We consider on-line density estimation with a parameterized density from the exponential family. The on-line algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss, which is the negative loglikelihood of the example with respect to the past parameter of the algorithm. An o-line algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the on-line algorithm over the total loss of the best o-line parameter. These relative loss bounds hold for an arbitrary sequence of examples. The goal is to design algorithms with the best possible relative loss bounds. We use a Bregman divergence to derive and analyze each algorithm. These divergences are relative entropies between two exponential distributions. We also use our methods to prove relative loss bounds for linear regression.

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.

A radically new approach to statistical modelling, which combines mathematical techniques of Bayesian statistics with the philosophy of the theory of competitive on-line algorithms, has arisen over the last decade in computer science (to a large degree, under the influence of Dawid’s prequential statistics). In this approach, which we call “competitive on-line statistics”, it is not assumed that data are generated by some stochastic mechanism; the bounds derived for the performance of competitive on-line statistical procedures are guaranteed to hold (and not just hold with high probability or on the average). This paper reviews some results in this area; the new material in it includes the proofs for the performance of the Aggregating Algorithm in the problem of linear regression with square loss. Keywords: Bayes’s rule, competitive on-line algorithms, linear regression, prequential statistics, worst-case analysis.

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

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We consider on-line density estimation with a parameterized density from an exponential family. In each trial t the learner predicts a parameter t . Then it receives an instance x t chosen by the adversary and incurs loss ln p(x t j t ) which is the negative log-likelihood of x t w.r.t. the predicted density of the learner. The performance of the learner is measured by the regret dened as the total loss of the learner minus the total loss of the best parameter chosen o-line. We develop an algorithm called the Last-step Minimax Algorithm that predicts with the minimax optimal parameter assuming that the current trial is the last one. For one-dimensional exponential families, we give an explicit form of the prediction of the Last-step Minimax Algorithm and show that its regret is O(ln T ), where T is the number of trials. In particular, for Bernoulli density estimation the Last-step Minimax Algorithm is slightly better than the standard Laplace estimator. This work was done while...

. Foster and Vovk proved relative loss bounds for linear regression where the total loss of the on-line algorithm minus the total loss of the best linear predictor (chosen in hindsight) grows logarithmically with the number of trials. We give similar bounds for temporal-difference learning. Learning takes place in a sequence of trials where the learner tries to predict discounted sums of future reinforcement signals. The quality of the prediction is measured with the square loss and we bound the total loss of the on-line algorithm minus the total loss of the best linear predictor for the whole sequence of trials. Again the difference of the losses grows logarithmic with the number of trials. The bounds hold for an arbitrary (worst-case) sequence of examples. We also give a bound on the expected difference for the case when the instances are chosen from an unknown distribution. For linear regression a corresponding lower bound shows that this expected bound cannot be improved substantia...