationally emulates the behavior of natural immune systems is effective in reducing false positive rates. We first describe a simplified scenario of immune response in natural systems based on danger theory and then, convert it to a computational model as a network protocol. In our protocol, we define several immune signals and model cell signaling via message passing between agents that emulate cells. Most messages include application-specific patterns that must be meaningfully extracted from various system properties. We show how to model these messages in practice by performing a case study on the problem of detecting distributed denial-of-service attacks in wireless sensor networks. We conduct a set of systematic experiments to find a set of performance metrics that can accurately distinguish malicious patterns. The results indicate that the system can be efficiently used to detect malicious patterns with a high level of accuracy.