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null (Ed.)We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p_1, p_2, …, p_s, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the p_i’s are uniform or εfar from being uniform in ℓ_1distance (2) Identity Testing: Testing whether all the p_i’s are equal to an explicitly given distribution q or εfar from q in ℓ_1distance, and (3) Closeness Testing: Testing whether all the p_i’s are equal to a distribution q which we have sample access to, or εfar from q in ℓ_1distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.more » « less

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most (1+ϵ)n edges (where n is the number of vertices and ϵ is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge e belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of e. The challenge is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general boundeddegree graphs, the query complexity of any such algorithm must be Ω(n−−√). This lower bound holds for constantdegree graphs that have high expansion. Next we design an algorithm for (boundeddegree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence nonexpanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of this algorithm is poly(1/ϵ,h) (independent of n and the maximum degree), where h is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a highlevel there are several similarities, and we highlight both the similarities and the differences.more » « less

null (Ed.)Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "onthefly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sublinear and thus, sampling the entire object up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the ErdösRényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous localaccess implementations for random graphs, we support VertexPair and NextNeighbor queries. In addition, we introduce a new RandomNeighbor query. Next, we give the first localaccess implementation for AllNeighbors queries in the (sparse and directed) Kleinberg’s SmallWorld model. Our implementations require no preprocessing time, and answer each query using O(poly(log n)) time, random bits, and additional space. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always nonnegative). Here, we support Height queries to find the location of the walk, and FirstReturn queries to find the time when the walk returns to a specified location. This in turn can be used to implement NextNeighbor queries on random rooted ordered trees, and MatchingBracket queries on random well bracketed expressions (the Dyck language). Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid qcolorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sublinear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sublinear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memoryless, and can maintain consistency with noncommunicating copies of itself.more » « less

null (Ed.)In this paper we study the smooth convexconcave saddle point problem. Specifically, we analyze the last iterate convergence properties of the Extragradient (EG) algorithm. It is well known that the ergodic (averaged) iterates of EG converge at a rate of O(1/T) (Nemirovski, 2004). In this paper, we show that the last iterate of EG converges at a rate of O(1/T‾‾√). To the best of our knowledge, this is the first paper to provide a convergence rate guarantee for the last iterate of EG for the smooth convexconcave saddle point problem. Moreover, we show that this rate is tight by proving a lower bound of Ω(1/T‾‾√) for the last iterate. This lower bound therefore shows a quadratic separation of the convergence rates of ergodic and last iterates in smooth convexconcave saddle point problems.more » « less

null (Ed.)Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution q over the subsets of a ground set, we aim to distinguish whether q is a DPP distribution or ϵfar from all DPP distributions in ℓ1distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of logsubmodular distributionsmore » « less

null (Ed.)Generative neural networks have been empirically found very promising in providing effective structural priors for compressed sensing, since they can be trained to span lowdimensional data manifolds in highdimensional signal spaces. Despite the nonconvexity of the resulting optimization problem, it has also been shown theoretically that, for neural networks with random Gaussian weights, a signal in the range of the network can be efficiently, approximately recovered from a few noisy measurements. However, a major bottleneck of these theoretical guarantees is a network expansivity condition: that each layer of the neural network must be larger than the previous by a logarithmic factor. Our main contribution is to break this strong expansivity assumption, showing that constant expansivity suffices to get efficient recovery algorithms, besides it also being informationtheoretically necessary. To overcome the theoretical bottleneck in existing approaches we prove a novel uniform concentration theorem for random functions that might not be Lipschitz but satisfy a relaxed notion which we call "pseudoLipschitzness." Using this theorem we can show that a matrix concentration inequality known as the Weight Distribution Condition (WDC), which was previously only known to hold for Gaussian matrices with logarithmic aspect ratio, in fact holds for constant aspect ratios too. Since the WDC is a fundamental matrix concentration inequality in the heart of all existing theoretical guarantees on this problem, our tighter bound immediately yields improvements in all known results in the literature on compressed sensing with deep generative priors, including onebit recovery, phase retrieval, lowrank matrix recovery, and more.more » « less

null (Ed.)We identify the first static credible mechanism for multiitem additive auctions that achieves a constant factor of the optimal revenue. This is one instance of a more general framework for designing twopart tariff auctions, adapting the duality framework of Cai et al [CDW16]. Given a (not necessarily incentive compatible) auction format A satisfying certain technical conditions, our framework augments the auction with a personalized entry fee for each bidder, which must be paid before the auction can be accessed. These entry fees depend only on the prior distribution of bidder types, and in particular are independent of realized bids. Our framework can be used with many common auction formats, such as simultaneous firstprice, simultaneous secondprice, and simultaneous allpay auctions. If allpay auctions are used, we prove that the resulting mechanism is credible in the sense that the auctioneer cannot benefit by deviating from the stated mechanism after observing agent bids. If secondprice auctions are used, we obtain a truthful O(1)approximate mechanism with fixed entry fees that are amenable to tuning via online learning techniques. Our results for first price and allpay are the first revenue guarantees of nontruthful mechanisms in multidimensional environments; an open question in the literature [RST17].more » « less

null (Ed.)Generative adversarial networks (GANs) are a widely used framework for learning generative models. Wasserstein GANs (WGANs), one of the most successful variants of GANs, require solving a minmax optimization problem to global optimality, but are in practice successfully trained using stochastic gradient descentascent. In this paper, we show that, when the generator is a onelayer network, stochastic gradient descentascent converges to a global solution with polynomial time and sample complexity.more » « less

null (Ed.)We obtain global, nonasymptotic convergence guarantees for independent learning algorithms in competitive reinforcement learning settings with two agents (i.e., zerosum stochastic games). We consider an episodic setting where in each episode, each player independently selects a policy and observes only their own actions and rewards, along with the state. We show that if both players run policy gradient methods in tandem, their policies will converge to a minmax equilibrium of the game, as long as their learning rates follow a twotimescale rule (which is necessary). To the best of our knowledge, this constitutes the first finitesample convergence result for independent policy gradient methods in competitive RL; prior work has largely focused on centralized, coordinated procedures for equilibrium computation.more » « less

null (Ed.)We consider the classical problem of selling a single item to a single bidder whose value for the item is drawn from a regular distribution F, in a "datapoor'' regime where Fis not known to the seller, and very few samples from Fare available. Prior work [Dhangwatnotai et al '10] has shown that one sample from Fcan be used to attain a 1/2factor approximation to the optimal revenue, but it has been challenging to improve this guarantee when more samples from Fare provided, even when two samples from Fare provided. In this case, the best approximation known to date is 0.509, achieved by the Empirical Revenue Maximizing (ERM) mechanism Babaioff et al. '18]. We improve this guarantee to 0.558, and provide a lower bound of 0.65. Our results are based on a general framework, based on factorrevealing Semidefinite Programming relaxations aiming to capture as tight as possible a superset of product measures of regular distributions, the challenge being that neither regularity constraints nor product measures are convex constraints. The framework is general and can be applied in more abstract settings to evaluate the performance of a policy chosen using independent samples from a distribution and applied on a fresh sample from that same distribution.more » « less