[Paper Review] Dual Representation of Minimum Divergence Under Integral Constraints

Shubhanshu Shekhar, Shubhada Agrawal|arXiv (Cornell University)|Mar 22, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

The paper develops a two-stage discretization-based method to derive dual representations for minimum divergence under integral constraints for distributions on [0,1]^K, extending KL_inf and f-divergences with applications to sequential inference.

ABSTRACT

Minimum divergence problems under integral constraints appear throughout statistics and probability, including sequential inference, bandit theory, and distributionally robust optimization. In many such settings, dual representations are the key step that convert information-theoretic lower bounds into computationally tractable (and often near-optimal) algorithms. In this paper, we present a general two-stage recipe for deriving dual representations of constrained minimum divergence (in the second argument) for distributions supported on $[0,1]^K$. The first stage derives a dual representation for finitely-supported distributions using classical finite-dimensional convex duality techniques, while the second establishes an abstract interchange argument that lifts this discretized dual to arbitrary distributions. We begin with the simplest case of mean-constrained minimum relative entropy, commonly called $\mathrm{KL}_{\inf}$, and generalize an existing argument from multi-armed bandits literature for $K=1$ to arbitrary dimensions. Our main contribution is to significantly expand the scope of this approach to a broad class of $f$-divergences (beyond relative entropy) and to general integral constraint functionals (beyond the mean constraint). Finally, we illustrate the statistical implications of our results by constructing optimal procedures for sequential testing, estimation, and change detection with observations in $[0,1]^K$.

Motivation & Objective

• Develop dual representations for minimum divergence in the second argument under integral constraints for distributions on [0,1]^K.
• Extend duality results from KL_inf to a broad class of f-divergences and general integral constraint functionals.
• Provide a constructive discretization-based pipeline and lifting argument to general distributions via limiting arguments.
• Apply the resulting dual representations to sequential inference problems such as testing, estimation, and change detection.

Proposed method

• Derive duality for finitely supported distributions using classical convex duality.
• Introduce a two-stage discretization pipeline: (i) discretize to finite support with exact constraint satisfaction, (ii) lift the dual to arbitrary distributions via a limiting argument leveraging DPI and lower semicontinuity.
• Extend the framework beyond relative entropy to a general class of f-divergences through an abstract limiting argument.
• Use mean-preserving discretization channels to preserve mean constraints exactly in the KL_inf setting, and generalize to approximate constraint satisfaction for general constraints.
• Establish conditions under which discretized duals converge to the continuous-dual and formulate an abstract dual representation for I(P,g,C) under these conditions.
• Discuss statistical applications to sequential testing, estimation, and change detection with [0,1]^K-valued observations.

Experimental results

Research questions

• RQ1 How can we obtain a dual representation for I(P,g,C) for distributions on [0,1]^K under integral constraints?
• RQ2 Can a two-stage discretization approach (finite-support duals followed by a limiting argument) extend to general f-divergences and constraints beyond the mean?
• RQ3 What verifiable conditions ensure convergence of discretized duals to the continuous dual and allow lifting from finite to general distributions?
• RQ4 What are the resulting dual forms for KL_inf with general integral constraints and for other divergences like Hellinger and Chi-squared?
• RQ5 How can these dual representations inform optimal procedures for sequential inference tasks on [0,1]^K data?

Key findings

• A dual representation for KL_inf is derived in the finite-support case and extended to general distributions via a limiting argument.
• A mean-preserving discretization channel is introduced to preserve mean constraints exactly in discretizations.
• The two-stage approach is extended from KL_inf to a broader class of f-divergences and to general continuous constraints.
• An abstract dual representation framework is developed for I(P,g,C) under mild regularity and continuity assumptions, enabling broader applicability.
• The results are connected to sequential inference tasks, illustrating optimal procedures for testing, estimation, and change detection with [0,1]^K observations.