A Framework for Adversarially Robust Streaming Algorithms

Abstract

We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems.

Separating Adaptive Streaming from Oblivious Streaming

“…The adversarial streaming model, in various forms, was considered by [19,11,12,1,2,14,6,5,16,23]. We give here the formulation presented by Ben-Eliezer et al [5].…”

“…The adversarial streaming model, in various forms, was considered by [19,11,12,1,2,14,6,5,16,23]. We give here the formulation presented by Ben-Eliezer et al [5]. The adversarial setting is modeled by a two-player game between a (randomized) StreamingAlgorithm and an Adversary.…”

“…We refer to this setting as the oblivious setting. Recently, there has been a growing interest in streaming algorithms that maintain utility even when the choice of stream items depends on previous answers given by the streaming algorithm, and can hence depend on the internal state of the algorithm [19,11,12,1,2,14,6,5,16,23]. Such streaming algorithms are said to be adversarially robust.…”

“…1 This result does not rule out non-linear algorithms. Indeed, strong positive results were shown by [5,16,23] who constructed (non-linear) adversarially robust algorithms for many problems of interest, with small overhead compared to the oblivious setting. These results leave open the question whether adversarial robustness can come "for free" in terms of the additional costs to memory.…”

On Adaptive Distance Estimation

“…In the context of streaming algorithms (i.e. sublinear memory), the very recent work of Ben-Eliezer et al [BEJWY20] considers streaming algorithms with both adaptive queries and updates. One key difference is they considered the insertion-only model of streaming, which does not allow one to model computing some function of the difference of two vectors (e.g.…”