A Convergent Primal-Dual Algorithm for Computing Rate-Distortion-Perception Functions

Chunhui Chen; Hao Wu; Linyi Chen; Xueyan Niu

REVIEW 5 major objections 5 minor 33 references

A Convergent Primal-Dual Algorithm for Computing Rate-Distortion-Perception Functions

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper establishes an $O(1/n)$ convergent primal-dual algorithm for the information rate-distortion-perception function under KL and entropy-regularized Wasserstein perception measures.

desk verdict The outer-loop O(1/n) result is a real advance for an exact alternating scheme, but the paper overstates it as a guarantee for the implemented algorithms, and the Wasserstein pseudocode has a marginal-constraint bug. read the letter →

arxiv 2508.13486 v1 pith:PYBYBAHX submitted 2025-08-19 cs.IT math.IT

Chunhui Chen , Linyi Chen , Xueyan Niu , Hao Wu This is my paper · ORCID

classification cs.ITmath.IT MSC 65K0590C2594A34

keywords rate-distortion-perceptionfunctionoptimaltransportWassersteindistanceKLdivergenceprimal-dualalgorithmO(1/n)convergenceentropicregularizationalternatingminimization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper targets the information rate-distortion-perception (RDP) function, a three-way trade-off among compression rate, distortion, and perceptual similarity of reconstructions. Its central move is to relax the hard constraint on the reconstruction distribution and instead alternate optimization over the channel and over that distribution directly, a reformulation that preserves the optimal value (Theorem 1). The main claim is that the resulting primal-dual algorithm, named IAS-II, converges in objective value at rate $O(1/n)$ for both KL-divergence perception and entropy-regularized Wasserstein perception. If true, this supplies the first rigorous convergence-rate guarantee for computing RDP functions, turning a problem previously handled by heuristic adversarial training or empirically convergent optimal-transport schemes into one with a worst-case accuracy certificate.

What carries the argument

The load-bearing mechanism is the alternating minimization of a convex dual potential built from the channel, the reconstruction marginal, and (in the Wasserstein case) an auxiliary coupling $\Pi$. The key object is the exact marginal update $r_j=\sum_i p_i w_{ij}$, which gives the reconstruction distribution in closed form and turns the perceptual constraint into an optimization variable. For the Wasserstein case, the entropy term $\varepsilon H(\Pi)$ makes the inner problem strictly convex, and entropic optimal transport convergence justifies taking $\varepsilon\to0$. The convergence proof rests on Lemmas 2 and 6, which show that each outer step decreases the objective by at least a marginal KL divergence, and on monotonicity (Lemmas 4 and 8); a telescoping sum converts those decreases into the $O(1/n)$ bounds of Theorems 3 and 5.

What would settle it

Under the paper's exact-inner-solve assumption, take a Bernoulli source with $p=0.1$, Hamming distortion, and total-variation perception, compute the RDP value by exhaustive enumeration over small alphabets, and run the algorithm with increasingly exact inner solves and a range of outer iterations; the gap $f(w_n,r_n)-f(w^*,r^*)$ should follow a $C/n$ envelope. A plateau above that envelope would contradict the $O(1/n)$ claim; separately, comparing the entropy-regularized Wasserstein output to the unregularized value as $\varepsilon\to0$ tests Theorem 5.

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Extended reading notes

Core claim

The paper's central claim is that the discrete information RDP problem (2) can be solved by a relaxed primal-dual algorithm with a provable objective-error rate. The perceptual constraint $d(p,r)\le P$ is replaced by an optimization over the reconstruction marginal $r$, so the problem alternates between updating the channel $w$ and updating $r$; Theorem 1 states that the optimum is unchanged. Under KL perception, Theorem 3 shows $f(w_n,r_n)-f(w^*,r^*)=O(1/n)$ when each inner subproblem is solved exactly. Under entropy-regularized Wasserstein perception, Theorem 5 gives $g(w_n,\Pi_n,r_n)-g(w^*,\Pi^*,r^*)\le \varepsilon C/((1+\varepsilon)^n-1)$, and choosing $\varepsilon$ optimally for a fixed $n$ keeps the rate $O(1/n)$. The reconstruction marginal $r_n$ is always the output marginal $\sum_i p_i w_{n,i}$, so it is updated in closed form in both cases.

Load-bearing premise

Every $O(1/n)$ theorem assumes the inner optimization is solved exactly at each outer step, while the implemented algorithms only run a finite inner loop with no stated stopping criterion, so the proof does not directly cover the practical code.

Editorial extensions

If this is right

For KL-divergence perception, the objective gap $f(w_n,r_n)-f(w^*,r^*)$ decays as $O(1/n)$ with exact inner solves, so the number of outer iterations needed for a fixed accuracy grows linearly in the inverse tolerance.
For entropy-regularized Wasserstein perception, the bound $g(w_n,\Pi_n,r_n)-g(w^*,\Pi^*,r^*)\le \varepsilon C/((1+\varepsilon)^n-1)$ holds, and choosing $\varepsilon$ with $n$ gives the same $O(1/n)$ rate while $\varepsilon\to0$ recovers the original Wasserstein RDP value.
Because the relaxation preserves the optimal value (Theorem 1), the computed curve is the information RDP function itself rather than an approximation of a nearby problem.
Total-variation perception is a special case of the Wasserstein formulation with cost $c_{ij}=\mathbf{1}_{i\neq j}$, so the $O(1/n)$ guarantee also covers TV perception.
In the reported experiments, IAS-II matches the accuracy of the earlier IAS algorithm while running faster, with speed-ups from about 1.2x to 13x in the reported settings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An unstated but natural corollary is that the outer-rate proof would transfer to any perception measure whose reconstruction marginal can be updated in closed form from the channel; the paper does not test that generalization.
The theorems assume exact inner solves, so a practical stopping rule for the inner updates would be needed to turn the $O(1/n)$ bound into a runtime guarantee; the paper leaves that unspecified.
A concrete testable extension would be a schedule $\varepsilon=\Theta(1/n)$ in the Wasserstein algorithm; the bound then gives an explicit iteration count for a target tolerance, but the paper does not derive such a schedule or report its empirical cost.
The constants in the bounds inherit the KL divergence $D(r^*\|r_1)$ and may grow with alphabet size; analyzing that dependence would decide whether the rate guarantee is useful for high-dimensional discretized sources.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this one with the stress-test concern in mind: it is real. The paper has a genuine new idea—relaxing the RDP constraint and proving an outer-loop O(1/n) rate for an exact alternating scheme—but the convergence theorem does not cover the algorithm as implemented. The abstract overstates the guarantee.

What is actually new: Theorem 1 gives an equivalent (or entropy-regularized) reformulation that avoids the non-convex perceptual constraint, and Lemmas 2–3 plus Theorem 3 give a clean telescoping argument for the KL case. The Wasserstein analogue in Theorem 5 is a genuine extension, with an epsilon-dependent factor that still yields O(1/n). The proof is self-contained convex analysis; the rate is not fitted, and there is no circularity beyond relying on the authors' prior CBA machinery. The numerical tables show consistent speedups over IAS, though only against their own predecessor and with no code shipped.

Soft spots, in order of severity. First, the load-bearing gap: Theorems 3 and 5 define wtilde(r) as an exact minimizer of the inner problem, while Algorithms 1 and 2 run a finite inner loop with a vaguely specified stopping rule and no rate. The telescoping argument does not control the quantity actually computed. If inner suboptimality contributes (1/n) times the sum of per-step errors, the paper needs either an inner complexity bound or a tolerance schedule. This must be fixed before the O(1/n) claim can be accepted. Second, Algorithm 2's update for what the text calls phi-hat_i should be p_i divided by the sum of exponentials, not one divided by that sum, because the constraint is sum_j Pi_ij = p_i. As written, the beta update does not enforce the source marginal. This looks like a typo, since the derivation in the text has the right form, but the pseudocode will not reproduce the theory. Third, the entropy-regularization bias is not quantified: Theorem 5 is for the regularized problem, and the citation to entropic OT convergence supplies no rate or explicit bound back to the original RDP value. Fourth, smaller issues: the proof of Theorem 1 has a P/D typo, there are repeated 'Kulback' and indexing typos, and no reproducibility details are included.

The underlying approach is sound and the result is plausibly repairable. I would send this to serious peer review and require major revision. I would not cite the O(1/n) claim as it currently stands.

Referee Report

5 major / 5 minor

Summary. The paper proposes two primal-dual algorithms, IAS-II for KL-divergence perception and for entropy-regularized Wasserstein perception, to compute the information rate-distortion-perception (RDP) function. The authors reformulate the RDP problem by optimizing over the reconstruction distribution rather than imposing hard perceptual constraints, derive explicit dual updates for the resulting subproblems, and claim an O(1/n) convergence rate to the global optimum in both cases. Numerical comparisons against the authors' earlier IAS algorithm on binary and Gaussian sources report speed-ups and small objective differences. The central theoretical claims, however, are proved only for an idealized alternating-minimization scheme with exact inner solves and, in the Wasserstein case, for the entropy-regularized objective rather than the original RDP problem.

Significance. If the O(1/n) claim were fully established, this would be a notable advance: a rigorous, non-asymptotic convergence guarantee for RDP computation would close a well-recognized gap, and the relaxation idea of optimizing over the reconstruction marginal is a useful structural contribution. The outer-loop telescoping argument is elegant, the monotone decrease lemmas are self-contained, and the experimental comparisons are a practical strength. However, the advertised guarantee is currently not supported for the algorithms as implemented, for the specific reasons detailed in the major comments. The paper therefore provides a promising framework whose central theorem needs substantial additional work before the result can be considered established.

major comments (5)

[Section 4 (Theorems 3 and 5), Algorithms 1 and 2] The O(1/n) proofs assume that each outer step uses exact inner minimizers w_n = \tilde w(r_n) and Pi_n = \tilde Pi(r_n), as defined by problems (6) and (15). Both algorithms, however, run an inner 'while the inner iteration is sufficient' loop with no tolerance, no stopping rule, and no inner-loop convergence rate. If the inner solve at outer step k is only delta_k-suboptimal, the proof acquires an additional term (1/n) sum_k delta_k; the paper supplies no mechanism, such as delta_k <= O(1/k^2), to make this term vanish. Consequently the telescoping arguments in Theorems 3 and 5 control an idealized quantity that is not what Algorithms 1 and 2 return. The authors must either prove an inner-loop rate with an explicit stopping criterion that makes the accumulated inner error vanish in the outer bound, or explicitly restate the main claim as a rate for an exact-arithmetic alternating-minimization scheme.
[Section 3.2, Algorithm 2 line 6] The update \hat phi_i <- 1 / sum_j exp(-beta_j/epsilon - lambda c_ij/epsilon) does not enforce the source marginal. Since Pi_ij = exp(-beta_j/epsilon - lambda c_ij/epsilon) \hat phi_i, this update normalizes sum_j Pi_ij to 1, whereas constraint (15c) requires sum_j Pi_ij = p_i. The correct update is \hat phi_i <- p_i / sum_j exp(-beta_j/epsilon - lambda c_ij/epsilon). As written, even a perfectly converged inner loop in Algorithm 2 does not solve the inner problem (15), so Lemma 6 and Theorem 5 cannot be applied to Algorithm 2 without correcting this error.
[Section 3.2, Eq. (13), and Theorem 5] The Wasserstein convergence result is stated for the entropy-regularized problem (13) with a fixed regularization coefficient epsilon, not for the original Wasserstein RDP problem (12)/(2). The paragraph citing [20] asserts only qualitative convergence as epsilon goes to 0; no quantitative bound of the form |g_epsilon^* - g_0^*| <= rho(epsilon) is derived. The experiments fix epsilon = 0.01, yet the paper presents the computed values as the RDP function. Without a quantified bias in epsilon, the numerical results in Tables 1 and 2 are for an unquantified approximation of the quantity that the paper claims to compute.
[Section 4.1, Lemma 2 and Theorem 3] The KL convergence proof assumes that \tilde w(r) is a global minimizer of subproblem (6) and that it admits the representation (8) in terms of Lagrange multipliers. However, constraint (6d), sum_j p_j log(sum_i p_i w_ij) >= T, is a lower bound on a concave function, so the feasible set of (6) is generally nonconvex. The paper does not establish convexity of (6), nor KKT sufficiency, nor strong duality for this nonconvex problem. Without such an argument, the fundamental inequality in Lemma 2 that drives Theorem 3 is not justified, and Theorem 3 cannot be read as a proof of global convergence to f(w*, r*).
[Section 4.1, Theorem 2, and Section 4.2, Theorem 4] The proofs that the inner loops converge are incomplete as stated. The authors assert that the implemented updates coincide with alternating minimization of F1 and F2, but no detailed correspondence is given, and the derivation in Theorem 2 contains an apparent error: the b-update is written with log(-sum_j r_j e^{...}), which has a negative argument and is undefined. In addition, the Newton root-finding steps for a_j, gamma, and lambda are stated without local or global convergence analysis. Since the inner convergence is a premise for Theorems 3 and 5, this is a load-bearing gap: the inner loop as pseudocoded is not yet proven to converge to the exact inner minimizer.

minor comments (5)

[Section 3.2, proof of Theorem 1] In the first line of the proof, the inequality W(p, r*) <= sum_{i,j} Pi*_{ij} c_{ij} <= D should read <= P, because the perception constraint is (12e), not the distortion constraint. The typo is confusing in a proof that is otherwise about feasibility.
[Throughout, in particular Lemmas 2, 3, and Theorem 3] The symbol D is used both for the distortion budget and for the KL divergence D(. || .). These two uses appear in the same equations and should be renamed (for example, use an uppercase script or a different letter for the divergence).
[Section 4, Theorems 3 and 5] The notation 'simeq O(1/n)' in Theorem 3 and 'O(1/n)' in Theorem 5 should be replaced by a precise big-O statement with an explicit constant, since the proof actually yields a bound with a finite constant C depending on the initial KL divergence and, in the Wasserstein case, a bound of the form epsilon C/((1+epsilon)^n - 1).
[Section 5, Eq. (19)] The Gaussian discretization parameters S and delta are fixed without a convergence study; the paper does not report how the computed RDP value depends on S and delta or whether the results are in the asymptotic regime of the discretization.
[Algorithms 1 and 2] The output lines return sum_{i,j} w_ij p_i (log w_ij - log r_j), which is the objective value rather than 'Minimal value'; this wording should be clarified so that the reader knows whether the reported quantity is the objective at the final iterate or a computed lower/upper bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the O(1/n) convergence proof is a self-contained convex-optimization derivation; the identified discrepancies are proof gaps, not circular reductions.

full rationale

The paper's central claim is an O(1/n) convergence rate for the IAS-II primal-dual algorithm. The proof chain (Lemmas 2-4 and 6-8, Theorems 3 and 5) is an explicit alternating-minimization argument: it defines wtilde(r) and Pitilde(r) as exact minimizers of inner subproblems (6) and (15), proves a per-step decrease lower bound using convexity and KL divergence inequalities, and telescopes these decreases to obtain the rate. No parameter is fitted to the target value, no empirical quantity is renamed as a prediction, and the rate is not assumed as an input. The self-citations to the authors' prior CBA and WBM-RDP/IAS work [7,10] are used as design antecedents; the key inequalities and telescoping are proved in the present paper and do not reduce to those citations. The main weaknesses are correctness gaps rather than circularity: Theorem 3 applies Lemma 2 under equality constraints that a global optimizer need not satisfy, and Theorems 3 and 5 require exact inner solves while Algorithms 1 and 2 use an unspecified 'while the inner iteration is sufficient' loop with no tolerance or inner-rate control. These are mismatches between the idealized analyzed scheme and the implemented algorithm, not cases where a claimed prediction is equivalent to its inputs by definition. No circular step can be quoted from the paper.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central proof rests on exact inner solves, KKT existence, convergence of alternating minimization, root-finding by Newton's method, and entropic OT convergence. The only hand-chosen numerical constant is epsilon, plus discretization parameters for the Gaussian experiment. No new physical or mathematical entities are postulated.

free parameters (2)

Entropy regularization coefficient epsilon = 0.01
Chosen by hand for the Wasserstein experiments. It controls the gap between problems (13) and (12), so the reported O(1/n) applies to a regularized objective and the error due to epsilon is not quantified.
Gaussian discretization parameters S and delta = S=8, delta=0.5
The continuous Gaussian source is truncated to [-S,S] and discretized with spacing delta in Eq. (19). The induced approximation error in the RDP value is not analyzed.

assumptions (6)

domain assumption The inner subproblems can be solved exactly at each outer iteration, with wtilde(r) and Pitilde(r) being exact minimizers.
Theorems 3 and 5 bound outer iterations using the exact inner optimal solution. The algorithms only run a finite inner loop with no convergence rate or stopping criterion.
domain assumption Lagrange multipliers a, b, lambda, gamma and alpha, alphahat, beta, lambda, gamma exist and satisfy the KKT conditions for the inner problems.
Constraint qualification is not stated in the paper. The update formulas and Lemma 2 and Lemma 6 assume optimal dual variables exist.
standard math Alternating minimization of a convex function converges to its global minimum.
Used in Theorems 2 and 4 to conclude that the inner iterations converge. No rate or regularity conditions are given.
standard math Newton's method finds the unique positive roots of f_j, g, G, and F.
The algorithm relies on monotonicity for g, G, and F, but the root-finding for f_j is asserted without a proof of global convergence.
standard math Entropic optimal transport solutions converge to the unregularized solution as epsilon goes to 0.
Invoked via reference [20] to justify that problem (13) approximates (12). No rate in epsilon is provided.
domain assumption For the KL case, the source and reconstruction alphabets coincide, making p_j log r_j a valid KL divergence.
Equation (4d) sums p_j log(sum_i p_i w_ij) over the same index j. The paper does not state that M=N in Section 3.1.

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Pith. "Pith review of A Convergent Primal-Dual Algorithm for Computing Rate-Distortion-Perception Functions." pith.science (2026). https://pith.science/paper/PYBYBAHX

@misc{pith2026250813486,
  author       = {Pith},
  title        = {Pith review of: A Convergent Primal-Dual Algorithm for Computing Rate-Distortion-Perception Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PYBYBAHX}},
  note         = {Machine review of arXiv:2508.13486}
}

abstract

Recent advances in Rate-Distortion-Perception (RDP) theory highlight the importance of balancing compression level, reconstruction quality, and perceptual fidelity. While previous work has explored numerical approaches to approximate the information RDP function, the lack of theoretical guarantees remains a major limitation, especially in the presence of complex perceptual constraints that introduce non-convexity and computational intractability. Inspired by our previous constrained Blahut-Arimoto algorithm for solving the rate-distortion function, in this paper, we present a new theoretical framework for computing the information RDP function by relaxing the constraint on the reconstruction distribution and replacing it with an alternative optimization approach over the reconstruction distribution itself. This reformulation significantly simplifies the optimization and enables a rigorous proof of convergence. Based on this formulation, we develop a novel primal-dual algorithm with provable convergence guarantees. Our analysis establishes, for the first time, a rigorous convergence rate of $O(1/n)$ for the computation of RDP functions. The proposed method not only bridges a key theoretical gap in the existing literature but also achieves competitive empirical performance in representative settings. These results lay the groundwork for more reliable and interpretable optimization in RDP-constrained compression systems. Experimental results demonstrate the efficiency and accuracy of the proposed algorithm.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [20]

Probability Theory and Related Fields 184(1), 401–424 (2022)

Nutz, M., Wiesel, J.: Entropic Optimal Transport: Convergence of Potentials. Probability Theory and Related Fields 184(1), 401–424 (2022)

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