The problem of optimal and automatic design of a detector cascade is considered. A novel mathematical model is introduced for a cascaded detector. This model is analytically tractable, leads to recursive computation, and accounts for both classification and complexity. A boosting algorithm, FCBoost, is proposed for fully automated cascade design. It exploits the new cascade model, minimizes a Lagrangian cost that accounts for both classification risk and complexity. It searches the space of cascade configurations to automatically determine the optimal number of stages and their predictors, and is compatible with bootstrapping of negative examples and cost sensitive learning. Experiments show that the resulting cascades have state-of-the-art performance in various computer vision problems.

Despite the ubiquity of clustering as a tool in unsupervised learning, there is not yet a consensus on a formal theory, and the vast majority of work in this direction has focused on unsupervised clustering. We study a recently proposed framework for supervised clustering where there is access to a teacher. We give an improved generic algorithm to cluster any concept class in that model. Our algorithm is query-efficient in the sense that it involves only a small amount of interaction with the teacher. We also present and study two natural generalizations of the model. The model assumes that the teacher response to the algorithm is perfect. We eliminate this limitation by proposing a noisy model and give an algorithm for clustering the class of intervals in this noisy model. We also propose a dynamic model where the teacher sees a random subset of the points. Finally, for datasets satisfying a spectrum of weak to strong properties, we give query bounds, and show that a class of clustering functions containing Single-Linkage will find the target clustering under the strongest property.

We consider the problem of reinforcement learning in high-dimensional spaces when the number of features is bigger than the number of samples. In particular, we study the least-squares temporal difference (LSTD) learning algorithm when a space of low dimension is generated with a random projection from a high-dimensional space. We provide a thorough theoretical analysis of the LSTD with random projections and derive performance bounds for the resulting algorithm. We also show how the error of LSTD with random projections is propagated through the iterations of a policy iteration algorithm and provide a performance bound for the resulting least-squares policy iteration (LSPI) algorithm.

The Random Projection Tree (RPTree) structures proposed in [Dasgupta-Freund-STOC-08] are space partitioning data structures that automatically adapt to various notions of intrinsic dimensionality of data. We prove new results for both the RPTree-Max and the RPTree-Mean data structures. Our result for RPTree-Max gives a near-optimal bound on the number of levels required by this data structure to reduce the size of its cells by a factor s >= 2. We also prove a packing lemma for this data structure. Our final result shows that low-dimensional manifolds possess bounded Local Covariance Dimension. As a consequence we show that RPTree-Mean adapts to manifold dimension as well.

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.

Consider a convex relaxation $\hat f$ of a pseudo-boolean function $f$. We say that the relaxation is {\em totally half-integral} if $\hat f(\bx)$ is a polyhedral function with half-integral extreme points $\bx$, and this property is preserved after adding an arbitrary combination of constraints of the form $x_i=x_j$, $x_i=1-x_j$, and $x_i=\gamma$ where $\gamma\in\{0,1,\frac{1}{2}\}$ is a constant. A well-known example is the {\em roof duality} relaxation for quadratic pseudo-boolean functions $f$. We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations $\hat f$ by establishing a one-to-one correspondence with {\em bisubmodular functions}. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality.

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.