SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Tomas Vaskevicius and Lenaic Chizat
Institute of Mathematics
École Polytechnique Fédérale de Lausanne (EPFL) Station Z
CH-1015 Lausanne
We study the computation of doubly regularized Wasserstein barycenters, a recently introduced family of entropic barycenters governed by inner and outer regularization strengths. Previous research has demonstrated that various regularization parameter choices unify several notions of entropy-penalized barycenters while also revealing new ones, including a special case of debiased barycenters. In this paper, we propose and analyze an algorithm for computing doubly regularized Wasserstein barycenters. Our procedure builds on damped Sinkhorn iterations followed by exact maximization/minimization steps and guarantees convergence for any choice of regularization parameters. An inexact variant of our algorithm, implementable using approximate Monte Carlo sampling, offers the first non-asymptotic convergence guarantees for approximating Wasserstein barycenters between discrete point clouds in the free-support/grid-free setting.
The Wasserstein distance between two probability distributions measures the least amount of effort needed to reconfigure one measure into the other. Unlike other notions of distances based solely on the numerical values taken by the distribution functions (e.g., the Kullback-Leibler divergence), the Wasserstein distance incorporates an additional layer of complexity by considering pairwise distances between distinct points, measured by some predetermined cost function. As a result, the Wasserstein distances can be seen to lift the geometry of the underlying space where the probability measures are defined to the space of the probability measures itself. This allows for a more thorough and geometrically nuanced understanding of the relationships between different probability measures, which proved to be a versatile tool of increasing importance in a broad spectrum of areas.
Given a collection of probability measures and an associated set of positive weights that sum to one, the corresponding Wasserstein barycenter minimizes the weighted sum of Wasserstein distances to the given measures. In the special case of two measures and the squared Euclidean cost function, Wasserstein barycenters coincide with the notion of McCann's displacement interpolation introduced in the seminal paper [42]. The general case, encompassing an arbitrary number of measures, was first studied by Agueh and Carlier [1], where they also demonstrated a close link between Wasserstein barycenters and the multi-marginal optimal transport problem [29]. Recent years have witnessed an increasing number of applications of Wasserstein barycenters across various scientific disciplines. See, for instance, the following sample of works in economics [16, 11], statistics [8], image processing [50], and machine learning [19], among other areas. For further background and references we point the interested reader to the introductory surveys [47, 45] and the textbooks [58, 59, 54, 46, 28].
Despite their compelling theoretical characteristics, the computation of Wasserstein barycenters poses significant computational challenges, particularly in large-scale applications. While Wasserstein barycenters can be computed in polynomial time for fixed dimensions [3], the approximation of Wasserstein barycenters is known to be NP-hard [4]. Currently employed methods for approximating Wasserstein barycenters are predominantly based on space discretizations. Unfortunately, such strategies are only computationally practical for problems of relatively modest scale. Although there are a handful of grid-free techniques available for approximating Wasserstein barycenters (e.g., [18, 35, 23, 40]), we are not aware of any existing methods that provide bounds on computational complexity. One contribution of the present paper is to introduce a method that in some regimes can provably approximate Wasserstein barycenters without relying on space discretizations, but instead employing approximate Monte Carlo sampling.
More broadly, the difficulties associated with computation of the optimal transport cost has prompted the exploration of computationally efficient alternatives, leading to the consideration of regularized Wasserstein distances. Among these, the entropic penalty has emerged as one of the most successful in applications. The practical success of entropic penalization can be attributed to Sinkhorn's algorithm [56], which enables efficient and highly parallelizable computation, an algorithm that gained substantial traction in the machine learning community following the work of Cuturi [20]. It is worth noting that entropic Wasserstein distances are of intrinsic interest, beyond their approximation capabilities. Indeed, they hold a rich historical connection to the Schrödinger bridge problem [55, 60, 26], as highlighted in the recent surveys [39, 14]. Furthermore, they increasingly serve as an analytically convenient tool for studying the unregularized optimal transport problem (see, e.g., [38, 31, 27, 15]) and they underlie some favorable statistical properties that are currently under active investigation; see the works [43, 30, 24, 48, 52, 49] and the references therein.
Let us now define the entropic optimal transport cost. Consider two probability measures,  $\mu$  and  $\nu$ , both supported on  $\mathcal{X}$ , and let  $c: \mathcal{X} \times \mathcal{X} \to [0, \infty)$  be a cost function. The entropic Wasserstein distance with a regularization level  $\lambda > 0$  is defined as
$$
T _ {\lambda} (\mu , \nu) = \inf  _ {\gamma \in \Pi (\mu , \nu)} \mathbf {E} _ {(X, Y) \sim \gamma} [ c (X, Y) ] + \lambda \mathrm {K L} (\gamma , \mu \otimes \nu), \tag {1}
$$
where  $\Pi (\mu ,\nu)$  denotes the set of probability measures on  $\mathcal{X}\times \mathcal{X}$  with marginal distributions equal to  $\mu$  and  $\nu$ , and  $\mathrm{KL}(\cdot ,\cdot)$  is the Kullback-Leibler divergence. When  $\lambda \rightarrow 0$ , the regularized cost  $T_{\lambda}(\mu ,\nu)$  converges to the unregularized Wasserstein distance. Various properties of entropic optimal transport can be found in the recent lecture notes by Léonard [39].
To develop efficiently computable approximations for Wasserstein barycenters, a natural approach is to replace the unregularized Wasserstein cost with the minimizer of the weighted sum of entropy-regularized costs. This method was first explored by Cuturi and Doucet [21] and it has gained additional traction in the recent years. There is some flexibility in the definition of (1), which arises from substituting the reference product measure  $\mu \otimes \nu$  with alternatives such as the Lebesgue measure. Consequently, various notions of entropic barycenters have emerged in the literature, which can be unified through the following optimization problem:
$$
\min  _ {\mu} \sum_ {j = 1} ^ {k} w _ {j} T _ {\lambda} \left(\mu , \nu^ {j}\right) + \tau \mathrm {K L} \left(\mu , \pi_ {\text {r e f}}\right). \tag {2}
$$
Here  $\nu^1, \ldots, \nu^k$  are the probability measures whose barycenter we wish to compute and  $w_1, \ldots, w_k$  are positive weights that sum to one. The inner regularization strength is denoted by  $\lambda > 0$  while  $\tau > 0$  is the outer regularization strength. The measure  $\pi_{\mathrm{ref}}$  is an arbitrary reference measure, the support of which dictates the support of the computed barycenter. For instance, if we take  $\pi_{\mathrm{ref}}$  to be a uniform measure on a particular discretization of the underlying space, we are dealing with a fixed-support setup. On the other hand, letting  $\pi_{\mathrm{ref}}$  be the Lebesgue measure puts us in the free-support setup. We shall refer to the minimizer of (2) as the  $(\lambda, \tau)$ -barycenter, which exists and is unique due to the strict convexity of the outer regularization penalty; however, uniqueness may no longer hold when  $\tau = 0$ .
The objective (2) was recently studied in [17]; it also appeared earlier in [5] for the special case  $\tau \geq \lambda$ , where stochastic approximation algorithms were considered for the computation of fixed-support barycenters. In [17, Section 1.3], it is discussed how various choices of  $(\lambda, \tau)$  relate to Barycenters previously explored in the literature. To provide a brief overview,  $(0,0)$  are the unregularized
Wasserstein barycenters studied in [1]. Inner-regularized barycenters  $(\lambda, 0)$  introduce a shrinking bias; this can be seen already when  $k = 1$ , in which case the solution computes a maximum-likelihood deconvolution [53]. The  $(\lambda, \lambda)$ -barycenters were considered in [21, 7, 22, 10, 36]; they introduce a blurring bias. Likewise, blurring bias is introduced by the outer-regularized barycenters  $(0, \tau)$ , studied in [9, 12]. The only case not covered via the formulation (2) appears to be the one of debiased Sinkhorn barycenters [51, 33], for which an algorithm exists but without computational guarantees. Of particular interest are the  $(\lambda, \lambda/2)$  barycenters: the choice  $\tau = \lambda/2$  for smooth densities yields approximation bias of order  $\lambda^2$ , while the choice  $\tau = \lambda$  results in bias of order  $\lambda$ , which is significantly larger than  $\lambda^2$  in the regimes of interest. This is a new notion of entropic barycenters that was unveiled in the analysis of [17]. We provide the first convergence guarantees for this type of barycenters.
The regularity, stability, approximation, and statistical sample complexity properties of  $(\lambda, \tau)$ -barycenters were investigated in [17]. However, the question of obtaining non-asymptotic convergence guarantees for the computation of  $(\lambda, \tau)$ -barycenters with arbitrary regularization parameters was not addressed therein. In particular, the  $(\lambda, \lambda/2)$  case, which has stood out due to its compelling mathematical features, has not yet been addressed in the existing literature. This gap is addressed by the present paper; we summarize our contributions in the following section.
The remainder of this paper is organized as follows: Section 2 provides the necessary background on entropic optimal transport and a particular dual problem of the doubly regularized entropic objective (2). Section 3 introduces a damped Sinkhorn iteration scheme and complements it with convergence guarantees. An approximate version of the algorithm together with convergence results and implementation details is discussed in Section 4. We summarize our key contributions:
1. Lemma 1, presented in Section 3, demonstrates that bounds on the dual suboptimality gap for the dual problem (8), defined in Section 2.2, can be translated into Kullback-Leibler divergence bounds between the  $(\lambda, \tau)$ -barycenter and the barycenters corresponding to dual-feasible variables. This translation enables us to formulate all our subsequent results in terms of optimizing the dual objective (8).
2. In Section 3, we introduce a damped Sinkhorn scheme (Algorithm 1) that can be employed to optimize  $(\lambda, \tau)$ -barycenters for any choice of regularization parameters. The damping factor  $\min(1, \tau / \lambda)$  accommodates the degrading smoothness properties of the dual objective (8) as a function of decreasing outer regularization parameter  $\tau$ . The introduced damping of the Sinkhorn iterations is, in fact, necessary and it is one of our core contributions: undamped exact scheme can be experimentally shown to diverge as soon as  $\tau < \lambda / 2$ .
3. The main result of this paper is Theorem 1 proved in Section 3. It provides convergence guarantees for Algorithm 1 with arbitrary choice of regularization parameters  $\lambda, \tau > 0$ . This, in particular, results in the first algorithm with guarantees for computing  $(\lambda, \lambda/2)$  barycenters. For smooth densities, these barycenters incur a bias of order  $\lambda^2$  in contrast to the predominantly studied  $(\lambda, \lambda)$  barycenters that incur bias of order  $\lambda$ .
4. In Section 4, we describe Algorithm 2, an extension of Algorithm 1 that allows us to perform inaccurate updates. We formulate sufficient conditions on the inexact updates oracle under which the errors in the convergence analysis do not accumulate. Section 4.1 details an implementation of this inexact oracle, based on approximate Monte Carlo sampling.
5. Theorem 2 proved in Section 4 furnishes convergence guarantees for Algorithm 2. When combined with the implementation of the inexact oracle described in Section 4.1, this yields a provably convergent scheme for a grid-free computation of entropic Wasserstein barycenters between discrete distributions, provided sufficient regularity on the domain  $\mathcal{X}$  and the cost function  $c$ .
6. Appendix F complements our theoretical results with numerical experiments. Our simulations experimentally confirm the necessity of damping when  $\tau < \lambda / 2$ . They also provide experimental support for the suggested damping factor in Algorithms 1 and 2.
This section provides the background material on doubly regularized entropic Wasserstein barycenters and introduces the notation used throughout the paper. In the remainder of the paper, let  $\mathcal{X}$  be a compact and convex subset of  $\mathbb{R}^d$  with a non-empty interior. Let  $\mathcal{P}(\mathcal{X})$  denote the set of probability measures on  $\mathcal{X}$  endowed with Borel sigma-algebra. Let  $c: \mathcal{X} \times \mathcal{X} \to [0, \infty)$  be a cost function such that  $c_{\infty}(\mathcal{X}) = \sup_{x, x' \in \mathcal{X}} c(x, x') < \infty$ . We denote by  $\mathrm{KL}(\cdot, \cdot)$  the Kullback-Leibler divergence,  $\| \cdot \|_{\mathrm{TV}}$  is the total-variation norm, and  $\| f \|_{\mathrm{osc}} = \sup_x f(x) - \inf_{x'} f(x')$  is the oscillation norm. Given two measures  $\nu, \nu'$ , the notation  $\nu \ll \nu'$  denotes that  $\nu$  is absolutely continuous with respect to the measure  $\nu'$ ; in this case  $d\nu / d\nu'$  denotes the Radon-Nikodym derivative of  $\nu$  with respect to  $\nu'$ . Finally, throughout the paper  $w$  denotes a vector of  $k$  strictly positive elements that sum to one.
2.1 Entropic Optimal Transport
For any  $\mu, \nu \in \mathcal{P}(\mathcal{X})$  define the entropy regularized optimal transport problem by
$$
T _ {\lambda} (\mu , \nu) = \inf  _ {\gamma \in \Pi (\mu , \nu)} \mathbf {E} _ {(X, Y) \sim \gamma} [ c (X, Y) ] + \lambda \mathrm {K L} (\gamma , \mu \otimes \nu), \tag {3}
$$
where KL is the Kullback-Leibler divergence and  $\Pi (\mu ,\nu)\subseteq \mathcal{P}(\mathcal{X}\otimes \mathcal{X})$  is the set of probability measures such that for any  $\gamma \in \Pi (\mu ,\nu)$  and any Borel subset  $A$  of  $\mathcal{X}$  it holds that  $\gamma (A\times \mathcal{X}) = \mu (A)$  and  $\gamma (\mathcal{X}\times A) = \nu (A)$ .
Let  $E_{\lambda}^{\mu ,\nu}:L_{1}(\mu)\times L_{1}(\nu)\to \mathbb{R}$  be the function defined by
$$
\begin{array}{l} E _ {\lambda} ^ {\mu , \nu} (\phi , \psi) = \mathbf {E} _ {X \sim \mu} [ \phi (X) ] + \mathbf {E} _ {Y \sim \nu} [ \psi (Y) ] \\ + \lambda \left(1 - \int_ {\mathcal {X}} \int_ {\mathcal {X}} \exp \left(\frac {\phi (x) + \psi (y) - c (x , y)}{\lambda}\right) \nu (d y) \mu (d x)\right). \\ \end{array}
$$
The entropic optimal transport problem (3) admits the following dual representation:
$$
T _ {\lambda} (\mu , \nu) = \max  _ {\phi , \psi} E _ {\lambda} ^ {\mu , \nu} (\phi , \psi). \tag {4}
$$
For any  $\psi$  define
$$
\phi_ {\psi} \in \operatorname {a r g m a x} _ {\phi \in L _ {1} (\mu)} E _ {\lambda} ^ {\mu , \nu} (\phi , \psi).
$$
The solution is unique  $\mu$ -almost everywhere up to a constant; we fix a particular choice
$$
\phi_ {\psi} (x) = - \lambda \log \left(\int_ {\mathcal {X}} \exp \left(\frac {\psi (y) - c (x , y)}{\lambda}\right) \nu (d y)\right).
$$
Likewise, we denote  $\psi_{\phi} = \operatorname{argmax}_{\psi \in L_1(\nu)} E_{\lambda}^{\mu, \nu}(\phi, \psi)$  with the analogous expression to the one given above, interchanging the roles of  $\phi$  and  $\psi$ . Then, the maximum in (4) is attained by any pair  $(\phi^{*}, \psi^{*})$  such that  $\phi^{*} = \phi_{\psi^{*}}$  and  $\psi^{*} = \psi_{\phi^{*}}$ ; such a pair is said to solve the Schrödinger system and it is unique up to translations  $(\phi^{*} + a, \psi^{*} - a)$  by any constant  $a \in \mathbb{R}$ . The optimal coupling that solves the primal problem (3) can be obtained from the pair  $(\phi^{*}, \psi^{*})$  via the primal-dual relation
$$
\gamma^ {*} (d x, d y) = \exp \left(\frac {\phi^ {*} (x) + \psi^ {*} (y) - c (x , y)}{\lambda}\right) \mu (d x) \nu (d y).
$$
We conclude this section by listing two properties of functions of the form  $\phi_{\psi}$ . These properties will be used repeatedly throughout this paper. First, for any  $\psi$  we have
$$
\int_ {\mathcal {X}} \int_ {\mathcal {X}} \exp \left(\frac {\phi_ {\psi} (x) + \psi (y) - c (x , y)}{\lambda}\right) \nu (d y) \mu (d x) = 1,
$$
which means, in particular, that for any  $\psi$  we have
$$
E _ {\lambda} ^ {\mu , \nu} (\phi_ {\psi}, \psi) = \mathbf {E} _ {X \sim \mu} [ \phi_ {\psi} (X) ] + \mathbf {E} _ {Y \sim \nu} [ \psi (Y) ]. \tag {5}
$$
The second property of interest is that for any  $\psi$  and any  $x, x' \in \mathcal{X}$  it holds that
$$
\begin{array}{l} \phi_ {\psi} (x) - \phi_ {\psi} (x ^ {\prime}) = - \lambda \log \frac {\int \exp \left(\frac {\psi (y) - c (x , y)}{\lambda}\right) \nu (d y)}{\int \exp \left(\frac {\psi (y) - c (x ^ {\prime} , y)}{\lambda}\right) \nu (d y)} \\ = - \lambda \log \frac {\int \exp \left(\frac {\psi (y) - c \left(x ^ {\prime} , y\right) + c \left(x ^ {\prime} , y\right) - c (x , y)}{\lambda}\right) \nu (d y)}{\int \exp \left(\frac {\psi (y) - c \left(x ^ {\prime} , y\right)}{\lambda}\right) \nu (d y)} \\ \leq \sup  _ {y \in \mathcal {X}} c (x ^ {\prime}, y) - c (x, y) \leq c _ {\infty} (\mathcal {X}). \\ \end{array}
$$
In particular, for any  $\psi$  we have
$$
\left\| \phi_ {\psi} \right\| _ {\mathrm {o s c}} = \sup  _ {x} \phi_ {\psi} (x) - \inf  _ {x ^ {\prime}} \phi_ {\psi} \left(x ^ {\prime}\right) \leq c _ {\infty} (\mathcal {X}). \tag {6}
$$
2.2 Doubly Regularized Entropic Barycenters
Let  $\pmb{\nu} = (\nu^{1},\dots,\nu^{k})\in \mathcal{P}(\mathcal{X})^{k}$  be  $k$  probability measures and let  $w\in \mathbb{R}^k$  be a vector of positive numbers that sum to one. Given the inner regularization strength  $\lambda >0$  and the outer regularization strength  $\tau >0$ , the  $(\lambda ,\tau)$  barycenter  $\mu_{\lambda ,\tau}\in \mathcal{P}(\mathcal{X})$  of probability measures  $\pmb{\nu}$  with respect to the weights vector  $w$  is defined as the unique solution to the following optimization problem:
$$
\mu_ {\lambda , \tau} = \operatorname {a r g m i n} _ {\mu \in \mathcal {P} (\mathcal {X})} \sum_ {j = 1} ^ {k} w _ {j} T _ {\lambda} \left(\mu , \nu^ {j}\right) + \tau \mathrm {K L} \left(\mu , \pi_ {\text {r e f}}\right), \tag {7}
$$
where  $\pi_{\mathrm{ref}}\in \mathcal{P}(\mathcal{X})$  is a reference probability measure.
We will now describe how to obtain a concave dual maximization problem to the primal problem (7), following along the lines of Chizat [17, Section 2.3], where the interested reader will find a comprehensive justification of all the claims made in the rest of this section.
First, using the semi-dual formulation of entropic optimal transport problem (5), we have, for each  $j \in \{1, \ldots, k\}$
$$
T _ {\lambda} (\mu , \nu^ {j}) = \sup  _ {\psi^ {j} \in L _ {1} (\nu^ {j})} \mathbf {E} _ {X \sim \mu} [ \phi_ {\psi^ {j}} (X) ] + \mathbf {E} _ {Y \sim \nu^ {j}} [ \psi^ {j} (Y) ].
$$
Denote  $\psi = (\psi^1, \ldots, \psi^j) \in L_1(\nu)$ . Then, we may rewrite the primal problem (7) by
$$
\min  _ {\mu \in \mathcal {P} (X)} \max  _ {\psi \in L _ {1} (\nu)} \sum_ {j = 1} ^ {k} w _ {j} \mathbf {E} _ {Y \sim \nu^ {j}} \left[ \psi^ {j} (Y) \right] + \mathbf {E} _ {X \sim \mu} \left[ \sum_ {j = 1} ^ {k} w _ {j} \phi_ {\psi^ {j}} (X) \right] + \tau \mathrm {K L} (\mu , \pi_ {\mathrm {r e f}}).
$$
Interchanging min and max, which is justified using compactness of  $\mathcal{X}$  as detailed in [17], we obtain the dual optimization objective  $E_{\lambda ,\tau}^{\nu ,w}:L_1(\nu)\to \mathbb{R}$  defined by
$$
\begin{array}{l} E _ {\lambda , \tau} ^ {\nu , w} (\psi) = \min  _ {\mu \in \mathcal {P} (X)} \sum_ {j = 1} ^ {k} w _ {j} \mathbf {E} _ {Y \sim \nu^ {j}} [ \psi^ {j} (Y) ] + \mathbf {E} _ {X \sim \mu} [ \sum_ {j = 1} ^ {k} w _ {j} \phi_ {\psi^ {j}} (X) ] + \tau \mathrm {K L} (\mu , \pi_ {\mathrm {r e f}}). \\ = \sum_ {j = 1} ^ {k} w _ {j} \mathbf {E} _ {Y \sim \nu^ {j}} \left[ \psi^ {j} (Y) \right] - \tau \log \int \exp \left(\frac {- \sum_ {j = 1} ^ {k} w _ {j} \phi_ {\psi^ {j}} (x)}{\tau}\right) \pi_ {\text {r e f}} (d x). \tag {8} \\ \end{array}
$$
The infimum above is attained by the measure
$$
\mu_ {\psi} (d x) = Z _ {\psi} ^ {- 1} \exp \left(\frac {- \sum_ {j = 1} ^ {k} \phi_ {\psi^ {j}} (x)}{\tau}\right) \pi_ {\text {r e f}} (d x), \quad Z _ {\psi} = \int \exp \left(\frac {- \sum_ {j = 1} ^ {k} \phi_ {\psi^ {j}} (x)}{\tau}\right) \pi_ {\text {r e f}} (d x).
$$
To each dual variable  $\psi$  we associate the marginal measures  $\nu_{\psi}^{j}(dy)$  defined for  $j = 1,\ldots ,k$  by
$$
\nu_ {\psi} ^ {j} (d y) = \nu^ {j} (d y) \int \exp \left(\frac {\phi_ {\psi^ {j}} (x) + \psi^ {j} (y) - c (x , y)}{\lambda}\right) \mu_ {\psi} (d x). \tag {9}
$$
Finally, we mention that the objective  $E_{\lambda,\tau}^{\nu,w}$  is concave and for any  $\psi, \psi'$  it holds that
$$
\lim  _ {h \rightarrow 0} \frac {E _ {\lambda , \tau} ^ {\nu , w} (\boldsymbol {\psi} + h \boldsymbol {\psi} ^ {\prime}) - E _ {\lambda , \tau} ^ {\nu , w} (\boldsymbol {\psi})}{h} = \sum_ {j = 1} ^ {k} w _ {j} \left(\mathbf {E} _ {\nu^ {j}} [ (\psi^ {\prime}) ^ {j} ] - \mathbf {E} _ {\nu_ {\psi} ^ {j}} [ (\psi^ {\prime}) ^ {j} ]\right).
$$
In particular, fixing any optimal dual variable  $\psi^{*}$ , for any  $\psi$  it holds using concavity of  $E_{\lambda,\tau}^{\nu,w}$  that
$$
0 \leq E _ {\lambda , \tau} ^ {\boldsymbol {\nu}, w} (\boldsymbol {\psi} ^ {*}) - E _ {\lambda , \tau} ^ {\boldsymbol {\nu}, w} (\boldsymbol {\psi}) \leq \sum_ {j = 1} ^ {k} w _ {k} \left(\mathbf {E} _ {\nu^ {j}} \left[ (\psi^ {*}) ^ {j} - \psi^ {j} \right] - \mathbf {E} _ {\nu_ {\psi} ^ {j}} \left[ (\psi^ {*}) ^ {j} - \psi^ {j} \right]\right). \tag {10}
$$
This concludes our overview of the background material on  $(\lambda, \tau)$ -barycenters.
3 Damped Sinkhorn Scheme
This section introduces a damped Sinkhorn-based optimization scheme (Algorithm 1) and provides guarantees for its convergence (Theorem 1). Before describing the algorithm, we make a quick detour to the following lemma, proved in Appendix A, which shows that the sub-optimality gap bounds on the dual objective (8) can be transformed into corresponding bounds on relative entropy between the  $(\lambda, \tau)$ -barycenter and the barycenter associated to a given dual variable.
Lemma 1. Fix any  $\lambda, \tau > 0$  and  $\pmb{\nu}, w$ . Let  $\psi^{*}$  be the maximizer of dual problem  $E_{\lambda,\tau}^{\pmb{\nu},w}$  and let  $\mu_{\psi^{*}}$  be the corresponding minimizer of the primal objective (7). Then, for any  $\pmb{\psi} \in L_1(\pmb{\nu})$  we have
$$
\operatorname {K L} \left(\mu_ {\psi^ {*}}, \mu_ {\psi}\right) \leq \tau^ {- 1} \left(E _ {\lambda , \tau} ^ {\boldsymbol {\nu}, w} \left(\psi^ {*}\right) - E _ {\lambda , \tau} ^ {\boldsymbol {\nu}, w} (\psi)\right).
$$
We now turn to describing an iterative scheme that ensures convergence of the dual suboptimality gap to zero. Let  $\psi_t$  be an iterate at time  $t$ . Then, we have
$$
E _ {\lambda , \tau} ^ {\nu , w} (\psi_ {t}) = L (\psi_ {t}, \phi_ {t}, \mu_ {t}) = \sum_ {j = 1} ^ {k} w _ {j} \mathbf {E} _ {\nu^ {j}} [ \psi_ {t} ^ {j} ] - \mathbf {E} _ {\mu_ {t}} [ \phi_ {t} ^ {j} ] + \tau \mathrm {K L} (\mu_ {t}, \pi_ {\mathrm {r e f}}),
$$
$$
\phi^ {j} = \operatorname {a r g m a x} _ {\phi} E _ {\lambda} ^ {\mu_ {t - 1}, \nu^ {j}} (\phi , \psi_ {t} ^ {j}) \quad \text {a n d} \quad \mu_ {t} = \operatorname {a r g m i n} _ {\mu} \left\{\mathbf {E} _ {\mu} \left[ \sum_ {j} w _ {j} \phi_ {t} ^ {j} \right] + \tau \mathrm {K L} (\mu , \pi_ {\mathrm {r e f}}) \right\}. \tag {11}
$$