High-Probability Interactive Fano’s Method

• The method replaces expectation-based minimax risk with transcript-dependent, quantile-focused lower bounds to better capture tail events in adaptive settings.
• It leverages the interactive statistical decision framework to transform loss distributions via f-divergence and Bernoulli reduction, ensuring robust risk guarantees.
• The approach extends to transform-based and CVaR formulations, providing actionable insights for bandit problems and privacy-preserving analyses.

Searching arXiv for the cited papers to ground the article in current metadata and ensure accurate citation coverage. arXiv search query: High-probability interactive Fano minimax quantile interactive statistical decision making High-probability interactive Fano’s method is a converse framework for interactive statistical decision making (ISDM) that replaces expectation-based minimax risk lower bounds by confidence-indexed lower bounds on loss quantiles or strict tail probabilities. In ISDM, the law of the observed outcome or full transcript depends jointly on the model and the adaptive algorithm, so lower bounds are stated in terms of algorithm-dependent transcript laws $\mathbb P^{M,ALG}$ rather than passive sample distributions. Recent formulations make this shift explicit by lower-bounding minimax quantiles through transcript-level $f$-divergences and a Bernoulli reduction of the event $\{L(M,X)\le \Delta\}$, while closely related work broadens the method to bounded transforms of the loss and to Bayesian CVaR (Bongole et al., 22 Jun 2026, Bongole et al., 17 Jan 2026).

1. Interactive setting and quantile-based objective

The method is formulated in the ISDM framework of Chen et al., where an instance is a quadruple $(\mathcal X,\mathcal M,\mathcal D,L)$. Here $\mathcal M$ is the model or environment class, $\mathcal D$ is the admissible algorithm class, $\mathcal X$ is the outcome or transcript space, and $L:\mathcal M\times\mathcal X\to[0,\infty)$ is the loss. For each pair $(M,ALG)$, interaction induces a law $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$. This is the defining departure from passive estimation: the data law itself depends on the adaptive decision rule. In bandits, for example, the final law is the law of the full history $f$0, not merely the law of exogenous observations (Bongole et al., 22 Jun 2026).

The motivation for a high-probability formulation is that the classical minimax criterion

$f$1

controls only the mean of the loss. The recent ISDM quantile literature states this explicitly: minimax risk and regret are expectation-based criteria and do not capture rare but consequential failures, especially in safety-critical bandits, reinforcement learning, and adaptive experimentation. High-probability interactive Fano therefore targets loss quantiles and strict tail events rather than only $f$2 (Bongole et al., 7 Oct 2025).

This shift fits into a broader non-interactive development of minimax quantiles. The non-interactive framework of “High-probability minimax lower bounds” introduced minimax quantiles, high-probability versions of Le Cam and Fano, and a conversion from local minimax risk lower bounds to lower bounds on minimax quantiles. The interactive ISDM results can be read as the adaptive-protocol analogue of that quantile program (Ma et al., 2024).

2. Minimax quantiles and lower minimax quantiles

For $f$3, the $f$4-quantile of the loss under $f$5 is defined by

$f$6

The corresponding strict minimax quantile is

$f$7

while the lower minimax quantile is

$f$8

The lower minimax quantile is the object most directly accessed by Fano- and Le Cam-type converses, because one proves impossibility by showing that every algorithm has tail probability above $f$9 at radius $\{L(M,X)\le \Delta\}$0 (Bongole et al., 22 Jun 2026).

The structural relations are central. For every $\{L(M,X)\le \Delta\}$1,

$\{L(M,X)\le \Delta\}$2

so quantile lower bounds dominate expectation lower bounds. Moreover, for every $\{L(M,X)\le \Delta\}$3 and every $\{L(M,X)\le \Delta\}$4,

$\{L(M,X)\le \Delta\}$5

and consequently $\{L(M,X)\le \Delta\}$6 for all $\{L(M,X)\le \Delta\}$7 except a countable set. This justifies the standard workflow in the interactive literature: prove a lower bound for $\{L(M,X)\le \Delta\}$8, then transfer it to $\{L(M,X)\le \Delta\}$9 (Bongole et al., 22 Jun 2026).

A direct antecedent appears in the non-interactive quantile framework, where the same lower-minimax-versus-strict-minimax relation was established for general statistical models. The interactive theory retains the same architecture but replaces fixed observation laws by transcript laws generated through adaptive interaction (Ma et al., 2024).

3. Core high-probability interactive Fano inequality

The direct ISDM formulation fixes an $(\mathcal X,\mathcal M,\mathcal D,L)$0-divergence $(\mathcal X,\mathcal M,\mathcal D,L)$1, a prior $(\mathcal X,\mathcal M,\mathcal D,L)$2, a threshold $(\mathcal X,\mathcal M,\mathcal D,L)$3, and, for each algorithm $(\mathcal X,\mathcal M,\mathcal D,L)$4, a reference law $(\mathcal X,\mathcal M,\mathcal D,L)$5. It then defines the reference success probability

$(\mathcal X,\mathcal M,\mathcal D,L)$6

and the Bernoulli divergence threshold

$(\mathcal X,\mathcal M,\mathcal D,L)$7

The algorithm-uniform threshold is

$(\mathcal X,\mathcal M,\mathcal D,L)$8

The theorem then states that for every $(\mathcal X,\mathcal M,\mathcal D,L)$9,

$\mathcal M$0

This is the canonical high-probability interactive Fano statement in the recent ISDM literature (Bongole et al., 22 Jun 2026).

Its proof is a Bernoulli reduction. For fixed $\mathcal M$1, one defines

$\mathcal M$2

constructs two joint laws on $\mathcal M$3,

$\mathcal M$4

and compresses the problem through the indicator

$\mathcal M$5

Under $\mathcal M$6, $\mathcal M$7; under $\mathcal M$8, $\mathcal M$9. Data processing yields

$\mathcal D$0

Comparing the right-hand side with $\mathcal D$1 forces $\mathcal D$2, hence

$\mathcal D$3

Uniformizing this over all algorithms gives the lower bound on $\mathcal D$4 (Bongole et al., 22 Jun 2026).

A closely related risk-level-dependent formulation appeared earlier in ISDM. That version defined a uniform threshold $\mathcal D$5 using

$\mathcal D$6

and emphasized a strict-tail calibration: replacing the success event $\mathcal D$7 by the non-strict event $\mathcal D$8 yields lower bounds on $\mathcal D$9, which match the strict quantile definition exactly (Bongole et al., 7 Oct 2025).

4. Bernoulli compression, transform-based extensions, and CVaR

The direct quantile theorem compresses the loss through the hard event $\mathcal X$0. A broader development replaces this hard success bit by a randomized one-bit statistic that encodes an arbitrary bounded transform of the loss. In “Generalizing the Fano inequality further,” the key object is

$\mathcal X$1

where $\mathcal X$2 is bounded measurable. Since

$\mathcal X$3

the Bernoulli mean equals a transformed risk rather than merely a success probability. The resulting theorem states

$\mathcal X$4

where

$\mathcal X$5

Inverting the Bernoulli $\mathcal X$6-ball yields a two-sided interval

$\mathcal X$7

Choosing $\mathcal X$8 recovers the earlier interactive high-probability or tail-probability statement as a special case (Bongole et al., 17 Jan 2026).

The same transform-based framework yields Bayesian prior-predictive CVaR lower bounds. For bounded losses $\mathcal X$9, one sets

$L:\mathcal M\times\mathcal X\to[0,\infty)$0

uses the Rockafellar–Uryasev representation

$L:\mathcal M\times\mathcal X\to[0,\infty)$1

and derives lower bounds on $L:\mathcal M\times\mathcal X\to[0,\infty)$2 from the Bernoulli inversion. In the KL-plus-mixture specialization, the divergence budget becomes mutual information,

$L:\mathcal M\times\mathcal X\to[0,\infty)$3

and Pinsker yields explicit lower bounds through

$L:\mathcal M\times\mathcal X\to[0,\infty)$4

The paper is explicit that this is best understood as a tail-sensitive, transform-based generalization of interactive Fano rather than a standalone high-probability interactive Fano paper in the narrow sense (Bongole et al., 17 Jan 2026).

A later instantiation paper made these abstract CVaR corollaries concrete through a two-point Hellinger–CVaR template. With a hard pair $L:\mathcal M\times\mathcal X\to[0,\infty)$5, a balanced pointwise loss-separation condition $L:\mathcal M\times\mathcal X\to[0,\infty)$6, and a squared Hellinger bound on the interactive laws, it derived explicit Bayesian CVaR lower bounds for Gaussian mean estimation and two-armed Gaussian bandits. This does not output a threshold-specific minimax quantile theorem, but it operationalizes the same Bernoulli inversion architecture in a tail-sensitive form (Bongole et al., 14 Apr 2026).

5. Instantiations in bandits and privacy

The clearest direct use of the interactive Fano theorem is the $L:\mathcal M\times\mathcal X\to[0,\infty)$7-armed Gaussian bandit lower bound. The model class is

$L:\mathcal M\times\mathcal X\to[0,\infty)$8

the outcome is the transcript

$L:\mathcal M\times\mathcal X\to[0,\infty)$9

and the loss is pseudo-regret

$(M,ALG)$0

The Fano construction uses a reference model $(M,ALG)$1 with all means zero and alternatives $(M,ALG)$2 in which only arm $(M,ALG)$3 has mean $(M,ALG)$4. With the uniform prior on $(M,ALG)$5 and threshold

$(M,ALG)$6

success under $(M,ALG)$7 implies $(M,ALG)$8. Under the all-zero reference model, the events $(M,ALG)$9 are disjoint, so the reference success probability satisfies

$\mathbb P^{M,ALG}\in\Delta(\mathcal X)$0

Choosing $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$1, so that $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$2, and taking $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$3, one obtains

$\mathbb P^{M,ALG}\in\Delta(\mathcal X)$4

The resulting theorem is

$\mathbb P^{M,ALG}\in\Delta(\mathcal X)$5

which gives $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$6-type scaling. The paper is explicit that this multi-way Fano construction captures the exploration cost across multiple possible best arms, a feature not available from a two-point method alone (Bongole et al., 22 Jun 2026).

The same ISDM quantile framework also produces two-point lower bounds for Gaussian mean estimation and two-armed Gaussian bandits via high-probability interactive Le Cam. Those bounds scale respectively as $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$7 for squared-error Gaussian mean estimation and $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$8 for two-armed bounded-mean Gaussian bandits. In the two-armed case this is the expected high-confidence rate, but the paper uses Le Cam rather than Fano because two hypotheses suffice (Bongole et al., 22 Jun 2026).

Privacy is incorporated by restricting the admissible class from $\mathbb P^{M,ALG}\in\Delta(\mathcal X)$9 to a private class $f$00. The private high-probability interactive Fano corollary has the same structure as the non-private theorem, but with transcript laws replaced by released-output laws and the supremum taken over private algorithms only. For coordinatewise Gaussian MI privacy, the same paper derives a variance-inflation factor

$f$01

which enters the Le Cam instantiations for Gaussian mean estimation and two-armed Gaussian bandits by weakening distinguishability through increased effective noise (Bongole et al., 22 Jun 2026).

6. Precursors, alternatives, and limitations

Several earlier lines of work anticipate parts of the method. “Distance-based and continuum Fano inequalities” extended classical Fano from exact recovery to events of the form $f$02 and from finite cardinality to a volume ratio

$f$03

thereby providing direct high-probability lower bounds in non-interactive estimation. The logic is model-agnostic once one can upper bound mutual information, so these inequalities are readily portable to interactive settings at the level of endpoint bounds, although they do not themselves analyze adaptive transcripts (Duchi et al., 2013).

“Fano’s inequality for random variables” generalized Fano to arbitrary measurable events and even arbitrary $f$04-valued random variables. Its central Bernoulli reduction

$f$05

is a conceptual precursor to the later transform-based interactive theory. The paper is not a genuine high-probability interactive theorem, but it supplied the bounded-functional viewpoint that later became central in interactive Bernoulli compressions (Gerchinovitz et al., 2017).

A different alternative replaces Fano altogether by binary testing. “A strong converse bound for multiple hypothesis testing” compared the true joint law of parameter and data to an artificial independence law and derived stronger minimax lower bounds, including in active learning where the query strategy depends on the past. The paper explicitly handles an adaptive observation model through

$f$06

and shows that the adaptive query kernel cancels in the Radon–Nikodym derivative when the reference law preserves the querying mechanism. This makes it highly relevant as a high-probability interactive Fano replacement, even though it is not framed as a general interactive Fano theorem (Venkataramanan et al., 2017).

The present literature also distinguishes direct minimax-quantile converses from broader transform-based and Bayesian extensions. The generalized transform paper and its CVaR instantiations produce prior-predictive Bayesian CVaR lower bounds rather than minimax CVaR bounds, and they emphasize that CVaR is obtained through bounded hinge transforms rather than through a threshold-specific tail theorem. The interactive minimax-quantile papers, by contrast, are explicitly strict-tail and $f$07-explicit, but they first lower-bound $f$08 and then invoke the structural relation to $f$09 outside a countable set of confidence levels (Bongole et al., 17 Jan 2026, Bongole et al., 14 Apr 2026, Bongole et al., 22 Jun 2026).

Taken together, these developments define high-probability interactive Fano’s method as a modern converse toolkit for adaptive statistical decision problems: it compresses an interactive loss or transformed loss to a Bernoulli object, bounds the distinguishability of transcript laws from a reference experiment, and converts that bound into a statement about strict tails, minimax quantiles, or, in broader transform-based variants, Bayesian CVaR.