Web crawling is the problem of keeping a cache of webpages fresh, i.e., having the most recent copy available when a page is requested. This problem is usually coupled with the natural restriction that the bandwidth available to the web crawler is limited. The corresponding optimization problem was solved optimally by Azar et al. [2018] under the assumption that, for each webpage, both the elapsed time between two changes and the elapsed time between two requests follow a Poisson distribution with known parameters. In this paper, we study the same control problem but under the assumption that the change rates are unknown a priori, and thus we need to estimate them in an online fashion using only partial observations (i.e., single-bit signals indicating whether the page has changed since the last refresh). As a point of departure, we characterise the conditions under which one can solve the problem with such partial observability. Next, we propose a practical estimator and compute confidence intervals for it in terms of the elapsed time between the observations. Finally, we show that the explore-and-commit algorithm achieves an $\mathcal{O}(\sqrt{T})$ regret with a carefully chosen exploration horizon. Our simulation study shows that our online policy scales well and achieves close to optimal performance for a wide range of the parameters.

We revisit isotonic regression on linear orders, the problem of fitting monotonic functions to best explain the data, in an online setting. It was previously shown that online isotonic regression is unlearnable in a fully adversarial model, which lead to its study in the fixed design model. Here, we instead develop the more practical random permutation model. We show that the regret is bounded above by the excess leave-one-out loss for which we develop efficient algorithms and matching lower bounds. We also analyze the class of simple and popular forward algorithms and recommend where to look for algorithms for online isotonic regression on partial orders.

We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the optimal strategy can be calculated without knowing the time horizon in advance. They put forward the question what families have exchangeable SNML strategies. This paper fully answers this open problem for one-dimensional exponential families. The exchangeability can happen only for three classes of natural exponential family distributions, namely the Gaussian, Gamma, and the Tweedie exponential family of order 3/2. Keywords: SNML Exchangeability, Exponential Family, Online Learning, Logarithmic Loss, Bayesian Strategy, Jeffreys Prior, Fisher Information1

We consider the problem of rank loss minimization in the setting of multilabel classification, which is usually tackled by means of convex surrogate losses defined on pairs of labels. Very recently, this approach was put into question by a negative result showing that commonly used pairwise surrogate losses, such as exponential and logistic losses, are inconsistent. In this paper, we show a positive result which is arguably surprising in light of the previous one: the simpler univariate variants of exponential and logistic surrogates (i.e., defined on single labels) are consistent for rank loss minimization. Instead of directly proving convergence, we give a much stronger result by deriving regret bounds and convergence rates. The proposed losses suggest efficient and scalable algorithms, which are tested experimentally.

We extend the classical problem of predicting a sequence of outcomes from a finite alphabet to the matrix domain. In this extension, the alphabet of $n$ outcomes is replaced by the set of all dyads, i.e. outer products $\u\u^\top$ where $\u$ is a vector in $\R^n$ of unit length. Whereas in the classical case the goal is to learn (i.e. sequentially predict as well as) the best multinomial distribution, in the matrix case we desire to learn the density matrix that best explains the observed sequence of dyads. We show how popular online algorithms for learning a multinomial distribution can be extended to learn density matrices. Intuitively, learning the $n^2$ parameters of a density matrix is much harder than learning the $n$ parameters of a multinomial distribution. Completely surprisingly, we prove that the worst-case regrets of certain classical algorithms and their matrix generalizations are identical. The reason is that the worst-case sequence of dyads share a common eigensystem, i.e. the worst case regret is achieved in the classical case. So these matrix algorithms learn the eigenvectors without any regret.

Pattern recognition is a central topic in Learning Theory with numerous applications such as voice and text recognition, image analysis, computer diagnosis. The statistical set-up in classification is the following: we are given an i.i.d. training set $(X_{1},Y_{1}),... (X_{n},Y_{n})$ where $X_{i}$ represents a feature and $Y_{i}\in \{0,1\}$ is a label attached to that feature. The underlying joint distribution of $(X,Y)$ is unknown, but we can learn about it from the training set and we aim at devising low error classifiers $f:X\to Y$ used to predict the label of new incoming features. Here we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown qubit states. Given a number of `training' copies from each of the states, we would like to `learn' about them by performing a measurement on the training set. The outcome is then used to design mesurements for the classification of future systems with unknown labels. We find the asymptotically optimal classification strategy and show that typically, it performs strictly better than a plug-in strategy based on state estimation. The figure of merit is the excess risk which is the difference between the probability of error and the probability of error of the optimal measurement when the states are known, that is the Helstrom measurement. We show that the excess risk has rate $n^{-1}$ and compute the exact constant of the rate.