We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worst-case situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We then show howthis leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently known in this context. We also compare our analysis to the case in which log loss is used instead of the expected number of mistakes.

We consider adaptive sequential prediction of arbitrary binary sequences when the performance is evaluated using a general loss function. The goal is to predict on each individual sequence nearly as well as the best prediction strategy in a given comparison class of (possibly adaptive) prediction strategies, called experts. By using a general loss function, we generalize previous work on universal prediction, forecasting, and data compression. However, here we restrict ourselves to the case when the comparison class is finite. For a given sequence, we define the regret as the total loss on the entire sequence suffered by the adaptive sequential predictor, minus the total loss suffered by the predictor in the comparison class that performs best on that particular sequence. We show that for a large class of loss functions, the minimax regret is either \Theta(log N) or \Omega\Gamma p ` log N ), depending on the loss function, where N is the number of predictors in the comparison class a...

this paper is somewhat different from the one just described. Assume that there are N experts E i , i = 1; : : : ; N , each trying to predict the outcomes y t as best they can. Let x t;i be the prediction of the ith expert E i about the

This paper is based on research carried out jointly with James H. Stock, who I thank for comments on this paper. I thank Jean-Philippe Laforte for research assistance. This research was supported by the National Science Foundation (SBR-9730489). (Version WC_2b) 1

The rate equations are applied to investigate the structure of growing networks. Within this framework, the degree distribution of a network in which nodes are introduced sequentially and attach to an earlier node of degree k with rate Ak k is computed. Very dierent behaviors arise for < 1, = 1, and > 1. The rate equation approach is extended to determine the joint order-degree distribution, the degree correlations of neighboring nodes, as well as basic global properties. The complete solution for the degree distribution of a finite-size network is outlined. Some unusual properties associated with the most popular node are discussed; these follow simply from the order-degree distribution. Finally, a toy protein interaction network model is investigated, where the network grows by the processes of node duplication and particular form of random mutations. This system exhibits an infinite-order percolation transition, giant sample-specific fluctuations, and a non-universal degree distribution.

Abstract. The extremal characteristics of random structures, including trees, graphs, and networks, are discussed. A statistical physics approach is employed in which extremal properties are obtained through suitably defined rate equations. A variety of unusual time dependences and system-size dependences for basic extremal properties are obtained. 1