Bandit Online Optimization Over the Permutahedron

The permutahedron is the convex polytope with vertex set consisting of the vectors $(\pi(1),\dots, \pi(n))$ for all permutations (bijections) $\pi$ over $\{1,\dots, n\}$. We study a bandit game in which, at each step $t$, an adversary chooses a hidden weight weight vector $s_t$, a player chooses a vertex $\pi_t$ of the permutahedron and suffers an observed loss of $\sum_{i=1}^n \pi(i) s_t(i)$. A previous algorithm CombBand of Cesa-Bianchi et al (2009) guarantees a regret of $O(n\sqrt{T \log n})$ for a time horizon of $T$. Unfortunately, CombBand requires at each step an $n$-by-$n$ matrix permanent approximation to within improved accuracy as $T$ grows, resulting in a total running time that is super linear in $T$, making it impractical for large time horizons. We provide an algorithm of regret $O(n^{3/2}\sqrt{T})$ with total time complexity $O(n^3T)$. The ideas are a combination of CombBand and a recent algorithm by Ailon (2013) for online optimization over the permutahedron in the full information setting. The technical core is a bound on the variance of the Plackett-Luce noisy sorting process's "pseudo loss". The bound is obtained by establishing positive semi-definiteness of a family of 3-by-3 matrices generated from rational functions of exponentials of 3 parameters

Leading strategies in competitive on-line prediction

We start from a simple asymptotic result for the problem of on-line regression with the quadratic loss function: the class of continuous limited-memory prediction strategies admits a "leading prediction strategy", which not only asymptotically performs at least as well as any continuous limited-memory strategy but also satisfies the property that the excess loss of any continuous limited-memory strategy is determined by how closely it imitates the leading strategy. More specifically, for any class of prediction strategies constituting a reproducing kernel Hilbert space we construct a leading strategy, in the sense that the loss of any prediction strategy whose norm is not too large is determined by how closely it imitates the leading strategy. This result is extended to the loss functions given by Bregman divergences and by strictly proper scoring rules.Comment: 20 pages; a conference version is to appear in the ALT'2006 proceeding

Optimal dynamic portfolio selection with earnings-at-risk

In this paper we investigate a continuous-time portfolio selection problem. Instead of using the classical variance as usual, we use earnings-at-risk (EaR) of terminal wealth as a measure of risk. In the settings of Black-Scholes type financial markets and constantly-rebalanced portfolio (CRP) investment strategies, we obtain closed-form expressions for the best CRP investment strategy and the efficient frontier of the mean-EaR problem, and compare our mean-EaR analysis to the classical mean-variance analysis and to the mean-CaR (capital-at-risk) analysis. We also examine some economic implications arising from using the mean-EaR model. © 2007 Springer Science+Business Media, LLC.postprin

The development, educational stratification and decomposition of mothers' and fathers' childcare time in Germany: an update for 2001-2013

"This study updates empirical knowledge about the development,(the educational stratification, and the decomposition of mothers' and fathers' childcare time in Germany with the most recent time use data. Using time series data from the German Time Use Study 2001/2002 and 2012/ 2013, we analyze time budgets for total childcare and six specific childcare activities on weekdays and weekends and estimate OLS regressions and Oaxaca decompositions. The study found that total childcare time has increased for mothers and fathers between 2001 and 2013 and that this change is predominantly due to increased time for basic childcare. It also found consistent evidence of an education gradient only for reading time with children. If there is significant change of time budgets between 2001 and 2013, this change seems to be driven by behavioral change rather than changing demographics. Our empirical findings on childcare time in Germany do not provide evidence of dynamics and stratification but rather of stability and similarity across parents’ educational levels. Besides the updates on German parents' development, stratification and decomposition of time use for childcare, these analyses show that change in total childcare is not due to a proportional change over all single activities but due to changes in a few activities only." (author's abstract)"Diese Studie aktualisiert das empirische Wissen über die Entwicklung, die Bildungsstratifizierung und die Dekomposition der Zeitverwendung von Müttern und Vätern für Kinderbetreuung mit den aktuellen Zeitbudgetdaten für Deutschland. Auf Basis der der letzten beiden Erhebungen der Deutschen Zeitverwendungsstudie 2001/2002 und 2012/2013 werden die Zeitbudgets für die Gesamtzeit für Kinderbetreuung sowie sechs Einzeltätigkeiten mit OLS-Regressionen und Oaxaca- Dekompositionen untersucht. Die Studie zeigt, dass die Zeit für Kinderbetreuung von Müttern und Vätern zwischen 2001 und 2013 angestiegen ist, es einen Bildungsgradienten für Vorlesen gibt und signifikante Veränderungen in den Zeitbudgets nicht auf Kompositionsveränderung der Bevölkerung zurückgeführt werden können. Insgesamt belegt die Studie weniger die Dynamik als vielmehr die Stabilität und die geringe Bildungsdifferenzierung der Zeitverwendung für Kinderbetreuung. Darüber hinaus wird gezeigt, dass die Veränderungen in der Gesamtzeit für Kinderbetreuung nicht auf proportionale Veränderungen in allen, sondern nur auf Veränderungen in wenigen Einzeltätigkeiten zurückgeführt werden können." (Autorenreferat

Strong entropy concentration, game theory and algorithmic randomness

We give a characterization of Maximum Entropy/Minimum Relative Entropy inference by providing two ‘strong entropy concentration’ theorems. These theorems unify and generalize Jaynes’ ‘concentration phenomenon’ and Van Campenhout and Cover’s ‘conditional limit theorem’. The theorems characterize exactly in what sense a ‘prior’ distribution Q conditioned on a given constraint and the distribution P minimizing D(P//Q) over all P satisfyingthe constraint are ‘close’ to each other. We show how our theorems are related to ‘universal models’ for exponential families, thereby establishinga link with Rissanen’s MDL/stochastic complexity. We then apply our theorems to establish the relationship (A) between entropy concentration and a game-theoretic characterization of Maximum Entropy Inference due to Topsøe and others; (B) between maximum entropy distributions and sequences that are random (in the sense of Martin-Löf/Kolmogorov) with respect to the given constraint. These two applications have strong implications for the use of Maximum Entropy distributions in sequential prediction tasks, both for the logarithmic loss and for general loss functions. We identify circumstances under which Maximum Entropy predictions are almost optimal

Learning Probability Distributions over Permutations by Means of Fourier Coefficients

A large and increasing number of data mining domains consider data that can be represented as permutations. Therefore, it is important to devise new methods to learn predictive models over datasets of permutations. However, maintaining models, such as probability distributions, over the space of permutations is a hard task since there are n! permutations of n elements. Recently the Fourier transform has been successfully generalized to functions over permutations and offers an attractive way to represent uncertainty over the space of permutations. One of its main advantages is that the Fourier transform compactly summarizes approximations to functions by discarding high order marginals information. Moreover, a lately proposed framework for making inference completely in the Fourier domain has opened new doors for efficiently reasoning over a space of permutations. In this paper, we present a method to learn a probability distribution that approximates the generating distribution of a given sample of permutations. Particularly, this method learns the Fourier domain information representing this probability distribution