Optimization problems with set submodular objective functions have many real-world applications. In discrete scenarios, where the same item can be selected more than once, the domain is generalized from a 2-element set to a bounded integer lattice. In this work, we consider the problem of maximizing a monotone submodular function on the bounded integer lattice subject to a cardinality constraint. In particular, we focus on maximizing DR-submodular functions, i.e., functions defined on the integer lattice that exhibit the diminishing returns property. Given any epsilon > 0, we present a randomized algorithm with probabilistic guarantees of O(1 - 1/e - epsilon) approximation, using a framework inspired by a Stochastic Greedy algorithm developed for set submodular functions by Mirzasoleiman et al. We then show that, on synthetic DR-submodular functions, applying our proposed algorithm on the integer lattice is faster than the alternatives, including reducing a target problem to the set domain and then applying the fastest known set submodular maximization algorithm.
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The saddlepoint approximation gives an approximation to the density of a random variable in terms of its moment generating function. When the underlying random variable is itself the sum of $n$ unobserved i.i.d. terms, the basic classical result is that the relative error in the density is of order $1/n$. If instead the approximation is interpreted as a likelihood and maximised as a function of model parameters, the result is an approximation to the maximum likelihood estimate (MLE) that can be much faster to compute than the true MLE. This paper proves the analogous basic result for the approximation error between the saddlepoint MLE and the true MLE: subject to certain explicit identifiability conditions, the error has asymptotic size $O(1/n^2)$ for some parameters, and $O(1/n^{3/2})$ or $O(1/n)$ for others. In all three cases, the approximation errors are asymptotically negligible compared to the inferential uncertainty. The proof is based on a factorisation of the saddlepoint likelihood into an exact and approximate term, along with an analysis of the approximation error in the gradient of the log-likelihood. This factorisation also gives insight into alternatives to the saddlepoint approximation, including a new and simpler saddlepoint approximation, for which we derive analogous error bounds. As a corollary of our results, we also obtain the asymptotic size of the MLE error approximation when the saddlepoint approximation is replaced by the normal approximation.
We study the off-policy evaluation (OPE) problem in an infinite-horizon Markov decision process with continuous states and actions. We recast the $Q$-function estimation into a special form of the nonparametric instrumental variables (NPIV) estimation problem. We first show that under one mild condition the NPIV formulation of $Q$-function estimation is well-posed in the sense of $L^2$-measure of ill-posedness with respect to the data generating distribution, bypassing a strong assumption on the discount factor $\gamma$ imposed in the recent literature for obtaining the $L^2$ convergence rates of various $Q$-function estimators. Thanks to this new well-posed property, we derive the first minimax lower bounds for the convergence rates of nonparametric estimation of $Q$-function and its derivatives in both sup-norm and $L^2$-norm, which are shown to be the same as those for the classical nonparametric regression (Stone, 1982). We then propose a sieve two-stage least squares estimator and establish its rate-optimality in both norms under some mild conditions. Our general results on the well-posedness and the minimax lower bounds are of independent interest to study not only other nonparametric estimators for $Q$-function but also efficient estimation on the value of any target policy in off-policy settings.
In a wide variety of applications including online advertising, contractual hiring, and wireless scheduling, the controller is constrained by a stringent budget constraint on the available resources, which are consumed in a random amount by each action, and a stochastic feasibility constraint that may impose important operational limitations on decision-making. In this work, we consider a general model to address such problems, where each action returns a random reward, cost, and penalty from an unknown joint distribution, and the decision-maker aims to maximize the total reward under a budget constraint $B$ on the total cost and a stochastic constraint on the time-average penalty. We propose a novel low-complexity algorithm based on Lyapunov optimization methodology, named ${\tt LyOn}$, and prove that for $K$ arms it achieves $O(\sqrt{K B\log B})$ regret and zero constraint-violation when $B$ is sufficiently large. The low computational cost and sharp performance bounds of ${\tt LyOn}$ suggest that Lyapunov-based algorithm design methodology can be effective in solving constrained bandit optimization problems.
Running machine learning algorithms on large and rapidly growing volumes of data is often computationally expensive, one common trick to reduce the size of a data set, and thus reduce the computational cost of machine learning algorithms, is \emph{probability sampling}. It creates a sampled data set by including each data point from the original data set with a known probability. Although the benefit of running machine learning algorithms on the reduced data set is obvious, one major concern is that the performance of the solution obtained from samples might be much worse than that of the optimal solution when using the full data set. In this paper, we examine the performance loss caused by probability sampling in the context of adaptive submodular maximization. We consider a simple probability sampling method which selects each data point with probability at least $r\in[0,1]$. If we set $r=1$, our problem reduces to finding a solution based on the original full data set. We define sampling gap as the largest ratio between the optimal solution obtained from the full data set and the optimal solution obtained from the samples, over independence systems. Our main contribution is to show that if the sampling probability of each data point is at least $r$ and the utility function is policywise submodular, then the sampling gap is both upper bounded and lower bounded by $1/r$. We show that the property of policywise submodular can be found in a wide range of real-world applications, including pool-based active learning and adaptive viral marketing.
In this work, two problems associated with a downlink multi-user system are considered with the aid of intelligent reflecting surface (IRS): weighted sum-rate maximization and weighted minimal-rate maximization. For the first problem, a novel DOuble Manifold ALternating Optimization (DOMALO) algorithm is proposed by exploiting the matrix manifold theory and introducing the beamforming matrix and reflection vector using complex sphere manifold and complex oblique manifold, respectively, which incorporate the inherent geometrical structure and the required constraint. A smooth double manifold alternating optimization (S-DOMALO) algorithm is then developed based on the Dinkelbach-type algorithm and smooth exponential penalty function for the second problem. Finally, possible cooperative beamforming gain between IRSs and the IRS phase shift with limited resolution is studied, providing a reference for practical implementation. Numerical results show that our proposed algorithms can significantly outperform the benchmark schemes.
Strategic behavior in two-sided matching markets has been traditionally studied in a "one-sided" manipulation setting where the agent who misreports is also the intended beneficiary. Our work investigates "two-sided" manipulation of the deferred acceptance algorithm where the misreporting agent and the manipulator (or beneficiary) are on different sides. Specifically, we generalize the recently proposed accomplice manipulation model (where a man misreports on behalf of a woman) along two complementary dimensions: (a) the two for one model, with a pair of misreporting agents (man and woman) and a single beneficiary (the misreporting woman), and (b) the one for all model, with one misreporting agent (man) and a coalition of beneficiaries (all women). Our main contribution is to develop polynomial-time algorithms for finding an optimal manipulation in both settings. We obtain these results despite the fact that an optimal one for all strategy fails to be inconspicuous, while it is unclear whether an optimal two for one strategy satisfies the inconspicuousness property. We also study the conditions under which stability of the resulting matching is preserved. Experimentally, we show that two-sided manipulations are more frequently available and offer better quality matches than their one-sided counterparts.
This paper addresses theory and applications of $\ell_p$-based Laplacian regularization in semi-supervised learning. The graph $p$-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised learning problems with very few labels, where Laplacian learning is degenerate. In the first part of the paper we prove new discrete to continuum convergence results for $p$-Laplace problems on $k$-nearest neighbor ($k$-NN) graphs, which are more commonly used in practice than random geometric graphs. Our analysis shows that, on $k$-NN graphs, the $p$-Laplacian retains information about the data distribution as $p\to \infty$ and Lipschitz learning ($p=\infty$) is sensitive to the data distribution. This situation can be contrasted with random geometric graphs, where the $p$-Laplacian forgets the data distribution as $p\to \infty$. We also present a general framework for proving discrete to continuum convergence results in graph-based learning that only requires pointwise consistency and monotonicity. In the second part of the paper, we develop fast algorithms for solving the variational and game-theoretic $p$-Laplace equations on weighted graphs for $p>2$. We present several efficient and scalable algorithms for both formulations, and present numerical results on synthetic data indicating their convergence properties. Finally, we conduct extensive numerical experiments on the MNIST, FashionMNIST and EMNIST datasets that illustrate the effectiveness of the $p$-Laplacian formulation for semi-supervised learning with few labels. In particular, we find that Lipschitz learning ($p=\infty$) performs well with very few labels on $k$-NN graphs, which experimentally validates our theoretical findings that Lipschitz learning retains information about the data distribution (the unlabeled data) on $k$-NN graphs.
The Extended Randomized Kaczmarz method is a well known iterative scheme which can find the Moore-Penrose inverse solution of a possibly inconsistent linear system and requires only one additional column of the system matrix in each iteration in comparison with the standard randomized Kaczmarz method. Also, the Sparse Randomized Kaczmarz method has been shown to converge linearly to a sparse solution of a consistent linear system. Here, we combine both ideas and propose an Extended Sparse Randomized Kaczmarz method. We show linear expected convergence to a sparse least squares solution in the sense that an extended variant of the regularized basis pursuit problem is solved. Moreover, we generalize the additional step in the method and prove convergence to a more abstract optimization problem. We demonstrate numerically that our method can find sparse least squares solutions of real and complex systems if the noise is concentrated in the complement of the range of the system matrix and that our generalization can handle impulsive noise.
The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As non-parametric model, it offers a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We describe an efficient algorithm to find the maximum likelihood estimator. The method is contrasted with the ordinary least squares approach in a simulation study, and the method is illustrated on a realistic data set.
Despite their overwhelming capacity to overfit, deep neural networks trained by specific optimization algorithms tend to generalize well to unseen data. Recently, researchers explained it by investigating the implicit regularization effect of optimization algorithms. A remarkable progress is the work (Lyu&Li, 2019), which proves gradient descent (GD) maximizes the margin of homogeneous deep neural networks. Except GD, adaptive algorithms such as AdaGrad, RMSProp and Adam are popular owing to their rapid training process. However, theoretical guarantee for the generalization of adaptive optimization algorithms is still lacking. In this paper, we study the implicit regularization of adaptive optimization algorithms when they are optimizing the logistic loss on homogeneous deep neural networks. We prove that adaptive algorithms that adopt exponential moving average strategy in conditioner (such as Adam and RMSProp) can maximize the margin of the neural network, while AdaGrad that directly sums historical squared gradients in conditioner can not. It indicates superiority on generalization of exponential moving average strategy in the design of the conditioner. Technically, we provide a unified framework to analyze convergent direction of adaptive optimization algorithms by constructing novel adaptive gradient flow and surrogate margin. Our experiments can well support the theoretical findings on convergent direction of adaptive optimization algorithms.
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