arXiv subjects

Yuval Dagan

Publications and source records attributed to Yuval Dagan.

39 records · Page 3Linked to original sources

arXiv2018

A Better Resource Allocation Algorithm with Semi-Bandit Feedback

We study a sequential resource allocation problem between a fixed number of arms. On each iteration the algorithm distributes a resource among the arms in order to maximize the expected success rate. Allocating more of the resource to a given arm increases the probability that it succeeds, yet with a cut-off. We follow Lattimore et al. (2014) and assume that the probability increases linearly until it equals one, after which allocating more of the resource is wasteful. These cut-off values are fixed and unknown to the learner. We present an algorithm for this problem and prove a regret upper bound of $O(\log n)$ improving over the best known bound of $O(\log^2 n)$. Lower bounds we prove show that our upper bound is tight. Simulations demonstrate the superiority of our algorithm.

cs.LG↗

arXiv2017

Twenty (simple) questions

A basic combinatorial interpretation of Shannon's entropy function is via the "20 questions" game. This cooperative game is played by two players, Alice and Bob: Alice picks a distribution $π$ over the numbers $\{1,\ldots,n\}$, and announces it to Bob. She then chooses a number $x$ according to $π$, and Bob attempts to identify $x$ using as few Yes/No queries as possible, on average. An optimal strategy for the "20 questions" game is given by a Huffman code for $π$: Bob's questions reveal the codeword for $x$ bit by bit. This strategy finds $x$ using fewer than $H(π)+1$ questions on average. However, the questions asked by Bob could be arbitrary. In this paper, we investigate the following question: Are there restricted sets of questions that match the performance of Huffman codes, either exactly or approximately? Our first main result shows that for every distribution $π$, Bob has a strategy that uses only questions of the form "$x < c$?" and "$x = c$?", and uncovers $x$ using at most $H(π)+1$ questions on average, matching the performance of Huffman codes in this sense. We also give a natural set of $O(rn^{1/r})$ questions that achieve a performance of at most $H(π)+r$, and show that $Ω(rn^{1/r})$ questions are required to achieve such a guarantee. Our second main result gives a set $\mathcal{Q}$ of $1.25^{n+o(n)}$ questions such that for every distribution $π$, Bob can implement an optimal strategy for $π$ using only questions from $\mathcal{Q}$. We also show that $1.25^{n-o(n)}$ questions are needed, for infinitely many $n$. If we allow a small slack of $r$ over the optimal strategy, then roughly $(rn)^{Θ(1/r)}$ questions are necessary and sufficient.

cs.DM↗

arXiv2016

Trading information complexity for error

We consider the standard two-party communication model. The central problem studied in this article is how much one can save in information complexity by allowing an error of $ε$. For arbitrary functions, we obtain lower bounds and upper bounds indicating a gain that is of order $Ω(h(ε))$ and $O(h(\sqrtε))$. Here $h$ denotes the binary entropy function. We analyze the case of the two-bit AND function in detail to show that for this function the gain is $Θ(h(ε))$. This answers a question of [M. Braverman, A. Garg, D. Pankratov, and O. Weinstein, From information to exact communication (extended abstract), STOC'13]. We obtain sharp bounds for the set disjointness function of order $n$. For the case of the distributional error, we introduce a new protocol that achieves a gain of $Θ(\sqrt{h(ε)})$ provided that $n$ is sufficiently large. We apply these results to answer another of question of Braverman et al. regarding the randomized communication complexity of the set disjointness function. Answering a question of [Mark Braverman, Interactive information complexity, STOC'12], we apply our analysis of the set disjointness function to establish a gap between the two different notions of the prior-free information cost. This implies that amortized randomized communication complexity is not necessarily equal to the amortized distributional communication complexity with respect to the hardest distribution.

cs.CC↗