We study the design of energy-efficient algorithms for the LOCAL and CONGEST models. Specifically, as a measure of complexity, we consider the maximum, taken over all the edges, or over all the nodes, of the number of rounds at which an edge, or a node, is active in the algorithm. We first show that every Turing-computable problem has a CONGEST algorithm with constant node-activation complexity, and therefore constant edge-activation complexity as well. That is, every node (resp., edge) is active in sending (resp., transmitting) messages for only $O(1)$ rounds during the whole execution of the algorithm. In other words, every Turing-computable problem can be solved by an algorithm consuming the least possible energy. In the LOCAL model, the same holds obviously, but with the additional feature that the algorithm runs in $O(\mbox{poly}(n))$ rounds in $n$-node networks. However, we show that insisting on algorithms running in $O(\mbox{poly}(n))$ rounds in the CONGEST model comes with a severe cost in terms of energy. Namely, there are problems requiring $\Omega(\mbox{poly}(n))$ edge-activations (and thus $\Omega(\mbox{poly}(n))$ node-activations as well) in the CONGEST model whenever solved by algorithms bounded to run in $O(\mbox{poly}(n))$ rounds. Finally, we demonstrate the existence of a sharp separation between the edge-activation complexity and the node-activation complexity in the CONGEST model, for algorithms bounded to run in $O(\mbox{poly}(n))$ rounds. Specifically, under this constraint, there is a problem with $O(1)$ edge-activation complexity but $\tilde{\Omega}(n^{1/4})$ node-activation complexity.
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Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.
The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is the most prominent multi-objective evolutionary algorithm for real-world applications. While it performs evidently well on bi-objective optimization problems, empirical studies suggest that it is less effective when applied to problems with more than two objectives. A recent mathematical runtime analysis confirmed this observation by proving the NGSA-II for an exponential number of iterations misses a constant factor of the Pareto front of the simple 3-objective OneMinMax problem. In this work, we provide the first mathematical runtime analysis of the NSGA-III, a refinement of the NSGA-II aimed at better handling more than two objectives. We prove that the NSGA-III with sufficiently many reference points -- a small constant factor more than the size of the Pareto front, as suggested for this algorithm -- computes the complete Pareto front of the 3-objective OneMinMax benchmark in an expected number of O(n log n) iterations. This result holds for all population sizes (that are at least the size of the Pareto front). It shows a drastic advantage of the NSGA-III over the NSGA-II on this benchmark. The mathematical arguments used here and in previous work on the NSGA-II suggest that similar findings are likely for other benchmarks with three or more objectives.
We present Self-Driven Strategy Learning ($\textit{sdsl}$), a lightweight online learning methodology for automated reasoning tasks that involve solving a set of related problems. $\textit{sdsl}$ does not require offline training, but instead automatically constructs a dataset while solving earlier problems. It fits a machine learning model to this data which is then used to adjust the solving strategy for later problems. We formally define the approach as a set of abstract transition rules. We describe a concrete instance of the sdsl calculus which uses conditional sampling for generating data and random forests as the underlying machine learning model. We implement the approach on top of the Kissat solver and show that the combination of Kissat+$\textit{sdsl}$ certifies larger bounds and finds more counter-examples than other state-of-the-art bounded model checking approaches on benchmarks obtained from the latest Hardware Model Checking Competition.
The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most prominent algorithms to solve multi-objective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. In this work, we give the first proven performance guarantees for a classic optimization problem, the NP-complete bi-objective minimum spanning tree problem. More specifically, we show that the NSGA-II with population size $N \ge 4((n-1) w_{\max} + 1)$ computes all extremal points of the Pareto front in an expected number of $O(m^2 n w_{\max} \log(n w_{\max}))$ iterations, where $n$ is the number of vertices, $m$ the number of edges, and $w_{\max}$ is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGA-II observed empirically. It also shows that mathematical analyses of this algorithm are not only possible for synthetic benchmark problems, but also for more complex combinatorial optimization problems. As a side result, we also obtain a new analysis of the performance of the global SEMO algorithm on the bi-objective minimum spanning tree problem, which improves the previous best result by a factor of $|F|$, the number of extremal points of the Pareto front, a set that can be as large as $n w_{\max}$. The main reason for this improvement is our observation that both multi-objective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs.
Federated learning (FL) allows a large number of clients to collaboratively train machine learning (ML) models by sending only their local gradients to a central server for aggregation in each training iteration, without sending their raw training data. Unfortunately, recent attacks on FL demonstrate that local gradients may leak information about local training data. In response to such attacks, Bonawitz \textit{et al.} (CCS 2017) proposed a secure aggregation protocol that allows a server to compute the sum of clients' local gradients in a secure manner. However, their secure aggregation protocol requires at least 4 rounds of communication between each client and the server in each training iteration. The number of communication rounds is closely related not only to the total communication cost but also the ML model accuracy, as the number of communication rounds affects client dropouts. In this paper, we propose FSSA, a 3-round secure aggregation protocol, that is efficient in terms of computation and communication, and resilient to client dropouts. We prove the security of FSSA in honest-but-curious setting and show that the security can be maintained even if an arbitrarily chosen subset of clients drop out at any time. We evaluate the performance of FSSA and show that its computation and communication overhead remains low even on large datasets. Furthermore, we conduct an experimental comparison between FSSA and Bonawitz \textit{et al.}'s protocol. The comparison results show that, in addition to reducing the number of communication rounds, FSSA achieves a significant improvement in computational efficiency.
Non-stationary multi-armed bandit (NS-MAB) problems have recently received significant attention. NS-MAB are typically modelled in two scenarios: abruptly changing, where reward distributions remain constant for a certain period and change at unknown time steps, and smoothly changing, where reward distributions evolve smoothly based on unknown dynamics. In this paper, we propose Discounted Thompson Sampling (DS-TS) with Gaussian priors to address both non-stationary settings. Our algorithm passively adapts to changes by incorporating a discounted factor into Thompson Sampling. DS-TS method has been experimentally validated, but analysis of the regret upper bound is currently lacking. Under mild assumptions, we show that DS-TS with Gaussian priors can achieve nearly optimal regret bound on the order of $\tilde{O}(\sqrt{TB_T})$ for abruptly changing and $\tilde{O}(T^{\beta})$ for smoothly changing, where $T$ is the number of time steps, $B_T$ is the number of breakpoints, $\beta$ is associated with the smoothly changing environment and $\tilde{O}$ hides the parameters independent of $T$ as well as logarithmic terms. Furthermore, empirical comparisons between DS-TS and other non-stationary bandit algorithms demonstrate its competitive performance. Specifically, when prior knowledge of the maximum expected reward is available, DS-TS has the potential to outperform state-of-the-art algorithms.
We present an effective framework for improving the breakdown point of robust regression algorithms. Robust regression has attracted widespread attention due to the ubiquity of outliers, which significantly affect the estimation results. However, many existing robust least-squares regression algorithms suffer from a low breakdown point, as they become stuck around local optima when facing severe attacks. By expanding on the previous work, we propose a novel framework that enhances the breakdown point of these algorithms by inserting a prior distribution in each iteration step, and adjusting the prior distribution according to historical information. We apply this framework to a specific algorithm and derive the consistent robust regression algorithm with iterative local search (CORALS). The relationship between CORALS and momentum gradient descent is described, and a detailed proof of the theoretical convergence of CORALS is presented. Finally, we demonstrate that the breakdown point of CORALS is indeed higher than that of the algorithm from which it is derived. We apply the proposed framework to other robust algorithms, and show that the improved algorithms achieve better results than the original algorithms, indicating the effectiveness of the proposed framework.
Recent work on deep clustering has found new promising methods also for constrained clustering problems. Their typically pairwise constraints often can be used to guide the partitioning of the data. Many problems however, feature cluster-level constraints, e.g. the Capacitated Clustering Problem (CCP), where each point has a weight and the total weight sum of all points in each cluster is bounded by a prescribed capacity. In this paper we propose a new method for the CCP, Neural Capacited Clustering, that learns a neural network to predict the assignment probabilities of points to cluster centers from a data set of optimal or near optimal past solutions of other problem instances. During inference, the resulting scores are then used in an iterative k-means like procedure to refine the assignment under capacity constraints. In our experiments on artificial data and two real world datasets our approach outperforms several state-of-the-art mathematical and heuristic solvers from the literature. Moreover, we apply our method in the context of a cluster-first-route-second approach to the Capacitated Vehicle Routing Problem (CVRP) and show competitive results on the well-known Uchoa benchmark.
Benefiting from powerful convolutional neural networks (CNNs), learning-based image inpainting methods have made significant breakthroughs over the years. However, some nature of CNNs (e.g. local prior, spatially shared parameters) limit the performance in the face of broken images with diverse and complex forms. Recently, a class of attention-based network architectures, called transformer, has shown significant performance on natural language processing fields and high-level vision tasks. Compared with CNNs, attention operators are better at long-range modeling and have dynamic weights, but their computational complexity is quadratic in spatial resolution, and thus less suitable for applications involving higher resolution images, such as image inpainting. In this paper, we design a novel attention linearly related to the resolution according to Taylor expansion. And based on this attention, a network called $T$-former is designed for image inpainting. Experiments on several benchmark datasets demonstrate that our proposed method achieves state-of-the-art accuracy while maintaining a relatively low number of parameters and computational complexity. The code can be found at \href{//github.com/dengyecode/T-former_image_inpainting}{github.com/dengyecode/T-former\_image\_inpainting}
Graph convolutional network (GCN) has been successfully applied to many graph-based applications; however, training a large-scale GCN remains challenging. Current SGD-based algorithms suffer from either a high computational cost that exponentially grows with number of GCN layers, or a large space requirement for keeping the entire graph and the embedding of each node in memory. In this paper, we propose Cluster-GCN, a novel GCN algorithm that is suitable for SGD-based training by exploiting the graph clustering structure. Cluster-GCN works as the following: at each step, it samples a block of nodes that associate with a dense subgraph identified by a graph clustering algorithm, and restricts the neighborhood search within this subgraph. This simple but effective strategy leads to significantly improved memory and computational efficiency while being able to achieve comparable test accuracy with previous algorithms. To test the scalability of our algorithm, we create a new Amazon2M data with 2 million nodes and 61 million edges which is more than 5 times larger than the previous largest publicly available dataset (Reddit). For training a 3-layer GCN on this data, Cluster-GCN is faster than the previous state-of-the-art VR-GCN (1523 seconds vs 1961 seconds) and using much less memory (2.2GB vs 11.2GB). Furthermore, for training 4 layer GCN on this data, our algorithm can finish in around 36 minutes while all the existing GCN training algorithms fail to train due to the out-of-memory issue. Furthermore, Cluster-GCN allows us to train much deeper GCN without much time and memory overhead, which leads to improved prediction accuracy---using a 5-layer Cluster-GCN, we achieve state-of-the-art test F1 score 99.36 on the PPI dataset, while the previous best result was 98.71 by [16]. Our codes are publicly available at //github.com/google-research/google-research/tree/master/cluster_gcn.
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