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arxiv: 2606.25492 · v1 · pith:G7DQ2PLInew · submitted 2026-06-24 · 🧮 math.ST · stat.TH

Closed-form solutions to some generalized variational inference problems

Hien Duy Nguyen , Jacob Westerhout This is my paper

Pith reviewed 2026-06-25 20:21 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords variational inferencef-divergencesclosed-form solutionsmeasure-valued optimizationKullback-Leibler divergenceRényi divergenceBregman divergence
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The pith

Generalized variational inference admits explicit measure-level solutions for f-divergences, Bregman divergences, and Rényi penalties.

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

The paper studies the unrestricted problem of choosing a probability measure Q to minimize the expected loss under Q plus a scaled divergence from a fixed prior P. It derives explicit formulas for the optimal Q when the divergence belongs to the f-divergence family, the Bregman family between densities, or the Rényi family of order greater than one. These formulas are obtained by reducing the infinite-dimensional problem to a scalar equation through separability of the objective. A reader would care because most generalized variational inference work restricts Q to a parametric family; closed forms remove that restriction for the cases treated.

Core claim

For f-divergence penalties the optimal density satisfies a scalar inverse-gradient formula obtained from a one-dimensional dual identity; the Kullback-Leibler, Cressie-Read, and squared-Hellinger cases are worked out explicitly. Bregman penalties between densities yield solutions containing a scalar mass multiplier. Rényi penalties of order r greater than 1 produce a normalized truncated-power density together with a threshold equation satisfied by every global optimizer. Reverse f-divergences and mixed forward/reverse Kullback-Leibler penalties follow from the same separable-integral reduction.

What carries the argument

The separable integral principle that reduces the measure-valued objective to a scalar equation solved by an inverse-gradient density formula or a one-dimensional dual identity.

If this is right

  • Kullback-Leibler, Cressie-Read, and squared-Hellinger penalties each produce an explicit optimal measure.
  • Bregman penalties between densities admit density-space solutions containing only a scalar mass multiplier.
  • Rényi penalties of order r>1 are solved by a normalized truncated-power density satisfying a threshold equation.
  • Reverse f-divergences and mixed forward/reverse Kullback-Leibler penalties are obtained from the same separable-integral reduction.
  • Finite model-weight formulas and conjugate Bayesian examples realize the closed forms in practice.

Where Pith is reading between the lines

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

  • The closed forms allow direct comparison of inference results across different divergence choices inside the same model without numerical optimization.
  • Whenever a new divergence class satisfies the separability condition, the same reduction technique may supply closed forms.
  • The explicit solutions make it possible to study how the choice of divergence affects posterior concentration without confounding effects from variational approximation error.

Load-bearing premise

The variational objective separates over the underlying space so that the measure-valued problem reduces to a scalar equation via the inverse-gradient or dual identity for the chosen divergence class.

What would settle it

A concrete loss function, prior, and divergence (for example squared-Hellinger) for which the proposed density formula fails to attain the minimum value of the variational objective.

Figures

Figures reproduced from arXiv: 2606.25492 by Hien Duy Nguyen, Jacob Westerhout.

Figure 1
Figure 1. Figure 1: Normal–normal one-observation illustration. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Poisson–gamma one-observation illustration. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
read the original abstract

The Donsker--Varadhan formula characterizes the ordinary Bayesian posterior as the solution of an unrestricted $\mathsf{KL}$-regularized variational problem. Generalized variational inference replaces this regularizer by other divergences, but the resulting measure-valued optimization problem is often studied only after restriction to a parametric variational family. This paper studies the unrestricted measure-level problem. Given a measurable space $(\mathcal{Z},\mathfrak{Z})$, a prior probability measure $P$, a measurable loss $\ell:\mathcal{Z}\to(-\infty,\infty]$, a regularization strength $\alpha>0$, and a divergence $\mathsf{D}(Q\Vert P)$, we seek probability measures in \[ \underset{Q\in\mathcal{P}(\mathcal{Z})}{\mathrm{arg\,min}}\left\{\int_{\mathcal{Z}} \ell\,\mathrm{d}Q+\alpha\mathsf{D}(Q\Vert P)\right\}. \] For $f$-divergence penalties we derive a scalar inverse-gradient density formula and a one-dimensional dual identity; the Kullback--Leibler, Cressie--Read, and squared-Hellinger problems are treated as examples. Reverse $f$-divergences and mixed forward/reverse Kullback--Leibler penalties follow from the same separable integral principle. For Bregman divergences between densities we obtain a density-space solution with a scalar mass multiplier, including least-squares, density-power, and Burg/Itakura--Saito examples. For R\'enyi penalties of order $r>1$ we derive a normalized truncated-power characterization and a threshold equation for every global optimizer. Finite model-weight formulas and simple conjugate Bayesian model illustrations show how these closed forms are realized in practice and differ from the traditional solutions.

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.

Referee Report

2 major / 2 minor

Summary. The paper derives closed-form solutions to the unrestricted measure-valued variational problem min_Q ∫ ℓ dQ + α D(Q||P) for f-divergences (including KL, Cressie-Read, squared-Hellinger), reverse f-divergences, mixed KL penalties, Bregman divergences between densities (least-squares, density-power, Burg), and Rényi penalties of order r>1, yielding explicit scalar formulas such as inverse-gradient densities, dual identities, mass multipliers, and normalized truncated-power characterizations, together with finite-model illustrations that differ from standard posteriors.

Significance. If the derivations are valid under the stated conditions, the explicit closed forms provide exact, non-parametric characterizations of the optimizers for several common regularizers, enabling direct comparison with Bayesian posteriors and exact solutions in conjugate finite-model settings; this strengthens the theoretical foundation of generalized variational inference beyond parametric restrictions.

major comments (2)
  1. [treatment of the unrestricted measure-valued problem (abstract and §3)] The central reductions for f-divergences (scalar inverse-gradient density formula and one-dimensional dual identity) and Bregman cases (density-space solution with scalar mass multiplier) rest on the assumption that the integral objective separates sufficiently to permit reduction to a scalar equation for arbitrary measurable ℓ and general measurable spaces (Z, Z). This separability is invoked without an explicit statement of the required conditions on ℓ (e.g., measurability, lower semicontinuity, or atomicity of the space) that guarantee the global optimizer is captured by the derived scalar multiplier; if the assumption fails for some ℓ, the claimed closed forms would not characterize the argmin.
  2. [Rényi penalties section] §4 (Rényi penalties): the normalized truncated-power characterization and threshold equation are stated to hold for every global optimizer, but the derivation does not address whether multiple solutions to the threshold equation can exist or how to select among them when the loss ℓ induces non-unique mass allocations; this leaves open whether the formula always yields the global minimizer or only candidate points.
minor comments (2)
  1. [Introduction] Notation for the divergence D(Q||P) and the loss ℓ is introduced without an early table or list clarifying the domain and range assumptions (e.g., whether ℓ can take +∞).
  2. [finite model-weight formulas] The finite-model illustrations are presented as differing from traditional solutions, but the explicit numerical comparison (e.g., posterior weights) is only summarized rather than tabulated for the conjugate examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [treatment of the unrestricted measure-valued problem (abstract and §3)] The central reductions for f-divergences (scalar inverse-gradient density formula and one-dimensional dual identity) and Bregman cases (density-space solution with scalar mass multiplier) rest on the assumption that the integral objective separates sufficiently to permit reduction to a scalar equation for arbitrary measurable ℓ and general measurable spaces (Z, Z). This separability is invoked without an explicit statement of the required conditions on ℓ (e.g., measurability, lower semicontinuity, or atomicity of the space) that guarantee the global optimizer is captured by the derived scalar multiplier; if the assumption fails for some ℓ, the claimed closed forms would not characterize the argmin.

    Authors: We agree that an explicit statement of conditions is warranted. The manuscript already assumes ℓ is measurable (as stated in the problem setup), which permits the pointwise separation of the integral objective used in the derivations. However, to guarantee existence of a minimizer and that the scalar formula captures the global argmin in general measurable spaces, lower semicontinuity of the objective functional is typically needed. We will add a dedicated remark in §3 clarifying these conditions and noting the settings (e.g., Polish spaces with continuous ℓ) under which the reductions are rigorous. revision: yes

  2. Referee: [Rényi penalties section] §4 (Rényi penalties): the normalized truncated-power characterization and threshold equation are stated to hold for every global optimizer, but the derivation does not address whether multiple solutions to the threshold equation can exist or how to select among them when the loss ℓ induces non-unique mass allocations; this leaves open whether the formula always yields the global minimizer or only candidate points.

    Authors: The derivation shows that any global optimizer must take the normalized truncated-power form and satisfy the threshold equation. When the threshold equation admits multiple roots, these yield distinct candidate measures; the global minimizer is identified by evaluating the original objective at each candidate and selecting the lowest value. We will revise §4 to state this selection procedure explicitly and note that the formula therefore supplies all candidate points rather than automatically returning the unique global minimizer in every case. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations follow from separability and standard divergence identities without reduction to inputs by construction

full rationale

The paper derives scalar inverse-gradient formulas, dual identities, and normalized characterizations for f-divergences, Bregman divergences, and Rényi penalties by reducing the measure-valued optimization under the assumption that the integral objective separates. These steps invoke standard properties of the divergences (e.g., the form of D(Q||P)) and the separability of the loss integral, without the resulting closed forms being equivalent to fitted parameters or self-defined quantities. No self-citation chains, uniqueness theorems from the authors, or renamings of known results are load-bearing. The central claims remain independent mathematical reductions under the stated assumptions, consistent with the reader's assessment of low circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rest on standard mathematical properties of f-divergences, Bregman divergences, and Rényi divergences (convexity, differentiability, separability of integrals) plus the existence of the argmin over probability measures; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption f-divergences, Bregman divergences, and Rényi divergences admit the separable integral representations and dual identities used to obtain the scalar formulas.
    Invoked to reduce the measure-valued optimization to a one-dimensional problem.
  • domain assumption The loss function ℓ is measurable and the optimization problem over P(Z) admits a solution of the claimed form.
    Required for the argmin to exist and match the derived density or truncated-power expressions.

pith-pipeline@v0.9.1-grok · 5835 in / 1394 out tokens · 25245 ms · 2026-06-25T20:21:05.647615+00:00 · methodology

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