We describe a method for predicting a classification of an object given classifications of the objects in the training set, assuming that the pairs object/classification are generated by an i.i.d. process from a continuous probability distribution. Our method is a modification of Vapnik's support-vector machine; its main novelty is that it gives not only the prediction itself but also a practicable measure of the evidence found in support of that prediction. We also describe a procedure for assigning degrees of confidence to predictions made by the support vector machine. Some experimental results are presented, and possible extensions of the algorithms are discussed.

We apply a general algorithm for merging prediction strategies (the Aggregating Algorithm) to the problem of linear regression with the square loss; our main assumption is that the response variable is bounded. It turns out that for this particular problem the Aggregating Algorithm resembles, but is slightly different from, the wellknown ridge estimation procedure. From general results about the Aggregating Algorithm we deduce a guaranteed bound on the difference between our algorithm's performance and the best, in some sense, linear regression function's performance. We show that the AA attains the optimal constant in our bound, whereas the constant attained by the ridge regression procedure in general can be 4 times worse. 1 INTRODUCTION The usual approach to regression problems is to assume that the data are generated by some stochastic mechanism and make some, typically very restrictive, assumptions about that stochastic mechanism. In recent years, however, a different approach to this kind of problems was developed (see, e.g., DeSantis et al. [2], Littlestone and Warmuth [7]): in our context, that approach sets the goal of finding an online algorithm that performs not much worse than the best regression function found off-line; in other words, it replaces the usual statistical analyses by the competitive analysis of online algorithms. DeSantis et al. [2] performed a competitive analysis of the Bayesian merging scheme for the log-loss prediction game; later Littlestone and Warmuth [7] and Vovk [10] introduced an online algorithm (called the Weighted Majority Algorithm by the Competitive Online Linear Regression 365 former authors) for the simple binary prediction game. These two algorithms (the Bayesian merging scheme and the Weighted Majority Algorithm) are special cases of the Aggregating Algorithm (AA) proposed in [9, 11]. The AA is a member of a wide family of algorithms called "multiplicative weight" or "exponential weight" algorithms. Closer to the topic of this paper, Cesa-Bianchi et al. [1) performed a competitive analysis, under the square loss, of the standard Gradient Descent Algorithm and Kivinen and Warmuth [6] complemented it by a competitive analysis of a modification of the Gradient Descent, which they call the Exponentiated Gradient Algorithm.

We apply a general algorithm for merging prediction strategies (the Aggregating Algorithm) to the problem of linear regression with the square loss; our main assumption is that the response variable is bounded. It turns out that for this particular problem the Aggregating Algorithm resembles, but is slightly different from, the wellknown ridge estimation procedure. From general results about the Aggregating Algorithm we deduce a guaranteed bound on the difference between our algorithm's performance and the best, in some sense, linear regression function's performance. We show that the AA attains the optimal constant in our bound, whereas the constant attained by the ridge regression procedure in general can be 4 times worse. 1 INTRODUCTION The usual approach to regression problems is to assume that the data are generated by some stochastic mechanism and make some, typically very restrictive, assumptions about that stochastic mechanism. In recent years, however, a different approach to this kind of problems was developed (see, e.g., DeSantis et al. [2], Littlestone and Warmuth [7]): in our context, that approach sets the goal of finding an online algorithm that performs not much worse than the best regression function found off-line; in other words, it replaces the usual statistical analyses by the competitive analysis of online algorithms. DeSantis et al. [2] performed a competitive analysis of the Bayesian merging scheme for the log-loss prediction game; later Littlestone and Warmuth [7] and Vovk [10] introduced an online algorithm (called the Weighted Majority Algorithm by the Competitive Online Linear Regression 365 former authors) for the simple binary prediction game. These two algorithms (the Bayesian merging scheme and the Weighted Majority Algorithm) are special cases of the Aggregating Algorithm (AA) proposed in [9, 11]. The AA is a member of a wide family of algorithms called "multiplicative weight" or "exponential weight" algorithms. Closer to the topic of this paper, Cesa-Bianchi et al. [1) performed a competitive analysis, under the square loss, of the standard Gradient Descent Algorithm and Kivinen and Warmuth [6] complemented it by a competitive analysis of a modification of the Gradient Descent, which they call the Exponentiated Gradient Algorithm.

We apply a general algorithm for merging prediction strategies (the Aggregating Algorithm) to the problem of linear regression with the square loss; our main assumption is that the response variable is bounded. It turns out that for this particular problem the Aggregating Algorithmresembles, but is slightly different from, the wellknown ridgeestimation procedure. From general results about the Aggregating Algorithm we deduce a guaranteed bound on the difference betweenour algorithm's performance and the best, in some sense, linear regression function's performance. We show that the AA attains the optimal constant in our bound, whereas the constant attainedby the ridge regression procedure in general can be 4 times worse. 1 INTRODUCTION The usual approach to regression problems is to assume that the data are generated bysome stochastic mechanism and make some, typically very restrictive, assumptions about that stochastic mechanism. In recent years, however, a different approach to this kind of problems was developed (see, e.g., DeSantis et al. [2], Littlestone andWarmuth [7]): in our context, that approach sets the goal of finding an online algorithm that performs not much worse than the best regression function foundoff-line; in other words, it replaces the usual statistical analyses by the competitive analysis of online algorithms. DeSantis et al. [2] performed a competitive analysis of the Bayesian merging scheme for the log-loss prediction game; later Littlestone and Warmuth [7] and Vovk [10] introduced an online algorithm (called the Weighted Majority Algorithm by the Competitive Online Linear Regression 365 former authors) for the simple binary prediction game. These two algorithms (the Bayesian merging scheme and the Weighted Majority Algorithm) are special cases of the Aggregating Algorithm (AA) proposed in [9, 11]. The AA is a member of a wide family of algorithms called "multiplicative weight" or "exponential weight" algorithms. Closerto the topic of this paper, Cesa-Bianchi et al. [1) performed a competitive analysis, under the square loss, of the standard Gradient Descent Algorithm and Kivinen and Warmuth [6] complemented it by a competitive analysis of a modification ofthe Gradient Descent, which they call the Exponentiated Gradient Algorithm.