arXiv subjects

Samson Zhou

Publications and source records attributed to Samson Zhou.

At least 37 records · Page 2Linked to original sources

arXiv2025

On Fine-Grained Distinct Element Estimation

We study the problem of distributed distinct element estimation, where $α$ servers each receive a subset of a universe $[n]$ and aim to compute a $(1+\varepsilon)$-approximation to the number of distinct elements using minimal communication. While prior work establishes a worst-case bound of $Θ\left(α\log n+\fracα{\varepsilon^2}\right)$ bits, these results rely on assumptions that may not hold in practice. We introduce a new parameterization based on the number $C = \fracβ{\varepsilon^2}$ of pairwise collisions, i.e., instances where the same element appears on multiple servers, and design a protocol that uses only $\mathcal{O}\left(α\log n+\frac{\sqrtβ}{\varepsilon^2} \log n\right)$ bits, breaking previous lower bounds when $C$ is small. We further improve our algorithm under assumptions on the number of distinct elements or collisions and provide matching lower bounds in all regimes, establishing $C$ as a tight complexity measure for the problem. Finally, we consider streaming algorithms for distinct element estimation parameterized by the number of items with frequency larger than $1$. Overall, our results offer insight into why statistical problems with known hardness results can be efficiently solved in practice.

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arXiv2025

Private Training & Data Generation by Clustering Embeddings

Deep neural networks often use large, high-quality datasets to achieve high performance on many machine learning tasks. When training involves potentially sensitive data, this process can raise privacy concerns, as large models have been shown to unintentionally memorize and reveal sensitive information, including reconstructing entire training samples. Differential privacy (DP) provides a robust framework for protecting individual data and in particular, a new approach to privately training deep neural networks is to approximate the input dataset with a privately generated synthetic dataset, before any subsequent training algorithm. We introduce a novel principled method for DP synthetic image embedding generation, based on fitting a Gaussian Mixture Model (GMM) in an appropriate embedding space using DP clustering. Our method provably learns a GMM under separation conditions. Empirically, a simple two-layer neural network trained on synthetically generated embeddings achieves state-of-the-art (SOTA) classification accuracy on standard benchmark datasets. Additionally, we demonstrate that our method can generate realistic synthetic images that achieve downstream classification accuracy comparable to SOTA methods. Our method is quite general, as the encoder and decoder modules can be freely substituted to suit different tasks. It is also highly scalable, consisting only of subroutines that scale linearly with the number of samples and/or can be implemented efficiently in distributed systems.

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arXiv2025

Learning-Augmented Hierarchical Clustering

Hierarchical clustering (HC) is an important data analysis technique in which the goal is to recursively partition a dataset into a tree-like structure while grouping together similar data points at each level of granularity. Unfortunately, for many of the proposed HC objectives, there exist strong barriers to approximation algorithms with the hardness of approximation. Thus, we consider the problem of hierarchical clustering given auxiliary information from natural oracles. Specifically, we focus on a *splitting oracle* which, when provided with a triplet of vertices $(u,v,w)$, answers (possibly erroneously) the pairs of vertices whose lowest common ancestor includes all three vertices in an optimal tree, i.e., identifying which vertex ``splits away'' from the others. Using such an oracle, we obtain the following results: - A polynomial-time algorithm that outputs a hierarchical clustering tree with $O(1)$-approximation to the Dasgupta objective (Dasgupta [STOC'16]). - A near-linear time algorithm that outputs a hierarchical clustering tree with $(1-o(1))$-approximation to the Moseley-Wang objective (Moseley and Wang [NeurIPS'17]). Under the plausible Small Set Expansion Hypothesis, no polynomial-time algorithm can achieve any constant approximation for Dasgupta's objective or $(1-C)$-approximation for the Moseley-Wang objective for some constant $C>0$. As such, our results demonstrate that the splitting oracle enables algorithms to outperform standard HC approaches and overcome hardness constraints. Furthermore, our approaches extend to sublinear settings, in which we show new streaming and PRAM algorithms for HC with improved guarantees.

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arXiv2025

On the Price of Differential Privacy for Hierarchical Clustering

Hierarchical clustering is a fundamental unsupervised machine learning task with the aim of organizing data into a hierarchy of clusters. Many applications of hierarchical clustering involve sensitive user information, therefore motivating recent studies on differentially private hierarchical clustering under the rigorous framework of Dasgupta's objective. However, it has been shown that any privacy-preserving algorithm under edge-level differential privacy necessarily suffers a large error. To capture practical applications of this problem, we focus on the weight privacy model, where each edge of the input graph is at least unit weight. We present a novel algorithm in the weight privacy model that shows significantly better approximation than known impossibility results in the edge-level DP setting. In particular, our algorithm achieves $O(\log^{1.5}n/\varepsilon)$ multiplicative error for $\varepsilon$-DP and runs in polynomial time, where $n$ is the size of the input graph, and the cost is never worse than the optimal additive error in existing work. We complement our algorithm by showing if the unit-weight constraint does not apply, the lower bound for weight-level DP hierarchical clustering is essentially the same as the edge-level DP, i.e. $Ω(n^2/\varepsilon)$ additive error. As a result, we also obtain a new lower bound of $\tildeΩ(1/\varepsilon)$ additive error for balanced sparsest cuts in the weight-level DP model, which may be of independent interest. Finally, we evaluate our algorithm on synthetic and real-world datasets. Our experimental results show that our algorithm performs well in terms of extra cost and has good scalability to large graphs.

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arXiv2025

Fast, Space-Optimal Streaming Algorithms for Clustering and Subspace Embeddings

We show that both clustering and subspace embeddings can be performed in the streaming model with the same asymptotic efficiency as in the central/offline setting. For $(k, z)$-clustering in the streaming model, we achieve a number of words of memory which is independent of the number $n$ of input points and the aspect ratio $Δ$, yielding an optimal bound of $\tilde{\mathcal{O}}\left(\frac{dk}{\min(\varepsilon^4,\varepsilon^{z+2})}\right)$ words for accuracy parameter $\varepsilon$ on $d$-dimensional points. Additionally, we obtain amortized update time of $d\,\log(k)\cdot\text{polylog}(\log(nΔ))$, which is an exponential improvement over the previous $d\,\text{poly}(k,\log(nΔ))$. Our method also gives the fastest runtime for $(k,z)$-clustering even in the offline setting. For subspace embeddings in the streaming model, we achieve $\mathcal{O}(d)$ update time and space-optimal constructions, using $\tilde{\mathcal{O}}\left(\frac{d^2}{\varepsilon^2}\right)$ words for $p\le 2$ and $\tilde{\mathcal{O}}\left(\frac{d^{p/2+1}}{\varepsilon^2}\right)$ words for $p>2$, showing that streaming algorithms can match offline algorithms in both space and time complexity.

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arXiv2025

On Approximability of $\ell_2^2$ Min-Sum Clustering

The $\ell_2^2$ min-sum $k$-clustering problem is to partition an input set into clusters $C_1,\ldots,C_k$ to minimize $\sum_{i=1}^k\sum_{p,q\in C_i}\|p-q\|_2^2$. Although $\ell_2^2$ min-sum $k$-clustering is NP-hard, it is not known whether it is NP-hard to approximate $\ell_2^2$ min-sum $k$-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the $\ell_2^2$ min-sum $k$-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than $1.056$ and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a nearly linear time parameterized PTAS for $\ell_2^2$ min-sum $k$-clustering running in time $O\left(n^{1+o(1)}d\cdot \exp((k\cdot\varepsilon^{-1})^{O(1)})\right)$, where $d$ is the underlying dimension of the input dataset. Finally, we consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label $i\in[k]$ for input point, thereby implicitly partitioning the input dataset into $k$ clusters that induce an approximately optimal solution, up to some amount of adversarial error $α\in\left[0,\frac{1}{2}\right)$. We give a polynomial-time algorithm that outputs a $\frac{1+γα}{(1-α)^2}$-approximation to $\ell_2^2$ min-sum $k$-clustering, for a fixed constant $γ>0$.

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arXiv2025

Perfect Sampling in Turnstile Streams Beyond Small Moments

Given a vector $x \in \mathbb{R}^n$ induced by a turnstile stream $S$, a non-negative function $G: \mathbb{R} \to \mathbb{R}$, a perfect $G$-sampler outputs an index $i$ with probability $\frac{G(x_i)}{\sum_{j\in[n]} G(x_j)}+\frac{1}{\text{poly}(n)}$. Jayaram and Woodruff (FOCS 2018) introduced a perfect $L_p$-sampler, where $G(z)=|z|^p$, for $p\in(0,2]$. In this paper, we solve this problem for $p>2$ by a sampling-and-rejection method. Our algorithm runs in $n^{1-2/p} \cdot \text{polylog}(n)$ bits of space, which is tight up to polylogarithmic factors in $n$. Our algorithm also provides a $(1+\varepsilon)$-approximation to the sampled item $x_i$ with high probability using an additional $\varepsilon^{-2} n^{1-2/p} \cdot \text{polylog}(n)$ bits of space. Interestingly, we show our techniques can be generalized to perfect polynomial samplers on turnstile streams, which is a class of functions that is not scale-invariant, in contrast to the existing perfect $L_p$ samplers. We also achieve perfect samplers for the logarithmic function $G(z)=\log(1+|z|)$ and the cap function $G(z)=\min(T,|z|^p)$. Finally, we give an application of our results to the problem of norm/moment estimation for a subset $\mathcal{Q}$ of coordinates of a vector, revealed only after the data stream is processed, e.g., when the set $\mathcal{Q}$ represents a range query, or the set $n\setminus\mathcal{Q}$ represents a collection of entities who wish for their information to be expunged from the dataset.

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arXiv2025

Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

We introduce a novel technique for ``lifting'' dimension lower bounds for linear sketches in the real-valued setting to dimension lower bounds for linear sketches with polynomially-bounded integer entries when the input is a polynomially-bounded integer vector. Using this technique, we obtain the first optimal sketching lower bounds for discrete inputs in a data stream, for classical problems such as approximating the frequency moments, estimating the operator norm, and compressed sensing. Additionally, we lift the adaptive attack of Hardt and Woodruff (STOC, 2013) for breaking any real-valued linear sketch via a sequence of real-valued queries, and show how to obtain an attack on any integer-valued linear sketch using integer-valued queries. This shows that there is no linear sketch in a data stream with insertions and deletions that is adversarially robust for approximating any $L_p$ norm of the input, resolving a central open question for adversarially robust streaming algorithms. To do so, we introduce a new pre-processing technique of independent interest which, given an integer-valued linear sketch, increases the dimension of the sketch by only a constant factor in order to make the orthogonal lattice to its row span smooth. This pre-processing then enables us to leverage results in lattice theory on discrete Gaussian distributions and reason that efficient discrete sketches imply efficient continuous sketches. Our work resolves open questions from the Banff '14 and '17 workshops on Communication Complexity and Applications, as well as the STOC '21 and FOCS '23 workshops on adaptivity and robustness.

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arXiv2025

Learning-Augmented Search Data Structures

We study the integration of machine learning advice to improve upon traditional data structure designed for efficient search queries. Although there has been recent effort in improving the performance of binary search trees using machine learning advice, e.g., Lin et. al. (ICML 2022), the resulting constructions nevertheless suffer from inherent weaknesses of binary search trees, such as complexity of maintaining balance across multiple updates and the inability to handle partially-ordered or high-dimensional datasets. For these reasons, we focus on skip lists and KD trees in this work. Given access to a possibly erroneous oracle that outputs estimated fractional frequencies for search queries on a set of items, we construct skip lists and KD trees that provably provides the optimal expected search time, within nearly a factor of two. In fact, our learning-augmented skip lists and KD trees are still optimal up to a constant factor, even if the oracle is only accurate within a constant factor. We also demonstrate robustness by showing that our data structures achieves an expected search time that is within a constant factor of an oblivious skip list/KD tree construction even when the predictions are arbitrarily incorrect. Finally, we empirically show that our learning-augmented search data structures outperforms their corresponding traditional analogs on both synthetic and real-world datasets.

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arXiv2025

Fair Clustering in the Sliding Window Model

We study streaming algorithms for proportionally fair clustering, a notion originally suggested by Chierichetti et. al. (2017), in the sliding window model. We show that although there exist efficient streaming algorithms in the insertion-only model, surprisingly no algorithm can achieve finite multiplicative ratio without violating the fairness constraint in the sliding window. Hence, the problem of fair clustering is a rare separation between the insertion-only streaming model and the sliding window model. On the other hand, we show that if the fairness constraint is relaxed by a multiplicative $(1+\varepsilon)$ factor, there exists a $(1 + \varepsilon)$-approximate sliding window algorithm that uses $\text{poly}(k\varepsilon^{-1}\log n)$ space. This achieves essentially the best parameters (up to degree in the polynomial) provided the aforementioned lower bound. We also implement a number of empirical evaluations on real datasets to complement our theoretical results.

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arXiv2024

Adversarially Robust Dense-Sparse Tradeoffs via Heavy-Hitters

In the adversarial streaming model, the input is a sequence of adaptive updates that defines an underlying dataset and the goal is to approximate, collect, or compute some statistic while using space sublinear in the size of the dataset. In 2022, Ben-Eliezer, Eden, and Onak showed a dense-sparse trade-off technique that elegantly combined sparse recovery with known techniques using differential privacy and sketch switching to achieve adversarially robust algorithms for $L_p$ estimation and other algorithms on turnstile streams. In this work, we first give an improved algorithm for adversarially robust $L_p$-heavy hitters, utilizing deterministic turnstile heavy-hitter algorithms with better tradeoffs. We then utilize our heavy-hitter algorithm to reduce the problem to estimating the frequency moment of the tail vector. We give a new algorithm for this problem in the classical streaming setting, which achieves additive error and uses space independent in the size of the tail. We then leverage these ingredients to give an improved algorithm for adversarially robust $L_p$ estimation on turnstile streams.

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arXiv2024

On Socially Fair Low-Rank Approximation and Column Subset Selection

Low-rank approximation and column subset selection are two fundamental and related problems that are applied across a wealth of machine learning applications. In this paper, we study the question of socially fair low-rank approximation and socially fair column subset selection, where the goal is to minimize the loss over all sub-populations of the data. We show that surprisingly, even constant-factor approximation to fair low-rank approximation requires exponential time under certain standard complexity hypotheses. On the positive side, we give an algorithm for fair low-rank approximation that, for a constant number of groups and constant-factor accuracy, runs in $2^{\text{poly}(k)}$ time rather than the naïve $n^{\text{poly}(k)}$, which is a substantial improvement when the dataset has a large number $n$ of observations. We then show that there exist bicriteria approximation algorithms for fair low-rank approximation and fair column subset selection that run in polynomial time.

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arXiv2024

A Strong Separation for Adversarially Robust $\ell_0$ Estimation for Linear Sketches

The majority of streaming problems are defined and analyzed in a static setting, where the data stream is any worst-case sequence of insertions and deletions that is fixed in advance. However, many real-world applications require a more flexible model, where an adaptive adversary may select future stream elements after observing the previous outputs of the algorithm. Over the last few years, there has been increased interest in proving lower bounds for natural problems in the adaptive streaming model. In this work, we give the first known adaptive attack against linear sketches for the well-studied $\ell_0$-estimation problem over turnstile, integer streams. For any linear streaming algorithm $\mathcal{A}$ that uses sketching matrix $\mathbf{A}\in \mathbb{Z}^{r \times n}$ where $n$ is the size of the universe, this attack makes $\tilde{\mathcal{O}}(r^8)$ queries and succeeds with high constant probability in breaking the sketch. We also give an adaptive attack against linear sketches for the $\ell_0$-estimation problem over finite fields $\mathbb{F}_p$, which requires a smaller number of $\tilde{\mathcal{O}}(r^3)$ queries. Finally, we provide an adaptive attack over $\mathbb{R}^n$ against linear sketches $\mathbf{A} \in \mathbb{R}^{r \times n}$ for $\ell_0$-estimation, in the setting where $\mathbf{A}$ has all nonzero subdeterminants at least $\frac{1}{\textrm{poly}(r)}$. Our results provide an exponential improvement over the previous number of queries known to break an $\ell_0$-estimation sketch.

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arXiv2024

Fair Submodular Cover

Submodular optimization is a fundamental problem with many applications in machine learning, often involving decision-making over datasets with sensitive attributes such as gender or age. In such settings, it is often desirable to produce a diverse solution set that is fairly distributed with respect to these attributes. Motivated by this, we initiate the study of Fair Submodular Cover (FSC), where given a ground set $U$, a monotone submodular function $f:2^U\to\mathbb{R}_{\ge 0}$, a threshold $τ$, the goal is to find a balanced subset of $S$ with minimum cardinality such that $f(S)\geτ$. We first introduce discrete algorithms for FSC that achieve a bicriteria approximation ratio of $(\frac{1}ε, 1-O(ε))$. We then present a continuous algorithm that achieves a $(\ln\frac{1}ε, 1-O(ε))$-bicriteria approximation ratio, which matches the best approximation guarantee of submodular cover without a fairness constraint. Finally, we complement our theoretical results with a number of empirical evaluations that demonstrate the effectiveness of our algorithms on instances of maximum coverage.

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arXiv2024

Streaming Algorithms with Few State Changes

In this paper, we study streaming algorithms that minimize the number of changes made to their internal state (i.e., memory contents). While the design of streaming algorithms typically focuses on minimizing space and update time, these metrics fail to capture the asymmetric costs, inherent in modern hardware and database systems, of reading versus writing to memory. In fact, most streaming algorithms write to their memory on every update, which is undesirable when writing is significantly more expensive than reading. This raises the question of whether streaming algorithms with small space and number of memory writes are possible. We first demonstrate that, for the fundamental $F_p$ moment estimation problem with $p\ge 1$, any streaming algorithm that achieves a constant factor approximation must make $Ω(n^{1-1/p})$ internal state changes, regardless of how much space it uses. Perhaps surprisingly, we show that this lower bound can be matched by an algorithm that also has near-optimal space complexity. Specifically, we give a $(1+\varepsilon)$-approximation algorithm for $F_p$ moment estimation that uses a near-optimal $\widetilde{\mathcal{O}}_\varepsilon(n^{1-1/p})$ number of state changes, while simultaneously achieving near-optimal space, i.e., for $p\in[1,2]$, our algorithm uses $\text{poly}\left(\log n,\frac{1}{\varepsilon}\right)$ bits of space, while for $p>2$, the algorithm uses $\widetilde{\mathcal{O}}_\varepsilon(n^{1-2/p})$ space. We similarly design streaming algorithms that are simultaneously near-optimal in both space complexity and the number of state changes for the heavy-hitters problem, sparse support recovery, and entropy estimation. Our results demonstrate that an optimal number of state changes can be achieved without sacrificing space complexity.

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arXiv2024

Private Vector Mean Estimation in the Shuffle Model: Optimal Rates Require Many Messages

We study the problem of private vector mean estimation in the shuffle model of privacy where $n$ users each have a unit vector $v^{(i)} \in\mathbb{R}^d$. We propose a new multi-message protocol that achieves the optimal error using $\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)$ messages per user. Moreover, we show that any (unbiased) protocol that achieves optimal error requires each user to send $Ω(\min(n\varepsilon^2,d)/\log(n))$ messages, demonstrating the optimality of our message complexity up to logarithmic factors. Additionally, we study the single-message setting and design a protocol that achieves mean squared error $\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})$. Moreover, we show that any single-message protocol must incur mean squared error $Ω(dn^{d/(d+2)})$, showing that our protocol is optimal in the standard setting where $\varepsilon = Θ(1)$. Finally, we study robustness to malicious users and show that malicious users can incur large additive error with a single shuffler.

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arXiv2024

Selective experience replay compression using coresets for lifelong deep reinforcement learning in medical imaging

Selective experience replay is a popular strategy for integrating lifelong learning with deep reinforcement learning. Selective experience replay aims to recount selected experiences from previous tasks to avoid catastrophic forgetting. Furthermore, selective experience replay based techniques are model agnostic and allow experiences to be shared across different models. However, storing experiences from all previous tasks make lifelong learning using selective experience replay computationally very expensive and impractical as the number of tasks increase. To that end, we propose a reward distribution-preserving coreset compression technique for compressing experience replay buffers stored for selective experience replay. We evaluated the coreset compression technique on the brain tumor segmentation (BRATS) dataset for the task of ventricle localization and on the whole-body MRI for localization of left knee cap, left kidney, right trochanter, left lung, and spleen. The coreset lifelong learning models trained on a sequence of 10 different brain MR imaging environments demonstrated excellent performance localizing the ventricle with a mean pixel error distance of 12.93 for the compression ratio of 10x. In comparison, the conventional lifelong learning model localized the ventricle with a mean pixel distance of 10.87. Similarly, the coreset lifelong learning models trained on whole-body MRI demonstrated no significant difference (p=0.28) between the 10x compressed coreset lifelong learning models and conventional lifelong learning models for all the landmarks. The mean pixel distance for the 10x compressed models across all the landmarks was 25.30, compared to 19.24 for the conventional lifelong learning models. Our results demonstrate that the potential of the coreset-based ERB compression method for compressing experiences without a significant drop in performance.

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arXiv2023

Near-Optimal $k$-Clustering in the Sliding Window Model

Clustering is an important technique for identifying structural information in large-scale data analysis, where the underlying dataset may be too large to store. In many applications, recent data can provide more accurate information and thus older data past a certain time is expired. The sliding window model captures these desired properties and thus there has been substantial interest in clustering in the sliding window model. In this paper, we give the first algorithm that achieves near-optimal $(1+\varepsilon)$-approximation to $(k,z)$-clustering in the sliding window model, where $z$ is the exponent of the distance function in the cost. Our algorithm uses $\frac{k}{\min(\varepsilon^4,\varepsilon^{2+z})}\,\text{polylog}\frac{nΔ}{\varepsilon}$ words of space when the points are from $[Δ]^d$, thus significantly improving on works by Braverman et. al. (SODA 2016), Borassi et. al. (NeurIPS 2021), and Epasto et. al. (SODA 2022). Along the way, we develop a data structure for clustering called an online coreset, which outputs a coreset not only for the end of a stream, but also for all prefixes of the stream. Our online coreset samples $\frac{k}{\min(\varepsilon^4,\varepsilon^{2+z})}\,\text{polylog}\frac{nΔ}{\varepsilon}$ points from the stream. We then show that any online coreset requires $Ω\left(\frac{k}{\varepsilon^2}\log n\right)$ samples, which shows a separation from the problem of constructing an offline coreset, i.e., constructing online coresets is strictly harder. Our results also extend to general metrics on $[Δ]^d$ and are near-optimal in light of a $Ω\left(\frac{k}{\varepsilon^{2+z}}\right)$ lower bound for the size of an offline coreset.

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