Sparse coding has recently become a popular approach in computer vision to learn dictionaries of natural images. In this paper we extend sparse coding to learn interpretable spatio-temporal primitives of human motion. We cast the problem of learning spatio-temporal primitives as a tensor factorization problem and introduce constraints to learn interpretable primitives. In particular, we use group norms over those tensors, diagonal constraints on the activations as well as smoothness constraints that are inherent to human motion. We demonstrate the effectiveness of our approach to learn interpretable representations of human motion from motion capture data, and show that our approach outperforms recently developed matching pursuit and sparse coding algorithms.

We provide new theoretical results for apprenticeship learning, a variant of reinforcement learning in which the true reward function is unknown, and the goal is to perform well relative to an observed expert. We study a common approach to learning from expert demonstrations: using a classification algorithm to learn to imitate the expert's behavior. Although this straightforward learning strategy is widely-used in practice, it has been subject to very little formal analysis. We prove that, if the learned classifier has error rate $\eps$, the difference between the value of the apprentice's policy and the expert's policy is $O(\sqrt{\eps})$. Further, we prove that this difference is only $O(\eps)$ when the expert's policy is close to optimal. This latter result has an important practical consequence: Not only does imitating a near-optimal expert result in a better policy, but far fewer demonstrations are required to successfully imitate such an expert. This suggests an opportunity for substantial savings whenever the expert is known to be good, but demonstrations are expensive or difficult to obtain.

Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of constrained optimization problems with nonquadratic objective and constraints can be understood as nonlinear eigenproblems. We derive a generalization of the inverse power method which is guaranteed to converge to a nonlinear eigenvector. We apply the inverse power method to 1-spectral clustering and sparse PCA which can naturally be formulated as nonlinear eigenproblems. In both applications we achieve state-of-the-art results in terms of solution quality and runtime. Moving beyond the standard eigenproblem should be useful also in many other applications and our inverse power method can be easily adapted to new problems.

The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis clearly separates the convergence of the optimization algorithm itself from the effects of communication constraints arising from the network structure. We show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks.

We extend logistic regression by using t-exponential families which were introduced recently in statistical physics. This gives rise to a regularized risk minimization problem with a non-convex loss function. An efficient block coordinate descent optimization scheme can be derived for estimating the parameters. Because of the nature of the loss function, our algorithm is tolerant to label noise. Furthermore, unlike other algorithms which employ non-convex loss functions, our algorithm is fairly robust to the choice of initial values. We verify both these observations empirically on a number of synthetic and real datasets.

Recent experimental work has suggested that the neural firing rate can be interpreted as a fractional derivative, at least when signal variation induces neural adaptation. Here, we show that the actual neural spike-train itself can be considered as the fractional derivative, provided that the neural signal is approximated by a sum of power-law kernels. A simple standard thresholding spiking neuron suffices to carry out such an approximation, given a suitable refractory response. Empirically, we find that the online approximation of signals with a sum of power-law kernels is beneficial for encoding signals with slowly varying components, like long-memory self-similar signals. For such signals, the online power-law kernel approximation typically required less than half the number of spikes for similar SNR as compared to sums of similar but exponentially decaying kernels. As power-law kernels can be accurately approximated using sums or cascades of weighted exponentials, we demonstrate that the corresponding decoding of spike-trains by a receiving neuron allows for natural and transparent temporal signal filtering by tuning the weights of the decoding kernel.

In some stochastic environments the well-known reinforcement learning algorithm Q-learning performs very poorly. This poor performance is caused by large overestimations of action values. These overestimations result from a positive bias that is introduced because Q-learning uses the maximum action value as an approximation for the maximum expected action value. We introduce an alternative way to approximate the maximum expected value for any set of random variables. The obtained double estimator method is shown to sometimes underestimate rather than overestimate the maximum expected value. We apply the double estimator to Q-learning to construct Double Q-learning, a new off-policy reinforcement learning algorithm. We show the new algorithm converges to the optimal policy and that it performs well in some settings in which Q-learning performs poorly due to its overestimation.

Dimensionality reduction is commonly used in the setting of multi-label supervised classification to control the learning capacity and to provide a meaningful representation of the data. We introduce a simple forward probabilistic model which is a multinomial extension of reduced rank regression; we show that this model provides a probabilistic interpretation of discriminative clustering methods with added benefits in terms of number of hyperparameters and optimization. While expectation-maximization (EM) algorithm is commonly used to learn these models, its optimization usually leads to local minimum because it relies on a non-convex cost function with many such local minima. To avoid this problem, we introduce a local approximation of this cost function, which leads to a quadratic non-convex optimization problem over a product of simplices. In order to minimize such functions, we propose an efficient algorithm based on convex relaxation and low-rank representation of our data, which allows to deal with large instances. Experiments on text document classification show that the new model outperforms other supervised dimensionality reduction methods, while simulations on unsupervised clustering show that our probabilistic formulation has better properties than existing discriminative clustering methods.

Remarkably easy implementation and guaranteed convergence has made the EM algorithm one of the most used algorithms for mixture modeling. On the downside, the E-step is linear in both the sample size and the number of mixture components, making it impractical for large-scale data. Based on the variational EM framework, we propose a fast alternative that uses component-specific data partitions to obtain a sub-linear E-step in sample size, while the algorithm still maintains provable convergence. Our approach builds on previous work, but is significantly faster and scales much better in the number of mixture components. We demonstrate this speedup by experiments on large-scale synthetic and real data.

The CUR decomposition provides an approximation of a matrix X that has low reconstruction error and that is sparse in the sense that the resulting approximation lies in the span of only a few columns of X. In this regard, it appears to be similar to many sparse PCA methods. However, CUR takes a randomized algorithmic approach whereas most sparse PCA methods are framed as convex optimization problems. In this paper, we try to understand CUR from a sparse optimization viewpoint. In particular, we show that CUR is implicitly optimizing a sparse regression objective and, furthermore, cannot be directly cast as a sparse PCA method. We observe that the sparsity attained by CUR possesses an interesting structure, which leads us to formulate a sparse PCA method that achieves a CUR-like sparsity.