We present simple algorithms for batched stochastic multi-armed bandit and batched stochastic linear bandit problems. We prove bounds for their expected regrets that improve and extend the best known regret bounds of Gao, Han, Ren, and Zhou (NeurIPS 2019), for any number of batches. In particular, our algorithms in both settings achieve the optimal expected regrets by using only a logarithmic number of batches. We also study the batched adversarial multi-armed bandit problem for the first time and provide the optimal regret, up to logarithmic factors, of any algorithm with predetermined batch sizes.

We study a multiplayer stochastic multi-armed bandit problem in which players cannot communicate, and if two or more players pull the same arm, a collision occurs and the involved players receive zero reward. We consider the challenging heterogeneous setting, in which different arms may have different means for different players, and propose a new and efficient algorithm that combines the idea of leveraging forced collisions for implicit communication and that of performing matching eliminations. We present a finite-time analysis of our algorithm, giving the first sublinear minimax regret bound for this problem, and prove that if the optimal assignment of players to arms is unique, our algorithm attains the optimal O(ln(T)) regret, solving an open question raised at NeurIPS 2018 by Bistritz and Leshem (2018).

We propose RandUCB, a bandit strategy that uses theoretically derived confidence intervals similar to upper confidence bound (UCB) algorithms, but akin to Thompson sampling (TS), uses randomization to trade off exploration and exploitation. In the $K$-armed bandit setting, we show that there are infinitely many variants of RandUCB, all of which achieve the minimax-optimal $\widetilde{O}(\sqrt{K T})$ regret after $T$ rounds. Moreover, in a specific multi-armed bandit setting, we show that both UCB and TS can be recovered as special cases of RandUCB. For structured bandits, where each arm is associated with a $d$-dimensional feature vector and rewards are distributed according to a linear or generalized linear model, we prove that RandUCB achieves the minimax-optimal $\widetilde{O}(d \sqrt{T})$ regret even in the case of infinite arms. We demonstrate the practical effectiveness of RandUCB with experiments in both multi-armed and structured bandit settings. We show that RandUCB matches the empirical performance of TS while matching the theoretically optimal bounds of UCB algorithms, thus achieving the best of both worlds.

We introduce a novel technique for distribution learning based on a notion of sample compression . Any class of distributions that allows such a compression scheme can be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. As an application of this technique, we prove that ˜Θ( kd 2 /ε 2 ) samples are necessary and sufficient for learning a mixture of k Gaussians in R d , up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that Õ( kd /ε 2 ) samples suffice, matching a known lower bound. Moreover, these results hold in an agnostic learning (or robust estimation) setting, in which the target distribution is only approximately a mixture of Gaussians. Our main upper bound is proven by showing that the class of Gaussians in R d admits a small compression scheme.

We prove new upper and lower bounds on the VC-dimension of deep neural networks with the ReLU activation function. These bounds are tight for almost the entire range of parameters. Letting $W$ be the number of weights and $L$ be the number of layers, we prove that the VC-dimension is $O(W L \log(W))$, and provide examples with VC-dimension $\Omega( W L \log(W/L) )$. This improves both the previously known upper bounds and lower bounds. In terms of the number $U$ of non-linear units, we prove a tight bound $\Theta(W U)$ on the VC-dimension. All of these bounds generalize to arbitrary piecewise linear activation functions, and also hold for the pseudodimensions of these function classes. Combined with previous results, this gives an intriguing range of dependencies of the VC-dimension on depth for networks with different non-linearities: there is no dependence for piecewise-constant, linear dependence for piecewise-linear, and no more than quadratic dependence for general piecewise-polynomial.

We consider PAC learning of probability distributions (a.k.a. density estimation), where we are given an i.i.d. sample generated from an unknown target distribution, and want to output a distribution that is close to the target in total variation distance. Let F be an arbitrary class of probability distributions, and let Fk denote the class of k-mixtures of elements of F. Assuming the existence of a method for learning F with sample complexity m(ε), we provide a method for learning Fk with sample complexity O((k.log k .m(ε))/(ε2)). Our mixture learning algorithm has the property that, if the F-learner is proper and agnostic, then the Fk-learner would be proper and agnostic as well. This general result enables us to improve the best known sample complexity upper bounds for a variety of important mixture classes. First, we show that the class of mixtures of k axis-aligned Gaussians in Rd is PAC-learnable in the agnostic setting with O((kd)/(ε4)) samples, which is tight in k and d up to logarithmic factors. Second, we show that the class of mixtures of k Gaussians in Rd is PAC-learnable in the agnostic setting with sample complexity Õ((kd2)/(ε4)), which improves the previous known bounds of Õ((k3.d2)/(ε4)) and Õ(k4.d4/ε2) in its dependence on k and d. Finally, we show that the class of mixtures of k log-concave distributions over Rd is PAC-learnable using Õ(k.d((d+5)/2)ε(-(d+9)/2)) samples.

We prove that ϴ(k d^2 / ε^2) samples are necessary and sufficient for learning a mixture of k Gaussians in R^d, up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that O(k d / ε^2) samples suffice, matching a known lower bound. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in R^d has an efficient sample compression.

For a rumor spreading protocol, the spread time is defined as the first time that everyone learns the rumor. We compare the synchronous push&pull rumor spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by O(n^{1/3} log^{2/3} n). This improves the O(sqrt n) upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph.

We prove new upper and lower bounds on the VC-dimension of deep neural networks with the ReLU activation function. These bounds are tight for almost the entire range of parameters. Letting $W$ be the number of weights and $L$ be the number of layers, we prove that the VC-dimension is $O(W L \log(W))$, and provide examples with VC-dimension $Ω( W L \log(W/L) )$. This improves both the previously known upper bounds and lower bounds. In terms of the number $U$ of non-linear units, we prove a tight bound $Θ(W U)$ on the VC-dimension. All of these results generalize to arbitrary piecewise linear activation functions.

We consider the following balls-into-bins process with n bins and m balls: Each ball is equipped with a mutually independent exponential clock of rate 1. Whenever a ball's clock rings, the ball samples a random bin and moves there if the number of balls in the sampled bin is smaller than in its current bin. This simple process models a typical load balancing problem where users (balls) seek a selfish improvement of their assignment to resources (bins). From a game theoretic perspective, this is a randomized approach to the well-known KP model [1], while it is known as Randomized Local Search (RLS) in load balancing literature [2], [3]. Up to now, the best bound on the expected time to reach perfect balance was O((ln n)2+ln(n).n2/m) due to [3]. We improve this to an asymptotically tight O(ln(n)+n2/m). Our analysis is based on the crucial observation that performing destructive moves (reversals of RLS moves) cannot decrease the balancing time. This allows us to simplify problem instances and to ignore “inconvenient moves” in the analysis.

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumor in a graph $G$, is defined as follows. Independent exponential clocks of rate 1 are associated with the vertices of $G$, one to each vertex. Initially, one vertex of $G$ knows the rumor. Whenever the clock of a vertex $x$ rings, it calls a random neighbor $y$: if $x$ knows the rumor and $y$ does not, then $x$ tells $y$ the rumor (a push operation), and if $x$ does not know the rumor and $y$ knows it, $y$ tells $x$ the rumor (a pull operation). The average spread time of $G$ is the expected time it takes for all vertices to know the rumor, and the guaranteed spread time of $G$ is the smallest time $t$ such that with probability at least $1 - 1/n$, after time $t$ all vertices know the rumor. The synchronous variant of this protocol, in which each clock rings precisely at times $1,2,\dots$, has been studied extensively. We prove the following results for any $n$-vertex graph: In either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, so it is $O(n \log n)$. In the asynchronous version, both the average and guaranteed spread times are $\Omega(\log n)$. We give examples of graphs illustrating that these bounds are best possible up to constant factors. We also prove the first analytical relationships between the guaranteed spread times in the two versions. First, in all graphs the guaranteed spread time in the asynchronous version is within an $O(\log n)$ factor of that in the synchronous version, and this is tight. Next, we find examples of graphs whose asynchronous spread times are logarithmic, but the synchronous versions are polynomially large. Finally, we show for any graph that the ratio of the guaranteed synchronous spread time to the guaranteed asynchronous spread time is $O\big(n^{2/3}\big)$.

Rumor spreading is a protocol for modeling the spread of information through a network via user-to-user interaction. The spatial preferred attachment (SPA) model is a random graph model for complex networks: Vertices are placed in a metric space, and the link probability depends on the metric distance between vertices and on their degree. We show that the SPA model typically produces graphs that have small effective diameter, i.e., $O(\log^2 n)$, while rumor spreading is relatively slow, namely, polynomial in $n$.

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph G, is defined as follows. Independent exponential clocks of rate 1 are associated with the vertices of G, one to each vertex. Initially, one vertex of G knows the rumour. Whenever the clock of a vertex x rings, it calls a random neighbour y: if x knows the rumour and y does not, then x tells y the rumour (a push operation), and if x does not know the rumour and y knows it, y tells x the rumour (a pull operation). The average spread time of G is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of G is the smallest time t such that with probability at least 1 - 1/n, after time t all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times 1,2,..., has been studied extensively.

We introduce a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In this model, each link has a unit price, and each agent tries to minimize its cost, which is either its eccentricity or its total distance to other players in the underlying (undirected) graph of the created network. Two versions of the game are studied: In the MAX version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and, in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players’ budgets is n − 1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Θ( n ) and Θ(log n ) in MAX and SUM versions, respectively. When each vertex has a unit budget (i.e., can establish a link to just one vertex), the diameter of any equilibrium graph in either version is Θ(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Ω(√log n ). This interesting (and perhaps counterintuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2 O (√log n ) . Finally, we show that if the budget of each player is at least k , then every equilibrium graph in the SUM version is k -connected or has a diameter smaller than 4.

We prove logarithmic upper bounds for the diameters of the random-surfer Webgraph model and the PageRank-based selection Webgraph model, confirming the small-world phenomenon holds for them. In the special case when the generated graph is a tree, we get close lower and upper bounds for the diameters of both models.

A unique-path labeling of a simple, finite graph is a labeling of its edges with real numbers such that for every ordered pair of vertices $(u,v)$, there is at most one nondecreasing path from $u$ to $v$. In this paper we prove that any graph on $n$ vertices that admits a unique-path labeling has at most $n \log_2(n)/2$ edges and that this bound is tight for infinitely many values of $n$. Thus we significantly improve on the previously best known bounds. The main tool of the proof is a combinatorial lemma which might be of independent interest. For every $n$ we also construct an $n$-vertex graph that admits a unique-path labeling and has $n\log_2(n)/2 - O(n)$ edges.

A good edge-labeling of a simple graph is a labeling of its edges with real numbers such that, for any ordered pair of vertices $(u,v)$, there is at most one nondecreasing path from $u$ to $v$. Say a graph is good if it admits a good edge-labeling, and is bad otherwise. Our main result is that any good $n$-vertex graph whose maximum degree is within a constant factor of its average degree (in particular, any good regular graph) has at most $n^{1+o(1)}$ edges. As a corollary, we show that there are bad graphs with arbitrarily large girth, answering a question of Bode, Farzad, and Theis [Good edge-labelings and graphs with girth at least five, preprint, available online at arXiv:1109.1125]. We also prove that for any $\Delta$, there is a $g$ such that any graph with maximum degree at most $\Delta$ and girth at least $g$ is good.

We consider a network creation game in which, each player (vertex) has a limited budget to establish links to other players. In our model, each link has a unit cost and each agent tries to minimize its cost which is its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two variants of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between that vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between that vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard.