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.
The Hong Kong University of Science and Technology（香港科技大学）；Shanghai Jiao Tong University（上海交通大学）；Peking University（北京大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
We study the streaming model for approximate matrix multiplication (AMM). We are interested in the scenario that the algorithm can only take one pass over the data with limited memory. The state-of-the-art deterministic sketching algorithm for streaming AMM is the co-occurring directions (COD), which has much smaller approximation errors than randomized algorithms and outperforms other deterministic sketching methods empirically. In this paper, we provide a tighter error bound for COD whose leading term considers the potential approximate low-rank structure and the correlation of input matrice…
Count-sketch is a popular matrix sketching algorithm that can produce a sketch of an input data matrix X in O(nnz(X))time where nnz(X) denotes the number of non-zero entries in X. The sketched matrix will be much smaller than X while preserving most of its properties. Therefore, count-sketch is widely used for addressing high-dimensionality challenge in machine learning. However, there are two main limitations of count-sketch: (1) The sketching matrix used count-sketch is generated randomly which does not consider any intrinsic data properties of X. This data-oblivious matrix sketching method …
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Carnegie Mellon University（卡内基梅隆大学）
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,…
University of California, Berkeley（加利福尼亚大学伯克利分校）；Johns Hopkins University（约翰斯·霍普金斯大学）；Amazon（亚马逊）
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 …
Max-Planck-Institut für Informatik（马克斯·普朗克信息学研究所）；Yale University（耶鲁大学）；Tel Aviv University（特拉维夫大学）；KAIST（韩国科学技术院）
The Neural Tangent Kernel (NTK) characterizes the behavior of infinitely-wide neural networks trained under least squares loss by gradient descent. Recent works also report that NTK regression can outperform finitely-wide neural networks trained on small-scale datasets. However, the computational complexity of kernel methods has limited its use in large-scale learning tasks. To accelerate learning with NTK, we design a near input-sparsity time approximation algorithm for NTK, by sketching the polynomial expansions of arc-cosine kernels: our sketch for the convolutional counterpart of NTK (CNTK…
Norwegian University of Science and Technology（挪威科技大学）
加权最小哈希是一种标准的降维技术，可应用于相似性搜索和大规模核机器。我们介绍了一个简单的算法，采用加权集$x \in \mathbb{R}_{\geq 0}^{d}$并在预期时间$O(k \log k + \Vert x \Vert_{0}\log( \Vert x \Vert_1 + 1/\Vert x \Vert_1))$内计算$k$独立minhash，改进了最先进的BagMinHash算法（KDD '18），并代表了稀疏数据的最快加权minhash算法。我们的实验表明，与ICWS（ICDM '10）和BagMinhash相比，$k$和$\Vert x \Vert_0$的运行时间扩展更好，在常见用例中获得$10$ x加速。我们的方法还产生了一种技术，用于计算完全独立的局部敏感的哈希值$(L, K)$ -参数化的近似近邻搜索加权Jaccard相似性下的最佳预期时间$O(LK + \Vert x \Vert_0)$，改善以前的工作，即使在未加权集的情况下。
University of California, Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）；University of Pennsylvania（宾夕法尼亚大学）
For a tall $n\times d$ matrix $A$ and a random $m\times n$ sketching matrix $S$, the sketched estimate of the inverse covariance matrix $(A^\top A)^{-1}$ is typically biased: $E[(\tilde A^\top\tilde A)^{-1}]\ne(A^\top A)^{-1}$, where $\tilde A=SA$. This phenomenon, which we call inversion bias, arises, e.g., in statistics and distributed optimization, when averaging multiple independently constructed estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(ε,δ)$-unbiased estimator for random matric…
This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm whose complexity is significantly lower than the number of entries of A, i.e., it runs without accessing all entries of [f(Aij)] explicitly. The main idea underlying our method is to combine a polynomial approximation of f with the existing tensor sketch scheme for approximating…
Mean field theory has provided theoretical insights into various algorithms by letting the problem size tend to infinity. We argue that the applications of mean-field theory go beyond theoretical insights as it can inspire the design of practical algorithms. Leveraging mean-field analyses in physics, we propose a novel algorithm for sparse measure recovery. For sparse measures over $\mathbb{R}$, we propose a polynomial-time recovery method from Fourier moments that improves upon convex relaxation methods in a specific parameter regime; then, we demonstrate the application of our results for th…
Low-rank tensors are an established framework for high-dimensional least-squares problems. We propose to extend this framework by including the concept of block-sparsity. In the context of polynomial regression each sparsity pattern corresponds to some subspace of homogeneous multivariate polynomials. This allows us to adapt the ansatz space to align better with known sample complexity results. The resulting method is tested in numerical experiments and demonstrates improved computational resource utilization and sample efficiency.
