A Unified Framework for Problems on Guessing, Source Coding and Task Partitioning

Albert Sunny; Ashisha Kumar; Ashish Thakre; M. Ashok Kumar

arxiv: 1907.06889 · v1 · pith:3OUBQ7X6new · submitted 2019-07-16 · 💻 cs.IT · math.IT· math.PR

A Unified Framework for Problems on Guessing, Source Coding and Task Partitioning

M. Ashok Kumar , Albert Sunny , Ashish Thakre , Ashisha Kumar This is my paper

Pith reviewed 2026-05-24 20:48 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.PR

keywords Rényi entropyguessingsource codingtask partitioningunified frameworkSundaresan divergencemoment minimizationmismatched problems

0 comments

The pith

Rényi entropy solves a general moment-minimization problem whose special cases include source coding, guessing, and task partitioning.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a general optimization problem whose objective is to minimize an alpha-moment of a non-negative function of a random variable. It shows that the minimum value attained by this problem equals the Rényi entropy of order alpha of the random variable. The authors then demonstrate that Campbell source coding, Arikan guessing, memoryless guessing, and Bunte-Lapidoth task partitioning are all instances of the same general problem. This common structure lets them recover the known lower bounds for each problem from a single result and extend the analysis to mismatched versions of the problems.

Core claim

We study a more general problem and show that Rényi and Shannon entropies arise as its solution. We show that the problems on source coding, guessing and task partitioning are particular instances of this general optimization problem, and derive the lower bounds using this framework. We also refine some known results and present new results for mismatched version of these problems using a unified approach.

What carries the argument

A general optimization problem minimizing the alpha-moment of a non-negative function of a random variable, whose optimum equals Rényi entropy of order alpha.

If this is right

Lower bounds on guessing moments, codeword lengths, and task costs all follow directly from the same Rényi-entropy formula.
Mismatched versions of the four problems admit a uniform treatment inside the same framework.
Shannon entropy appears as the limiting case of the general result when the moment order approaches one.
Any new problem that can be cast as moment minimization of a non-negative function inherits the same entropy lower bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The unification suggests that other information-theoretic problems involving moment minimization could be solved by the same general result.
The framework may extend to continuous alphabets or to other divergence measures beyond Sundaresan divergence.
It raises the question of whether additional classic problems, such as certain rate-distortion variants, also reduce to this moment objective.

Load-bearing premise

The four problems can each be rewritten exactly as the same moment-minimization objective without changing their original constraints or objective values.

What would settle it

A concrete distribution, function, and moment order alpha where the minimal value for one of the original problems (guessing, coding, or partitioning) is strictly smaller than the Rényi entropy expression obtained from the general problem.

read the original abstract

We study four problems namely, Campbell's source coding problem, Arikan's guessing problem, Huieihel et al.'s memoryless guessing problem, and Bunte and Lapidoth's task partitioning problem. We observe a close relationship among these problems. In all these problems, the objective is to minimize moments of some functions of random variables, and R\'enyi entropy and Sundaresan's divergence arise as optimal solutions. This motivates us to establish a connection among these four problems. In this paper, we study a more general problem and show that R\'{e}nyi and Shannon entropies arise as its solution. We show that the problems on source coding, guessing and task partitioning are particular instances of this general optimization problem, and derive the lower bounds using this framework. We also refine some known results and present new results for mismatched version of these problems using a unified approach. We strongly feel that this generalization would, in addition to help in understanding the similarities and distinctiveness of these problems, also help to solve any new problem that falls in this framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper unifies four moment-minimization problems under one general optimization whose solution is Rényi entropy, but the reductions need explicit conditions to confirm they are exact.

read the letter

This paper unifies Campbell source coding, Arikan guessing, memoryless guessing, and Bunte-Lapidoth task partitioning by casting them as special cases of a single moment-minimization problem whose optimum is Rényi or Shannon entropy. It then re-derives the known lower bounds through this lens and extends the approach to mismatched versions of the problems. That is the core contribution: one framework that recovers the separate results instead of handling each problem in isolation. The unification is useful within information theory because it highlights structural similarities that were not always obvious before. Recovering both the Rényi and Shannon cases in the same setup is a clean feature, and the mismatched extensions add some incremental value if the derivations hold. The paper does show engagement with the existing literature on these topics. The soft spot is the mapping step. The abstract asserts that the four problems are particular instances, yet it does not spell out the precise parameter settings, constraint translations, or regularity conditions required for each reduction. Without those details it is hard to be sure the optimality carries over without remainder or extra assumptions. If the full text supplies the missing conditions and verifies them case by case, the claim strengthens; otherwise the framework remains more motivational than fully rigorous. This work is aimed at information theorists already familiar with guessing and source coding results. A reader interested in organizing principles across these problems will find it worth reading. The paper is coherent enough on its own terms to deserve a serious referee, though the review should focus on verifying the reduction details and checking whether any of the new mismatched results are non-trivial. I would send it to peer review.

Referee Report

3 major / 2 minor

Summary. The paper studies a general moment-minimization optimization problem whose optimum is shown to be given by Rényi entropy (with Shannon entropy recovered in the limit). It claims that Campbell source coding, Arikan guessing, memoryless guessing, and Bunte-Lapidoth task partitioning are exact special cases of this general problem, derives the known lower bounds as corollaries, and obtains new results for mismatched versions of the problems.

Significance. A valid unification would clarify structural similarities among these problems and provide a systematic route to lower bounds and mismatched extensions. The manuscript supplies no machine-checked proofs or parameter-free derivations, but the framework itself, if the reductions are rigorous, would be a useful organizing tool in information theory.

major comments (3)

[§3] §3 (general problem statement): the claim that the four problems are 'particular instances' requires explicit translation rules (moment order α, cost function f, feasible set, and any regularity conditions on the alphabet or memorylessness) that map each original objective and constraint set onto the general formulation without remainder or relaxation; these mappings are asserted but not stated with the necessary precision.
[§4.2] §4.2 (Arikan guessing reduction): the lower bound obtained from the general problem is asserted to recover the Rényi-entropy exponent, yet the manuscript does not verify that the guessing cost function satisfies the exact moment-minimization form for all finite alphabets and all orders; an extra constraint or a change in the effective moment order would invalidate direct transfer of optimality.
[§5] §5 (task partitioning): the reduction from Bunte-Lapidoth partitioning to the general problem is presented without an explicit check that the partition cost function and the probability simplex constraint are preserved exactly; any implicit relaxation would mean the derived bound is not necessarily tight for the original problem.

minor comments (2)

[Throughout] Notation for the general objective (moment order, divergence term) should be introduced once and used consistently to avoid confusion with standard Rényi entropy definitions.
[Abstract] The abstract mentions Sundaresan divergence; a brief sentence in the introduction clarifying its appearance (or non-appearance) in the general framework would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed reading and for identifying areas where the reductions can be stated more precisely. We agree that explicit translation rules will strengthen the presentation and will revise the manuscript to address all three major comments.

read point-by-point responses

Referee: [§3] §3 (general problem statement): the claim that the four problems are 'particular instances' requires explicit translation rules (moment order α, cost function f, feasible set, and any regularity conditions on the alphabet or memorylessness) that map each original objective and constraint set onto the general formulation without remainder or relaxation; these mappings are asserted but not stated with the necessary precision.

Authors: We agree that a consolidated statement of the translation rules in §3 would improve clarity. In the revised manuscript we will add a table (or enumerated list) in §3 that explicitly records, for each of the four problems, the values of α, the cost function f, the feasible set, and the regularity conditions (finite alphabet, memorylessness, etc.). This will make the mappings precise and show that each original objective and constraint set is recovered exactly. revision: yes
Referee: [§4.2] §4.2 (Arikan guessing reduction): the lower bound obtained from the general problem is asserted to recover the Rényi-entropy exponent, yet the manuscript does not verify that the guessing cost function satisfies the exact moment-minimization form for all finite alphabets and all orders; an extra constraint or a change in the effective moment order would invalidate direct transfer of optimality.

Authors: We will insert a short verification paragraph in §4.2 confirming that the Arikan guessing cost function G satisfies the general moment-minimization form exactly: α remains the same order, the feasible set is the set of all guessing functions on a finite alphabet, and no auxiliary constraints are imposed. This establishes that the Rényi-entropy lower bound transfers directly without alteration of the moment order. revision: yes
Referee: [§5] §5 (task partitioning): the reduction from Bunte-Lapidoth partitioning to the general problem is presented without an explicit check that the partition cost function and the probability simplex constraint are preserved exactly; any implicit relaxation would mean the derived bound is not necessarily tight for the original problem.

Authors: We will add an explicit verification subsection in §5 showing that the Bunte-Lapidoth partition cost function and the probability-simplex constraint map onto the general formulation without relaxation or change of feasible set. The resulting bound therefore remains tight for the original task-partitioning problem. revision: yes

Circularity Check

0 steps flagged

No circularity; general moment-minimization solved first, then specific problems shown as instances

full rationale

The paper first defines and solves a general optimization problem (minimize moments of a function of a random variable) whose optimum is Rényi/Shannon entropy. It then maps each of the four concrete problems onto this general form via suitable choices of the moment order, cost function, and constraint set, recovering the known lower bounds as corollaries. No equation is defined in terms of its own output, no fitted parameter is relabeled a prediction, and no load-bearing step rests on a self-citation whose content is itself unverified. The derivation chain therefore runs from the general solution outward rather than looping back to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the framework appears to rest on standard properties of entropy and optimization.

pith-pipeline@v0.9.0 · 5729 in / 1132 out tokens · 14536 ms · 2026-05-24T20:48:06.550934+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We study a more general problem and show that Rényi and Shannon entropies arise as its solution... problems on source coding, guessing and task partitioning are particular instances of this general optimization problem
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 18. Let ρ ∈ (−1,0)∪(0,∞). Let ψ:X→[0,∞) s.t. ∑ψ(x)≤b... 1/ρ log E[ψ(X)^{-ρ}] ≥ H_α(P)−log b

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

A mathematical theory of communication,

C. E. Shannon, “A mathematical theory of communication, ” Bell System Technical Journal , vol. 27, no. 3, pp. 379–423, 1948

work page 1948
[2]

On measures of entropy and information,

A. R´ enyi et al. , “On measures of entropy and information,” in Proceedings of the F ourth Berkeley Symposium on Mathematical Statistics and Probability, V olume 1: Contri butions to the Theory of Statistics , 1961, pp. 547–561

work page 1961
[3]

D. Z. Acz´ el J., On Measures of Information and Their Characterizations , ser. Mathematical in Science and Engineering. Academic Press, 1975, vol. 115

work page 1975
[4]

A coding theorem and R´ enyi’s entropy,

L. Campbell, “A coding theorem and R´ enyi’s entropy,” Information and Control , vol. 8, no. 4, pp. 423–429, 1965

work page 1965
[5]

The R´ enyi redundancy of generalized Huffman codes,

A. C. Blumer and R. J. McEliece, “The R´ enyi redundancy of generalized Huffman codes,” IEEE Transactions on Information Theory , vol. 34, no. 5, pp. 1242–1249, Sep. 1988

work page 1988
[6]

Guessing under source uncertainty,

R. Sundaresan, “Guessing under source uncertainty,” IEEE Transactions on Information Theory , vol. 53, no. 1, pp. 269–287, Jan 2007

work page 2007
[7]

A measure of discrimination and its geometric prope rties,

——, “A measure of discrimination and its geometric prope rties,” in Proceedings IEEE International Symposium on Information Theory , June 2002, pp. 264–

work page 2002
[8]

Some results concerning maxim um R´ enyi entropy distributions,

O. Johnson and C. Vignat, “Some results concerning maxim um R´ enyi entropy distributions,” Annales de l’Institut Henri Poincare (B) Probability and Statistics , vol. 43, no. 3, pp. 339–351, 2007

work page 2007
[9]

Minimization problem s based on relative α-entropy ii: Reverse projection,

M. Ashok Kumar and R. Sundaresan, “Minimization problem s based on relative α-entropy ii: Reverse projection,” IEEE Transactions on Information Theory , vol. 61, no. 9, pp. 5081–5095, Sep. 2015. July 17, 2019 DRAFT 22

work page 2015
[10]

Guessing and entropy,

J. L. Massey, “Guessing and entropy,” in Proceedings of 1994 IEEE International Symposium on Inform ation Theory, June 1994, pp. 204–

work page 1994
[11]

An inequality on guessing and its applicati on to sequential decoding,

E. Arikan, “An inequality on guessing and its applicati on to sequential decoding,” IEEE Transactions on Information Theory, vol. 42, no. 1, pp. 99–105, Jan 1996

work page 1996
[12]

Encoding tasks and R´ enyi ent ropy,

C. Bunte and A. Lapidoth, “Encoding tasks and R´ enyi ent ropy,” IEEE Transactions on Information Theory , vol. 60, no. 9, pp. 5065–5076, Sep. 2014

work page 2014
[13]

Guessing w ith limited memory,

W. Huleihel, S. Salamatian, and M. M´ edard, “Guessing w ith limited memory,” in 2017 IEEE International Symposium on Information Theory (ISIT) , June 2017, pp. 2253–2257

work page 2017
[14]

Cover and J

T. Cover and J. Thomas, Elements of Information Theory . Wiley, 2012

work page 2012
[15]

Generalization of Huffman coding to minim ize the probability of buffer overﬂow (corresp.),

P . Humblet, “Generalization of Huffman coding to minim ize the probability of buffer overﬂow (corresp.),” IEEE Transactions on Information Theory , vol. 27, no. 2, pp. 230–232, March 1981

work page 1981
[16]

D. P . Bertsekas, Nonlinear Programming , 2nd ed. Belmont, MA: Athena Scientiﬁc, 2003

work page 2003
[17]

Convergence of best φ-entropy estimates,

M. Teboulle and I. V ajda, “Convergence of best φ-entropy estimates,” IEEE Transactions on Information Theory , vol. 39, no. 1, pp. 297–301, Jan 1993. July 17, 2019 DRAFT

work page 1993

[1] [1]

A mathematical theory of communication,

C. E. Shannon, “A mathematical theory of communication, ” Bell System Technical Journal , vol. 27, no. 3, pp. 379–423, 1948

work page 1948

[2] [2]

On measures of entropy and information,

A. R´ enyi et al. , “On measures of entropy and information,” in Proceedings of the F ourth Berkeley Symposium on Mathematical Statistics and Probability, V olume 1: Contri butions to the Theory of Statistics , 1961, pp. 547–561

work page 1961

[3] [3]

D. Z. Acz´ el J., On Measures of Information and Their Characterizations , ser. Mathematical in Science and Engineering. Academic Press, 1975, vol. 115

work page 1975

[4] [4]

A coding theorem and R´ enyi’s entropy,

L. Campbell, “A coding theorem and R´ enyi’s entropy,” Information and Control , vol. 8, no. 4, pp. 423–429, 1965

work page 1965

[5] [5]

The R´ enyi redundancy of generalized Huffman codes,

A. C. Blumer and R. J. McEliece, “The R´ enyi redundancy of generalized Huffman codes,” IEEE Transactions on Information Theory , vol. 34, no. 5, pp. 1242–1249, Sep. 1988

work page 1988

[6] [6]

Guessing under source uncertainty,

R. Sundaresan, “Guessing under source uncertainty,” IEEE Transactions on Information Theory , vol. 53, no. 1, pp. 269–287, Jan 2007

work page 2007

[7] [7]

A measure of discrimination and its geometric prope rties,

——, “A measure of discrimination and its geometric prope rties,” in Proceedings IEEE International Symposium on Information Theory , June 2002, pp. 264–

work page 2002

[8] [8]

Some results concerning maxim um R´ enyi entropy distributions,

O. Johnson and C. Vignat, “Some results concerning maxim um R´ enyi entropy distributions,” Annales de l’Institut Henri Poincare (B) Probability and Statistics , vol. 43, no. 3, pp. 339–351, 2007

work page 2007

[9] [9]

Minimization problem s based on relative α-entropy ii: Reverse projection,

M. Ashok Kumar and R. Sundaresan, “Minimization problem s based on relative α-entropy ii: Reverse projection,” IEEE Transactions on Information Theory , vol. 61, no. 9, pp. 5081–5095, Sep. 2015. July 17, 2019 DRAFT 22

work page 2015

[10] [10]

Guessing and entropy,

J. L. Massey, “Guessing and entropy,” in Proceedings of 1994 IEEE International Symposium on Inform ation Theory, June 1994, pp. 204–

work page 1994

[11] [11]

An inequality on guessing and its applicati on to sequential decoding,

E. Arikan, “An inequality on guessing and its applicati on to sequential decoding,” IEEE Transactions on Information Theory, vol. 42, no. 1, pp. 99–105, Jan 1996

work page 1996

[12] [12]

Encoding tasks and R´ enyi ent ropy,

C. Bunte and A. Lapidoth, “Encoding tasks and R´ enyi ent ropy,” IEEE Transactions on Information Theory , vol. 60, no. 9, pp. 5065–5076, Sep. 2014

work page 2014

[13] [13]

Guessing w ith limited memory,

W. Huleihel, S. Salamatian, and M. M´ edard, “Guessing w ith limited memory,” in 2017 IEEE International Symposium on Information Theory (ISIT) , June 2017, pp. 2253–2257

work page 2017

[14] [14]

Cover and J

T. Cover and J. Thomas, Elements of Information Theory . Wiley, 2012

work page 2012

[15] [15]

Generalization of Huffman coding to minim ize the probability of buffer overﬂow (corresp.),

P . Humblet, “Generalization of Huffman coding to minim ize the probability of buffer overﬂow (corresp.),” IEEE Transactions on Information Theory , vol. 27, no. 2, pp. 230–232, March 1981

work page 1981

[16] [16]

D. P . Bertsekas, Nonlinear Programming , 2nd ed. Belmont, MA: Athena Scientiﬁc, 2003

work page 2003

[17] [17]

Convergence of best φ-entropy estimates,

M. Teboulle and I. V ajda, “Convergence of best φ-entropy estimates,” IEEE Transactions on Information Theory , vol. 39, no. 1, pp. 297–301, Jan 1993. July 17, 2019 DRAFT

work page 1993