Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependencies. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing, and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently non-convex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing non-convex programs, we are able to both analyze the statistical error associated with any global optimum, and prove that a simple projected gradient descent algorithm will converge in polynomial time to a small neighborhood of the set of global minimizers. On the statistical side, we provide non-asymptotic bounds that hold with high probability for the cases of noisy, missing, and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm will converge at geometric rates to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing agreement with the predicted scalings.

A hallmark of modern machine learning is its ability to deal with high dimensional problems by exploiting structural assumptions that limit the degrees of freedom in the underlying model. A deep understanding of the capabilities and limits of high dimensional learning methods under specific assumptions such as sparsity, group sparsity, and low rank has been attained. Efforts (Negahban et al., 2010, Chandrasekaran et al., 2010} are now underway to distill this valuable experience by proposing general unified frameworks that can achieve the twin goals of summarizing previous analyses and enabling their application to notions of structure hitherto unexplored. Inspired by these developments, we propose and analyze a general computational scheme based on a greedy strategy to solve convex optimization problems that arise when dealing with structurally constrained high-dimensional problems. Our framework not only unifies existing greedy algorithms by recovering them as special cases but also yields novel ones. Finally, we extend our results to infinite dimensional problems by using interesting connections between smoothness of norms and behavior of martingales in Banach spaces.

The development of statistical models for continuous-time longitudinal network data is of increasing interest in machine learning and social science. Leveraging ideas from survival and event history analysis, we introduce a continuous-time regression modeling framework for network event data that can incorporate both time-dependent network statistics and time-varying regression coefficients. We also develop an efficient inference scheme that allows our approach to scale to large networks. On synthetic and real-world data, empirical results demonstrate that the proposed inference approach can accurately estimate the coefficients of the regression model, which is useful for interpreting the evolution of the network; furthermore, the learned model has systematically better predictive performance compared to standard baseline methods.

Transfer reinforcement learning (RL) methods leverage on the experience collected on a set of source tasks to speed-up RL algorithms. A simple and effective approach is to transfer samples from source tasks and include them in the training set used to solve a target task. In this paper, we investigate the theoretical properties of this transfer method and we introduce novel algorithms adapting the transfer process on the basis of the similarity between source and target tasks. Finally, we report illustrative experimental results in a continuous chain problem.

Motor prostheses aim to restore function to disabled patients. Despite compelling proof of concept systems, barriers to clinical translation remain. One challenge is to develop a low-power, fully-implantable system that dissipates only minimal power so as not to damage tissue. To this end, we implemented a Kalman-filter based decoder via a spiking neural network (SNN) and tested it in brain-machine interface (BMI) experiments with a rhesus monkey. The Kalman filter was trained to predict the arm's velocity and mapped on to the SNN using the Neural Engineering Framework(NEF). A 2,000-neuron embedded Matlab SNN implementation runs in real-time and its closed-loop performance is quite comparable to that of the standard Kalman filter. The success of this closed-loop decoder holds promise for hardware SNN implementations of statistical signal processing algorithms on neuromorphic chips,which may offer power savings necessary to overcome a major obstacle to the successful clinical translation of neural motor prostheses. Present: Research Fellow F.R.S.-FNRS, Systmod Unit, University of Liege, Belgium.

Kernel-based reinforcement-learning (KBRL) is a method for learning a decision policy from a set of sample transitions which stands out for its strong theoretical guarantees. However, the size of the approximator grows with the number of transitions, which makes the approach impractical for large problems. In this paper we introduce a novel algorithm to improve the scalability of KBRL. We resort to a special decomposition of a transition matrix, called stochastic factorization, to fix the size of the approximator while at the same time incorporating all the information contained in the data. The resulting algorithm, kernel-based stochastic factorization (KBSF), is much faster but still converges to a unique solution. We derive a theoretical upper bound for the distance between the value functions computed by KBRL and KBSF. The effectiveness of our method is illustrated with computational experiments on four reinforcement-learning problems, including a difficult task in which the goal is to learn a neurostimulation policy to suppress the occurrence of seizures in epileptic rat brains. We empirically demonstrate that the proposed approach is able to compress the information contained in KBRL's model. Also, on the tasks studied, KBSF outperforms two of the most prominent reinforcement-learning algorithms, namely least-squares policy iteration and fitted Q-iteration.

A sizable literature has focused on the problem of estimating a low-dimensional feature space capturing a neuron's stimulus sensitivity. However, comparatively little work has addressed the problem of estimating the nonlinear function from feature space to a neuron's output spike rate. Here, we use a Gaussian process (GP) prior over the infinite-dimensional space of nonlinear functions to obtain Bayesian estimates of the "nonlinearity" in the linear-nonlinear-Poisson (LNP) encoding model. This offers flexibility, robustness, and computational tractability compared to traditional methods (e.g., parametric forms, histograms, cubic splines). Most importantly, we develop a framework for optimal experimental design based on uncertainty sampling. This involves adaptively selecting stimuli to characterize the nonlinearity with as little experimental data as possible, and relies on a method for rapidly updating hyperparameters using the Laplace approximation. We apply these methods to data from color-tuned neurons in macaque V1. We estimate nonlinearities in the 3D space of cone contrasts, which reveal that V1 combines cone inputs in a highly nonlinear manner. With simulated experiments, we show that optimal design substantially reduces the amount of data required to estimate this nonlinear combination rule.

Sparse coding, a method of explaining sensory data with as few dictionary bases as possible, has attracted much attention in computer vision. For visual object category recognition, L1 regularized sparse coding is combined with spatial pyramid representation to obtain state-of-the-art performance. However, because of its iterative optimization, applying sparse coding onto every local feature descriptor extracted from an image database can become a major bottleneck. To overcome this computational challenge, this paper presents "Generalized Lasso based Approximation of Sparse coding" (GLAS). By representing the distribution of sparse coefficients with slice transform, we fit a piece-wise linear mapping function with generalized lasso. We also propose an efficient post-refinement procedure to perform mutual inhibition between bases which is essential for an overcomplete setting. The experiments show that GLAS obtains comparable performance to L1 regularized sparse coding, yet achieves significant speed up demonstrating its effectiveness for large-scale visual recognition problems.

Latent tree graphical models are natural tools for expressing long range and hierarchical dependencies among many variables which are common in computer vision, bioinformatics and natural language processing problems. However, existing models are largely restricted to discrete and Gaussian variables due to computational constraints; furthermore, algorithms for estimating the latent tree structure and learning the model parameters are largely restricted to heuristic local search. We present a method based on kernel embeddings of distributions for latent tree graphical models with continuous and non-Gaussian variables. Our method can recover the latent tree structures with provable guarantees and perform local-minimum free parameter learning and efficient inference. Experiments on simulated and real data show the advantage of our proposed approach.

Many species show avoidance reactions in response to looming object approaches. In locusts, the corresponding escape behavior correlates with the activity of the lobula giant movement detector (LGMD) neuron. During an object approach, its firing rate was reported to gradually increase until a peak is reached, and then it declines quickly. The $\eta$-function predicts that the LGMD activity is a product between an exponential function of angular size $\exp(-\Theta)$ and angular velocity $\dot{\Theta}$, and that peak activity is reached before time-to-contact (ttc). The $\eta$-function has become the prevailing LGMD model because it reproduces many experimental observations, and even experimental evidence for the multiplicative operation was reported. Several inconsistencies remain unresolved, though. Here we address these issues with a new model ($\psi$-model), which explicitly connects $\Theta$ and $\dot{\Theta}$ to biophysical quantities. The $\psi$-model avoids biophysical problems associated with implementing $\exp(\cdot)$, implements the multiplicative operation of $\eta$ via divisive inhibition, and explains why activity peaks could occur after ttc. It consistently predicts response features of the LGMD, and provides excellent fits to published experimental data, with goodness of fit measures comparable to corresponding fits with the $\eta$-function.