Minimum average-case queries of q + 1 -ary search game with small sets

Given a search space S={1,2,...,n}, an unknown element x*∈S and fixed integers ℓ≥1 and q≥1, a q+1-ary ℓ-restricted query is of the following form: which one of the set {A 0,A 1,...,A q} is the x* in?, where (A 0,A 1,...,A q) is a partition of S and | Ai|≤ℓ for i=1,2,...,q. The problem of finding x* from S with q+1-ary size-restricted queries is called as a q+1-ary search game with small sets. In this paper, we consider sequential algorithms for the above problem, and establish the minimum number of average-case sequential queries when x* satisfies the uniform distribution on S. © 2011 Elsevier B.V. All rights reserved

Unbiased Black-Box Complexities of Jump Functions

We analyze the unbiased black-box complexity of jump functions with small, medium, and large sizes of the fitness plateau surrounding the optimal solution. Among other results, we show that when the jump size is $(1/2 - \varepsilon)n$, that is, only a small constant fraction of the fitness values is visible, then the unbiased black-box complexities for arities $3$ and higher are of the same order as those for the simple \textsc{OneMax} function. Even for the extreme jump function, in which all but the two fitness values $n/2$ and $n$ are blanked out, polynomial-time mutation-based (i.e., unary unbiased) black-box optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a $\Theta(n^{-1/2})$ fraction) is a plateau of constant fitness. To prove these results, we introduce new tools for the analysis of unbiased black-box complexities, for example, selecting the new parent individual not by comparing the fitnesses of the competing search points, but also by taking into account the (empirical) expected fitnesses of their offspring.Comment: This paper is based on results presented in the conference versions [GECCO 2011] and [GECCO 2014

An Efficient Streaming Algorithm for the Submodular Cover Problem

We initiate the study of the classical Submodular Cover (SC) problem in the data streaming model which we refer to as the Streaming Submodular Cover (SSC). We show that any single pass streaming algorithm using sublinear memory in the size of the stream will fail to provide any non-trivial approximation guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only seek to find a partial cover. We design the first Efficient bicriteria Submodular Cover Streaming (ESC-Streaming) algorithm for this problem, and provide theoretical guarantees for its performance supported by numerical evidence. Our algorithm finds solutions that are competitive with the near-optimal offline greedy algorithm despite requiring only a single pass over the data stream. In our numerical experiments, we evaluate the performance of ESC-Streaming on active set selection and large-scale graph cover problems.Comment: To appear in NIPS'1

HopSkipJumpAttack: A Query-Efficient Decision-Based Attack

The goal of a decision-based adversarial attack on a trained model is to generate adversarial examples based solely on observing output labels returned by the targeted model. We develop HopSkipJumpAttack, a family of algorithms based on a novel estimate of the gradient direction using binary information at the decision boundary. The proposed family includes both untargeted and targeted attacks optimized for $\ell_2$ and $\ell_\infty$ similarity metrics respectively. Theoretical analysis is provided for the proposed algorithms and the gradient direction estimate. Experiments show HopSkipJumpAttack requires significantly fewer model queries than Boundary Attack. It also achieves competitive performance in attacking several widely-used defense mechanisms. (HopSkipJumpAttack was named Boundary Attack++ in a previous version of the preprint.

Cryptography based on the Hardness of Decoding

This thesis provides progress in the fields of for lattice and coding based cryptography. The first contribution consists of constructions of IND-CCA2 secure public key cryptosystems from both the McEliece and the low noise learning parity with noise assumption. The second contribution is a novel instantiation of the lattice-based learning with errors problem which uses uniform errors

Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with $|E(G)| = O(n)$, for which we show a lowerbound of $\frac{n}{2^{O(\sqrt{\log n})}}$ for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size $O(\frac{n}{RS(n)^{c}})$ for some $0 < c < 1$, where $RS(n)$ is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to $\frac{n}{2^{(\log n)^{o(1)}}}$ would require a breakthrough in the study of lower bounds on $RS(n)$, which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of $\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$, where $SumIndex(n)$ is the communication complexity of the Sum-Index problem over $Z_n$. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be $\Theta(\frac{n}{2^{(\log n)^c}})$ for some $0<c < 1$