Variational methods are widely used for approximate posterior inference. However, their use is typically limited to families of distributions that enjoy particular conjugacy properties. To circumvent this limitation, we propose a family of variational approximations inspired by nonparametric kernel density estimation. The locations of these kernels and their bandwidth are treated as variational parameters and optimized to improve an approximate lower bound on the marginal likelihood of the data. Using multiple kernels allows the approximation to capture multiple modes of the posterior, unlike most other variational approximations. We demonstrate the efficacy of the nonparametric approximation with a hierarchical logistic regression model and a nonlinear matrix factorization model. We obtain predictive performance as good as or better than more specialized variational methods and sample-based approximations. The method is easy to apply to more general graphical models for which standard variational methods are difficult to derive.

We show that the variational representations for f-divergences currently used in the literature can be tightened. This has implications to a number of methods recently proposed based on this representation. As an example application we use our tighter representation to derive a general f-divergence estimator based on two i.i.d. samples and derive the dual program for this estimator that performs well empirically. We also point out a connection between our estimator and MMD.

In many real-world applications of machine learning classifiers, it is essential to predict the probability of an example belonging to a particular class. This paper proposes a simple technique for predicting probabilities based on optimizing a ranking loss, followed by isotonic regression. This semi-parametric technique offers both good ranking and regression performance, and models a richer set of probability distributions than statistical workhorses such as logistic regression. We provide experimental results that show the effectiveness of this technique on real-world applications of probability prediction.

We present a max-margin nonparametric latent feature relational model, which unites the ideas of max-margin learning and Bayesian nonparametrics to discover discriminative latent features for link prediction and automatically infer the unknown latent social dimension. By minimizing a hinge-loss using the linear expectation operator, we can perform posterior inference efficiently without dealing with a highly nonlinear link likelihood function; by using a fully-Bayesian formulation, we can avoid tuning regularization constants. Experimental results on real datasets appear to demonstrate the benefits inherited from max-margin learning and fully-Bayesian nonparametric inference.

We present local discriminative Gaussian (LDG) dimensionality reduction, a supervised dimensionality reduction technique for classification. The LDG objective function is an approximation to the leave-one-out training error of a local quadratic discriminant analysis classifier, and thus acts locally to each training point in order to find a mapping where similar data can be discriminated from dissimilar data. While other state-of-the-art linear dimensionality reduction methods require gradient descent or iterative solution approaches, LDG is solved with a single eigen-decomposition. Thus, it scales better for datasets with a large number of feature dimensions or training examples. We also adapt LDG to the transfer learning setting, and show that it achieves good performance when the test data distribution differs from that of the training data.

Random projections have been applied in many machine learning algorithms. However, whether margin is preserved after random projection is non-trivial and not well studied. In this paper we analyse margin distortion after random projection, and give the conditions of margin preservation for binary classification problems. We also extend our analysis to margin for multiclass problems, and provide theoretical bounds on multiclass margin on the projected data.

In real supervised learning scenarios, it is not uncommon that the training and test sample follow different probability distributions, thus rendering the necessity to correct the sampling bias. Focusing on a particular covariate shift problem, we derive high probability confidence bounds for the kernel mean matching (KMM) estimator, whose convergence rate turns out to depend on some regularity measure of the regression function and also on some capacity measure of the kernel. By comparing KMM with the natural plug-in estimator, we establish the superiority of the former hence provide concrete evidence/understanding to the effectiveness of KMM under covariate shift.

We present a comprehensive framework for structured sparse coding and modeling extending the recent ideas of using learnable fast regressors to approximate exact sparse codes. For this purpose, we develop a novel block-coordinate proximal splitting method for the iterative solution of hierarchical sparse coding problems, and show an efficient feed forward architecture derived from its iteration. This architecture faithfully approximates the exact structured sparse codes with a fraction of the complexity of the standard optimization methods. We also show that by using different training objective functions, learnable sparse encoders are no longer restricted to be mere approximants of the exact sparse code for a pre-given dictionary, as in earlier formulations, but can be rather used as full-featured sparse encoders or even modelers. A simple implementation shows several orders of magnitude speedup compared to the state-of-the-art at minimal performance degradation, making the proposed framework suitable for real time and large-scale applications.

Stochastic neighbor embedding (SNE) and related nonlinear manifold learning algorithms achieve high-quality low-dimensional representations of similarity data, but are notoriously slow to train. We propose a generic formulation of embedding algorithms that includes SNE and other existing algorithms, and study their relation with spectral methods and graph Laplacians. This allows us to define several partial-Hessian optimization strategies, characterize their global and local convergence, and evaluate them empirically. We achieve up to two orders of magnitude speedup over existing training methods with a strategy (which we call the spectral direction) that adds nearly no overhead to the gradient and yet is simple, scalable and applicable to several existing and future embedding algorithms.

Convex regression is a promising area for bridging statistical estimation and deterministic convex optimization. New piecewise linear convex regression methods are fast and scalable, but can have instability when used to approximate constraints or objective functions for optimization. Ensemble methods, like bagging, smearing and random partitioning, can alleviate this problem and maintain the theoretical properties of the underlying estimator. We empirically examine the performance of ensemble methods for prediction and optimization, and then apply them to device modeling and constraint approximation for geometric programming based circuit design.