Policy gradient (PG) reinforcement learning algorithms have strong (local) convergence guarantees, but their learning performance is typically limited by a large variance in the estimate of the gradient. In this paper, we formulate the variance reduction problem by describing a signal-to-noise ratio (SNR) for policy gradient algorithms, and evaluate this SNR carefully for the popular Weight Perturbation (WP) algorithm. We confirm that SNR is a good predictor of long-term learning performance, and that in our episodic formulation, the cost-to-go function is indeed the optimal baseline. We then propose two modifications to traditional model-free policy gradient algorithms in order to optimize the SNR. First, we examine WP using anisotropic sampling distributions, which introduces a bias into the update but increases the SNR; this bias can be interpretted as following the natural gradient of the cost function. Second, we show that non-Gaussian distributions can also increase the SNR, and argue that the optimal isotropic distribution is a ‘shell’ distribution with a constant magnitude and uniform distribution in direction. We demonstrate that both modifications produce substantial improvements in learning performance in challenging policy gradient experiments.

The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. Despite its far-reaching application, there is almost no work on applying stochastic approximation to learning problems with constraints. The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. We propose that interior-point methods are a natural solution. We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization.

For undiscounted reinforcement learning in Markov decision processes (MDPs) we consider the total regret of a learning algorithm with respect to an optimal policy. In order to describe the transition structure of an MDP we propose a new parameter: An MDP has diameter D if for any pair of states s1,s2 there is a policy which moves from s1 to s2 in at most D steps (on average). We present a reinforcement learning algorithm with total regret O(DSAT) after T steps for any unknown MDP with S states, A actions per state, and diameter D. This bound holds with high probability. We also present a corresponding lower bound of Omega(DSAT) on the total regret of any learning algorithm. Both bounds demonstrate the utility of the diameter as structural parameter of the MDP.

While many advances have already been made in hierarchical classification learning, wetake a step back and examine how a hierarchical classification problem should be formally defined. We pay particular attention to the fact that many arbitrary decisionsgo into the design of the label taxonomy that is given with the training data. Moreover, many hand-designed taxonomies are unbalanced and misrepresent the class structure in the underlying data distribution. We attempt to correct these problems by using the data distribution itself to calibrate the hierarchical classificationloss function. This distribution-based correction must be done with care, to avoid introducing unmanageable statistical dependencies into the learning problem. This leads us off the beaten path of binomial-type estimation andinto the unfamiliar waters of geometric-type estimation. In this paper, we present a new calibrated definition of statistical risk for hierarchical classification, anunbiased estimator for this risk, and a new algorithmic reduction from hierarchical classification to cost-sensitive classification.

The linear correlation coefficient is typically used to characterize and analyze dependencies of neural spike counts. Here, we show that the correlation coefficient is in general insufficient to characterize these dependencies. We construct two neuron spike count models with Poisson-like marginals and vary their dependence structure using copulas. To this end, we construct a copula that allows to keep the spike counts uncorrelated while varying their dependence strength. Moreover, we employ a network of leaky integrate-and-fire neurons to investigate whether weakly correlated spike counts with strong dependencies are likely to occur in real networks. We find that the entropy of uncorrelated but dependent spike count distributions can deviate from the corresponding distribution with independent components by more than 25% and that weakly correlated but strongly dependent spike counts are very likely to occur in biological networks. Finally, we introduce a test for deciding whether the dependence structure of distributions with Poisson-like marginals is well characterized by the linear correlation coefficient and verify it for different copula-based models.

Kernel Principal Component Analysis (KPCA) is a popular generalization of linear PCA that allows non-linear feature extraction. In KPCA, data in the input space is mapped to higher (usually) dimensional feature space where the data can be linearly modeled. The feature space is typically induced implicitly by a kernel function, and linear PCA in the feature space is performed via the kernel trick. However, due to the implicitness of the feature space, some extensions of PCA such as robust PCA cannot be directly generalized to KPCA. This paper presents a technique to overcome this problem, and extends it to a unified framework for treating noise, missing data, and outliers in KPCA. Our method is based on a novel cost function to perform inference in KPCA. Extensive experiments, in both synthetic and real data, show that our algorithm outperforms existing methods.

We describe a new algorithmic framework for inference in probabilistic models, and apply it to inference for latent Dirichlet allocation. Our framework adopts the methodology of variational inference, but unlike existing variational methods such as mean field and expectation propagation it is not restricted to tractable classes of approximating distributions. Our approach can also be viewed as a sequential Monte Carlo (SMC) method, but unlike existing SMC methods there is no need to design the artificial sequence of distributions. Notably, our framework offers a principled means to exchange the variance of an importance sampling estimate for the bias incurred through variational approximation. Experiments on a challenging inference problem in population genetics demonstrate improvements in stability and accuracy over existing methods, and at a comparable cost.

One of the original goals of computer vision was to fully understand a natural scene. This requires solving several problems simultaneously, including object detection, labeling of meaningful regions, and 3d reconstruction. While great progress has been made in tackling each of these problems in isolation, only recently have researchers again been considering the difficult task of assembling various methods to the mutual benefit of all. We consider learning a set of such classification models in such a way that they both solve their own problem and help each other. We develop a framework known as Cascaded Classification Models (CCM), where repeated instantiations of these classifiers are coupled by their input/output variables in a cascade that improves performance at each level. Our method requires only a limited “black box” interface with the models, allowing us to use very sophisticated, state-of-the-art classifiers without having to look under the hood. We demonstrate the effectiveness of our method on a large set of natural images by combining the subtasks of scene categorization, object detection, multiclass image segmentation, and 3d scene reconstruction.

Covariance estimation for high dimensional vectors is a classically difficult problem in statistical analysis and machine learning due to limited sample size. In this paper, we propose a new approach to covariance estimation, which is based on constrained maximum likelihood (ML) estimation of the covariance. Specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy minimization of the log likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The estimator obtained using this method is always positive definite and well-conditioned even with limited sample size. Experiments on hyperspectral data show that SMT covariance estimation results in consistently better estimates of the covariance for a variety of different classes and sample sizes compared to traditional shrinkage estimators.

Extensive games are often used to model the interactions of multiple agents within an environment. Much recent work has focused on increasing the size of an extensive game that can be feasibly solved. Despite these improvements, many interesting games are still too large for such techniques. A common approach for computing strategies in these large games is to first employ an abstraction technique to reduce the original game to an abstract game that is of a manageable size. This abstract game is then solved and the resulting strategy is used in the original game. Most top programs in recent AAAI Computer Poker Competitions use this approach. The trend in this competition has been that strategies found in larger abstract games tend to beat strategies found in smaller abstract games. These larger abstract games have more expressive strategy spaces and therefore contain better strategies. In this paper we present a new method for computing strategies in large games. This method allows us to compute more expressive strategies without increasing the size of abstract games that we are required to solve. We demonstrate the power of the approach experimentally in both small and large games, while also providing a theoretical justification for the resulting improvement.