We present a parallel algorithm for the (1 — ɛ) -approximate maximum flow problem in capacitated, undirected graphs with n vertices and m edges, achieving O(ɛ-3 polylog n) depth and O(mɛ-3 polylog n) work in the PRAM model. Although near-linear time sequential algorithms for this problem have been known for almost a decade, no parallel algorithms that simultaneously achieved polylogarithmic depth and near-linear work were known.

We consider the problem of finding top-$k$ items from a set of $n$ items using actively chosen pairwise comparisons. This problem has been widely studied in machine learning and has widespread applications in recommendation systems, sports, social choice etc. Motivated by applications where there can be a substantial delay between requesting comparisons and receiving feedback, we consider an active/adaptive learning setting where the algorithm uses limited rounds of parallel interaction with the feedback generating oracle. We study this problem under the strong stochastic transitivity (SST) noise model which is a widely studied ranking model and captures many applications. A special case of this model is the noisy comparison model for which it was recently shown that $O(n \log k)$ comparisons and $\log^* n$ rounds of adaptivity are sufficient to find the set of top-$k$ items (Cohen-Addad et al., 2020; Braverman et al., 2019). Under the more general SST model, it is known that $O(n)$ comparisons and $O(n)$ rounds are sufficient to find a PAC top-1 item (Falahatgar et al., 2017a,b), however, not much seems to be known for general $k$, even given unbounded rounds of adaptivity. We first show that $\Omega (nk)$ comparisons are necessary for PAC top-$k$ identification under SST even with unbounded adaptivity, establishing that this problem is strictly harder under SST than it is for the noisy comparison model. Our main contribution is to show that the 2-round query complexity for this problem is $\widetilde{\Theta} (n^{4/3} + nk)$, and to show that just 3 rounds are sufficient to obtain a nearly optimal query complexity of $\widetilde{\Theta}(nk)$. We further show that our 3-round result can be improved by a $\log (n)$ factor using $2 \log^* n + 4$ rounds.

The stochastic $K$-armed bandit problem has been studied extensively due to its applications in various domains ranging from online advertising to clinical trials. In practice however, the number of arms can be very large resulting in large memory requirements for simultaneously processing them. In this paper we consider a streaming setting where the arms are presented in a stream and the algorithm uses limited memory to process these arms. Here, the goal is not only to minimize regret, but also to do so in minimal memory. Previous algorithms for this problem operate in one of the two settings: they either use $\Omega(\log \log T)$ passes over the stream \citep{rathod2021reducing, ChaudhuriKa20, Liau+18}, or just a single pass \citep{Maiti+21}. In this paper we study the trade-off between memory and regret when $B$ passes over the stream are allowed, for any $B \geq 1$, and establish \emph{tight} regret upper and lower bounds for any $B$-pass algorithm. Our results uncover a surprising \emph{sharp transition phenomenon}: $O(1)$ memory is sufficient to achieve $\widetilde\Theta\paren{T^{\half + \frac{1}{2^{B+2}-2}}}$ regret in $B$ passes, and increasing the memory to any quantity that is $o(K)$ has almost no impact on further reducing this regret, unless we use $\Omega(K)$ memory. Our main technical contribution is our lower bound which requires the use of \emph{information-theoretic techniques} as well as ideas from \emph{round elimination} to show that the \emph{residual problem} remains challenging over subsequent passes.

Hierarchical clustering over graphs is a fundamental task in data mining and machine learning with applications in many domains including phylogenetics, social network analysis, and information retrieval. Specifically, we consider the recently popularized objective function for hierarchical clustering due to Dasgupta~\cite{Dasgupta16}, namely, minimum cost hierarchical partitioning. Previous algorithms for (approximately) minimizing this objective function require linear time/space complexity. In many applications the underlying graph can be massive in size making it computationally challenging to process the graph even using a linear time/space algorithm. As a result, there is a strong interest in designing algorithms that can perform global computation using only sublinear resources (space, time, and communication). The focus of this work is to study hierarchical clustering for massive graphs under three well-studied models of sublinear computation which focus on space, time, and communication, respectively, as the primary resources to optimize: (1) (dynamic) streaming model where edges are presented as a stream, (2) query model where the graph is queried using neighbor and degree queries, (3) massively parallel computation (MPC) model where the edges of the graph are partitioned over several machines connected via a communication channel.We design sublinear algorithms for hierarchical clustering in all three models above. At the heart of our algorithmic results is a view of the objective in terms of cuts in the graph, which allows us to use a relaxed notion of cut sparsifiers to do hierarchical clustering while introducing only a small distortion in the objective function. Our main algorithmic contributions are then to show how cut sparsifiers of the desired form can be efficiently constructed in the query model and the MPC model. We complement our algorithmic results by establishing nearly matching lower bounds that rule out the possibility of designing algorithms with better performance guarantees in each of these models.

The dueling bandits problem has received a lot of attention in recent years due to its applications in recommendation systems and information retrieval. However, due to the prevalence of malicious users in these systems, it is becoming increasingly important to design dueling bandit algorithms that are robust to corruptions introduced by these malicious users. In this paper we study dueling bandits in the presence of an adversary that can corrupt some of the feedback received by the learner. We propose an algorithm for this problem that is agnostic to the amount of corruption introduced by the adversary: its regret degrades gracefully with the amount of corruption, and in case of no corruption, it essentially matches the optimal regret bounds achievable in the purely stochastic dueling bandits setting.

Rank aggregation from pairwise preferences has widespread applications in recommendation systems and information retrieval. Given the enormous economic and societal impact of these applications, and the consequent incentives for malicious players to manipulate ranking outcomes in their favor, an important challenge is to make rank aggregation algorithms robust to adversarial manipulations in data. In this paper, we initiate the study of robustness in rank aggregation under the popular Bradley-Terry-Luce (BTL) model for pairwise comparisons. We consider a setting where pairwise comparisons are initially generated according to a BTL model, but a fraction of these comparisons are corrupted by an adversary prior to being reported to us. We consider a strong contamination model, where an adversary having complete knowledge of the initial truthful data and the underlying true BTL parameters, can subsequently corrupt the truthful data by inserting, deleting, or changing data points. The goal is to estimate the true score/weight of each item under the BTL model, even in the presence of these corruptions. We characterize the extent of adversarial corruption under which the true BTL parameters are uniquely identifiable. We also provide a novel pruning algorithm that provably cleans the data of adversarial corruption under reasonable conditions on data generation and corruption. We corroborate our theory with experiments on both synthetic as well as real data showing that previous algorithms are vulnerable to even small amounts of corruption, whereas our algorithm can clean a reasonably high amount of corruption.

The problem of rank aggregation from pairwise and multiway comparisons has a wide range of implications, ranging from recommendation systems to sports rankings to social choice. Some of the most popular algorithms for this problem come from the class of spectral ranking algorithms; these include the rank centrality (RC) algorithm for pairwise comparisons, which returns consistent estimates under the Bradley-Terry-Luce (BTL) model for pairwise comparisons (Negahban et al., 2017), and its generalization, the Luce spectral ranking (LSR) algorithm, which returns consistent estimates under the more general multinomial logit (MNL) model for multiway comparisons (Maystre & Grossglauser, 2015). In this paper, we design a provably faster spectral ranking algorithm, which we call accelerated spectral ranking (ASR), that is also consistent under the MNL/BTL models. Our accelerated algorithm is achieved by designing a random walk that has a faster mixing time than the random walks associated with previous algorithms. In addition to a faster algorithm, our results yield improved sample complexity bounds for recovery of the MNL/BTL parameters: to the best of our knowledge, we give the first general sample complexity bounds for recovering the parameters of the MNL model from multiway comparisons under any (connected) comparison graph (and improve significantly over previous bounds for the BTL model for pairwise comparisons). We also give a message-passing interpretation of our algorithm, which suggests a decentralized distributed implementation. Our experiments on several real-world and synthetic datasets confirm that our new ASR algorithm is indeed orders of magnitude faster than existing algorithms.