We propose a new approach to value function approximation which combines linear temporal difference reinforcement learning with subspace identification. In practical applications, reinforcement learning (RL) is complicated by the fact that state is either high-dimensional or partially observable. Therefore, RL methods are designed to work with features of state rather than state itself, and the success or failure of learning is often determined by the suitability of the selected features. By comparison, subspace identification (SSID) methods are designed to select a feature set which preserves as much information as possible about state. In this paper we connect the two approaches, looking at the problem of reinforcement learning with a large set of features, each of which may only be marginally useful for value function approximation. We introduce a new algorithm for this situation, called Predictive State Temporal Difference (PSTD) learning. As in SSID for predictive state representations, PSTD finds a linear compression operator that projects a large set of features down to a small set that preserves the maximum amount of predictive information. As in RL, PSTD then uses a Bellman recursion to estimate a value function. We discuss the connection between PSTD and prior approaches in RL and SSID. We prove that PSTD is statistically consistent, perform several experiments that illustrate its properties, and demonstrate its potential on a difficult optimal stopping problem.
The drawback of all of the approaches enumerated above is that they first assume that the dynamical system model is known, and only then give us a way of finding a compact representation and a value function. In practice, we would like to be able to find a good set of features, . Kolter and Ng @cite_18 contend with this problem from a sparse feature selection standpoint. Given a large set of possibly-relevant features of observations, they proposed augmenting LSTD by applying an @math penalty to the coefficients, forcing LSTD to select a sparse set of features for value function estimation. The resulting algorithm, LARS-TD, works well in certain situations (for example, see ), but only if our original large set of features contains a small subset of highly-relevant features.
@cite_18
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We introduce a new adaptive method for analyzing nonlinear and nonstationary data. This method is inspired by the empirical mode decomposition (EMD) method and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form a(t )c os(θ(t)) ,w herea ≥ 0i s assumed to be smoother than cos(θ(t)) and θ is a piecewise smooth increasing function. We formulate this problem as a nonlinear L 1 optimization problem. Further, we propose an iterative algorithm to solve this nonlinear optimization problem recursively. We also introduce an adaptive filter method to decompose data with noise. Numerical examples are given to demonstrate the robustness of our method and comparison is made with the EMD method. One advantage of performing such a decomposition is to preserve some intrinsic physical property of the signal, such as trend and instantaneous frequency. Our method shares many important properties of the original EMD method. Because our method is based on a solid mathematical formulation, its performance does not depend on numerical parameters such as the number of shifting or stop criterion, which seem to have a major effect on the original EMD method. Our method is also less sensitive to noise perturbation and the end effect compared with the original EMD method. Summary Variable selection for recovering sparsity in nonadditive and nonparametric models with high-dimensional variables has been challenging. This problem becomes even more difficult due to complications in modeling unknown interaction terms among high-dimensional variables. There is currently no variable selection method to overcome these limitations. Hence, in this article we propose a variable selection approach that is developed by connecting a kernel machine with the nonparametric regression model. The advantages of our approach are that it can: (i) recover the sparsity; (ii) automatically model unknown and complicated interactions; (iii) connect with several existing approaches including linear nonnegative garrote and multiple kernel learning; and (iv) provide flexibility for both additive and nonadditive nonparametric models. Our approach can be viewed as a nonlinear version of a nonnegative garrote method. We model the smoothing function by a Least Squares Kernel Machine (LSKM) and construct the nonnegative garrote objective function as the function of the sparse scale parameters of kernel machine to recover sparsity of input variables whose relevances to the response are measured by the scale parameters. We also provide the asymptotic properties of our approach. We show that sparsistency is satisfied with consistent initial kernel function coefficients under certain conditions. An efficient coordinate descent backfitting algorithm is developed. A resampling procedure for our variable selection methodology is also proposed to improve the power. This paper considers constrained lscr1 minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both lscr1 minimization with an lscrinfin constraint (Dantzig selector) and lscr1 minimization under an llscr2 constraint are considered. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. In particular, our results illustrate the relationship between lscr1 minimization with an llscr2 constraint and lscr1 minimization with an lscrinfin constraint. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg, and Tao (2006), Candes and Tao (2007), and Donoho, Elad, and Temlyakov (2006) are extended. Recommender problems with large and dynamic item pools are ubiquitous in web applications like content optimization, online advertising and web search. Despite the availability of rich item meta-data, excess heterogeneity at the item level often requires inclusion of item-specific "factors" (or weights) in the model. However, since estimating item factors is computationally intensive, it poses a challenge for time-sensitive recommender problems where it is important to rapidly learn factors for new items (e.g., news articles, event updates, tweets) in an online fashion. In this paper, we propose a novel method called FOBFM (Fast Online Bilinear Factor Model) to learn item-specific factors quickly through online regression. The online regression for each item can be performed independently and hence the procedure is fast, scalable and easily parallelizable. However, the convergence of these independent regressions can be slow due to high dimensionality. The central idea of our approach is to use a large amount of historical data to initialize the online models based on offline features and learn linear projections that can effectively reduce the dimensionality. We estimate the rank of our linear projections by taking recourse to online model selection based on optimizing predictive likelihood. Through extensive experiments, we show that our method significantly and uniformly outperforms other competitive methods and obtains relative lifts that are in the range of 10-15 in terms of predictive log-likelihood, 200-300 for a rank correlation metric on a proprietary My Yahoo! dataset; it obtains 9 reduction in root mean squared error over the previously best method on a benchmark MovieLens dataset using a time-based train test data split. It has been known for a while that l1-norm relaxation can in certain cases solve an under-determined system of linear equations. Recently, E. Candes ("Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information," IEEE Trans. Information Theory, vol. 52, no. 12, pp. 489-509, Dec. 2006) and D. Donoho ("High-dimensional centrally symmetric polytopes with neighborlines proportional to dimension," Disc. Comput. Geometry, vol. 35, no. 4, pp. 617-652, 2006) proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of nonzero elements of the unknown vector) also proportional to the length of the unknown vector such that l1-norm relaxation succeeds in solving the system. In this paper, in a large dimensional and statistical context, we determine sharp lower bounds on the values of allowable sparsity for any given number (proportional to the length of the unknown vector) of equations for the case of the so-called block-sparse unknown vectors considered in "On the reconstruction of block-sparse signals with an optimal number of measurements," (M. , IEEE Trans, Signal Processing, submitted for publication. Recently, a number of researchers have proposed spectral algorithms for learning models of dynamical systems—for example, Hidden Markov Models (HMMs), Partially Observable Markov Decision Processes (POMDPs), and Transformed Predictive State Representations (TPSRs). These algorithms are attractive since they are statistically consistent and not subject to local optima. However, they are batch methods: they need to store their entire training data set in memory at once and operate on it as a large matrix, and so they cannot scale to extremely large data sets (either many examples or many features per example). In turn, this restriction limits their ability to learn accurate models of complex systems. To overcome these limitations, we propose a new online spectral algorithm, which uses tricks such as incremental Singular Value Decomposition (SVD) and random projections to scale to much larger data sets and more complex systems than previous methods. We demonstrate the new method on an inertial measurement prediction task and a high-bandwidth video mapping task and we illustrate desirable behaviors such as "closing the loop," where the latent state representation changes suddenly as the learner recognizes that it has returned to a previously known place. In this paper, we address the problem of 3D object instance recognition and pose estimation of localized objects in cluttered environments using convolutional neural networks. Inspired by the descriptor learning approach of , we propose a method that introduces the dynamic margin in the manifold learning triplet loss function. Such a loss function is designed to map images of different objects under different poses to a lower-dimensional, similarity-preserving descriptor space on which efficient nearest neighbor search algorithms can be applied. Introducing the dynamic margin allows for faster training times and better accuracy of the resulting low-dimensional manifolds. Furthermore, we contribute the following: adding in-plane rotations (ignored by the baseline method) to the training, proposing new background noise types that help to better mimic realistic scenarios and improve accuracy with respect to clutter, adding surface normals as another powerful image modality representing an object surface leading to better performance than merely depth, and finally implementing an efficient online batch generation that allows for better variability during the training phase. We perform an exhaustive evaluation to demonstrate the effects of our contributions. Additionally, we assess the performance of the algorithm on the large BigBIRD dataset to demonstrate good scalability properties of the pipeline with respect to the number of models. Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (lscr 2) error term added to a sparsity-inducing (usually lscr1) regularizater. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (i.e., separable in the unknowns) plus the original sparsity-inducing regularizer; our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. Under mild conditions (namely convexity of the regularizer), we prove convergence of the proposed iterative algorithm to a minimum of the objective function. In addition to solving the standard lscr2-lscr1 case, our framework yields efficient solution techniques for other regularizers, such as an lscrinfin norm and group-separable regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard lscr2-lscr1 problem, as well as being efficient on problems with other separable regularization terms. Patch-based low-rank models have shown effective in exploiting spatial redundancy of natural images especially for the application of image denoising. However, two-dimensional low-rank model can not fully exploit the spatio-temporal correlation in larger data sets such as multispectral images and 3D MRIs. In this work, we propose a novel low-rank tensor approximation framework with Laplacian Scale Mixture (LSM) modeling for multi-frame image denoising. First, similar 3D patches are grouped to form a tensor of d-order and high-order Singular Value Decomposition (HOSVD) is applied to the grouped tensor. Then the task of multiframe image denoising is formulated as a Maximum A Posterior (MAP) estimation problem with the LSM prior for tensor coefficients. Both unknown sparse coefficients and hidden LSM parameters can be efficiently estimated by the method of alternating optimization. Specifically, we have derived closed-form solutions for both subproblems. Experimental results on spectral and dynamic MRI images show that the proposed algorithm can better preserve the sharpness of important image structures and outperform several existing state-of-the-art multiframe denoising methods (e.g., BM4D and tensor dictionary learning). The goal of sparse recovery is to recover a k-sparse signal x e Rn from (possibly noisy) linear measurements of the form y = Ax, where A e Rmxn describes the measurement process. Standard results in compressive sensing show that it is possible to recover the signal x from m = O(k log(n k)) measurements, and that this bound is tight. The framework of model-based compressive sensing [BCDH10] overcomes the lower bound and reduces the number of measurements further to O(k) by limiting the supports of x to a subset M of the (nk) possible supports. This has led to many measurement-efficient algorithms for a wide variety of signal models, including block-sparsity and tree-sparsity. Unfortunately, extending the framework to other, more general models has been stymied by the following obstacle: for the framework to apply, one needs an algorithm that, given a signal x, solves the following optimization problem exactly: [EQUATION] (here x[n] denotes the projection of x on coordinates not in Ω). However, an approximation algorithm for this optimization task is not sufficient. Since many problems of this form are not known to have exact polynomial-time algorithms, this requirement poses an obstacle for extending the framework to a richer class of models. In this paper, we remove this obstacle and show how to extend the model-based compressive sensing framework so that it requires only approximate solutions to the aforementioned optimization problems. Interestingly, our extension requires the existence of approximation algorithms for both the maximization and the minimization variants of the optimization problem. Further, we apply our framework to the Constrained Earth Mover's Distance (CEMD) model introduced in [SHI13], obtaining a sparse recovery scheme that uses significantly less than O(k log(n k)) measurements. This is the first non-trivial theoretical bound for this model, since the validation of the approach presented in [SHI13] was purely empirical. The result is obtained by designing a novel approximation algorithm for the maximization version of the problem and proving approximation guarantees for the minimization algorithm described in [SHI13]. We consider the problem of state-estimation of a linear dynamical system when some of the sensor measurements are corrupted by an adversarial attacker. The errors injected by the attacker in the sensor measurements can be arbitrary and are not assumed to follow a specific model (in particular they can be of arbitrary magnitude). We first characterize the number of attacked sensors that can be tolerated so that the state of the system can still be correctly recovered by any decoding algorithm. We then propose a specific computationally feasible decoding algorithm and we give a characterization of the number of errors this decoder can correct. For this we use ideas from compressed sensing and error correction over the reals and we exploit the dynamical nature of the problem. We show using numerical simulations that this decoder performs very well in practice and allows to correct a large number of errors. Many pattern analysis and data mining problems have witnessed high-dimensional data represented by a large number of features, which are often redundant and noisy. Feature selection is one main technique for dimensionality reduction that involves identifying a subset of the most useful features. In this paper, a novel unsupervised feature selection algorithm, named clustering-guided sparse structural learning (CGSSL), is proposed by integrating cluster analysis and sparse structural analysis into a joint framework and experimentally evaluated. Nonnegative spectral clustering is developed to learn more accurate cluster labels of the input samples, which guide feature selection simultaneously. Meanwhile, the cluster labels are also predicted by exploiting the hidden structure shared by different features, which can uncover feature correlations to make the results more reliable. Row-wise sparse models are leveraged to make the proposed model suitable for feature selection. To optimize the proposed formulation, we propose an efficient iterative algorithm. Finally, extensive experiments are conducted on 12 diverse benchmarks, including face data, handwritten digit data, document data, and biomedical data. The encouraging experimental results in comparison with several representative algorithms and the theoretical analysis demonstrate the efficiency and effectiveness of the proposed algorithm for feature selection. Consider the problem of joint parameter estimation and prediction in a Markov random field: that is, the model parameters are estimated on the basis of an initial set of data, and then the fitted model is used to perform prediction (e.g., smoothing, denoising, interpolation) on a new noisy observation. Working under the restriction of limited computation, we analyze a joint method in which the same convex variational relaxation is used to construct an M-estimator for fitting parameters, and to perform approximate marginalization for the prediction step. The key result of this paper is that in the computation-limited setting, using an inconsistent parameter estimator (i.e., an estimator that returns the "wrong" model even in the infinite data limit) is provably beneficial, since the resulting errors can partially compensate for errors made by using an approximate prediction technique. En route to this result, we analyze the asymptotic properties of M-estimators based on convex variational relaxations, and establish a Lipschitz stability property that holds for a broad class of convex variational methods. This stability result provides additional incentive, apart from the obvious benefit of unique global optima, for using message-passing methods based on convex variational relaxations. We show that joint estimation prediction based on the reweighted sum-product algorithm substantially outperforms a commonly used heuristic based on ordinary sum-product. Two problems arise when using distant supervision for relation extraction. First, in this method, an already existing knowledge base is heuristically aligned to texts, and the alignment results are treated as labeled data. However, the heuristic alignment can fail, resulting in wrong label problem. In addition, in previous approaches, statistical models have typically been applied to ad hoc features. The noise that originates from the feature extraction process can cause poor performance. In this paper, we propose a novel model dubbed the Piecewise Convolutional Neural Networks (PCNNs) with multi-instance learning to address these two problems. To solve the first problem, distant supervised relation extraction is treated as a multi-instance problem in which the uncertainty of instance labels is taken into account. To address the latter problem, we avoid feature engineering and instead adopt convolutional architecture with piecewise max pooling to automatically learn relevant features. Experiments show that our method is effective and outperforms several competitive baseline methods. Discriminative methods for learning structured models have enabled wide-spread use of very rich feature representations. However, the computational cost of feature extraction is prohibitive for large-scale or time-sensitive applications, often dominating the cost of inference in the models. Significant efforts have been devoted to sparsity-based model selection to decrease this cost. Such feature selection methods control computation statically and miss the opportunity to fine-tune feature extraction to each input at run-time. We address the key challenge of learning to control fine-grained feature extraction adaptively, exploiting non-homogeneity of the data. We propose an architecture that uses a rich feedback loop between extraction and prediction. The run-time control policy is learned using efficient value-function approximation, which adaptively determines the value of information of features at the level of individual variables for each input. We demonstrate significant speedups over state-of-the-art methods on two challenging datasets. For articulated pose estimation in video, we achieve a more accurate state-of-the-art model that is also faster, with similar results on an OCR task. We consider the task of reinforcement learning with linear value function approximation. Temporal difference algorithms, and in particular the Least-Squares Temporal Difference (LSTD) algorithm, provide a method for learning the parameters of the value function, but when the number of features is large this algorithm can over-fit to the data and is computationally expensive. In this paper, we propose a regularization framework for the LSTD algorithm that overcomes these difficulties. In particular, we focus on the case of l1 regularization, which is robust to irrelevant features and also serves as a method for feature selection. Although the l1 regularized LSTD solution cannot be expressed as a convex optimization problem, we present an algorithm similar to the Least Angle Regression (LARS) algorithm that can efficiently compute the optimal solution. Finally, we demonstrate the performance of the algorithm experimentally. We present an algorithm that infers the model structure of a mixture of factor analysers using an efficient and deterministic variational approximation to full Bayesian integration over model parameters. This procedure can automatically determine the optimal number of components and the local dimensionality of each component (i.e. the number of factors in each factor analyser). Alternatively it can be used to infer posterior distributions over number of components and dimensionalities. Since all parameters are integrated out the method is not prone to overfitting. Using a stochastic procedure for adding components it is possible to perform the variational optimisation incrementally and to avoid local maxima. Results show that the method works very well in practice and correctly infers the number and dimensionality of nontrivial synthetic examples. By importance sampling from the variational approximation we show how to obtain unbiased estimates of the true evidence, the exact predictive density, and the KL divergence between the variational posterior and the true posterior, not only in this model but for variational approximations in general. We describe a novel unsupervised method for learning sparse, overcomplete features. The model uses a linear encoder, and a linear decoder preceded by a sparsifying non-linearity that turns a code vector into a quasi-binary sparse code vector. Given an input, the optimal code minimizes the distance between the output of the decoder and the input patch while being as similar as possible to the encoder output. Learning proceeds in a two-phase EM-like fashion: (1) compute the minimum-energy code vector, (2) adjust the parameters of the encoder and decoder so as to decrease the energy. The model produces "stroke detectors" when trained on handwritten numerals, and Gabor-like filters when trained on natural image patches. Inference and learning are very fast, requiring no preprocessing, and no expensive sampling. Using the proposed unsupervised method to initialize the first layer of a convolutional network, we achieved an error rate slightly lower than the best reported result on the MNIST dataset. Finally, an extension of the method is described to learn topographical filter maps. Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on ℝN with smoothness of order s can in general be captured with accuracy at most O(n−s N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x)=g(a⋅x) when both a∈ℝN and g∈C[0,1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g∈Cs[0,1]. We also study the role of sparsity or compressibility of a in such query problems. The set of features used by a learning algorithm can have a dramatic impact on the performance of that algorithm. Including extraneous features can make the learning problem harder by adding useless, noisy dimensions that lead to over-fitting and increased computational complexity. Conversely, leaving out useful features can deprive the model of important signals. The problem of feature selection is to find a subset of features that allows the learning algorithm to learn the “best” model in terms of measures such as accuracy or model simplicity. The problem of feature selection continues to grow in both importance and difficulty as extremely high-dimensional data sets become the standard in real-world machine learning tasks. Scalability can become a problem for even simple approaches. For example, common feature selection approaches that evaluate each new feature by training a new model containing that feature require a learning a linear number of models each time they add a new feature. This computational cost can add up quickly when we are iteratively adding many new features. Even techniques that use relatively computationally inexpensive tests of a feature’s value, such as mutual information, require at least linear time in the number of features being evaluated. As a simple illustrative example consider the task of classifying websites. In this case the data set could easily contain many millions of examples. Just including very basic features such as text unigrams on the page or HTML tags could easily provide many thousands of potential features for the model. Considering more complex attributes such as bigrams of words