We present a unified framework for learning continuous control policies usingbackpropagation. It supports stochastic control by treating stochasticity in theBellman equation as a deterministic function of exogenous noise. The productis a spectrum of general policy gradient algorithms that range from model-freemethods with value functions to model-based methods without value functions.We use learned models but only require observations from the environment insteadof observations from model-predicted trajectories, minimizing the impactof compounded model errors. We apply these algorithms first to a toy stochasticcontrol problem and then to several physics-based control problems in simulation.One of these variants, SVG(1), shows the effectiveness of learning models, valuefunctions, and policies simultaneously in continuous domains.

Independent Component Analysis (ICA) is a popular model for blind signal separation. The ICA model assumes that a number of independent source signals are linearly mixed to form the observed signals. We propose a new algorithm, PEGI (for pseudo-Euclidean Gradient Iteration), for provable model recovery for ICA with Gaussian noise. The main technical innovation of the algorithm is to use a fixed point iteration in a pseudo-Euclidean (indefinite “inner product”) space. The use of this indefinite “inner product” resolves technical issues common to several existing algorithms for noisy ICA. This leads to an algorithm which is conceptually simple, efficient and accurate in testing.Our second contribution is combining PEGI with the analysis of objectives for optimal recovery in the noisy ICA model. It has been observed that the direct approach of demixing with the inverse of the mixing matrix is suboptimal for signal recovery in terms of the natural Signal to Interference plus Noise Ratio (SINR) criterion. There have been several partial solutions proposed in the ICA literature. It turns out that any solution to the mixing matrix reconstruction problem can be used to construct an SINR-optimal ICA demixing, despite the fact that SINR itself cannot be computed from data. That allows us to obtain a practical and provably SINR-optimal recovery method for ICA with arbitrary Gaussian noise.

We introduce Embed to Control (E2C), a method for model learning and control of non-linear dynamical systems from raw pixel images. E2C consists of a deep generative model, belonging to the family of variational autoencoders, that learns to generate image trajectories from a latent space in which the dynamics is constrained to be locally linear. Our model is derived directly from an optimal control formulation in latent space, supports long-term prediction of image sequences and exhibits strong performance on a variety of complex control problems.

We consider an adversarial formulation of the problem ofpredicting a time series with square loss. The aim is to predictan arbitrary sequence of vectors almost as well as the bestsmooth comparator sequence in retrospect. Our approach allowsnatural measures of smoothness such as the squared norm ofincrements. More generally, we consider a linear time seriesmodel and penalize the comparator sequence through the energy ofthe implied driving noise terms. We derive the minimax strategyfor all problems of this type and show that it can be implementedefficiently. The optimal predictions are linear in the previousobservations. We obtain an explicit expression for the regret interms of the parameters defining the problem. For typical,simple definitions of smoothness, the computation of the optimalpredictions involves only sparse matrices. In the case ofnorm-constrained data, where the smoothness is defined in termsof the squared norm of the comparator's increments, we show thatthe regret grows as $T/\sqrt{\lambda_T}$, where $T$ is the lengthof the game and $\lambda_T$ is an increasing limit on comparatorsmoothness.

Estimating distributions over large alphabets is a fundamental machine-learning tenet. Yet no method is known to estimate all distributions well. For example, add-constant estimators are nearly min-max optimal but often perform poorly in practice, and practical estimators such as absolute discounting, Jelinek-Mercer, and Good-Turing are not known to be near optimal for essentially any distribution.We describe the first universally near-optimal probability estimators. For every discrete distribution, they are provably nearly the best in the following two competitive ways. First they estimate every distribution nearly as well as the best estimator designed with prior knowledge of the distribution up to a permutation. Second, they estimate every distribution nearly as well as the best estimator designed with prior knowledge of the exact distribution, but as all natural estimators, restricted to assign the same probability to all symbols appearing the same number of times.Specifically, for distributions over $k$ symbols and $n$ samples, we show that for both comparisons, a simple variant of Good-Turing estimator is always within KL divergence of $(3+o(1))/n^{1/3}$ from the best estimator, and that a more involved estimator is within $\tilde{\mathcal{O}}(\min(k/n,1/\sqrt n))$. Conversely, we show that any estimator must have a KL divergence $\ge\tilde\Omega(\min(k/n,1/ n^{2/3}))$ over the best estimator for the first comparison, and $\ge\tilde\Omega(\min(k/n,1/\sqrt{n}))$ for the second.

This paper studies the generalization performance of multi-class classification algorithms, for which we obtain, for the first time, a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis. The theoretical analysis motivates us to introduce a new multi-class classification machine based on lp-norm regularization, where the parameter p controls the complexity of the corresponding bounds. We derive an efficient optimization algorithm based on Fenchel duality theory. Benchmarks on several real-world datasets show that the proposed algorithm can achieve significant accuracy gains over the state of the art.

Normalized random measures (NRMs) provide a broad class of discrete random measures that are often used as priors for Bayesian nonparametric models. Dirichlet process is a well-known example of NRMs. Most of posterior inference methods for NRM mixture models rely on MCMC methods since they are easy to implement and their convergence is well studied. However, MCMC often suffers from slow convergence when the acceptance rate is low. Tree-based inference is an alternative deterministic posterior inference method, where Bayesian hierarchical clustering (BHC) or incremental Bayesian hierarchical clustering (IBHC) have been developed for DP or NRM mixture (NRMM) models, respectively. Although IBHC is a promising method for posterior inference for NRMM models due to its efficiency and applicability to online inference, its convergence is not guaranteed since it uses heuristics that simply selects the best solution after multiple trials are made. In this paper, we present a hybrid inference algorithm for NRMM models, which combines the merits of both MCMC and IBHC. Trees built by IBHC outlinespartitions of data, which guides Metropolis-Hastings procedure to employ appropriate proposals. Inheriting the nature of MCMC, our tree-guided MCMC (tgMCMC) is guaranteed to converge, and enjoys the fast convergence thanks to the effective proposals guided by trees. Experiments on both synthetic and real world datasets demonstrate the benefit of our method.

Learning of low dimensional structure in multidimensional data is a canonical problem in machine learning. One common approach is to suppose that the observed data are close to a lower-dimensional smooth manifold. There are a rich variety of manifold learning methods available, which allow mapping of data points to the manifold. However, there is a clear lack of probabilistic methods that allow learning of the manifold along with the generative distribution of the observed data. The best attempt is the Gaussian process latent variable model (GP-LVM), but identifiability issues lead to poor performance. We solve these issues by proposing a novel Coulomb repulsive process (Corp) for locations of points on the manifold, inspired by physical models of electrostatic interactions among particles. Combining this process with a GP prior for the mapping function yields a novel electrostatic GP (electroGP) process. Focusing on the simple case of a one-dimensional manifold, we develop efficient inference algorithms, and illustrate substantially improved performance in a variety of experiments including filling in missing frames in video.

Within a statistical learning setting, we propose and study an iterative regularization algorithmfor least squares defined by an incremental gradient method. In particular, we show that, if all other parameters are fixed a priori, the number of passes over the data (epochs) acts as a regularization parameter, and prove strong universal consistency, i.e. almost sure convergence of the risk, as well as sharp finite sample bounds for the iterates. Our results are a step towards understanding the effect of multiple epochs in stochastic gradient techniques in machine learning and rely on integrating statistical and optimization results.

We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the noise inherent to the stochastic approximation scheme dominates any noise from asynchrony. We also give empirical evidence demonstrating the strong performance of asynchronous, parallel stochastic optimization schemes, demonstrating that the robustness inherent to stochastic approximation problems allows substantially faster parallel and asynchronous solution methods. In short, we show that for many stochastic approximation problems, as Freddie Mercury sings in Queen's \emph{Bohemian Rhapsody}, ``Nothing really matters.''