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Register a new userMisspecified Nonconvex Statistical Optimization for Phase Retrieval
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Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.
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We propose a new variant of AMSGrad, a popular adaptive gradient based optimization algorithm widely used for training deep neural networks. Our algorithm adds prior knowledge about the sequence of consecutive mini-batch gradients and leverages its underlying structure making the gradients sequentially predictable. By exploiting the predictability and ideas from optimistic online learning, the proposed algorithm can accelerate the convergence and increase sample efficiency. After establishing a tighter upper bound under some convexity conditions on the regret, we offer a complimentary view of our algorithm which generalizes the offline and stochastic version of nonconvex optimization. In the nonconvex case, we establish a non-asymptotic convergence bound independently of the initialization. We illustrate the practical speedup on several deep learning models via numerical experiments.
Large graphs abound in machine learning, data mining, and several related areas. A useful step towards analyzing such graphs is that of obtaining certain summary statistics - e.g., or the expected length of a shortest path between two nodes, or the expected weight of a minimum spanning tree of the graph, etc. These statistics provide insight into the structure of a graph, and they can help predict global properties of a graph. Motivated thus, we propose to study statistical properties of structured subgraphs (of a given graph), in particular, to estimate the expected objective function value of a combinatorial optimization problem over these subgraphs. The general task is very difficult, if not unsolvable; so for concreteness we describe a more specific statistical estimation problem based on spanning trees. We hope that our position paper encourages others to also study other types of graphical structures for which one can prove nontrivial statistical estimates.
Bayesian coresets have emerged as a promising approach for implementing scalable Bayesian inference. The Bayesian coreset problem involves selecting a (weighted) subset of the data samples, such that the posterior inference using the selected subset closely approximates the posterior inference using the full dataset. This manuscript revisits Bayesian coresets through the lens of sparsity constrained optimization. Leveraging recent advances in accelerated optimization methods, we propose and analyze a novel algorithm for coreset selection. We provide explicit convergence rate guarantees and present an empirical evaluation on a variety of benchmark datasets to highlight our proposed algorithms superior performance compared to state-of-the-art on speed and accuracy.
We study a bandit version of phase retrieval where the learner chooses actions $(A_t)_{t=1}^n$ in the $d$-dimensional unit ball and the expected reward is $langle A_t, theta_starrangle^2$ where $theta_star in mathbb R^d$ is an unknown parameter vector. We prove that the minimax cumulative regret in this problem is $smash{tilde Theta(d sqrt{n})}$, which improves on the best known bounds by a factor of $smash{sqrt{d}}$. We also show that the minimax simple regret is $smash{tilde Theta(d / sqrt{n})}$ and that this is only achievable by an adaptive algorithm. Our analysis shows that an apparently convincing heuristic for guessing lower bounds can be misleading and that uniform bounds on the information ratio for information-directed sampling are not sufficient for optimal regret.
We consider the problem of reconstructing an $n$-dimensional $k$-sparse signal from a set of noiseless magnitude-only measurements. Formulating the problem as an unregularized empirical risk minimization task, we study the sample complexity performance of gradient descent with Hadamard parametrization, which we call Hadamard Wirtinger flow (HWF). Provided knowledge of the signal sparsity $k$, we prove that a single step of HWF is able to recover the support from $k(x^*_{max})^{-2}$ (modulo logarithmic term) samples, where $x^*_{max}$ is the largest component of the signal in magnitude. This support recovery procedure can be used to initialize existing reconstruction methods and yields algorithms with total runtime proportional to the cost of reading the data and improved sample complexity, which is linear in $k$ when the signal contains at least one large component. We numerically investigate the performance of HWF at convergence and show that, while not requiring any explicit form of regularization nor knowledge of $k$, HWF adapts to the signal sparsity and reconstructs sparse signals with fewer measurements than existing gradient based methods.
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