arXiv subjects

Honghao Lin

Publications and source records attributed to Honghao Lin.

28 records · Page 2Linked to original sources

arXiv2024

Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that $(1 \pm ε)$-approximates cut values in graphs with $n$ vertices. For arbitrary directed graphs, such a data structure requires $Ω(n^2)$ bits even for constant $ε$. To circumvent this, recent works study $β$-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most $β$ times that in the other direction. We consider two models: the {\em for-each} model, where the goal is to approximate each cut with constant probability, and the {\em for-all} model, where all cuts must be preserved simultaneously. We improve the previous $Ω(n \sqrt{β/ε})$ lower bound to $\tildeΩ(n \sqrtβ/ε)$ in the for-each model, and we improve the previous $Ω(n β/ε)$ lower bound to $Ω(n β/ε^2)$ in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous $Ω\bigl(\frac{m}{k}\bigr)$ query complexity lower bound to $Ω\bigl(\min\{m, \frac{m}{ε^2 k}\}\bigr)$ for this problem, where $m$ is the number of edges, $k$ is the size of the minimum cut, and we seek a $(1+ε)$-approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors.

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arXiv2024

Learning the Positions in CountSketch

We consider sketching algorithms which first compress data by multiplication with a random sketch matrix, and then apply the sketch to quickly solve an optimization problem, e.g., low-rank approximation and regression. In the learning-based sketching paradigm proposed by~\cite{indyk2019learning}, the sketch matrix is found by choosing a random sparse matrix, e.g., CountSketch, and then the values of its non-zero entries are updated by running gradient descent on a training data set. Despite the growing body of work on this paradigm, a noticeable omission is that the locations of the non-zero entries of previous algorithms were fixed, and only their values were learned. In this work, we propose the first learning-based algorithms that also optimize the locations of the non-zero entries. Our first proposed algorithm is based on a greedy algorithm. However, one drawback of the greedy algorithm is its slower training time. We fix this issue and propose approaches for learning a sketching matrix for both low-rank approximation and Hessian approximation for second order optimization. The latter is helpful for a range of constrained optimization problems, such as LASSO and matrix estimation with a nuclear norm constraint. Both approaches achieve good accuracy with a fast running time. Moreover, our experiments suggest that our algorithm can still reduce the error significantly even if we only have a very limited number of training matrices.

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arXiv2024

The $\ell_p$-Subspace Sketch Problem in Small Dimensions with Applications to Support Vector Machines

In the $\ell_p$-subspace sketch problem, we are given an $n\times d$ matrix $A$ with $n>d$, and asked to build a small memory data structure $Q(A,ε)$ so that, for any query vector $x\in\mathbb{R}^d$, we can output a number in $(1\pmε)\|Ax\|_p^p$ given only $Q(A,ε)$. This problem is known to require $\tildeΩ(dε^{-2})$ bits of memory for $d=Ω(\log(1/ε))$. However, for $d=o(\log(1/ε))$, no data structure lower bounds were known. We resolve the memory required to solve the $\ell_p$-subspace sketch problem for any constant $d$ and integer $p$, showing that it is $Ω(ε^{-2(d-1)/(d+2p)})$ bits and $\tilde{O} (ε^{-2(d-1)/(d+2p)})$ words. This shows that one can beat the $Ω(ε^{-2})$ lower bound, which holds for $d = Ω(\log(1/ε))$, for any constant $d$. We also show how to implement the upper bound in a single pass stream, with an additional multiplicative $\operatorname{poly}(\log \log n)$ factor and an additive $\operatorname{poly}(\log n)$ cost in the memory. Our bounds can be applied to point queries for SVMs with additive error, yielding an optimal bound of $\tildeΘ(ε^{-2d/(d+3)})$ for every constant $d$. This is a near-quadratic improvement over the $Ω(ε^{-(d+1)/(d+3)})$ lower bound of (Andoni et al. 2020). Our techniques rely on a novel connection to low dimensional techniques from geometric functional analysis.

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arXiv2023

$\ell_p$-Regression in the Arbitrary Partition Model of Communication

We consider the randomized communication complexity of the distributed $\ell_p$-regression problem in the coordinator model, for $p\in (0,2]$. In this problem, there is a coordinator and $s$ servers. The $i$-th server receives $A^i\in\{-M, -M+1, \ldots, M\}^{n\times d}$ and $b^i\in\{-M, -M+1, \ldots, M\}^n$ and the coordinator would like to find a $(1+ε)$-approximate solution to $\min_{x\in\mathbb{R}^n} \|(\sum_i A^i)x - (\sum_i b^i)\|_p$. Here $M \leq \mathrm{poly}(nd)$ for convenience. This model, where the data is additively shared across servers, is commonly referred to as the arbitrary partition model. We obtain significantly improved bounds for this problem. For $p = 2$, i.e., least squares regression, we give the first optimal bound of $\tildeΘ(sd^2 + sd/ε)$ bits. For $p \in (1,2)$,we obtain an $\tilde{O}(sd^2/ε+ sd/\mathrm{poly}(ε))$ upper bound. Notably, for $d$ sufficiently large, our leading order term only depends linearly on $1/ε$ rather than quadratically. We also show communication lower bounds of $Ω(sd^2 + sd/ε^2)$ for $p\in (0,1]$ and $Ω(sd^2 + sd/ε)$ for $p\in (1,2]$. Our bounds considerably improve previous bounds due to (Woodruff et al. COLT, 2013) and (Vempala et al., SODA, 2020).

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arXiv2022

Streaming Algorithms with Large Approximation Factors

We initiate a broad study of classical problems in the streaming model with insertions and deletions in the setting where we allow the approximation factor $α$ to be much larger than $1$. Such algorithms can use significantly less memory than the usual setting for which $α= 1+ε$ for an $ε\in (0,1)$. We study large approximations for a number of problems in sketching and streaming and the following are some of our results. For the $\ell_p$ norm/quasinorm $\|x\|_p$ of an $n$-dimensional vector $x$, $0 < p \le 2$, we show that obtaining a $\poly(n)$-approximation requires the same amount of memory as obtaining an $O(1)$-approximation for any $M = n^{Θ(1)}$. For estimating the $\ell_p$ norm, $p > 2$, we show an upper bound of $O(n^{1-2/p} (\log n \allowbreak \log M)/α^{2})$ bits for an $α$-approximation, and give a matching lower bound, for almost the full range of $α\geq 1$ for linear sketches. For the $\ell_2$-heavy hitters problem, we show that the known lower bound of $Ω(k \log n\log M)$ bits for identifying $(1/k)$-heavy hitters holds even if we are allowed to output items that are $1/(αk)$-heavy, for almost the full range of $α$, provided the algorithm succeeds with probability $1-O(1/n)$. We also obtain a lower bound for linear sketches that is tight even for constant probability algorithms. For estimating the number $\ell_0$ of distinct elements, we give an $n^{1/t}$-approximation algorithm using $O(t\log \log M)$ bits of space, as well as a lower bound of $Ω(t)$ bits, both excluding the storage of random bits.

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arXiv2022

Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra

We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression that are comparable to their quantum analogues. De-quantizing such algorithms has received a flurry of attention in recent years; we obtain sharper bounds for these problems. More significantly, we achieve these improvements by arguing that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise; these are powerful and standard techniques in randomized numerical linear algebra. With this recognition, we are able to employ the large body of work in numerical linear algebra to obtain algorithms for these problems that are simpler or faster (or both) than existing approaches. Our experiments demonstrate that the proposed data structures also work well on real-world datasets.

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arXiv2022

Learning Augmented Binary Search Trees

A treap is a classic randomized binary search tree data structure that is easy to implement and supports O(\log n) expected time access. However, classic treaps do not take advantage of the input distribution or patterns in the input. Given recent advances in algorithms with predictions, we propose pairing treaps with machine advice to form a learning-augmented treap. We are the first to propose a learning-augmented data structure that supports binary search tree operations such as range-query and successor functionalities. With the assumption that we have access to advice from a frequency estimation oracle, we assign learned priorities to the nodes to better improve the treap's structure. We theoretically analyze the learning-augmented treap's performance under various input distributions and show that under those circumstances, our learning-augmented treap has stronger guarantees than classic treaps and other classic tree-based data structures. Further, we experimentally evaluate our learned treap on synthetic datasets and demonstrate a performance advantage over other search tree data structures. We also present experiments on real world datasets with known frequency estimation oracles and show improvements as well.

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arXiv2022

Triangle and Four Cycle Counting with Predictions in Graph Streams

We propose data-driven one-pass streaming algorithms for estimating the number of triangles and four cycles, two fundamental problems in graph analytics that are widely studied in the graph data stream literature. Recently, (Hsu 2018) and (Jiang 2020) applied machine learning techniques in other data stream problems, using a trained oracle that can predict certain properties of the stream elements to improve on prior "classical" algorithms that did not use oracles. In this paper, we explore the power of a "heavy edge" oracle in multiple graph edge streaming models. In the adjacency list model, we present a one-pass triangle counting algorithm improving upon the previous space upper bounds without such an oracle. In the arbitrary order model, we present algorithms for both triangle and four cycle estimation with fewer passes and the same space complexity as in previous algorithms, and we show several of these bounds are optimal. We analyze our algorithms under several noise models, showing that the algorithms perform well even when the oracle errs. Our methodology expands upon prior work on "classical" streaming algorithms, as previous multi-pass and random order streaming algorithms can be seen as special cases of our algorithms, where the first pass or random order was used to implement the heavy edge oracle. Lastly, our experiments demonstrate advantages of the proposed method compared to state-of-the-art streaming algorithms.

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arXiv2021

Learning-Augmented Sketches for Hessians

Sketching is a dimensionality reduction technique where one compresses a matrix by linear combinations that are chosen at random. A line of work has shown how to sketch the Hessian to speed up each iteration in a second order method, but such sketches usually depend only on the matrix at hand, and in a number of cases are even oblivious to the input matrix. One could instead hope to learn a distribution on sketching matrices that is optimized for the specific distribution of input matrices. We show how to design learned sketches for the Hessian in the context of second order methods. We prove that a smaller sketching dimension of the column space of a tall matrix is possible, given an oracle that can predict the indices of the rows of large leverage score. We design such an oracle for various datasets, and this leads to a faster convergence of the well-studied iterative Hessian sketch procedure, which applies to a wide range of problems in convex optimization. We show empirically that learned sketches, compared with their "non-learned" counterparts, do improve the approximation accuracy for important problems, including LASSO and matrix estimation with nuclear norm constraints.

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arXiv2021

Robust Learning of Fixed-Structure Bayesian Networks in Nearly-Linear Time

We study the problem of learning Bayesian networks where an $ε$-fraction of the samples are adversarially corrupted. We focus on the fully-observable case where the underlying graph structure is known. In this work, we present the first nearly-linear time algorithm for this problem with a dimension-independent error guarantee. Previous robust algorithms with comparable error guarantees are slower by at least a factor of $(d/ε)$, where $d$ is the number of variables in the Bayesian network and $ε$ is the fraction of corrupted samples. Our algorithm and analysis are considerably simpler than those in previous work. We achieve this by establishing a direct connection between robust learning of Bayesian networks and robust mean estimation. As a subroutine in our algorithm, we develop a robust mean estimation algorithm whose runtime is nearly-linear in the number of nonzeros in the input samples, which may be of independent interest.

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