Title: Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference
Venue: NeurIPS 2025
Accepted: yes
Keywords: variational inference

variational Bayes

mean-field approximation

stochastic optimization

SGD

Bayesian inference
Abstract: We prove that, given a mean-field location-scale variational family, black-box variational inference (BBVI) with the reparametrization gradient converges at a rate that is nearly independent of explicit dimension dependence. Specifically, for a $d$-dimensional strongly log-concave and log-smooth target, the number of iterations for BBVI with a sub-Gaussian family to obtain a solution $\epsilon$-close to the global optimum has a dimension dependence of $\mathrm{O}(\log d)$. This is a significant improvement over the $\mathrm{O}(d)$ dependence of full-rank location-scale families. For heavy-tailed families, we prove a weaker $\mathrm{O}(d^{2/k})$ dependence, where $k$ is the number of finite moments of the family. Additionally, if the Hessian of the target log-density is constant, the complexity is free of any explicit dimension dependence. We also prove that our bound on the gradient variance, which is key to our result, cannot be improved using only spectral bounds on the Hessian of the target log-density.

Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference

Abstract

Weprove that, given a mean-field location-scale variational family, black-box variational inference (BBVI) with the reparametrization gradient converges at a rate that is nearly independent of any explicit dimension dependence. Specifically, for a d -dimensional strongly log-concave and log-smooth target, the number of iterations for BBVI with a sub-Gaussian family to obtain a solution ϵ -close to the global optimum has an explicit dimension dependence no larger than O(log d ) . This is a significant improvement over the O( d ) dependence of full-rank locationscale families. For heavy-tailed families, we prove a weaker O( d 2 /k ) dependence, where k is the number of finite moments of the family. Additionally, if the Hessian of the target log-density is constant, the complexity is free of any explicit dimension dependence. We also prove that our bound on the gradient variance, which is key to our result, cannot be improved using only spectral bounds on the Hessian of the target log-density.

1 Introduction

Variational inference (VI; Blei et al., 2017; Hinton and van Camp, 1993; Jordan et al., 1999; Peterson and Hartman, 1989) is an effective method for approximating intractable high-dimensional distributions and models with tall datasets. Among various VI algorithms, black-box VI (BBVI; Kucukelbir et al., 2017; Ranganath et al., 2014; Titsias and Lázaro-Gredilla, 2014; Wingate and Weber, 2013), which minimizes the exclusive KL divergence (Kullback and Leibler, 1951) via stochastic gradient descent (SGD; Bottou et al., 2018; Robbins and Monro, 1951) in the space of parameters, is widely used due to its flexibility to apply to a wide range of variational families with only minor modifications (Bingham et al., 2019; Carpenter et al., 2017; Fjelde et al., 2025; Ge et al., 2018; Patil et al., 2010). Specifically, location-scale variational families-in which a base distribution is mutated by an affine transformation-remain a popular choice, encompassing those with diagonal scale matrices (the 'mean-field' approximation; Hinton and van Camp, 1993; Peterson and Hartman, 1989), as well as scale matrices with low rank (Ong et al., 2018; Rezende et al., 2014; Tomczak et al., 2020) and full-rank (Kucukelbir et al., 2017; Titsias and Lázaro-Gredilla, 2014) factors.

The choice of the variational family is generally known to affect the convergence speed of BBVI, where families that are more 'expressive,' those that contain more complex distributions, result in slower convergence. For example, in location-scale families, it has been empirically observed that mean-field families often provide faster convergence to an accurate posterior approximation than full-rank families (Agrawal et al., 2020; Giordano et al., 2018, 2024; Ko et al., 2024; Zhang et al., 2022). This is because full-rank families often require running SGD with a smaller step size and

for longer; even given a large computation budget, BBVI on a full-rank family may not converge adequately (Ko et al., 2024). Therefore, choosing the expressiveness of the family corresponds to trading statistical accuracy for computational efficiency (Bhatia et al., 2022). In order to control this trade-off for our benefit, a clear theoretical understanding of the relationship between convergence speed and expressiveness is needed.

Formally, consider the setting of approximating a µ -strongly log-concave and L -log-smooth target, where κ ≜ L/µ is the condition number. For BBVI with the reparametrization gradient (Kingma and Welling, 2014; Rezende et al., 2014; Titsias and Lázaro-Gredilla, 2014) on a full-rank locationscale family, an ϵ -close solution to the global optimum in squared distance in parameter space can be obtained after at least O( dκ 2 ϵ -1 ) iterations (Domke, 2019; Kim et al., 2023a). For mean-field location-scale families, on the other hand, the iteration complexity improves to O( √ d κ 2 ϵ -1 ) (Kim et al., 2023a). While this is clearly better than the O( d ) explicit dimension dependence of full-rank families, it has been conjectured that a better dependence is more likely (Kim et al., 2023a).

In this work, we positively resolve this conjecture by obtaining stronger convergence guarantees for BBVI on mean-field location-scale families (Section 3). In particular, under the conditions stated above, we prove that BBVI with a mean-field location-scale family with sub-Gaussian tails can obtain an ϵ -accurate solution in squared distance after O((log d ) κ 2 ϵ -1 ) iterations. Heavier-tailed families achieve a weaker O( d 2 /k κ 2 ϵ -1 ) iteration complexity guarantee, where k is the number of finite moments of the variational family. For the Studentt variational family with a high-enough degrees of freedom ν , this corresponds to a O( d 2 / ( ν -2) ) explicit dimension dependence. In addition, if the Hessian of the target log-density is constant, any mean-field location-scale family attains a O( κ 2 ϵ -1 ) iteration complexity without any explicit dependence on d .

The key element of the proof is a careful probabilistic analysis of the variance of the reparametrization gradient (Section 4): In general, the reparametrization gradient of the scale parameters contains heavy-tailed components that grow not-so-slowly in d . However, for mean-field families, only a single random coordinate turns out to be heavy-tailed. Through a probabilistic decomposition, the influence of this heavy-tailed component can be averaged out over all d coordinates. Then the lighter-tailed components of the gradient dominate as d increases, resulting in a benign dimension dependence (Lemma 4.1). We also provide a lower bound (Proposition 4.2) showing that our analysis cannot be improved when using only spectral bounds on the Hessian of the target log-density.

2 Preliminaries

Notation We denote random variables in sans serif ( e.g. , u , U ). S d ≻ 0 ⊂ R d × d denotes the set of d × d positive definite (PD) matrices, D d ⊂ R d × d denotes the set of diagonal matrices, and D d ≻ 0 ⊂ D d ∩ S d × d ≻ 0 is its positive definite subset. 〈· , ·〉 and ‖·‖ 2 denote the Euclidean inner product and norm. For a matrix A ∈ R d × d , ‖ A ‖ F = √ tr( A ⊤ A ) is the Frobenius norm, ‖ A ‖ 2 = σ max ( A ) is the ℓ 2 operator norm, where σ max ( · ) and σ min ( · ) are the largest and smallest singular values.

2.1 Problem Setup

Our problem of interest is an optimization problem over some space Λ ⊆ R p of the form of

ℓ : R d → R is a measurable function we refer to as the 'target function', h : Λ → R is a potentially non-smooth convex regularizer, and the expectation E z ∼ q λ ℓ ( z ) is assumed to be intractable.

BBVI is a special case of Eq. (1) where ℓ = -log π is the negative (unnormalized) log-density of some distribution π with respect to the Lebesgue measure and h ( λ ) = -H [ q λ ] is the negative differential entropy of q λ . Then F is the exclusive Kullback-Leibler divergence D KL (Kullback and Leibler, 1951) up to an additive constant (Jordan et al., 1999), where Eq. (1) reduces to

We assume π is supported on R d , which, unless discrete-valued variables are involved, is often valid after appropriate support transformations (Kim et al., 2023a, §2.2). Such a setup for BBVI has been proposed by Kucukelbir et al. (2017), and now encompasses most practical use of BBVI with the

reparametrization gradient as implemented in Stan (Carpenter et al., 2017), PyMC (Patil et al., 2010), Pyro (Bingham et al., 2019), and Turing (Fjelde et al., 2025; Ge et al., 2018).

For the purpose of a quantitative theoretical analysis, we will consider the following properties:

Definition (Smoothness) . For some ϕ : R d → R , we say ϕ is L-(Lipschitz )smooth if there exists some L ∈ (0 , + ∞ ) such that, for all z, z ′ ∈ R d ,

Definition (Strong Convexity) . For some ϕ : R d → R , we say ϕ is µ -strongly convex if there exists some constant µ ∈ (0 , L ] such that, for all z, z ′ ∈ R d ,

In the context of BBVI, assuming that ℓ = -log π is both µ -strongly convex and L -smooth is equivalent to assuming π is µ -strongly log-concave and L -log-Lipschitz smooth, respectively, which is common in the analysis of MCMC (Chewi, 2024) and VI (Arnese and Lacker, 2024; Diao et al., 2023; Domke et al., 2023; Kim et al., 2023a; Lambert et al., 2022; Lavenant and Zanella, 2024).

2.2 Variational Family

We consider the location-scale family (Casella and Berger, 2001, §3.5):

Definition 2.1 (Location-Scale Variational Family) . A family of distributions Q is referred to as a location-scale variational family if there exists some univariate distribution φ dominated by the Lebesgue measure such that each member of Q indexed by λ = ( m,C ) ∈ R d ×C , where C ⊂ R d × d and q λ ∈ Q , satisfies where

and d = is equivalence in distribution. Then T λ is referred to as the 'reparametrization function, ' while m and C are referred to as the location and scale parameters, respectively.

In addition, we impose mild regularity assumptions on the moments of the base distribution:

Assumption 2.2. φ satisfies the following: (i) It is standardized such that E u i = 0 and E u 2 i = 1 , (ii) symmetric such that E u 3 i = 0 , and (iii) its kurtosis is finite such that E u 4 i = r 4 < ∞ .

The location-scale family with Assumption 2.2 encompasses many variational families used in practice, such as Gaussians, Studentt with a high-enough degrees of freedom ν , Laplace, and so on, and enables the use of the reparametrization gradient.

While the choice of φ gives control over the tail behavior of the family, the choice of the structure of the scale matrix C gives control over how much correlation between coordinates of ℓ the variational approximation can represent. This ability to represent correlations is often referred to as the 'expressiveness' of a variational family, where the most expressive choice is the following:

Definition 2.3 (Full-Rank Location-Scale Family) . We say Q is a full-rank location-scale family if it satisfies Definition 2.1 and, for any C ∈ C , C is invertible and the squared C s, CC ⊤ , span the whole space of dense R d × d positive definite matrices as { CC ⊤ | C ∈ C} = S d ≻ 0 .

Typically, full-rank location-scale families are formed by setting C to be the set of invertible triangular matrices (the 'Cholesky factor parametrization'; Kucukelbir et al., 2017; Titsias and LázaroGredilla, 2014) or the set of symmetric square roots (Domke, 2020; Domke et al., 2023). Adding further restrictions on C forms various subsets of the broader location-scale family. In this work, we focus on the case where C ∈ C is restricted to be diagonal such that C ⊂ D d , which is known as the mean-field approximation (Hinton and van Camp, 1993; Peterson and Hartman, 1989):

Definition 2.4 (Mean-Field Location-Scale Family) . We say Q is a mean-field location-scale family if it satisfies Definition 2.1 and all C ∈ C are diagonal such that C ⊂ D d .

2.3 Algorithm Setup

Recall that BBVI is essentially SGD in the space of parameters of the variational distribution. Therefore, we have to define the space of parameters. For this, we use the 'linear' parametrization:

Under this parametrization, the desirable properties of ℓ easily transfer to f . For instance, if ℓ is µ -strongly convex and L -smooth, f is also µ -strongly convex and L -smooth (Domke, 2020). This contrasts with 'non-linear parametrizations' commonly used in practice, such as making the diagonal positive by C ii = exp( λ C ii ) . Such practice rules out transfer of strong convexity and smoothness (Kim et al., 2023a) unless constraints such as C ii ≥ δ for some δ > 0 are enforced (Hotti et al., 2024). (Though they can sometimes be beneficial by reducing gradient variance; Hotti et al., 2024; Kim et al., 2023b.) The flip side of using the linear parametrization is that we must now enforce the constraint C glyph[follows] 0 . Furthermore, h then becomes non-smooth with respect to C :

h ( λ ) = -H ( q λ ) = -log | det C | -d H ( φ ) = -∑ d i =1 log | C ii | -d H ( φ ) . (4) This corresponds to log-barrier functions (Parikh and Boyd, 2014, §6.7.5), which are non-smooth. Thus, the optimization algorithm must somehow deal with these difficulties (Domke, 2020).

In this work, we will rely on the proximal variant of stochastic gradient descent (SGD; Bottou, 1999; Bottou et al., 2018; Nemirovski et al., 2009; Robbins and Monro, 1951; Shalev-Shwartz et al., 2011), often referred to as stochastic proximal gradient descent (SPGD; Nemirovski et al., 2009). Proximal methods are a family of methods that rely on proximal operators (Parikh and Boyd, 2014), which are well defined as long as the following hold:

Assumption 2.5. h : Λ → R ∪ { + ∞} is convex, bounded below, and lower semi-continuous.

The non-smoothness of h and the domain constraint are handled by the proximal operator while the intractability of f is handled through stochastic estimates of ∇ f (Definition 2.6). For a step size schedule ( γ t ) t ≥ 0 , ̂ ∇ f , an unbiased estimator of ∇ f ( λ t ) = E ̂ ∇ f ( λ t ; u ) , and a sequence of i.i.d. noise ( u t ) t ≥ 0 , for each t ≥ 0 , SPGD iterates

λ t +1 = prox γ t h ( λ t -γ t ̂ ∇ f ( λ ; u t ) ) . In the case of BBVI with a mean-field location-scale family, the proximal operator of Eq. (4) is identical to that of log-barrier functions (Parikh and Boyd, 2014, §6.7.5):

Instead of using SPGD, one can also use projected SGD, where C is projected to a subset where F is smooth (Domke, 2020) and use the 'closed-form entropy' gradient ̂ ∇ F ≜ ̂ ∇ f + ∇ h (Kucukelbir et al., 2017; Titsias and Lázaro-Gredilla, 2014). However, the resulting theoretical guarantees are indistinguishable (Domke et al., 2023), and the need for setting a closed domain of C is inconvenient. Therefore, we only consider SPGD. But our results can easily be applied to projected SGD.

For ̂ ∇ f , we will use the classic reparametrization gradient (Ho and Cao, 1983; Rubinstein, 1992): Definition 2.6 (Reparametrization Gradient) . For a differentiable function ℓ : R d → R , is an unbiased estimator of ∇ f such that ∇ λ E z ∼ q λ ℓ ( z ) = ∇ f ( λ ) .

The reparametrization gradient, also known as the push-in gradient or pathwise gradient, was introduced to VI by Kingma and Welling (2014); Rezende et al. (2014); Titsias and Lázaro-Gredilla (2014). (See also the reviews by Glasserman 1991; Mohamed et al. 2020; Pflug 1996.) It is empirically observed to outperform alternatives (Kucukelbir et al., 2017; Mohamed et al., 2020) such as the score gradient (Glynn, 1990; Williams, 1992) and de facto standard whenever ℓ is differentiable. (Though theoretical evidence of this superiority is limited to the quadratic setting; Xu et al., 2019.)

2.4 General Analysis of Stochastic Proximal Gradient Descent

Analyzing the convergence of BBVI corresponds to analyzing the convergence of SPGD (or more broadly, of SGD) for the class of problems that corresponds to BBVI. For this, we will first discuss sufficient conditions for the convergence of SPGD and the resulting consequences.

Assumption 2.7 (Lipschitz Gradients in Expectation) . There exists some constant L ∈ [0 , ∞ ) such that, for all λ, λ ′ ∈ Λ ,

Assumption 2.8 (Bounded Variance) . There exists some constant σ ∈ [0 , ∞ ) such that, for all λ ∗ ∈ arg min λ ∈ Λ F ( λ ) ,

E ‖ ̂ ∇ f ( λ ∗ ; u ) ‖ 2 2 ≤ σ 2 . Both assumptions were initially used by Bach and Moulines (2011, Assumptions H2 and H4) to analyze the convergence of SGD. Here, Assumption 2.7 serves as an analog of L -smoothness, and thus determines the largest stepsize we can use. The strategy of combining Assumptions 2.7 and 2.8 is referred to as 'variance transfer' (Garrigos and Gower, 2023, §4.3.3). Previously, for analyzing BBVI, a slightly different assumption called quadratically-bounded variance (QV)-which assumes the existence of α, β ∈ [0 , + ∞ ) such that, for all λ ∈ Λ , E ‖ ̂ ∇ f ( λ ; u ) ‖ 2 2 ≤ α ‖ λ -λ ∗ ‖ 2 2 + β holdshas been commonly used (Domke, 2019; Domke et al., 2023; Kim et al., 2024b). While similar, our assumptions result in a constant-factor improvement in the resulting bounds.

For the analysis, we will use a two-stage step size schedule (Gower et al., 2019, Theorem 3.2):

This operates by first maintaining a fixed step size γ 0 until some switching time t ∗ ∈ { 0 , . . . , T } , and then switches to the 1 /t schedule of Lacoste-Julien et al. (2012) with an offset τ ≥ 0 .

Under Assumptions 2.7 and 2.8, we can now provide a complexity guarantee for solving Eq. (1) via SPGD. Since BBVI consists of a subset of Eq. (1), establishing Assumptions 2.7 and 2.8 and invoking the following result will constitute our complexity guarantee for BBVI.

Proposition 2.9. Suppose f is µ -strongly convex, h satisfies Assumption 2.5, and ̂ ∇ f satisfies Assumptions 2.7 and 2.8. Then, for the global optimum λ ∗ = arg min λ ∈ Λ F ( λ ) , ∆ ≜ ‖ λ 0 -λ ∗ ‖ 2 , some t ∗ , τ , and γ 0 (explicit in the proof), SPGD with the step size schedule in Eq. (5) guarantees

Proof . See the full proof

This result is a slight improvement over past analysis of SPGD with Eq. (5) (Domke et al., 2023, Theorem 7). In particular, the dependence on the initialization ∆ has been improved to be logarithmic instead of polynomial. Furthermore, it encompasses the case where we have 'interpolation' ( σ 2 = 0 ; Kim et al., 2024b; Schmidt and Roux, 2013; Vaswani et al., 2019) automatically resulting in a O(log 1 /ϵ ) complexity. A similar result for vanilla SGD, where the dependence on both ϵ and ∆ is optimized simultaneously, was reported by Stich (2019). However, this result required a schedule that depends on the maximum number of iterations T . Compared with this, our result provides an any-time guarantee that holds for any number of iterations.

For a non-strongly convex f , using the strategy of Domke et al. (2023, Theorem 8 and 11) should yield a corresponding O ( 1 /ϵ 2 ) complexity guarantee under the same set of assumptions. However, this requires fixing the horizon T in advance, and it is currently unknown how to obtain an anytime O(1 / √ T ) convergence bound for SGD under Assumptions 2.7 and 2.8 or QV. If one moves away from the canonical SGD update by incorporating Halpern iterations (Halpern, 1967), it is possible to obtain any-time convergence under a QV-like assumption (Alacaoglu et al., 2025).

3 Main Results

3.1 General Result

For our results, we impose an additional assumption that is a generalization of L -smoothness under twice differentiability of ℓ .

Assumption 3.1. ℓ is twice differentiable and, for all z ∈ R d , there exist some matrix H ∈ R d × d and constant δ ∈ [0 , ∞ ) satisfying

Notably, if ℓ is twice differentiable, µ -strongly convex, L -smooth, it already satisfies Assumption 3.1 with H = L + µ 2 I d and δ = L -µ 2 . If ℓ is only L -smooth, it satisfies it with H = 0 d × d and δ = L . The key advantage of this assumption, however, is that it characterizes Hessians that are not necessarily well-conditioned, but almost constant. This crucially affects the dimension dependence.

Given our assumptions on the target function ℓ , variational family Q , and our choice of gradient estimator, we can guarantee that SPGD applied to a problem structure corresponding to BBVI (Eq. (1)) achieves a given level of accuracy ϵ after O( g ( d, H, δ, µ, φ ) ϵ -1 ) number of iterations:

Theorem 3.2. Suppose the following hold:

ℓ is µ -strongly convex and satisfies Assumption 3.1 and µ ≤ σ min ( H ) ≤ σ max ( H ) ≤ L .

h satisfies Assumption 2.5.

Q is a mean-field location-scale family, where Assumption 2.2 holds.

T ≥ O { g ( d, H, δ, µ, φ ) ( σ 2 ∗ ϵ -1 + σ ∗ log ( ‖ λ 0 -λ ∗ ‖ 2 2 ) ϵ -1 / 2 )} ⇒ E ‖ λ T -λ ∗ ‖ 2 2 ≤ ϵ , where g ( d, H, δ, µ, φ ) ≜ 2 (1 + r 4 ) ( ‖ H ‖ 2 2 /µ 2 ) +4 ( δ 2 /µ 2 ) ( (1 / 2) + r 4 + E max j =1 ,...,d u 2 j ) . Proof. The full proof can be found in Appendix B.2.1, p. 25.

4. ̂ ∇ f is the reparametrization gradient. Denote the global optimum λ ∗ = ( m ∗ , C ∗ ) = arg min λ ∈ Λ F ( λ ) , the irreducible gradient noise as σ 2 ∗ ≜ ‖ m ∗ -¯ z ‖ 2 2 + ‖ C ∗ ‖ 2 F , and the stationary point of ℓ as ¯ z ≜ arg min z ∈ R d ℓ ( z ) . Then, for some t ∗ , τ , and γ 0 (explicit in the proof), SPGD with the step size schedule in Eq. (5) guarantees

Due to the identity ‖ λ -λ ′ ‖ 2 2 = E u ∼ φ ⊗ d ‖T λ ( u ) - T λ ′ ( u ) ‖ 2 2 (Lemma A.3), which is the squared cost of a coupling between q λ T and q λ ∗ , our guarantee also translates to a guarantee in Wasserstein2 distance: E ‖ λ T -λ ∗ ‖ 2 2 ≤ ϵ ⇒ E W 2 ( q λ T , q λ ∗ ) 2 ≤ ϵ . In the general case where δ > 0 , the dimension dependence enters through E max j =1 ,...,d u 2 j , which depends on the order-statistics of the base distribution φ . In case ℓ is a quadratic, corresponding to π being a Gaussian target distribution in the BBVI context, there exists some H such that ∇ 2 ℓ ( z ) = H for all z ∈ R d . Thus, Assumption 3.1 holds with δ = 0 , implying a dimension-independent convergence rate. We will present additional special cases with more explicit choices of φ in the next section.

In case we do not want to assume Assumption 3.1 and only assume that ℓ is µ -strongly convex and L -smooth instead, we can replace them with the generic choices of H = L + µ 2 I d and δ = L -µ 2 , which hold for all ℓ s that are µ -strongly convex, L -smooth, and twice differentiable. This then makes the role of the condition number κ ≜ L/µ more explicit.

Corollary 3.3. Suppose ℓ is is twice differentiable, µ -strongly convex, and L -smooth. Then, denoting the condition number as κ ≜ L/µ , Theorem 3.2 holds with

This makes the O( κ 2 ) condition number dependence explicit, but the downside is that we lose dimension independence in the case of ill-conditioned quadratic ℓ s. This fact suggests that dimension dependence is more fundamentally related to how close the Hessian is to a constant rather than how well-conditioned it is.

3.2 Special Cases with Benign Dimension Dependence

We now present some special cases of Theorem 3.2, which has yet to exhibit an explicit dependence on dimensionality. As mentioned in the previous section, dimension dependence depends on the order statistics of φ , which is related to the tail behavior of φ .

Variational Families with Sub-Gaussian Tails. The most commonly used variational family in practice is the Gaussian variational family. More broadly, for sub-Gaussian variational families, u 2 i is sub-exponential and therefore admits a moment generating function (MGF) (Wainwright, 2019, Theorem 2.6), which leads to a O(log d ) explicit dimension dependence.

Proposition 3.4. Suppose there exists some t > 0 such that the MGF of u 2 i satisfies M u 2 i ( t ) < ∞ . Then

For example, if φ is a standard Gaussian, then

Proof. The full proof can be found in Appendix B.2.2, p. 26.

Variational Families with Finite Higher Moments. For families with tails heavier than subGaussian, however, u 2 i may not have an MGF. While we then lose the O(log d ) dependence, we may still obtain a polynomial dependence that can be better than O( √ d ) obtained in previous works (Kim et al., 2023b). In particular, the result that will follow states that the highest order of the available moments determines the order of dimension dependence. For Studentt families, this implies that using a high-enough degree of freedom ν can make the dimension dependence benign.

For example, if φ is a Studentt with ν > 4 degrees of freedom and unit variance, then

Proof . See the full proof in Appendix B.2.3, p. 28.

4 Analysis of Gradient Variance

4.1 Overview

The key technical contribution of this work is analyzing the gradient variance and thus establishing the constants L (Assumption 2.7) and σ 2 (Assumption 2.8), which boils down to analyzing

∥ ∥ where the equality follows from the fact that the Jacobian ∂ T λ ( u ) / ∂λ does not depend on λ . For mean-field location-scale variational families, the squared Jacobian follows as

where U ≜ diag( u 1 , . . . , u d ) (Kim et al., 2023b). This implies that

E ‖ ̂ ∇ f ( λ ; u ) -̂ ∇ f ( λ ′ ; u ) ‖ 2 2 = E ‖∇ ℓ ( T λ ( u )) -∇ ℓ ( T λ ′ ( u )) ‖ 2 2 ︸ ︷︷ ︸ ≜ V loc + E ‖ U ( ∇ ℓ ( T λ ( u )) -∇ ℓ ( T λ ′ ( u ))) ‖ 2 2 ︸ ︷︷ ︸ ≜ V scale . (6) Our goal is to bound each term by ‖ λ -λ ′ ‖ 2 2 .

In order to solve the expectations, we need to simplify the ∇ ℓ terms. For instance, for the gradient of the location V loc, assuming that ℓ is L -smooth allows for a quadratic approximation. That is,

where the last equality is by Lemma A.3.

Now, it is tempting to use the same quadratic approximation strategy for the gradient of the scale V scale . Indeed, this strategy was used by Domke (2019) to bound the gradient variance of full-rank location-scale variational families and by Ko et al. (2024) for structured location-scale variational families. Unfortunately, this strategy does not immediately apply to mean-field families due to the matrix U . We somehow have to decouple ∇ ℓ ( T λ ( u )) -∇ ℓ ( T λ ′ ( u )) and U , but in a way that does not lose the correlation between the two; the correlation leads to cancellations critical to obtaining a tight bound. Kim et al. (2023b) used the inequality

which resulted in a dimension dependence of O( r 4 √ d ) after solving the expectation. The key question is whether this dimension dependence can be improved. Due to the ordering of norms ‖·‖ 2 ≤ ‖·‖ F , it is natural to consider the tighter inequality

(This step corresponds to Eq. (8) in the proof sketch of the upcoming result.) The main challenge, however, is solving the resulting expectation in a way that is also tight with respect to d . We will see that this requires a careful probabilistic analysis.

4.2 Upper Bound on Gradient Variance

We now formally state our upper bound on the gradient variance. In the context of proving Theorem 3.2, the following lemma implies both Assumption 2.7 and Assumption 2.2. (See the proof of Theorem 3.2.) We provide a corresponding unimprovability result in Section 4.3.

Lemma 4.1. Suppose Assumptions 2.2 and 3.1 hold, Q is a mean-field location-family, and ̂ ∇ f is the reparametrization gradient. Then, for any λ, λ ′ ∈ R d × D d .

Proof Sketch. For the proof sketch, we will assume that ℓ is L -smooth instead of taking Assumption 3.1. This will vastly simplify the analysis and let us focus on the key elements.

Recall V scale in Eq. (6). Applying the operator norm and the L -smoothness of ℓ yields

It remains to solve the expectation over u , . . . , u

̸

where i ∗ = arg max i =1 ,...,d u 2 i is the coordinate of maximum magnitude. Here, u 4 i ∗ = max i =1 ,...,d u 4 i is a heavy-tailed quantity that generally grows fast in d , unlike u 2 i ∗ . ( e.g. , for a Gaussian u i , u 2 i ∗ has an MGF but u 4 i ∗ does not.) Therefore, a benign dimension dependence might appear futile. Notice, however, that the problematic term only affects a single dimension: the coordinate indicated by i ∗ . A probabilistic analysis reveals that as d increases, the effect of u 4 i ∗ becomes averaged out and the effect of the remaining term involving u 2 i dominates. More formally, recall that T λ ( u ) = Cu + m (Definition 2.1), and notice that, since U is a diagonal matrix, ‖ U ‖ 2 2 = max i =1 ,...,d u 2 j . Then we can rewrite Eq. (8) as

The problematic term is

where

glyph[negationslash]

Since the maximum of d random variable is always smaller than their sum, the probability of the maximally random event, P [ i ∗ = i ] = 1 /d , kills off the dimensional growth of u 4 i ∗ . In fact, using the crude bound E u 4 i ∗ ≤ E ∑ d i =1 u 4 i = dr 4 , where the last equality is due to Assumption 2.2, is enough to make this term independent of d . The remaining dimension dependence comes from u 2 i ∗ :

glyph[negationslash]

E [ u 2 i ∗ u 2 i 1 { i ∗ = i } ] = E [ max j = i u 2 j u 2 i 1 { i ∗ = i } ] ≤ E [ max j =1 ,...,d -1 u 2 j ] E [ u 2 i ] = E max j =1 ,...,d -1 u 2 j , where the last equality follows from Assumption 2.2. Therefore, we finally obtain

̸

glyph[negationslash]

The full proof performs an analogous analysis under the more general Assumption 3.1. See the full proof in Appendix B.3.1, p. 30.

4.3 Unimprovability

Wealso demonstrate a lower bound, which implies that Lemma 4.1 cannot be improved when relying on the spectral bounds on ∇ 2 ℓ . From Eq. (6) and the fundamental theorem of calculus,

E ‖ ̂ ∇ f ( λ ; u ) ‖ 2 2 ≥ E ‖ U ( ∇ ℓ ( z ) -∇ ℓ (¯ z )) ‖ 2 2 = E ∥ ∥ U ∫ 1 0 ∇ 2 ℓ ( z w )( z -¯ z )d w ∥ ∥ 2 2 , (9) where ¯ z ∈ { z | ∇ ℓ ( z ) = 0 } is a stationary point of ℓ , z ≜ T λ ( u ) , and z w ≜ w z +(1 -w )¯ z . There exists a matrix-valued function with bounded singular values that lower-bounds this quantity:

Proposition 4.2. Suppose Assumption 2.2 holds and Q is a mean-field location-scale family. Then, for any t > 0 , d > 0 , µ, L ∈ (0 , + ∞ ) satisfying µ ≤ L , there exists a matrix-valued function H ( z ) : R d → S d ≻ 0 satisfying µ I d glyph[precedesequal] H glyph[precedesequal] L I d almost surely and a set of parameters λ = ( m,C ) ∈ R d × D d ≻ 0 such that

For Gaussians, E max i u 4 i is upper bounded as O( √ d ) (Gumbel, 1954, Eq. 1.6), which means the negative term vanishes at a 1 / √ d rate. Furthermore, E max i u 2 i ≥ ( E max i u i ) 2 = Ω(log d ) by the well-known lower bound on the expected maximum of i.i.d. Gaussians (Wainwright, 2019, Exercise 2.11.(b)). Combining these facts with Proposition 4.2 yield a Ω ( L 2 log d ) bound on Eq. (9).

Remark 4.3. It is not obvious that the rows of our worst-case example H worst form conservative vector fields. This means that Proposition 4.2 does not assert the existence of a function ℓ that satisfies ∇ 2 ℓ = H worst . However, it does suggest that one cannot improve Lemma 4.1 by relying only on spectral bounds on the Hessian.

5 Discussion

5.1 Related Works

Early results analyzing VI had to rely on assumptions that either: (i) do not hold on Gaussian targets, (ii) are difficult to verify, or (iii) require bounds on the domain (Alquier and Ridgway, 2020; Buchholz et al., 2018; Fan et al., 2015; Fujisawa and Sato, 2021; Khan et al., 2016; Liu and Owen, 2021; Nguyen et al., 2025; Regier et al., 2017). coordinate-ascent VI (CAVI), in particular, was studied on specific models (Ghorbani et al., 2019; Zhang and Zhou, 2020) only. Under general and verifiable assumptions, Xu and Campbell (2022) obtained asymptotic convergence guarantees, while partial results, such as bounds on the gradient variance (Domke, 2019; Fan et al., 2015; Kim et al., 2023b), or regularity of the ELBO (Challis and Barber, 2013; Domke, 2020; Titsias and LázaroGredilla, 2014), were known.

It was only recently that non-asymptotic quantitative convergence under realizable and verifiable assumptions was established. For BBVI specifically, Hoffman and Ma (2020) first proved convergence on Gaussian targets (quadratic ℓ ), while Domke et al. (2023); Kim et al. (2023a, 2024b) proved the first results on strongly convex and smooth functions with location-scale families. Surendran et al. (2025) extended these results to non-convex smooth functions and more complex variational family parametrizations, and Cheng et al. (2024) analyzed a variant of semi-implicit VI. The results of Domke et al.; Hoffman and Ma; Kim et al., who focused on full-rank families, suggest a O( d ) dimension dependence in the iteration complexity. On the other hand, Kim et al. (2023a) reported a O( √ d ) dimension dependence for mean-field location-scale families, while conjecturing O(log d ) dependence, based on the partial result of Kim et al. (2023b). For targets with a diagonal Hessian structure, Ko et al. (2024, Corollary 1) show that mean-field families are dimension-independent.

Apart from BBVI, Wasserstein VI algorithms-which minimize the KL divergence on the Wasserstein geometry-provide non-asymptotic convergence guarantees. In particular, the algorithms by Diao et al. (2023); Lambert et al. (2022) optimize over the full-rank Gaussian family, while that of Jiang et al. (2025) optimizes over all mean-field families with bounded second moments. To guarantee E W 2 ( q λ T , q λ ∗ ) 2 ≤ ϵ on strongly log-concave and log-smooth targets, they all report an iteration complexity of O( dϵ -1 log ϵ -1 ) . Meanwhile, under the same conditions, Arnese and Lacker

(2024); Lavenant and Zanella (2024) analyzed (block) CAVI, and reported an iteration complexity of O( d log ϵ -1 ) . Bhattacharya et al. (2025) provides a concurrent result on CAVI, but relies on an assumption that departs from log-concavity and smoothness. Finally, Bhatia et al. (2022) analyzes a specialized algorithm optimizing over only the scale of Gaussians, which has a gradient query complexity of O( dkϵ -3 ) , where k is the user-chosen number of rank-1 factors in the scale matrix.

5.2 Conclusion

In this work, we proved that BBVI with mean-field location-scale families is able to converge with an iteration complexity with only a O(log d ) dimension dependence, as long as the tails of the family are sub-Gaussian. For high-dimensional targets, this suggests a substantial speed advantage over BBVI with full-rank families. In practice, the mean-field approximation can be combined with other design elements such as control variates (Boustati et al., 2020; Geffner and Domke, 2018, 2020; Miller et al., 2017; Roeder et al., 2017; Wang et al., 2024) and data-point subsampling (Kucukelbir et al., 2017; Titsias and Lázaro-Gredilla, 2014). Our analysis strategy should easily be combined with existing analyses (Kim et al., 2024a,b) for such design elements.

For a target distribution π with a condition number of κ and a target accuracy level ϵ , we now know how to improve the dependence on d and ϵ in the iteration complexity: Using less-expressive families such as mean-field (Theorem 3.2) or structured (Ko et al., 2024) families improves the dependence on d , while applying control variates to gradient estimators (Kim et al., 2024b) improves the dependence on ϵ . However, it is currently unclear whether the dependence on κ is tight or improvable. If it is tight, it would be worth investigating whether this can be provably improved through algorithmic modifications, for example, via stochastic second-order optimization methods (Byrd et al., 2016; Fan et al., 2015; Liu and Owen, 2021; Meng et al., 2020; Regier et al., 2017).

Another future direction would be to develop methods that are able to adaptively adjust the computational cost between O(log d ) and O( d ) by trading statistical accuracy akin to the method of Bhatia et al. (2022). Existing BBVI schemes with 'low-rank(-plus-diagonal)' families (Ong et al., 2018; Rezende et al., 2014; Tomczak et al., 2020) result in a non-smooth, non-Lipschitz, and non-convex landscape. This not only rules out typical theoretical convergence guarantees but also exhibits unstable and slow convergence in practice (Modi et al., 2025). Furthermore, understanding the statistical side of this trade-off will be an important direction. As of now, our understanding is restricted to either mean-field or full-rank families (Katsevich and Rigollet, 2024; Margossian and Saul, 2023, 2025; Wang and Blei, 2019a,b; Yang et al., 2020; Zhang and Gao, 2020) with little in between except for the work of Bhatia et al. (2022).

Acknowledgments and Disclosure of Funding

The authors thank Anton Xue for helpful discussions and the reviewers for helpful suggestions.

K. Kim, J. R. Gardner were supported through the NSF award [IIS2145644]; Y.-A. Ma was supported by the NSF Award CCF-2112665 (TILOS), the DARPA AIE program, and the CDC-RFAFT-23-0069; T. Campbell was supported by the NSERC Discovery Grant RGPIN-2025-04208.

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Contents

A Auxiliary Lemmas

Lemma A.1. Suppose Assumption 2.2 holds. Then r 4 = E u 4 i ≥ 1 .

Lemma A.2. Suppose Assumption 2.2 holds and denote U = diag( u 1 , . . . , u d ) . Then we have the following identities: (i) E uu ⊤ = I d , (ii) E U 2 = I d .

Proof. From Assumption 2.2, we know that E u 2 i = 1 . Then (i) follows from glyph[negationslash]

glyph[negationslash]

For (ii), we only need to focus on the diagonal since the off-diagonal is already zero.

Lemma A.3. Suppose Assumption 2.2 holds, T λ is the reparametrization function for a locationscale family, and the linear parametrization is used. Then

Proof. ′ ′ ′

Lastly,

Combining this with Eq. (10) yields the result.

Lemma A.4. Suppose f is µ -strongly convex and Assumption 2.7 holds. Then f is L -Lipschitz smooth, while the constants satisfy the ordering

Proof. For all λ, λ ′ ∈ Λ , the unbiasedness of ̂ ∇ f and Jensen's inequality states that ‖∇ f ( λ ) -∇ f ( λ ′ ) ‖ 2 2 = ‖ E ̂ ∇ f ( λ ; u ) -E ̂ ∇ f ( λ ′ ; u ) ‖ 2 2 ≤ E ‖ ̂ ∇ f ( λ ; u ) -̂ ∇ f ( λ ′ ; u ) ‖ 2 2 . Then the µ -strong convexity of f and Assumption 2.7 yields the inequality

µ 2 ‖ λ -λ ′ ‖ 2 2 ≤ ‖∇ f ( λ ) -∇ f ( λ ′ ) ‖ 2 2 ≤ E ‖ ̂ ∇ f ( λ ; u ) -̂ ∇ f ( λ ′ ; u ) ‖ 2 2 ≤ L 2 ‖ λ -λ ′ ‖ 2 2 , from which the statement follows immediately.

B Proofs

B.1 Proofs of Results in Section 2

B.1.1 Proof of Proposition 2.9

Under the stated assumptions, we first establish a convergence bound which bounds E ‖ λ T -λ ∗ ‖ 2 2 after T iterations under a given step size schedule. We will invert this convergence bound into a complexity guarantee by identifying the conditions on T , t ∗ , τ , and γ 0 that guarantee E ‖ λ T -λ ∗ ‖ 2 2 ≤ ϵ for a given ϵ > 0 .

Lemma B.1. Suppose f is µ -strongly convex, h satisfies Assumption 2.5, and ̂ ∇ f satisfies Assumptions 2.7 and 2.8. Then, for the global optimum λ ∗ = arg min λ ∈ Λ F ( λ ) , and the step size schedule in Eq. (5) with any t ∗ ≥ 0 and τ ≥ 4 L 2 /µ 2 , the contraction coefficient ρ ≜ 1 -µγ 0 satisfies ρ ∈ (0 , 1) and the last iterate of SPGD after T iterations, λ T , satisfies

Proof . The full proof is deferred to Appendix B.1.2, p. 22.

This is a slight generalization of the result by Domke et al. (2023, Theorem 7), where the switching time t ∗ was fixed to some t ∗ ∝ L 2 /µ 2 . While the choice of t ∗ ∝ L 2 /µ 2 results in the typical O(1 /ϵ ) asymptotic complexity, it suffers from a suboptimal polynomial dependence on the initialization error ∆ = ‖ λ 0 -λ ∗ ‖ 2 . Picking an alternative t ∗ , which is what we do in the proof, improves the iteration complexity to O ( 1 /ϵ +1 / √ ϵ log ∆ 2 +log(∆ 2 /ϵ ) ) . Now, a similar O(1 /ϵ ) result for vanilla SGD was demonstrated by Stich (2019), where the dependence on ∆ is also logarithmic. Their step size schedule, however, requires the maximum number of iterations T , which means T must be fixed before running the algorithm. Our step size, on the other hand, does not require T and is therefore an any-time result.

Proposition 2.9. Suppose f is µ -strongly convex, h satisfies Assumption 2.5, and ̂ ∇ f satisfies Assumptions 2.7 and 2.8. Then, for the global optimum λ ∗ = arg min λ ∈ Λ F ( λ ) , ∆ ≜ ‖ λ 0 -λ ∗ ‖ 2 , some t ∗ , τ , and γ 0 (explicit in the proof), SPGD with the step size schedule in Eq. (5) guarantees

Proof. Since f is strongly convex and h is convex, F is also strongly convex. This implies that, by the property of strictly convex functions, F has a unique global optimum, which we denote as λ ∗ . Then we can invoke Lemma B.1. We will optimize the upper bound over the parameters t ∗ , τ , γ 0 , and T so that we can ensure the ϵ -accuracy guarantee E ‖ λ T -λ ∗ ‖ 2 2 ≤ ϵ .

Consider the choice

We will separately analyze the cases of t ∗ ≥ T and t ∗ < T . When t ∗ ≥ T , the second-stage never kicks in. Therefore, we can apply Eq. (27) with t ∗ = T . Furthermore,

is true. An immediate implication is that

Considering this fact, applying Eq. (27) with t ∗ = T ,

The number of required steps for achieving the ϵ -accuracy requirement follows from

For the case t ∗ < T ,

This implies

Substituting for this in Lemma B.1,

which is a quadratic function of 1 /T with the coefficients

Achieving the ϵ -accuracy guarantee is equivalent to finding the largest x = 1 /T satisfying the inequalities x > 0 and

By the quadratic formula, this is equivalent to finding the largest x satisfying

is sufficient to obtain an ϵ -accurate solution. To make the bound more interpretable, after defining α = 4 aϵ and β = b , we can use the inequality (Symbol-1, 2022)

Then

where we used the inequality √ a + b ≤ √ a + √ b . Thus, we have

Substituting t ∗ and γ 0 with the expressions in Eq. (11),

Considering this, the sufficient condition for E ‖ λ T -λ ∗ ‖ 2 2 ≤ ϵ is now

Combining both cases, that is, Eqs. (13) and (17), we have

This implies the stated result.

Therefore, picking any

B.1.2 Proof of Lemma B.1

The proof closely mirrors the strategy of Garrigos and Gower (2023, Theorem 12.9), which is a combination of previous analyses of SPGD (Gorbunov et al., 2020; Khaled et al., 2023) with the analysis of SGD strongly convex objectives with a decreasing step size schedule (Gower et al., 2019). The main difference is that Garrigos and Gower utilize a different condition on the gradient variance instead of Assumption 2.7. Specificially, they assume that, for all λ, λ ′ ∈ Λ , there exists some function of L ( u ) : supp( u ) → [0 , ∞ ) such that, for each u ∈ supp( u ) , the function ̂ ∇ f ( λ ; u ) : Λ → R p is L ( u ) -smooth with respect to λ . This then enables the use of the 'convex expected smoothness' (Gorbunov et al., 2020; Khaled et al., 2023) condition, which postulates that, for all λ ∈ Λ , there exists some L < ∞ such that where

is the Bregman divergence associated with f . Note that Assumption 2.7 and the µ -strong convexity of f implies Eq. (18). Therefore, under our assumptions, one can invoke the results that assume Eq. (18), which was the strategy by some previous analyses of BBVI (Kim et al., 2023a; Ko et al., 2024). Here, we will take a more straightforward approach that uses Assumption 2.7 directly in the convergence proof, but the results are identical to the indirect approach of establishing Eq. (18).

Lemma B.1. Suppose f is µ -strongly convex, h satisfies Assumption 2.5, and ̂ ∇ f satisfies Assumptions 2.7 and 2.8. Then, for the global optimum λ ∗ = arg min λ ∈ Λ F ( λ ) , and the step size schedule in Eq. (5) with any t ∗ ≥ 0 and τ ≥ 4 L 2 /µ 2 , the contraction coefficient ρ ≜ 1 -µγ 0 satisfies ρ ∈ (0 , 1) and the last iterate of SPGD after T iterations, λ T , satisfies

Proof. Since f is strongly convex and h is convex, F is also strongly convex. This implies that F has a unique global optimum, which we denote as λ ∗ . Furthermore, under the stated assumptions on h , the proximal operator prox γh ( · ) is non-expansive for any γ ∈ (0 , ∞ ) (Garrigos and Gower, 2023, Lemma 8.17) and any λ, λ ′ ∈ R p such that

and λ ∗ is the fixed-point of the deterministic proximal gradient descent step (Garrigos and Gower, 2023, Lemma 8.18) such that

Using these facts,

‖ λ t +1 -λ ∗ ‖ 2 2 ≤ ‖ λ t -λ ∗ ‖ 2 2 -2 γ t 〈 ̂ ∇ f ( λ t ; u ) -∇ f ( λ ∗ ) , λ t -λ ∗ 〉 + γ 2 t ‖ ̂ ∇ f ( λ t ; u ) -∇ f ( λ ∗ ) ‖ 2 2 . Denoting the filtration of the σ -field of the iterates generated up to iteration t as F t , where the equality follows from the fact that ̂ ∇ f is unbiased conditional on any λ t ∈ Λ . From the µ -strong convexity of f ,

The gradient variance at λ t , on the other hand, can be compared against the gradient variance at λ ∗ through the variance transfer strategy as

Applying Eqs. (23) and (24) to Eq. (22),

Taking expectation over all randomness, we obtain our general partial contraction bound

Due to the form of the step size schedule, SPGD operates in two different regimes: the first stage with a fixed step size γ t = γ 0 ( t ∈ { 0 , . . . , t ∗ -1 } ) and the second stage with a decreasing step size γ t +1 < γ t ( t ∈ { t ∗ , . . . , T } ). In the first stage, γ t = γ 0 ≤ µ 2 L 2 . Then the Bregman divergence term in Eq. (25) is negative such that

Unrolling the recursion yields

From Lemma A.4, we deduce that γ 0 µ = µ 2 / (2 L 2 ) ≤ 1 / 2 , which implies ρ ∈ (0 , 1) .

We now turn to the second stage, where the step size starts decreasing. Notice that, for any τ ≥ 4 L 2 /µ 2 , Eq. (5) satisfies

Therefore, γ t ≤ µ 2 L 2 for all t ≥ 0 . Again, the Bregman term in Eq. (25) is negative such that

Subtituting γ t with the choice in Eq. (5), we obtain

Multiplying ( t + τ +1) 2 to both sides,

Let us choose the Lyapunov function V t ≜ ( t + τ +1) 2 E ‖ λ t +1 -λ ∗ ‖ 2 2 . Then the discrete derivative of the Lyapunov,

shows that the energy is increasing only by a constant. By integrating the Lyapunov over the time interval t = t ∗ , . . . , T -1 ,

Substuting ‖ λ t ∗ -λ ∗ ‖ 2 2 with the error in Eq. (27),

which is our stated result.

B.2 Proofs of Results in Section 3

B.2.1 Proof of Theorem 3.2

Theorem 3.2. Suppose the following hold:

ℓ is µ -strongly convex and satisfies Assumption 3.1 and µ ≤ σ min ( H ) ≤ σ max ( H ) ≤ L .

h satisfies Assumption 2.5.

Q is a mean-field location-scale family, where Assumption 2.2 holds.

4. ̂ ∇ f is the reparametrization gradient. Denote the global optimum λ ∗ = ( m ∗ , C ∗ ) = arg min λ ∈ Λ F ( λ ) , the irreducible gradient noise as σ 2 ∗ ≜ ‖ m ∗ -¯ z ‖ 2 2 + ‖ C ∗ ‖ 2 F , and the stationary point of ℓ as ¯ z ≜ arg min z ∈ R d ℓ ( z ) . Then, for some t ∗ , τ , and γ 0 (explicit in the proof), SPGD with the step size schedule in Eq. (5) guarantees

Proof. The proof consists of establishing the sufficient conditions of Proposition 2.9 as follows:

(i) ℓ is µ -strongly convex ⇒ f is µ -strongly convex.

(ii) Assumption 3.1 ⇒ Assumptions 2.7 and 2.8.

Under the linear parametrization, (i) was established by Domke (2020, Thm. 9). It remains to establish (ii). Therefore, the proof focuses on analyzing the variance of the gradient estimator ̂ ∇ f . Since Assumption 3.1 holds, Lemma 4.1 states that, for all λ, λ ′ ∈ R d × D d , the inequality holds. Since Λ ⊂ R d × D d under the linear parametrization, this implies we satisfy Assumption 2.7 with

Furthermore, For the specific choice of λ ∗ = ( m ∗ , C ∗ ) = arg min λ ∈ Λ F ( λ ) and ¯ λ = (¯ z, 0 d × d ) (which is not part of Λ ), we have the equality

E ‖ ̂ ∇ f ( λ ∗ ; u ) -̂ ∇ f ( ¯ λ ; u ) ‖ 2 2 = E ‖ ̂ ∇ f ( λ ∗ ; u ) -̂ ∇ f (¯ z ; u ) ‖ 2 2 = E ‖ ̂ ∇ f ( λ ∗ ; u ) ‖ 2 2 . This means Lemma 4.1 also implies Assumption 2.8 with the constant

We are now able to invoke Proposition 2.9. Substituting L and σ 2 in Eq. (28) with the expressions above, we obtain the condition

Finally, substituting for Eq. (29) yields our stated result.

B.2.2 Proof of Proposition 3.4

The result follows from a well-known bound on the expected maximum of sub-exponential random variables. We state the proof for completeness.

Lemma B.2. Let x 1 , . . . , x d be i.i.d. random variables. Suppose there exists some t > 0 such that their moment-generating function (MGF) satisfies M x i ( t ) < ∞ . Then

Proof.

Dividing both sides by t yields the statement.

Applying Lemma B.2 to u 2 i yields the result.

Proposition 3.4. Suppose there exists some t > 0 such that the MGF of u 2 i satisfies M u 2 i ( t ) < ∞ . Then

For example, if φ is a standard Gaussian, then

Proof. The first part of the statement is a re-statement of Lemma B.2.

For the special case of u i ∼ N (0 , 1) , we know that u 2 i ∼ χ 2 1 (Johnson et al., 1995, Eq. 29.1), which is the χ 2 distribution with 1 degree of freedom. The MGF of χ 2 1 is given as

for t ∈ (0 , 1 / 2) . Then we can invoke Lemma B.2, which suggests

Any fixed choice of t ∈ (0 , 1 / 2) is a valid upper bound. Picking t = 1 2 ( 1 -1 e ) ≥ 1 4 yields

Furthermore, the kurtosis of the standard Gaussian is r 4 = 3 (Johnson et al., 1994, Eq. 13.11). Plugging r 4 and Eq. (31) into g in Theorem 3.2 yields the statement.

B.2.3 Proof of Proposition 3.5

The result follows from the following moment-based bound on the expected maximum of random variables, which is a non-asymptotic refinement of the proof by Rana (2017).

Lemma B.3. Let x 1 , . . . , x d be i.i.d. non-negative random variables where, for k ≥ 2 , their k th moment is finite. That is, E x k i = r k < ∞ . Then

Proof. For any ϵ > 0 , we have

(Decreased lower limit of integral) .

Now, from the definition of moments, we know that

Therefore,

The bound is minimized when setting

Then

If the k th moment of u 2 i is finite, this then immediately implies a polynomial O( d 1 /k ) bound on g . Proposition 3.5. Suppose, for k ≥ 2 , the k th moment of u 2 i is finite as r 2 k = E u 2 k i < ∞ . Then E max u 2 i √ 2 d 1 /k r 1 /k 2 k .

For example, if φ is a Studentt with ν > 4 degrees of freedom and unit variance, then g ( d, H, δ, µ, φ ) ≤ 8( ‖ H ‖ 2 2 /µ 2 ) + ( δ 2 /µ 2 ) ( 16 + √ 2 ν 3 d 2 ν -2 ) .

Proof. The first part of the statement directly follows from Lemma B.3, where we simplified ( k / k -1 ) ( k -1) /k . In particular, for k ≥ 2 , ( k / k -1 ) ( k -1) /k is monotonically decreasing. Since an order k ≥ 2 moment exists by the assumption on the degrees of freedom, ( k / k -1 ) ( k -1) /k ≤ √ 2 .

Let's turn to the second part of the statement. We will denote a Studentt distribution with ν -degrees of freedom as t ν . Since t ν does not have unit variance (Johnson et al., 1995, Eq. 28.7a), we have to set the sampling process from φ to be

Now, it is known that v 2 i d = w i ∼ FDist(1 , ν 2 ) (Johnson et al., 1995, §28.7), where FDist( ν 1 , ν 2 ) is Fisher's F -distribution with ( ν 1 , ν 2 ) degrees of freedom. The k th raw moment of FDist( ν 1 , ν 2 ) , denoted as m k ≜ E w k i , exists up to 2 k < ν 2 = ν and is given as

This means that we can invoke Lemma B.3 as

For m 1 /k k , we can use the fact that the gamma function satisfies the recursion Γ( z +1) = z Γ( z ) , which implies Γ( a/ 2 + k ) = Γ( a/ 2) ∏ k -1 i =0 ( a/ 2 + i ) for any a > 0 . Therefore,

(geometric series sum formula)

Applying this bound to a = ν 2 = ν and a = ν 1 = 1 respectively,

Also, choosing k = glyph[ceilingleft] ν/ 2 -1 glyph[ceilingright] , we have d 1 /k ≤ d 2 / ( ν -2) . This yields

Lastly, the kurtosis of u i = ( ν -2) /ν v i follows as (Johnson et al., 1995, Eq. 28.5)

Plugging the bound in Eq. (32) and the value of r 4 into g in Theorem 3.2 yields the statement.

B.3 Proofs of Results in Section 4

B.3.1 Proof of Lemma 4.1

Under the assumption that ∇ 2 ℓ glyph[precedesequal] L I d and twice differentiability, it is well known that ∇ 2 ℓ glyph[precedesequal] L I d ⇒ ℓ is L -smooth. We will prove a supporting result analogous to this under Assumption 3.1, which will allow us to bound the relative growth of ∇ ℓ .

Lemma B.4. Suppose ℓ : R d → R satisfies Assumption 3.1. Then, for any W ∈ R d × d satisfying ‖ W ‖ 2 < ∞ ,

Proof. The full proof is deferred to Appendix B.3.2, p. 32.

Using this, we can now simplify the ∇ ℓ terms in Eq. (6). Applying Lemma B.4 to V loc with W = I d and Young's inequality,

V const corresponds to the constant component of the Hessian ∇ 2 ℓ , whereas V non-const corresponds to the non-constant residual. Denote the location and scale parameters of λ and λ ′ as

For V const, we can use the following lemma:

Lemma B.5. Suppose T λ is the reparameterization operator of a mean-field location-family and Assumption 2.2 holds. Then, for any matrix H ∈ R d × d and any λ, λ ′ ∈ R d × D d ,

See the full proof in Appendix B.3.3, p. 33.

The remaining part of the proof closely resembles the proof sketch of Lemma 4.1. For convenience, we first restate Lemma 4.1 and then proceed to the full proof.

Lemma 4.1. Suppose Assumptions 2.2 and 3.1 hold, Q is a mean-field location-family, and ̂ ∇ f is the reparametrization gradient. Then, for any λ, λ ′ ∈ R d × D d .

Proof. Recall Eq. (34). The proof consists of bounding the two terms V const and V non-const . First, for V const, under Assumption 3.1,

It remains to bound V non-const, which is our main challenge.

Denote ¯ m ≜ m -m ′ and ¯ C ≜ C -C ′ such that

Then

glyph[negationslash]

We will focus on the first term. Denoting i ∗ = arg max i =1 ,...,d u 2 i , the coordinate of maximum magnitude, we can decompose the expectation by the contribution of the event i ∗ = i and i ∗ = i . That is, glyph[negationslash]

The expectation of the event i ∗ = i follows as

On the other hand, for the event i ∗ = i , glyph[negationslash]

glyph[negationslash]

Therefore, we finally obtain

Combining Eqs. (6), (33) to (35) and (37) yields the statement.

̸

̸

̸

glyph[negationslash]

glyph[negationslash]

B.3.2 Proof of Lemma B.4

Lemma B.4. Suppose ℓ : R d → R satisfies Assumption 3.1. Then, for any W ∈ R d × d satisfying ‖ W ‖ 2 < ∞ ,

Proof. From twice differentiability of ℓ (Assumption 3.1) and the fundamental theorem of calculus, we know that

Denoting z t ≜ tz +(1 -t ) z ′ for clarity,

B.3.3 Proof of Lemma B.5

Lemma B.5. Suppose T λ is the reparameterization operator of a mean-field location-family and Assumption 2.2 holds. Then, for any matrix H ∈ R d × d and any λ, λ ′ ∈ R d × D d ,

Proof. For clarity, let us denote ¯ C ≜ C -C ′ and ¯ m ≜ m -m ′ such that

Then

V loc and V cross are straightforward. Under Assumption 2.2, it immediately follows that

On the other hand,

The expectation follows as

̸

Thus, the cross term V cross vanishes.

V scale requires careful elementwise inspection in order to apply Assumption 2.2. That is, V scale = E ‖ U H ¯ C u ‖ 2 2

̸

̸

̸

̸

̸

Combining everything,

From the property of the Frobenius norm, for any matrix A ∈ R d × d , we can decompose

̸

where off( A ) is a function that zeroes-out the diagonal of A . Then from Eq. (38),

which is the stated result.

̸

̸

̸

̸

B.3.4 Proof of Proposition 4.2

For any µ, L ∈ (0 , ∞ ) such that µ ≤ L , our goal is to obtain a matrix-valued function H worst : R d → S d ≻ 0 satisfying

that, under the choice H = H worst , maximizes the quantity

∥ ∥ where ¯ z ∈ { z | ∇ ℓ ( z ) = 0 } is any stationary point of ℓ , z ≜ T λ ( u ) , and z w ≜ w z + (1 -w )¯ z . Given the norm constraint, the worst-case example that maximizes Eq. (39) will be the matrix-valued function that approximately results in for any realization of u on R d . For this, we will establish the relations

The first equality in Eq. (40) follows from identifying the conditions where H ( z w ) is independent of the value of w . For the specific choice of

H ( z w ) is independent of w if it only depends on the quantities

That is, with some abuse of notation, H ( z w ) = H (ˆ z w , i ∗ ) .

Lemma B.6. Suppose m = ¯ z = 0 d , and for any δ > 0 , C = diag( δ, . . . , δ ) . If H ( z w ) is a function of only i ∗ and ˆ z w , then H ( z w ) is constant with respect to w ∈ [0 , 1] .

Proof. It suffices to show that, under the stated conditions, the values of i ∗ and ˆ z w are invariant to w . For ˆ z w , this trivially follows from the assumption that ¯ z = 0 as

For i ∗ , we use the fact that the diagonal matrix C is isotropic as

From H ( z w ) = H (ˆ z w , i ∗ ) , the integral in Eq. (39) can be solved as

It remains to construct H in a way that depends only on ˆ z w and i ∗ such that

Recalling the spectral constraints, this is equivalent to, for all z ∈ R d , H solving the equation

Notice the equivalence

Thus, ˆ z w and i ∗ contain all the information we need. The following matrix-valued function almost solves Eq. (42):

This function is reminiscent of a Householder reflector (Trefethen and Bau, 1997, Eq. 10.4), where some modifications were made to make it satisfy the eigenvalue constraint. From the fact that both e i ∗ and ˆ z have a unit norm, it is apparent that this matrix satisfies Assumption 3.1 with H = α I d and δ = β . Furthermore, by setting the constants as

the triangle inequality asserts that the eigenvalue constraint µ I d ≤ H worst ≤ L I d is satisfied almost surely.

Given the specific form of H worst , we are now ready to formally prove Proposition 4.2. Let us first restate the proposition for convenience and then proceed to the proof.

Proposition 4.2. Suppose Assumption 2.2 holds and Q is a mean-field location-scale family. Then, for any t > 0 , d > 0 , µ, L ∈ (0 , + ∞ ) satisfying µ ≤ L , there exists a matrix-valued function H ( z ) : R d → S d ≻ 0 satisfying µ I d glyph[precedesequal] H glyph[precedesequal] L I d almost surely and a set of parameters λ = ( m,C ) ∈ R d × D d ≻ 0 such that

Proof. Recall H worst in Eq. (43). By inspection, we know that H worst ( z w ) only depends on the quantities i ∗ and z w . Then Lemma B.6 states that w ↦→ H worst ( z w ) is a constant function. Therefore,

Here, the first term β 2 / 4 u 2 i ∗ ‖ z ‖ 2 2 is the worst-case behavior we expect from solving Eq. (42). The remaining terms V 1 and V 2 are the error caused by inexactly solving Eq. (42). It suffices to show that β 2 / 4 u 2 i ∗ ‖ z ‖ 2 2 dominates lower bounds on V 1 and V 2 asymptotically in L and d .

V 1 ≥ 0 trivially holds and can immediately be lower-bounded. V 2 , on the other hand, is not necessarily non-negative. Therefore, we will use the bound V 2 ≥ -| E V 2 | .

For V 3 , we can use an argument similar to Eq. (36) where we distribute the influence of the maximum coordinate over the d coordinates.

The last equality follows by applying Lemma A.2 to the identity E ‖ z ‖ 2 2 = E u ⊤ C ⊤ C u . By applying Eq. (48) into Eq. (47), we can now notice that V 2 decreases by a factor of E u 4 i ∗ /d .

It is clear that V 2 vanishes as d →∞ .

Applying the lower bound on V 2 into Eqs. (45) and (46), we have

It remains to solve the expectation.

glyph[negationslash]

Let us decompose the events where the i th coordinate attains the maximum ( i ∗ = i ) or not ( i ∗ = i ) as done in Lemma 4.1.

̸

glyph[negationslash]

̸

̸

By introducing a free variable t > 0 , we can break up the indicator

̸

̸

̸

̸

This then allows the expectation to break up between terms depending on u 2 i and max j = i u j , which is the independence that we were after. That is, applying Eq. (51) to Eq. (50),

̸

̸

Notice that the function ( t, φ ) ↦→ ∫ t 0 P [ u 2 i > s ] d s is strictly positive as long as t > 0 and only dependent on t and the base distribution φ .

̸

We are left with the expectation over the event i ∗ = i . For the upper bound in Lemma 4.1, the expectation was solved by noticing that u 2 i and u 2 i ∗ can be made independent after upper bounding the indicator. For a lower bound, however, breaking up the expectation for u 2 i and u 2 i ∗ is more involved.

̸

̸

̸

We now obtain our final result by combining the results into Eq. (49). With explicit constants,

NeurIPS Paper Checklist

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The full details can be provided either with the code, in appendix, or as supplemental material.

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Answer: [NA]

Justification: The paper does not contain any experiments.

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The answer NA means that the paper does not include experiments.

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Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

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If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

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Justification: The paper does not involve real data or models.

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The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

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We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

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The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper analyzes the rate of convergence of a particular optimization algorithm (stochastic proximal gradient descent using reparameterization gradients and a particular step size schedule) applied to mean-field variational inference for problems with strongly log-concave and log-smooth likelihoods using location-scale variational families. The main result is that the rate of convergence only depends weakly on the dimension of the problem: for variational families generated by sub-exponential random variables the dependence is $O(\\log d)$, for variational families with $k$ finite moments it is $O(d^{2/k})$.  The authors then provide some justification that similar proof techniques could not result in a better bound.  The main ingredient in the proof is a careful analysis of the variance of the gradient in this setup, but the authors also use a number of additional clever tricks like a carefully chosen step size schedule.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* Overall I thought that the paper was really clear. The exposition and proofs were generally easy to follow (although see below) and well-motivated.\n* The main result is very interesting and suggests that mean-field BBVI should scale to higher dimensional problems more gracefully (at least computationally).\n* The results could be useful in practice (e.g., setting step sizes or choosing a variational family to balance the ability to fit heavier tails with a worse scaling in dimension).\n\nWeaknesses (all minor):\n* There is a lot of emphasis on the scaling being nearly dimension-free, which -- unless I'm missing something --  obscures that some of the assumptions likely do not scale sub-linearly in the dimension.  For example, it would be reasonable to expect the Frobenius norm of the optimal scale matrix, $\\Vert C_* \\Vert_F^2$, to grow linearly in the dimension.  I don't think this is a major issue, but some discussion of how the other parameters in the bound are likely to grow with dimension in practice is warranted.\n* It would be nice if the authors could discuss how relevant (at least in terms of the intuition) the authors think their results are to practical implementations.  For example, not all implementations use proximal SGD, many implementations use parameterizations other than the linear parameterization, the optimal step size and step size schedule depend on generally unknown parameters and so cannot be set like this in practice and so on.  Proofs in these other cases are obviously outside the scope of the paper, but it would be helpful if the authors could discuss which of their assumptions are for technical purposes and which they think are likely actually required of algorithms.\n* I found the proof of Proposition 4.2 hard to follow.  In particular, it was unclear which parts of the proof are just for developing intuition for how they chose $H_{worst}$ and which parts of the proof are actually rigorous.  Relatedly, why is there a $z^w$ in the middle of the display after line 911? (And similarly in the second line of the display after line 921)?\n\nAdditional comments and typos:\n* I think the step size schedule has a nice interpretation -- $t_*$ is chosen so that for the first $t_*$ steps the noise in the gradient is negligible relative to the ``signal''.  As a result, one can use a constant step size as if the gradients were noise free, this also allows one to geometrically converge from the initialization to $\\lambda$ that are close enough to the optimum such that the noise in the gradients begins to dominate.  After this, one is in a regime that is more SGD-like and so a step-size like $1/t$ is required (c.f., Robbins-Monro).\n* I found the phrasing around the formal Assumptions in the main text to be somewhat confusing on my first read.  For example \"Assumptions\" 2.1 and 2.2 read more like definitions, and the actual assumption is that various functions have those properties.  Relatedly, it was unclear throughout when \"Assumptions\" were being stated in the general case vs. being actual assumptions about the problem at hand.  For example, I think Assumption 2.7 should always hold for mean-field VI with location-scale families with linear parameterizations. I understand that it is being used to set up a more general result, but some changes to the wording might make that clearer.\n* It's up to the authors, but Equation (16) appears in, I think, 3 different proofs, and I think it would be cleaner if that were made to be another Lemma in Appendix A instead of needing to refer back to an equation in the middle of a proof.\n* Line 37: “parameterization gradient” —> “reparameterization gradient”\n* Line 95: Shouldn't \"diag\" should be \"vec\"? $z$ is vector valued and $C$ is a matrix, so $Cu$ as defined would also be a $d\\times d$ matrix, I think.\n* Line 109: Latex typo $Q$ --> $\\mathcal{Q}$\n* Line 222: $q_T$ --> $q_{\\lambda_T}$\n* Line 253: missing reference in \"of the result by , states that the highest\"\n* Line 290: second $\\lambda$ should be $\\lambda'$\n* Line 326: \"difficult to verify\" --> \"are difficult to verify\"\n* Line 362: \"should easily be combined\" --> \"should be easy to combine\"\n* Line 669: second $\\lambda$ should be $\\lambda'$\n* Line 735: \"straightly\" --> \"straightforwardly\"\n* Not that it matters, but I believe the first line on p. 21 can be replaced with an equality because it's just algebra\n* Statement of Lemma B.1: assumption should be on $\\widehat{\\nabla f}$ not $\\nabla f$ (appears in both statements of Lemma B.1)\n* \"Equality\" after line 704 should be $\\le$\n* After line 714 is presumably missing a $\\le \\epsilon$\n* I believe the equations after lines 796 and 797 are missing terms that correspond to the function $g$ defined in the Theorem.\n* Second line in the display after line 822: should be $X_i > t$ not $X_i > s$ in the domain of the indicator\n* Norm on right hand side of first line of display after line 891 is missing a subscript 2.\n* In both statements of Proposition 4.2 \"location-family\" --> \"location-scale family\"\n\nQUESTIONS:\nn/a\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors developed a new theoretical convergence guarantee for black-box variational inference when using mean-field location-scale families. They show that by using stochastic proximal gradient descent, it can achieve a convergence rate that is nearly independent of dimension for target distributions that are log-smooth and strongly log-concave. Moreover, authors have analyzed the variance of the reparameterization gradient and showed that the bound on the gradient variance cannot be improved using only spectral bounds on the Hessian of the target log-density.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThis paper is well-written and clearly structured. The inclusion of proof sketches is helpful for readers. The paper demonstrates that the iteration complexity of mean-field location-scale families, which are commonly used in practice, can be improved to nearly a dimension-independent rate, representing an advancement over previous work. I believe this is a significant contribution to the variational inference community.\n\nWeaknesses:\nThe paper is highly theoretical and lacks numerical experiments or synthetic illustrations. I think including illustrations would help the reader follow the concepts more easily. While the paper focuses on the reparameterization gradient and provides a technical justification for its use, it does not adequately explain why this estimator is prioritized over alternatives like the score function gradient. Providing additional context or justification could strengthen the relevance of the paper's contributions.\n\nQUESTIONS:\n1. Typo in lines 144 & 185: It looks like the acronym \"PSGD\" is a typo. Should it be SPGD instead?\n2. Should the matrix $C$ in Definition 2.6 be required to be positive definite and diagonal?\n\nLIMITATIONS:\nAll limitations are addressed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work the authors investigate the problem of optimizing black box variational inference, in particular regularized problems which use location (a $d$-dimensional vector $m$ designating the \"center\" of the distribution) and scale ( a $d\\times d$-dimensional matrix $C$, which linearly transforms the distribution, may be enforced to be diagonal, low rank plus identity, etc) parameters, which are contained in $\\lambda$. In such problems,  it is theoretically known to that estimate to $\\epsilon$ accuracy is known to take order $d\\kappa^2 \\epsilon^-1$. It has been conjectured that this dependence on $d$ is no so bad. Indeed for the *mean field* setting where the matrix is enforced to be diagonal, this improves to order $\\sqrt{d}\\kappa^2\\epsilon^-1$, yet it is thought that this dependence on $d$ can be further improved.\n\nThis paper presents results showing that estimating $\\lambda$ using a proximal stochastic gradient descent algorithm with correct step sizes can achieve much better dependences on $d$, with $\\log(d)$ dependence in the case of a sub-Gaussian family, and $d^{2/k}$ when $k$ moments exist for the distribution (so $d^{2/(2-\\nu)}$ for Student-t with $\\nu$ degrees of freedom).\n\nSTRENGTHS AND WEAKNESSES:\nThis paper shines light on a very simple yet fundamental technique in a statistical method that is quite popular recently. Understanding how these estimators work in high dimensions is of interest. I found that paper pleasant to read, even though I feel as though I couldn't fully grasp every part, which is probably unavoidable due to the topicality.\n\nI've scanned the proofs and they also look very nice. They seem well organized with clear justifications provided at every step and signposts for the reader to help guide the proof logic.\n\nFor weaknesses, I don't have any significant issues with the work. The paper is purely theoretical result on a phenomenon that was already suspected, on a topic that isn't as mainstream, as some at NeurIPS, with no clear practical impact. Hence I am a bit reluctant to give a strong accept. This line of work decently outside of my field of expertise so its possible that I'm missing something here.\n\nSmall errors:\n\n* line 144: PSGD is never defined, and the sentence \"Instead of using PSGD, one can also use projected SGD\" is a bit confusing since its easy to think that PSGD refers to projected SGD. I think this may actually be an issue in other parts of the paper. Please just make sure the nomenclature is clear and consistent.\n\n* Prop 2.11 (and elsewhere): I don't really like the $T \\ge O(...)$. Maybe $\\Omega$ is better?\n\n* line 193 and 196: \"anytime\" vs \"any-time\"\n\n* line 253: missing citation\n\nQUESTIONS:\nI don't have any particularly deep questions regarding the paper.\n\n* I wonder if it would be possible to verify the results experimentally?\n* Are there any concrete useable/practical lessons from this paper, other than one need not worry so much about dimensionality when the data is expected to be not heavy tailed.\n* Again the main point keeping me from a strong accept is my lack of certainty regarding whether the impact, originality or usefulness is good enough to warrant the top reviewing score. This may just be because I am not super comfortable with the topic. Anything you could do here to elaborate for a non-expert would be helpful.\n\nLIMITATIONS:\nI think the limitations are pretty clear from the exposition in the paper that indicate where their results apply.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proves that mean field VI converges at an $O(log d)$, where $d$ is the dimension of the parameters, improving on an $O(d)$ rate for full-rank VI. Previously, it was known that the rate could at most be on the order of the square root of d, and faster convergence was speculated but not proven. If the Hessian of the target density is constant, there is no dimension dependence. To obtain this result, there are assumptions about the regularity of the target density (log-concave and log-smooth), the variational family (mean field location-scale family with sub-Gaussian tails), optimization (proximal SGD), and the gradient estimator (reparameterization trick). The assumption of sub-Gaussian tails can be relaxed, leading to, for example in the student-t case, a $O(d^2/(\\nu-2))$ dependence, where $\\nu$ is the degrees of freedom.\n\nSTRENGTHS AND WEAKNESSES:\nThis is an exceptionally well-written paper, with a detailed overview of related work and rigorous proofs of all claims. The assumptions are standard and discussed carefully. The contribution is important because it may justify the use of mean field VI for high-dimensional problems. \n\nSection 4, and the proof sketch of lemma 4.1, provides a nice explanation of the challenges in the proof and why past works obtained looser bounds. \n\nI think the lack of experiments is fine for this kind of paper, but it would be nice to understand the practical significance for common problems. \n\nAs the authors remark, it had been speculated that a faster convergence happens in the mean field case, so one weakness is that the result doesn’t seem surprising, but there is value in identifying and proving the rate as this paper does. The proof strategy also seems like a nice contribution.\n\nQUESTIONS:\n- Theorem 3.2 states that there exists a switching time and initial step size for which the result holds. Can you comment on the practical significance of having to choose these values? \n\nSmall:\n- Missing citation on line 253\n- Should be $\\lambda’ = (m’, C’)$ in line 290\n- In the equation after line 290, I believe it should be an inequality right after $V_{scale}$\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}