A Happiness Maximizing Set (HMS) is a useful concept in which a smaller subset of a database is selected while mostly preserving the best scores along every possible utility function. In this paper, we study the Average Happiness Maximizing Sets (AHMS) and $k$-Happiness Maximizing Sets ($k$-HMS) problems. Specifically, AHMS maximizes the average of this ratio within a distribution of utility functions. Meanwhile, $k$-HMS selects $r$ records from the database such that the minimum happiness ratio between the $k$-th best score in the database and the best score in the selected records for any possible utility function is maximized. AHMS and $k$-HMS seek the same optimal solutions as the more established Average Regret Minimizing Sets (ARMS) and $k$-Regret Minimizing Sets ($k$-RMS) problems, respectively, but the use of the happiness metric allows for the derivation of stronger theoretical results and more natural approximation schemes. We provide approximation algorithms for AHMS with better approximation ratios and time complexities than known algorithms for ARMS. Next, we show that the problem of approximating $k$-HMS within any finite factor is NP-Hard when the dimensionality of the database is unconstrained and extend the result to an inapproximability proof of $k$-RMS. Finally, we provide dataset reduction schemes which can be used to reduce the runtime of existing heuristic based algorithms, as well as to derive polynomial-time approximation schemes for both $k$-HMS when dimensionality is fixed. We further provide experimental validation showing that our AHMS algorithm achieves the same happiness as the existing Greedy Shrink FAM algorithm while running faster by over 2 orders of magnitude on even small datasets while our reduction scheme was able to reduce runtimes of existing $k$-RMS solvers by up to 92\%.
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In this work, we propose a globally optimal joint successive interference cancellation (SIC) ordering and power allocation (JSPA) algorithm for the sum-rate maximization problem in downlink multi-cell non-orthogonal multiple access (NOMA) systems. The proposed algorithm is based on the exploration of base stations (BSs) power consumption, and closed-form of optimal powers obtained for each cell. Although the optimal JSPA algorithm scales well with larger number of users, it is still exponential in the number of cells. For any suboptimal decoding order, we propose a low-complexity near-optimal joint rate and power allocation (JRPA) strategy in which the complete rate region of users is exploited. Furthermore, we design a near-optimal semi-centralized JSPA framework for a two-tier heterogeneous network such that it scales well with larger number of small-BSs and users. Numerical results show that JRPA highly outperforms the case that the users are enforced to achieve their channel capacity by imposing the well-known SIC necessary condition on power allocation. Moreover, the proposed semi-centralized JSPA framework significantly outperforms the fully distributed framework, where all the BSs operate in their maximum power budget. Therefore, the centralized JRPA and semi-centralized JSPA algorithms with near-optimal performances are good choices for larger number of cells and users.
We consider the problem of online learning in Linear Quadratic Control systems whose state transition and state-action transition matrices $A$ and $B$ may be initially unknown. We devise an online learning algorithm and provide guarantees on its expected regret. This regret at time $T$ is upper bounded (i) by $\widetilde{O}((d_u+d_x)\sqrt{d_xT})$ when $A$ and $B$ are unknown, (ii) by $\widetilde{O}(d_x^2\log(T))$ if only $A$ is unknown, and (iii) by $\widetilde{O}(d_x(d_u+d_x)\log(T))$ if only $B$ is unknown and under some mild non-degeneracy condition ($d_x$ and $d_u$ denote the dimensions of the state and of the control input, respectively). These regret scalings are minimal in $T$, $d_x$ and $d_u$ as they match existing lower bounds in scenario (i) when $d_x\le d_u$ [SF20], and in scenario (ii) [lai1986]. We conjecture that our upper bounds are also optimal in scenario (iii) (there is no known lower bound in this setting). Existing online algorithms proceed in epochs of (typically exponentially) growing durations. The control policy is fixed within each epoch, which considerably simplifies the analysis of the estimation error on $A$ and $B$ and hence of the regret. Our algorithm departs from this design choice: it is a simple variant of certainty-equivalence regulators, where the estimates of $A$ and $B$ and the resulting control policy can be updated as frequently as we wish, possibly at every step. Quantifying the impact of such a constantly-varying control policy on the performance of these estimates and on the regret constitutes one of the technical challenges tackled in this paper.
We present the first formal verification of approximation algorithms for NP-complete optimization problems: vertex cover, independent set, set cover, center selection, load balancing, and bin packing. We uncover incompletenesses in existing proofs and improve the approximation ratio in one case. All proofs are uniformly invariant based.
For optimal control problems constrained by a initial-valued parabolic PDE, we have to solve a large scale saddle point algebraic system consisting of considering the discrete space and time points all together. A popular strategy to handle such a system is the Krylov subspace method, for which an efficient preconditioner plays a crucial role. The matching-Schur-complement preconditioner has been extensively studied in literature and the implementation of this preconditioner lies in solving the underlying PDEs twice, sequentially in time. In this paper, we propose a new preconditioner for the Schur complement, which can be used parallel-in-time (PinT) via the so called diagonalization technique. We show that the eigenvalues of the preconditioned matrix are low and upper bounded by positive constants independent of matrix size and the regularization parameter. The uniform boundedness of the eigenvalues leads to an optimal linear convergence rate of conjugate gradient solver for the preconditioned Schur complement system. To the best of our knowledge, it is the first time to have an optimal convergence analysis for a PinT preconditioning technique of the optimal control problem. Numerical results are reported to show that the performance of the proposed preconditioner is robust with respect to the discretization step-sizes and the regularization parameter.
When the sizes of the state and action spaces are large, solving MDPs can be computationally prohibitive even if the probability transition matrix is known. So in practice, a number of techniques are used to approximately solve the dynamic programming problem, including lookahead, approximate policy evaluation using an m-step return, and function approximation. In a recent paper, (Efroni et al. 2019) studied the impact of lookahead on the convergence rate of approximate dynamic programming. In this paper, we show that these convergence results change dramatically when function approximation is used in conjunction with lookout and approximate policy evaluation using an m-step return. Specifically, we show that when linear function approximation is used to represent the value function, a certain minimum amount of lookahead and multi-step return is needed for the algorithm to even converge. And when this condition is met, we characterize the finite-time performance of policies obtained using such approximate policy iteration. Our results are presented for two different procedures to compute the function approximation: linear least-squares regression and gradient descent.
We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [2021] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called corner-clipped rectangles (CCRs), without intersecting certain special horizontal line segments called fences. In this paper, we present a $(2+\epsilon)$-approximation algorithm for MISR which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio to 6. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to $2+\epsilon$. In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to $O(1/\epsilon)$ line segments each. This partitioning routine and our other new ideas may be useful for future work towards a PTAS for MISR.
We obtain explicit $p$-Wasserstein distance error bounds between the distribution of the multi-parameter MLE and the multivariate normal distribution. Our general bounds are given for possibly high-dimensional, independent and identically distributed random vectors. Our general bounds are of the optimal $\mathcal{O}(n^{-1/2})$ order. Explicit numerical constants are given when $p\in(1,2]$, and in the case $p>2$ the bounds are explicit up to a constant factor that only depends on $p$. We apply our general bounds to derive Wasserstein distance error bounds for the multivariate normal approximation of the MLE in several settings; these being single-parameter exponential families, the normal distribution under canonical parametrisation, and the multivariate normal distribution under non-canonical parametrisation. In addition, we provide upper bounds with respect to the bounded Wasserstein distance when the MLE is implicitly defined.
Due to the outstanding capability of capturing underlying data distributions, deep learning techniques have been recently utilized for a series of traditional database problems. In this paper, we investigate the possibilities of utilizing deep learning for cardinality estimation of similarity selection. Answering this problem accurately and efficiently is essential to many data management applications, especially for query optimization. Moreover, in some applications the estimated cardinality is supposed to be consistent and interpretable. Hence a monotonic estimation w.r.t. the query threshold is preferred. We propose a novel and generic method that can be applied to any data type and distance function. Our method consists of a feature extraction model and a regression model. The feature extraction model transforms original data and threshold to a Hamming space, in which a deep learning-based regression model is utilized to exploit the incremental property of cardinality w.r.t. the threshold for both accuracy and monotonicity. We develop a training strategy tailored to our model as well as techniques for fast estimation. We also discuss how to handle updates. We demonstrate the accuracy and the efficiency of our method through experiments, and show how it improves the performance of a query optimizer.
We develop a computationally tractable method for estimating the optimal map between two distributions over $\mathbb{R}^d$ with rigorous finite-sample guarantees. Leveraging an entropic version of Brenier's theorem, we show that our estimator -- the barycentric projection of the optimal entropic plan -- is easy to compute using Sinkhorn's algorithm. As a result, unlike current approaches for map estimation, which are slow to evaluate when the dimension or number of samples is large, our approach is parallelizable and extremely efficient even for massive data sets. Under smoothness assumptions on the optimal map, we show that our estimator enjoys comparable statistical performance to other estimators in the literature, but with much lower computational cost. We showcase the efficacy of our proposed estimator through numerical examples. Our proofs are based on a modified duality principle for entropic optimal transport and on a method for approximating optimal entropic plans due to Pal (2019).
While Generative Adversarial Networks (GANs) have empirically produced impressive results on learning complex real-world distributions, recent work has shown that they suffer from lack of diversity or mode collapse. The theoretical work of Arora et al.~\cite{AroraGeLiMaZh17} suggests a dilemma about GANs' statistical properties: powerful discriminators cause overfitting, whereas weak discriminators cannot detect mode collapse. In contrast, we show in this paper that GANs can in principle learn distributions in Wasserstein distance (or KL-divergence in many cases) with polynomial sample complexity, if the discriminator class has strong distinguishing power against the particular generator class (instead of against all possible generators). For various generator classes such as mixture of Gaussians, exponential families, and invertible neural networks generators, we design corresponding discriminators (which are often neural nets of specific architectures) such that the Integral Probability Metric (IPM) induced by the discriminators can provably approximate the Wasserstein distance and/or KL-divergence. This implies that if the training is successful, then the learned distribution is close to the true distribution in Wasserstein distance or KL divergence, and thus cannot drop modes. Our preliminary experiments show that on synthetic datasets the test IPM is well correlated with KL divergence, indicating that the lack of diversity may be caused by the sub-optimality in optimization instead of statistical inefficiency.
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