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This paper analyzes a necessary and sufficient condition for the change-making problem to be solvable with a greedy algorithm. The change-making problem is to minimize the number of coins used to pay a given value in a specified currency system. This problem is NP-hard, and therefore the greedy algorithm does not always yield an optimal solution. Yet for almost all real currency systems, the greedy algorithm outputs an optimal solution. A currency system for which the greedy algorithm returns an optimal solution for any value of payment is called a canonical system. Canonical systems with at most five types of coins have been characterized in previous studies. In this paper, we give characterization of canonical systems with six types of coins, and we propose a partial generalization of characterization of canonical systems.

Sparse fusion is a compile-time loop transformation and runtime scheduling implemented as a domain-specific code generator. Sparse fusion generates efficient parallel code for the combination of two sparse matrix kernels where at least one of the kernels has loop-carried dependencies. Available implementations optimize individual sparse kernels. When optimized separately, the irregular dependence patterns of sparse kernels create synchronization overheads and load imbalance, and their irregular memory access patterns result in inefficient cache usage, which reduces parallel efficiency. Sparse fusion uses a novel inspection strategy with code transformations to generate parallel fused code for sparse kernel combinations that is optimized for data locality and load balance. Code generated by Sparse fusion outperforms the existing implementations ParSy and MKL on average 1.6X and 5.1X respectively and outperforms the LBC and DAGP coarsening strategies applied to a fused data dependence graph on average 5.1X and 7.2X respectively for various kernel combinations.

Many problems in signal processing and machine learning can be formalized as weak submodular optimization tasks. For such problems, a simple greedy algorithm (\textsc{Greedy}) is guaranteed to find a solution achieving the objective with a value no worse than $1-e^{-1/c}$ of the optimal, where $c$ is the multiplicative weak-submodularity constant. Due to the high cost of querying large-scale systems, the complexity of \textsc{Greedy} becomes prohibitive in contemporary applications. In this work, we study the tradeoff between performance and complexity when one resorts to random sampling strategies to reduce the query complexity of \textsc{Greedy}. Specifically, we quantify the effect of uniform sampling strategies on \textsc{Greedy}'s performance through two metrics: (i) probability of identifying an optimal subset, and (ii) suboptimality with respect to the optimal solution. The latter implies that uniform sampling strategies with a fixed sampling size achieve a non-trivial approximation factor; however, we show that with overwhelming probability, these methods fail to find the optimal subset. Our analysis shows that the failure of uniform sampling strategies with fixed sample size can be circumvented by successively increasing the size of the search space. Building upon this insight, we propose a simple progressive stochastic greedy algorithm and study its approximation guarantees. Moreover, we demonstrate effectiveness of the proposed method in dimensionality reduction applications and feature selection tasks for clustering and object tracking.

This paper studies a scalar Gaussian wiretap channel where instead of an average input power constraint, we consider a peak amplitude constraint on the input. The goal is to obtain insights into the secrecy-capacity and the structure of the secrecy-capacity-achieving distribution. Capitalizing on the recent theoretical progress on the structure of the secrecy-capacity-achieving distribution, this paper develops a numerical procedure, based on the gradient ascent algorithm and a version of the Blahut-Arimoto algorithm, for computing the secrecy-capacity and the secrecy-capacity-achieving input and output distributions.

Despite the recent success of graph neural networks (GNN), common architectures often exhibit significant limitations, including sensitivity to oversmoothing, long-range dependencies, and spurious edges, e.g., as can occur as a result of graph heterophily or adversarial attacks. To at least partially address these issues within a simple transparent framework, we consider a new family of GNN layers designed to mimic and integrate the update rules of two classical iterative algorithms, namely, proximal gradient descent and iterative reweighted least squares (IRLS). The former defines an extensible base GNN architecture that is immune to oversmoothing while nonetheless capturing long-range dependencies by allowing arbitrary propagation steps. In contrast, the latter produces a novel attention mechanism that is explicitly anchored to an underlying end-toend energy function, contributing stability with respect to edge uncertainty. When combined we obtain an extremely simple yet robust model that we evaluate across disparate scenarios including standardized benchmarks, adversarially-perturbated graphs, graphs with heterophily, and graphs involving long-range dependencies. In doing so, we compare against SOTA GNN approaches that have been explicitly designed for the respective task, achieving competitive or superior node classification accuracy.

Deep learning methods for graphs achieve remarkable performance on many node-level and graph-level prediction tasks. However, despite the proliferation of the methods and their success, prevailing Graph Neural Networks (GNNs) neglect subgraphs, rendering subgraph prediction tasks challenging to tackle in many impactful applications. Further, subgraph prediction tasks present several unique challenges, because subgraphs can have non-trivial internal topology, but also carry a notion of position and external connectivity information relative to the underlying graph in which they exist. Here, we introduce SUB-GNN, a subgraph neural network to learn disentangled subgraph representations. In particular, we propose a novel subgraph routing mechanism that propagates neural messages between the subgraph's components and randomly sampled anchor patches from the underlying graph, yielding highly accurate subgraph representations. SUB-GNN specifies three channels, each designed to capture a distinct aspect of subgraph structure, and we provide empirical evidence that the channels encode their intended properties. We design a series of new synthetic and real-world subgraph datasets. Empirical results for subgraph classification on eight datasets show that SUB-GNN achieves considerable performance gains, outperforming strong baseline methods, including node-level and graph-level GNNs, by 12.4% over the strongest baseline. SUB-GNN performs exceptionally well on challenging biomedical datasets when subgraphs have complex topology and even comprise multiple disconnected components.

CRF has been used as a powerful model for statistical sequence labeling. For neural sequence labeling, however, BiLSTM-CRF does not always lead to better results compared with BiLSTM-softmax local classification. This can be because the simple Markov label transition model of CRF does not give much information gain over strong neural encoding. For better representing label sequences, we investigate a hierarchically-refined label attention network, which explicitly leverages label embeddings and captures potential long-term label dependency by giving each word incrementally refined label distributions with hierarchical attention. Results on POS tagging, NER and CCG supertagging show that the proposed model not only improves the overall tagging accuracy with similar number of parameters, but also significantly speeds up the training and testing compared to BiLSTM-CRF.

Efficient exploration remains a major challenge for reinforcement learning. One reason is that the variability of the returns often depends on the current state and action, and is therefore heteroscedastic. Classical exploration strategies such as upper confidence bound algorithms and Thompson sampling fail to appropriately account for heteroscedasticity, even in the bandit setting. Motivated by recent findings that address this issue in bandits, we propose to use Information-Directed Sampling (IDS) for exploration in reinforcement learning. As our main contribution, we build on recent advances in distributional reinforcement learning and propose a novel, tractable approximation of IDS for deep Q-learning. The resulting exploration strategy explicitly accounts for both parametric uncertainty and heteroscedastic observation noise. We evaluate our method on Atari games and demonstrate a significant improvement over alternative approaches.

Traditional methods for link prediction can be categorized into three main types: graph structure feature-based, latent feature-based, and explicit feature-based. Graph structure feature methods leverage some handcrafted node proximity scores, e.g., common neighbors, to estimate the likelihood of links. Latent feature methods rely on factorizing networks' matrix representations to learn an embedding for each node. Explicit feature methods train a machine learning model on two nodes' explicit attributes. Each of the three types of methods has its unique merits. In this paper, we propose SEAL (learning from Subgraphs, Embeddings, and Attributes for Link prediction), a new framework for link prediction which combines the power of all the three types into a single graph neural network (GNN). GNN is a new type of neural network which directly accepts graphs as input and outputs their labels. In SEAL, the input to the GNN is a local subgraph around each target link. We prove theoretically that our local subgraphs also reserve a great deal of high-order graph structure features related to link existence. Another key feature is that our GNN can naturally incorporate latent features and explicit features. It is achieved by concatenating node embeddings (latent features) and node attributes (explicit features) in the node information matrix for each subgraph, thus combining the three types of features to enhance GNN learning. Through extensive experiments, SEAL shows unprecedentedly strong performance against a wide range of baseline methods, including various link prediction heuristics and network embedding methods.