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93
Tracking the best expert
 In Proceedings of the 12th International Conference on Machine Learning
, 1995
"... Abstract. We generalize the recent relative loss bounds for online algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments, and the goal is to bound th ..."
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Cited by 246 (20 self)
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Abstract. We generalize the recent relative loss bounds for online algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments, and the goal is to bound the additional loss of the algorithm over the sum of the losses of the best experts for each segment. This is to model situations in which the examples change and different experts are best for certain segments of the sequence of examples. In the single segment case, the additional loss is proportional to log n, where n is the number of experts and the constant of proportionality depends on the loss function. Our algorithms do not produce the best partition; however the loss bound shows that our predictions are close to those of the best partition. When the number of segments is k +1and the sequence is of length ℓ, we can bound the additional loss of our algorithm over the best partition by O(k log n + k log(ℓ/k)). For the case when the loss per trial is bounded by one, we obtain an algorithm whose additional loss over the loss of the best partition is independent of the length of the sequence. The additional loss becomes O(k log n + k log(L/k)), where L is the loss of the best partition with k +1segments. Our algorithms for tracking the predictions of the best expert are simple adaptations of Vovk’s original algorithm for the single best expert case. As in the original algorithms, we keep one weight per expert, and spend O(1) time per weight in each trial.
Universal prediction
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... This paper consists of an overview on universal prediction from an informationtheoretic perspective. Special attention is given to the notion of probability assignment under the selfinformation loss function, which is directly related to the theory of universal data compression. Both the probabili ..."
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Cited by 186 (16 self)
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This paper consists of an overview on universal prediction from an informationtheoretic perspective. Special attention is given to the notion of probability assignment under the selfinformation loss function, which is directly related to the theory of universal data compression. Both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings.
On the Generalization Ability of Online Learning Algorithms
 IEEE Transactions on Information Theory
, 2001
"... In this paper we show that online 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 concentrationofmeasure arguments and they hold for arbitrary onlin ..."
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Cited by 184 (8 self)
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In this paper we show that online 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 concentrationofmeasure arguments and they hold for arbitrary online learning algorithms. Furthermore, when applied to concrete online algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds.
Relative Loss Bounds for Online Density Estimation with the Exponential Family of Distributions
 MACHINE LEARNING
, 2000
"... We consider online density estimation with a parameterized density from the exponential family. The online 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 n ..."
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Cited by 153 (14 self)
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We consider online density estimation with a parameterized density from the exponential family. The online 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 oline algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the online algorithm over the total loss of the best oline 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.
Competitive online statistics
 International Statistical Review
, 1999
"... A radically new approach to statistical modelling, which combines mathematical techniques of Bayesian statistics with the philosophy of the theory of competitive online algorithms, has arisen over the last decade in computer science (to a large degree, under the influence of Dawid’s prequential sta ..."
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Cited by 97 (15 self)
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A radically new approach to statistical modelling, which combines mathematical techniques of Bayesian statistics with the philosophy of the theory of competitive online 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 online statistics”, it is not assumed that data are generated by some stochastic mechanism; the bounds derived for the performance of competitive online 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 online algorithms, linear regression, prequential statistics, worstcase analysis.
Approximate Solutions to Markov Decision Processes
, 1999
"... One of the basic problems of machine learning is deciding how to act in an uncertain world. For example, if I want my robot to bring me a cup of coffee, it must be able to compute the correct sequence of electrical impulses to send to its motors to navigate from the coffee pot to my office. In fact, ..."
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Cited by 83 (10 self)
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One of the basic problems of machine learning is deciding how to act in an uncertain world. For example, if I want my robot to bring me a cup of coffee, it must be able to compute the correct sequence of electrical impulses to send to its motors to navigate from the coffee pot to my office. In fact, since the results of its actions are not completely predictable, it is not enough just to compute the correct sequence; instead the robot must sense and correct for deviations from its intended path. In order for any machine learner to act reasonably in an uncertain environment, it must solve problems like the above one quickly and reliably. Unfortunately, the world is often so complicated that it is difficult or impossible to find the optimal sequence of actions to achieve a given goal. So, in order to scale our learners up to realworld problems, we usually must settle for approximate solutions. One representation for a learner's environment and goals is a Markov decision process or MDP. ...
Tracking the Best Linear Predictor
 Journal of Machine Learning Research
, 2001
"... In most online learning research the total online loss of the algorithm is compared to the total loss of the best offline predictor u from a comparison class of predictors. We call such bounds static bounds. The interesting feature of these bounds is that they hold for an arbitrary sequence of ex ..."
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Cited by 76 (13 self)
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In most online learning research the total online loss of the algorithm is compared to the total loss of the best offline predictor u from a comparison class of predictors. We call such bounds static bounds. The interesting feature of these bounds is that they hold for an arbitrary sequence of examples. Recently some work has been done where the predictor u t at each trial t is allowed to change with time, and the total online loss of the algorithm is compared to the sum of the losses of u t at each trial plus the total "cost" for shifting to successive predictors. This is to model situations in which the examples change over time, and different predictors from the comparison class are best for different segments of the sequence of examples. We call such bounds shifting bounds. They hold for arbitrary sequences of examples and arbitrary sequences of predictors. Naturally shifting bounds are much harder to prove. The only known bounds are for the case when the comparison class consists of a sequences of experts or boolean disjunctions. In this paper we develop the methodology for lifting known static bounds to the shifting case. In particular we obtain bounds when the comparison class consists of linear neurons (linear combinations of experts). Our essential technique is to project the hypothesis of the static algorithm at the end of each trial into a suitably chosen convex region. This keeps the hypothesis of the algorithm wellbehaved and the static bounds can be converted to shifting bounds.
Tracking a Small Set of Experts by Mixing Past Posteriors
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2002
"... In this paper, we examine online learning problems in which the target concept is allowed to change over time. In each trial a master algorithm receives predictions from a large set of n experts. Its goal is to predict almost as well as the best sequence of such experts chosen offline by partit ..."
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Cited by 75 (11 self)
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In this paper, we examine online learning problems in which the target concept is allowed to change over time. In each trial a master algorithm receives predictions from a large set of n experts. Its goal is to predict almost as well as the best sequence of such experts chosen offline by partitioning the training sequence into k + 1 sections and then choosing the best expert for each section. We build on methods developed by Herbster and Warmuth and consider an open problem posed by Freund where the experts in the best partition are from a small pool of size m. Since k >> m, the best expert shifts back and forth between the experts of the small pool. We propose algorithms that solve this open problem by mixing the past posteriors maintained by the master algorithm. We relate the number of bits needed for encoding the best partition to the loss bounds of the algorithms. Instead of paying log n for choosing the best expert in each section we first pay log bits in the bounds for identifying the pool of m experts and then log m bits per new section. In the bounds we also pay twice for encoding the boundaries of the sections.
Averaging Expert Predictions
 Computational Learning Theory: 4th European Conference (EuroCOLT ’99
, 1999
"... We consider algorithms for combining advice from a set of experts. In each trial, the algorithm receives the predictions of the experts and produces its own prediction. A loss function is applied to measure the discrepancy between the predictions and actual observations. ..."
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Cited by 72 (15 self)
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We consider algorithms for combining advice from a set of experts. In each trial, the algorithm receives the predictions of the experts and produces its own prediction. A loss function is applied to measure the discrepancy between the predictions and actual observations.
Adaptive Regression by Mixing
 Journal of American Statistical Association
"... Adaptation over different procedures is of practical importance. Different procedures perform well under different conditions. In many practical situations, it is rather hard to assess which conditions are (approximately) satisfied so as to identify the best procedure for the data at hand. Thus auto ..."
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Cited by 67 (11 self)
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Adaptation over different procedures is of practical importance. Different procedures perform well under different conditions. In many practical situations, it is rather hard to assess which conditions are (approximately) satisfied so as to identify the best procedure for the data at hand. Thus automatic adaptation over various scenarios is desirable. A practically feasible method, named Adaptive Regression by Mixing (ARM) is proposed to convexly combine general candidate regression procedures. Under mild conditions, the resulting estimator is theoretically shown to perform optimally in rates of convergence without knowing which of the original procedures work the best. Simulations are conducted in several settings, including comparing a parametric model with nonparametric alternatives, comparing a neural network with a projection pursuit in multidimensional regression, and combining bandwidths in kernel regression. The results clearly support the theoretical property of ARM. The ARM ...