We present a family of margin based online learning algorithms for various prediction tasks. In particular we derive and analyze algorithms for binary and multiclass categorization, regression, uniclass prediction and sequence prediction. The update steps of our different algorithms are all based on analytical solutions to simple constrained optimization problems. This unified view allows us to prove worst-case loss bounds for the different algorithms and for the various decision problems based on a single lemma. Our bounds on the cumulative loss of the algorithms are relative to the smallest loss that can be attained by any fixed hypothesis, and as such are applicable to both realizable and unrealizable settings. We demonstrate some of the merits of the proposed algorithms in a series of experiments with synthetic and real data sets.

In this paper we show that on-line algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentration-of-measure arguments and they hold for arbitrary on-line learning algorithms. Furthermore, when applied to concrete on-line algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds.

Abstract. The problem of learning linear-discriminant concepts can be solved by various mistake-driven update procedures, including the Winnow family of algorithms and the well-known Perceptron algorithm. In this paper we define the general class of “quasi-additive ” algorithms, which includes Perceptron and Winnow as special cases. We give a single proof of convergence that covers a broad subset of algorithms in this class, including both Perceptron and Winnow, but also many new algorithms. Our proof hinges on analyzing a generic measure of progress construction that gives insight as to when and how such algorithms converge. Our measure of progress construction also permits us to obtain good mistake bounds for individual algorithms. We apply our unified analysis to new algorithms as well as existing algorithms. When applied to known algorithms, our method “automatically ” produces close variants of existing proofs (recovering similar bounds)—thus showing that, in a certain sense, these seemingly diverse results are fundamentally isomorphic. However, we also demonstrate that the unifying principles are more broadly applicable, and analyze a new class of algorithms that smoothly interpolate between the additive-update behavior of Perceptron and the multiplicative-update behavior of Winnow.

A new incremental learning algorithm is described which approximates the maximal margin hyperplane w.r.t. norm p 2 for a set of linearly separable data. Our algorithm, called alma p (Approximate Large Margin algorithm w.r.t. norm p), takes O (p 1) 2 2 corrections to separate the data with p-norm margin larger than (1 ) , where is the (normalized) p-norm margin of the data. alma p avoids quadratic (or higher-order) programming methods. It is very easy to implement and is as fast as on-line algorithms, such as Rosenblatt's Perceptron algorithm. We performed extensive experiments on both real-world and artificial datasets. We compared alma 2 (i.e., alma p with p = 2) to standard Support vector Machines (SVM) and to two incremental algorithms: the Perceptron algorithm and Li and Long's ROMMA. The accuracy levels achieved by alma 2 are superior to those achieved by the Perceptron algorithm and ROMMA, but slightly inferior to SVM's. On the other hand, alma 2 is quite faster and easier to implement than standard SVM training algorithms. When learning sparse target vectors, alma p with p > 2 largely outperforms Perceptron-like algorithms, such as alma 2 .

We study on-line generalized linear regression with multidimensional outputs, i.e., neural networks with multiple output nodes but no hidden nodes. We allow at the final layer transfer functions such as the softmax function that need to consider the linear activations to all the output neurons. The weight vectors used to produce the linear activations are represented indirectly by maintaining separate parameter vectors. We get the weight vector by applying a particular parameterization function to the parameter vector. Updating the parameter vectors upon seeing new examples is done additively, as in the usual gradient descent update. However, by using a nonlinear parameterization function between the parameter vectors and the weight vectors, we can make the resulting update of the weight vector quite different from a true gradient descent update. To analyse such updates, we define a notion of a matching loss function and apply it both to the transfer function and to the parameterization function. The loss function that matches the transfer function is used to measure the goodness of the predictions of the algorithm. The loss function that matches the parameterization function can be used both as a measure of divergence between models in motivating the update rule of the algorithm and as a measure of progress in analyzing its relative performance compared to an arbitrary fixed model. As a result, we have a unified treatment that generalizes earlier results for the gradient descent and exponentiated gradient algorithms to multidimensional outputs, including multiclass logistic regression.

Kernels are typically applied to linear algorithms whose weight vector is a linear combination of the feature vectors of the examples. On-line versions of these algorithms are sometimes called "additive updates" because they add a multiple of the last feature vector to the current weight vector.

We study on-line learning in the linear regression framework. Most of the performance bounds for on-line algorithms in this framework assume a constant learning rate. To achieve these bounds the learning rate must be optimized based on a posteriori information. This information depends on the whole sequence of examples and thus it is not available to any strictly on-line algorithm. We introduce new techniques for adaptively tuning the learning rate as the data sequence is progressively revealed. Our techniques allow us to prove essentially the same bounds as if we knew the optimal learning rate in advance. Moreover, such techniques apply to a wide class of on-line algorithms, including p-norm algorithms for generalized linear regression and Weighted Majority for linear regression with absolute loss. Our adaptive tunings are radically dierent from previous techniques, such as the so-called doubling trick. Whereas the doubling trick restarts the on-line algorithm several ti...

Kernel-based linear-threshold algorithms, such as support vector machines and Perceptron-like algorithms, are among the best available techniques for solving pattern classification problems. In this paper, we describe an extension of the classical Perceptron algorithm, called second-order Perceptron, and analyze its performance within the mistake bound model of on-line learning. The bound achieved by our algorithm depends on the sensitivity to second-order data information and is the best known mistake bound for (efficient) kernel-based linear-threshold classifiers to date. This mistake bound, which strictly generalizes the well-known Perceptron bound, is expressed in terms of the eigenvalues of the empirical data correlation matrix and depends on a parameter controlling the sensitivity of the algorithm to the distribution of these eigenvalues. Since the optimal setting of this parameter is not known a priori, we also analyze two variants of the second-order Perceptron algorithm: one that adaptively sets the value of the parameter in terms of the number of mistakes made so far, and one that is parameterless, based on pseudoinverses.

We describe a new boosting algorithm which generates only smooth distributions which do not assign too much weight to any single example. We show that this new boosting algorithm can be used to construct efficient PAC learning algorithms which tolerate relatively high rates of malicious noise. In particular, we use the new smooth boosting algorithm to construct malicious noise tolerant versions of the PACmodel p-norm linear threshold learning algorithms described in [23]. The bounds on sample complexity and malicious noise tolerance of these new PAC algorithms closely correspond to known bounds for the online p-...

A selective sampling algorithm is a learning algorithm for classification that, based on the past observed data, decides whether to ask the label of each new instance to be classified. In this paper, we introduce a general technique for turning linear-threshold classification algorithms from the general additive family into randomized selective sampling algorithms. For the most popular algorithms in this family we derive mistake bounds that hold for individual sequences of examples. These bounds