There has been a surge of interest in learning nonlinear manifold models to approximate high-dimensional data. Both for computational complexity reasons and for generalization capability, sparsity is a desired feature in such models. This usually means dimensionality reduction, which naturally implies estimating the intrinsic dimension, but it can also mean selecting a subset of the data to use as landmarks, which is especially important because many existing algorithms have quadratic complexity in the number of observations.

We extend radial basis function (RBF) networks to the scenario in which multiple correlated tasks are learned simultaneously, and present the corresponding learning algorithms. We develop the algorithms for learning the network structure, in either a supervised or unsupervised manner. Training data may also be actively selected to improve the network's generalization to test data. Experimental results based on real data demonstrate the advantage of the proposed algorithms and support our conclusions.

We present a competitive analysis of some nonparametric Bayesian algorithms in a worst-case online learning setting, where no probabilistic assumptions about the generation of the data are made. We consider models which use a Gaussian process prior (over the space of all functions) and provide bounds on the regret (under the log loss) for commonly used nonparametric Bayesian algorithms -- including Gaussian regression and logistic regression -- which show how these algorithms can perform favorably under rather general conditions. These bounds explicitly handle the infinite dimensionality of these nonparametric classes in a natural way. We also make formal connections to the minimax and minimum description length (MDL) framework. Here, we show precisely how Bayesian Gaussian regression is a minimax strategy.

It is well-known that everything that is learnable in the difficult online setting, where an arbitrary sequences of examples must be labeled one at a time, is also learnable in the batch setting, where examples are drawn independently from a distribution. We show a result in the opposite direction. We give an efficient conversion algorithm from batch to online that is transductive: it uses future unlabeled data. This demonstrates the equivalence between what is properly and efficiently learnable in a batch model and a transductive online model.

The goal of this study is to combine neuronal mechanisms with biomechanics to obtain very fast speed and the online learning of circuit parameters. Our controller is built with biologically inspired sensor-and motor-neuron models, including local reflexes and not employing any kind of position or trajectory-tracking control algorithm. Instead, this reflexive controller allows RunBot to exploit its own natural dynamics during critical stages of its walking gait cycle. To our knowledge, this is the first time that dynamic biped walking is achieved using only a pure reflexive controller. In addition, this structure allows using a policy gradient reinforcement learning algorithm to tune the parameters of the reflexive controller in real-time during walking. This way RunBot can reach a relative speed of 3.5 leg-lengths per second after a few minutes of online learning, which is faster than that of any other biped robot, and is also comparable to the fastest relative speed of human walking. In addition, the stability domain of stable walking is quite large supporting this design strategy.

Online learning algorithms are typically fast, memory efficient, and simple to implement. However, many common learning problems fit more naturally in the batch learning setting. The power of online learning algorithms can be exploited in batch settings by using online-to-batch conversions techniques which build a new batch algorithm from an existing online algorithm. We first give a unified overview of three existing online-to-batch conversion techniques which do not use training data in the conversion process. We then build upon these data-independent conversions to derive and analyze data-driven conversions. Our conversions find hypotheses with a small risk by explicitly minimizing datadependent generalization bounds. We experimentally demonstrate the usefulness of our approach and in particular show that the data-driven conversions consistently outperform the data-independent conversions.

The Perceptron algorithm, despite its simplicity, often performs well on online classification tasks. The Perceptron becomes especially effective when it is used in conjunction with kernels. However, a common difficulty encountered when implementing kernel-based online algorithms is the amount of memory required to store the online hypothesis, which may grow unboundedly. In this paper we present and analyze the Forgetron algorithm for kernel-based online learning on a fixed memory budget. To our knowledge, this is the first online learning algorithm which, on one hand, maintains a strict limit on the number of examples it stores while, on the other hand, entertains a relative mistake bound. In addition to the formal results, we also present experiments with real datasets which underscore the merits of our approach.

This paper proposes an algorithm to convert a T -stage stochastic decision problem with a continuous state space to a sequence of supervised learning problems. The optimization problem associated with the trajectory tree and random trajectory methods of Kearns, Mansour, and Ng, 2000, is solved using the Gauss-Seidel method. The algorithm breaks a multistage reinforcement learning problem into a sequence of single-stage reinforcement learning subproblems, each of which is solved via an exact reduction to a weighted-classification problem that can be solved using off-the-self methods. Thus the algorithm converts a reinforcement learning problem into simpler supervised learning subproblems. It is shown that the method converges in a finite number of steps to a solution that cannot be further improved by componentwise optimization. The implication of the proposed algorithm is that a plethora of classification methods can be applied to find policies in the reinforcement learning problem.

We present a series of theoretical arguments supporting the claim that a large class of modern learning algorithms that rely solely on the smoothness prior - with similarity between examples expressed with a local kernel - are sensitive to the curse of dimensionality, or more precisely to the variability of the target. Our discussion covers supervised, semisupervised and unsupervised learning algorithms. These algorithms are found to be local in the sense that crucial properties of the learned function at x depend mostly on the neighbors of x in the training set. This makes them sensitive to the curse of dimensionality, well studied for classical nonparametric statistical learning. We show in the case of the Gaussian kernel that when the function to be learned has many variations, these algorithms require a number of training examples proportional to the number of variations, which could be large even though there may exist short descriptions of the target function, i.e. their Kolmogorov complexity may be low. This suggests that there exist non-local learning algorithms that at least have the potential to learn about such structured but apparently complex functions (because locally they have many variations), while not using very specific prior domain knowledge.

We propose a fast manifold learning algorithm based on the methodology ofdomain decomposition. Starting with the set of sample points partitioned into two subdomains, we develop the solution of the interface problemthat can glue the embeddings on the two subdomains into an embedding on the whole domain. We provide a detailed analysis to assess the errors produced by the gluing process using matrix perturbation theory.Numerical examples are given to illustrate the efficiency and effectiveness of the proposed methods.