Quadratic programming is a ubiquitous prototype in convex programming. Many combinatorial optimizations on graphs and machine learning problems can be formulated as quadratic programming; for example, Support Vector Machines (SVMs). Linear and kernel SVMs have been among the most popular models in machine learning over the past three decades, prior to the deep learning era. Generally, a quadratic program has an input size of $\Theta(n^2)$, where $n$ is the number of variables. Assuming the Strong Exponential Time Hypothesis ($\textsf{SETH}$), it is known that no $O(n^{2-o(1)})$ algorithm exists (Backurs, Indyk, and Schmidt, NIPS'17). However, problems such as SVMs usually feature much smaller input sizes: one is given $n$ data points, each of dimension $d$, with $d \ll n$. Furthermore, SVMs are variants with only $O(1)$ linear constraints. This suggests that faster algorithms are feasible, provided the program exhibits certain underlying structures. In this work, we design the first nearly-linear time algorithm for solving quadratic programs whenever the quadratic objective has small treewidth or admits a low-rank factorization, and the number of linear constraints is small. Consequently, we obtain a variety of results for SVMs: * For linear SVM, where the quadratic constraint matrix has treewidth $\tau$, we can solve the corresponding program in time $\widetilde O(n\tau^{(\omega+1)/2}\log(1/\epsilon))$; * For linear SVM, where the quadratic constraint matrix admits a low-rank factorization of rank-$k$, we can solve the corresponding program in time $\widetilde O(nk^{(\omega+1)/2}\log(1/\epsilon))$; * For Gaussian kernel SVM, where the data dimension $d = \Theta(\log n)$ and the squared dataset radius is small, we can solve it in time $O(n^{1+o(1)}\log(1/\epsilon))$. We also prove that when the squared dataset radius is large, then $\Omega(n^{2-o(1)})$ time is required.
相關內容
Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related linear systems to estimate their transition matrices more accurately. To address this problem, the current paper investigates methods for jointly estimating the transition matrices of multiple systems. It is assumed that the transition matrices are unknown linear functions of some unknown shared basis matrices. We establish finite-time estimation error rates that fully reflect the roles of trajectory lengths, dimension, and number of systems under consideration. The presented results are fairly general and show the significant gains that can be achieved by pooling data across systems in comparison to learning each system individually. Further, they are shown to be robust against model misspecifications. To obtain the results, we develop novel techniques that are of interest for addressing similar joint-learning problems. They include tightly bounding estimation errors in terms of the eigen-structures of transition matrices, establishing sharp high probability bounds for singular values of dependent random matrices, and capturing effects of misspecified transition matrices as the systems evolve over time.
Marker code is an effective coding scheme to protect data from insertions and deletions. It has potential applications in future storage systems, such as DNA storage and racetrack memory. When decoding marker codes, perfect channel state information (CSI), i.e., insertion and deletion probabilities, are required to detect insertion and deletion errors. Sometimes, the perfect CSI is not easy to obtain or the accurate channel model is unknown. Therefore, it is deserved to develop detecting algorithms for marker code without the knowledge of perfect CSI. In this paper, we propose two CSI-agnostic detecting algorithms for marker code based on deep learning. The first one is a model-driven deep learning method, which deep unfolds the original iterative detecting algorithm of marker code. In this method, CSI become weights in neural networks and these weights can be learned from training data. The second one is a data-driven method which is an end-to-end system based on the deep bidirectional gated recurrent unit network. Simulation results show that error performances of the proposed methods are significantly better than that of the original detection algorithm with CSI uncertainty. Furthermore, the proposed data-driven method exhibits better error performances than other methods for unknown channel models.
We present an elementary yet general proof of duality for Wasserstein distributionally robust optimization. The duality holds for any arbitrary Kantorovich transport cost, measurable loss function, and nominal probability distribution, provided that an interchangeability principle holds, which is equivalent to certain measurability conditions. To illustrate the broader applicability of our approach, we provide a rigorous treatment of duality results in distributionally robust Markov decision processes and distributionally robust multistage stochastic programming. Furthermore, we extend the result to other problems including infinity-Wasserstein distributionally robust optimization, risk-averse optimization, and globalized distributionally robust counterpart.
The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.
Object detection is a fundamental task in computer vision and image processing. Current deep learning based object detectors have been highly successful with abundant labeled data. But in real life, it is not guaranteed that each object category has enough labeled samples for training. These large object detectors are easy to overfit when the training data is limited. Therefore, it is necessary to introduce few-shot learning and zero-shot learning into object detection, which can be named low-shot object detection together. Low-Shot Object Detection (LSOD) aims to detect objects from a few or even zero labeled data, which can be categorized into few-shot object detection (FSOD) and zero-shot object detection (ZSD), respectively. This paper conducts a comprehensive survey for deep learning based FSOD and ZSD. First, this survey classifies methods for FSOD and ZSD into different categories and discusses the pros and cons of them. Second, this survey reviews dataset settings and evaluation metrics for FSOD and ZSD, then analyzes the performance of different methods on these benchmarks. Finally, this survey discusses future challenges and promising directions for FSOD and ZSD.
Traffic forecasting is an important factor for the success of intelligent transportation systems. Deep learning models including convolution neural networks and recurrent neural networks have been applied in traffic forecasting problems to model the spatial and temporal dependencies. In recent years, to model the graph structures in the transportation systems as well as the contextual information, graph neural networks (GNNs) are introduced as new tools and have achieved the state-of-the-art performance in a series of traffic forecasting problems. In this survey, we review the rapidly growing body of recent research using different GNNs, e.g., graph convolutional and graph attention networks, in various traffic forecasting problems, e.g., road traffic flow and speed forecasting, passenger flow forecasting in urban rail transit systems, demand forecasting in ride-hailing platforms, etc. We also present a collection of open data and source resources for each problem, as well as future research directions. To the best of our knowledge, this paper is the first comprehensive survey that explores the application of graph neural networks for traffic forecasting problems. We have also created a public Github repository to update the latest papers, open data and source resources.
Knowledge graph completion aims to predict missing relations between entities in a knowledge graph. While many different methods have been proposed, there is a lack of a unifying framework that would lead to state-of-the-art results. Here we develop PathCon, a knowledge graph completion method that harnesses four novel insights to outperform existing methods. PathCon predicts relations between a pair of entities by: (1) Considering the Relational Context of each entity by capturing the relation types adjacent to the entity and modeled through a novel edge-based message passing scheme; (2) Considering the Relational Paths capturing all paths between the two entities; And, (3) adaptively integrating the Relational Context and Relational Path through a learnable attention mechanism. Importantly, (4) in contrast to conventional node-based representations, PathCon represents context and path only using the relation types, which makes it applicable in an inductive setting. Experimental results on knowledge graph benchmarks as well as our newly proposed dataset show that PathCon outperforms state-of-the-art knowledge graph completion methods by a large margin. Finally, PathCon is able to provide interpretable explanations by identifying relations that provide the context and paths that are important for a given predicted relation.
It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.
We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191