Online retailers execute a very large number of price updates when compared to brick-and-mortar stores. Even a few mis-priced items can have a significant business impact and result in a loss of customer trust. Early detection of anomalies in an automated real-time fashion is an important part of such a pricing system. In this paper, we describe unsupervised and supervised anomaly detection approaches we developed and deployed for a large-scale online pricing system at Walmart. Our system detects anomalies both in batch and real-time streaming settings, and the items flagged are reviewed and actioned based on priority and business impact. We found that having the right architecture design was critical to facilitate model performance at scale, and business impact and speed were important factors influencing model selection, parameter choice, and prioritization in a production environment for a large-scale system. We conducted analyses on the performance of various approaches on a test set using real-world retail data and fully deployed our approach into production. We found that our approach was able to detect the most important anomalies with high precision.

The l1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix even under high-dimensional settings. However, it requires solving a difficult non-smooth log-determinant program with number of parameters scaling quadratically with the number of Gaussian variables. State-of-the-art methods thus do not scale to problems with more than 20,000 variables. In this paper, we develop an algorithm BigQUIC, which can solve 1 million dimensional l1-regularized Gaussian MLE problems (which would thus have 1000 billion parameters) using a single machine, with bounded memory. In order to do so, we carefully exploit the underlying structure of the problem. Our innovations include a novel block-coordinate descent method with the blocks chosen via a clustering scheme to minimize repeated computations; and allowing for inexact computation of specific components. In spite of these modifications, we are able to theoretically analyze our procedure and show that BigQUIC can achieve super-linear or even quadratic convergence rates.

One of the enduring challenges in Markov chain Monte Carlo methodology is the development of proposal mechanisms to make moves distant from the current point, that are accepted with high probability and at low computational cost. The recent introduction of locally adaptive MCMC methods based on the natural underlying Riemannian geometry of such models goes some way to alleviating these problems for certain classes of models for which the metric tensor is analytically tractable, however computational efficiency is not assured due to the necessity of potentially high-dimensional matrix operations at each iteration. In this paper we firstly investigate a sampling-based approach for approximating the metric tensor and suggest a valid MCMC algorithm that extends the applicability of Riemannian Manifold MCMC methods to statistical models that do not admit an analytically computable metric tensor. Secondly, we show how the approximation scheme we consider naturally motivates the use of l1 regularisation to improve estimates and obtain a sparse approximate inverse of the metric, which enables stable and sparse approximations of the local geometry to be made. We demonstrate the application of this algorithm for inferring the parameters of a realistic system of ordinary differential equations using a biologically motivated robust student-t error model, for which the expected Fisher Information is analytically intractable.

We introduce a method of partitioning for massively-parallel hardware accelerated functional verification. Our approach augments classical hypergraph partitioning to model temporal dependencies that maximize parallelization within the instruction memories of the machine. Simulation depth is further reduced by optimizing path criticality and cut directionality. Our techniques are demonstrated on an industrial accelerator containing 262,144 parallel processors, and benchmarked across designs containing up to 200 million gates.

The L_1 regularized Gaussian maximum likelihood estimator has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to other state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modifications that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and also present experimental results using synthetic and real application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods.

Performance models are typically written by hand for a new model or assembled piece-meal from the prior simulation code of an old model. In either case, many man-months of work may be required to write the new model and validate design details against a prior or current design. In reality, the majority of information about the performance of the design already exists in the design structure of either the old hardware model or the new model or both. To harvest this information and eliminate the significant duplicate coding and validation efforts, we propose that a performance model be automatically synthesized from a prior or current hardware design using a bottom-up, design-oriented approach. We demarcate the performance-critical boundaries of the design and perform backward-trace cone analysis to identify logic to include in the performance model. We then abstract specific components for design changes and expend modeling effort only on the few functions relevant to a particular design study. Engineering effort then becomes focused on workload selection and quality, defining and projecting new designs, and assessing design tradeoffs and sensitivities - the small set of tasks with the highest potential to improve design performance.

Kernel learning plays an important role in many machine learning tasks. However, algorithms for learning a kernel matrix often scale poorly, with running times that are cubic in the number of data points. In this paper, we propose efficient algorithms for learning low-rank kernel matrices; our algorithms scale linearly in the number of data points and quadratically in the rank of the kernel. We introduce and employ Bregman matrix divergences for rank-deficient matrices---these divergences are natural for our problem since they preserve the rank as well as positive semi-definiteness of the kernel matrix. Special cases of our framework yield faster algorithms for various existing kernel learning problems. Experimental results demonstrate the effectiveness of our algorithms in learning both low-rank and full-rank kernels.