Robust Streaming Against Low-Memory Adversaries

Abstract

Robust streaming, the study of streaming algorithms that provably work when the stream is generated by an adaptive adversary, has seen tremendous progress in recent years. However, fundamental barriers remain: the best known algorithm for turnstile $F_p$-estimation in the robust streaming setting is exponentially worse than in the oblivious setting, and closing this gap seems difficult. Arguably, one possible cause of this barrier is the adversarial model, which may be too strong: unlike the space-bounded streaming algorithm, the adversary can memorize the entire history of the interaction with the algorithm. Can we then close the exponential gap if we insist that the adversary itself is an adaptive but low-memory entity, roughly as powerful as (or even weaker than) the algorithm? In this work we present the first set of models and results aimed towards this question. We design efficient robust streaming algorithms against adversaries that are fully adaptive but have no long-term memory ("memoryless") or very little memory of the history of interaction. Roughly speaking, a memoryless adversary only sees, at any given round, the last output of the algorithm (and does not even know the current time) and can generate an unlimited number of independent coin tosses. A low-memory adversary is similar, but maintains an additional small buffer. While these adversaries may seem quite limited at first glance, we show that this adversarial model is strong enough to produce streams that have high flip number and density in the context of $F_2$-estimation, which rules out most of known robustification techniques. We then design a new simple approach, similar to the computation paths framework, to obtain efficient algorithms against memoryless and low-memory adversaries for a wide class of order-invariant problems.