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icml26/32a9a1bf-fc3e-433d-855e-5d1a0149a10b/assets.json

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icml26/32a9a1bf-fc3e-433d-855e-5d1a0149a10b/main_body_chunks.jsonl

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+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0000", "section": "Abstract", "page_start": 1, "page_end": 1, "type": "Text", "text": "For approximating a target distribution given only its unnormalized log-density, stochastic gradientbased variational inference (VI) algorithms are a popular approach. For example, Wasserstein VI (WVI) and black-box VI (BBVI) perform gradient descent in measure space (Bures-Wasserstein space) and parameter space, respectively. Previously, for the Gaussian variational family, convergence guarantees for WVI have shown superiority over existing results for black-box VI with the reparametrization gradient, suggesting the measure space approach might provide some unique benefits. In this work, however, we close this gap by obtaining identical state-of-the-art iteration complexity guarantees for both. In particular, we identify that WVI's superiority stems from the specific gradient estimator it uses, which BBVI can also leverage with minor modifications. The estimator in question is usually associated with Price's theorem and utilizes second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient. On the flip side, WVI can be made more widely applicable by using the reparametrization gradient, which requires only gradients of the log-density. We empirically demonstrate that the use of Price's gradient is the major source of performance improvement.", "source": "marker_v2", "marker_block_id": "/page/0/Text/8"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0001", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "Variational inference (VI; Jordan et al., 1998; Blei et al., 2017; Peterson & Hartman, 1989; Hinton & van Camp, 1993) is a collection of algorithms for approximating a target distribution π over some family of parametric distributions Q when only the unnormalized density of π, denoted by π e, is available. When π is supported on R d such that its", "source": "marker_v2", "marker_block_id": "/page/0/Text/10"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0002", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "associated potential function U = − log π e is in R d → R, it is common to leverage stochastic gradient-based algorithms, as they only require local information of U (Graves, 2011; Salimans & Knowles, 2013; Wingate & Weber, 2013; Titsias & Lázaro-Gredilla, 2014; Ranganath et al., 2014; Kingma & Welling, 2014; Rezende et al., 2014) . Most VI algorithms are designed to minimize the variational free energy, also known as the negative evidence lower bound (Jordan et al., 1999) , defined as F(q) ≜ E(q) + H(q), where E is the energy functional associated with U, while H is the Boltzmann entropy. That is, we solve", "source": "marker_v2", "marker_block_id": "/page/0/Text/13"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0003", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Equation", "text": "\\underset{q \\in \\mathcal{Q}}{\\text{minimize}} \\left\\{ \\mathcal{F}(q) = \\text{KL}(q, \\pi) + \\mathcal{F}(\\pi) \\right\\}", "source": "marker_v2", "marker_block_id": "/page/0/Equation/14"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0004", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "through only zeroth-, first-, and second-order information of U. Since F is equal to the exclusive KL divergence (Kull back & Leibler, 1951) between q and π up to the constant F(π), this also minimizes q 7→ KL(q, π).", "source": "marker_v2", "marker_block_id": "/page/0/Text/15"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0005", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "A common approach for minimizing F is, informally speaking, to leverage some sort of stochastic gradient descent (SGD; Robbins & Monro, 1951; Bottou, 1999; Bottou et al., 2018; Bach & Moulines, 2011; Nemirovski et al., 2009; Shalev-Shwartz et al., 2011) scheme, where intractable terms (such as the gradient of the energy E) are stochastically estimated. There are two popular ways to realize this conceptual algorithm. The most widely used approach in practice is to represent each q λ ∈ Q via a Euclidean vector of variational parameters λ ∈ Λ, and run gradient descent on the Euclidean space of parameters Λ ⊆ R p (Wingate & Weber, 2013; Kucukelbir et al., 2017; Titsias & Lázaro- Gredilla, 2014; Ranganath et al., 2014; Salimans & Knowles, 2013) . This is now referred to as black-box variational in ference (BBVI). The other approach is to define a tractable notion of measure-valued derivatives, which can directly perform gradient descent in measure space. In particular, recent advances in our understanding of the Wasserstein geometry (Villani, 2009; Chewi et al., 2025) and gradient flows (Ambrosio et al., 2005; Jordan et al., 1998) have contributed to the development of (parametric 1 ) Wasserstein variational inference (WVI; Lambert et al., 2022; Diao et al., 2023; Huix et al., 2024; Talamon et al., 2025) algorithms", "source": "marker_v2", "marker_block_id": "/page/0/Text/16"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0006", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Footnote", "text": "1 We focus on WVI on parametric families, which excludes particle-based WVI methods.", "source": "marker_v2", "marker_block_id": "/page/0/Footnote/17"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0007", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Table", "text": "Table 1. Overview of Main Theoretical Results. Algorithm Space Gradient Estimator Maximum Step Size Iterations Complexity Reference SPGD Λ Reparam. 1/(dL\\kappa) d\\kappa^2 {\\rm tr}(\\mu \\Sigma_*)^1\\!/\\epsilon Domke et al. (2023) Kim et al. (2023) SPBWGD \\mathrm{BW}(\\mathbb{R}^d) Bonnet-Price 1/(L\\kappa^2) d\\kappa \\frac{1}{\\epsilon} \\log \\frac{1}{\\epsilon} + \\kappa^3 \\log(\\Delta^2 \\frac{1}{\\epsilon}) Diao et al. (2023) SPGD \\Lambda Bonnet-Price 1/(L\\kappa) d\\kappa \\frac{1}{\\epsilon} + \\sqrt{d} \\kappa^{3/2} \\log(\\kappa \\Delta^2) \\frac{1}{\\sqrt{\\epsilon}} + \\kappa^2 \\log \\frac{1}{\\epsilon} Theorem 3.3 SPBWGD \\mathrm{BW}(\\mathbb{R}^d) Bonnet-Price 1/(L\\kappa) d\\kappa \\frac{1}{\\epsilon} + \\sqrt{d} \\kappa^{3/2} \\log(\\kappa \\Delta^2) \\frac{1}{\\sqrt{\\epsilon}} + \\kappa^2 \\log \\frac{1}{\\epsilon} Theorem 3.2", "source": "marker_v2", "marker_block_id": "/page/1/Table/2"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0008", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "Note: The complexity statements assume \\mu -strong convexity and L-smoothness of U (Assumption 3.1); \\epsilon > 0 is the target accuracy level for ensuring \\mu \\mathbb{E} W_2(q_T, q_*)^2 \\le \\epsilon , where q_T is the last iterate and q_* is the global minimizer of \\mathcal{F} ; \\kappa = L/\\mu is the condition number; \\Delta is the distance between the initialization and the optimum.", "source": "marker_v2", "marker_block_id": "/page/1/Text/3"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0009", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "that take this route. We also note that natural gradient VI algorithms (Khan & Rue, 2023; Khan & Nielsen, 2018; Lin et al., 2019; Tan, 2025) utilize parameter gradients while measuring distance using the KL pseudo-metric. However, NGVI will not be the focus of this work.", "source": "marker_v2", "marker_block_id": "/page/1/Text/4"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0010", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "Given that the WVI methods utilize a proper metric (Villani, 2009) between measures—the Wasserstein-2 metric W_2 —it is natural to expect them to outperform BBVI. Indeed, current theoretical evidence suggests that this is the case. Consider the Gaussian variational family, also known as the Bures-Wasserstein space (Bures, 1969; Bhatia et al., 2019), set as", "source": "marker_v2", "marker_block_id": "/page/1/Text/5"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0011", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Equation", "text": "Q = \\mathrm{BW}(\\mathbb{R}^d) \\triangleq \\{ \\mathrm{Normal}(m, \\Sigma) \\mid m \\in \\mathbb{R}^d, \\Sigma \\in \\mathbb{S}^d_{\\succ 0} \\} .", "source": "marker_v2", "marker_block_id": "/page/1/Equation/6"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0012", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "Here, (\\mathrm{BW}(\\mathbb{R}^d), \\mathrm{W}_2) forms a metric space. Denoting the global minimizer as q_* = \\mathrm{Normal}(m_*, \\Sigma_*) = \\mathrm{arg} \\min_{q \\in \\mathcal{Q}} \\mathcal{F}(q) , for a \\mu -strongly convex and L-smooth potential U, the algorithm by Diao et al. (2023) requires \\mathrm{O}(d\\kappa\\epsilon^{-1}\\log\\epsilon^{-1}+\\kappa^3\\log\\epsilon^{-1}) steps to ensure \\mu\\mathbb{E}[\\mathrm{W}_2(q,q_*)^2] \\leq \\epsilon . In contrast, the BBVI equivalent with a certain covariance parametrization, the reparametrization gradient estimator (Ho & Cao, 1983; Rubinstein, 1992), and stochastic proximal gradient descent (SPGD; Nemirovski et al., 2009) requires \\mathrm{O}(d\\kappa^2\\mathrm{tr}(\\mu\\Sigma_*)\\epsilon^{-1}) steps (Kim et al., 2023; Domke et al., 2023). It might therefore appear that the guarantees for BBVI are weaker in the limit \\epsilon \\to 0 .", "source": "marker_v2", "marker_block_id": "/page/1/Text/7"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0013", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "In this work, we demonstrate that the difference in theoretical guarantees originates from the specific gradient estimator for the scale parameter used in stochastic implementations of WVI rather than the geometry being used. The estimator in question can be derived via Price's theorem (Price, 1958) and leverages second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient estimator. Through Stein (Liu, 1994; Stein, 1981) or Price's theorem, BBVI with SPGD can also make use of essentially the same gradient estimator, resulting in an iteration complexity of O(d\\kappa\\epsilon^{-1} + \\sqrt{d}\\,\\kappa^{3/2}\\log(\\kappa\\Delta^2)\\epsilon^{-1/2} + \\kappa^2\\log\\epsilon^{-1}) . Furthermore, to ensure a fair comparison, we present a refined analysis of the WVI counterpart of SPGD (Diao et al.,", "source": "marker_v2", "marker_block_id": "/page/1/Text/8"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0014", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "2023), resulting in an iteration complexity of O(d\\kappa\\epsilon^{-1} + \\sqrt{d} \\kappa^{3/2} \\log(\\kappa \\Delta^2) \\epsilon^{-1/2} + \\kappa^2 \\log \\epsilon^{-1}) . These results suggest that the specific implementation of BBVI studied here and WVI might not be that different after all. While this has been suggested by Yi & Liu (2023); Hoffman & Ma (2020), this work further supports this fact by contributing a non-asymptotic discrete-time analysis. Our theoretical results are organized in Table 1.", "source": "marker_v2", "marker_block_id": "/page/1/Text/9"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0015", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "In addition, we demonstrate that WVI can also leverage the reparametrization gradient traditionally used in BBVI (Titsias & Lázaro-Gredilla, 2014; Kingma & Welling, 2014; Rezende et al., 2014). Unlike Price's gradient previously used in WVI, the reparametrization gradient only requires first-order information ( \\nabla U ). Thus, the resulting WVI algorithm should be more widely applicable in practice. Given this fact, we empirically compare the performance of BBVI and WVI, both with the Hessian-based gradient estimators and the reparametrization gradient. Results demonstrate that a large fraction of the performance difference stems from the use of Hessian-based gradients, supporting the claim that the gradient estimator is the main source of performance.", "source": "marker_v2", "marker_block_id": "/page/1/Text/10"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0016", "section": "2. Background", "page_start": 2, "page_end": 2, "type": "Text", "text": "Notation. For any x,y \\in \\mathbb{R}^d , we denote the Euclidean inner product and norm as \\langle x,y \\rangle = x^\\top y and \\|x\\|_2 \\triangleq \\sqrt{\\langle x,x \\rangle} . For any matrix A,B \\in \\mathbb{R}^{d \\times d} , \\operatorname{tr}(A) \\triangleq \\sum_{i=1}^d A_{ii} , \\langle A,B \\rangle_F \\triangleq \\operatorname{tr}(A^\\top B) , \\|A\\|_F \\triangleq \\sqrt{\\langle A,A \\rangle_F} , and the \\ell_2 operator norm of A is denoted as \\|A\\|_2 . The symmetric and positive definite subsets of \\mathbb{R}^{d \\times d} will be denoted as \\mathbb{S}^d and \\mathbb{S}^d_{\\succ 0} , while \\mathbb{L}^d_{\\succ 0} denotes the set of lower-triangular matrices with strictly positive eigenvalues. We will represent a measure and its density with the same symbol. For the space of square-integrable measures \\mathcal{P}_2(\\mathcal{X}) \\triangleq \\{q \\mid \\int_{\\mathcal{X}} \\|x\\|^2 \\mathrm{d}q(x) < +\\infty\\} , for some q \\in \\mathcal{P}_2(\\mathbb{R}^d) , the set of integrable functions is denoted as \\mathrm{L}^2(q) \\triangleq \\{f \\mid \\int \\|f\\|_2^2 \\mathrm{d}q < +\\infty\\} . For any two probability measures p,q \\in \\mathcal{P}_2(\\mathbb{R}^d) , we denote the set of couplings between the two as \\Psi(p,q) . Then the squared Wasserstein-2 distance between p and q is \\mathrm{W}_2(p,q)^2 \\triangleq", "source": "marker_v2", "marker_block_id": "/page/1/Text/12"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0017", "section": "2. Background", "page_start": 3, "page_end": 3, "type": "Text", "text": "\\inf_{\\psi\\in\\Psi(p,q)}\\int_{\\mathbb{R}^d\\times\\mathbb{R}^d}\\|x-y\\|_2^2\\,\\mathrm{d}\\psi(x,y). For some measurable map M:\\mathbb{R}^d\\to\\mathbb{R}^d and measure q supported on \\mathbb{R}^d , M_{\\#q} denotes the corresponding push-forward measure. Unless stated otherwise, the coupling attaining the infimum of W_2,\\,\\psi^*\\in\\Psi(p,q) , is referred to as \"the optimal coupling,\" which is guaranteed to exist (Villani, 2009, Theorem 4.1) and is unique by Brenier's Theorem (Brenier, 1991).", "source": "marker_v2", "marker_block_id": "/page/2/Text/1"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0018", "section": "2.1. Problem Setup", "page_start": 3, "page_end": 3, "type": "Text", "text": "Our focus will be on first-order stochastic optimization algorithms for solving the problem", "source": "marker_v2", "marker_block_id": "/page/2/Text/3"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0019", "section": "2.1. Problem Setup", "page_start": 3, "page_end": 3, "type": "Equation", "text": "\\underset{q \\in \\mathcal{Q}}{\\text{minimize}} \\ \\left\\{ \\mathcal{F}(q) \\triangleq \\mathcal{E}(q) + \\mathcal{H}(q) \\right\\},", "source": "marker_v2", "marker_block_id": "/page/2/Equation/4"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0020", "section": "2.1. Problem Setup", "page_start": 3, "page_end": 3, "type": "Equation", "text": "\\begin{array}{ll} \\text{where} & \\mathcal{E}(q) & \\triangleq \\int_{\\mathbb{R}^d} U(z) \\, q(\\mathrm{d}z) & \\text{(Energy)} \\\\ & \\mathcal{H}(q) & \\triangleq \\int_{\\mathbb{R}^d} \\log q(z) \\, q(\\mathrm{d}z) \\; . & \\text{(Boltzmann entropy)} \\end{array}", "source": "marker_v2", "marker_block_id": "/page/2/Equation/5"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0021", "section": "2.1. Problem Setup", "page_start": 3, "page_end": 3, "type": "Text", "text": "We consider the \"non-conjugate\" setup, where \\mathcal E is intractable due to the expectation over q. Suppose we can parametrize each q_\\lambda \\in \\mathcal Q with a Euclidean vector of parameters \\lambda \\in \\Lambda . Then it is equivalent to minimize \\lambda \\mapsto \\mathcal F(q_\\lambda) over the Euclidean parameter space \\Lambda \\subseteq \\mathbb R^p .", "source": "marker_v2", "marker_block_id": "/page/2/Text/6"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0022", "section": "2.1. Problem Setup", "page_start": 3, "page_end": 3, "type": "Text", "text": "Informally speaking, when U is \"regular,\" \\mathcal{E} also tends to be regular. For instance, if U is Lipschitz-smooth, then \\mathcal{E} also tends to exhibit appropriate notion of Lipschitz-smoothness (Domke, 2020; Diao et al., 2023; Lambert et al., 2022). The entropy term \\mathcal{H} , however, does not enjoy Lipschitz smoothness in general. For instance, for Gaussians q=\\operatorname{Normal}(m,\\Sigma) , \\mathcal{H}(q) blows up as the covariance \\Sigma becomes singular. Typically, in optimization, such non-smoothness is remedied by relying on proximal gradient algorithms (Wright & Recht, 2021, §9.3). Indeed, for minimizing \\mathcal{F} , stochastic proximal gradient algorithms have been proposed for both the Bures-Wasserstein (Diao et al., 2023) and Euclidean parameter spaces (Domke, 2020). In the following sections, we will introduce these algorithms.", "source": "marker_v2", "marker_block_id": "/page/2/Text/7"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0023", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "Since the seminal work of Jordan et al. (1998), it is known that \\mathcal{F} can be minimized by simulating its Wasserstein gradient flow. The forward-backward discretization of this flow results in the Wasserstein-analog of proximal gradient descent (Wibisono, 2018; Bernton, 2018; Salim et al., 2020) operating on the metric space (\\mathcal{P}_2(\\mathbb{R}^d), W_2) . This algorithm, however, is not directly implementable. Recently, Lambert et al. (2022); Diao et al. (2023) demonstrated that, by constraining optimization to the Bures -Wasserstein manifold \\mathrm{BW}(\\mathbb{R}^d) \\subset \\mathcal{P}_2(\\mathbb{R}^d) (Bures, 1969; Bhatia et al., 2019), the algorithm becomes implementable. In particular, the proximal Bures-Wasserstein gradient descent scheme iter-", "source": "marker_v2", "marker_block_id": "/page/2/Text/9"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0024", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "ates, for each t \\geq 0 ,", "source": "marker_v2", "marker_block_id": "/page/2/Text/10"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0025", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Equation", "text": "q_{t+1/2} = (\\operatorname{Id} - \\gamma_t \\nabla_{\\mathrm{BW}} \\mathcal{E}(q_t))_{\\#q_t} q_{t+1} = \\operatorname{JKO}_{\\gamma_t \\mathcal{H}}(q_{t+1/2}), (1)", "source": "marker_v2", "marker_block_id": "/page/2/Equation/11"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0026", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "where, for any q = \\text{Normal}(m, \\Sigma) \\in \\text{BW}(\\mathbb{R}^d) , the Bures-Wasserstein gradient of \\mathcal{E} can be derived (Lambert et al., 2022, Appendix C) as", "source": "marker_v2", "marker_block_id": "/page/2/Text/12"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0027", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Equation", "text": "\\nabla_{\\rm BW} \\mathcal{E}(q) \\triangleq x \\mapsto \\nabla_m \\mathcal{E}(q) + 2\\nabla_{\\Sigma} \\mathcal{E}(q)(x-m) ,", "source": "marker_v2", "marker_block_id": "/page/2/Equation/13"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0028", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "while the Wasserstein-analog of the proximal operator, commonly referred to as the \"JKO operator,\" is defined as, for any functional \\mathcal{G}: \\mathrm{BW}(\\mathbb{R}^d) \\to \\mathbb{R} \\cup \\{+\\infty\\} satisfying some mild regularity conditions,", "source": "marker_v2", "marker_block_id": "/page/2/Text/14"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0029", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Equation", "text": "\\mathrm{JKO}_{\\mathcal{G}}(q) \\triangleq \\operatorname*{arg\\,min}_{p \\in \\mathrm{BW}(\\mathbb{R}^d)} \\left\\{ \\mathcal{G}(p) + (1/2) \\mathrm{W}_2(p,q)^2 \\right\\} \\,.", "source": "marker_v2", "marker_block_id": "/page/2/Equation/15"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0030", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "For the Bures-Wasserstein space, the proximal operator has a tractable closed-form expression (Wibisono, 2018, Example 7), which is key to implementing the algorithm.", "source": "marker_v2", "marker_block_id": "/page/2/Text/16"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0031", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "Still, the Bures-Wasserstein gradient involves expectations over q=\\operatorname{Normal}(\\mu,\\Sigma) that are generally not tractable. Therefore, these have to be replaced with stochastic estimates (Lambert et al., 2022) of \\nabla_m \\mathcal{E} and \\nabla_\\Sigma \\mathcal{E} resulting in an estimator of the Bures-Wasserstein gradient", "source": "marker_v2", "marker_block_id": "/page/2/Text/17"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0032", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Equation", "text": "\\widehat{\\nabla}_{\\mathrm{BW}} \\mathcal{E}(q; \\epsilon) \\triangleq x \\mapsto \\widehat{\\nabla}_m \\widehat{\\mathcal{E}}(q; \\epsilon) + 2\\widehat{\\nabla}_{\\Sigma} \\widehat{\\mathcal{E}}(q; \\epsilon)(x - m)", "source": "marker_v2", "marker_block_id": "/page/2/Equation/18"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0033", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "where \\epsilon \\sim \\varphi = \\mathrm{Normal}(0_d, \\mathrm{I}_d) is standard Gaussian noise. Replacing \\nabla_{\\mathrm{BW}} \\mathcal{E} in Eq. (1) with \\widehat{\\nabla_{\\mathrm{BW}}} \\mathcal{E} results in stochastic proximal Bures-Wasserstein gradient descent (SPBWGD; Diao et al., 2023). Lambert et al.; Diao et al. rely on the estimators", "source": "marker_v2", "marker_block_id": "/page/2/Text/19"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0034", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Equation", "text": "\\widehat{\\nabla_{m}^{\\text{poinet}}} \\mathcal{E}(q; \\epsilon) \\triangleq \\nabla U(Z) \\widehat{\\nabla_{\\Sigma}^{\\text{price}}} \\mathcal{E}(q; \\epsilon) \\triangleq (1/2) \\nabla^{2} U(Z) , \\qquad (2)", "source": "marker_v2", "marker_block_id": "/page/2/Equation/20"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0035", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "where Z={\\rm cholesky}(\\Sigma)\\epsilon+\\mu . The fact that these estimators are unbiased follows from the theorems by Bonnet (1964) and Price (1958) or Riemannian geometry (Altschuler et al., 2021, Appendix B.3). For each t\\geq 0 , the resulting update rule for the iterate q_t={\\rm Normal}(m_t,\\Sigma_t) is", "source": "marker_v2", "marker_block_id": "/page/2/Text/21"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0036", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Equation", "text": "m_{t+1} = m_t - \\gamma_t \\widehat{\\nabla_{m_t} \\mathcal{E}}(q_t; \\epsilon_t) M_{t+1} = I_d - 2\\gamma_t \\widehat{\\nabla_{\\Sigma_t} \\mathcal{E}}(q_t; \\epsilon_t) \\Sigma_{t+1/2} = M_{t+1} \\Sigma_t M_{t+1}^{\\top} \\Sigma_{t+1} = \\frac{1}{2} \\Big( \\Sigma_{t+1/2} + 2\\gamma_t I_d + \\big( \\Sigma_{t+1/2} \\big( \\Sigma_{t+1/2} + 4\\gamma_t I_d \\big) \\big)^{1/2} \\Big) ,", "source": "marker_v2", "marker_block_id": "/page/2/Equation/22"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0037", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 3, "page_end": 3, "type": "Text", "text": "where the standard Gaussian noise sequence (\\epsilon_t)_{t\\geq 0} is sampled as \\epsilon_t \\overset{\\text{i.i.d.}}{\\sim} \\varphi . Note the update rule for \\Sigma_{t+1/2} is different from the one originally presented by Diao et al. (2023); a transpose has been added to M_{t+1} . This change will become necessary later in Section 4 when we replace \\widehat{\\nabla_\\Sigma \\mathcal{E}}", "source": "marker_v2", "marker_block_id": "/page/2/Text/23"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0038", "section": "2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "with an estimator that is not almost surely symmetric.", "source": "marker_v2", "marker_block_id": "/page/3/Text/1"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0039", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "An alternative to SBWPGD is to optimize over the Euclidean space of parameters \\Lambda . Recall, in this case, each q_{\\lambda} \\in \\Lambda is assumed to be associated with a Euclidean vector \\lambda \\in \\Lambda . Then, if we have access to an unbiased estimator of \\nabla_{\\lambda} \\mathcal{E}(q_{\\lambda}) , denoted as \\widehat{\\nabla_{\\lambda} \\mathcal{E}}(q_{\\lambda}) , \\mathcal{F} can be minimized via SPGD, which, for each t \\geq 0 , updates the variational parameters \\lambda_t as", "source": "marker_v2", "marker_block_id": "/page/3/Text/3"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0040", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\lambda_{t+1/2} = \\lambda_t - \\gamma_t \\widehat{\\nabla_{\\lambda_t} \\mathcal{E}}(q_{\\lambda_t}; \\epsilon_t) \\lambda_{t+1} = \\operatorname{prox}_{\\lambda \\mapsto \\gamma_t \\mathcal{H}(q_{\\lambda})} (\\lambda_{t+1/2}) ,", "source": "marker_v2", "marker_block_id": "/page/3/Equation/4"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0041", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "where \\epsilon_t \\overset{\\text{i.i.d.}}{\\sim} \\varphi is some randomness source and prox is the canonical Euclidean proximal operator (Parikh & Boyd, 2014) defined as, for any proper lower semi-continuous convex function g: \\mathbb{R}^p \\to \\mathbb{R} \\cup \\{+\\infty\\} ,", "source": "marker_v2", "marker_block_id": "/page/3/Text/5"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0042", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\mathrm{prox}_g(\\lambda) \\triangleq \\underset{\\lambda' \\in \\Lambda}{\\mathrm{arg\\,min}} \\big\\{ g(\\lambda') + (1/2) \\|\\lambda' - \\lambda\\|_2^2 \\big\\} \\;.", "source": "marker_v2", "marker_block_id": "/page/3/Equation/6"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0043", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "For some classes of variational families Q and parametrizations, the proximal operator can be made tractable (Domke, 2020). Before that, however, we must come up with an unbiased estimator of the parameter gradient \\nabla_{\\lambda} \\mathcal{E}(q_{\\lambda}) .", "source": "marker_v2", "marker_block_id": "/page/3/Text/7"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0044", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "Suppose the variational family Q and the parametrization \\lambda \\mapsto q_{\\lambda} satisfy the following:", "source": "marker_v2", "marker_block_id": "/page/3/Text/8"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0045", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "Definition 2.1. For some \\Lambda \\subset \\mathbb{R}^p , the variational family \\mathcal{Q} = \\{q_{\\lambda} \\mid \\lambda \\in \\Lambda\\} is referred to as a reparameterizable family if there exists some bijective map \\phi_{\\lambda} : \\mathbb{R}^d \\to \\mathbb{R}^d differentiable with respect to \\lambda and a base distribution \\varphi \\in \\mathcal{P}_2(\\mathbb{R}^d) such that, for all \\lambda \\in \\Lambda ,", "source": "marker_v2", "marker_block_id": "/page/3/Text/9"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0046", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "Z \\sim q_{\\lambda} \\quad \\Leftrightarrow \\quad Z \\stackrel{\\mathrm{d}}{=} \\phi_{\\lambda}(\\epsilon) \\; ; \\; \\epsilon \\sim \\varphi \\; .", "source": "marker_v2", "marker_block_id": "/page/3/Equation/10"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0047", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "Here, \\stackrel{\\mathrm{d}}{=} is equivalence in distribution. Then an immediate option for estimating \\nabla_{\\lambda}\\mathcal{E}(q_{\\lambda}) is to use the reparametrization gradient (Ho & Cao, 1983; Rubinstein, 1992; see also Mohamed et al., 2020)", "source": "marker_v2", "marker_block_id": "/page/3/Text/11"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0048", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\widehat{\\nabla_{\\lambda}^{\\text{rep}} \\mathcal{E}}(q_{\\lambda}) \\triangleq \\nabla_{\\lambda} \\phi_{\\lambda}(\\epsilon) \\nabla U(\\phi_{\\lambda}(\\epsilon)) , \\qquad (3)", "source": "marker_v2", "marker_block_id": "/page/3/Equation/12"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0049", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "which can be derived by combining the law of the unconscious statistician with the Leibniz integral rule. This combination of SGD with the reparametrization gradient—commonly referred to as BBVI—is widely used in practice through probabilistic programming frameworks such as Stan (Carpenter et al., 2017), Turing (Fjelde et al., 2025), Pyro (Bingham et al., 2019), and PyMC (Patil et al., 2010).", "source": "marker_v2", "marker_block_id": "/page/3/Text/13"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0050", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "The wide adoption of BBVI in practice is partly due to its flexibility: Definition 2.1 applies to a very wide range of families from Gaussians (Titsias & Lázaro-Gredilla, 2014) to normalizing flows (Rezende & Mohamed, 2015). Fur-", "source": "marker_v2", "marker_block_id": "/page/3/Text/14"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0051", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "thermore, Eq. (3) uses only gradients of U, which can be efficiently computed via automatic differentiation (Kucukelbir et al., 2017). In this work, however, we will further restrict our attention to the Gaussian variational family with a specific parametrization:", "source": "marker_v2", "marker_block_id": "/page/3/Text/15"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0052", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "Assumption 2.2. The variational family Q is the Gaussian variational family, where each member q_{\\lambda} = \\text{Normal}(m, CC^{\\top}) \\in Q is parametrized as", "source": "marker_v2", "marker_block_id": "/page/3/Text/16"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0053", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\Lambda = \\left\\{ \\lambda = (m, \\text{vec}(C)) \\mid m \\in \\mathbb{R}^d, C \\in \\mathbb{L}^d_{\\succ 0} \\right\\} \\subset \\mathbb{R}^p while the reparametrization function is set as", "source": "marker_v2", "marker_block_id": "/page/3/Equation/17"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0054", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\phi_{\\lambda}(\\epsilon) = C\\epsilon + m \\quad and \\quad \\varphi = \\text{Normal}(0_d, I_d) .", "source": "marker_v2", "marker_block_id": "/page/3/Equation/18"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0055", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "Under this parametrization, Eq. (3) reduces to", "source": "marker_v2", "marker_block_id": "/page/3/Text/19"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0056", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\widehat{\\nabla_{\\lambda}^{\\text{rep}}\\mathcal{E}}(q_{\\lambda};\\epsilon) = \\left[\\widehat{\\nabla_{m}^{\\text{bonnet}}\\mathcal{E}}(q_{\\lambda};\\epsilon)\\right] = \\left[\\nabla U(\\phi_{\\lambda}(\\epsilon))\\right] \\cdot \\left[\\epsilon \\nabla U(\\phi_{\\lambda}(\\epsilon))\\right]. \\tag{4}", "source": "marker_v2", "marker_block_id": "/page/3/Equation/20"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0057", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "Furthermore, the proximal operator for the entropy has the closed-form solution (Domke, 2020; Domke et al., 2023)", "source": "marker_v2", "marker_block_id": "/page/3/Text/21"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0058", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\begin{split} (m,C') &= \\operatorname{prox}_{\\lambda \\mapsto \\gamma_t \\mathcal{H}(q_{\\lambda_t})}((m,C)), \\text{ where} \\\\ [C']_{ij} &= \\begin{cases} (1/2) \\left( C_{ii} + \\sqrt{C_{ii} + 4\\gamma_t} \\right) & \\text{if } i = j \\\\ C_{ij} & \\text{if } i \\neq j \\end{cases}. \\end{split}", "source": "marker_v2", "marker_block_id": "/page/3/Equation/22"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0059", "section": "2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent", "page_start": 4, "page_end": 4, "type": "Text", "text": "Compared to alternative ways to parametrize Gaussians, this \"linear\" parametrization is particularly well-behaved (Kim et al., 2023) and also computationally efficient: each step of BBVI only needs O(d^2) operations except for evaluating \\nabla U . Furthermore, it has been shown that the Bures-Wasserstein gradient \\nabla_{\\rm BW} \\mathcal{F} is equal to the parameter gradient \\nabla_{\\lambda} \\mathcal{F}(q_{\\lambda}) under the parametrization of Assumption 2.2 up to a coordinate transformation (Yi & Liu, 2023).", "source": "marker_v2", "marker_block_id": "/page/3/Text/23"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0060", "section": "2.4. Price Gradient Estimators", "page_start": 4, "page_end": 4, "type": "Text", "text": "A crucial point here is that, for the Gaussian variational family, the estimators traditionally used in WVI (Eq. (5)) and BBVI (Eq. (3)) both target the same quantities up to constant factor adjustments and are therefore interchangeable.", "source": "marker_v2", "marker_block_id": "/page/3/Text/25"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0061", "section": "2.4. Price Gradient Estimators", "page_start": 4, "page_end": 4, "type": "Text", "text": "Proposition 2.3. For any twice-differentiable function f, Gaussian q = \\text{Normal}(m, \\Sigma) , and assuming the expectations exist,", "source": "marker_v2", "marker_block_id": "/page/3/Text/26"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0062", "section": "2.4. Price Gradient Estimators", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\nabla_{\\Sigma} \\mathbb{E}_q f = \\frac{1}{2} \\mathbb{E}_q \\nabla^2 f = \\frac{1}{2} \\Sigma^{-1} \\mathbb{E}_{X \\sim q} \\left[ (X - m) \\nabla f(X)^{\\top} \\right].", "source": "marker_v2", "marker_block_id": "/page/3/Equation/27"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0063", "section": "2.4. Price Gradient Estimators", "page_start": 4, "page_end": 4, "type": "Text", "text": "Proof. The first equality is Price's theorem (Price, 1958), while the second equality is Stein's identity (Stein, 1981; Liu, 1994). □", "source": "marker_v2", "marker_block_id": "/page/3/Text/28"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0064", "section": "2.4. Price Gradient Estimators", "page_start": 4, "page_end": 4, "type": "Text", "text": "Denoting \\Sigma = CC^{\\top} , an immediate corollary is", "source": "marker_v2", "marker_block_id": "/page/3/Text/29"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0065", "section": "2.4. Price Gradient Estimators", "page_start": 4, "page_end": 4, "type": "Equation", "text": "\\nabla_{C} \\mathcal{E}(q_{\\lambda}) = C^{\\top} \\mathbb{E}_{q} \\nabla^{2} U = C^{-1} \\mathbb{E}_{X \\sim q} \\left[ (X - m) \\nabla U(X)^{\\top} \\right], \\quad (5)", "source": "marker_v2", "marker_block_id": "/page/3/Equation/30"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0066", "section": "2.4. Price Gradient Estimators", "page_start": 5, "page_end": 5, "type": "Text", "text": "where, when restricting C \\in \\mathbb{L}^d_{\\succ 0} , the gradient only needs to be projected to the lower-triangular subspace (tril). Under Assumption 2.2, C^{-1}(X-m)\\nabla U(X)^{\\top} exactly corresponds to the reparametrization gradient in Eq. (3). At the same time, Eq. (5) points towards an analog of \\nabla^{\\text{price}}_{\\Sigma} for the scale parameter C:", "source": "marker_v2", "marker_block_id": "/page/4/Text/1"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0067", "section": "2.4. Price Gradient Estimators", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\widehat{\\nabla_C^{\\text{price}}} \\mathcal{E}(q_\\lambda;\\epsilon) = C^\\top \\nabla^2 U(X) \\;, \\text{ where } X = \\phi_\\lambda(\\epsilon) \\;. \\eqno(6)", "source": "marker_v2", "marker_block_id": "/page/4/Equation/2"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0068", "section": "2.4. Price Gradient Estimators", "page_start": 5, "page_end": 5, "type": "Text", "text": "Conveniently, this estimator also stays unbiased when \\nabla^2 U is replaced with an unbiased estimator of \\nabla^2 U , enabling doubly stochastic optimization (Titsias & Lázaro-Gredilla, 2014). Similarly, Proposition 2.3 points towards a gradient estimator that could be used in WVI,", "source": "marker_v2", "marker_block_id": "/page/4/Text/3"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0069", "section": "2.4. Price Gradient Estimators", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\widehat{\\nabla_{\\Sigma}^{\\text{rep}}\\mathcal{E}}(q_{\\lambda};\\epsilon) = \\Sigma^{-1}(X-m)\\nabla U(X)^{\\top}, \\text{ where } X = \\phi_{\\lambda}(\\epsilon).", "source": "marker_v2", "marker_block_id": "/page/4/Equation/4"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0070", "section": "2.4. Price Gradient Estimators", "page_start": 5, "page_end": 5, "type": "Text", "text": "Note that similar remarks have already been made by Rezende et al. (2014); Lin et al. (2025); Graves (2011); Opper & Archambeau (2009). Therefore, the use of these estimators is by no means new. However, these Hessian-based estimators have not been widely adopted in BBVI, nor have they been analyzed in detail.", "source": "marker_v2", "marker_block_id": "/page/4/Text/5"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0071", "section": "2.4. Price Gradient Estimators", "page_start": 5, "page_end": 5, "type": "Text", "text": "A natural question here is how much the choice of gradient estimator affects the performance of different algorithms. Past experience in stochastic gradient-based VI has shown that the choice of gradient estimator crucially affects performance both in practice (Kucukelbir et al., 2017; Geffner & Domke, 2020a; 2021; 2018; 2020b; Agrawal et al., 2020; Miller et al., 2017; Wang et al., 2024; Fujisawa & Sato, 2021; Buchholz et al., 2018) and in theory (Kim et al., 2024; Xu et al., 2019; Luu et al., 2025). Indeed, in our theoretical analysis, we will demonstrate that, once we use the same gradient estimator, the state-of-the-art iteration complexities of SPBWGD and SPGD become the same.", "source": "marker_v2", "marker_block_id": "/page/4/Text/6"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0072", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "For our theoretical analysis, we assume the following regularity conditions on {\\cal U}.", "source": "marker_v2", "marker_block_id": "/page/4/Text/9"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0073", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "Assumption 3.1. The potential U: \\mathbb{R}^d \\to \\mathbb{R} is twice differentiable and there exists some \\mu \\in (0, +\\infty) and L \\in [0, +\\infty) such that, for all z \\in \\mathbb{R}^d , the following holds:", "source": "marker_v2", "marker_block_id": "/page/4/Text/10"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0074", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\mu I_d \\quad \\prec \\quad \\nabla^2 U(z) \\quad \\prec \\quad L I_d .", "source": "marker_v2", "marker_block_id": "/page/4/Equation/11"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0075", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "This assumption corresponds to assuming that the density of \\pi is \\mu -log-concave and L-log-smooth, and has been widely used to establish the iteration complexity of stochastic gradient-based VI (Kim et al., 2023; Domke et al., 2023; Lambert et al., 2022; Diao et al., 2023) and sampling algorithms (Chewi, 2024). Crucially, the energy \\mathcal E is now well behaved: On the Bures-Wasserstein geometry, it is", "source": "marker_v2", "marker_block_id": "/page/4/Text/12"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0076", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "\\mu -geodesically convex and L-geodesically smooth (Diao et al., 2023). Similarly, under Assumption 2.2, \\lambda \\mapsto \\mathcal{E}(q_{\\lambda}) is \\mu -strongly convex and L-smooth (Domke, 2020).", "source": "marker_v2", "marker_block_id": "/page/4/Text/13"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0077", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "For stochastic first-order optimization algorithms, the choice of step size schedule is crucial for obtaining tight bounds (Bach & Moulines, 2011). In all cases, we will consider a two-stage step size schedule (Gower et al., 2019; Stich, 2019) of the form of, for some base step size \\gamma_0 \\in (0, +\\infty) , switching time t_* \\geq 0 , and offset \\tau \\geq 0 ,", "source": "marker_v2", "marker_block_id": "/page/4/Text/14"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0078", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\gamma_t = \\begin{cases} \\gamma_0 & \\text{if } t < t_* \\\\ \\frac{1}{\\mu} \\frac{2(t+\\tau)+1}{(t+\\tau+1)^2} & \\text{if } t \\ge t_* \\end{cases} (7)", "source": "marker_v2", "marker_block_id": "/page/4/Equation/15"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0079", "section": "3.1. Theoretical Setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "This two-stage schedule holds the step size constant for a certain period ( t \\in \\{0,\\dots,t_*-1\\} ) and then starts decreasing the step size at a rate of \\gamma_t \\asymp 1/(\\mu t) . The choice of asymptote 1/(\\mu t) ensures an optimal asymptotic convergence rate of \\mathrm{O}(1/T) for strongly convex objectives (Lacoste-Julien et al., 2012; Shamir & Zhang, 2013).", "source": "marker_v2", "marker_block_id": "/page/4/Text/16"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0080", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Text", "text": "We now present the iteration-complexity guarantees for VI with SPBWGD and SPGD using Price's gradient for the scale/covariance component. First, the Bonnet-Price estimator of the Bures-Wasserstein gradient is formally defined in functional form as, for any q = \\text{Normal}(m, \\Sigma) ,", "source": "marker_v2", "marker_block_id": "/page/4/Text/18"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0081", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\nabla_{\\mathrm{BW}}^{\\widehat{\\mathrm{bonnet-price}}} \\mathcal{E}(q; \\epsilon) \\triangleq x \\mapsto \\widehat{\\nabla_{m}^{\\mathrm{bonnet}}} \\mathcal{E}(q; \\epsilon) + 2\\widehat{\\nabla_{\\Sigma}^{\\mathrm{price}}} \\mathcal{E}(q; \\epsilon)(x - m) = x \\mapsto \\nabla U(Z) + \\nabla^{2}U(Z)(x - m) ,", "source": "marker_v2", "marker_block_id": "/page/4/Equation/19"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0082", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Text", "text": "where Z = \\text{cholesky}(\\Sigma)\\epsilon + \\mu and \\epsilon \\sim \\varphi = \\text{Normal}(0_d, I_d) . Then we obtain the following iteration complexity:", "source": "marker_v2", "marker_block_id": "/page/4/Text/20"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0083", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Text", "text": "Theorem 3.2 (SPBWGD). Suppose Assumption 3.1 holds and the gradient estimator \\nabla_{\\rm BW}^{\\rm bonnet-price} \\mathcal{E} is used. Then, for any \\epsilon > 0 , there exists some t_* and \\tau (shown explicitly in the proof) such that running stochastic proximal Bures-Wasserstein gradient descent with the step size schedule in Eq. (7) with \\gamma_0 = 1/(10L\\kappa) guarantees", "source": "marker_v2", "marker_block_id": "/page/4/Text/21"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0084", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Equation", "text": "T \\gtrsim d\\kappa \\frac{1}{\\epsilon} + \\sqrt{d} \\,\\kappa^{3/2} \\log(\\kappa \\Delta^2) \\frac{1}{\\sqrt{\\epsilon}} + \\kappa^2 \\log\\left(\\Delta^2 \\frac{1}{\\epsilon}\\right) \\Rightarrow \\quad \\mu \\mathbb{E}[W_2(q_T, q_*)^2] \\le \\epsilon ,", "source": "marker_v2", "marker_block_id": "/page/4/Equation/22"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0085", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Equation", "text": "where \\Delta^2 = \\mu W(q_0, q_*)^2 .", "source": "marker_v2", "marker_block_id": "/page/4/Equation/23"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0086", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Text", "text": "Proof. The full proof is deferred to Section D.1.3. \\Box", "source": "marker_v2", "marker_block_id": "/page/4/Text/24"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0087", "section": "3.2. Main Results", "page_start": 5, "page_end": 5, "type": "Text", "text": "This improves over the O(d\\kappa\\epsilon^{-1}\\log\\epsilon^{-1} + \\kappa^3\\log(\\Delta\\epsilon^{-1})) complexity obtained by Diao et al. (2023, Thm 5.8). In particular, our result allows for step sizes larger by a factor of \\kappa . Consequently, the dependence on \\kappa in the non-asymptotic term (\\log 1/\\epsilon) is improved by a factor of \\kappa . Furthermore,", "source": "marker_v2", "marker_block_id": "/page/4/Text/25"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0088", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "the asymptotic complexity in \\epsilon \\to 0 is improved by a factor of \\log 1/\\epsilon .", "source": "marker_v2", "marker_block_id": "/page/5/Text/1"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0089", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "Let's now compare this result against the iteration complexity of SPGD. Formally, we define the Bonnet-Price gradient estimator", "source": "marker_v2", "marker_block_id": "/page/5/Text/2"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0090", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\nabla_{\\lambda}^{\\widehat{\\text{bonnet-price}}} \\mathcal{E}(q_{\\lambda}; \\epsilon) \\triangleq \\begin{bmatrix} \\widehat{\\nabla_{m}^{\\text{bonnet}}} \\mathcal{E}(q_{\\lambda}; \\epsilon) \\\\ \\widehat{\\nabla_{C}^{\\text{price}}} \\mathcal{E}(q_{\\lambda}; \\epsilon) \\end{bmatrix} = \\begin{bmatrix} \\nabla U(Z) \\\\ C^{\\top} \\nabla^{2} U(Z) \\end{bmatrix} \\ .", "source": "marker_v2", "marker_block_id": "/page/5/Equation/3"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0091", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "Using this estimator in BBVI with PSGD results in the following iteration complexity.", "source": "marker_v2", "marker_block_id": "/page/5/Text/4"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0092", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "Theorem 3.3 (SPGD). Suppose Assumption 3.1 holds and the gradient estimator \\nabla_{\\lambda}^{\\text{bonnet-price}} \\mathcal{E} is used. Then, for any \\epsilon > 0 , there exists some t_* and \\tau (stated explicitly in the proof) such that running stochastic proximal gradient descent with the step size schedule in Eq. (7) with \\gamma_0 = 1/(10L\\kappa) guarantees", "source": "marker_v2", "marker_block_id": "/page/5/Text/5"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0093", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Equation", "text": "T \\gtrsim d\\kappa \\frac{1}{\\epsilon} + \\sqrt{d} \\,\\kappa^{3/2} \\log(\\kappa \\Delta^2) \\frac{1}{\\sqrt{\\epsilon}} + \\kappa^2 \\log\\left(\\Delta^2 \\frac{1}{\\epsilon}\\right) \\Rightarrow \\quad \\mu \\mathbb{E}[W_2(q_T, q_*)^2] \\le \\epsilon ,", "source": "marker_v2", "marker_block_id": "/page/5/Equation/6"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0094", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "where \\Delta^2 = \\mu \\|\\lambda_0 - \\lambda_*\\|^2 .", "source": "marker_v2", "marker_block_id": "/page/5/Text/7"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0095", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "Proof. The full proof is deferred to Section D.2.1. \\Box", "source": "marker_v2", "marker_block_id": "/page/5/Text/8"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0096", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "Previously, for Gaussian variational families with a dense covariance (the \"full-rank\" Gaussian family), in the limit of \\epsilon \\to 0 , Kim et al. (2023); Domke et al. (2023) reported an iteration complexity of O(d\\kappa^2\\operatorname{tr}(\\mu\\Sigma_*)\\epsilon^{-1}) , which used the canonical reparametrization gradient (Eq. (4)). Compared to this, Price's gradient improves the iteration complexity by a factor of \\kappa\\operatorname{tr}(\\mu\\Sigma_*) . (Note that d/L \\le \\operatorname{tr}(\\Sigma_*) \\le d/\\mu .) This is comparable to the complexity of BBVI with the mean-field Gaussian family (diagonal covariance), which is O((\\log d)\\kappa^2\\operatorname{tr}(\\mu\\Sigma_*)\\epsilon^{-1}) (Kim et al., 2025). This suggests that, with Price's gradient, BBVI on a full-rank Gaussian family can be as fast as using a mean-field Gaussian family and the reparametrization gradient.", "source": "marker_v2", "marker_block_id": "/page/5/Text/9"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0097", "section": "3.2. Main Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "An immediate implication of Theorems 3.2 and 3.3 is that the gap between the best known iteration complexity bounds between the two algorithms has now been closed. In addition, Section 3.3 that follows will explain that this resemblance is unsurprising, as the convergence analyses of both algorithms rely on nearly the same properties. Though, since we lack matching lower bounds, we cannot yet claim that the two algorithms behave exactly the same. However, our results do provide evidence towards this fact along with the continuous-time results of Yi & Liu (2023); Hoffman & Ma (2020). This again reinforces the intuition that, for stochastic optimization algorithms, the quality of the gradient estimator has the largest impact on the performance.", "source": "marker_v2", "marker_block_id": "/page/5/Text/10"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0098", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "The overall structure of the proofs for both SPBWGD and SPGD is identical. If we had access to exact gradients instead of stochastic estimates, under Assumption 3.1, \\|\\lambda_t - \\lambda_*\\|_2 or W_2(q_t, q_*) would contract exponentially in t. When dealing with stochastic gradients, however, the noise in the estimates perturbs the iterates. We thus need to show that the variance of the noise is bounded and the contraction is strong enough such that controlling the step size schedule \\gamma_t can neutralize the perturbations.", "source": "marker_v2", "marker_block_id": "/page/5/Text/12"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0099", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "First, under Assumption 3.1, we can define a Bregman divergence associated with U,", "source": "marker_v2", "marker_block_id": "/page/5/Text/13"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0100", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Equation", "text": "D_U(x,y) \\triangleq U(x) - U(y) - \\langle \\nabla U(y), x - y \\rangle.", "source": "marker_v2", "marker_block_id": "/page/5/Equation/14"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0101", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "For both SPBWGD and SPGD, we establish gradient variance bounds involving \\mathrm{D}_U .", "source": "marker_v2", "marker_block_id": "/page/5/Text/15"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0102", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "Lemma 3.4. Suppose Assumption 3.1 holds and q_* = \\operatorname{Normal}(\\mu_*, \\Sigma_*) \\in \\arg\\min_{q \\in \\operatorname{BW}(\\mathbb{R}^d)} \\mathcal{F}(q) . Then, for any q \\in \\operatorname{BW}(\\mathbb{R}^d) , and any coupling \\psi \\in \\Psi(q, q_*) ,", "source": "marker_v2", "marker_block_id": "/page/5/Text/16"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0103", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\mathbb{E}_{(X,X_*) \\sim \\psi, \\epsilon \\sim \\varphi} \\left[ \\|\\nabla_{\\mathrm{BW}}^{\\text{bonnet-price}} \\mathcal{E}(q;\\epsilon)(X) - \\nabla \\mathcal{E}(q_*)(X_*) \\|_2^2 \\right] \\\\ \\leq 10 L \\kappa \\, \\mathbb{E}_{(X,X_*) \\sim \\psi} \\left[ \\mathrm{D}_U(X,X_*) \\right] + 10 dL \\; .", "source": "marker_v2", "marker_block_id": "/page/5/Equation/17"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0104", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "Proof. The proof is deferred to Section D.1.4. \\Box", "source": "marker_v2", "marker_block_id": "/page/5/Text/18"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0105", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "This is a refinement of Lemma 5.6 by Diao et al. (2023). Specifically, instead of upper-bounding the gradient variance with the squared Wasserstein distance W_2(q, q_*)^2 , we bound it with the Bregman divergence D_U . In fact, for the optimal coupling \\psi^* \\in \\Psi(q, q_*) , we have", "source": "marker_v2", "marker_block_id": "/page/5/Text/19"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0106", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\mathbb{E}_{(X,X_*)\\sim\\psi^*}[D_U(X,X_*)] \\le LW_2(q,q_*)^2", "source": "marker_v2", "marker_block_id": "/page/5/Equation/20"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0107", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "(Eq. (11) in Section C.1.) Therefore, the use of the Bregman divergence avoids paying for an extra factor of \\kappa=L/\\mu . The corresponding bound for parameter-space SPGD has the exact same constants.", "source": "marker_v2", "marker_block_id": "/page/5/Text/21"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0108", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "Lemma 3.5. Suppose Assumptions 3.1 and 2.2 holds, and \\lambda_* \\in \\arg\\min_{\\lambda \\in \\Lambda} \\mathcal{F}(q_{\\lambda}) . Then, for any \\lambda \\in \\Lambda and any coupling \\psi \\in \\Psi(q_{\\lambda}, q_{\\lambda_*}) ,", "source": "marker_v2", "marker_block_id": "/page/5/Text/22"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0109", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\begin{split} \\mathbb{E}_{\\epsilon \\sim \\varphi} \\big[ \\| \\nabla_{\\lambda}^{\\widehat{\\text{bonnet-price}}} \\mathcal{E}(q_{\\lambda}; \\epsilon) - \\nabla \\mathcal{E}(q_{\\lambda_*}) \\|_2^2 \\big] \\\\ & \\leq 10 L \\kappa \\, \\mathbb{E}_{(X, X_*) \\sim \\psi} \\big[ \\mathcal{D}_U(X, X_*) \\big] + 10 dL \\; . \\end{split}", "source": "marker_v2", "marker_block_id": "/page/5/Equation/23"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0110", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "Proof. The proof is deferred to Section D.2.2. \\Box", "source": "marker_v2", "marker_block_id": "/page/5/Text/24"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0111", "section": "3.3. Proof Sketch", "page_start": 6, "page_end": 6, "type": "Text", "text": "The bounds Lemmas 3.4 and 3.5, however, are not immediately usable for a convergence analysis. The \"growth\" or \"multiplicative noise\" term \\mathbb{E}[\\mathrm{D}_U(X,X_*)] needs to be related to the growth of \\mathcal{E} . Notice that both Lemmas 3.4 and 3.5 hold for any coupling in \\Psi(q,q_*) . By specifying the coupling \\psi , we will invoke properties of the geometry associated with \\psi induced by each SPGD and SPBWGD, which will allow us to relate \\mathbb{E}[\\mathrm{D}_U(X,X_*)] with the appropriate notion of growth of \\mathcal{E} .", "source": "marker_v2", "marker_block_id": "/page/5/Text/25"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0112", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Let's start with SPBWGD. For the optimal coupling \\psi^* \\in \\Psi(q,q_*) , we will define the Wasserstein analog of the Bregman divergence", "source": "marker_v2", "marker_block_id": "/page/6/Text/1"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0113", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "D_{\\mathcal{E}}(q, q_*) \\triangleq \\mathcal{E}(q) - \\mathcal{E}(q_*) - \\mathbb{E}_{(X, X_*) \\sim \\psi_*} \\langle \\nabla \\mathcal{E}(q_*)(X_*), X - X_* \\rangle \\ge 0. (8)", "source": "marker_v2", "marker_block_id": "/page/6/Equation/2"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0114", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "The non-negativity of D_{\\mathcal{E}} follows from the fact that \\mathcal{E} is geodesically convex under Assumption 3.1 (Lemma D.1). Then the Bures-Wasserstein geometry yields the following:", "source": "marker_v2", "marker_block_id": "/page/6/Text/3"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0115", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Lemma 3.6. For any p, q \\in BW(\\mathbb{R}^d) , denote the coupling optimal for the squared Euclidean cost between p and q as \\psi_* \\in \\Psi(p,q) . Then", "source": "marker_v2", "marker_block_id": "/page/6/Text/4"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0116", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "\\mathbb{E}_{(X,Y)\\sim\\psi_*}[D_U(X,Y)] = D_{\\mathcal{E}}(p,q) .", "source": "marker_v2", "marker_block_id": "/page/6/Equation/5"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0117", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Proof. The proof is deferred to Section D.1.5. \\Box", "source": "marker_v2", "marker_block_id": "/page/6/Text/6"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0118", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "For SPGD, first consider some \\lambda, \\lambda' \\in \\Lambda , where \\lambda = (m, \\text{vec } C) and \\lambda' = (m', \\text{vec } C') and the map", "source": "marker_v2", "marker_block_id": "/page/6/Text/7"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0119", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "M_{q_{\\lambda}\\mapsto q_{\\lambda'}}^{\\text{rep}}(z) \\triangleq C'C^{-1}(z-m) + m' .", "source": "marker_v2", "marker_block_id": "/page/6/Equation/8"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0120", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "This is, in fact, a transport map from q_{\\lambda} to q_{\\lambda'} . Then, under Assumption 2.2, the identities", "source": "marker_v2", "marker_block_id": "/page/6/Text/9"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0121", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "\\|\\lambda - \\lambda'\\|_2^2 = \\mathbb{E}_{Z \\sim q} \\|Z - M_{q_{\\lambda} \\mapsto q_{\\lambda'}}^{\\text{rep}}(Z)\\|_2^2 (9)", "source": "marker_v2", "marker_block_id": "/page/6/Equation/10"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0122", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "and", "source": "marker_v2", "marker_block_id": "/page/6/Text/11"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0123", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "353354", "source": "marker_v2", "marker_block_id": "/page/6/Text/53"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0124", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "\\langle \\nabla_{\\lambda} \\mathcal{E}(q_{\\lambda'}), \\lambda - \\lambda' \\rangle = \\mathbb{E}_{Z' \\sim q_{\\lambda'}} \\langle \\nabla U(Z'), M_{q_{\\lambda'} \\mapsto q_{\\lambda}}^{\\text{rep}}(Z') - Z' \\rangle \\quad (10)", "source": "marker_v2", "marker_block_id": "/page/6/Equation/12"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0125", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "hold. Also, from the fact that the Wasserstein distance is the cost of the optimal coupling, we have the ordering \\|\\lambda-\\lambda'\\|_2 \\geq W_2(q_\\lambda,q_{\\lambda'}) . That is, the metric associated with parameter space is a coupling-based distance measure. Now, under Assumptions 2.2 and 3.1, \\lambda\\mapsto \\mathcal{F}(q_\\lambda) is \\mu -strongly convex. Then it is well known that the gradient flow minimizes \\|\\lambda_t-\\lambda_*\\|_2^2 exponentially in time. The identity Eq. (9) implies that this flow is also minimizing a coupling distance, and in turn the Wasserstein distance. Back to the proof sketch, Eq. (10) implies the following:", "source": "marker_v2", "marker_block_id": "/page/6/Text/13"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0126", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Lemma 3.7. Suppose Assumption 2.2 hold. Then, for any \\lambda, \\lambda' \\in \\Lambda , denote the coupling induced by the transport map M_{q_{\\lambda} \\mapsto q_{\\lambda'}}^{\\text{rep}} as \\psi^{\\text{rep}} . Then", "source": "marker_v2", "marker_block_id": "/page/6/Text/14"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0127", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "\\mathbb{E}_{(X,X')\\sim\\psi^{\\text{rep}}}[D_U(X,X')] = D_{\\lambda\\mapsto\\mathcal{E}(q_\\lambda)}(\\lambda,\\lambda') .", "source": "marker_v2", "marker_block_id": "/page/6/Equation/15"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0128", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Proof. The proof is deferred to Section D.2.3. \\Box", "source": "marker_v2", "marker_block_id": "/page/6/Text/16"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0129", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Therefore, the optimal coupling \\psi^* and the coupling associated with Assumption 2.2 \\psi^{\\rm rep} respectively retrieve a relationship with the growth of \\mathcal{E} .", "source": "marker_v2", "marker_block_id": "/page/6/Text/17"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0130", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Lastly, we need to show that the contraction is able to counteract the \"growth\" of the gradient variance. Typically, the contraction of first-order methods follows from the coercivity of the gradient operator. For our proof, however, instead of obtaining a full contraction, we establish a slight generalization of coercivity (previously developed by Gorbunov", "source": "marker_v2", "marker_block_id": "/page/6/Text/18"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0131", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "et al. 2020), that allow us to control the Bregman terms D_{\\lambda \\mapsto \\mathcal{E}(q_{\\lambda})} and D_{\\mathcal{E}} . For SPBWGD, our result reads:", "source": "marker_v2", "marker_block_id": "/page/6/Text/19"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0132", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Lemma 3.8. Suppose Assumption 3.1 holds. Then, for any q \\in BW(\\mathbb{R}^d) and q_* = \\arg\\min_{q \\in BW(\\mathbb{R}^d)} \\mathcal{F}(q) , where we denote their coupling \\psi_* \\in \\Psi(q, q_*) optimal in terms of squared Euclidean distance,", "source": "marker_v2", "marker_block_id": "/page/6/Text/20"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0133", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "\\mathbb{E}_{(X,Y)\\sim\\psi_*}\\langle\\nabla_{\\mathrm{BW}}\\mathcal{E}(p)(X) - \\nabla_{\\mathrm{BW}}\\mathcal{E}(q)(Y), X - Y\\rangle \\geq \\frac{\\mu}{2} \\operatorname{W}_2(q_t, q_*)^2 + \\operatorname{D}_{\\mathcal{E}}(q_t, q_*).", "source": "marker_v2", "marker_block_id": "/page/6/Equation/21"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0134", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Proof. The proof is deferred to Section D.1.6.", "source": "marker_v2", "marker_block_id": "/page/6/Text/22"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0135", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "The corresponding result for parameter space SPGD is:", "source": "marker_v2", "marker_block_id": "/page/6/Text/23"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0136", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Lemma 3.9. Suppose Assumptions 3.1 and 2.2 hold. Then, for any \\lambda \\in \\Lambda and \\lambda_* = \\arg\\min_{\\lambda \\in \\Lambda} \\mathcal{F}(q_{\\lambda}) ,", "source": "marker_v2", "marker_block_id": "/page/6/Text/24"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0137", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Equation", "text": "\\begin{split} \\langle \\nabla_{\\lambda_t} \\mathcal{E}(q_{\\lambda_t}) - \\nabla_{\\lambda_*} \\mathcal{E}(q_{\\lambda_*}), \\lambda_t - \\lambda_* \\rangle \\\\ &\\geq \\frac{\\mu}{2} \\|\\lambda - \\lambda_*\\|_2^2 + D_{\\lambda \\mapsto \\mathcal{E}(q_{\\lambda})}(\\lambda, \\lambda_*) \\;. \\end{split}", "source": "marker_v2", "marker_block_id": "/page/6/Equation/25"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0138", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "Proof. The proof is deferred to Section D.2.4. \\Box", "source": "marker_v2", "marker_block_id": "/page/6/Text/26"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0139", "section": "3.3. Proof Sketch", "page_start": 7, "page_end": 7, "type": "Text", "text": "The extra \"Bregman term\" on the right-hand sides directly allows control over the Bregman term in the gradient variance bounds.", "source": "marker_v2", "marker_block_id": "/page/6/Text/27"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0140", "section": "4. Empirical Analysis", "page_start": 7, "page_end": 7, "type": "Text", "text": "For the empirical evaluation, we will compare the performance of VI with SPGD, SPBWGD, and NGVI with the reparametrization and Price estimators.", "source": "marker_v2", "marker_block_id": "/page/6/Text/29"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0141", "section": "4. Empirical Analysis", "page_start": 7, "page_end": 7, "type": "Text", "text": "Setup and Implementation. For SPGD and SPBWGD, we follow the implementation described in Section 2.3 and Section 2.2, respectively. As mentioned in Section 2.2, the reparamtrization gradient \\widehat{\\nabla_{\\Sigma}^{\\text{rep}}\\mathcal{E}} is not almost surely symmetric. Therefore, the modified SPBWGD implementation in Section 2.2 is used. All algorithms were implemented in the Julia language (Bezanson et al., 2017) and the AdvancedVI.jl library (v0.6.1), which is part of the Turing probabilistic programming ecosystem (Ge et al., 2018; Fjelde et al., 2025)<sup>2</sup>. The experimental problems were taken from the PosteriorDB (Magnusson et al., 2025) benchmark suite of Stan models (Carpenter et al., 2017), which was made accessible from Julia through the BridgeStan interface (Roualdes et al., 2023). The benchmark problems are described in more detail in Section A. All methods are initialized at q_0 = \\text{Normal}(0_d, 0.34 \\, \\text{I}_d) . The gradients were estimated using 8 Monte Carlo samples in all cases, while for evaluation, \\mathcal{F} was estimated using 2^{12} samples. We run both SPGD and SPBWGD with a fixed step size \\gamma and estimate the free energy \\mathcal{F}(q_t) of the iterate q_t at each iteration.", "source": "marker_v2", "marker_block_id": "/page/6/Text/30"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0142", "section": "4. Empirical Analysis", "page_start": 7, "page_end": 7, "type": "Footnote", "text": "& lt;sup>2</sup>All of the code needed to reproduce the results is available in the following repository:", "source": "marker_v2", "marker_block_id": "/page/6/Footnote/31"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0143", "section": "4. Empirical Analysis", "page_start": 8, "page_end": 8, "type": "FigureGroup", "text": "Figure 1. Variational free energy ( \\mathcal{F} ) at T=4000 versus step size \\gamma . The top 8 problems with the largest dimensionality d are shown here, while the full set of results can be found in Section B. Refer to the main text for why the dotted lines are missing on Rats. The solid lines are the mean estimated over 32 independent repetitions, while the shaded regions are the 95% bootstrap confidence intervals.", "source": "marker_v2", "marker_block_id": "/page/7/FigureGroup/447"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0144", "section": "4. Empirical Analysis", "page_start": 8, "page_end": 8, "type": "Text", "text": "Results. Part of the results are shown in Fig. 1, while the full set of results can be found in Section B. First, when using Price's gradient (solid line), both SPGD and SPB-WGD achieve similar performance, except for Rats, where SPGD remains stable over a wider range of step sizes. On the other hand, when using the reparametrization gradient, both SPGD and SPBWGD perform poorly: they require smaller step sizes. In fact, on Rats, none of the methods using the reparametrization gradient converged for all step sizes between 10^{-8} and 10^{0} . In addition, on some problems, SPBWGD appears to perform worse than SPGD. For example, on Rats with Price's gradient, SPBWGD requires step sizes orders of magnitude smaller to prevent divergence. A possible explanation is that SPBWGD requires an estimator of \\nabla_{\\Sigma} \\mathcal{E}(q) , whereas SPGD uses an estimator for \\nabla_C \\mathcal{E}(q) . These two are related through the chain rule as (1/2)C^{-\\top}\\nabla_{\\Sigma}\\mathcal{E}(q) = \\nabla_{C}\\mathcal{E}(q) ; the extra scaling of C^{-\\top} could make an estimator for \\nabla_{\\Sigma} \\mathcal{E}(q) noisier than \\nabla_{C} \\mathcal{E}(q) .", "source": "marker_v2", "marker_block_id": "/page/7/Text/3"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0145", "section": "5. Discussions", "page_start": 8, "page_end": 8, "type": "Text", "text": "In this work, we theoretically analyzed stochastic gradient-based VI algorithms operating in the Euclidean space of parameters (SPGD) and Bures-Wasserstein space (SPBWGD). Our results improve upon the state-of-the-art complexity guarantees for both, closing the gap between them. For SPBWGD, we have technically improved the previous results by (Diao et al., 2023). Meanwhile, for SPGD, we have shown that the use of the Price gradient \\nabla_C^{\\text{price}} \\mathcal{E} achieves better theoretical guarantees than those obtained (Domke et al., 2023; Kim et al., 2023) under the reparametrization gradient \\nabla_C^{\\text{rep}} \\mathcal{E} . This shows that the previously observed advantage of SPBWGD was due to the choice of gradient", "source": "marker_v2", "marker_block_id": "/page/7/Text/5"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0146", "section": "5. Discussions", "page_start": 8, "page_end": 8, "type": "Text", "text": "estimator rather than the geometry.", "source": "marker_v2", "marker_block_id": "/page/7/Text/6"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0147", "section": "5. Discussions", "page_start": 8, "page_end": 8, "type": "Text", "text": "However, this doesn't completely rule out the possibility that measure-space algorithms can be more effective. NGVI, which uses the Fisher metric, yields a preconditioned update to the location parameter m_t reminiscent of Newton's method (Khan & Rue, 2023). Possibly due to this, empirical evidence suggests that NGVI methods can converge significantly faster than BBVI (Lin et al., 2019). However, our theoretical understanding of NGVI is still limited, where existing analyses assume conjugacy (Wu & Gardner, 2024) or assumptions much stronger than those considered in this work (Sun et al., 2025; Kumar et al., 2025).", "source": "marker_v2", "marker_block_id": "/page/7/Text/7"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0148", "section": "5. Discussions", "page_start": 8, "page_end": 8, "type": "Text", "text": "On a practical note, it isn't clear if Price's gradient is always better. For instance, at each iteration, SPGD with the reparametrization gradient requires \\Omega(d^2) operations (matrix-vector product for computing \\phi_{\\lambda} ). Moving to second-order increases the cost to \\Omega(d^3) operations (matrix-matrix product for computing C\\nabla^2 U ). Meanwhile, SPB-WGD requires \\Omega(d^3) in both cases. Therefore, when \\nabla U can be computed in O(d^2) time, as d\\to\\infty , BBVI with SPGD and the reparametrization gradient could be more efficient (\\Theta(d^2) versus \\Theta(d^3) ) depending on the conditioning \\kappa . In addition, WVI requires numerically sensitive operations, such as matrix square roots and Cholesky decompositions, which makes it less robust.", "source": "marker_v2", "marker_block_id": "/page/7/Text/8"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0149", "section": "Impact Statement", "page_start": 9, "page_end": 9, "type": "Text", "text": "This work studies the theoretical properties of variational inference, which is a collection of algorithms for approximating distributions. Therefore, our work is not expected to have direct societal consequences other than those of downstream applications of variational inference.", "source": "marker_v2", "marker_block_id": "/page/8/Text/2"}

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+      "title": "Stochastic Gradient Variational Inference with Price's Gradient Estimator\nfrom Bures-Wasserstein to Parameter Space",
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+      "title": "1. Introduction",
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+[p. 1 | section: Abstract | type: Text]
+For approximating a target distribution given only its unnormalized log-density, stochastic gradientbased variational inference (VI) algorithms are a popular approach. For example, Wasserstein VI (WVI) and black-box VI (BBVI) perform gradient descent in measure space (Bures-Wasserstein space) and parameter space, respectively. Previously, for the Gaussian variational family, convergence guarantees for WVI have shown superiority over existing results for black-box VI with the reparametrization gradient, suggesting the measure space approach might provide some unique benefits. In this work, however, we close this gap by obtaining identical state-of-the-art iteration complexity guarantees for both. In particular, we identify that WVI's superiority stems from the specific gradient estimator it uses, which BBVI can also leverage with minor modifications. The estimator in question is usually associated with Price's theorem and utilizes second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient. On the flip side, WVI can be made more widely applicable by using the reparametrization gradient, which requires only gradients of the log-density. We empirically demonstrate that the use of Price's gradient is the major source of performance improvement.
+
+[p. 1 | section: 1. Introduction | type: Text]
+Variational inference (VI; Jordan et al., 1998; Blei et al., 2017; Peterson & Hartman, 1989; Hinton & van Camp, 1993) is a collection of algorithms for approximating a target distribution π over some family of parametric distributions Q when only the unnormalized density of π, denoted by π e, is available. When π is supported on R d such that its
+
+[p. 1 | section: 1. Introduction | type: Text]
+associated potential function U = − log π e is in R d → R, it is common to leverage stochastic gradient-based algorithms, as they only require local information of U (Graves, 2011; Salimans & Knowles, 2013; Wingate & Weber, 2013; Titsias & Lázaro-Gredilla, 2014; Ranganath et al., 2014; Kingma & Welling, 2014; Rezende et al., 2014) . Most VI algorithms are designed to minimize the variational free energy, also known as the negative evidence lower bound (Jordan et al., 1999) , defined as F(q) ≜ E(q) + H(q), where E is the energy functional associated with U, while H is the Boltzmann entropy. That is, we solve
+
+[p. 1 | section: 1. Introduction | type: Equation]
+\underset{q \in \mathcal{Q}}{\text{minimize}} \left\{ \mathcal{F}(q) = \text{KL}(q, \pi) + \mathcal{F}(\pi) \right\}
+
+[p. 1 | section: 1. Introduction | type: Text]
+through only zeroth-, first-, and second-order information of U. Since F is equal to the exclusive KL divergence (Kull back & Leibler, 1951) between q and π up to the constant F(π), this also minimizes q 7→ KL(q, π).
+
+[p. 1 | section: 1. Introduction | type: Text]
+A common approach for minimizing F is, informally speaking, to leverage some sort of stochastic gradient descent (SGD; Robbins & Monro, 1951; Bottou, 1999; Bottou et al., 2018; Bach & Moulines, 2011; Nemirovski et al., 2009; Shalev-Shwartz et al., 2011) scheme, where intractable terms (such as the gradient of the energy E) are stochastically estimated. There are two popular ways to realize this conceptual algorithm. The most widely used approach in practice is to represent each q λ ∈ Q via a Euclidean vector of variational parameters λ ∈ Λ, and run gradient descent on the Euclidean space of parameters Λ ⊆ R p (Wingate & Weber, 2013; Kucukelbir et al., 2017; Titsias & Lázaro- Gredilla, 2014; Ranganath et al., 2014; Salimans & Knowles, 2013) . This is now referred to as black-box variational in ference (BBVI). The other approach is to define a tractable notion of measure-valued derivatives, which can directly perform gradient descent in measure space. In particular, recent advances in our understanding of the Wasserstein geometry (Villani, 2009; Chewi et al., 2025) and gradient flows (Ambrosio et al., 2005; Jordan et al., 1998) have contributed to the development of (parametric 1 ) Wasserstein variational inference (WVI; Lambert et al., 2022; Diao et al., 2023; Huix et al., 2024; Talamon et al., 2025) algorithms
+
+[p. 1 | section: 1. Introduction | type: Footnote]
+1 We focus on WVI on parametric families, which excludes particle-based WVI methods.
+
+[p. 2 | section: 1. Introduction | type: Table]
+Table 1. Overview of Main Theoretical Results. Algorithm Space Gradient Estimator Maximum Step Size Iterations Complexity Reference SPGD Λ Reparam. 1/(dL\kappa) d\kappa^2 {\rm tr}(\mu \Sigma_*)^1\!/\epsilon Domke et al. (2023) Kim et al. (2023) SPBWGD \mathrm{BW}(\mathbb{R}^d) Bonnet-Price 1/(L\kappa^2) d\kappa \frac{1}{\epsilon} \log \frac{1}{\epsilon} + \kappa^3 \log(\Delta^2 \frac{1}{\epsilon}) Diao et al. (2023) SPGD \Lambda Bonnet-Price 1/(L\kappa) d\kappa \frac{1}{\epsilon} + \sqrt{d} \kappa^{3/2} \log(\kappa \Delta^2) \frac{1}{\sqrt{\epsilon}} + \kappa^2 \log \frac{1}{\epsilon} Theorem 3.3 SPBWGD \mathrm{BW}(\mathbb{R}^d) Bonnet-Price 1/(L\kappa) d\kappa \frac{1}{\epsilon} + \sqrt{d} \kappa^{3/2} \log(\kappa \Delta^2) \frac{1}{\sqrt{\epsilon}} + \kappa^2 \log \frac{1}{\epsilon} Theorem 3.2
+
+[p. 2 | section: 1. Introduction | type: Text]
+Note: The complexity statements assume \mu -strong convexity and L-smoothness of U (Assumption 3.1); \epsilon > 0 is the target accuracy level for ensuring \mu \mathbb{E} W_2(q_T, q_*)^2 \le \epsilon , where q_T is the last iterate and q_* is the global minimizer of \mathcal{F} ; \kappa = L/\mu is the condition number; \Delta is the distance between the initialization and the optimum.
+
+[p. 2 | section: 1. Introduction | type: Text]
+that take this route. We also note that natural gradient VI algorithms (Khan & Rue, 2023; Khan & Nielsen, 2018; Lin et al., 2019; Tan, 2025) utilize parameter gradients while measuring distance using the KL pseudo-metric. However, NGVI will not be the focus of this work.
+
+[p. 2 | section: 1. Introduction | type: Text]
+Given that the WVI methods utilize a proper metric (Villani, 2009) between measures—the Wasserstein-2 metric W_2 —it is natural to expect them to outperform BBVI. Indeed, current theoretical evidence suggests that this is the case. Consider the Gaussian variational family, also known as the Bures-Wasserstein space (Bures, 1969; Bhatia et al., 2019), set as
+
+[p. 2 | section: 1. Introduction | type: Equation]
+Q = \mathrm{BW}(\mathbb{R}^d) \triangleq \{ \mathrm{Normal}(m, \Sigma) \mid m \in \mathbb{R}^d, \Sigma \in \mathbb{S}^d_{\succ 0} \} .
+
+[p. 2 | section: 1. Introduction | type: Text]
+Here, (\mathrm{BW}(\mathbb{R}^d), \mathrm{W}_2) forms a metric space. Denoting the global minimizer as q_* = \mathrm{Normal}(m_*, \Sigma_*) = \mathrm{arg} \min_{q \in \mathcal{Q}} \mathcal{F}(q) , for a \mu -strongly convex and L-smooth potential U, the algorithm by Diao et al. (2023) requires \mathrm{O}(d\kappa\epsilon^{-1}\log\epsilon^{-1}+\kappa^3\log\epsilon^{-1}) steps to ensure \mu\mathbb{E}[\mathrm{W}_2(q,q_*)^2] \leq \epsilon . In contrast, the BBVI equivalent with a certain covariance parametrization, the reparametrization gradient estimator (Ho & Cao, 1983; Rubinstein, 1992), and stochastic proximal gradient descent (SPGD; Nemirovski et al., 2009) requires \mathrm{O}(d\kappa^2\mathrm{tr}(\mu\Sigma_*)\epsilon^{-1}) steps (Kim et al., 2023; Domke et al., 2023). It might therefore appear that the guarantees for BBVI are weaker in the limit \epsilon \to 0 .
+
+[p. 2 | section: 1. Introduction | type: Text]
+In this work, we demonstrate that the difference in theoretical guarantees originates from the specific gradient estimator for the scale parameter used in stochastic implementations of WVI rather than the geometry being used. The estimator in question can be derived via Price's theorem (Price, 1958) and leverages second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient estimator. Through Stein (Liu, 1994; Stein, 1981) or Price's theorem, BBVI with SPGD can also make use of essentially the same gradient estimator, resulting in an iteration complexity of O(d\kappa\epsilon^{-1} + \sqrt{d}\,\kappa^{3/2}\log(\kappa\Delta^2)\epsilon^{-1/2} + \kappa^2\log\epsilon^{-1}) . Furthermore, to ensure a fair comparison, we present a refined analysis of the WVI counterpart of SPGD (Diao et al.,
+
+[p. 2 | section: 1. Introduction | type: Text]
+2023), resulting in an iteration complexity of O(d\kappa\epsilon^{-1} + \sqrt{d} \kappa^{3/2} \log(\kappa \Delta^2) \epsilon^{-1/2} + \kappa^2 \log \epsilon^{-1}) . These results suggest that the specific implementation of BBVI studied here and WVI might not be that different after all. While this has been suggested by Yi & Liu (2023); Hoffman & Ma (2020), this work further supports this fact by contributing a non-asymptotic discrete-time analysis. Our theoretical results are organized in Table 1.
+
+[p. 2 | section: 1. Introduction | type: Text]
+In addition, we demonstrate that WVI can also leverage the reparametrization gradient traditionally used in BBVI (Titsias & Lázaro-Gredilla, 2014; Kingma & Welling, 2014; Rezende et al., 2014). Unlike Price's gradient previously used in WVI, the reparametrization gradient only requires first-order information ( \nabla U ). Thus, the resulting WVI algorithm should be more widely applicable in practice. Given this fact, we empirically compare the performance of BBVI and WVI, both with the Hessian-based gradient estimators and the reparametrization gradient. Results demonstrate that a large fraction of the performance difference stems from the use of Hessian-based gradients, supporting the claim that the gradient estimator is the main source of performance.
+
+[p. 2 | section: 2. Background | type: Text]
+Notation. For any x,y \in \mathbb{R}^d , we denote the Euclidean inner product and norm as \langle x,y \rangle = x^\top y and \|x\|_2 \triangleq \sqrt{\langle x,x \rangle} . For any matrix A,B \in \mathbb{R}^{d \times d} , \operatorname{tr}(A) \triangleq \sum_{i=1}^d A_{ii} , \langle A,B \rangle_F \triangleq \operatorname{tr}(A^\top B) , \|A\|_F \triangleq \sqrt{\langle A,A \rangle_F} , and the \ell_2 operator norm of A is denoted as \|A\|_2 . The symmetric and positive definite subsets of \mathbb{R}^{d \times d} will be denoted as \mathbb{S}^d and \mathbb{S}^d_{\succ 0} , while \mathbb{L}^d_{\succ 0} denotes the set of lower-triangular matrices with strictly positive eigenvalues. We will represent a measure and its density with the same symbol. For the space of square-integrable measures \mathcal{P}_2(\mathcal{X}) \triangleq \{q \mid \int_{\mathcal{X}} \|x\|^2 \mathrm{d}q(x) < +\infty\} , for some q \in \mathcal{P}_2(\mathbb{R}^d) , the set of integrable functions is denoted as \mathrm{L}^2(q) \triangleq \{f \mid \int \|f\|_2^2 \mathrm{d}q < +\infty\} . For any two probability measures p,q \in \mathcal{P}_2(\mathbb{R}^d) , we denote the set of couplings between the two as \Psi(p,q) . Then the squared Wasserstein-2 distance between p and q is \mathrm{W}_2(p,q)^2 \triangleq
+
+[p. 3 | section: 2. Background | type: Text]
+\inf_{\psi\in\Psi(p,q)}\int_{\mathbb{R}^d\times\mathbb{R}^d}\|x-y\|_2^2\,\mathrm{d}\psi(x,y). For some measurable map M:\mathbb{R}^d\to\mathbb{R}^d and measure q supported on \mathbb{R}^d , M_{\#q} denotes the corresponding push-forward measure. Unless stated otherwise, the coupling attaining the infimum of W_2,\,\psi^*\in\Psi(p,q) , is referred to as "the optimal coupling," which is guaranteed to exist (Villani, 2009, Theorem 4.1) and is unique by Brenier's Theorem (Brenier, 1991).
+
+[p. 3 | section: 2.1. Problem Setup | type: Text]
+Our focus will be on first-order stochastic optimization algorithms for solving the problem
+
+[p. 3 | section: 2.1. Problem Setup | type: Equation]
+\underset{q \in \mathcal{Q}}{\text{minimize}} \ \left\{ \mathcal{F}(q) \triangleq \mathcal{E}(q) + \mathcal{H}(q) \right\},
+
+[p. 3 | section: 2.1. Problem Setup | type: Equation]
+\begin{array}{ll} \text{where} & \mathcal{E}(q) & \triangleq \int_{\mathbb{R}^d} U(z) \, q(\mathrm{d}z) & \text{(Energy)} \\ & \mathcal{H}(q) & \triangleq \int_{\mathbb{R}^d} \log q(z) \, q(\mathrm{d}z) \; . & \text{(Boltzmann entropy)} \end{array}
+
+[p. 3 | section: 2.1. Problem Setup | type: Text]
+We consider the "non-conjugate" setup, where \mathcal E is intractable due to the expectation over q. Suppose we can parametrize each q_\lambda \in \mathcal Q with a Euclidean vector of parameters \lambda \in \Lambda . Then it is equivalent to minimize \lambda \mapsto \mathcal F(q_\lambda) over the Euclidean parameter space \Lambda \subseteq \mathbb R^p .
+
+[p. 3 | section: 2.1. Problem Setup | type: Text]
+Informally speaking, when U is "regular," \mathcal{E} also tends to be regular. For instance, if U is Lipschitz-smooth, then \mathcal{E} also tends to exhibit appropriate notion of Lipschitz-smoothness (Domke, 2020; Diao et al., 2023; Lambert et al., 2022). The entropy term \mathcal{H} , however, does not enjoy Lipschitz smoothness in general. For instance, for Gaussians q=\operatorname{Normal}(m,\Sigma) , \mathcal{H}(q) blows up as the covariance \Sigma becomes singular. Typically, in optimization, such non-smoothness is remedied by relying on proximal gradient algorithms (Wright & Recht, 2021, §9.3). Indeed, for minimizing \mathcal{F} , stochastic proximal gradient algorithms have been proposed for both the Bures-Wasserstein (Diao et al., 2023) and Euclidean parameter spaces (Domke, 2020). In the following sections, we will introduce these algorithms.
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+Since the seminal work of Jordan et al. (1998), it is known that \mathcal{F} can be minimized by simulating its Wasserstein gradient flow. The forward-backward discretization of this flow results in the Wasserstein-analog of proximal gradient descent (Wibisono, 2018; Bernton, 2018; Salim et al., 2020) operating on the metric space (\mathcal{P}_2(\mathbb{R}^d), W_2) . This algorithm, however, is not directly implementable. Recently, Lambert et al. (2022); Diao et al. (2023) demonstrated that, by constraining optimization to the Bures -Wasserstein manifold \mathrm{BW}(\mathbb{R}^d) \subset \mathcal{P}_2(\mathbb{R}^d) (Bures, 1969; Bhatia et al., 2019), the algorithm becomes implementable. In particular, the proximal Bures-Wasserstein gradient descent scheme iter-
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+ates, for each t \geq 0 ,
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Equation]
+q_{t+1/2} = (\operatorname{Id} - \gamma_t \nabla_{\mathrm{BW}} \mathcal{E}(q_t))_{\#q_t} q_{t+1} = \operatorname{JKO}_{\gamma_t \mathcal{H}}(q_{t+1/2}), (1)
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+where, for any q = \text{Normal}(m, \Sigma) \in \text{BW}(\mathbb{R}^d) , the Bures-Wasserstein gradient of \mathcal{E} can be derived (Lambert et al., 2022, Appendix C) as
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Equation]
+\nabla_{\rm BW} \mathcal{E}(q) \triangleq x \mapsto \nabla_m \mathcal{E}(q) + 2\nabla_{\Sigma} \mathcal{E}(q)(x-m) ,
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+while the Wasserstein-analog of the proximal operator, commonly referred to as the "JKO operator," is defined as, for any functional \mathcal{G}: \mathrm{BW}(\mathbb{R}^d) \to \mathbb{R} \cup \{+\infty\} satisfying some mild regularity conditions,
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Equation]
+\mathrm{JKO}_{\mathcal{G}}(q) \triangleq \operatorname*{arg\,min}_{p \in \mathrm{BW}(\mathbb{R}^d)} \left\{ \mathcal{G}(p) + (1/2) \mathrm{W}_2(p,q)^2 \right\} \,.
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+For the Bures-Wasserstein space, the proximal operator has a tractable closed-form expression (Wibisono, 2018, Example 7), which is key to implementing the algorithm.
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+Still, the Bures-Wasserstein gradient involves expectations over q=\operatorname{Normal}(\mu,\Sigma) that are generally not tractable. Therefore, these have to be replaced with stochastic estimates (Lambert et al., 2022) of \nabla_m \mathcal{E} and \nabla_\Sigma \mathcal{E} resulting in an estimator of the Bures-Wasserstein gradient
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Equation]
+\widehat{\nabla}_{\mathrm{BW}} \mathcal{E}(q; \epsilon) \triangleq x \mapsto \widehat{\nabla}_m \widehat{\mathcal{E}}(q; \epsilon) + 2\widehat{\nabla}_{\Sigma} \widehat{\mathcal{E}}(q; \epsilon)(x - m)
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+where \epsilon \sim \varphi = \mathrm{Normal}(0_d, \mathrm{I}_d) is standard Gaussian noise. Replacing \nabla_{\mathrm{BW}} \mathcal{E} in Eq. (1) with \widehat{\nabla_{\mathrm{BW}}} \mathcal{E} results in stochastic proximal Bures-Wasserstein gradient descent (SPBWGD; Diao et al., 2023). Lambert et al.; Diao et al. rely on the estimators
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Equation]
+\widehat{\nabla_{m}^{\text{poinet}}} \mathcal{E}(q; \epsilon) \triangleq \nabla U(Z) \widehat{\nabla_{\Sigma}^{\text{price}}} \mathcal{E}(q; \epsilon) \triangleq (1/2) \nabla^{2} U(Z) , \qquad (2)
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+where Z={\rm cholesky}(\Sigma)\epsilon+\mu . The fact that these estimators are unbiased follows from the theorems by Bonnet (1964) and Price (1958) or Riemannian geometry (Altschuler et al., 2021, Appendix B.3). For each t\geq 0 , the resulting update rule for the iterate q_t={\rm Normal}(m_t,\Sigma_t) is
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Equation]
+m_{t+1} = m_t - \gamma_t \widehat{\nabla_{m_t} \mathcal{E}}(q_t; \epsilon_t) M_{t+1} = I_d - 2\gamma_t \widehat{\nabla_{\Sigma_t} \mathcal{E}}(q_t; \epsilon_t) \Sigma_{t+1/2} = M_{t+1} \Sigma_t M_{t+1}^{\top} \Sigma_{t+1} = \frac{1}{2} \Big( \Sigma_{t+1/2} + 2\gamma_t I_d + \big( \Sigma_{t+1/2} \big( \Sigma_{t+1/2} + 4\gamma_t I_d \big) \big)^{1/2} \Big) ,
+
+[p. 3 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+where the standard Gaussian noise sequence (\epsilon_t)_{t\geq 0} is sampled as \epsilon_t \overset{\text{i.i.d.}}{\sim} \varphi . Note the update rule for \Sigma_{t+1/2} is different from the one originally presented by Diao et al. (2023); a transpose has been added to M_{t+1} . This change will become necessary later in Section 4 when we replace \widehat{\nabla_\Sigma \mathcal{E}}
+
+[p. 4 | section: 2.2. Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent | type: Text]
+with an estimator that is not almost surely symmetric.
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+An alternative to SBWPGD is to optimize over the Euclidean space of parameters \Lambda . Recall, in this case, each q_{\lambda} \in \Lambda is assumed to be associated with a Euclidean vector \lambda \in \Lambda . Then, if we have access to an unbiased estimator of \nabla_{\lambda} \mathcal{E}(q_{\lambda}) , denoted as \widehat{\nabla_{\lambda} \mathcal{E}}(q_{\lambda}) , \mathcal{F} can be minimized via SPGD, which, for each t \geq 0 , updates the variational parameters \lambda_t as
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+\lambda_{t+1/2} = \lambda_t - \gamma_t \widehat{\nabla_{\lambda_t} \mathcal{E}}(q_{\lambda_t}; \epsilon_t) \lambda_{t+1} = \operatorname{prox}_{\lambda \mapsto \gamma_t \mathcal{H}(q_{\lambda})} (\lambda_{t+1/2}) ,
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+where \epsilon_t \overset{\text{i.i.d.}}{\sim} \varphi is some randomness source and prox is the canonical Euclidean proximal operator (Parikh & Boyd, 2014) defined as, for any proper lower semi-continuous convex function g: \mathbb{R}^p \to \mathbb{R} \cup \{+\infty\} ,
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+\mathrm{prox}_g(\lambda) \triangleq \underset{\lambda' \in \Lambda}{\mathrm{arg\,min}} \big\{ g(\lambda') + (1/2) \|\lambda' - \lambda\|_2^2 \big\} \;.
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+For some classes of variational families Q and parametrizations, the proximal operator can be made tractable (Domke, 2020). Before that, however, we must come up with an unbiased estimator of the parameter gradient \nabla_{\lambda} \mathcal{E}(q_{\lambda}) .
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+Suppose the variational family Q and the parametrization \lambda \mapsto q_{\lambda} satisfy the following:
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+Definition 2.1. For some \Lambda \subset \mathbb{R}^p , the variational family \mathcal{Q} = \{q_{\lambda} \mid \lambda \in \Lambda\} is referred to as a reparameterizable family if there exists some bijective map \phi_{\lambda} : \mathbb{R}^d \to \mathbb{R}^d differentiable with respect to \lambda and a base distribution \varphi \in \mathcal{P}_2(\mathbb{R}^d) such that, for all \lambda \in \Lambda ,
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+Z \sim q_{\lambda} \quad \Leftrightarrow \quad Z \stackrel{\mathrm{d}}{=} \phi_{\lambda}(\epsilon) \; ; \; \epsilon \sim \varphi \; .
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+Here, \stackrel{\mathrm{d}}{=} is equivalence in distribution. Then an immediate option for estimating \nabla_{\lambda}\mathcal{E}(q_{\lambda}) is to use the reparametrization gradient (Ho & Cao, 1983; Rubinstein, 1992; see also Mohamed et al., 2020)
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+\widehat{\nabla_{\lambda}^{\text{rep}} \mathcal{E}}(q_{\lambda}) \triangleq \nabla_{\lambda} \phi_{\lambda}(\epsilon) \nabla U(\phi_{\lambda}(\epsilon)) , \qquad (3)
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+which can be derived by combining the law of the unconscious statistician with the Leibniz integral rule. This combination of SGD with the reparametrization gradient—commonly referred to as BBVI—is widely used in practice through probabilistic programming frameworks such as Stan (Carpenter et al., 2017), Turing (Fjelde et al., 2025), Pyro (Bingham et al., 2019), and PyMC (Patil et al., 2010).
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+The wide adoption of BBVI in practice is partly due to its flexibility: Definition 2.1 applies to a very wide range of families from Gaussians (Titsias & Lázaro-Gredilla, 2014) to normalizing flows (Rezende & Mohamed, 2015). Fur-
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+thermore, Eq. (3) uses only gradients of U, which can be efficiently computed via automatic differentiation (Kucukelbir et al., 2017). In this work, however, we will further restrict our attention to the Gaussian variational family with a specific parametrization:
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+Assumption 2.2. The variational family Q is the Gaussian variational family, where each member q_{\lambda} = \text{Normal}(m, CC^{\top}) \in Q is parametrized as
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+\Lambda = \left\{ \lambda = (m, \text{vec}(C)) \mid m \in \mathbb{R}^d, C \in \mathbb{L}^d_{\succ 0} \right\} \subset \mathbb{R}^p while the reparametrization function is set as
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+\phi_{\lambda}(\epsilon) = C\epsilon + m \quad and \quad \varphi = \text{Normal}(0_d, I_d) .
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+Under this parametrization, Eq. (3) reduces to
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+\widehat{\nabla_{\lambda}^{\text{rep}}\mathcal{E}}(q_{\lambda};\epsilon) = \left[\widehat{\nabla_{m}^{\text{bonnet}}\mathcal{E}}(q_{\lambda};\epsilon)\right] = \left[\nabla U(\phi_{\lambda}(\epsilon))\right] \cdot \left[\epsilon \nabla U(\phi_{\lambda}(\epsilon))\right]. \tag{4}
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+Furthermore, the proximal operator for the entropy has the closed-form solution (Domke, 2020; Domke et al., 2023)
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Equation]
+\begin{split} (m,C') &= \operatorname{prox}_{\lambda \mapsto \gamma_t \mathcal{H}(q_{\lambda_t})}((m,C)), \text{ where} \\ [C']_{ij} &= \begin{cases} (1/2) \left( C_{ii} + \sqrt{C_{ii} + 4\gamma_t} \right) & \text{if } i = j \\ C_{ij} & \text{if } i \neq j \end{cases}. \end{split}
+
+[p. 4 | section: 2.3. Black-Box Variational Inference via Stochastic Proximal Gradient Descent | type: Text]
+Compared to alternative ways to parametrize Gaussians, this "linear" parametrization is particularly well-behaved (Kim et al., 2023) and also computationally efficient: each step of BBVI only needs O(d^2) operations except for evaluating \nabla U . Furthermore, it has been shown that the Bures-Wasserstein gradient \nabla_{\rm BW} \mathcal{F} is equal to the parameter gradient \nabla_{\lambda} \mathcal{F}(q_{\lambda}) under the parametrization of Assumption 2.2 up to a coordinate transformation (Yi & Liu, 2023).
+
+[p. 4 | section: 2.4. Price Gradient Estimators | type: Text]
+A crucial point here is that, for the Gaussian variational family, the estimators traditionally used in WVI (Eq. (5)) and BBVI (Eq. (3)) both target the same quantities up to constant factor adjustments and are therefore interchangeable.
+
+[p. 4 | section: 2.4. Price Gradient Estimators | type: Text]
+Proposition 2.3. For any twice-differentiable function f, Gaussian q = \text{Normal}(m, \Sigma) , and assuming the expectations exist,
+
+[p. 4 | section: 2.4. Price Gradient Estimators | type: Equation]
+\nabla_{\Sigma} \mathbb{E}_q f = \frac{1}{2} \mathbb{E}_q \nabla^2 f = \frac{1}{2} \Sigma^{-1} \mathbb{E}_{X \sim q} \left[ (X - m) \nabla f(X)^{\top} \right].
+
+[p. 4 | section: 2.4. Price Gradient Estimators | type: Text]
+Proof. The first equality is Price's theorem (Price, 1958), while the second equality is Stein's identity (Stein, 1981; Liu, 1994). □
+
+[p. 4 | section: 2.4. Price Gradient Estimators | type: Text]
+Denoting \Sigma = CC^{\top} , an immediate corollary is
+
+[p. 4 | section: 2.4. Price Gradient Estimators | type: Equation]
+\nabla_{C} \mathcal{E}(q_{\lambda}) = C^{\top} \mathbb{E}_{q} \nabla^{2} U = C^{-1} \mathbb{E}_{X \sim q} \left[ (X - m) \nabla U(X)^{\top} \right], \quad (5)
+
+[p. 5 | section: 2.4. Price Gradient Estimators | type: Text]
+where, when restricting C \in \mathbb{L}^d_{\succ 0} , the gradient only needs to be projected to the lower-triangular subspace (tril). Under Assumption 2.2, C^{-1}(X-m)\nabla U(X)^{\top} exactly corresponds to the reparametrization gradient in Eq. (3). At the same time, Eq. (5) points towards an analog of \nabla^{\text{price}}_{\Sigma} for the scale parameter C:
+
+[p. 5 | section: 2.4. Price Gradient Estimators | type: Equation]
+\widehat{\nabla_C^{\text{price}}} \mathcal{E}(q_\lambda;\epsilon) = C^\top \nabla^2 U(X) \;, \text{ where } X = \phi_\lambda(\epsilon) \;. \eqno(6)
+
+[p. 5 | section: 2.4. Price Gradient Estimators | type: Text]
+Conveniently, this estimator also stays unbiased when \nabla^2 U is replaced with an unbiased estimator of \nabla^2 U , enabling doubly stochastic optimization (Titsias & Lázaro-Gredilla, 2014). Similarly, Proposition 2.3 points towards a gradient estimator that could be used in WVI,
+
+[p. 5 | section: 2.4. Price Gradient Estimators | type: Equation]
+\widehat{\nabla_{\Sigma}^{\text{rep}}\mathcal{E}}(q_{\lambda};\epsilon) = \Sigma^{-1}(X-m)\nabla U(X)^{\top}, \text{ where } X = \phi_{\lambda}(\epsilon).
+
+[p. 5 | section: 2.4. Price Gradient Estimators | type: Text]
+Note that similar remarks have already been made by Rezende et al. (2014); Lin et al. (2025); Graves (2011); Opper & Archambeau (2009). Therefore, the use of these estimators is by no means new. However, these Hessian-based estimators have not been widely adopted in BBVI, nor have they been analyzed in detail.
+
+[p. 5 | section: 2.4. Price Gradient Estimators | type: Text]
+A natural question here is how much the choice of gradient estimator affects the performance of different algorithms. Past experience in stochastic gradient-based VI has shown that the choice of gradient estimator crucially affects performance both in practice (Kucukelbir et al., 2017; Geffner & Domke, 2020a; 2021; 2018; 2020b; Agrawal et al., 2020; Miller et al., 2017; Wang et al., 2024; Fujisawa & Sato, 2021; Buchholz et al., 2018) and in theory (Kim et al., 2024; Xu et al., 2019; Luu et al., 2025). Indeed, in our theoretical analysis, we will demonstrate that, once we use the same gradient estimator, the state-of-the-art iteration complexities of SPBWGD and SPGD become the same.
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Text]
+For our theoretical analysis, we assume the following regularity conditions on {\cal U}.
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Text]
+Assumption 3.1. The potential U: \mathbb{R}^d \to \mathbb{R} is twice differentiable and there exists some \mu \in (0, +\infty) and L \in [0, +\infty) such that, for all z \in \mathbb{R}^d , the following holds:
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Equation]
+\mu I_d \quad \prec \quad \nabla^2 U(z) \quad \prec \quad L I_d .
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Text]
+This assumption corresponds to assuming that the density of \pi is \mu -log-concave and L-log-smooth, and has been widely used to establish the iteration complexity of stochastic gradient-based VI (Kim et al., 2023; Domke et al., 2023; Lambert et al., 2022; Diao et al., 2023) and sampling algorithms (Chewi, 2024). Crucially, the energy \mathcal E is now well behaved: On the Bures-Wasserstein geometry, it is
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Text]
+\mu -geodesically convex and L-geodesically smooth (Diao et al., 2023). Similarly, under Assumption 2.2, \lambda \mapsto \mathcal{E}(q_{\lambda}) is \mu -strongly convex and L-smooth (Domke, 2020).
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Text]
+For stochastic first-order optimization algorithms, the choice of step size schedule is crucial for obtaining tight bounds (Bach & Moulines, 2011). In all cases, we will consider a two-stage step size schedule (Gower et al., 2019; Stich, 2019) of the form of, for some base step size \gamma_0 \in (0, +\infty) , switching time t_* \geq 0 , and offset \tau \geq 0 ,
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Equation]
+\gamma_t = \begin{cases} \gamma_0 & \text{if } t < t_* \\ \frac{1}{\mu} \frac{2(t+\tau)+1}{(t+\tau+1)^2} & \text{if } t \ge t_* \end{cases} (7)
+
+[p. 5 | section: 3.1. Theoretical Setup | type: Text]
+This two-stage schedule holds the step size constant for a certain period ( t \in \{0,\dots,t_*-1\} ) and then starts decreasing the step size at a rate of \gamma_t \asymp 1/(\mu t) . The choice of asymptote 1/(\mu t) ensures an optimal asymptotic convergence rate of \mathrm{O}(1/T) for strongly convex objectives (Lacoste-Julien et al., 2012; Shamir & Zhang, 2013).
+
+[p. 5 | section: 3.2. Main Results | type: Text]
+We now present the iteration-complexity guarantees for VI with SPBWGD and SPGD using Price's gradient for the scale/covariance component. First, the Bonnet-Price estimator of the Bures-Wasserstein gradient is formally defined in functional form as, for any q = \text{Normal}(m, \Sigma) ,
+
+[p. 5 | section: 3.2. Main Results | type: Equation]
+\nabla_{\mathrm{BW}}^{\widehat{\mathrm{bonnet-price}}} \mathcal{E}(q; \epsilon) \triangleq x \mapsto \widehat{\nabla_{m}^{\mathrm{bonnet}}} \mathcal{E}(q; \epsilon) + 2\widehat{\nabla_{\Sigma}^{\mathrm{price}}} \mathcal{E}(q; \epsilon)(x - m) = x \mapsto \nabla U(Z) + \nabla^{2}U(Z)(x - m) ,
+
+[p. 5 | section: 3.2. Main Results | type: Text]
+where Z = \text{cholesky}(\Sigma)\epsilon + \mu and \epsilon \sim \varphi = \text{Normal}(0_d, I_d) . Then we obtain the following iteration complexity:
+
+[p. 5 | section: 3.2. Main Results | type: Text]
+Theorem 3.2 (SPBWGD). Suppose Assumption 3.1 holds and the gradient estimator \nabla_{\rm BW}^{\rm bonnet-price} \mathcal{E} is used. Then, for any \epsilon > 0 , there exists some t_* and \tau (shown explicitly in the proof) such that running stochastic proximal Bures-Wasserstein gradient descent with the step size schedule in Eq. (7) with \gamma_0 = 1/(10L\kappa) guarantees
+
+[p. 5 | section: 3.2. Main Results | type: Equation]
+T \gtrsim d\kappa \frac{1}{\epsilon} + \sqrt{d} \,\kappa^{3/2} \log(\kappa \Delta^2) \frac{1}{\sqrt{\epsilon}} + \kappa^2 \log\left(\Delta^2 \frac{1}{\epsilon}\right) \Rightarrow \quad \mu \mathbb{E}[W_2(q_T, q_*)^2] \le \epsilon ,
+
+[p. 5 | section: 3.2. Main Results | type: Equation]
+where \Delta^2 = \mu W(q_0, q_*)^2 .
+
+[p. 5 | section: 3.2. Main Results | type: Text]
+Proof. The full proof is deferred to Section D.1.3. \Box
+
+[p. 5 | section: 3.2. Main Results | type: Text]
+This improves over the O(d\kappa\epsilon^{-1}\log\epsilon^{-1} + \kappa^3\log(\Delta\epsilon^{-1})) complexity obtained by Diao et al. (2023, Thm 5.8). In particular, our result allows for step sizes larger by a factor of \kappa . Consequently, the dependence on \kappa in the non-asymptotic term (\log 1/\epsilon) is improved by a factor of \kappa . Furthermore,
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+the asymptotic complexity in \epsilon \to 0 is improved by a factor of \log 1/\epsilon .
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+Let's now compare this result against the iteration complexity of SPGD. Formally, we define the Bonnet-Price gradient estimator
+
+[p. 6 | section: 3.2. Main Results | type: Equation]
+\nabla_{\lambda}^{\widehat{\text{bonnet-price}}} \mathcal{E}(q_{\lambda}; \epsilon) \triangleq \begin{bmatrix} \widehat{\nabla_{m}^{\text{bonnet}}} \mathcal{E}(q_{\lambda}; \epsilon) \\ \widehat{\nabla_{C}^{\text{price}}} \mathcal{E}(q_{\lambda}; \epsilon) \end{bmatrix} = \begin{bmatrix} \nabla U(Z) \\ C^{\top} \nabla^{2} U(Z) \end{bmatrix} \ .
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+Using this estimator in BBVI with PSGD results in the following iteration complexity.
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+Theorem 3.3 (SPGD). Suppose Assumption 3.1 holds and the gradient estimator \nabla_{\lambda}^{\text{bonnet-price}} \mathcal{E} is used. Then, for any \epsilon > 0 , there exists some t_* and \tau (stated explicitly in the proof) such that running stochastic proximal gradient descent with the step size schedule in Eq. (7) with \gamma_0 = 1/(10L\kappa) guarantees
+
+[p. 6 | section: 3.2. Main Results | type: Equation]
+T \gtrsim d\kappa \frac{1}{\epsilon} + \sqrt{d} \,\kappa^{3/2} \log(\kappa \Delta^2) \frac{1}{\sqrt{\epsilon}} + \kappa^2 \log\left(\Delta^2 \frac{1}{\epsilon}\right) \Rightarrow \quad \mu \mathbb{E}[W_2(q_T, q_*)^2] \le \epsilon ,
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+where \Delta^2 = \mu \|\lambda_0 - \lambda_*\|^2 .
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+Proof. The full proof is deferred to Section D.2.1. \Box
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+Previously, for Gaussian variational families with a dense covariance (the "full-rank" Gaussian family), in the limit of \epsilon \to 0 , Kim et al. (2023); Domke et al. (2023) reported an iteration complexity of O(d\kappa^2\operatorname{tr}(\mu\Sigma_*)\epsilon^{-1}) , which used the canonical reparametrization gradient (Eq. (4)). Compared to this, Price's gradient improves the iteration complexity by a factor of \kappa\operatorname{tr}(\mu\Sigma_*) . (Note that d/L \le \operatorname{tr}(\Sigma_*) \le d/\mu .) This is comparable to the complexity of BBVI with the mean-field Gaussian family (diagonal covariance), which is O((\log d)\kappa^2\operatorname{tr}(\mu\Sigma_*)\epsilon^{-1}) (Kim et al., 2025). This suggests that, with Price's gradient, BBVI on a full-rank Gaussian family can be as fast as using a mean-field Gaussian family and the reparametrization gradient.
+
+[p. 6 | section: 3.2. Main Results | type: Text]
+An immediate implication of Theorems 3.2 and 3.3 is that the gap between the best known iteration complexity bounds between the two algorithms has now been closed. In addition, Section 3.3 that follows will explain that this resemblance is unsurprising, as the convergence analyses of both algorithms rely on nearly the same properties. Though, since we lack matching lower bounds, we cannot yet claim that the two algorithms behave exactly the same. However, our results do provide evidence towards this fact along with the continuous-time results of Yi & Liu (2023); Hoffman & Ma (2020). This again reinforces the intuition that, for stochastic optimization algorithms, the quality of the gradient estimator has the largest impact on the performance.
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+The overall structure of the proofs for both SPBWGD and SPGD is identical. If we had access to exact gradients instead of stochastic estimates, under Assumption 3.1, \|\lambda_t - \lambda_*\|_2 or W_2(q_t, q_*) would contract exponentially in t. When dealing with stochastic gradients, however, the noise in the estimates perturbs the iterates. We thus need to show that the variance of the noise is bounded and the contraction is strong enough such that controlling the step size schedule \gamma_t can neutralize the perturbations.
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+First, under Assumption 3.1, we can define a Bregman divergence associated with U,
+
+[p. 6 | section: 3.3. Proof Sketch | type: Equation]
+D_U(x,y) \triangleq U(x) - U(y) - \langle \nabla U(y), x - y \rangle.
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+For both SPBWGD and SPGD, we establish gradient variance bounds involving \mathrm{D}_U .
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+Lemma 3.4. Suppose Assumption 3.1 holds and q_* = \operatorname{Normal}(\mu_*, \Sigma_*) \in \arg\min_{q \in \operatorname{BW}(\mathbb{R}^d)} \mathcal{F}(q) . Then, for any q \in \operatorname{BW}(\mathbb{R}^d) , and any coupling \psi \in \Psi(q, q_*) ,
+
+[p. 6 | section: 3.3. Proof Sketch | type: Equation]
+\mathbb{E}_{(X,X_*) \sim \psi, \epsilon \sim \varphi} \left[ \|\nabla_{\mathrm{BW}}^{\text{bonnet-price}} \mathcal{E}(q;\epsilon)(X) - \nabla \mathcal{E}(q_*)(X_*) \|_2^2 \right] \\ \leq 10 L \kappa \, \mathbb{E}_{(X,X_*) \sim \psi} \left[ \mathrm{D}_U(X,X_*) \right] + 10 dL \; .
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+Proof. The proof is deferred to Section D.1.4. \Box
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+This is a refinement of Lemma 5.6 by Diao et al. (2023). Specifically, instead of upper-bounding the gradient variance with the squared Wasserstein distance W_2(q, q_*)^2 , we bound it with the Bregman divergence D_U . In fact, for the optimal coupling \psi^* \in \Psi(q, q_*) , we have
+
+[p. 6 | section: 3.3. Proof Sketch | type: Equation]
+\mathbb{E}_{(X,X_*)\sim\psi^*}[D_U(X,X_*)] \le LW_2(q,q_*)^2
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+(Eq. (11) in Section C.1.) Therefore, the use of the Bregman divergence avoids paying for an extra factor of \kappa=L/\mu . The corresponding bound for parameter-space SPGD has the exact same constants.
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+Lemma 3.5. Suppose Assumptions 3.1 and 2.2 holds, and \lambda_* \in \arg\min_{\lambda \in \Lambda} \mathcal{F}(q_{\lambda}) . Then, for any \lambda \in \Lambda and any coupling \psi \in \Psi(q_{\lambda}, q_{\lambda_*}) ,
+
+[p. 6 | section: 3.3. Proof Sketch | type: Equation]
+\begin{split} \mathbb{E}_{\epsilon \sim \varphi} \big[ \| \nabla_{\lambda}^{\widehat{\text{bonnet-price}}} \mathcal{E}(q_{\lambda}; \epsilon) - \nabla \mathcal{E}(q_{\lambda_*}) \|_2^2 \big] \\ & \leq 10 L \kappa \, \mathbb{E}_{(X, X_*) \sim \psi} \big[ \mathcal{D}_U(X, X_*) \big] + 10 dL \; . \end{split}
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+Proof. The proof is deferred to Section D.2.2. \Box
+
+[p. 6 | section: 3.3. Proof Sketch | type: Text]
+The bounds Lemmas 3.4 and 3.5, however, are not immediately usable for a convergence analysis. The "growth" or "multiplicative noise" term \mathbb{E}[\mathrm{D}_U(X,X_*)] needs to be related to the growth of \mathcal{E} . Notice that both Lemmas 3.4 and 3.5 hold for any coupling in \Psi(q,q_*) . By specifying the coupling \psi , we will invoke properties of the geometry associated with \psi induced by each SPGD and SPBWGD, which will allow us to relate \mathbb{E}[\mathrm{D}_U(X,X_*)] with the appropriate notion of growth of \mathcal{E} .
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Let's start with SPBWGD. For the optimal coupling \psi^* \in \Psi(q,q_*) , we will define the Wasserstein analog of the Bregman divergence
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+D_{\mathcal{E}}(q, q_*) \triangleq \mathcal{E}(q) - \mathcal{E}(q_*) - \mathbb{E}_{(X, X_*) \sim \psi_*} \langle \nabla \mathcal{E}(q_*)(X_*), X - X_* \rangle \ge 0. (8)
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+The non-negativity of D_{\mathcal{E}} follows from the fact that \mathcal{E} is geodesically convex under Assumption 3.1 (Lemma D.1). Then the Bures-Wasserstein geometry yields the following:
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Lemma 3.6. For any p, q \in BW(\mathbb{R}^d) , denote the coupling optimal for the squared Euclidean cost between p and q as \psi_* \in \Psi(p,q) . Then
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+\mathbb{E}_{(X,Y)\sim\psi_*}[D_U(X,Y)] = D_{\mathcal{E}}(p,q) .
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Proof. The proof is deferred to Section D.1.5. \Box
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+For SPGD, first consider some \lambda, \lambda' \in \Lambda , where \lambda = (m, \text{vec } C) and \lambda' = (m', \text{vec } C') and the map
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+M_{q_{\lambda}\mapsto q_{\lambda'}}^{\text{rep}}(z) \triangleq C'C^{-1}(z-m) + m' .
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+This is, in fact, a transport map from q_{\lambda} to q_{\lambda'} . Then, under Assumption 2.2, the identities
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+\|\lambda - \lambda'\|_2^2 = \mathbb{E}_{Z \sim q} \|Z - M_{q_{\lambda} \mapsto q_{\lambda'}}^{\text{rep}}(Z)\|_2^2 (9)
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+and
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+353354
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+\langle \nabla_{\lambda} \mathcal{E}(q_{\lambda'}), \lambda - \lambda' \rangle = \mathbb{E}_{Z' \sim q_{\lambda'}} \langle \nabla U(Z'), M_{q_{\lambda'} \mapsto q_{\lambda}}^{\text{rep}}(Z') - Z' \rangle \quad (10)
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+hold. Also, from the fact that the Wasserstein distance is the cost of the optimal coupling, we have the ordering \|\lambda-\lambda'\|_2 \geq W_2(q_\lambda,q_{\lambda'}) . That is, the metric associated with parameter space is a coupling-based distance measure. Now, under Assumptions 2.2 and 3.1, \lambda\mapsto \mathcal{F}(q_\lambda) is \mu -strongly convex. Then it is well known that the gradient flow minimizes \|\lambda_t-\lambda_*\|_2^2 exponentially in time. The identity Eq. (9) implies that this flow is also minimizing a coupling distance, and in turn the Wasserstein distance. Back to the proof sketch, Eq. (10) implies the following:
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Lemma 3.7. Suppose Assumption 2.2 hold. Then, for any \lambda, \lambda' \in \Lambda , denote the coupling induced by the transport map M_{q_{\lambda} \mapsto q_{\lambda'}}^{\text{rep}} as \psi^{\text{rep}} . Then
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+\mathbb{E}_{(X,X')\sim\psi^{\text{rep}}}[D_U(X,X')] = D_{\lambda\mapsto\mathcal{E}(q_\lambda)}(\lambda,\lambda') .
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Proof. The proof is deferred to Section D.2.3. \Box
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Therefore, the optimal coupling \psi^* and the coupling associated with Assumption 2.2 \psi^{\rm rep} respectively retrieve a relationship with the growth of \mathcal{E} .
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Lastly, we need to show that the contraction is able to counteract the "growth" of the gradient variance. Typically, the contraction of first-order methods follows from the coercivity of the gradient operator. For our proof, however, instead of obtaining a full contraction, we establish a slight generalization of coercivity (previously developed by Gorbunov
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+et al. 2020), that allow us to control the Bregman terms D_{\lambda \mapsto \mathcal{E}(q_{\lambda})} and D_{\mathcal{E}} . For SPBWGD, our result reads:
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Lemma 3.8. Suppose Assumption 3.1 holds. Then, for any q \in BW(\mathbb{R}^d) and q_* = \arg\min_{q \in BW(\mathbb{R}^d)} \mathcal{F}(q) , where we denote their coupling \psi_* \in \Psi(q, q_*) optimal in terms of squared Euclidean distance,
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+\mathbb{E}_{(X,Y)\sim\psi_*}\langle\nabla_{\mathrm{BW}}\mathcal{E}(p)(X) - \nabla_{\mathrm{BW}}\mathcal{E}(q)(Y), X - Y\rangle \geq \frac{\mu}{2} \operatorname{W}_2(q_t, q_*)^2 + \operatorname{D}_{\mathcal{E}}(q_t, q_*).
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Proof. The proof is deferred to Section D.1.6.
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+The corresponding result for parameter space SPGD is:
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Lemma 3.9. Suppose Assumptions 3.1 and 2.2 hold. Then, for any \lambda \in \Lambda and \lambda_* = \arg\min_{\lambda \in \Lambda} \mathcal{F}(q_{\lambda}) ,
+
+[p. 7 | section: 3.3. Proof Sketch | type: Equation]
+\begin{split} \langle \nabla_{\lambda_t} \mathcal{E}(q_{\lambda_t}) - \nabla_{\lambda_*} \mathcal{E}(q_{\lambda_*}), \lambda_t - \lambda_* \rangle \\ &\geq \frac{\mu}{2} \|\lambda - \lambda_*\|_2^2 + D_{\lambda \mapsto \mathcal{E}(q_{\lambda})}(\lambda, \lambda_*) \;. \end{split}
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+Proof. The proof is deferred to Section D.2.4. \Box
+
+[p. 7 | section: 3.3. Proof Sketch | type: Text]
+The extra "Bregman term" on the right-hand sides directly allows control over the Bregman term in the gradient variance bounds.
+
+[p. 7 | section: 4. Empirical Analysis | type: Text]
+For the empirical evaluation, we will compare the performance of VI with SPGD, SPBWGD, and NGVI with the reparametrization and Price estimators.
+
+[p. 7 | section: 4. Empirical Analysis | type: Text]
+Setup and Implementation. For SPGD and SPBWGD, we follow the implementation described in Section 2.3 and Section 2.2, respectively. As mentioned in Section 2.2, the reparamtrization gradient \widehat{\nabla_{\Sigma}^{\text{rep}}\mathcal{E}} is not almost surely symmetric. Therefore, the modified SPBWGD implementation in Section 2.2 is used. All algorithms were implemented in the Julia language (Bezanson et al., 2017) and the AdvancedVI.jl library (v0.6.1), which is part of the Turing probabilistic programming ecosystem (Ge et al., 2018; Fjelde et al., 2025)<sup>2</sup>. The experimental problems were taken from the PosteriorDB (Magnusson et al., 2025) benchmark suite of Stan models (Carpenter et al., 2017), which was made accessible from Julia through the BridgeStan interface (Roualdes et al., 2023). The benchmark problems are described in more detail in Section A. All methods are initialized at q_0 = \text{Normal}(0_d, 0.34 \, \text{I}_d) . The gradients were estimated using 8 Monte Carlo samples in all cases, while for evaluation, \mathcal{F} was estimated using 2^{12} samples. We run both SPGD and SPBWGD with a fixed step size \gamma and estimate the free energy \mathcal{F}(q_t) of the iterate q_t at each iteration.
+
+[p. 7 | section: 4. Empirical Analysis | type: Footnote]
+& lt;sup>2</sup>All of the code needed to reproduce the results is available in the following repository:
+
+[p. 8 | section: 4. Empirical Analysis | type: FigureGroup]
+Figure 1. Variational free energy ( \mathcal{F} ) at T=4000 versus step size \gamma . The top 8 problems with the largest dimensionality d are shown here, while the full set of results can be found in Section B. Refer to the main text for why the dotted lines are missing on Rats. The solid lines are the mean estimated over 32 independent repetitions, while the shaded regions are the 95% bootstrap confidence intervals.
+
+[p. 8 | section: 4. Empirical Analysis | type: Text]
+Results. Part of the results are shown in Fig. 1, while the full set of results can be found in Section B. First, when using Price's gradient (solid line), both SPGD and SPB-WGD achieve similar performance, except for Rats, where SPGD remains stable over a wider range of step sizes. On the other hand, when using the reparametrization gradient, both SPGD and SPBWGD perform poorly: they require smaller step sizes. In fact, on Rats, none of the methods using the reparametrization gradient converged for all step sizes between 10^{-8} and 10^{0} . In addition, on some problems, SPBWGD appears to perform worse than SPGD. For example, on Rats with Price's gradient, SPBWGD requires step sizes orders of magnitude smaller to prevent divergence. A possible explanation is that SPBWGD requires an estimator of \nabla_{\Sigma} \mathcal{E}(q) , whereas SPGD uses an estimator for \nabla_C \mathcal{E}(q) . These two are related through the chain rule as (1/2)C^{-\top}\nabla_{\Sigma}\mathcal{E}(q) = \nabla_{C}\mathcal{E}(q) ; the extra scaling of C^{-\top} could make an estimator for \nabla_{\Sigma} \mathcal{E}(q) noisier than \nabla_{C} \mathcal{E}(q) .
+
+[p. 8 | section: 5. Discussions | type: Text]
+In this work, we theoretically analyzed stochastic gradient-based VI algorithms operating in the Euclidean space of parameters (SPGD) and Bures-Wasserstein space (SPBWGD). Our results improve upon the state-of-the-art complexity guarantees for both, closing the gap between them. For SPBWGD, we have technically improved the previous results by (Diao et al., 2023). Meanwhile, for SPGD, we have shown that the use of the Price gradient \nabla_C^{\text{price}} \mathcal{E} achieves better theoretical guarantees than those obtained (Domke et al., 2023; Kim et al., 2023) under the reparametrization gradient \nabla_C^{\text{rep}} \mathcal{E} . This shows that the previously observed advantage of SPBWGD was due to the choice of gradient
+
+[p. 8 | section: 5. Discussions | type: Text]
+estimator rather than the geometry.
+
+[p. 8 | section: 5. Discussions | type: Text]
+However, this doesn't completely rule out the possibility that measure-space algorithms can be more effective. NGVI, which uses the Fisher metric, yields a preconditioned update to the location parameter m_t reminiscent of Newton's method (Khan & Rue, 2023). Possibly due to this, empirical evidence suggests that NGVI methods can converge significantly faster than BBVI (Lin et al., 2019). However, our theoretical understanding of NGVI is still limited, where existing analyses assume conjugacy (Wu & Gardner, 2024) or assumptions much stronger than those considered in this work (Sun et al., 2025; Kumar et al., 2025).
+
+[p. 8 | section: 5. Discussions | type: Text]
+On a practical note, it isn't clear if Price's gradient is always better. For instance, at each iteration, SPGD with the reparametrization gradient requires \Omega(d^2) operations (matrix-vector product for computing \phi_{\lambda} ). Moving to second-order increases the cost to \Omega(d^3) operations (matrix-matrix product for computing C\nabla^2 U ). Meanwhile, SPB-WGD requires \Omega(d^3) in both cases. Therefore, when \nabla U can be computed in O(d^2) time, as d\to\infty , BBVI with SPGD and the reparametrization gradient could be more efficient (\Theta(d^2) versus \Theta(d^3) ) depending on the conditioning \kappa . In addition, WVI requires numerically sensitive operations, such as matrix square roots and Cholesky decompositions, which makes it less robust.
+
+[p. 9 | section: Impact Statement | type: Text]
+This work studies the theoretical properties of variational inference, which is a collection of algorithms for approximating distributions. Therefore, our work is not expected to have direct societal consequences other than those of downstream applications of variational inference.

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+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0153", "section": "References", "page_start": 10, "page_end": 10, "type": "ListGroup", "text": "Fujisawa, M. and Sato, I. Multilevel Monte Carlo variational inference. Journal of Machine Learning Research , 22 (278):1–44, 2021. (Not cited.) Garrigos, G. and Gower, R. M. Handbook of convergence theorems for (stochastic) gradient methods. arXiv Preprint arXiv:2301.11235, 2023. (Not cited.) Ge, H., Xu, K., and Ghahramani, Z. Turing: A language for flexible probabilistic inference. In Proceedings of the In ternational Conference on Machine Learning , volume 84 of PMLR , pp. 1682–1690. JMLR, 2018. (Not cited.) Geffner, T. and Domke, J. Using large ensembles of control variates for variational inference. In Advances in Neural Information Processing Systems , volume 31, pp. 9960– 9970. Curran Associates, Inc., 2018. (Not cited.) Geffner, T. and Domke, J. Approximation based variance reduction for reparameterization gradients. In Advances in Neural Information Processing Systems , volume 33, pp. 2397–2407. Curran Associates, Inc., 2020a. (Not cited.) Geffner, T. and Domke, J. A rule for gradient estimator selection, with an application to variational inference. In Proceedings of the International Conference on Artificial Intelligence and Statistics , volume 108 of PMLR , pp. 1803–1812. JMLR, 2020b. (Not cited.) Geffner, T. and Domke, J. On the difficulty of unbiased alpha divergence minimization. In Proceedings of the International Conference on Machine Learning , volume 139 of PMLR , pp. 3650–3659. JMLR, 2021. (Not cited.) Gelfand, A. E., Hills, S. E., Racine-Poon, A., and Smith, A. F. M. Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association , 85(412):972–985, 1990. (Not cited.) Gelman, A. and Hill, J. Data Analysis Using Regression and Multilevel/Hierarchical Models . Analytical Methods for Social Research. Cambridge Univ. Press, Cambridge, 23rd printing edition, 2021. (Not cited.) Gelman, A., Carlin, J., Stern, H., Dunson, D., Vehtari, A., and Rubin, D. Bayesian Data Analysis . Chapman & Hall/CRC Texts in Statistical Science. CRC Press, Boca Raton, 3 edition, 2014. (Not cited.) Gorbunov, E., Hanzely, F., and Richtarik, P. A unified theory of SGD: Variance reduction, sampling, quantization and coordinate descent. In Proceedings of the Interna tional Conference on Artificial Intelligence and Statistics , volume 108 of PMLR , pp. 680–690. JMLR, 2020. (Not cited.)", "source": "marker_v2", "marker_block_id": "/page/9/ListGroup/548"}
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+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0157", "section": "References", "page_start": 12, "page_end": 12, "type": "Text", "text": "605 606 607 608 609 Lambert, M., Chewi, S., Bach, F., Bonnabel, S., and Rigollet, P. Variational inference via Wasserstein gradient flows. In Advances in Neural Information Processing Systems , volume 35, pp. 14434–14447. Curran Associates, Inc., 2022. (Not cited.)", "source": "marker_v2", "marker_block_id": "/page/11/Text/582"}
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+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0162", "section": "References", "page_start": 13, "page_end": 13, "type": "ListGroup", "text": "Villani, C. Optimal Transport , volume 338 of Grundlehren Der Mathematischen Wissenschaften . Springer Berlin Heidelberg, Berlin, Heidelberg, 2009. (Not cited.) Wang, X., Geffner, T., and Domke, J. Joint control variate for faster black-box variational inference. In Proceedings of the International Conference on Artificial Intelligence and Statistics , volume 238 of PMLR , pp. 1639–1647. JMLR, 2024. (Not cited.) Wibisono, A. Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem. In Proceedings of the Conference On Learning Theory , volume 75 of PMLR , pp. 2093–3027. JMLR, 2018. (Not cited.) Wilson, A. Lyapunov Arguments in Optimization . PhD thesis, University of California, Berkeley, Berkeley, California, 2018. (Not cited.) Wingate, D. and Weber, T. Automated variational inference in probabilistic programming. arXiv Preprint arXiv:1301.1299, 2013. (Not cited.) Wright, S. J. and Recht, B. Optimization for Data Analysis . Cambridge University Press, New York, 2021. (Not cited.) Wu, K. and Gardner, J. R. Understanding Stochastic Natural Gradient Variational Inference. In Proceedings of the 41st International Conference on Machine Learning , volume 235 of PMLR , pp. 53398–53421. JMLR, 2024. (Not cited.) Xu, M., Quiroz, M., Kohn, R., and Sisson, S. A. Variance reduction properties of the reparameterization trick. In Proceedings of the International Conference on Artificial Intelligence and Statistics , volume 89 of PMLR , pp. 2711– 2720. JMLR, 2019. (Not cited.) Yi, M. and Liu, S. Bridging the gap between variational inference and Wasserstein gradient flows. arXiv Preprint arXiv:2310.20090, 2023. (Not cited.)", "source": "marker_v2", "marker_block_id": "/page/12/ListGroup/550"}
+{"paper_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b", "chunk_id": "32a9a1bf-fc3e-433d-855e-5d1a0149a10b:0163", "section": "References", "page_start": 14, "page_end": 14, "type": "TableOfContents", "text": "Introduction 2 Background 2.1 Problem Setup 2.2 Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent . 2.3 Black-Box Variational Inference via Stochastic Proximal Gradient Descent 2.4 Price Gradient Estimators Theoretical Analysis 3.1 Theoretical Setup 3.2 Main Results 3.3 Proof Sketch Empirical Analysis Discussions Benchmark Problems Additional Experimental Results C Auxiliary Results C.1 Properties of the Bregman Divergence C.2 Miscellaneous Results C.3 Stationarity Condition C.4 Bound on the Covariance-Weighted Hessian Norm C.5 Variance Bound for Bonnet's Gradient Estimator C.5.1 Lemma C.7 C.6 Lyapunov Convergence Lemma C.6.1 Proposition C.8 C.6.2 Proof of Proposition C.8 C.7 General Convergence Analysis of Proximal Bures-Wasserstein Gradient Descent C.7.1 Proposition C.12 C.7.2 Proof of Proposition C.12 C.8 General Convergence Analysis of Proximal Gradient Descent C.8.1 Proposition C.18 C.8.2 Proof of Proposition C.18 D Deferred Proofs D.1 Stochastic Proximal Bures-Wasserstein Gradient Descent D.1.1 Stationary Point of SPBWGD (Proof of Lemma C.14) D.1.2 Non-Expansiveness of the JKO Operator (Proof of Lemma C.15) D.1.3 Iteration Complexity (Proof of Theorem 3.2) D.1.4 Variance Bound on the Bures-Wasserstein Gradient Estimator (Proof of Lemma 3.4) D.1.5 Wasserstein Bregman Divergence Identity (Proof of Lemma 3.6) D.1.6 Coercivity of Bures-Wasserstein Gradient (Proof of Lemma 3.8) D.2 Parameter Space Proximal Gradient Descent D.2.1 Iteration Complexity (Proof of Theorem 3.3) D.2.2 Variance Bound on the Parameter Gradient Estimator (Proof of Lemma 3.5) D.2.3 Bregman Divergence Identity (Proof of Lemma 3.7) D.2.4 Coercivity of the Parameter Gradient (Proof of Lemma 3.9) TABLE OF CONTENTS 1 3 4 5 A B", "source": "marker_v2", "marker_block_id": "/page/13/TableOfContents/1"}

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+[p. 14 | section: References | type: TableOfContents]
+Introduction 2 Background 2.1 Problem Setup 2.2 Wasserstein Variational Inference via Stochastic Proximal Bures-Wasserstein Gradient Descent . 2.3 Black-Box Variational Inference via Stochastic Proximal Gradient Descent 2.4 Price Gradient Estimators Theoretical Analysis 3.1 Theoretical Setup 3.2 Main Results 3.3 Proof Sketch Empirical Analysis Discussions Benchmark Problems Additional Experimental Results C Auxiliary Results C.1 Properties of the Bregman Divergence C.2 Miscellaneous Results C.3 Stationarity Condition C.4 Bound on the Covariance-Weighted Hessian Norm C.5 Variance Bound for Bonnet's Gradient Estimator C.5.1 Lemma C.7 C.6 Lyapunov Convergence Lemma C.6.1 Proposition C.8 C.6.2 Proof of Proposition C.8 C.7 General Convergence Analysis of Proximal Bures-Wasserstein Gradient Descent C.7.1 Proposition C.12 C.7.2 Proof of Proposition C.12 C.8 General Convergence Analysis of Proximal Gradient Descent C.8.1 Proposition C.18 C.8.2 Proof of Proposition C.18 D Deferred Proofs D.1 Stochastic Proximal Bures-Wasserstein Gradient Descent D.1.1 Stationary Point of SPBWGD (Proof of Lemma C.14) D.1.2 Non-Expansiveness of the JKO Operator (Proof of Lemma C.15) D.1.3 Iteration Complexity (Proof of Theorem 3.2) D.1.4 Variance Bound on the Bures-Wasserstein Gradient Estimator (Proof of Lemma 3.4) D.1.5 Wasserstein Bregman Divergence Identity (Proof of Lemma 3.6) D.1.6 Coercivity of Bures-Wasserstein Gradient (Proof of Lemma 3.8) D.2 Parameter Space Proximal Gradient Descent D.2.1 Iteration Complexity (Proof of Theorem 3.3) D.2.2 Variance Bound on the Parameter Gradient Estimator (Proof of Lemma 3.5) D.2.3 Bregman Divergence Identity (Proof of Lemma 3.7) D.2.4 Coercivity of the Parameter Gradient (Proof of Lemma 3.9) TABLE OF CONTENTS 1 3 4 5 A B

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