Active classification with comparison queries

We study an extension of active learning in which the learning algorithm may ask the annotator to compare the distances of two examples from the boundary of their label-class. For example, in a recommendation system application (say for restaurants), the annotator may be asked whether she liked or disliked a specific restaurant (a label query); or which one of two restaurants did she like more (a comparison query). We focus on the class of half spaces, and show that under natural assumptions, such as large margin or bounded bit-description of the input examples, it is possible to reveal all the labels of a sample of size $n$ using approximately $O(\log n)$ queries. This implies an exponential improvement over classical active learning, where only label queries are allowed. We complement these results by showing that if any of these assumptions is removed then, in the worst case, $\Omega(n)$ queries are required. Our results follow from a new general framework of active learning with additional queries. We identify a combinatorial dimension, called the \emph{inference dimension}, that captures the query complexity when each additional query is determined by $O(1)$ examples (such as comparison queries, each of which is determined by the two compared examples). Our results for half spaces follow by bounding the inference dimension in the cases discussed above.Comment: 23 pages (not including references), 1 figure. The new version contains a minor fix in the proof of Lemma 4.

Predicting a binary sequence almost as well as the optimal biased coin

AbstractWe apply the exponential weight algorithm, introduced and Littlestone and Warmuth [26] and by Vovk [35] to the problem of predicting a binary sequence almost as well as the best biased coin. We first show that for the case of the logarithmic loss, the derived algorithm is equivalent to the Bayes algorithm with Jeffrey’s prior, that was studied by Xie and Barron [38] under probabilistic assumptions. We derive a uniform bound on the regret which holds for any sequence. We also show that if the empirical distribution of the sequence is bounded away from 0 and from 1, then, as the length of the sequence increases to infinity, the difference between this bound and a corresponding bound on the average case regret of the same algorithm (which is asymptotically optimal in that case) is only 1/2. We show that this gap of 1/2 is necessary by calculating the regret of the min–max optimal algorithm for this problem and showing that the asymptotic upper bound is tight. We also study the application of this algorithm to the square loss and show that the algorithm that is derived in this case is different from the Bayes algorithm and is better than it for prediction in the worst-case

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum