We show convergence rates of ensemble learning methods such as AdaBoost [10], Logistic Regression (LR) [11, 5] and the Least-Square (LS) regression algorithm called LS-Boost [12].

We compare discriminative and generative learning as typified by logistic regression and naive Bayes. We show, contrary to a widely(cid:173) held belief that discriminative classifiers are almost always to be preferred, that there can often be two distinct regimes of per(cid:173) formance as the training set size is increased, one in which each algorithm does better. This stems from the observation- which is borne out in repeated experiments- that while discriminative learning has lower asymptotic error, a generative classifier may also approach its (higher) asymptotic error much faster.

The support vector machine (SVM) is known for its good performance in binary classification, but its extension to multi-class classification is still an on-going research issue. In this paper, we propose a new approach for classification, called the import vector machine (IVM), which is built on kernel logistic regression (KLR). We show that the IVM not only per- forms as well as the SVM in binary classification, but also can naturally be generalized to the multi-class case. Furthermore, the IVM provides an estimate of the underlying probability. Similar to the "support points" of the SVM, the IVM model uses only a fraction of the training data to index kernel basis functions, typically a much smaller fraction than the SVM.

We formulate the regression problem as one of maximizing the mini- mum probability, symbolized by (cid:10), that future predicted outputs of the regression model will be within some (cid:6)" bound of the true regression function. Our formulation is unique in that we obtain a direct estimate of this lower probability bound (cid:10). The proposed framework, minimax probability machine regression (MPMR), is based on the recently de- scribed minimax probability machine classification algorithm [Lanckriet et al.] and uses Mercer Kernels to obtain nonlinear regression models. MPMR is tested on both toy and real world data, verifying the accuracy of the (cid:10) bound, and the efficacy of the regression models.

We introduce the Generalized2 Linear2 Model, a statistical estima(cid:173) tor which combines features of nonlinear regression and factor anal(cid:173) ysis. Here A and Bare low-rank matrices, while j, g, and h are link functions. They also include new and interesting special cases, one of which we describe below. We also present an iterative procedure which optimizes the parameters of a (GL)2M. This procedure reduces to well-known algorithms for some of the special cases listed above; for other special cases, it is new.

We devise and experiment with a dynamical kernel-based system for tracking hand movements from neural activity. The state of the system corresponds to the hand location, velocity, and acceleration, while the system's input are the instantaneous spike rates. The system's state dy- namics is defined as a combination of a linear mapping from the previous estimated state and a kernel-based mapping tailored for modeling neural activities. In contrast to generative models, the activity-to-state mapping is learned using discriminative methods by minimizing a noise-robust loss function. We use this approach to predict hand trajectories on the basis of neural activity in motor cortex of behaving monkeys and find that the proposed approach is more accurate than both a static approach based on support vector regression and the Kalman filter. The paper focuses on the problem of tracking hand movements, which constitute smooth spatial trajectories, from spike trains of a neural population. We do so by devising a dynam- ical system which employs a tailored kernel for spike trains along with a linear mapping corresponding to the states' dynamics.

The problem of learning a sparse conic combination of kernel functions or kernel matrices for classification or regression can be achieved via the regularization by a block 1-norm [1]. In this paper, we present an al- gorithm that computes the entire regularization path for these problems. The path is obtained by using numerical continuation techniques, and involves a running time complexity that is a constant times the complex- ity of solving the problem for one value of the regularization parameter. Working in the setting of kernel linear regression and kernel logistic re- gression, we show empirically that the effect of the block 1-norm reg- ularization differs notably from the (non-block) 1-norm regularization commonly used for variable selection, and that the regularization path is of particular value in the block case.

We present a competitive analysis of Bayesian learning algorithms in the online learning setting and show that many simple Bayesian algorithms (such as Gaussian linear regression and Bayesian logistic regression) per- form favorably when compared, in retrospect, to the single best model in the model class. The analysis does not assume that the Bayesian algo- rithms' modeling assumptions are "correct," and our bounds hold even if the data is adversarially chosen. For Gaussian linear regression (us- ing logloss), our error bounds are comparable to the best bounds in the online learning literature, and we also provide a lower bound showing that Gaussian linear regression is optimal in a certain worst case sense. We also give bounds for some widely used maximum a posteriori (MAP) estimation algorithms, including regularized logistic regression.

Lasso regression tends to assign zero weights to most irrelevant or redundant features, and hence is a promising technique for feature selection. Its limitation, however, is that it only offers solutions to linear models. Kernel machines with feature scaling techniques have been studied for feature selection with non-linear models. However, such approaches require to solve hard non-convex optimization problems. This paper proposes a new approach named the Feature Vector Machine (FVM). It reformulates the standard Lasso regression into a form isomorphic to SVM, and this form can be easily extended for feature selection with non-linear models by introducing kernels defined on feature vectors.

An increasing number of projects in neuroscience requires the sta- tistical analysis of high dimensional data sets, as, for instance, in predicting behavior from neural firing or in operating artificial de- vices from brain recordings in brain-machine interfaces. Linear analysis techniques remain prevalent in such cases, but classical linear regression approaches are often numerically too fragile in high dimensions. In this paper, we address the question of whether EMG data collected from arm movements of monkeys can be faith- fully reconstructed with linear approaches from neural activity in primary motor cortex (M1). To achieve robust data analysis, we develop a full Bayesian approach to linear regression that auto- matically detects and excludes irrelevant features in the data, reg- ularizing against overfitting. In comparison with ordinary least squares, stepwise regression, partial least squares, LASSO regres- sion and a brute force combinatorial search for the most predictive input features in the data, we demonstrate that the new Bayesian method offers a superior mixture of characteristics in terms of reg- ularization against overfitting, computational efficiency and ease of use, demonstrating its potential as a drop-in replacement for other linear regression techniques.