Optimal Single-Choice Prophet Inequalities from Samples

We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant $\varepsilon > 0$, $O(n)$ samples from the distribution suffice to achieve the optimal competitive ratio ($\approx 0.745$) within $(1+\varepsilon)$, resolving an open problem of Correa, D\"utting, Fischer, and Schewior.Comment: Appears in Innovations in Theoretical Computer Science (ITCS) 202

Non-Malleable Codes for Small-Depth Circuits

We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by small-depth circuits. For constant-depth circuits of polynomial size (i.e. $\mathsf{AC^0}$ tampering functions), our codes have codeword length $n = k^{1+o(1)}$ for a $k$-bit message. This is an exponential improvement of the previous best construction due to Chattopadhyay and Li (STOC 2017), which had codeword length $2^{O(\sqrt{k})}$. Our construction remains efficient for circuit depths as large as $\Theta(\log(n)/\log\log(n))$ (indeed, our codeword length remains $n\leq k^{1+\epsilon})$, and extending our result beyond this would require separating $\mathsf{P}$ from $\mathsf{NC^1}$. We obtain our codes via a new efficient non-malleable reduction from small-depth tampering to split-state tampering. A novel aspect of our work is the incorporation of techniques from unconditional derandomization into the framework of non-malleable reductions. In particular, a key ingredient in our analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC 2013), a derandomization of the influential switching lemma from circuit complexity; the randomness-efficiency of this switching lemma translates into the rate-efficiency of our codes via our non-malleable reduction.Comment: 26 pages, 4 figure

Detecting communities is Hard (And Counting Them is Even Harder)

We consider the algorithmic problem of community detection in networks. Given an undirected friendship graph G, a subset S of vertices is an (a,b)-community if: * Every member of the community is friends with an (a)-fraction of the community; and * every non-member is friends with at most a (b)-fraction of the community. [Arora, Ge, Sachdeva, Schoenebeck 2012] gave a quasi-polynomial time algorithm for enumerating all the (a,b)-communities for any constants a>b. Here, we prove that, assuming the Exponential Time Hypothesis (ETH), quasi-polynomial time is in fact necessary - and even for a much weaker approximation desideratum. Namely, distinguishing between: * G contains an (1,o(1))-community; and * G does not contain a (b,b+o(1))-community for any b. We also prove that counting the number of (1,o(1))-communities requires quasi-polynomial time assuming the weaker #ETH

Private Learning Implies Online Learning: An Efficient Reduction

We study the relationship between the notions of differentially private learning and online learning in games. Several recent works have shown that differentially private learning implies online learning, but an open problem of Neel, Roth, and Wu \cite{NeelAaronRoth2018} asks whether this implication is {\it efficient}. Specifically, does an efficient differentially private learner imply an efficient online learner? In this paper we resolve this open question in the context of pure differential privacy. We derive an efficient black-box reduction from differentially private learning to online learning from expert advice

When users control the algorithms: Values expressed in practices on the twitter platform

Recent interest in ethical AI has brought a slew of values, including fairness, into conversations about technology design. Research in the area of algorithmic fairness tends to be rooted in questions of distribution that can be subject to precise formalism and technical implementation. We seek to expand this conversation to include the experiences of people subject to algorithmic classification and decision-making. By examining tweets about the “Twitter algorithm” we consider the wide range of concerns and desires Twitter users express. We find a concern with fairness (narrowly construed) is present, particularly in the ways users complain that the platform enacts a political bias against conservatives. However, we find another important category of concern, evident in attempts to exert control over the algorithm. Twitter users who seek control do so for a variety of reasons, many well justified. We argue for the need for better and clearer definitions of what constitutes legitimate and illegitimate control over algorithmic processes and to consider support for users who wish to enact their own collective choices

Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?