We present a novel algorithm (Principal Sensitivity Analysis; PSA) to analyze the knowledge of the classifier obtained from supervised machine learning techniques. In particular, we define principal sensitivity map (PSM) as the direction on the input space to which the trained classifier is most sensitive, and use analogously defined k -th PSM to define a basis for the input space. We train neural networks with artificial data and real data, and apply the algorithm to the obtained supervised classifiers. We then visualize the PSMs to demonstrate the PSA's ability to decompose the knowledge acquired by the trained classifiers.

The recently proposed Minimal Complexity Machine (MCM) finds a hyperplane classifier by minimizing an exact bound on the Vapnik-Chervonenkis (VC) dimension. The VC dimension measures the capacity of a learning machine, and a smaller VC dimension leads to improved generalization. On many benchmark datasets, the MCM generalizes better than SVMs and uses far fewer support vectors than the number used by SVMs. In this paper, we describe a neural network based on a linear dynamical system, that converges to the MCM solution. The proposed MCM dynamical system is conducive to an analogue circuit implementation on a chip or simulation using Ordinary Differential Equation (ODE) solvers. Numerical experiments on benchmark datasets from the UCI repository show that the proposed approach is scalable and accurate, as we obtain improved accuracies and fewer number of support vectors (upto 74.3% reduction) with the MCM dynamical system.Keywords.

Our results reveal an interesting phase transition behavior universal to this class of high dimensional problems. In the sparse regime when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression, and therefore there is no additional cost to entertain a nonparametric model. Otherwise, in the so-called smooth regime, the rates coincide with the optimal rates for estimating a univariate function, and therefore they are immune to the "curse of dimensionality". Key words: Convergence rate, method of regularization, minimax optimality, phase transition, reproducing kernel Hilbert space, Sobolev space. 2 1 Introduction With the recent advances in science and technology, high dimensional regression problems have become ubiquitous in a multitude of areas - genomics, medical imaging, and finance are a few well known examples. Considerable amount of research effort has been devoted to the understanding of challenges brought about by the high dimensionality, and development of statistical methodology to counter them.

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 descent. In our experiments, we show that our algorithm can achieve the same level of prediction accuracy as Gibbs sampling an order of magnitude faster. We also show that our method reduces the prediction error as fast as distributed stochastic gradient descent, achieving a 4.1% improvement in RMSE for the Netflix dataset and an 1.8% for the Yahoo music dataset.

Low rank tensor learning, such as tensor completion and multilinear multitask learning, has received much attention in recent years. In this paper, we propose higher order matching pursuit for low rank tensor learning problems with a convex or a nonconvex cost function, which is a generalization of the matching pursuit type methods. At each iteration, the main cost of the proposed methods is only to compute a rank-one tensor, which can be done efficiently, making the proposed methods scalable to large scale problems. Moreover, storing the resulting rank-one tensors is of low storage requirement, which can help to break the curse of dimensionality. The linear convergence rate of the proposed methods is established in various circumstances. Along with the main methods, we also provide a method of low computational complexity for approximately computing the rank-one tensors, with provable approximation ratio, which helps to improve the efficiency of the main methods and to analyze the convergence rate. Experimental results on synthetic as well as real datasets verify the efficiency and effectiveness of the proposed methods. Tensors, appearing as the higher order generalization of vectors and matrices, make it possible to represent data that have intrinsically many dimensions, and give a better understanding of the relationship behind the information from a higher order perspective. In many machine learning problems such as tensor completion [1]-[4], multilinear multitask learning (MLMTL) [5]-[7] and tensor regression [8], one often aims at learning a tensor that has low rankness. For example, in tensor completion, the goal is to learn a low rank tensor provided that only partial observations are available. In the context of MLMTL, to allow for common information shared between tasks to pursuit better generalization, by learning several tasks simultaneously, where each task is indexed by more than two indices, all the tasks can be represented by a tensor assumed to lie in a low dimensional spaces. In tensor regression, to better understand the information behind high dimensionality data, the weight vector is represented by a low rank tensor. These applications give rise to low rank tensor learning. Commonly speaking, to learn a low rank tensor, tensor learning minimizes a real-valued cost functionF: T R subject to some constraints or with regularizations to encourage the low rank property of the learned tensor.

Multivariate categorical data occur in many applications of machine learning. One of the main difficulties with these vectors of categorical variables is sparsity. The number of possible observations grows exponentially with vector length, but dataset diversity might be poor in comparison. Recent models have gained significant improvement in supervised tasks with this data. These models embed observations in a continuous space to capture similarities between them. Building on these ideas we propose a Bayesian model for the unsupervised task of distribution estimation of multivariate categorical data. We model vectors of categorical variables as generated from a non-linear transformation of a continuous latent space. Non-linearity captures multi-modality in the distribution. The continuous representation addresses sparsity. Our model ties together many existing models, linking the linear categorical latent Gaussian model, the Gaussian process latent variable model, and Gaussian process classification. We derive inference for our model based on recent developments in sampling based variational inference. We show empirically that the model outperforms its linear and discrete counterparts in imputation tasks of sparse data.

Sales pipeline win-propensity prediction is fundamental to effective sales management. In contrast to using subjective human rating, we propose a modern machine learning paradigm to estimate the win-propensity of sales leads over time. A profile-specific two-dimensional Hawkes processes model is developed to capture the influence from seller's activities on their leads to the win outcome, coupled with lead's personalized profiles. It is motivated by two observations: i) sellers tend to frequently focus their selling activities and efforts on a few leads during a relatively short time. This is evidenced and reflected by their concentrated interactions with the pipeline, including login, browsing and updating the sales leads which are logged by the system; ii) the pending opportunity is prone to reach its win outcome shortly after such temporally concentrated interactions. Our model is deployed and in continual use to a large, global, B2B multinational technology enterprize (Fortune 500) with a case study. Due to the generality and flexibility of the model, it also enjoys the potential applicability to other real-world problems.

Bayesian reinforcement learning (BRL) provides a formal framework for optimal exploration-exploitation tradeoff in reinforcement learning. Unfortunately, it is generally intractable to find the Bayes-optimal behavior except for restricted cases. As a consequence, many BRL algorithms, model-based approaches in particular, rely on approximated models or real-time search methods. In this paper, we present potential-based shaping for improving the learning performance in model-based BRL. We propose a number of potential functions that are particularly well suited for BRL, and are domain-independent in the sense that they do not require any prior knowledge about the actual environment. By incorporating the potential function into real-time heuristic search, we show that we can significantly improve the learning performance in standard benchmark domains.

However, implementing such a strategy requires combining the VFI framework with policy parameterization, rebalancing continually as assets prices fluctuate, the proposed ADP method enjoys complementary advantages and therefore will lead to high or even infinite transaction of low approximation errors from VFI and high computational costs. Since then researchers have tried to address this issue efficiency from policy parameterization. Briefly, by solving Merton's portfolio problem in the presence the components from VFI pave the way for effectively parameterizing of transaction costs. Thereinto, the proportional transaction a complex policy in a high-dimensional space; costs model, as a suitable model for brokerage commissions the components from policy parameterization provide a and bid-ask spread costs, typifies the common situation pathway to efficiently evaluating the strategy and bypassing for normal investors (Brandt 2010; Cvitanic 2001; the issue of error amplification.

The expressive power of a Gaussian process (GP) model comes at a cost of poor scalability in the data size. To improve its scalability, this paper presents a low-rank-cum-Markov approximation (LMA) of the GP model that is novel in leveraging the dual computational advantages stemming from complementing a low-rank approximate representation of the full-rank GP based on a support set of inputs with a Markov approximation of the resulting residual process; the latter approximation is guaranteed to be closest in the Kullback-Leibler distance criterion subject to some constraint and is considerably more refined than that of existing sparse GP models utilizing low-rank representations due to its more relaxed conditional independence assumption (especially with larger data). As a result, our LMA method can trade off between the size of the support set and the order of the Markov property to (a) incur lower computational cost than such sparse GP models while achieving predictive performance comparable to them and (b) accurately represent features/patterns of any scale. Interestingly, varying the Markov order produces a spectrum of LMAs with PIC approximation and full-rank GP at the two extremes. An advantage of our LMA method is that it is amenable to parallelization on multiple machines/cores, thereby gaining greater scalability. Empirical evaluation on three real-world datasets in clusters of up to 32 computing nodes shows that our centralized and parallel LMA methods are significantly more time-efficient and scalable than state-of-the-art sparse and full-rank GP regression methods while achieving comparable predictive performances.