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Wasserstein Contraction of Coordinate Ascent Variational Inference

Wasserstein Contraction of Coordinate Ascent Variational Inference Rocco Caprio†

Adrien Corenflos‡

Sam Power§

May 29, 2026

arXiv:2605.30253v1 [stat.ML] 28 May 2026

Abstract We study the contraction in Wasserstein distance of the coordinate ascent variational inference algorithm. This is shown to hold under a transport-information inequality at the fixed points and a functional smoothness condition. The results are general and sharp, allow for local convergence guarantees, hold for general smooth manifolds, and also in some non-smooth spaces. We consider applications to Bayesian Gaussian Mixture Models, and high-dimensional Bayesian Probit Regression, and Logistic Regression with Polya-Gamma random variables (i.e. Jaakkola-Jordan’s algorithm). Keywords: Bayesian computation, functional inequalities, cross-smoothness, Wasserstein distance, Gaussian mixture models, high-dimensional regression, data augmentation.

1

Introduction

Finding approximations to an intractable probability distribution π of interest (usually known only up to a normalizing constant) is a key problem in scientific computing. Variational Inference stands out as a particularly attractive tool for this task, owing to its statistical and computational efficiency, and it has been the framework underlying many advances in computational statistics over the past half century (Parisi, 1980; Hinton and Van Camp, 1993; Jordan et al., 1999; Bishop and Nasrabadi, 2006). The central idea is to seek a tractable approximation to π within a chosen family of tractable distributions Q by minimizing a divergence to π over that ‘variational’ family. Often, it is convenient or well-motivated to work with the family of product (or tensor, or factorized) distributions Q = P ⊗m , and define optimality through minimisation of the Kullback–Leibler (KL) divergence (also ‘relative entropy’)  min KL(ϱ||π) : ϱ ∈ P ⊗m . A key practical aspect of working with this particular loss function is that in solving the associated optimisation problem, one is only required to compute expectations under the tractable variational distribution ϱ, rather than under the intractable target distribution π. In Bayesian statistics, π typically represents the joint posterior distribution of latent variables z ∈ Z and some parameters β ∈ B given observed data y ∈ Y. In these cases, we often choose m = 2 and seek the best variational approximation µ(dz) ⊗ ν(dβ) to π to solve  min KL(µ ⊗ ν||π) : µ ∈ P(Z), ν ∈ P(B) . The coordinate ascent variational inference algorithm (CAVI, Bishop and Nasrabadi, 2006; Blei et al., 2017) solves this problem by iteratively minimizing the Kullback–Leibler divergence with respect to one element at a time: given a starting point ν0 , it iterates µk := arg min KL(µ ⊗ νk−1 ||π), µ∈P(Z)

νk := arg min KL(µk ⊗ ν||π). ν∈P(B)

† Department of Statistics, University of Warwick. [email protected] ‡ Department of Statistics, University of Warwick. [email protected] § School of Mathematics, University of Bristol. [email protected]

1

Contributions. In this work, we study the contraction, in the sense of the Wasserstein distance, of the coordinate ascent variational inference iterates (µk , νk )k≥0 towards the solution of the variational inference problem, arg min KL(µ ⊗ ν|π). The contraction is established under two conditions: a functional smoothness assumption, and a transport-information inequality at the limiting fixed points of (µk ) and (νk ). The results are general, sharp, and the assumptions are verifiable. The underlying spaces Z and B are allowed to be general smooth Riemannian manifolds, and one of them may also be non-smooth. The analysis allows for multiple limiting points depending on the initialization of the algorithm. In the setting of data augmentation, where Z houses auxiliary variables introduced solely for computational purposes and statistical interest lies only in (νk ), we show that convergence of (νk ) alone can be established under a weaker functional smoothness condition, and a transportinformation inequality on (νk )’s fixed point only. This allows us to avoid making any direct requirement on the nature of Z or the fixed point of (µk ). This is particularly valuable when Z is non-smooth or discrete, as in Gaussian Mixture Models, or when verifying transport-information conditions on (µk )’s fixed point is difficult. We apply our convergence results to study the coordinate ascent variational inference in three different Bayesian models. Unbalanced Gaussian Mixture Models, where we show CAVI’s convergence is governed by a posterior phase transition, which explicitly depends on the posterior modes separation; Bayesian Probit and Logistic Bayesian Regression (the Jaakkola-Jordan’s algorithm), where we also prove the high-dimensional scalings of the obtained rates and bounds. Related works. While coordinate ascent variational inference is a fundamental algorithm in statistics and machine learning, convergence guarantees have only recently started to emerge. The first such guarantee for the two-block setting appears in Bhattacharya et al. (2025), who prove contraction in the symmetrized KL divergence under a novel notion of a generalized correlation coefficient. Shortly after, an alternative line of research adopted an optimizationstyle perspective; under the assumption of strong log-concavity and block-smoothness, Arnese and Lacker (2026) and Lavenant and Zanella (2024) analyse systematic and random-scan CAVI for general m-blocks, and established convergence of the CAVI iterates in relative entropy. The approach based on log-concavity and block-smoothness has since been extended to structured variational inference (Sheng et al., 2025). Some model or purpose-specific analyses of CAVI have also appeared, including Datta et al. (2025) for Bayesian principal component analysis and Pati (2026) for asymptotics of CAVI, with a focus on the comparison between sequential and parallel implementations. Our results are instead phrased in terms of functional inequalities and functional smoothness, and we prove direct contraction in the Wasserstein distance of the coordinate ascent variational inference iterates. We provide a sharp, direct, and short proof of convergence. The connections between our assumptions and the standard functional inequalities literature provide a direct and sharp methods for verifying that they hold in practice. The relaxation to functional inequalitybased assumptions on the fixed points is also crucial for practical applicability: standard models (including the logistic regression and mixture models we consider in this paper) yield target distributions that are not strongly log-concave. Moreover, our results accommodate the possibility of multiple limiting points, local convergence, and analysis in discrete spaces, features which are both excluded by strong log-concavity assumptions and prevalent in practice. On the other hand, our current analysis is restricted to the two-block (m = 2) setting; we leave the multi-block dynamics for future work. Finally, in terms of closely related algorithms, we mention Yao et al. (2024) and Jiang et al. (2025), who analyse the global convergence of some mean-field variational algorithms under a quadratic growth assumption on the relative entropy and strong log-concavity, respectively; as well as Caprio and Johansen (2025) and Ascolani and Zanella (2025), who analyse the global convergence of EM and the Gibbs algorithms, under some novel functional inequalities and strong log-concavity.

2

2

Coordinate Ascent Variational Inference

2.1

Preliminaries

Let π = e−f ∈ P(Z ×B) be a target probability distribution we wish to approximate. Coordinate ascent variational inference attempts to solve the problem min{KL(µ ⊗ ν||π) : µ ∈ P(Z), ν ∈ P(B)} via a coordinate descent algorithm for the relative entropy on the space of factorized, or product, distributions: for a starting distribution ν0 ∈ P(B), µk := arg min KL(µ ⊗ νk−1 ||π), µ∈P(Z)

νk := arg min KL(µk ⊗ ν||π).

(1)

ν∈P(B)

Here, KL denotes the relative entropy functional, or Kullback-Leibler divergence, defined as   Z dϱ (x) ϱ(dx) if ϱ ≪ γ and KL(ϱ||γ) = ∞ otherwise. KL(ϱ||γ) = log dγ We require the following analytical tools and definitions. Let X be a smooth, complete, Riemannian manifold (for instance, Rd , or a closed smooth subset of it), with geodesic distance d. We let Pp (X ) be the space of probability measures on X that are absolutely continuous with respect to the standard volume measure dx on X , and that possess finite p moments. For two probability measures γ, ϱ ∈ Pp (X ), we define the relative Fisher Information functional as Z 2 dϱ (2) ∇ log (x) ϱ(dx) if ϱ ≪ γ and I(ϱ||γ) = ∞ otherwise. I(ϱ||γ) = dγ In the above, |∇f | denotes the norm of the Riemannian gradient. This work leverages the Wasserstein distance and the related geometry of optimal transport (Villani, 2009). For two Riemannian manifolds X and Y and (ϱ, γ) ∈ Pp (X ) × Pp (Y), we denote Γ(ϱ, γ) the set of couplings of ϱ and γ, i.e. of joint distributions on X × Y admitting ϱ and γ as marginals. We define the Wasserstein-p distance on the space of probability distributions Pp (X ): Z Wp (ϱ, γ) := inf

2.2

1/p d(x, y) ρ(dx, dy) : ρ ∈ Γ(ϱ, γ) . p

Wasserstein Contraction

Define the operators µ⋆ [·] : P(B) 7→ P(Z) and ν⋆ [·] : P(Z) 7→ P(B) by

µ⋆ [ν](·) ∝ exp −Eν f (·, β) , ν⋆ [µ](·) ∝ exp −Eµ f (Z, ·) .

(3)

These operators are connected to the optimality conditions of (1): µk = µ⋆ [νk−1 ] and νk = ν⋆ [µk ], and the coordinate ascent variational inference algorithm is equivalently expressed as (ν0 ) 7→ (ν0 , µ⋆ [ν0 ]) 7→ (ν⋆ [µ⋆ [ν0 ]], µ⋆ [ν0 ]) 7→ (ν⋆ [µ⋆ [ν0 ]], µ⋆ [ν⋆ [µ⋆ [ν0 ]]]) . . . We study the contraction, in the sense of the Wasserstein distance, towards fixed points satisfying µ⋆ = µ⋆ [ν⋆ ] = µ⋆ [ν⋆ [µ⋆ [· · · ]]],

µ0

µ1

µ⋆ [·]

ν⋆ [·] µ⋆ [·]

ν0

ν1

ν⋆ = ν⋆ [µ⋆ ] = ν⋆ [µ⋆ [ν⋆ [· · · ]]].

···

···

ν⋆ [·]

µ⋆ µ⋆ [·] ν⋆ [·]

···

···

ν⋆

Figure 1: Illustration of the coordinate ascent variational inference algorithm

3

(4)

In general, these fixed points are not unique, and convergence towards either depends on the initialization ν0 ; we leave this dependence implicit. Our main result says that coordinate ascent variational inference converges rapidly if the optimality maps µ⋆ : P(B) 7→ P(Z) and ν⋆ : P(Z) 7→ P(B) are smooth around the fixed points µ⋆ = µ⋆ [ν⋆ [µ⋆ [. . . ]]] ∈ P(Z), and ν⋆ = ν⋆ [µ⋆ [ν⋆ [. . . ]]] ∈ P(B), and these fixed point distributions satisfy a Transport-Information inequality. Let p ≥ 1 be given. We define smoothness for an operator between probability distributions as follows. Definition (Fisher-smoothness). Let X and Y be smooth Riemannian manifolds. An operator γ⋆ [·] : P(X ) 7→ P(Y) is Fisher-smooth with constant L at B ⊆ P(X ) if, for all ϱ, ϱ′ ∈ B such that ϱ, ϱ′ ≪ γ, (5)

I(γ⋆ [ϱ]∥γ⋆ [ϱ′ ]) ≤ L2 W2p (ϱ, ϱ′ ).

Definition (Transport-Information inequality). Let X be a smooth Riemannian manifold. A probability measure γ ∈ Pp (X ) satisfies a Transport-Information inequality with constant λ if ∀ϱ ∈ B such that ϱ ≪ γ, it holds that (6)

λ2 W2p (ϱ, γ) ≤ I(ϱ∥γ).

Theorem 1. Let Z and B be smooth Riemannian manifolds, and let p ≥ 1. Consider π = e−f ∈ Pp (Z × B) and define B(εµ ) := {µ ∈ Pp (Z) : Wp (µ, µ⋆ ) < εµ } and B(εν ) := {ν ∈ Pp (B) : Wp (ν, ν⋆ ) < εν }. Assume the following. • The optimality maps µ⋆ [·] : P(B) 7→ P(Z) and ν⋆ [·] : P(Z) 7→ P(B) are Fisher-smooth with constants L1|2 , L2|1 on B(εµ ), B(εν ), respectively. • The fixed points µ⋆ := µ⋆ [ν⋆ [µ⋆ [. . . ]]] and ν⋆ := ν⋆ [µ⋆ [ν⋆ [. . . ]]] satisfy a Transport-Information inequality with constants λ1|2 , λ2|1 , respectively. Suppose that L1|2 L2|1 < λ1|2 λ2|1 . Then, for all µk ∈ B(εµ ) and νk ∈ B(εν ), we have Wp (νk+1 , ν⋆ ) ≤

L1|2 L2|1 Wp (νk , ν⋆ ) λ1|2 λ2|1

and

Wp (µk+1 , µ⋆ ) ≤

L1|2 L2|1 Wp (µk , µ⋆ ). λ1|2 λ2|1

In particular, (µk+1 , νk+1 ) ∈ B(εµ ) × B(εν ) and the coordinate ascent variational inference L L2|1 algorithm (1) converges exponentially fast to (µ⋆ , ν⋆ ) in Wasserstein-p distance with rate λ1|2 . 1|2 λ2|1 Proof. Since ν⋆ = ν⋆ [µ⋆ ], νk = ν⋆ [µk ] ∈ B(εν ) and µk ∈ B(εµ ), using the Fisher-smoothness assumptions and the transport-information inequality of ν⋆ , I(µ⋆ [ν⋆ [µk ]]||µ⋆ ) ≤ L21|2 W2p (ν⋆ [µk ], ν⋆ ) ≤

L21|2 λ22|1

I(ν⋆ [µk ]||ν⋆ ) ≤

L22|1 L21|2 λ22|1

W2p (µk , µ⋆ ).

By an application of the Transport-Information inequality for µ⋆ we find W2p (µ⋆ [ν⋆ [µk ]], µ⋆ ) ≤

1 λ21|2

I(µ⋆ [ν⋆ [µk ]]||µ⋆ ) ≤

L21|2 L22|1 λ21|2 λ22|1

W2p (µk , µ⋆ ).

Since, L1|2 L2|1 < λ1|2 λ2|1 , this shows µk+1 = µ⋆ [ν⋆ [µk ]] ∈ B(εµ ) and the first result. Now we can mirror the argument for νk+1 = ν⋆ [µ⋆ [νk ]] and complete the proof. Theorem 1 characterizes the Wasserstein contraction of coordinate ascent variational inference to the limiting fixed point. If the cross-smoothness assumptions (5) and the TransportInformation Inequalities (6) are verified for all (µ0 , ν0 ) ∈ Pp (Z) × Pp (B) towards the same fixed point, the convergence is global. The convergence occurs if the small interaction condition L1|2 L2|1 < λ1|2 λ2|1 holds. This signals that the interaction terms within π (represented by L1|2 L2|1 ) need to be smaller than the curvature at the fixed points (λ1|2 λ2|1 ). For example, in L1|2 L2|1 the Gaussian case, λ1|2 λ2|1 , coincides with the spectral norm of the correlation matrix. It is also possible to consider functional smoothness and transport-information inequalities with more general transport distances beyond metric costs, and weak versions thereof (Gozlan 4

and Léonard, 2010). Moreover, it should be possible to consider multi-block algorithms, i.e. the setting where m > 2. We leave each of these extensions for future work. Our argument for coordinate ascent variational inference also suggests a novel convergence analysis for the standard coordinate descent algorithm under cross-smoothness and gradient quadratic growth conditions, which might be of independent interest; we give a more detailed discussion of this connection in Appendix A.

2.3

On the assumptions

We discuss some sufficient conditions and the interpretation of our Transport-Information and Fisher-smoothness assumptions in the geometry of the optimal transport. Curvature. The transport-information inequalities (6) are curvature conditions on the fixed points µ⋆ := µ⋆ [ν⋆ [µ⋆ [. . . ]]], ν⋆ := ν⋆ [µ⋆ [ν⋆ [. . . ]]] of the optimality operators µ⋆ [·] : P(B) 7→ P(Z) and ν⋆ [·] : P(Z) 7→ P(B). They are weaker than log-Sobolev inequalities (which are themselves weaker than strong log-concavity), and they tensorize, making them suitable for high-dimensional applications. Lemma 2 (Guillin et al. (2009)). Let X be a smooth Riemannian manifold with volume measure dx, and let γ = e−V (x) dx ∈ P(X ), V ∈ C 2 (X ). The following statements are true. • If γ satisfies a log-Sobolev inequality with constant λ: 2λKL(ϱ||γ) ≤ I(ϱ||γ)

∀ϱ ∈ P(X ),

then γ also satisfies the Transport-Information inequality for all p ≤ 2. • In particular, if ∇2 V + Ric ≥ λI, then the Transport-Information inequality is satisfied with constant λ. • Let {Xi }ni=1 be a collection of complete connected Riemannian manifolds with X = ×ni=1 Xi . If γ1 , . . . , γn ∈ P(Xi ) each satisfy a Transport-Information inequality with constants λ1 , . . . , λn , respectively, then γ = ⊗ni=1 γi ∈ P(X ) also satisfies a transport-information inequality, with constant λ = mini∈[n] λi . Remark 1. When X = Rd , in the applications we will work with the geometry induced by a positive definite matrix B ≻ 0, namely |v|2B = v ⊺ Bv. If, in this geometry, V is 1-strongly convex ∇2 V ⪰ B, then λ = 1 and the corresponding Fisher information uses the dual norm Z 2 dϱ (x) ϱ(dx) if ϱ ≪ γ and I(ϱ||γ) = ∞ otherwise. I(ϱ||γ) = B −1/2 ∇ log dγ 2 Smoothness. The Fisher-smoothness condition is a requirement on the optimality operators µ⋆ [·] : P(B) 7→ P(Z) and ν⋆ [·] : P(Z) 7→ P(B) being smooth in functional sense. Notice the analogy with the (log) smoothness of a function h on Rd |∇ log h(x) − ∇ log h(y)| ≤ L|x − y|. Fisher-smoothness may then be regarded as its infinite-dimensional analogue. If we decompose the potential of the target π = e−f as f (z, β) = f1 (z) + f2 (β) + ω(z, β) for some functions f1 : Z 7→ R, f2 : B 7→ R and ω : Z × B 7→ R, then the Fisher-smoothness of µ⋆ [·] and ν⋆ [·] is a requirement only on ω (which represent the interaction terms). In the case p = 1, this condition is related to cross-smoothness of f (or ω) as follows. Recall Kantorovich’s duality formula: for a metric space (X , d), the Lipschitz semi-norm of a mapping h : X 7→ R is   |h(x1 ) − h(x2 )| := : Lip(h) sup x1 ̸= x2 ∈ X , d(x1 , x2 ) and one has the following variational characterization of the Wasserstein-1 distance: Z  W1 (ϱ, γ) = sup h(x)(ϱ − γ)(dx) : Lip(h) ≤ 1 . 5

Lemma 3. Let π = e−f ∈ P1 (Z × B) and let p = 1. The operators µ⋆ [·] : P(B) 7→ P(Z) and ν⋆ [·] : P(Z) 7→ P(B) are Fisher-smooth on P1 (Z) and P1 (B) constants L1|2 := sup{Lipβ (∇z f (z, ·)) : z ∈ Z} and L2|1 := sup{Lipz (∇β f (·, β)) : β ∈ B}, respectively, whenever these quantities are finite. ′ Proof. Consider two arbitrary
ν, ν ∈ P(B). From the optimality conditions (3), we have µ⋆ [ν](z) ∝ exp −Eν [f (z, β)] . Taking the logarithm and computing the gradient with respect to z yields ∇z log µ⋆ [ν](z) = −∇z Eν [f (z, β)]. By the definition of L1|2 , the mapping β 7→ ∇z f (z, β) is L1|2 -Lipschitz. This implies the growth bound ∥∇z f (z, β)∥ ≤ ∥∇z f (z, β0 )∥ + L1|2 d(β, β0 ) for any fixed β0 ∈ B. Since ν ∈ P1 (B), this dominating function is ν-integrable. This justifies passing the derivative inside the integral: ∇z log µ⋆ [ν](z) = −Eν [∇z f (z, β)]. Substituting this into the Fisher Information (2), obtain that h i 2 I(µ⋆ [ν]∥µ⋆ [ν ′ ]) = Eµ⋆ [ν] Eν [∇z f (Z, β)] − Eν ′ [∇z f (Z, β ′ )]  Z  2 ′ = Eµ⋆ [ν] ∇z f (Z, β)(ν − ν )(dβ) .

For any fixed z ∈ Z, the mapping β 7→ ∇z f (z, β) has a uniformly bounded Lipschitz semi-norm, by L1|2 . The result follows by Kantorovich’s duality formula when ν ′ = ν⋆ . The bound for L2|1 follows symmetrically by exchanging the roles of z and β. In other words, when p = 1, (5) are related to Lipschitz assumptions on f ’s gradient in the opposite coordinate: they require ∇1 f to be uniformly Lipschitz in the second coordinate, or ∇2 f to be uniformly Lipschitz in the first. The assumptions of smoothness and curvature at the fixed points only of Theorem 1 should be contrasted to the requirements of gradient descent-type algorithms, where one requires global smoothness and global curvature (Dalalyan, 2016; Durmus and Moulines, 2017; Vempala and Wibisono, 2019; Caprio et al., 2025; Lacker, 2026). In the coordinate ascent variational inference algorithm, we evolve an approximation of π by alternating the operators µ⋆ [·] and ν⋆ [·]. In gradient descent type algorithms such as Langevin Monte Carlo, we evolve an approximation of π by repeated application of one kernel, say K. Somewhat analogously, in CAVI we require the operators µ⋆ [·] and ν⋆ [·] to be smooth, and we require curvature at the limiting points. In an algorithm such as Langevin Monte Carlo, we require the operator K to be smooth, and to exhibit curvature at the limiting point. The smoothness of the optimality operators µ⋆ [·] and ν⋆ [·] is related to cross-smoothness at the level of the potential f , whereas the smoothness of K is related to block smoothness of f . Remark 2 (Log-concavity). The transport-information inequalities for the fixed points are weaker than log-Sobolev inequality assumptions (Lemma 2), which are weaker than strong log-concavity for the fixed points (the Bakry–Émery Theorem), and which are in turn weaker than π = e−f ’s strong log-concavity when the underlying space is Euclidean. In fact, if ∇2 f ⪰ λIdz +dβ , then ∇22 f (z, β) ⪰ λIdβ ⇒ −∇2 log µ⋆ [ν⋆ ](z) = Eν⋆ [∇22 f (z, β)] ⪰ λIdβ implying λ1|2 ≥ λ, and similarly λ2|1 ≥ λ. π’s strong log-concavity assumptions are considered in many other works on CAVI (Arnese and Lacker, 2026; Lavenant and Zanella, 2024; Sheng et al., 2025), and on sampling algorithms (Dalalyan, 2016; Durmus and Moulines, 2017; Chewi, 2026). Other than being a strict and significant relaxation of the log-concavity of π, Theorems 1 and 4 also allow for the possibility of multiple fixed points and more general smooth and non-smooth Riemannian manifolds. Remark 3 (Block-smoothness and cross-smoothness). If f is twice differentiable, when p = 1, L1|2 and L2|1 are finite whenever the off-diagonal elements of f ’s Hessian are bounded, and they coincide with the respective upper bounds (Lemma 3). This should be contrasted with the classical block-smoothness assumptions, considered for CAVI in Arnese and Lacker (2026) and Lavenant and Zanella (2024), which are instead Lipschitz conditions on ∇i f with respect to the same i-th coordinate. These require, instead, the on-diagonal elements of f ’s Hessian to be bounded. Neither of these implies the other. 6

Remark 4 (Accuracy-convergence duality). The conditions under which Theorem 1 establishes the exponential convergence of CAVI also ensure that the product variational approximation to π is of a high quality, and vice-versa. In the case (Z, B) = (R, R), Lacker et al. (2024) shows that 0 ≤ inf KL(µ ⊗ ν|π) ≤ µ⊗ν

1 Eµ ⊗ν [|∂1,2 f |2 + |∂2,1 f |2 ] λ2 ⋆ ⋆

whenever π = e−f is λ-log-concave. As we have argued, L1|2 and L2|1 are upper bounds to |∂1,2 f | and |∂2,1 f |, and when π = e−f is λ-log-concave we always have λ1|2 , λ2|1 ≥ λ. Theorem 1’s rate in the Wasserstein-1 distance can be bounded by κ := (L21|2 + L22|1 )/2λ2 by Young’s inequality. A smaller κ thus implies both a fast CAVI and a better variational approximation. This phenomenon is the accuracy-convergence duality (Goplerud et al., 2025) between the convergence speed of CAVI and the quality of the accuracy of the variational approximation. Goplerud et al. (2025) introduces and identifies this fact for Gaussian distributions. Theorem 1 establishes this more generally. Remark 5 (Wasserstein geometry). The assumptions (5) and (6) under which coordinate ascent variational inference is shown to converge have a natural interpretation in the geometry of the Wasserstein space (Villani, 2009). The coordinate ascent variational inference algorithm minimizes the relative entropy across the factorized distributions (µ, ν) 7→ KL(µ ⊗ ν||π) coordinate per coordinate. The optimality gap satisfies Z KL(µ ⊗ ν||π) − inf KL(µ ⊗ ν||π) = KL(µ||µ⋆ ) + KL(ν||ν⋆ ) + f (z, β)(µ − µ⋆ )(dz)(ν − ν⋆ )(dβ) (Bhattacharya et al., 2025, Lemma 2). Moreover, in the Wasserstein geometry, the squared gradient norm of the relative entropy in the space of factorized distributions (µ ⊗ ν) 7→ KL(µ ⊗ ν||π) is equal to I(µ||µ⋆ [ν]) + I(ν||ν⋆ [µ]) (Lacker, 2026). This is a squared gradient norm of a function with two coordinates, and we think of the summands as the squared norm of the first and second component of the gradient vector, so as |∇f (µ, ν)|2 = |∇1 f (µ, ν)|2 + |∇2 f (µ, ν)|2 . When one coordinate is at the fixed point, say ν = ν⋆ , we observe that the optimality gap is equal to KL(µ||µ⋆ ), and that the norm of the first component of the gradient is I(µ||µ⋆ ). Hence, (6) says that the distance to the fixed point has to decrease at least as fast as the gradient’s first component vanishes: speaking informally, λ21|2 d(µ, µ⋆ )2 ≤ |∇1 f (µ, ν⋆ )|2 , in a sense. On the other hand, the (5) condition is a contraction requirement on the first component of the gradient: at the CAVI iterate µ = µ⋆ [ν], we require that the gradient is Lipschitz with respect to the other coordinate. Both these observations are very natural in light of the novel proof for coordinate descent we give in Appendix A. Example 1 (Gaussian models and sharpness of the rate). Theorem 1’s rates are sharp for Gaussian targets. Consider (Z, B) = (Rdz , Rdβ ) and π = N (0, Q−1 ) with precision matrix   Q11 Q12 Q= Q21 Q22 and which we take to be non-degenerate. The CAVI updates for this model are −1 µ⋆ [ν](dz) = N (dz; Q−1 11 Q12 Eν [β], Q11 ), −1/2

−1 ν⋆ [µ](dβ) = N (dβ; Q−1 22 Q21 Eµ [Z], Q22 ).

(7)

−1/2

Let A := Q11 Q12 Q22 be the correlation matrix associated with π. We aim to apply Theorem 1 with p = 2. It is convenient to work with the norm induced by Q11 on Z and by Q22 on −1/2 −1/2 B. From the CAVI updates equations, we have λ1|2 = λmin (Q11 Q11 Q11 ) = 1 and similarly λ2|1 = 1 in this norm (Lemma 2). The Fisher information functional between two Gaussians ϱ := N (b, B −1 ) and γ := N (c, B −1 ) with the same covariance is equal to I(ϱ∥γ) = ∥B(b − c)∥2 . In particular, since µ⋆ [ν] and µ⋆ have the same covariance, in the norm induced by Q11 , −1/2

I(µ⋆ [ν]∥µ⋆ ) = ∥Q11

1/2

2 ⊺ 2 Q11 Q−1 11 Q12 (Eν [Y ] − Eν⋆ [Y ])∥2 ≤ λmax (A A)∥Q22 (Eν [β] − Eν⋆ [β])∥2

7

Recalling that the Wasserstein-2 distance between a pair of Gaussian measures with same covariances and different means is straightforwardly computable in terms of their difference of 1/2 1/2 means, we obtain that W2 (ν, ν⋆ ) = ∥Q22 (Eν [β] − Eν⋆ [β])∥2 , L1|2 = λmax (A⊺ A), and similarly for L2|1 . Theorem 1 gives a convergence rate in Wasserstein-2 distance equal to λmax (A⊺ A), which is sharp; see e.g. Bhattacharya et al. (2025).

2.4

Data augmentation

Often, π(dz, dβ) represents a joint posterior distribution where z is an auxiliary random vector introduced for the sole purpose of facilitating computation, and the interest lies only in β. This framework is known as data augmentation (Tanner and Wong, 1987; Gelfand and Smith, 1990). In these cases, the following result is helpful. It shows that a smoothness condition for the composed optimality map ν⋆ [µ⋆ [·]] : P(B) 7→ P(B) and the Transport-Information inequality on ν⋆ are all we require to investigate the convergence of (νk ), when that is what we are interested in. Theorem 4. Let B be a smooth Riemannian manifold. For some p ≥ 1, consider π = e−f ∈ Pp (Z × B) and define B(εν ) := {ν ∈ Pp (B) : Wp (ν, ν⋆ ) < εν }. Assume the following. • The composed optimality map ν⋆ [µ⋆ [·]] : P(B) 7→ P(B) is Fisher-smooth with constant L2 on B(εν ). • The fixed point ν⋆ := ν⋆ [µ⋆ [ν⋆ [. . . ]]] satisfies a Transport-Information inequality with constants λ2|1 on B(εν ). Suppose that L2 < λ2|1 . Then, for all νk ∈ B(εν ), we have Wp (νk+1 , ν⋆ ) ≤

L2 Wp (νk , ν⋆ ). λ2|1

In particular, νk+1 ∈ B(εν ) and (νk ) converges exponentially fast to ν⋆ in Wasserstein-p distance 2 with rate λL2|1 . Proof. Applying the transport-information inequality for ν⋆ and then marginal smoothness, λ22|1 W2p (ν⋆ [µ⋆ [νk ]], ν⋆ ) ≤ I(ν⋆ [µ⋆ [νk ]]||ν⋆ ) ≤ L22 W2p (νk , ν⋆ ).

By inspecting the proof of Theorem 1, we notice that the cross-smoothness and the transportinformation inequality for µ⋆ imply Fisher-smoothness of the composed map ν⋆ [µ⋆ [·]] : P(B) 7→ P(B) with L2 = L1|2 L2|1 λ−1 1|2 . Theorem 4 is helpful when Z involves non-smooth Riemannian manifolds, such as discrete spaces, or when the transport-information inequality for µ⋆ does not hold, or it is complicated to verify. We will apply this result to Gaussian Mixture Models, where Z = {0, 1}n is the hypercube, and to Pólya-Gamma augmentations in Bayesian logistic regression. Theorem 4 concludes exponential convergence of (νk ) without requirements on the curvature of the augmentation fixed point µ⋆ . The augmentations z play a role only within the smoothness constant L2 , which captures how much correlated these are with β. Example 2 (Gaussian augmentations and correlation). Suppose, for example, that our interest lies in approximating N (0, Q−1 22 ) alone, and that we are considering the Gaussian augmentation −1/2 −1/2 −1 π = N (0, Q ) of Example 1. Let A := Q11 Q12 Q22 be the correlation matrix associated with π. We work with the norm induced by Q22 on B, so that λ2|1 = 1 also here. From the updates (7), we find −1 −1 −1 Eν⋆ [µ⋆ [ν]] [β] = −Q−1 22 Q21 (−Q11 Q12 Eν [β]) = Q22 Q21 Q11 Q12 Eν [β].

Therefore, from the Fisher information formula between Gaussians, −1/2

I(ν⋆ [µ⋆ [ν]]∥ν⋆ ) = ∥Q22

1/2

−1 2 2 Q22 Q−1 22 Q21 Q11 Q12 (Eν [β] − Eν⋆ [β])∥2 = ∥AQ22 (Eν [β] − Eν⋆ [β])∥2 1/2

≤ λmax (A⊺ A)∥Q22 (Eν [β] − Eν⋆ [β])∥22 , 8

and by Theorem 4, the coordinate ascent variational inference convergence rate for (νk ) is again equal to the spectral norm of the correlation matrix, λmax (A⊺ A). This shows that when considering data augmentation approaches, one should try to minimize the intensity of cross-coupling terms so that λmax (A⊺ A) is as small as possible. This general phenomenon is well-documented in Bayesian computation (Papaspiliopoulos et al., 2007).

3

Applications to some Bayesian models

3.1

Bayesian Gaussian Mixtures

We consider the unbalanced two-component Bayesian Gaussian Mixture Model Yi ∼ pN (β, τ −1 Id ) + (1 − p)N (−β, τ −1 Id ),

i = 1, . . . , n,

(8)

for some given p ∈ (0, 1). The latent auxiliary variables Zi ∈ {0, 1} indicate the mixture components, i.e. Yi |Zi = 1 ∼ N (β, τ −1 Id ) and Yi |Zi = 0 ∼ N (−β, τ −1 Id ). We posit a Gaussian prior on the mean β ∼ N (0; τ0−1 Id ). We focus on this class of simple models to facilitate comparison with some related results which are available for the expectation-maximization (EM) algorithm, and to more cleanly illustrate the arguments. Let z = (z1 , . . . , zn ), y = (y1 , . . . , yn ). We would like to approximate the intractable posterior n n o Y π(z, β) = pδ{zi =1} N (yi ; β, τ −1 Id ) + (1 − p)δ{zi =0} N (yi ; −β, τ −1 Id ) N (β, 0, τ0−1 Id ) i=1

Nn with a product distribution of the form µ(dz) ⊗ ν(dβ) = ( i=1 µi (dzi )) ⊗ ν(dβ) to be fit via the coordinate ascent variational inference algorithm (Bishop and Nasrabadi, 2006, Chapter 10). The CAVI updates for µ consist in updating Bernoulli allocation probabilities and the mean of the Gaussian distribution as n O Ber(ri,ν ), ν⋆ [µ] = N (mµ , (τ0 + nτ )−1 Id ), µ⋆ [ν] = (9) i=1

Pn where logit(ri,ν ) = log(p/(1 − p)) + 2τ yi⊺ Eν [β] and mµ := (τ0 + nτ )−1 τ i=1 Eµi [2Zi − 1]yi . To analyse CAVI’s exponential convergence for this class of models, we aim to apply Theorem 4 with p = 2, noting that our interest only lies in the component β. We emphasise that the auxiliary variables z are only introduced for computational reasons, and are of only secondary interest inferentially. Proposition 5. Suppose that the starting distribution satisfies ν0 ∈ {ν ∈ P2 (B) : W2 (ν, ν⋆ ) < ε}. Then, (νk ) converges exponentially fast in Wasserstein-2 distance with rate rε if ! n   X p τ2 ⊺ ⊺ 2 1 sup λmax sech log + τ yi x yi yi < 1. (10) rε := τ0 + τ n |x−Eν⋆ [β]|≤ε 2 1−p i=1 Proof. We apply Theorem 4. We readily have λ2|1 = nτ + τ0 , so we only need to find L2 . Apply the formula for the Fisher information between Gaussians: I(ν⋆ [µ⋆ [ν]]||ν⋆ ) = (τ n + τ0 )2 |mµ⋆ [ν] − mµ⋆ |2 Since ri,ν = σ(log(p/(1 − p)) + 2τ yi⊺ Eν [β]) and tanh(x) = 2σ(2x) − 1, we have 2ri [ν] − 1 = tanh(0.5 log(p/(1 − p)) + τ yi⊺ Eν [β]), hence I(ν⋆ [µ⋆ [ν]]||ν⋆ ) = |φ(Eν [β]) − φ(Eν⋆ [β]|2 ,

φ(x) := τ

n X i=1

tanh

1 2

log

 p + τ yi⊺ x yi ∈ Rd . 1−p

Computing the Jacobian of φ as Jφ(x) = τ 2

n X i=1

sech2

1 2

log

 p + τ yi⊺ x yi yi⊺ 1−p

and recalling that W22 (ν, ν⋆ ) = |Eν [β] − Eν⋆ [β]|2 for Gaussians, the result follows by the mean value inequality. 9

With these results in tow, we can make the following observations. Consider first the balanced case p = 1/2 for simplicity. • Suppose that ε → ∞, i.e. that we seek to characterize the global convergence of coordinate ascent variational inference. In this case, since 0 < sech2 (τ y ⊺ x) ≤ 1, we obtain n

rε → r :=

X  τ2 λmax yi yi⊺ τ0 + τ n i=1

and CAVI converges globally if r < 1. This is a posterior phase transition criterion. Since 2 2 τ the posterior satisfies p(yi |β) ∝ e− 2 (|yi | +|β| ) cosh(τ yi⊺ β), the log-posterior and its Hessian at β = 0 reads log p(β|y) = −

n nτ + τ0 2 X |β| + log cosh(τ yi⊺ β), 2 i=1

∇2 log p(β|y) = −(nτ + τ0 )Id + τ 2

n X

yi yi⊺

i=1

For β = 0 to be a strict local maximum, one requires indeed r < 1, in which case the posterior is also unimodal. In this case, the posterior landscape is very well-behaved, and CAVI is globally exponentially convergent, but the mixture elements are completely unidentifiable. If the model is well-specified, this happens very rarely as the number of data points n grows: letting β0 denote the true population parameter, in the data-rich limit, the law of large numbers ensures that n   τ2 X yi yi⊺ → λmax (Id + τ ββ ⊺ ) = 1 + τ λmax (β0 β0 ⊺ ) > 1, r = λmax τ0 + τ n i=1 almost-surely. As such, global convergence can only be expected with a large prior precision τ0 at zero, with few data points, or if the model is misspecified. • When r > 1, the symmetric stationary point β = 0 is not a strict local maximum. By symmetry, since the posterior is coercive, there must be a pair of symmetric maximizers. The variational posterior ν⋆ , however, can only be unimodal, and different initializations can converge to different Gaussian approximations, each centred around a different posterior mode. • Suppose now that ε < |Eν⋆ [β]|, that is, the variational posterior mean (or the posterior mode separation) is large, relative to our initial displacement from the optimum. In this case, the sech term is active, and it will force rε < 1 for a sufficiently large posterior mode separation |Eν⋆ [β]| or a sufficiently small ε, leading to local exponential convergence. As the posterior mode separation grows, the initialization can be taken further away. Moreover, the convergence speed of CAVI depends positively on the magnitude of the posterior modes separation, and starting closer yields provably faster convergence. When the precision τ is large, the gap |Eν⋆ [β]| − ε may be larger. • Suppose that the ball {x : |x − Eν⋆ [β]| ≤ ε} contains zero, i.e. that ε ≥ |Eν⋆ [β]|. This corresponds to a situation where we start the algorithm in a neighbourhood of the posterior mode that is so large it contains the saddle point at zero. In this case, rε = r, and fast convergence is possible only when it also happens globally. In the unbalanced general case p ∈ (0, 1), the argument within the sech will always be at least as large as the p = 1/2 case. Because sech decreases away from the origin, coordinate ascent variational inference can be expected to converge more rapidly in the asymmetric case, and the radius of exponential convergence will be wider. Unbalanced mixtures make the labels less ambiguous, facilitate inference, and typically lead to a faster Bayesian computation. Figure 2 illustrates these phenomena numerically. These types of results seem to be novel for coordinate ascent variational inference, and elicit parallels with some of the convergence theory of the expectation-maximization (EM) algorithm for Gaussian Mixture Models (Xu and Jordan, 1996; Balakrishnan et al., 2017; Daskalakis et al., 2017; Weinberger and Bresler, 2022). In this context, r behaves as an empirical, Bayesian, signal-to-noise ratio, and the variational posterior mean Eν⋆ [β] plays the role of the maximum likelihood estimator. 10

Figure 2: Log-Wasserstein-2 convergence of CAVI in Gaussian Mixture Models (d = 10, n = 200, average cross 100 simulations). Left Panel: Local exponential convergence across varying mean separations |Eν⋆ [β]| for a fixed ε and precisions τ = τ0 = 0.1. Greater mean separation accelerates convergence and induces an earlier onset of the exponential phase (characterized by linear trajectories). Right Panel: Local exponential convergence across varying precisions τ for a fixed mean separation and ε. Higher precision similarly yields faster convergence and an earlier transition to the exponential phase.

3.2

Bayesian classification

We assume that the observations are generated according to Yi |β ∼ Ber(Φ(x⊺i β)),

i = 1, . . . , n,

where Φ represents a link function of choice, and with xi indicating the i-th row of the design p×p matrix X ∈ Rn×p . We posit a standard Gaussian prior β ∼ N (m0 , Q−1 0 ), with Q0 ∈ R potentially depending on the design. Denote y = (y1 , . . . , yn ). The posterior distribution π(dβ|y) = N (dβ; m0 , Q−1 0 )

n Y

Φ(x⊺i β)yi (1 − Φ(x⊺i β))1−yi

(11)

i=1

of interest is typically intractable, and hard to approximate directly. A common strategy in Bayesian computation is data augmentation (Tanner and Wong, 1987; Gelfand and Smith, 1990), which introduces a collection of random variables (Zi ) and an augmented posterior distribution π(dz, dβ) admitting (11) as marginal, and that allow for standard Bayesian computation algorithms to be implemented directly. The choice of distribution of the auxiliary random variables (Zi ) used for augmentation depends on the link function. We consider two common choices. Probit Regression. Suppose that Φ is the cumulative distribution function of a Gaussian distribution. In this case, we may consider augmenting our parameter space with latent variables Zi ∼ N (x⊺i β, 1). This allows us to exhibit our posterior of interest as the β-marginal of π(dz, dβ) = N (dβ; m0 , Q−1 0 )N (dz; Xβ, In )

n Y

{yi 1(zi > 0) + (1 − yi )1(zi ≤ 0)}

i=1

(Albert and Chib, 1993). We consider a two-block coordinate ascent variational approach to approximate the augmented posterior. The CAVI updates are available in closed-form, and are given concretely by a Gaussian and a product-of-truncated-Gaussians, with explicit forms µ⋆ [ν](dz) ∝

n O

N (dzi ; x⊺i Eν [β], Ip ){yi 1(zi > 0) + (1 − yi )1(zi ≤ 0)},

i=1

ν⋆ [µ](dβ) = N (dβ; mµ , Q−1 ),

(12) where mµ = Q−1 (Q0 m0 + X ⊺ Eµ [Z]), Q = Q0 + X ⊺ X. We characterize the exponential convergence of CAVI in Bayesian Probit Regression problems (12) in terms of the Bayesian fraction of missing information λmax ((Q0 + X ⊺ X)−1 X ⊺ X) 11

Proposition 6. The iterates (µk , νk ) from the CAVI algorithm (12) for Bayesian Probit Regression converge globally exponentially fast in Wasserstein-2 distance with rate r = λmax ((Q0 + X ⊺ X)−1 X ⊺ X). Proof. This is almost analogous to the Gaussian example (Remark 1), taking p = 2 and working with the norm induced by Q0 + X ⊺ X on B instead. Let A := (Q0 + X ⊺ X)−1/2 X ⊺ , and notice that λmax ((Q0 + X ⊺ X)−1 X ⊺ X) = λmax (A⊺ A). Following the same computations, we find that 1/2 L1|2 = L2|1 = λmax (A⊺ A). Recalling that the log-Sobolev inequality implies the transportinformation inequality (Lemma 2) and that the transport-information inequality tensorizes, the strong log-concavity of µ⋆ [ν] and ν⋆ [µ] yields that λ1|2 = λ2|1 = 1 in this norm. We conclude by application of Theorem 1. The Bayesian fraction of missing information is a ratio of posterior certainty provided by the data X ⊺ X on the total Q0 + X ⊺ X. In particular, if there is little prior information and Q0 → 0 in a suitable sense, then all of the information comes from the data, and CAVI converges slowly as a consequence. Equally, if X ⊺ X is large in a suitable sense, then the data is very informative, and CAVI is slow. We remark that for this specific example, the results of Bhattacharya et al. (2025) are also applicable, and return the same convergence rate. In the random design setting, following the approach in Lee and Zhang (2024) and Ascolani and Zanella (2025) for Gibbs sampling, we can use results from random matrix theory to show the rate stays bounded away from 1 almost surely, even as n, p → ∞, provided that we scale the prior precision Q0 appropriately with p (or, alternatively, the design matrix). We consider two choices of importance in Bayesian statistics: (i) a direct scaling with p, Q0 = (p/c)Ip for some c > 0, common in the context of high-dimensional Bayesian regression (Simpson et al., 2017), and (ii) the generalized g-prior Q0 = (X ⊺ X/g + cIp ) (Zellner, 1986; Ascolani and Zanella, 2025), again for some c > 0. Corollary 7. The following statements hold true. • Let Q0 = (p/c)Ip . Then, r=

(c/p)λmax (X ⊺ X) < 1. 1 + (c/p)λmax (X ⊺ X)

In particular, suppose that Xi,j are i.i.d. with unit variance. If n, p → ∞ and n/p → a ∈ (0, ∞), we have √ c(1 + a)2 √ lim r = <1 1 + c(1 + a)2 • Let Q0 = (X ⊺ X/g + cIp ). Then, r=

λmax (X ⊺ X) < 1. (1 + 1/g)λmax (X ⊺ X) + c

In particular, suppose that Xi,j are i.i.d. with unit variance. If n, p → ∞ and n/p → a ∈ (0, ∞), we have lim r =

1 <1 (1 + 1/g)

Proof. Let Q0 = (p/c)Id and define h(A) := ((p/c)Id + A)−1 A. By the spectral mapping theorem (Higham2008), r = λmax (h(X ⊺ X)) =

max

λ∈σ(X ⊺ X)

h(λ),

(13)

where σ(A) denotes the spectrum of A. Because t 7→ h(t) is increasing, we have r=

λmax (X ⊺ X) (p/c) + λmax (X ⊺ X)

and the first result follows by rearranging this expression. The limiting expression when n, p → √ ∞ and n/p → a ∈ (0, ∞) follows from the fact that λmax (X ⊺ X)/p → (1 + a)2 (Bai and Yin, 1993, Theorem 2). For Q0 = (X ⊺ X/g + cIp ) we follow the same steps. 12

Figure 3: Log-Wasserstein-2 convergence of CAVI in Bayesian Probit Regression with a g-prior (average cross 100 simulations). Left Panel: Convergence behaviour for a fixed g across varying values of (n, p). The rate of convergence is independent of both sample size and dimensionality. Right Panel: Convergence behaviour across varying values of g at a fixed (n, p). The convergence rate exhibits an inverse relationship with the magnitude of g.

Figure 3 illustrates these results numerically. Logistic Regression. Suppose now that Φ(x) is the logistic function. In this case, state-ofthe-art methods consider Pólya–Gamma augmentations Zi ∼ PG(1, x⊺i β) (Polson et al., 2013). The resulting augmented posterior π(dz, dβ) admits a two-block CAVI scheme given by ν⋆ [µ](dβ) = N (dβ; mµ , Q−1 µ ),

µ⋆ [ν](dz)

=

n O

PG(dzi ; 1, ci,ν ),

(14)

i=1

1/2 with ci,ν := (Eν [(x⊺i β)2 ] mµ = Q−1 Qµ = X ⊺ diag(Eµ [Zi ])X + Q0 , m = (X ⊺ (y − µ m, 1/2)+Q0 m0 ). This coincides with the celebrated algorithm given in Jaakkola and Jordan (2000), see Durante and Rigon (2019, Lemma 1) for details on this equivalence. The target distribution π in this case is log-concave, but not strongly so. Nevertheless, our exponential convergence results apply. Recall that E[X] = ϕ(c) := tanh(c/2)(1/2c) when X ∼ PG(1, c) (Polson et al., 2013). Define the second moment matrix of ν⋆ [µ]: S(c) = Q(c)−1 + m(c)m(c)⊺ , where Q(c) = X ⊺ diag(ϕ(ci ))X + Q0 , m(c) = Q(c)−1 m. We have the following convergence result.   q q ⊺ −1 ⊺ −1 := Proposition 8. Let ε > 0, let Ji,ε (ci,⋆ − xi Q⋆ xi ε)+ , (ci,⋆ + xi Q⋆ xi ε) , and define −1/2 Aε := Q⋆ X ⊺ diag

supc∈×ni=1 Ji,ε |ϕ′ (c)|2 x⊺i S(c)xi ϕ(ci,⋆ )

! −1/2

XQ⋆

.

Suppose that the starting distribution satisfies ν0 ∈ B(ε) := {ν ∈ P2 (B) : W2 (ν, ν⋆ ) < ε}. Then, the iterate (νk ) from the CAVI algorithm (14) for Bayesian Logit Regression converges exponentially fast in Wasserstein-2 distance with rate rε if rε := λmax (Aε ) < 1.

(15)

Proof. We apply Theorem 4 with p = 2 throughout, working in the norm induced by Q⋆ on Rp . Log-concavity of ν⋆ and Lemma 2 give the transport-information inequality with constant λ2|1 = 1 in this norm. It remains to bound I(ν⋆ [µ⋆ [ν]]∥ν⋆ ) for ν ∈ B(ε). Write µ := µ⋆ [ν], ∆Q := Qµ − Q⋆ = X ⊺ diag(ϕ(ci,ν ) − ϕ(ci,⋆ ))X, and D∆ := diag(ϕ(ci,ν ) − ϕ(ci,⋆ )). Since Qµ mµ = Q⋆ m⋆ = m, the gradient of log(ν⋆ [µ]/ν⋆ ) at a point x is ∇x log

ν⋆ [µ] (x) = −Qµ (x − mµ ) + Q⋆ (x − m⋆ ) = −∆Qx. ν⋆

13

The relative Fisher information in the Q⋆ -metric is therefore  
−1 I(ν⋆ [µ⋆ [ν]]∥ν⋆ ) = Eν⋆ [µ] β ⊺ ∆QQ−1 ⋆ ∆Qβ = Tr Q⋆ ∆QS(cν )∆Q ⊺ where cν = (c1,ν , . . . , cn,ν ). Applying cyclicity of the trace and then the matrix bound XQ−1 ⋆ X ⪯ −1 D⋆ for D⋆ := diag(ϕ(ci,⋆ )), we obtain that n
X (ϕ(ci,ν ) − ϕ(ci,⋆ ))2 ⊺ ⊺ I(ν⋆ [µ⋆ [ν]]∥ν⋆ ) = Tr D∆ XS(cν )X ⊺ D∆ XQ−1 X ≤ xi S(cν )xi . ⋆ ϕ(ci,⋆ ) i=1

Now recall ci,ν := (Eν [(x⊺i β)2 ])1/2 . For any coupling of ν and ν⋆ , the L2 reverse triangle inequality gives |ci,ν − ci,⋆ | ≤ E[|x⊺i (β − β ′ )|2 ]1/2 . Optimising over couplings and applying the Cauchy–Schwarz inequality with respect to the Q⋆ metric, we obtain that 1/2 1/2 |ci,ν − ci,⋆ | ≤ (x⊺i Q−1 W2 (ν, ν⋆ ) < (x⊺i Q−1 ε, ⋆ xi ) ⋆ xi )

so that ci,ν lies in the interval Ji,ε for all ν ∈ B(ε). By the mean value theorem, ϕ(ci,ν )−ϕ(ci,⋆ ) = ϕ′ (ξi )(ci,ν − ci,⋆ ) for some ξi ∈ Ji,ε . Define then ai :=

|ϕ′ (c)|2 x⊺i S(c) xi , ϕ(ci,⋆ ) c∈×j Jj,ε

−1/2

Aε := Q⋆

sup

−1/2

X ⊺ diag(ai ) XQ⋆

.

Substituting the mean value bound into the preceding bound for I and taking the optimal coupling of ν and ν⋆ , we obtain that I(ν⋆ [µ⋆ [ν]]∥ν⋆ ) ≤

n X

    ai E |x⊺i (β − β ′ )|2 = E (β − β ′ )⊺ X ⊺ diag(ai )X(β − β ′ )

i=1

  ≤ λmax (Aε )E (β − β ′ )⊺ Q⋆ (β − β ′ ) = λmax (Aε )W22 (ν, ν⋆ ) as required. While the statement of Proposition 8 requires an ε-close start, it is also possible to take ε → ∞ to obtain global convergence guarantees. We stated the result with an ε-close start because it is rather easy and clean to study the asymptotic case where ε → 0, which makes clear that the convergence rate in Proposition 8 does not degrade when the number of data points n and/or the dimensionality p is large. In fact, since ⊺ c2i,⋆ = Eν⋆ [(x⊺i β)2 ] = (x⊺i m⋆ )2 + x⊺i Q−1 ⋆ xi = xi S(c⋆ )xi ,

we see that the implied asymptotic convergence rate as ε → 0 is equal to   ϕ′ (ci,⋆ )2 c2   i,⋆ −1/2 −1/2 r⋆ = λmax Q⋆ X ⊺ diag XQ⋆ . ϕ(ci,⋆ )

(16)

To our knowledge, this is the first available convergence rate result for the Jaakkola-Jordan algorithm. The closest result in spirit we are aware of is Lee and Zhang (2024)’s mixing time result for the Gibbs sampler. The following is an analogue of Corollary 7 for logistic regression, and examines two relevant choices for the prior precision in Bayesian statistics: the g-prior, and a direct scaling of the prior precision in p. Corollary 9. The following statements hold true. • Let Q0 = (p/c)Ip . Then, r⋆ ≤

(c/p)λmax (X ⊺ X) < 1. 4 + (c/p)λmax (X ⊺ X)

lim r⋆ ≤

√ c(1 + a)2 √ <1 4 + c(1 + a)2 14

Figure 4: Log-Wasserstein-2 convergence of CAVI in Bayesian Logistic Regression with a g-prior (average cross 100 simulations). Left Panel: Convergence behaviour for a fixed g across varying values of (n, p). The rate of convergence is independent of both sample size and dimensionality. Right Panel: Convergence behaviour across varying values of g at a fixed (n, p). The convergence rate exhibits an inverse relationship with the magnitude of g. The convergence speed of CAVI in logit regression is greater than in probit problems, as predicted by our results.

• Let Q0 = (X ⊺ X/g + cIp ). Then, r⋆ ≤

λmax (X ⊺ X) < 1. (1 + 4/g)λmax (X ⊺ X) + 4c

In particular, suppose that Xi,j are i.i.d. with unit variance. If n, p → ∞ and n/p → a ∈ (0, ∞), we have lim r⋆ ≤

1 <1 (1 + 4/g)

Proof. Let Q0 = (p/c)Id . From (16), use the inequality |ϕ′ (t)t| < ϕ(t) and the expression Q⋆ = X ⊺ diag(ϕ(ci,⋆ ))X + Q0 to obtain that   −1/2 −1/2 r⋆ ≤ λmax Q⋆ X ⊺ diag(ϕ(ci,⋆ ))XQ⋆   = λmax ((c/p)Id + X ⊺ diag(ϕ(ci,⋆ ))X)−1 X ⊺ diag(ϕ(ci,⋆ ))X Similarly to the probit case (13), we apply the spectral mapping Theorem with h(A) := ((c/p)Id + A)−1 A, obtaining that r⋆ ≤

max ⊺

λ∈σ(X diag(ϕ(ci,⋆ ))X)

h(λ) =

λmax (X ⊺ diag(ϕ(ci,⋆ ))X) . λmax (X ⊺ diag(ϕ(ci,⋆ ))X) + (p/c)

Since ϕ(x) ≤ 1/4, we have X ⊺ diag(ϕ(ci,⋆ ))X ⪯ X ⊺ X/4 and the first result readily follows. Similarly to Corollary 7, the limiting expression when n, p → ∞ and n/p → γ ∈ (0, ∞) follows √ from the fact that λmax (X ⊺ X)/p → (1 + γ)2 (Bai and Yin, 1993, Theorem 2). For Q0 = (X ⊺ X/g + cIp ) we apply the same reasoning. Notably, the bounds for CAVI in logit problems suggest that the algorithm is faster to fit binary regression models with logistic link than with probit link; Figure 4 corroborates these findings numerically.

Acknowledgements This work was supported by the ProbAI Hub (EP/Y028783/1). The first author is grateful to Giacomo Zanella for helpful discussions. 15

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A

Coordinate descent

Our proof for the coordinate ascent variational inference algorithm admits an alternating minimization algorithm analogue that seems to be new, seems to work under different assumptions than those considered in the analysis of coordinate descent algorithms literature, and yields sharp rates of convergence. The coordinate descent method is a fundamental optimization algorithm that minimizes an objective function by iterating minimization procedures on the different blocks. Its convergence properties are classical (Ortega and Rheinboldt, 1970; Bertsekas and Tsitsiklis, 1989). More recently, various works have studied non-asymptotic quantitative bounds for convergence. The algorithm with two blocks is known as alternating minimization. Beck and Tetruashvili (2013) establish error bounds for alternating minimization under a global strong convexity assumption on f (although a global Polyak–Łojasiewicz inequality seems to work just as well in the proofs) and block-wise smoothness assumption. Beck (2015) investigates sub-linear convergence under convexity. More recently, Both (2022) improves on Beck and Tetruashvili (2013)’s linear convergence results when considering a quasi-strong convex objective f and block-wise smoothness. Here, we prove linear convergence of alternating minimization, under cross-smoothness and a quadratic growth condition on the gradient at the two fixed points. We also allow for local convergence guarantees. Let X and Y be smooth Riemannian manifolds, and let f : X × Y 7→ R be a given objective function we wish to minimize. Consider the optimality operators x⋆ : Y 7→ X and y⋆ : X 7→ Y given by x⋆ [y] = arg min f (x, y)

y⋆ [x] = arg min f (x, y).

x∈X

y∈Y

For a starting point y0 , consider the coordinate descent algorithm y0 7→ (x⋆ [y0 ], y0 ) 7→ (x⋆ [y0 ], y⋆ [x⋆ [y0 ]]) 7→ ((x⋆ [y⋆ [x⋆ [y0 ]]], y⋆ [x⋆ [y0 ]]) 7→ . . .

(17)

Or, equivalently, iteratively setting xk = x⋆ [yk−1 ] and yk = y⋆ [xk ], xk = arg min f (x, yk−1 ),

yk = arg min f (xk , y).

x∈X

y∈Y

We wish to understand the convergence of this optimization algorithm towards the fixed points x⋆ = x⋆ [ν⋆ [µ⋆ [. . . ]]],

y⋆ = y⋆ [x⋆ [y⋆ [. . . ]]],

which generally depend on the initialization y0 . The following result is the analogue to Theorem 1, and the proof is similar. Let d denote the geodesic distance, ∇ and | · | the gradient and norms, either on X or Y as appropriate. Theorem 10. Let X and Y be smooth Riemannian manifolds. Define B(εx ) := {x ∈ X : d(x, x⋆ ) < εx } and B(εy ) := {y ∈ Y : d(y, y⋆ ) < εy }. Assume the following. • The function f is cross-smooth in B(εx ) × B(εy ): there exist constants L1|2 , L2|1 < ∞ such that, for all x ∈ B(εµ ) and y ∈ B(εν ), |∇1 f (x, y) − ∇1 f (x, y⋆ )|2 ≤ L21|2 d(y, y⋆ )2 ,

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|∇2 f (x, y) − ∇f (x⋆ , y)|2 ≤ L22|1 d(x, x⋆ )2 . (18)

• The optimal value functions x 7→ f (x, y⋆ ) and y 7→ f (x⋆ , y) both satisfy a gradient quadratic growth inequality with respect to the gradient: there exist constants λ1|2 , λ2|1 > 0 such that, for all x ∈ X and y ∈ Y, λ21|2 d(x, x⋆ )2 ≤ |∇1 f (x, y⋆ )|2 ,

λ22|1 d(y, y⋆ )2 ≤ |∇2 f (x⋆ , y)|2 .

(19)

Then, for all x0 ∈ B(εx ) and y0 ∈ B(εy ), we have d(x1 , x⋆ ) ≤

L1|2 L2|1 d(x0 , x⋆ ) λ1|2 λ2|1

and

d(y1 , x⋆ ) ≤

L1|2 L2|1 d(y0 , x⋆ ). λ1|2 λ2|1

In particular, if L1|2 L2|1 < λ1|2 λ2|1 , the coordinate ascent algorithm (17) satisfies (xk , yk ) ∈ B(εx ) × B(εy ) for all k ≥ 1 and converges exponentially fast to (x⋆ , y⋆ ) in geodesic distance. Proof. Because x0 ∈ B(εx ) and y0 ∈ B(εy ), from the cross-smoothness conditions and the gradient quadratic growth inequality of x 7→ f (x, y⋆ ), we have |∇1 f (x⋆ [y⋆ [x0 ]], y⋆ )|2 = |∇1 f (x⋆ [y⋆ [x0 ]], y⋆ ) + ∇1 f (x⋆ [y⋆ [x0 ]], y⋆ [x0 ])|2 ≤ L21|2 d(y⋆ [x0 ], y⋆ )2 ≤ ≤

L21|2 λ22|1

|∇2 f (x⋆ , y⋆ [x0 ])|2 =

L22|1 L21|2 λ22|1

L21|2 λ22|1

|∇2 f (x⋆ , y⋆ [x0 ]) + ∇2 f (x0 , y⋆ [x0 ])|2

d(x0 , x⋆ )

Now the gradient quadratic growth inequality of y 7→ f (x⋆ , y) gives d(x⋆ [y⋆ [x0 ]], x⋆ )2 ≤

1 λ21|2

|∇1 f (x⋆ [y⋆ [x0 ]], y⋆ )|2 ≤

L22|1 L21|2 λ21|2 λ22|1

d(x0 , x⋆ ).

This means that x1 = x⋆ [y⋆ [x0 ]] ∈ B(εx ), and now we can mirror the proof for y1 = y⋆ [x⋆ [y0 ]]. Remark 6. A function f is said to satisfy the Polyak–Łojasiewicz (PŁ) inequality with constant λ > 0 if, on its domain, 2 2λ(f − inf f ) ≤ ∥∇f ∥ . (20) The PŁ inequality is a cornerstone of modern optimization theory (Karimi et al., 2016; Garrigos and Gower, 2023), it is weaker than strong convexity of f , and it implies the inequality λd(u, arg min f )2 ≤ 2(f (u)−inf f ). The gradient quadratic growth (19) are weaker than Polyak– Łojasiewicz (PŁ) inequalities for the value functions x 7→ f (x, y⋆ ) and y 7→ f (x⋆ , y). In fact, these imply (19): λ21|2 d(x, x⋆ ) ≤ 2λ1|2 (f (x, y⋆ ) − inf f ) ≤ |∇1 f (x, y⋆ )|2 and similarly for y 7→ f (x⋆ , y). Moreover, PŁ inequalities for the value functions x 7→ f (x, y⋆ ) and y 7→ f (x⋆ , y) are weaker than f ’s strong convexity. See Remark 2. The gradient quadratic growth condition is also known as an error bound condition (Karimi et al., 2016). Remark 7. It is also possible to allow for either X or Y to be non-smooth, see Theorem 4. Theorem 10 allows to examine linear convergence for objective functions that violate classical strong convexity, PŁ, and block-wise smoothness assumptions (Beck and Tetruashvili, 2013; Both, 2022), while still being cross-smooth and satisfying the quadratic growth to the gradient at the fixed points (19). Following essentially the arguments of Example 1, it is possible to show that for quadratics the rates are sharp.

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B

Some remarks on Transport-Information inequalities

Here, we offer some additional background on transportation-information inequalities (hereafter TI), situating them within the broader landscape of functional inequalities, and giving some pointers to verify them. Throughout, write γ for a fixed probability measure of interest, L for the infinitesimal generator of the associated Langevin diffusion, and {Pt : t ≥ 0} for the associated transition kernels. 1. Within the Bogachev–Kolesnikov hierarchy of geometric functional inequalities ‘of Gaussian type’ (Section 3.5 of bogachev2012monge) TI inequalities lie in between Sobolevtype energy-entropy inequalities and transport(-entropy) inequalities, being quantitatively weaker than the Logarithmic Sobolev inequality and quantitatively stronger than Talagrandtype inequalities (Proposition 2.9.a of Guillin et al. (2009)). 2. Analogously to transportation-entropy inequalities, in the case p = 2, a TI inequality still implies an energy-entropy inequality ‘of exponential type’, i.e. the Poincaré inequality (Proposition 2.9.b of Guillin et al. (2009)). Moreover, for log-concave γ (or γ which deviates from log-concavity sufficiently mildly), the TI inequality even yields a full Logarithmic Sobolev inequality (Proposition 2.9.c of Guillin et al. (2009)). Examples are known which demonstrate that some condition of this form is necessary, i.e. that one can satisfy a TI inequality but not any Logarithmic Sobolev inequality; see Remark 2.6 of guillin2009ii. 3. In terms of verifying TI inequalities ‘from scratch’, a rather crude but general tool is the method of Lyapunov conditions. If γ is known a priori to satisfy a Poincaré inequality, then it is sufficient to exhibit a Lyapunov function W ≥ 1, a reference point x0 ∈ X , and positive constants b, c for which 2

LW ≤ b − c d (x, x0 ) W. Under these conditions, a TI inequality holds for γ (Theorem 5.3 of Guillin et al. (2009)). If the Poincaré inequality is not known a priori for γ, then hope is not lost, as liu2017new shows that it suffices to instead exhibit the stronger estimate   2 LW ≤ b − c d (x, x0 ) W for some W > 0. The same work also establishes that such a condition is necessary for a TI inequality to hold. 4. For quantitative purposes, it often works well to study the metric contraction properties of the Langevin diffusion. In wu2009gradient, it is established that granted e.g.1 hypocontractivity estimates of the form 

∀x, x′ ∈ X , W1 Pt (x, ·) , Pt x′ , · ≤ ℓ(t) · d x, x′ with ℓ : R+ → R+ integrable, one can deduce a TI inequality for γ with p = 1 (Corollary 2.2 of wu2009gradient), and the associated constant can be taken to depend only on R ∥ℓ∥L1 = R+ ℓ(t) dt. As such, provided that the Langevin diffusion is ‘eventually uniformly exponentially contractive-on-average’ in the Euclidean metric, a TI inequality follows. 5. Thus far, the main applications of TI inequalities have apparently been in establishing path-space concentration inequalities for additive functionals along trajectories of the Langevin diffusion, following the developments in, e.g., Theorem 4.1 of Guillin et al. (2009), and subsequently in gao2014bernstein.

1 Actually, wu2009gradient makes the slightly more abstract assumption of a ‘Lipschitzian Spectral Gap’ for L: for some c > 0 and for all sufficiently-regular f , it holds that ∥Lf ∥Lip ≥ c ∥f ∥Lip . It is well-known that this is implied by the condition in the main text, and indeed, it is often proved by exactly these means.

20