Valued constraint satisfaction problems (VCSPs) are a large class of computational optimisation problems. If the variables of a VCSP take values from a finite domain, then recent results in constraint satisfaction imply that the problem is in P or NP-complete, depending on the set of admitted cost functions. Here we study the larger class of cost functions over countably infinite domains that have an oligomorphic automorphism group. We present a hardness condition based on a generalisation of pp-constructability as known for (classical) CSPs. We also provide a universal-algebraic polynomial-time tractability condition, based on the concept of fractional polymorphisms. We apply our general theory to study the computational complexity of resilience problems in database theory (under bag semantics). We show how to construct, for every fixed conjunctive query (and more generally for every union of conjunctive queries), a set of cost functions with an oligomorphic automorphism group such that the resulting VCSP is polynomial-time equivalent to the resilience problem; we only require that the query is connected and show that this assumption can be made without loss of generality. For the case where the query is acylic, we obtain a complexity dichotomy of the resilience problem, based on the dichotomy for finite-domain VCSPs. To illustrate the utility of our methods, we exemplarily settle the complexity of a (non-acyclic) conjunctive query whose computational complexity remained open in the literature by verifying that it satisfies our tractability condition. We conjecture that for resilience problems, our hardness and tractability conditions match, which would establish a complexity dichotomy for resilience problems for (unions of) conjunctive queries.
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This paper studies optimal estimation of large-dimensional nonlinear factor models. The key challenge is that the observed variables are possibly nonlinear functions of some latent variables where the functional forms are left unspecified. A local principal component analysis method is proposed to estimate the factor structure and recover information on latent variables and latent functions, which combines $K$-nearest neighbors matching and principal component analysis. Large-sample properties are established, including a sharp bound on the matching discrepancy of nearest neighbors, sup-norm error bounds for estimated local factors and factor loadings, and the uniform convergence rate of the factor structure estimator. Under mild conditions our estimator of the latent factor structure can achieve the optimal rate of uniform convergence for nonparametric regression. The method is illustrated with a Monte Carlo experiment and an empirical application studying the effect of tax cuts on economic growth.
We address the reflection optimization problem for a reconfigurable intelligent surface (RIS), where the RIS elements feature a set of non-uniformly spaced discrete phase shifts. This is motivated by the actual behavior of practical RIS elements, where it is shown that a uniform phase shift assumption is not realistic. A problem is formulated to find the optimal refection amplitudes and reflection phase shifts of the RIS elements such that the channel capacity of the target user is maximized. We first prove that in the optimal configuration, each RIS element is either turned off or operates at maximum amplitude. We then develop a method that finds the optimal reflection amplitudes and phases with complexity linear in the number of RIS elements. Some new and interesting insight into the reflection optimization problem is also provided.
Fast inverse kinematics (IK) is a central component in robotic motion planning. For complex robots, IK methods are often based on root search and non-linear optimization algorithms. These algorithms can be massively sped up using a neural network to predict a good initial guess, which can then be refined in a few numerical iterations. Besides previous work on learning-based IK, we present a learning approach for the fundamentally more complex problem of IK with collision avoidance. We do this in diverse and previously unseen environments. From a detailed analysis of the IK learning problem, we derive a network and unsupervised learning architecture that removes the need for a sample data generation step. Using the trained network's prediction as an initial guess for a two-stage Jacobian-based solver allows for fast and accurate computation of the collision-free IK. For the humanoid robot, Agile Justin (19 DoF), the collision-free IK is solved in less than 10 milliseconds (on a single CPU core) and with an accuracy of 10^-4 m and 10^-3 rad based on a high-resolution world model generated from the robot's integrated 3D sensor. Our method massively outperforms a random multi-start baseline in a benchmark with the 19 DoF humanoid and challenging 3D environments. It requires ten times less training time than a supervised training method while achieving comparable results.
Federated Learning (FL) typically aggregates client model parameters using a weighting approach determined by sample proportions. However, this naive weighting method may lead to unfairness and degradation in model performance due to statistical heterogeneity and the inclusion of noisy data among clients. Theoretically, distributional robustness analysis has shown that the generalization performance of a learning model with respect to any shifted distribution is bounded. This motivates us to reconsider the weighting approach in federated learning. In this paper, we replace the aforementioned weighting method with a new strategy that considers the generalization bounds of each local model. Specifically, we estimate the upper and lower bounds of the second-order origin moment of the shifted distribution for the current local model, and then use these bounds disagreements as the aggregation proportions for weightings in each communication round. Experiments demonstrate that the proposed weighting strategy significantly improves the performance of several representative FL algorithms on benchmark datasets.
Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points. When these measures live in the same metric space, the ground cost often defaults to its distance. When instantiated across two different spaces, however, choosing that cost in the absence of aligned data is a conundrum. As a result, practitioners often resort to solving instead a quadratic Gromow-Wasserstein (GW) problem. We exploit in this work a parallel between GW and cost-regularized OT, the regularized minimization of a linear OT objective parameterized by a ground cost. We use this cost-regularized formulation to match measures across two different Euclidean spaces, where the cost is evaluated between transformed source points and target points. We show that several quadratic OT problems fall in this category, and consider enforcing structure in linear transform (e.g. sparsity), by introducing structure-inducing regularizers. We provide a proximal algorithm to extract such transforms from unaligned data, and demonstrate its applicability to single-cell spatial transcriptomics/multiomics matching tasks.
We propose a general method to break down a main complex task into a set of intermediary easier sub-tasks, which are formulated in natural language as binary questions related to the final target task. Our method allows for representing each example by a vector consisting of the answers to these questions. We call this representation Natural Language Learned Features (NLLF). NLLF is generated by a small transformer language model (e.g., BERT) that has been trained in a Natural Language Inference (NLI) fashion, using weak labels automatically obtained from a Large Language Model (LLM). We show that the LLM normally struggles for the main task using in-context learning, but can handle these easiest subtasks and produce useful weak labels to train a BERT. The NLI-like training of the BERT allows for tackling zero-shot inference with any binary question, and not necessarily the ones seen during the training. We show that this NLLF vector not only helps to reach better performances by enhancing any classifier, but that it can be used as input of an easy-to-interpret machine learning model like a decision tree. This decision tree is interpretable but also reaches high performances, surpassing those of a pre-trained transformer in some cases.We have successfully applied this method to two completely different tasks: detecting incoherence in students' answers to open-ended mathematics exam questions, and screening abstracts for a systematic literature review of scientific papers on climate change and agroecology.
Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
We consider the problem of explaining the predictions of graph neural networks (GNNs), which otherwise are considered as black boxes. Existing methods invariably focus on explaining the importance of graph nodes or edges but ignore the substructures of graphs, which are more intuitive and human-intelligible. In this work, we propose a novel method, known as SubgraphX, to explain GNNs by identifying important subgraphs. Given a trained GNN model and an input graph, our SubgraphX explains its predictions by efficiently exploring different subgraphs with Monte Carlo tree search. To make the tree search more effective, we propose to use Shapley values as a measure of subgraph importance, which can also capture the interactions among different subgraphs. To expedite computations, we propose efficient approximation schemes to compute Shapley values for graph data. Our work represents the first attempt to explain GNNs via identifying subgraphs explicitly and directly. Experimental results show that our SubgraphX achieves significantly improved explanations, while keeping computations at a reasonable level.
Image segmentation is an important component of many image understanding systems. It aims to group pixels in a spatially and perceptually coherent manner. Typically, these algorithms have a collection of parameters that control the degree of over-segmentation produced. It still remains a challenge to properly select such parameters for human-like perceptual grouping. In this work, we exploit the diversity of segments produced by different choices of parameters. We scan the segmentation parameter space and generate a collection of image segmentation hypotheses (from highly over-segmented to under-segmented). These are fed into a cost minimization framework that produces the final segmentation by selecting segments that: (1) better describe the natural contours of the image, and (2) are more stable and persistent among all the segmentation hypotheses. We compare our algorithm's performance with state-of-the-art algorithms, showing that we can achieve improved results. We also show that our framework is robust to the choice of segmentation kernel that produces the initial set of hypotheses.
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