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  1. 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 "data-poor'' 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 1/2-factor 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 factor-revealing 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 samplemore »from that same distribution.« less
  2. Generative neural networks have been empirically found very promising in providing effective structural priors for compressed sensing, since they can be trained to span low-dimensional data manifolds in high-dimensional signal spaces. Despite the non-convexity 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 information-theoretically 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 "pseudo-Lipschitzness." 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. Sincemore »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 one-bit recovery, phase retrieval, low-rank matrix recovery, and more.« less
  3. As in standard linear regression, in truncated linear regression, we are given access to observations (Ai,yi)i whose dependent variable equals yi=ATi⋅x∗+ηi, where x∗ is some fixed unknown vector of interest and ηi is independent noise; except we are only given an observation if its dependent variable yi lies in some "truncation set" S⊂ℝ. The goal is to recover x∗ under some favorable conditions on the Ai's and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering k-sparse n-dimensional vectors x∗ from m truncated samples, which attains an optimal ℓ2 reconstruction error of O((klogn)/m‾‾‾‾‾‾‾‾‾‾√). As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaptation of the LASSO optimization problem that accommodates truncation. This generalizes the works of both: (1) [Daskalakis et al. 2018], where no regularization is needed due to the low-dimensionality of the data, and (2) [Wainright 2009], where the objective function is simple due to the absence of truncation. In order to deal with bothmore »truncation and high-dimensionality at the same time, we develop new techniques that not only generalize the existing ones but we believe are of independent interest.« less
  4. We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a d-variate normal N(μ,Σ) means a samples is only revealed if it falls in some subset S⊆Rd; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean μ and covariance matrix Σ can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to S, and S has non-trivial measure under the unknown d-variate normal distribution. Additionally we show that without oracle access to S, any non-trivial estimation is impossible.