Infinite mixture models are commonly used for clustering. One can sample from the posterior of mixture assignments by Monte Carlo methods or find its maximum a posteriori solution by optimization. However, in some problems the posterior is diffuse and it is hard to interpret the sampled partitionings. In this paper, we introduce novel statistics based on block sizes for representing sample sets of partitionings and feature allocations. We develop an element-based definition of entropy to quantify segmentation among their elements. Then we propose a simple algorithm called entropy agglomeration (EA) to summarize and visualize this information. Experiments on various infinite mixture posteriors as well as a feature allocation dataset demonstrate that the proposed statistics are useful in practice.

Learning the joint dependence of discrete variables is a fundamental problem in machine learning, with many applications including prediction, clustering and dimensionality reduction. More recently, the framework of copula modeling has gained popularity due to its modular parametrization of joint distributions. Among other properties, copulas provide a recipe for combining flexible models for univariate marginal distributions with parametric families suitable for potentially high dimensional dependence structures. More radically, the extended rank likelihood approach of Hoff (2007) bypasses learning marginal models completely when such information is ancillary to the learning task at hand as in, e.g., standard dimensionality reduction problems or copula parameter estimation. The main idea is to represent data by their observable rank statistics, ignoring any other information from the marginals. Inference is typically done in a Bayesian framework with Gaussian copulas, and it is complicated by the fact this implies sampling within a space where the number of constraints increase quadratically with the number of data points. The result is slow mixing when using off-the-shelf Gibbs sampling. We present an efficient algorithm based on recent advances on constrained Hamiltonian Markov chain Monte Carlo that is simple to implement and does not require paying for a quadratic cost in sample size.

The sparse additive model for text modeling involves the sum-of-exp computing, with consuming costs for large scales. Moreover, the assumption of equal background across all classes/topics may be too strong. This paper extends to propose sparse additive model with low rank background (SAM-LRB), and simple yet efficient estimation. Particularly, by employing a double majorization bound, we approximate the log-likelihood into a quadratic lower-bound with the sum-of-exp terms absent. The constraints of low rank and sparsity are then simply embodied by nuclear norm and $\ell_1$-norm regularizers. Interestingly, we find that the optimization task in this manner can be transformed into the same form as that in Robust PCA. Consequently, parameters of supervised SAM-LRB can be efficiently learned using an existing algorithm for Robust PCA based on accelerated proximal gradient. Besides the supervised case, we extend SAM-LRB to also favor unsupervised and multifaceted scenarios. Experiments on real world data demonstrate the effectiveness and efficiency of SAM-LRB, showing state-of-the-art performances.

We propose a Learning from Demonstration (LfD) algorithm which leverages expert data,even if they are very few or inaccurate. We achieve this by using both expert data, as well as reinforcement signals gathered through trial-and-error interactions withthe environment. The key idea of our approach, Approximate Policy Iteration with Demonstration (APID), is that expert's suggestions are used to define linearconstraints which guide the optimization performed by Approximate Policy Iteration. We prove an upper bound on the Bellman error of the estimate computed by APID at each iteration. Moreover, we show empirically that APID outperforms pure Approximate Policy Iteration, a state-of-the-art LfD algorithm, and supervised learning in a variety of scenarios, including when very few and/or suboptimal demonstrations are available. Our experiments include simulations as well as a real robot path-finding task.

We give differentially private algorithms for a large class of online learning algorithms, inboth the full information and bandit settings. Our algorithms aim to minimize a convex loss function which is a sum of smaller convex loss terms, one for each data point. To design our algorithms, we modify the popular mirror descent approach, or rather a variant called follow the approximate leader. The technique leads to the first nonprivate algorithms for private online learning in the bandit setting. In the full information setting, our algorithms improve over the regret bounds of previous work (due to Dwork, Naor, Pitassi and Rothblum (2010) and Jain, Kothari and Thakurta (2012)). In many cases, our algorithms (in both settings) match the dependence on the input length, T, of the optimal nonprivate regret bounds up to logarithmic factors in T . Our algorithms require logarithmic space and update time.

Bilinear approximation of a matrix is a powerful paradigm of unsupervised learning. In some applications, however, there is a natural hierarchy of concepts that ought to be reflected in the unsupervised analysis. For example, in the neurosciences image sequence considered here, there are the semantic concepts of pixel $\rightarrow$ neuron $\rightarrow$ assembly that should find their counterpart in the unsupervised analysis. Driven by this concrete problem, we propose a decomposition of the matrix of observations into a product of more than two sparse matrices, with the rank decreasing from lower to higher levels. In contrast to prior work, we allow for both hierarchical and heterarchical relations of lower-level to higher-level concepts. In addition, we learn the nature of these relations rather than imposing them. Finally, we describe an optimization scheme that allows to optimize the decomposition over all levels jointly, rather than in a greedy level-by-level fashion. The proposed bilevel SHMF (sparse heterarchical matrix factorization) is the first formalism that allows to simultaneously interpret a calcium imaging sequence in terms of the constituent neurons, their membership in assemblies, and the time courses of both neurons and assemblies. Experiments show that the proposed model fully recovers the structure from difficult synthetic data designed to imitate the experimental data. More importantly, bilevel SHMF yields plausible interpretations of real-world Calcium imaging data.

Human eye movements provide a rich source of information into the human visual processing. The complex interplay between the task and the visual stimulus is believed to determine human eye movements, yet it is not fully understood. This has precluded the development of reliable dynamic eye movement prediction systems. Our work makes three contributions towards addressing this problem. First, we complement one of the largest and most challenging static computer vision datasets, VOC 2012 Actions, with human eye movement annotations collected under the task constraints of action and context recognition. Our dataset is unique among eyetracking datasets for still images in terms of its large scale (over 1 million fixations, 9157 images), task control and action from a single image emphasis. Second, we introduce models to automatically discover areas of interest (AOI) and introduce novel dynamic consistency metrics, based on them. Our method can automatically determine the number and spatial support of the AOIs, in addition to their locations. Based on such encodings, we show that, on unconstrained read-world stimuli, task instructions have significant influence on visual behavior. Finally, we leverage our large scale dataset in conjunction with powerful machine learning techniques and computer vision features, to introduce novel dynamic eye movement prediction methods which learn task-sensitive reward functions from eye movement data and efficiently integrate these rewards to plan future saccades based on inverse optimal control. We show that the propose methodology achieves state of the art scanpath modeling results.

Sparse Subspace Clustering (SSC) and Low-Rank Representation (LRR) are both considered as the state-of-the-art methods for {\em subspace clustering}. The two methods are fundamentally similar in that both are convex optimizations exploiting the intuition of Self-Expressiveness''. The main difference is that SSC minimizes the vector $\ell_1$ norm of the representation matrix to induce sparsity while LRR minimizes nuclear norm (aka trace norm) to promote a low-rank structure. Because the representation matrix is often simultaneously sparse and low-rank, we propose a new algorithm, termed Low-Rank Sparse Subspace Clustering (LRSSC), by combining SSC and LRR, and develops theoretical guarantees of when the algorithm succeeds. The results reveal interesting insights into the strength and weakness of SSC and LRR and demonstrate how LRSSC can take the advantages of both methods in preserving the "Self-Expressiveness Property'' and "Graph Connectivity'' at the same time."

We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on the minimization of the empirical risk. We focus on problems without strong convexity, for which all previously known algorithms achieve a convergence rate for function values of $O(1/\sqrt{n})$. We consider and analyze two algorithms that achieve a rate of $O(1/n)$ for classical supervised learning problems. For least-squares regression, we show that averaged stochastic gradient descent with constant step-size achieves the desired rate. For logistic regression, this is achieved by a simple novel stochastic gradient algorithm that (a) constructs successive local quadratic approximations of the loss functions, while (b) preserving the same running time complexity as stochastic gradient descent. For these algorithms, we provide a non-asymptotic analysis of the generalization error (in expectation, and also in high probability for least-squares), and run extensive experiments showing that they often outperform existing approaches.

We consider streaming, one-pass principal component analysis (PCA), in the high-dimensional regime, with limited memory. Here, $p$-dimensional samples are presented sequentially, and the goal is to produce the $k$-dimensional subspace that best approximates these points. Standard algorithms require $O(p^2)$ memory; meanwhile no algorithm can do better than $O(kp)$ memory, since this is what the output itself requires. Memory (or storage) complexity is most meaningful when understood in the context of computational and sample complexity. Sample complexity for high-dimensional PCA is typically studied in the setting of the {\em spiked covariance model}, where $p$-dimensional points are generated from a population covariance equal to the identity (white noise) plus a low-dimensional perturbation (the spike) which is the signal to be recovered. It is now well-understood that the spike can be recovered when the number of samples, $n$, scales proportionally with the dimension, $p$. Yet, all algorithms that provably achieve this, have memory complexity $O(p^2)$. Meanwhile, algorithms with memory-complexity $O(kp)$ do not have provable bounds on sample complexity comparable to $p$. We present an algorithm that achieves both: it uses $O(kp)$ memory (meaning storage of any kind) and is able to compute the $k$-dimensional spike with $O(p \log p)$ sample-complexity -- the first algorithm of its kind. While our theoretical analysis focuses on the spiked covariance model, our simulations show that our algorithm is successful on much more general models for the data.