Honghao Lin

Honghao Lin, Hoai-An Nguyen, David P. Woodruff

We study sketching trimmed statistics of a frequency vector, including the $F_p$ moment of the top-$k$ coordinates and of the trimmed-$k$ vector. Despite their natural role in robust analytics, this is the first time these problems have been studied in any sublinear space setting. For $p \in [0,2]$, we obtain $poly(\log n/\varepsilon)$-space algorithms for both tasks when $k$ is moderately large, and for general $k$ we identify a sharp structural threshold that characterizes exactly when sublinear space is possible: in particular, it is actually determined by the ratio between $a_k^2$ and $\|x_{-k}\|_2^2/k$. We extend these results to $p > 2$ and present several applications including algorithms for thresholded $F_p$ estimation and generalized impact indices. Notably, we improve the space bounds of Govindan, Monemizadeh, and Muthukrishnan (PODS 2017) for computing the $h$-index.

Justin Y. Chen, Talya Eden, Piotr Indyk et al.

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.

Yi Li, Honghao Lin, Simin Liu et al.

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.

Yi Li, Honghao Lin, David P. Woodruff

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).

Elena Gribelyuk, Honghao Lin, David P. Woodruff et al.

We study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While robust algorithms exist for insertion-only streams with only a polylogarithmic overhead in memory over non-robust algorithms, it was widely conjectured that turnstile streams of length polynomial in the universe size $n$ require space linear in $n$. We refute this conjecture, showing that robustness can be achieved using space which is significantly sublinear in $n$. Our framework combines multiple linear sketches in a novel estimator-corrector-learner framework, yielding the first insertion-deletion algorithms that approximate: (1) the second moment $F_2$ up to a $(1+\varepsilon)$-factor in polylogarithmic space, (2) any symmetric function $\cal{F}$ with an $\mathcal{O}(1)$-approximate triangle inequality up to a $2^{\mathcal{O}(C)}$ factor in $\tilde{\mathcal{O}}(n^{1/C}) \cdot S(n)$ bits of space, where $S$ is the space required to approximate $\cal{F}$ non-robustly; this includes a broad class of functions such as the $L_1$-norm, the support size $F_0$, and non-normed losses such as the $M$-estimators, and (3) $L_2$ heavy hitters. For the $F_2$ moment, our algorithm is optimal up to $\textrm{poly}((\log n)/\varepsilon)$ factors. Given the recent results of Gribelyuk et al. (STOC, 2025), this shows an exponential separation between linear sketches and non-linear sketches for achieving adversarial robustness in turnstile streams.

Honghao Lin, Hoai-An Nguyen, William Swartworth et al.

We study $\ell_p$ sampling and frequency moment estimation in a single-pass insertion-only data stream. For $p \in (0,2)$, we present a nearly space-optimal approximate $\ell_p$ sampler that uses $\widetilde{O}(\log n \log(1/δ))$ bits of space and for $p = 2$, we present a sampler with space complexity $\widetilde{O}(\log^2 n \log(1/δ))$. This space complexity is optimal for $p \in (0, 2)$ and improves upon prior work by a $\log n$ factor. We further extend our construction to a continuous $\ell_p$ sampler, which outputs a valid sample index at every point during the stream. Leveraging these samplers, we design nearly unbiased estimators for $F_p$ in data streams that include forget operations, which reset individual element frequencies and introduce significant non-linear challenges. As a result, we obtain near-optimal algorithms for estimating $F_p$ for all $p$ in this model, originally proposed by Pavan, Chakraborty, Vinodchandran, and Meel [PODS'24], resolving all three open problems they posed. Furthermore, we generalize this model to what we call the suffix-prefix deletion model, and extend our techniques to estimate entropy as a corollary of our moment estimation algorithms. Finally, we show how to handle arbitrary coordinate-wise functions during the stream, for any $g \in \mathbb{G}$, where $\mathbb{G}$ includes all (linear or non-linear) contraction functions.

Yi Li, Honghao Lin, David P. Woodruff

We study the problem of residual error estimation for matrix and vector norms using a linear sketch. Such estimates can be used, for example, to quickly assess how useful a more expensive low-rank approximation computation will be. The matrix case concerns the Frobenius norm and the task is to approximate the $k$-residual $\|A - A_k\|_F$ of the input matrix $A$ within a $(1+ε)$-factor, where $A_k$ is the optimal rank-$k$ approximation. We provide a tight bound of $Θ(k^2/ε^4)$ on the size of bilinear sketches, which have the form of a matrix product $SAT$. This improves the previous $O(k^2/ε^6)$ upper bound in (Andoni et al. SODA 2013) and gives the first non-trivial lower bound, to the best of our knowledge. In our algorithm, our sketching matrices $S$ and $T$ can both be sparse matrices, allowing for a very fast update time. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. For the vector case, we consider the $\ell_p$-norm for $p>2$, where the task is to approximate the $k$-residual $\|x - x_k\|_p$ up to a constant factor, where $x_k$ is the optimal $k$-sparse approximation to $x$. Such vector norms are frequently studied in the data stream literature and are useful for finding frequent items or so-called heavy hitters. We establish an upper bound of $O(k^{2/p}n^{1-2/p}\operatorname{poly}(\log n))$ for constant $ε$ on the dimension of a linear sketch for this problem. Our algorithm can be extended to the $\ell_p$ sparse recovery problem with the same sketching dimension, which seems to be the first such bound for $p > 2$. We also show an $Ω(k^{2/p}n^{1-2/p})$ lower bound for the sparse recovery problem, which is tight up to a $\mathrm{poly}(\log n)$ factor.

Hamed Shirzad, Honghao Lin, Balaji Venkatachalam et al.

Graph Transformers excel in long-range dependency modeling, but generally require quadratic memory complexity in the number of nodes in an input graph, and hence have trouble scaling to large graphs. Sparse attention variants such as Exphormer can help, but may require high-degree augmentations to the input graph for good performance, and do not attempt to sparsify an already-dense input graph. As the learned attention mechanisms tend to use few of these edges, such high-degree connections may be unnecessary. We show (empirically and with theoretical backing) that attention scores on graphs are usually quite consistent across network widths, and use this observation to propose a two-stage procedure, which we call Spexphormer: first, train a narrow network on the full augmented graph. Next, use only the active connections to train a wider network on a much sparser graph. We establish theoretical conditions when a narrow network's attention scores can match those of a wide network, and show that Spexphormer achieves good performance with drastically reduced memory requirements on various graph datasets.

Hamed Shirzad, Honghao Lin, Ameya Velingker et al.

Transductive tasks on graphs differ fundamentally from typical supervised machine learning tasks, as the independent and identically distributed (i.i.d.) assumption does not hold among samples. Instead, all train/test/validation samples are present during training, making them more akin to a semi-supervised task. These differences make the analysis of the models substantially different from other models. Recently, Graph Transformers have significantly improved results on these datasets by overcoming long-range dependency problems. However, the quadratic complexity of full Transformers has driven the community to explore more efficient variants, such as those with sparser attention patterns. While the attention matrix has been extensively discussed, the hidden dimension or width of the network has received less attention. In this work, we establish some theoretical bounds on how and under what conditions the hidden dimension of these networks can be compressed. Our results apply to both sparse and dense variants of Graph Transformers.

Yu Cheng, Honghao Lin

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.

Yi Li, Honghao Lin, David P. Woodruff

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.

Nadiia Chepurko, Kenneth L. Clarkson, Lior Horesh et al.

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.