Improved Algorithms for White-Box Adversarial Streams

TL;DR

This paper develops cryptographically robust streaming algorithms capable of handling adaptive, white-box adversaries, enabling efficient sparse recovery, low-rank approximation, and solving combinatorial problems under adversarial conditions.

Contribution

It introduces the first cryptographically secure algorithms for sparse and low-rank recovery in white-box adversarial streams, with applications to graph matchings and matrix rank estimation.

Findings

Efficient algorithms for sparse vector recovery under adversarial streams.

Robust PCA algorithms that detect non-sparse or non-low-rank inputs.

Improved approximation-memory tradeoffs for matrix rank and non-zero element estimation.

Abstract

We study streaming algorithms in the white-box adversarial stream model, where the internal state of the streaming algorithm is revealed to an adversary who adaptively generates the stream updates, but the algorithm obtains fresh randomness unknown to the adversary at each time step. We incorporate cryptographic assumptions to construct robust algorithms against such adversaries. We propose efficient algorithms for sparse recovery of vectors, low rank recovery of matrices and tensors, as well as low rank plus sparse recovery of matrices, i.e., robust PCA. Unlike deterministic algorithms, our algorithms can report when the input is not sparse or low rank even in the presence of such an adversary. We use these recovery algorithms to improve upon and solve new problems in numerical linear algebra and combinatorial optimization on white-box adversarial streams. For example, we give the first efficient algorithm for outputting a matching in a graph with insertions and deletions to its edges provided the matching size is small, and otherwise we declare the matching size is large. We also improve the approximation versus memory tradeoff of previous work for estimating the number of non-zero elements in a vector and computing the matrix rank.