SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In particular, when optimizing the dual objective  $E_{\lambda,\tau}^{\nu,w}$ , every time the variable  $\psi_t$  is updated, it automatically triggers the exact maximization/minimization steps defined in (11). It is thus a natural strategy to fix  $\phi_t$  and  $\mu_t$  and perform exact minimization on  $\psi$ , which can be done in closed form:
$$
\psi_ {t + 1} ^ {j} = \operatorname {a r g m a x} _ {\psi} E _ {\lambda} ^ {\mu_ {t}, \nu^ {j}} \left(\phi_ {t} ^ {j}, \psi\right) = \psi_ {t} ^ {j} - \lambda \log \frac {d \nu_ {t} ^ {j}}{d \nu^ {j}}, \tag {12}
$$
where  $\nu_{t}^{j}$  denotes the marginal distribution  $\nu_{\psi_t}^j$  defined in (9). The update (12) performs a Sinkhorn update on each block of variables  $\psi^j$ . Together, the update (12) followed by (11) results in the iterative Bregman projections algorithm introduced in [7]. In [36], it was shown that this scheme converges for the  $(\lambda ,\lambda)$ -barycenters. The analysis of [36] is built upon a different dual formulation from the one considered in our work; this alternative formulation is only available when  $\tau = \lambda$  [17, Section 2.3] and thus excludes the consideration of debiased barycenters  $(\lambda ,\lambda /2)$ .
We have observed empirically (see Appendix F) that the iterates of the iterative Bregman projections (i.e., the scheme of updates defined in (12) and (11)) diverge whenever  $\tau < \lambda / 2$ . Indeed, decreasing the outer regularization parameter  $\tau$  makes the minimization step in (11) less stable. As a result, the cumulative effect of performing the updates (12) and (11) may result in a decrease in the value of the optimization objective  $E_{\lambda, \tau}^{\nu, w}$ .
One of the main contributions of our work is to show that this bad behaviour can be mitigated by damping the exact Sinkhorn updates (12). This leads to Algorithm 1 for which convergence guarantees are provided in Theorem 1 stated below.
Algorithm 1: Exact Damped Sinkhorn Scheme
Input: regularization strengths  $\lambda, \tau > 0$ , reference measure  $\pi_{\mathrm{ref}}$ , number of iterations  $T$  and  $k$  marginal measures  $\nu^1, \ldots, \nu^k$  with positive weights  $w_1, \ldots, w_k$  such that  $\sum_{j=1}^{k} w_j = 1$ .
1. Set  $\eta = \min (1,\tau /\lambda)$  and initialize  $(\psi_0^j) = 0$  for  $j\in \{1,\dots ,k\}$
2. For  $t = 0,1\ldots ,T - 1$  do
(a)  $\phi_t^j (x)\gets -\lambda \log \int_{\mathcal{X}}\exp ((\psi_t^j (y) - c(x,y)) / \lambda)\nu^j (dy)$  for  $j\in \{1,\ldots ,k\}$
(b)  $V_{t}(x)\gets \sum_{j = 1}^{k}w_{j}\phi^{j}(t)(x)$
(c)  $Z_{t}\gets \int \exp (-V_{t}(x) / \tau)d\pi_{\mathrm{ref}}(dx)$
(d)  $\mu_t(dx) \gets Z_t^{-1} \exp(-V_t(x) / \tau) \pi_{\mathrm{ref}}(dx)$
(e)  $\frac{d\nu_t^j}{d\nu^j} (y)\gets \int \exp \left(\frac{\phi_t^j(x) + \psi_t^j(y) - c(x,y)}{\lambda}\right)\mu_t(dx)$  for  $j\in \{1,\ldots ,k\}$
(f)  $\psi_{t + 1}^{j}(y)\gets \psi_{t}^{j}(y) - \eta \lambda \log \frac{d\nu_{t}^{j}}{d\nu^{j}} (y)$  for  $j\in \{1,\ldots ,k\}$
3. Return  $(\phi_T^j,\psi_T^j)_{j = 1}^k$
Theorem 1. Fix any  $\lambda, \tau > 0$  and  $\nu, w$ . Let  $\psi^{*}$  be the maximizer of dual problem  $E_{\lambda, \tau}^{\nu, w}$ . Let  $(\psi_t)_{t \geq 0}$  be the sequence of iterates generated by Algorithm 1. Then, for any  $t \geq 1$  it holds that
$$
E _ {\lambda , \tau} ^ {\nu , w} (\boldsymbol {\psi} ^ {*}) - E _ {\lambda , \tau} ^ {\nu , w} (\boldsymbol {\psi} _ {t}) \leq \frac {2 c _ {\infty} (\mathcal {X}) ^ {2}}{\min (\lambda , \tau)} \frac {1}{t}.
$$
Our convergence analysis draws upon the existing analyses of Sinkhorn's algorithm [2, 25], which in turn are based on standard proof strategies in smooth convex optimization (e.g., [44, Theorem 2.1.14]). Concerning the proof of Theorem 1, the main technical contribution of our work lies in the following proposition proved in Appendix B.
Proposition 1. Consider the setup of Theorem 1. Then, for any integer  $t \geq 0$  it holds that
$$
E _ {\lambda , \tau} ^ {\nu , w} (\psi_ {t + 1}) - E _ {\lambda , \tau} ^ {\nu , w} (\psi_ {t}) \geq \min  (\tau , \lambda) \sum_ {j = 1} ^ {k} w _ {j} \mathrm {K L} (\nu^ {j}, \nu_ {t} ^ {j}).
$$
With Proposition 1 at hand, we are ready to prove Theorem 1.
Proof of Theorem 1. Denote  $\delta_t = E_{\lambda, \tau}^{\nu, w}(\psi^*) - E_{\lambda, \tau}^{\nu, w}(\psi_t)$ . We would like to relate the suboptimality gap  $\delta_t$  to the increment  $\delta_t - \delta_{t+1}$ . To do this, we will first show that the iterates  $\psi_t$  have their oscillation norm bounded uniformly in  $t$ . Indeed, for any  $j \in \{1, \dots, k\}$ , any  $t \geq 1$ , and any  $y \in \mathcal{X}$  we have
$$
\psi_ {t} ^ {j} (y) = (1 - \eta) \psi_ {t - 1} ^ {j} (y) + \eta \psi_ {\phi_ {t} ^ {j}} (y).
$$
By (6),  $\psi_{\phi_t^j}$  has oscillation norm bounded by  $c_{\infty}(\mathcal{X})$ . Because  $\psi_0^j = 0$  and  $\eta \in (0,1]$ , by induction on  $t$  it follows that  $\| \psi_t\|_{\mathrm{osc}}\leq c_{\infty}(\mathcal{X})$  for any  $t\geq 0$ . Combining the bound on the dual sub-optimality gap (10) with Pinsker's inequality yields
$$
\delta_ {t} \leq 2 c _ {\infty} (\mathcal {X}) \sum_ {j = 1} ^ {k} w _ {j} \| \nu^ {j} - \nu_ {t} ^ {j} \| _ {\mathrm {T V}} \leq \sqrt {2} c _ {\infty} \sum_ {j = 1} ^ {k} w _ {j} \sqrt {\mathrm {K L} (\nu^ {j} , \nu_ {t} ^ {j})}.
$$
Using concavity of the square root function, Proposition 1 yields for any  $t \geq 0$
$$
\delta_ {t} - \delta_ {t + 1} \geq \min  (\lambda , \tau) \sum_ {j = 1} ^ {k} w _ {j} \mathrm {K L} \left(\nu^ {j}, \nu_ {t} ^ {j}\right) \geq \frac {\min  (\lambda , \tau)}{2 c _ {\infty} (\mathcal {X}) ^ {2}} \delta_ {t} ^ {2}.
$$
By Proposition 1, the sequence  $\delta_t$  is non-increasing. Hence, dividing the above equality by  $\delta_t\delta_{t + 1}$  yields
$$
\frac {1}{\delta_ {t + 1}} - \frac {1}{\delta_ {t}} \geq \frac {\operatorname* {m i n} (\lambda , \tau)}{2 c _ {\infty} (\mathcal {X}) ^ {2}}.
$$
Telescoping the left hand side completes the proof.
4 Approximate Damped Sinkhorn Scheme
In this section, we extend the analysis of Algorithm 1 to an approximate version of the algorithm. Then, in Section 4.1, we describe how inexact updates may be implemented via approximate random sampling, thus enabling the computation of  $(\lambda, \tau)$ -barycenters in the free-support setting with convergence guarantees.
Algorithm 2 describes an inexact version of Algorithm 1. It replaces the damped Sinkhorn iterations of Algorithm 1 via approximate updates computed by an approximate Sinkhorn oracle – a procedure that satisfies the properties listed in Definition 1.
Definition 1 (Approximate Sinkhorn Oracle). An  $\varepsilon$ -approximate Sinkhorn oracle is a procedure that given any  $\psi$  and any index  $j \in \{1, \dots, k\}$ , returns a Radon-Nikodym derivative  $\frac{d\widetilde{\nu}_{\psi}^j}{d\nu^j}$  of a measure  $\widetilde{\nu}_{\psi}^j \ll \nu^j$  that satisfies the following properties:
1.  $\frac{d\widetilde{\nu}_{\psi}^{j}}{d\nu^{j}}$  is strictly positive on the support of  $\nu^{j}$
2.  $\| \widetilde{\nu}_{\psi}^{j} - \nu_{\psi}^{j}\|_{\mathrm{TV}}\leq \varepsilon /(2c_{\infty}(\mathcal{X}))$
3.  $\mathbf{E}_{Y\sim \nu^j}\left[\frac{d\nu_j^i}{d\tilde{\nu}_\psi^j} (Y)\right]\leq 1 + \varepsilon^2 /(2c_\infty (\mathcal{X})^2);$
4. For any  $\eta \in [0,1]$  and any  $j\in \{1,\dots ,k\}$  it holds that  $\| \psi^j +\eta \lambda \log (d\tilde{\nu}_{\pmb{\psi}}^j /d\nu^j)\|_{\mathrm{osc}}\leq$ $(1 - \eta)\| \psi^{j}\|_{\mathrm{osc}} + \eta c_{\infty}(\mathcal{X})$
Algorithm 2: Approximate Damped Sinkhorn Scheme
Input: error tolerance parameter  $\varepsilon > 0$ , a function "ApproximateSinkhornOracle" satisfying properties listed in Definition 1, regularization strengths  $\lambda, \tau > 0$ , reference measure  $\pi_{\mathrm{ref}}$ , number of iterations  $T$  and  $k$  marginal measures  $\nu^1, \ldots, \nu^k$  with positive weights  $w_1, \ldots, w_k$  such that  $\sum_{j=1}^{k} w_j = 1$ .
(a)  $\frac{d\widetilde{\nu}_t^j}{d\nu^j} (y)\gets \mathrm{ApproximateSinkhornOracle}(\pmb {\nu},\lambda ,\tau ,\pmb {\psi}_t,\varepsilon ,j)$  for  $j\in \{1,\ldots ,k\}$
(b)  $\psi_{t + 1}^{j}(y)\gets \psi_{t}^{j}(y) - \eta \lambda \log \frac{d\tilde{\nu}_{t}^{j}}{d\nu^{j}} (y)$  for  $j\in \{1,\ldots ,k\}$
The following theorem shows that Algorithm 2 enjoys the same convergence guarantees as Algorithm 1 up to the error tolerance of the procedure used to implement the approximate updates. A noteworthy aspect of the below theorem is that the error does not accumulate over the iterations.
Theorem 2. Fix any  $\lambda, \tau > 0$  and  $\pmb{\nu}, w$ . Let  $\psi^{*}$  be the maximizer of dual problem  $E_{\lambda,\tau}^{\pmb{\nu},w}$ . Let  $(\widetilde{\psi}_{t})_{t\geq 0}$  be the sequence of iterates generated by Algorithm 2 with the accuracy parameter  $\varepsilon \geq 0$ . Let  $T = \min \{t:E_{\lambda,\tau}^{\pmb{\nu},w}(\psi^{*}) - E_{\lambda,\tau}^{\pmb{\nu},w}(\widetilde{\psi}_{t})\leq 2\varepsilon\}$ . Then, for any  $t\leq T$  it holds that
$$
E _ {\lambda , \tau} ^ {\nu , w} (\psi^ {*}) - E _ {\lambda , \tau} ^ {\nu , w} (\widetilde {\psi} _ {t}) \leq 2 \varepsilon + \frac {2 c _ {\infty} (\mathcal {X}) ^ {2}}{\min (\lambda , \tau)} \frac {1}{t}.
$$
The proof of the above theorem can be found in Appendix C.
4.1 Implementing the Approximate Sinkhorn Oracle
In this section, we show that the approximate Sinkhorn oracle (see Definition 1) can be implemented using approximate random sampling when the marginal distributions  $\nu^{j}$  are discrete. To this end, fix
the regularization parameters  $\lambda, \tau > 0$ , the weight vector  $w$ , and consider a set of  $k$  discrete marginal distributions
$$
\nu^ {j} = \sum_ {l = 1} ^ {m _ {j}} \nu^ {j} (y _ {l} ^ {j}) \delta_ {y _ {l} ^ {j}},
$$
where  $\delta_{x}$  is the Dirac measure located at  $x$  and  $\nu^{j}(y_{l}^{j})$  is equal to the probability of sampling the point  $y_{l}^{j}$  from measure  $\nu^{j}$ . We denote the total cardinality of the support of all measures  $\nu^{j}$  by
$$
m = \sum_ {j = 1} ^ {m} m _ {j}.
$$
Fix any  $\psi \in L_1(\nu)$ . Suppose we are given access to  $n$  i.i.d. samples  $X_{1},\ldots ,X_{n}$  from a probability measure  $\mu_{\psi}^{\prime}$  that satisfies
$$
\| \mu_ {\psi} - \mu_ {\psi} ^ {\prime} \| _ {\mathrm {T V}} \leq \varepsilon_ {\mu}.
$$
Then, for  $j = 1, \ldots, k$  and  $l = 1, \ldots, m_j$  consider
$$
\widehat {\nu} ^ {j} (y _ {i} ^ {j}) = \nu^ {j} (y _ {i} ^ {j}) \frac {1}{n} \sum_ {i = 1} ^ {n} \exp \left(\frac {\phi_ {\psi^ {j}} (X _ {i}) + \psi^ {j} (y) - c (x , y)}{\lambda}\right)
$$
and for any parameter  $\zeta \in (0,1 / 2]$  define
$$
\widetilde {\nu} ^ {j} = (1 - \zeta) \widehat {\nu} ^ {j} + \zeta \nu^ {j}. \tag {13}
$$
We claim that  $\widetilde{\nu}^j$  implements the approximate Sinkhorn oracle with accuracy parameter of order  $O(\varepsilon_{\mu}^{1/4})$  provided that  $n$  is large enough. This is shown in the following lemma, the proof of which can be found in Appendix D.
Lemma 2. Fix any  $\delta \in (0,1)$  and consider the setup described above. With probability at least  $1 - \delta$ , for each  $j \in \{1,\dots,k\}$  it holds simultaneously that the measure  $\widetilde{\nu}^j$  defined in (13) satisfies all the properties listed in Definition 1 with accuracy parameter
$$
\varepsilon_ {j} \leq c _ {\infty} (\mathcal {X}) \left(2 \zeta + \frac {1}{\zeta} m _ {j} \varepsilon_ {\mu} + \frac {1}{\zeta} m _ {j} \sqrt {\frac {2 \log \left(\frac {2 m}{\delta}\right)}{n}}\right) ^ {1 / 2}.
$$
The above lemma shows that a step of Algorithm 2 can be implemented provided access to i.i.d. sampling from some measure  $\mu_{\psi}^{\prime}$  close to  $\mu_{\psi}$  in total variation norm, where  $\psi$  is an arbitrary iterate of Algorithm 2. The remainder of this section is dedicated to showing that this can be achieved by sampling via Langevin Monte Carlo.
Henceforth, fix  $\pi_{\mathrm{ref}}$  to be the Lebesgue measure on  $\mathcal{X}$ , which corresponds to the free-support barycenters setup. Then, for any  $\psi$  we have
$$
\mu_ {\psi} (d x) \propto \mathbb {1} _ {\mathcal {X}} \exp (- V _ {\psi} (x) / \tau) d x, \quad \text {w h e r e} \quad V _ {\psi} (x) = \sum_ {j = 1} ^ {k} w _ {j} \phi_ {\psi^ {j}} ^ {j},
$$
where  $\mathbb{1}_{\mathcal{X}}$  is equal to one on  $\mathcal{X}$  and zero everywhere else. It follows by (6) that  $\| V_{\psi}\|_{\mathrm{osc}}\leq c_{\infty}(\mathcal{X}) / \tau$ . Further, let  $\mathrm{diam}\mathcal{X} = \sup_{x,x'\in \mathcal{X}}\| x - x'\| _2$ . By the convexity of  $\mathcal{X}$ , the uniform measure on  $\mathcal{X}$  satisfies the logarithmic Sobolev inequality (LSI) with constant  $\mathrm{diam}(\mathcal{X})^2 /4$  (cf. [37]). Hence, by the Holley-Stroock perturbation argument [32], the measure  $\mu_{\psi}$  satisfies LSI with constant at most  $\exp (2c_{\infty}(\mathcal{X}) / \tau)\mathrm{diam}(\mathcal{X})^{2} / 4 < \infty$ .
It is well-established that Langevin Monte Carlo algorithms offer convergence guarantees for approximate sampling from a target measure subject to functional inequality constraints provided additional conditions hold such as the smoothness of the function  $V_{\psi}$ . However, such guarantees do not directly apply to the measure  $\mu_{\psi}$  due to its constrained support. Instead, it is possible to approximate  $\mu_{\psi}$  arbitrarily well in total variation norm by a family of measures  $(\mu_{\psi,\sigma})_{\sigma > 0}$  (see Appendix E for details) supported on all of  $\mathbb{R}^d$ . Tuning the parameter  $\sigma$  allows us to trade-off between the approximation quality of  $\mu_{\psi,\sigma}$  and its LSI constant. Crucially, standard sampling guarantees for Langevin Monte Carlo (e.g., [57]) apply to the regularized measures  $\mu_{\psi,\sigma}$ , which leads to provable
guarantees for an implementation of Algorithm 2, thus furnishing the first convergence guarantees for computation of Wasserstein barycenters in the free support setup; see Theorem 3 stated below.
The above approximation argument applies to any cost function  $c$  that is Lipschitz on  $\mathcal{X}$  and exhibits quadratic growth at infinity. For the sake of simplicity, we consider the quadratic cost  $c(x,y) = \| x - y\| _2^2$ . The exact problem setup where we are able to obtain computational guarantees for free-support barycenter computation via Langevin Sampling is formalized below.
Problem Setting 1. Consider the setting described at the beginning of Section 4.1. In addition, suppose that
1. the reference measure  $\pi_{\mathrm{ref}}(dx) = \mathbb{1}_{\mathcal{X}}dx$  is the Lebesgue measure supported on  $\mathcal{X}$  (free-support setup);
2. it holds that  $\mathcal{X} \subseteq \mathcal{B}_R = \{x \in \mathbb{R}^d : \| x \|_2 \leq R\}$  for some constant  $R < \infty$ ;
3. the cost function  $c:\mathbb{R}^d\times \mathbb{R}^d\to [0,\infty)$  is defined by  $c(x,y) = \| x - y\| _2^2$
4. for any  $\psi$  we have access to a stationary point  $x_{\psi}$  of  $V_{\psi}$  over  $\mathcal{X}$ .
The last condition can be implemented in polynomial time using a first order gradient method. For our purposes, this condition is needed to obtain a good initialization point for the Unadjusted Langevin Algorithm following the explanation in [57, Lemma 1]; see Appendix E for further details.
We now proceed to the main result of this section, the proof of which can be found in Appendix E. The following theorem provides the first provably convergent method for computing Wasserstein barycenters in the free-support setting. We remark that a stochastic approximation argument of a rather different flavor used to compute fixed-support Wasserstein barycenters (for  $\tau \geq \lambda$ ) has been previously analyzed in [5].
Theorem 3. Consider the setup described in Problem Setting 1. Then, for any confidence parameter  $\delta \in (0,1)$  and any accuracy parameter  $\varepsilon > 0$ , we can simulate a step of Algorithm 2 with success probability at least  $1 - \delta$  in time polynomial in
$$
\varepsilon^ {- 1}, d, R, \exp (R ^ {2} / \tau), (R d ^ {- 1 / 4}) ^ {d}, \tau^ {- 1}, \lambda^ {- 1}, d, m, \log (m / \delta).
$$
In particular, an  $\varepsilon$ -approximation of the  $(\lambda, \tau)$ -Barycenter can be obtained within the same computational complexity.
Comparing the above guarantee with the discussion following the statement of Lemma 2, we see an additional polynomial dependence on  $(Rd^{-1/4})^d$  (note that for  $R \leq d^{1/4}$  this term disappears). We believe this term to be an artefact of our analysis appearing due to the approximation argument described above. Considering the setup with  $R \leq d^{1/4}$ , the running time of our algorithm depends exponentially in  $R^2/\tau$ .
We conclude with two observations. First, since approximating Wasserstein barycenters is generally NP-hard [4], an algorithm with polynomial dependence on all problem parameters does not exist if  $\mathrm{P} \neq \mathrm{NP}$ . Second, notice that computing an  $\varepsilon$  approximation of  $(\lambda, \tau)$ -Barycenter can be done in time polynomial in  $\varepsilon^{-1}$ . This should be contrasted with numerical schemes based on discretizations of the set  $\mathcal{X}$ , which would, in general, result in computational complexity of order  $(R / \varepsilon)^d$  to reach the same accuracy.
We introduced algorithms to compute doubly regularized entropic Wasserstein barycenters and studied their computational complexity, both in the fixed-support and in the free-support settings. Although a naive adaptation of the usual alternate maximization scheme from [7] to our setting leads to diverging iterates (at least for small values of  $\tau$ ), our analysis shows that it is sufficient to damp these iterations to get a converging algorithm.
While we have focused on the problem of barycenters of measures, we note that the idea of entropic regularization is pervasive in other applications of optimal transport. There, the flexibility offered by the double entropic regularization may prove to be useful as well, and we believe that our damped algorithm could be adapted to these more general settings.
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