We study the problem of solving Packing Integer Programs (PIPs) in the online setting, where columns in $[0,1]^d$ of the constraint matrix are revealed sequentially, and the goal is to pick a subset of the columns that sum to at most $B$ in each coordinate while maximizing the objective. Excellent results are known in the secretary setting, where the columns are adversarially chosen, but presented in a uniformly random order. However, these existing algorithms are susceptible to adversarial attacks: they try to "learn" characteristics of a good solution, but tend to over-fit to the model, and hence a small number of adversarial corruptions can cause the algorithm to fail. In this paper, we give the first robust algorithms for Packing Integer Programs, specifically in the recently proposed Byzantine Secretary framework. Our techniques are based on a two-level use of online learning, to robustly learn an approximation to the optimal value, and then to use this robust estimate to pick a good solution. These techniques are general and we use them to design robust algorithms for PIPs in the prophet model as well, specifically in the Prophet-with-Augmentations framework. We also improve known results in the Byzantine Secretary framework: we make the non-constructive results algorithmic and improve the existing bounds for single-item and matroid constraints.
相關內容
Given its status as a classic problem and its importance to both theoreticians and practitioners, edit distance provides an excellent lens through which to understand how the theoretical analysis of algorithms impacts practical implementations. From an applied perspective, the goals of theoretical analysis are to predict the empirical performance of an algorithm and to serve as a yardstick to design novel algorithms that perform well in practice. In this paper, we systematically survey the types of theoretical analysis techniques that have been applied to edit distance and evaluate the extent to which each one has achieved these two goals. These techniques include traditional worst-case analysis, worst-case analysis parametrized by edit distance or entropy or compressibility, average-case analysis, semi-random models, and advice-based models. We find that the track record is mixed. On one hand, two algorithms widely used in practice have been born out of theoretical analysis and their empirical performance is captured well by theoretical predictions. On the other hand, all the algorithms developed using theoretical analysis as a yardstick since then have not had any practical relevance. We conclude by discussing the remaining open problems and how they can be tackled.
The link prediction task on knowledge graphs without explicit negative triples in the training data motivates the usage of rank-based metrics. Here, we review existing rank-based metrics and propose desiderata for improved metrics to address lack of interpretability and comparability of existing metrics to datasets of different sizes and properties. We introduce a simple theoretical framework for rank-based metrics upon which we investigate two avenues for improvements to existing metrics via alternative aggregation functions and concepts from probability theory. We finally propose several new rank-based metrics that are more easily interpreted and compared accompanied by a demonstration of their usage in a benchmarking of knowledge graph embedding models.
We consider the question of adaptive data analysis within the framework of convex optimization. We ask how many samples are needed in order to compute $\epsilon$-accurate estimates of $O(1/\epsilon^2)$ gradients queried by gradient descent, and we provide two intermediate answers to this question. First, we show that for a general analyst (not necessarily gradient descent) $\Omega(1/\epsilon^3)$ samples are required. This rules out the possibility of a foolproof mechanism. Our construction builds upon a new lower bound (that may be of interest of its own right) for an analyst that may ask several non adaptive questions in a batch of fixed and known $T$ rounds of adaptivity and requires a fraction of true discoveries. We show that for such an analyst $\Omega (\sqrt{T}/\epsilon^2)$ samples are necessary. Second, we show that, under certain assumptions on the oracle, in an interaction with gradient descent $\tilde \Omega(1/\epsilon^{2.5})$ samples are necessary. Our assumptions are that the oracle has only \emph{first order access} and is \emph{post-hoc generalizing}. First order access means that it can only compute the gradients of the sampled function at points queried by the algorithm. Our assumption of \emph{post-hoc generalization} follows from existing lower bounds for statistical queries. More generally then, we provide a generic reduction from the standard setting of statistical queries to the problem of estimating gradients queried by gradient descent. These results are in contrast with classical bounds that show that with $O(1/\epsilon^2)$ samples one can optimize the population risk to accuracy of $O(\epsilon)$ but, as it turns out, with spurious gradients.
Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.
Multi-scale problems, where variables of interest evolve in different time-scales and live in different state-spaces. can be found in many fields of science. Here, we introduce a new recursive methodology for Bayesian inference that aims at estimating the static parameters and tracking the dynamic variables of these kind of systems. Although the proposed approach works in rather general multi-scale systems, for clarity we analyze the case of a heterogeneous multi-scale model with 3 time-scales (static parameters, slow dynamic state variables and fast dynamic state variables). The proposed scheme, based on nested filtering methodology of P\'erez-Vieites et al. (2018), combines three intertwined layers of filtering techniques that approximate recursively the joint posterior probability distribution of the parameters and both sets of dynamic state variables given a sequence of partial and noisy observations. We explore the use of sequential Monte Carlo schemes in the first and second layers while we use an unscented Kalman filter to obtain a Gaussian approximation of the posterior probability distribution of the fast variables in the third layer. Some numerical results are presented for a stochastic two-scale Lorenz 96 model with unknown parameters.
Low-rank matrix estimation under heavy-tailed noise is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a novel Riemannian sub-gradient (RsGrad) algorithm which is not only computationally efficient with linear convergence but also is statistically optimal, be the noise Gaussian or heavy-tailed. Convergence theory is established for a general framework and specific applications to absolute loss, Huber loss, and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of dual-phase convergence. In phase one, RsGrad behaves as in a typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator which is already observed in the existing literature. Interestingly, during phase two, RsGrad converges linearly as if minimizing a smooth and strongly convex objective function and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Lastly, RsGrad is applicable for low-rank tensor estimation under heavy-tailed noise where a statistically optimal rate is attainable with the same phenomenon of dual-phase convergence, and a novel shrinkage-based second-order moment method is guaranteed to deliver a warm initialization. Numerical simulations confirm our theoretical discovery and showcase the superiority of RsGrad over prior methods.
One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.
We present a novel static analysis technique to derive higher moments for program variables for a large class of probabilistic loops with potentially uncountable state spaces. Our approach is fully automatic, meaning it does not rely on externally provided invariants or templates. We employ algebraic techniques based on linear recurrences and introduce program transformations to simplify probabilistic programs while preserving their statistical properties. We develop power reduction techniques to further simplify the polynomial arithmetic of probabilistic programs and define the theory of moment-computable probabilistic loops for which higher moments can precisely be computed. Our work has applications towards recovering probability distributions of random variables and computing tail probabilities. The empirical evaluation of our results demonstrates the applicability of our work on many challenging examples.
Graph Convolutional Network (GCN) has achieved extraordinary success in learning effective task-specific representations of nodes in graphs. However, regarding Heterogeneous Information Network (HIN), existing HIN-oriented GCN methods still suffer from two deficiencies: (1) they cannot flexibly explore all possible meta-paths and extract the most useful ones for a target object, which hinders both effectiveness and interpretability; (2) they often need to generate intermediate meta-path based dense graphs, which leads to high computational complexity. To address the above issues, we propose an interpretable and efficient Heterogeneous Graph Convolutional Network (ie-HGCN) to learn the representations of objects in HINs. It is designed as a hierarchical aggregation architecture, i.e., object-level aggregation first, followed by type-level aggregation. The novel architecture can automatically extract useful meta-paths for each object from all possible meta-paths (within a length limit), which brings good model interpretability. It can also reduce the computational cost by avoiding intermediate HIN transformation and neighborhood attention. We provide theoretical analysis about the proposed ie-HGCN in terms of evaluating the usefulness of all possible meta-paths, its connection to the spectral graph convolution on HINs, and its quasi-linear time complexity. Extensive experiments on three real network datasets demonstrate the superiority of ie-HGCN over the state-of-the-art methods.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191