We consider distributed inference using sequentially interactive protocols. We obtain lower bounds on the minimax sample complexity of interactive protocols under local information constraints, a broad family of resource constraints which captures communication constraints , local differential privacy , and noisy binary queries as special cases . We focus on the inference tasks of learning (density estimation) and identity testing ( goodness-of-fit ) for discrete distributionsunder total variation distance, and establish general lower bounds that take into account the local constraints modeled as a channel family. Our main technical contribution is an approach to handle the correlation that builds due to interactivity and quantifies how effectively one can exploit this correlation in spite of the local constraints. Using this, we fill gaps in some prior works and characterize the optimal sample complexity of learning and testing discrete distributions under total variation distance, for both communication and local differential privacy constraints. Prior to our work, this was known only for the problem of testing under local privacy constraints (Amin, Joseph, and Mao (2020); Berrett and Butucea (2020)). Our results show that interactivity does not help for learning or testing under these two constraints. Finally, we provide the first instance of a natural family of "leaky query" local constraints under which interactive protocols strictly outperform the noninteractive ones for distribution testing .
We study the role of interactivity in distributed statistical inference under information constraints, e.g., communication constraints and local differential privacy . We focus on the tasks of goodness-of-fit testing and estimation of discrete distributions. From prior work, these tasks are well understood under noninteractive protocols. Extending these approaches directly for interactive protocols is difficult due to correlations that can build due to interactivity ; in fact, gaps can be found in prior claims of tight bounds of distribution estimation using interactive protocols. We propose a new approach to handle this correlation and establish a unified method to establish lower bounds for both tasks. As an application, we obtain optimal bounds for both estimation and testing under local differential privacy and communication constraints. We also provide an example of a natural testing problem where interactivity helps .
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consider distributed inference using sequentially interactive protocols. We obtain lower bounds on the minimax sample complexity of interactive protocols under local
study the role of interactivity in distributed statistical inference under
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a broad family of resource constraints which captures communication constraints ,
e.g., communication constraints and
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, and noisy binary queries as special cases
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inference tasks of learning (density estimation) and identity testing (
tasks of
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) for discrete distributionsunder total variation distance, and establish general lower bounds that take into account the local constraints modeled as a channel family. Our main technical contribution is an approach to handle the correlation that builds
testing and estimation of discrete distributions. From prior work, these tasks are well understood under noninteractive protocols. Extending these approaches directly for interactive protocols is difficult due to correlations that can build
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and quantifies how effectively one can exploit this correlation in spite of the local constraints. Using this, we fill gaps in some prior works and characterize the optimal sample complexity of learning and testing discrete distributions under total variation distance, for both communication and local differential privacy constraints. Prior to our work, this was known only for the problem of
; in fact, gaps can be found in prior claims of tight bounds of distribution estimation using interactive protocols. We propose a new approach to handle this correlation and establish a unified method to establish lower bounds for both tasks. As an application, we obtain optimal bounds for both estimation and
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privacy constraints (Amin, Joseph, and Mao (2020); Berrett and Butucea (2020)). Our results show that interactivity does not help for learning or testing under these two constraints. Finally, we provide the first instance
differential privacy and communication constraints. We also provide an example
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family of "leaky query" local constraints under which interactive protocols strictly outperform the noninteractive ones for distribution testing
testing problem where interactivity helps