We investigate the sample-memory-pass trade-offs for pure exploration in multi-pass streaming multi-armed bandits (MABs) with the *a priori* knowledge of the optimality gap ?_[2]. Here, and throughout, the optimality gap ?_[i] is defined as the mean reward gap between the best and the i-th best arms. A recent line of results have shown that if there is no known ?_[2], a pass complexity of ̃?(log(1/?_[2])) is necessary and sufficient to obtain the *worst-case optimal* O(n/?²_[2]) sample complexity with a single-arm memory. However, our understanding of multi-pass algorithms with known ?_[2] is still limited. Here, the key open problem is how many passes are required to achieve the complexity, i.e., O( ∑ᵢ₌₂ⁿ1/?²_[i] log{n}) arm pulls, with a sublinear memory size. In this work, we show that the ``right answer'' for the question is ̃?(log{n}) passes. We first present a lower bound, showing that any algorithm that finds the best arm with slightly sublinear memory -- a memory of o(n/polylog(n)) arms -- and O( ∑ᵢ₌₂ⁿ1/?²_[i] log n) arm pulls has to make ?(log n/loglog n) passes over the stream. We then show a nearly-matching algorithm that assuming the knowledge of ?_[2], finds the best arm with O( ∑ᵢ₌₂ⁿ1/?²_[i] log n) arm pulls and a *single arm* memory.

In this paper, we study the collaborative learning model, which concerns the tradeoff between parallelism and communication overhead in multi-agent multi-armed bandits. For regret minimization in multi-armed bandits, we present the first set of tradeoffs between the number of rounds of communication between the agents and the regret of the collaborative learning process.

In this paper, we study the tradeoffs between the time and the number of communication rounds of the best arm identification problem in the heterogeneous collaborative learning model, where multiple agents interact with possibly different environments and they want to learn in parallel an objective function in the aggregated environment. By proving almost tight upper and lower bounds, we show that collaborative learning in the heterogeneous setting is inherently more difficult than that in the homogeneous setting in terms of the time-round tradeoff.

We investigate top-m arm identification, a basic problem in bandit theory, in a multi-agent learning model in which agents collaborate to learn an objective function. We are interested in designing collaborative learning algorithms that achieve maximum speedup (compared to single-agent learning algorithms) using minimum communication cost, as communication is frequently the bottleneck in multi-agent learning. We give both algorithmic and impossibility results, and conduct a set of experiments to demonstrate the effectiveness of our algorithms.

Motivated by real-world applications such as fast fashion retailing and online advertising, the Multinomial Logit Bandit (MNL-bandit) is a popular model in online learning and operations research, and has attracted much attention in the past decade. In this paper, we give efficient algorithms for pure exploration in MNL-bandit. Our algorithms achieve instance-sensitive pull complexities. We also complement the upper bounds by an almost matching lower bound.

This paper introduces SyncSignature, the first fully parallelizable algorithmic framework for tree similarity joins under edit distance. SyncSignature makes use of implicit-synchronized signature generation schemes, which allow for an efficient and parallelizable candidate-generation procedure via hash join. Our experiments on large real-world datasets show that the proposed algorithms under the SyncSignature framework significantly outperform the state-of-the-art algorithm in the parallel computation environment. For datasets with big trees, they also exceed the state-of-the-art algorithms by a notable margin in the centralized/single-thread computation environment. To complement and guide the experimental study, we also provide a thorough theoretical analysis for all proposed signature generation schemes.

We consider the following problem in this paper: given a set of n distributions, find the top- m ones with the largest means. This problem is also called top- m arm identifications in the literature of reinforcement learning, and has numerous applications. We study the problem in the collaborative learning model where we have multiple agents who can draw samples from the n distributions in parallel. Our goal is to characterize the tradeoffs between the running time of learning process and the number of rounds of interaction between agents, which is very expensive in various scenarios. We give optimal time-round tradeoffs, as well as demonstrate complexity separations between top-1 arm identification and top- m arm identifications for general m and between fixed-time and fixed-confidence variants. As a byproduct, we also give an algorithm for selecting the distribution with the m-th largest mean in the collaborative learning model.

We study the problem of coarse ranking in the multi-armed bandits (MAB) setting, where we have a set of arms each of which is associated with an unknown distribution. The task is to partition the arms into clusters of predefined sizes, such that the mean of any arm in the $i$-th cluster is larger than that of any arm in the $j$-th cluster for any $j > i$. Coarse ranking generalizes a number of basic problems in MAB (e.g., best arm identification) and has many real-world applications. We initiate the study of the problem in the batched model where we can only have a small number of policy changes. We study both the fixed budget and fixed confidence variants in MAB, and propose algorithms and prove impossibility results which together give almost tight tradeoffs between the total number of arms pulls and the number of policy changes. We have tested our algorithms in both real and synthetic data; our experimental results have demonstrated the efficiency of the proposed methods.

We study linear programming and general LP-type problems in several big data (streaming and distributed) models. We mainly focus on low dimensional problems in which the number of constraints is much larger than the number of variables. Low dimensional LP-type problems appear frequently in various machine learning tasks such as robust regression, support vector machines, and core vector machines. As supporting large-scale machine learning queries in database systems has become an important direction for database research, obtaining efficient algorithms for low dimensional LP-type problems on massive datasets is of great value. In this paper we give both upper and lower bounds for LP-type problems in distributed and streaming models. Our bounds are almost tight when the dimensionality of the problem is a fixed constant.

Given an edge-weighted graph G with a set $$Q$$ of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the size of minimum cut between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph G being either an arbitrary graph or coming from a specific graph class. We show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with k terminals that require $$2^{k-2}$$ edges in any mimicking network. This nearly matches an upper bound of $$\mathcal {O}(k 2^{2k})$$ of Krauthgamer and Rika (in: Khanna (ed) Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, SODA 2013, New Orleans, 2013) and is in sharp contrast with the upper bounds of $$\mathcal {O}(k^2)$$ and $$\mathcal {O}(k^4)$$ under the assumption that all terminals lie on a single face (Goranci et al., in: Pruhs and Sohler (eds) 25th Annual European symposium on algorithms (ESA 2017), 2017, arXiv:1702.01136 ; Krauthgamer and Rika in Refined vertex sparsifiers of planar graphs, 2017, arXiv:1702.05951 ). As a side result we show a tight example for double-exponential upper bounds given by Hagerup et al. (J Comput Syst Sci 57(3):366–375, 1998), Khan and Raghavendra (Inf Process Lett 114(7):365–371, 2014), and Chambers and Eppstein (J Gr Algorithms Appl 17(3):201–220, 2013).

We introduce a new edit distance measure between a pair of "clonal trees", each representing the progression and mutational heterogeneity of a tumor sample, constructed by the use of single cell or bulk high throughput sequencing data. In a clonal tree, each vertex represents a specific tumor clone, and is labeled with one or more mutations in a way that each mutation is assigned to the oldest clone that harbors it. Given two clonal trees, our multi-labeled tree edit distance (MLTED) measure is defined as the minimum number of mutation/label deletions, (empty) leaf deletions, and vertex (clonal) expansions, applied in any order, to convert each of the two trees to the maximal common tree. We show that the MLTED measure can be computed efficiently in polynomial time and it captures the similarity between trees of different clonal granularity well. We have implemented our algorithm to compute MLTED exactly and applied it to a variety of data sets successfully. The source code of our method can be found in: https://github.com/khaled-rahman/leafDelTED.