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Robust Streaming Against Low-Memory Adversaries

Omri Ben-Eliezer, Krzysztof Onak, Sandeep Silwal

TL;DR

The paper investigates robust streaming under adaptive adversaries with bounded memory, aiming to bridge the gap between oblivious and robust $F_p$-estimation by restricting adversarial memory. It introduces the bounded-memory adversary model and the $ au$-Stream Adversary Model, developing deterministic and randomized algorithms that remain accurate under memory constraints, using techniques such as $psilon$-nets and the median trick. Key results include efficient deterministic tracking against memoryless adversaries, demonstrations that memoryless adversaries can induce dense, high-flip streams, and reductions showing robust solutions against memory-bounded randomized adversaries for order-invariant problems, with applications to moment estimation and triangle counting. The work provides a framework for balancing adversarial power with algorithmic space, offering practical guidance and several open questions, including dependencies on persistent memory and stream parameters. Overall, the results advance robust streaming by proposing scalable models and strategies that tolerate realistic adversarial capabilities while maintaining polylogarithmic-space performance for a broad class of problems.

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.

Robust Streaming Against Low-Memory Adversaries

TL;DR

The paper investigates robust streaming under adaptive adversaries with bounded memory, aiming to bridge the gap between oblivious and robust -estimation by restricting adversarial memory. It introduces the bounded-memory adversary model and the -Stream Adversary Model, developing deterministic and randomized algorithms that remain accurate under memory constraints, using techniques such as -nets and the median trick. Key results include efficient deterministic tracking against memoryless adversaries, demonstrations that memoryless adversaries can induce dense, high-flip streams, and reductions showing robust solutions against memory-bounded randomized adversaries for order-invariant problems, with applications to moment estimation and triangle counting. The work provides a framework for balancing adversarial power with algorithmic space, offering practical guidance and several open questions, including dependencies on persistent memory and stream parameters. Overall, the results advance robust streaming by proposing scalable models and strategies that tolerate realistic adversarial capabilities while maintaining polylogarithmic-space performance for a broad class of problems.

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 -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 -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.

Paper Structure

This paper contains 25 sections, 14 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Model
  3. The bounded--memory adversarial model.
  4. Our Results
  5. Robust streaming against deterministic memoryless adversaries.
  6. Memoryless adversaries can generate high flip--number, high density streams.
  7. Handling memory--bounded randomized adversaries.
  8. Preliminaries
  9. The Bounded--Memory Adversary Model
  10. Deterministic Adversaries
  11. Memoryless Adversaries Can Generate Non-Trivial Streams
  12. When $\textnormal{Adversary}$ has one bit of persistent memory.
  13. When $\textnormal{Adversary}$ is memoryless.
  14. Randomized Adversaries
  15. The $\tau$--Stream Adversary Model
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exists a deterministic streaming algorithm that, for a fixed function $f$ of the underlying stream frequency vector with range $\{0\} \cup [1,\alpha]$, tracks the value of $f$ up to a multiplicative factor of $1+\epsilon$ against a deterministic memoryless adversary, using $O\left(\frac{\log(\

Figures (1)

  • Figure 1: A pictorial representation of our streaming model. Not shown: The estimate memory is overwritten by the estimate $y_{j-1}$ of the streaming algorithm, and the working memory is wiped clean after each update $(i_j, \Delta_j)$ is computed. We also don't show the inner mechanisms of the streaming algorithm or the potential strategy the adversary is following. See Definition \ref{['def:robust_streaming']} for the formal presentation.

Theorems & Definitions (43)

  • Theorem 1.1: Deterministic adversaries; informal version of Theorem \ref{['thm:deterministic_adversary']} and Corollary \ref{['thm:deterministic_adversary2']}
  • Theorem 1.2: Low--memory stream with high density and large flip number; informal version of Theorem \ref{['thm:lb_construction']}
  • Theorem 1.3: Randomized adversaries; informal version of Theorem \ref{['thm:robust_low_memory']}
  • Definition 2.1: Multiplicative approximation
  • Definition 2.2: Estimation problems
  • Definition 2.3: Frequency vectors and streams of updates
  • Definition 2.4: "Oblivious" streaming
  • Definition 2.5: Order invariant streaming algorithms li2014turnstile
  • Definition 2.6: Flip number advrob
  • Definition 2.7: Density/sparsity of a stream
  • ...and 33 more