We consider the estimation of Dirichlet Process Mixture Models (DPMMs) in distributed environments, where data are distributed across multiple computing nodes. A key advantage of Bayesian nonparametric models such as DPMMs is that they allow new components to be introduced on the fly as needed. This, however, posts an important challenge to distributed estimation -- how to handle new components efficiently and consistently. To tackle this problem, we propose a new estimation method, which allows new components to be created locally in individual computing nodes. Components corresponding to the same cluster will be identified and merged via a probabilistic consolidation scheme. In this way, we can maintain the consistency of estimation with very low communication cost. Experiments on large real-world data sets show that the proposed method can achieve high scalability in distributed and asynchronous environments without compromising the mixing performance.

In this paper, we develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches is becoming a golden standard in the machine learning community, because mini-batch settings stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new "double acceleration" technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the mini-batch efficiencies of previous accelerated stochastic methods, and essentially only needs size $\sqrt{n}$ mini-batches for achieving the optimal iteration complexities for both non-strongly and strongly convex objectives, where $n$ is the training set size. Further, we show that even in non-mini-batch settings, our method achieves the best known convergence rate for both non-strongly and strongly convex objectives.

We revisit the problem of estimating the parameters of a partially observed diffusion process, consisting of a hidden state process and an observed process, with a continuous time parameter. The estimation is to be done online, i.e. the parameter estimate should be updated recursively based on the observation filtration. Here, we use an old but under-exploited representation of the incomplete-data log-likelihood function in terms of the filter of the hidden state from the observations. By performing a stochastic gradient ascent, we obtain a fully recursive algorithm for the time evolution of the parameter estimate. We prove the convergence of the algorithm under suitable conditions regarding the ergodicity of the process consisting of state, filter, and tangent filter. Additionally, our parameter estimation is shown numerically to have the potential of improving suboptimal filters, and can be applied even when the system is not identifiable due to parameter redundancies. Online parameter estimation is a challenging problem that is ubiquitous in fields such as robotics, neuroscience, or finance in order to design adaptive filters and optimal controllers for unknown or changing systems.

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using ${\rm T{\small ENSOR}F{\small LOW}}$ in ${\rm P{\small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a $ 100 $-dimensional $ G $-Brownian motion.

We consider the problem of non-parametric Conditional Independence testing (CI testing) for continuous random variables. Given i.i.d samples from the joint distribution $f(x,y,z)$ of continuous random vectors $X,Y$ and $Z,$ we determine whether $X \perp Y | Z$. We approach this by converting the conditional independence test into a classification problem. This allows us to harness very powerful classifiers like gradient-boosted trees and deep neural networks. These models can handle complex probability distributions and allow us to perform significantly better compared to the prior state of the art, for high-dimensional CI testing. The main technical challenge in the classification problem is the need for samples from the conditional product distribution $f^{CI}(x,y,z) = f(x|z)f(y|z)f(z)$ -- the joint distribution if and only if $X \perp Y | Z.$ -- when given access only to i.i.d. samples from the true joint distribution $f(x,y,z)$. To tackle this problem we propose a novel nearest neighbor bootstrap procedure and theoretically show that our generated samples are indeed close to $f^{CI}$ in terms of total variational distance. We then develop theoretical results regarding the generalization bounds for classification for our problem, which translate into error bounds for CI testing. We provide a novel analysis of Rademacher type classification bounds in the presence of non-i.i.d near-independent samples. We empirically validate the performance of our algorithm on simulated and real datasets and show performance gains over previous methods.

We consider the problem of active feature acquisition, where we sequentially select the subset of features in order to achieve the maximum prediction performance in the most cost-effective way. In this work, we formulate this active feature acquisition problem as a reinforcement learning problem, and provide a novel framework for jointly learning both the RL agent and the classifier (environment). We also introduce a more systematic way of encoding subsets of features that can properly handle innate challenge with missing entries in active feature acquisition problems, that uses the orderless LSTM-based set encoding mechanism that readily fits in the joint learning framework. We evaluate our model on a carefully designed synthetic dataset for the active feature acquisition as well as several real datasets such as electric health record (EHR) datasets, on which it outperforms all baselines in terms of prediction performance as well feature acquisition cost.

ECENTLY, Vapnik and Vashist [1] provided a new learning paradigm termed learning using privileged information (LUPI), which is aimed at enhancing the generalization performance of learning algorithms. Generally speaking, in classical supervised learning paradigm, the training data and test data must come from the same distribution. Although in this new learning paradigm the training data is also considered an unbiased representation for the test data, the LUPI provides a set of additional information for the training data during the training stage, which is called privileged information. In the LUPI paradigm, we use the new training set containing privileged information to train a learning algorithm, while the privileged information is not available in the test stage. We note that the new learning paradigm is analogous to human learning process. In class, a teacher can provide some important and helpful information about this course for students, and these information provided by the teacher can help students acquire knowledge better. Therefore, a teacher plays an essential role in human leaning process. The LUPI paradigm resembling the classroom teaching model can achieve better generalization performance than the traditional learning paradigm. The author is with Department of Industrial Engineering and Logistics Management, School of Engineering, Hong Kong University of Science and Technology, Hong Kong 999077, China.(Email:

The main aim of this paper is to provide an analysis of gradient descent (GD) algorithms with gradient errors that do not necessarily vanish, asymptotically. In particular, sufficient conditions are presented for both stability (almost sure boundedness of the iterates) and convergence of GD with bounded, (possibly) non-diminishing gradient errors. In addition to ensuring stability, such an algorithm is shown to converge to a small neighborhood of the minimum set, which depends on the gradient errors. It is worth noting that the main result of this paper can be used to show that GD with asymptotically vanishing errors indeed converges to the minimum set. The results presented herein are not only more general when compared to previous results, but our analysis of GD with errors is new to the literature to the best of our knowledge. Our work extends the contributions of Mangasarian & Solodov, Bertsekas & Tsitsiklis and Tadic & Doucet. Using our framework, a simple yet effective implementation of GD using simultaneous perturbation stochastic approximations (SP SA), with constant sensitivity parameters, is presented. Another important improvement over many previous results is that there are no `additional' restrictions imposed on the step-sizes. In machine learning applications where step-sizes are related to learning rates, our assumptions, unlike those of other papers, do not affect these learning rates. Finally, we present experimental results to validate our theory.

Kernel-based learning algorithms are widely used in machine learning for problems that make use of the similarity between object pairs. Such algorithms first embed all data points into an alternative space, where the inner product between object pairs specifies their distance in the embedding space. Applying kernel methods to partially labeled datasets is a classical challenge in this regard, requiring that the distances between unlabeled pairs must somehow be learnt using the labeled data. In this independent study, I will summarize the work of G. Lanckriet et al.'s work on "Learning the Kernel Matrix with Semidefinite Programming" used in support vector machines (SVM) algorithms for the transduction problem. Throughout the report, I have provide alternative explanations / derivations / analysis related to this work which is designed to ease the understanding of the original article.

In recent years, a number of artificial intelligent services have been developed such as defect detection system or diagnosis system for customer services. Unfortunately, the core in these services is a black-box in which human cannot understand the underlying decision making logic, even though the inspection of the logic is crucial before launching a commercial service. Our goal in this paper is to propose an analytic method of a model explanation that is applicable to general classification models. To this end, we introduce the concept of a contribution matrix and an explanation embedding in a constraint space by using a matrix factorization. We extract a rule-like model explanation from the contribution matrix with the help of the nonnegative matrix factorization. To validate our method, the experiment results provide with open datasets as well as an industry dataset of a LTE network diagnosis and the results show our method extracts reasonable explanations.