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Register a new userSparse Polynomial Learning and Graph Sketching
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Added by
Karthikeyan Shanmugam
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Let $f:{-1,1}^n$ be a polynomial with at most $s$ non-zero real coefficients. We give an algorithm for exactly reconstructing f given random examples from the uniform distribution on ${-1,1}^n$ that runs in time polynomial in $n$ and $2s$ and succeeds if the function satisfies the unique sign property: there is one output value which corresponds to a unique set of values of the participating parities. This sufficient condition is satisfied when every coefficient of f is perturbed by a small random noise, or satisfied with high probability when s parity functions are chosen randomly or when all the coefficients are positive. Learning sparse polynomials over the Boolean domain in time polynomial in $n$ and $2s$ is considered notoriously hard in the worst-case. Our result shows that the problem is tractable for almost all sparse polynomials. Then, we show an application of this result to hypergraph sketching which is the problem of learning a sparse (both in the number of hyperedges and the size of the hyperedges) hypergraph from uniformly drawn random cuts. We also provide experimental results on a real world dataset.
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This paper considers the problem of recovering an unknown sparse ptimes p matrix X from an mtimes m matrix Y=AXB^T, where A and B are known m times p matrices with m << p. The main result shows that there exist constructions of the sketching matrices A and B so that even if X has O(p) non-zeros, it can be recovered exactly and efficiently using a convex program as long as these non-zeros are not concentrated in any single row/column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(sqrt{# nonzeros in X} times log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require non-standard techniques. Indeed our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a ptimes n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DD^{T} is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A in R^{mtimes p}.
We present a novel method for graph partitioning, based on reinforcement learning and graph convolutional neural networks. Our approach is to recursively partition coarser representations of a given graph. The neural network is implemented using SAGE graph convolution layers, and trained using an advantage actor critic (A2C) agent. We present two variants, one for finding an edge separator that minimizes the normalized cut or quotient cut, and one that finds a small vertex separator. The vertex separators are then used to construct a nested dissection ordering to permute a sparse matrix so that its triangular factorization will incur less fill-in. The partitioning quality is compared with partitions obtained using METIS and SCOTCH, and the nested dissection ordering is evaluated in the sparse solver SuperLU. Our results show that the proposed method achieves similar partitioning quality as METIS and SCOTCH. Furthermore, the method generalizes across different classes of graphs, and works well on a variety of graphs from the SuiteSparse sparse matrix collection.
Kernel methods are fundamental in machine learning, and faster algorithms for kernel approximation provide direct speedups for many core tasks in machine learning. The polynomial kernel is especially important as other kernels can often be approximated by the polynomial kernel via a Taylor series expansion. Recent techniques in oblivious sketching reduce the dependence in the running time on the degree $q$ of the polynomial kernel from exponential to polynomial, which is useful for the Gaussian kernel, for which $q$ can be chosen to be polylogarithmic. However, for more slowly growing kernels, such as the neural tangent and arc-cosine kernels, $q$ needs to be polynomial, and previous work incurs a polynomial factor slowdown in the running time. We give a new oblivious sketch which greatly improves upon this running time, by removing the dependence on $q$ in the leading order term. Combined with a novel sampling scheme, we give the fastest algorithms for approximating a large family of slow-growing kernels.
Existing approaches to federated learning suffer from a communication bottleneck as well as convergence issues due to sparse client participation. In this paper we introduce a novel algorithm, called FetchSGD, to overcome these challenges. FetchSGD compresses model updates using a Count Sketch, and then takes advantage of the mergeability of sketches to combine model updates from many workers. A key insight in the design of FetchSGD is that, because the Count Sketch is linear, momentum and error accumulation can both be carried out within the sketch. This allows the algorithm to move momentum and error accumulation from clients to the central aggregator, overcoming the challenges of sparse client participation while still achieving high compression rates and good convergence. We prove that FetchSGD has favorable convergence guarantees, and we demonstrate its empirical effectiveness by training two residual networks and a transformer model.
Graph Neural Networks (GNNs) have proved to be an effective representation learning framework for graph-structured data, and have achieved state-of-the-art performance on many practical predictive tasks, such as node classification, link prediction and graph classification. Among the variants of GNNs, Graph Attention Networks (GATs) learn to assign dense attention coefficients over all neighbors of a node for feature aggregation, and improve the performance of many graph learning tasks. However, real-world graphs are often very large and noisy, and GATs are prone to overfitting if not regularized properly. Even worse, the local aggregation mechanism of GATs may fail on disassortative graphs, where nodes within local neighborhood provide more noise than useful information for feature aggregation. In this paper, we propose Sparse Graph Attention Networks (SGATs) that learn sparse attention coefficients under an $L_0$-norm regularization, and the learned sparse attentions are then used for all GNN layers, resulting in an edge-sparsified graph. By doing so, we can identify noisy/task-irrelevant edges, and thus perform feature aggregation on most informative neighbors. Extensive experiments on synthetic and real-world graph learning benchmarks demonstrate the superior performance of SGATs. In particular, SGATs can remove about 50%-80% edges from large assortative graphs, while retaining similar classification accuracies. On disassortative graphs, SGATs prune majority of noisy edges and outperform GATs in classification accuracies by significant margins. Furthermore, the removed edges can be interpreted intuitively and quantitatively. To the best of our knowledge, this is the first graph learning algorithm that shows significant redundancies in graphs and edge-sparsified graphs can achieve similar or sometimes higher predictive performances than original graphs.
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