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 constant rebalanced portfolio is an investment strategy that keeps the same distribution of wealth among a set of stocks from day to day. There has been much work on Cover's Universal algorithm, which is competitive with the best constant rebalanced portfolio determined in hindsight (3, 9, 2, 8, 16, 4, 5, 6). While this algorithm has good performance guarantees, all known implementations are exponential in the number of stocks, restricting the number of stocks used in experiments (9, 4, 2, 5, 6). We present an efficient implementation of the Universal algorithm that is based on non-uniform random walks that are rapidly mixing (1, 14, 7). This same implementation also works for non-financial applications of the Universal algorithm, such as data compression (6) and language modeling (11).

This paper studies sequence prediction based on the monotone Kolmogorov complexity Km=-log m, i.e. based on universal MDL. m is extremely close to Solomonoff's prior M, the latter being an excellent predictor in deterministic as well as probabilistic environments, where performance is measured in terms of convergence of posteriors or losses. Despite this closeness to M, it is difficult to assess the prediction quality of m, since little is known about the closeness of their posteriors, which are the important quantities for prediction. We show that for deterministic computable environments, the "posterior" and losses of m converge, but rapid convergence could only be shown on-sequence; the off-sequence behavior is unclear. In probabilistic environments, neither the posterior nor the losses converge, in general.

The purpose of this paper is to introduce sequential investment strategies that guarantee an optimal rate of growth of the capital, under minimal assumptions on the behavior of the market. The new strategies are analyzed both theoretically and empirically. The theoretical results show that the asymptotic rate of growth matches the optimal one that one could achieve with a full knowledge of the statistical properties of the underlying process generating the market, under the only assumption that the market is stationary and ergodic. The empirical results show that the performance of the proposed investment strategies measured on past NYSE and currency exchange data is solid, and sometimes even spectacular. KEY WORDS: sequential investment, universal portfolios, kernel estimation 1.

A novel control mechanism was recently introduced based on Extended Markov Tracking (EMT) [9, 10]. In this paper, we present a study of its response to multiple interacting control goals. We show a simple extension that can be integrated into EMT-based control, and which provides it with the ability to handle several behavioral targets. Experimental support for the validity of this extension is provided. We also describe an experiment with a simulated robot, where EMT-based controllers interact and interfere indirectly via the environment. Experiments support the resilience of multiagent EMT-based team control to potential conflicts that may appear within a team. 1.

INTRODUCTION The subject of this note is another inductive principle, which can be regarded as a direct generalization of the minimum description length (MDL) and minimum message length (MML) principles. We will describe the work started at the Computer Learning Research Centre (Royal Holloway, University of London) related to this new principle, which we call the complexity approximation principle (CAP). Both MDL and MML principles can be interpreted as Kolmogorov complexity approximation principles (as explained in Rissanen [1, 2] and Wallace and Freeman [3]; see also [4]). It is shown in [5] and [6] that it is possible to generalize Kolmogorov complexity to describe the optimal performance in different `games of prediction'. Using this general notion, called predictive complexity,itis straightforward to extend the MDL and MML principles to our more general CAP. In Section 2 we define predictive complexity, in Section 3 several examples are given and in Section 4

On the surface, the three on-line machine learning problems analyzed in this thesis may seem unrelated. The first is an on-line investment strategy introduced by Tom Cover. We begin with a simple analysis that extends to the case of fixed-percentage transaction costs. We then describe an efficient implementation that runs in time polynomial in the number of stocks. The second problem is k-fold cross validation, a popular technique in machine learning for estimating the error of a learned hypothesis. We show that this is a valid technique by comparing it to the hold-out estimate. Finally, we discuss work towards a dynamically-optimal adaptive binary search tree algorithm. To my mother, Marilyn Kalai. May her PBSCT be as easy on her as my committee was on me. Acknowledgments It should be no surprise that my biggest thanks go to my parents, who somehow created me and gave me a very happy childhood. For as long as I can remember, my father has been teaching me about problem solving and research through puzzles and questions. If I end up with a fraction of his creativity and accomplishments, I will feel very lucky. Since I was a baby, I couldn't have asked for a better role model than my mother. Even if I could have talked at that age, I still wouldn't have asked for one. I came to CMU in large part because of Avrim Blum. After three advisors, I can say with full confidence that Avrim is the best advisor and teacher at CMU. I don't think I would have finished with anyone else. They often say that, by the time you're ready to graduate, you should know your area better than your advisor. If that was a requirement, I would never graduate. I'm moving from one great advisor to another. Next year I'll be at MIT under the supervision of Santosh Vempala. Many thanks to Santosh...

Abstract—A wide variety of problems in signal processing can be formulated such that decisions are made by sequentially taking convex combinations of vector-valued observations and these convex combinations are then multiplicatively compounded over time. A “universal ” approach to such problems might attempt to sequentially achieve the performance of the best fixed convex combination, as might be achievable noncausally, by observing all of the outcomes in advance. By permitting different piecewise-fixed strategies within contiguous regions of time, the best algorithm in this broader class would be able to switch between different fixed strategies to optimize performance to the changing behavior of each individual sequence of outcomes. Without knowledge of the data length or the number of switches necessary, the algorithms developed in this paper can achieve the performance of the best piecewise-fixed strategy that can choose both the partitioning of the sequence of outcomes in time as well as the best strategy within each time segment. We compete with an exponential number of such partitions, using only complexity linear in the data length and demonstrate that the regret with respect to the best such algorithm is at most (ln ()) in the exponent, where is the data length. Finally, we extend these results to include finite collections of candidate algorithms, rather than convex combinations and further investigate the use of an arbitrary side-information sequence. Index Terms—Convex combinations, portfolio, sequential decisions, side information, switching, universal. I.

In this paper, we consider online (sequential) portfolio selection in a competitive algorithm framework. We construct a sequential algorithm for portfolio investment that asymptotically achieves the wealth of the best piecewise constant rebalanced portfolio tuned to the underlying individual sequence of price relative vectors. Without knowledge of the investment duration, the algorithm can perform as well as the best investment algorithm that can choose both the partitioning of the sequence of the price relative vectors as well as the best constant rebalanced portfolio within each segment, based on knowledge of the sequence of price relative vectors in advance. We use a transition diagram similar to that in [1] to compete with an exponential number of switching investment strategies, using only linear complexity in the data length. The regret with respect to the best piecewise constant strategy at most O(ln(n)) in the exponent, where n is the investment duration. We also consider switching among a finite collection of candidate algorithms, including the case where such transitions are represented by an arbitrary side-information sequence. Here, we construct sequential portfolios that asymptotically achieve the performance of the algorithm with the best side-information sequence. I.

In this paper, we investigate investment strategies that can rebalance their target portfolio vectors at arbitrary investment periods. These strategies are called semi-constant rebalanced portfolios in Blum and Kalai (1998), Helmbold et al. (1998). Unlike a constant rebalanced portfolio, which must rebalance at every investment interval, a semi-constant rebalanced portfolio rebalances its portfolio only on selected instants. Hence, a semi-constant rebalanced portfolio may avoid rebalancing if the transaction costs outweigh the benefits of rebalancing. In a competitive algorithm framework, we compete against all such semi-constant portfolios with an arbitrary number of rebalancings and corresponding rebalancing instants. We investigate this framework with and without transaction costs and demonstrate sequential portfolios that asymptotically achieve the wealth of the best semi-constant rebalanced portfolios whose number of rebalancings and instants of rebalancings are tuned to the individual sequence of price relatives.