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Distance-based and continuum Fano inequalities with applications to statistical estimation

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arxiv 1311.2669 v2 pith:IW5EUQZ4 submitted 2013-11-12 cs.IT math.ITmath.STstat.TH

classification cs.ITmath.ITmath.STstat.TH
keywords fanoinequalityboundboundscontinuumestimationextendsinequalities
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abstract

In this technical note, we give two extensions of the classical Fano inequality in information theory. The first extends Fano's inequality to the setting of estimation, providing lower bounds on the probability that an estimator of a discrete quantity is within some distance $t$ of the quantity. The second inequality extends our bound to a continuum setting and provides a volume-based bound. We illustrate how these inequalities lead to direct and simple proofs of several statistical minimax lower bounds.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the Gradient Complexity of Private Optimization with Private Oracles

    cs.LG 2025-11 accept novelty 8.0 of 10

    For high-dimensional non-smooth convex losses, any optimizer using a private gradient oracle needs Ω(√d/α²) gradient evaluations—a √d dimension penalty absent without privacy.

  2. Contaminated Multi-task Learning with Heterogeneity: Fundamental Limits and Optimal Algorithms

    stat.ML 2026-07 accept novelty 7.5 of 10

    Filtering-based robust multi-task gradient descent matches minimax rates under task contamination and heterogeneity, removing the √d contamination barrier of regularization and score-based methods.

  3. Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy

    cs.LG 2026-06 unverdicted novelty 7.0 of 10

    Derives explicit minimax quantile lower bounds for Gaussian mean estimation and K-armed bandits under interactive decision making and MI privacy, with log(1/δ)/n and √(KT log(1/δ)) scalings.

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