Title: Robust Algorithms on Adaptive Inputs from Bounded Adversaries

We study dynamic algorithms robust to adaptive input generated from sources with bounded capabilities, such as sparsity or limited interaction. For example, we consider robust linear algebraic algorithms when the updates to the input are sparse but given by an adversary with access to a query oracle. We also study robust algorithms in the standard centralized setting, where an adversary queries an algorithm in an adaptive manner, but the number of interactions between the adversary and the algorithm is bounded. We first recall a unified framework of [HKM+20, BKM+22, ACSS23] which is roughly a quadratic improvement over the na ̈ıve implementation, and only incurs a logarithmic overhead in query time. Although the general framework has diverse applications in machine learning and data science, such as adaptive distance estimation, kernel density estimation, linear regression, range queries, and point queries and serves as a preliminary benchmark, we demonstrate even better algorithmic improvements for (1) reducing the pre-processing time for adaptive distance estimation and (2) permitting an unlimited number of adaptive queries for kernel density estimation. Finally, we complement our theoretical results with additional empirical evaluations. more »« less

Stoughton, Alley; Chen, Carol; Gaboardi, Marco; Qu, Weihao(
, 13th International Conference on Interactive Theorem Proving (ITP 2022))

Andronick, June; de Moura, Leonardo
(Ed.)

We use the EasyCrypt proof assistant to formalize the adversarial approach to proving lower bounds for computational problems in the query model. This is done using a lower bound game between an algorithm and adversary, in which the adversary answers the algorithm’s queries in a way that makes the algorithm issue at least the desired number of queries. A complementary upper bound game is used for proving upper bounds of algorithms; here the adversary incrementally and adaptively realizes an algorithm’s input. We prove a natural connection between the lower and upper bound games, and apply our framework to three computational problems, including searching in an ordered list and comparison-based sorting, giving evidence for the generality of our notion of algorithm and the usefulness of our framework.

Ajtai, M.; Braverman, V.; Jayram, T.S.; Silwal, S.; Sun, A.; Woodruff, D.P.; Zhou, S.(
, Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS 2022))

There has been a flurry of recent literature studying streaming algorithms for which the input stream is chosen adaptively by a black-box adversary who observes the output of the streaming algorithm at each time step. However, these algorithms fail when the adversary has access to the internal state of the algorithm, rather than just the output of the algorithm. We study streaming algorithms in the white-box adversarial model, where the stream is
chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the L1-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our L1-heavy hitters algorithm even further
and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for randomized algorithms robust to a white-box adversary. In particular, our results show that for all p ≥ 0, there exists a constant Cp > 1 such that any
Cp-approximation algorithm for Fp moment estimation in insertion-only streams with a white-box adversary requires Ω(n) space for a universe of size n. Similarly, there is a constant C > 1 such that any C-approximation algorithm in an insertion-only stream for matrix rank requires Ω(n) space with a white-box adversary. These results do not contradict our upper bounds since they assume the adversary has unbounded computational power. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. Finally, we prove a lower bound of Ω(log n) bits for the fundamental problem of deterministic approximate counting in a stream of 0’s and 1’s, which holds even if we know how many total stream updates we have seen so far at each point in the stream. Such a lower bound for approximate counting with additional information was previously unknown, and in our context, it shows a separation between multiplayer deterministic maximum communication and the white-box space complexity of a streaming algorithm

Chuzhoy, Julia; Zhang, Ruimin(
, Proceedings of the 55th Annual {ACM} Symposium on Theory of Computing, {STOC} June 2023)

We study the fully dynamic All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. Given an n-vertex graph G with non-negative edge lengths, that undergoes an online sequence of edge insertions and deletions, the goal is to support approximate distance queries and shortest-path queries. We provide a deterministic algorithm for this problem, that, for a given precision parameter є, achieves approximation factor (loglogn)2O(1/є3), and has amortized update time O(nєlogL) per operation, where L is the ratio of longest to shortest edge length. Query time for distance-query is O(2O(1/є)· logn· loglogL), and query time for shortest-path query is O(|E(P)|+2O(1/є)· logn· loglogL), where P is the path that the algorithm returns. To the best of our knowledge, even allowing any o(n)-approximation factor, no adaptive-update algorithms with better than Θ(m) amortized update time and better than Θ(n) query time were known prior to this work. We also note that our guarantees are stronger than the best current guarantees for APSP in decremental graphs in the adaptive-adversary setting.
In order to obtain these results, we consider an intermediate problem, called Recursive Dynamic Neighborhood Cover (RecDynNC), that was formally introduced in [Chuzhoy, STOC ’21]. At a high level, given an undirected edge-weighted graph G undergoing an online sequence of edge deletions, together with a distance parameter D, the goal is to maintain a sparse D-neighborhood cover of G, with some additional technical requirements. Our main technical contribution is twofolds. First, we provide a black-box reduction from APSP in fully dynamic graphs to the RecDynNC problem. Second, we provide a new deterministic algorithm for the RecDynNC problem, that, for a given precision parameter є, achieves approximation factor (loglogm)2O(1/є2), with total update time O(m1+є), where m is the total number of edges ever present in G. This improves the previous algorithm of [Chuzhoy, STOC ’21], that achieved approximation factor (logm)2O(1/є) with similar total update time. Combining these two results immediately leads to the deterministic algorithm for fully-dynamic APSP with the guarantees stated above.

Bakshi, Ainesh; Kacham, Praneeth; Indyk, Piotr; Silwal, Sandeep; Zhou, Samson(
, International Conference on Learning Representations)

Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency – given n input points, most kernel-based algorithms need to materialize the full n × n kernel matrix before performing any subsequent computation, thus incurring Ω(n^2) runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts.
We break the quadratic barrier and obtain subquadratic time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving lin- ear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recently developed Kernel Density Estimation framework, which (after preprocessing in time subquadratic in n) can return estimates of row/column sums of the kernel matrix. In particular, we de- velop efficient reductions from weighted vertex and weighted edge sampling on kernel graphs, simulating random walks on kernel graphs, and importance sampling on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in sublinear (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsi- fication, where we observe a 9x decrease in the number of kernel evaluations over baselines for LRA and a 41x reduction in the graph size for spectral sparsification.

Gudmundsson, Joachim; Mirzanezhad, Majid; Mohades, Ali; Wenk, Carola(
, International Journal of Computational Geometry & Applications)

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve.

Cherapanamjeri, Yeshwanth, Silwal, Sandeep, Woodruff, David P, Zhang, Fred, Zhang, Qiuyi, and Zhou, Samson. Robust Algorithms on Adaptive Inputs from Bounded Adversaries. Retrieved from https://par.nsf.gov/biblio/10511016. International Conference on Learning Representations .

Cherapanamjeri, Yeshwanth, Silwal, Sandeep, Woodruff, David P, Zhang, Fred, Zhang, Qiuyi, and Zhou, Samson.
"Robust Algorithms on Adaptive Inputs from Bounded Adversaries". International Conference on Learning Representations (). Country unknown/Code not available: International Conference on Learning Representations. https://par.nsf.gov/biblio/10511016.

@article{osti_10511016,
place = {Country unknown/Code not available},
title = {Robust Algorithms on Adaptive Inputs from Bounded Adversaries},
url = {https://par.nsf.gov/biblio/10511016},
abstractNote = {We study dynamic algorithms robust to adaptive input generated from sources with bounded capabilities, such as sparsity or limited interaction. For example, we consider robust linear algebraic algorithms when the updates to the input are sparse but given by an adversary with access to a query oracle. We also study robust algorithms in the standard centralized setting, where an adversary queries an algorithm in an adaptive manner, but the number of interactions between the adversary and the algorithm is bounded. We first recall a unified framework of [HKM+20, BKM+22, ACSS23] which is roughly a quadratic improvement over the na ̈ıve implementation, and only incurs a logarithmic overhead in query time. Although the general framework has diverse applications in machine learning and data science, such as adaptive distance estimation, kernel density estimation, linear regression, range queries, and point queries and serves as a preliminary benchmark, we demonstrate even better algorithmic improvements for (1) reducing the pre-processing time for adaptive distance estimation and (2) permitting an unlimited number of adaptive queries for kernel density estimation. Finally, we complement our theoretical results with additional empirical evaluations.},
journal = {International Conference on Learning Representations},
publisher = {International Conference on Learning Representations},
author = {Cherapanamjeri, Yeshwanth and Silwal, Sandeep and Woodruff, David P and Zhang, Fred and Zhang, Qiuyi and Zhou, Samson},
}

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