SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
y _ {t} = y _ {0} + \frac {\sum_ {s = 1} ^ {t - 1} c _ {\eta_ {s}} (y _ {s})}{t} \left(1 + \sum_ {s = 1} ^ {t - 1} \langle c _ {\eta_ {s}} (y _ {s}), y _ {s} - y _ {0} \rangle\right), x _ {t} = \nabla \phi^ {*} (x _ {t}), \tag {11}
$$
Algorithm 2 Coin MSVGD
input: target density  $\pi$ , kernel  $k$ , mirror function  $\phi$ , particles  $(x_0^i)_{i=1}^N \sim \mu_0$ .
$$
\text {F o r} i \in [ N ], y _ {t} ^ {i} = y _ {0} ^ {i} - \frac {\sum_ {s = 1} ^ {t - 1} P _ {\hat {\eta} _ {s} , k _ {\phi}} \nabla \log \left(\frac {\hat {\eta} _ {s}}{\nu}\right) \left(y _ {s} ^ {i}\right)}{t} \left(1 - \sum_ {s = 1} ^ {t - 1} \left\langle P _ {\hat {\eta} _ {s}, k _ {\phi}} \nabla \log \left(\frac {\hat {\eta} _ {s}}{\nu}\right) \left(y _ {s} ^ {i}\right), y _ {s} ^ {i} - y _ {0} ^ {i} \right\rangle\right).
$$
$$
\text {F o r} i \in [ N ], x _ {t} ^ {i} = \nabla \phi^ {*} \left(y _ {t} ^ {i}\right).
$$
$$
\text {r e t u r n} \left(x _ {T} ^ {i}\right) _ {i = 1} ^ {N}.
$$
where  $\eta_t = (y_t)_{\#} \eta_0$  and  $\mu_t = (\nabla \phi^*)_{\#} \eta_t$ , and where  $c_{\eta_s}(y_s) = -\nabla_{W_2} \mathcal{F}(\eta_s)(y_s)$ , or some variant thereof. We will refer to this approach as mirrored coin sampling, or mirrored coin Wasserstein gradient descent. In order to obtain an implementable algorithm, we proceed as follows. First, decide on a suitable functional  $\mathcal{F}$  and a suitable  $(c_t)_{t \in \mathbb{N}} := (c_{\eta_t}(y_t))_{t \in \mathbb{N}}$ . For example,  $\mathcal{F}(\eta) = \mathrm{KL}(\eta | \nu)$ , and  $(c_t)_{t \in \mathbb{N}} = (-\mathcal{P}_{\eta_t, k_\phi} \nabla_{W_2} \mathcal{F}(\eta_t)(y_t))_{t \in \mathbb{N}}$ . Then, substitute these into (11), and approximate the updates using a set of interacting particles, as in existing ParVIs algorithms.
Following these steps, we obtain a learning rate free analogue of MSVGD [82], which we will refer to as Coin MSVGD. This is summarised in Alg. 2. For different choices of  $(c_{\eta_t})_{t\in \mathbb{N}}$  in (11), we also obtain learning rate free analogues of two other mirrored ParVI algorithms introduced in App. B, namely mirrored LAWGD (App. B.3), and mirrored KSDD (App. B.4). These algorithms, which we term Coin MLAWGD and Coin MKSDD, are given in Alg. 6 and Alg. 7 in App. D.
Even in the unconstrained setting, establishing the convergence of coin sampling remains an open problem. In [81], the authors provide a technical sufficient condition under which it is possible to establish convergence to the target measure in the population limit; however, it is difficult to verify this condition in general. In the interest of completeness, in App. C, we provide an analogous convergence result for mirrored coin sampling, adapted appropriately to the constrained setting.
4.3 Alternative Approaches
In this section, we outline an alternative approach, based on a 'coinification' of the mollified interaction energy descent (MIED) method recently introduced in [60]. MIED is based on minimising a function known as the mollified interaction energy (MIE)  $\mathcal{E}_{\epsilon}:\mathcal{P}(\mathbb{R}^{d})\to [0,\infty ]$ . The idea is that minimising this function balances two forces: a repulsive force which ensures the measure is well spread, and an attractive force which ensures the measure is concentrated around regions of high density. In order to obtain a practical sampling algorithm, [60] propose to minimise a discrete version of the logarithmic MIE, using, e.g., Adagrad [31] or Adam [48]. That is, minimising the function
$$
\log E _ {\epsilon} \left(\omega^ {n}\right) := \log \frac {1}{N ^ {2}} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \phi_ {\epsilon} \left(x ^ {i} - x ^ {j}\right) \left(\pi \left(x ^ {i}\right) \pi \left(x ^ {j}\right)\right) ^ {- \frac {1}{2}}, \tag {12}
$$
where  $\omega^n = \{x^1, \ldots, x^n\}$ , and where  $(\phi_{\epsilon})_{\epsilon > 0}$  is a family of mollifiers [60, Definition 3.1]. This approach can be adapted to handle the constrained case, as outlined in [60]. In particular, if there exists a differentiable  $f: \mathbb{R}^d \to \mathcal{X}$  such that  $\mathcal{X} \setminus f(\mathbb{R}^d)$  has Lebesgue measure zero, then one can reduce the constrained problem to an unconstrained one by minimising  $\log E_{\epsilon}(f(w^n))$ , where  $f(\omega^n) = \{f(x^1), \ldots, f(x^n)\}$ . Meanwhile, if  $\mathcal{X} = \{x \in \mathbb{R}^d : g(x) \leq 0\}$  for some differentiable  $g: \mathbb{R}^d \to \mathbb{R}$ , then one can use a variant of the dynamic barrier method introduced in [37].
Here, as an alternative to the optimisation methods considered in [60], we instead propose to use a learning-rate free algorithm to minimise (12), based on the coin betting framework described in 4.1. We term the resulting method, summarised in Alg. 3, Coin MIED. In comparison to our mirrored coin sampling algorithm, this approach has the advantage that it only requires a differentiable (not necessarily bijective) map.
Sampling as Optimisation. The connection between sampling and optimisation has a long history, dating back to [46], which established that the evolution of the law of the Langevin diffusion
Algorithm 3 Coin MIED
Input: target density  $\pi$ , mollifier  $\phi_{\epsilon}$ , particles  $\omega_0^n = (x_0^i)_{i=1}^N$ .
$$
\text {F o r} i \in [ N ], x _ {t} ^ {i} = x _ {0} ^ {i} - \frac {\sum_ {s = 1} ^ {t - 1} \nabla_ {x _ {s} ^ {i}} \log E _ {\epsilon} \left(\omega_ {s} ^ {n}\right)}{t} \Big (1 - \sum_ {s = 1} ^ {t - 1} \langle \nabla_ {x _ {s} ^ {i}} \log E _ {\epsilon} \left(\omega_ {s} ^ {n}\right), x _ {s} ^ {i} - x _ {0} ^ {i} \rangle \Big).
$$
corresponds to the WGF of the KL divergence. In recent years, there has been renewed interest in this perspective [99]. In particular, the viewpoint of existing algorithms such as LMC [23, 24, 33, 34] and SVGD [32, 63] as discretisations of WGFs has proved fruitful in deriving sharp convergence rates [6, 34, 52, 78, 83, 85]. It has also resulted in the rapid development of new sampling algorithms, inspired by ideas from the optimisation literature such as proximal methods [100], coordinate descent [29, 30], Nesterov acceleration [18, 25, 67], Newton methods [68, 84, 98], and coin betting [80, 81].
Sampling in Constrained Domains. Recent interest in constrained sampling has resulted in a range of general-purpose algorithms for this task [e.g., 43, 60, 82, 103, 104]. In cases where it is possible to explicitly parameterise the target domain in a lower dimensional space, one can use variants of classical methods, including LMC [98], rejection sampling [28], Hamiltonian Monte Carlo (HMC) [13], and Riemannian manifold HMC [55, 73]. Several other approaches are based on the use of mirror map. In particular, [43] proposed mirrored Langevin dynamics and its first-order discretisation. [103], and an earlier draft of [43], introduced the closely related mirror Langevin diffusion and the mirror Langevin algorithm, which has since also been studied in [1, 19, 44, 61]. Along the same lines, [82] introduced two variants of SVGD suitable for constrained domains based on the use of a mirror map, and established convergence of these schemes to the target distribution; see also [86]. In certain cases [e.g., 56, 58] one cannot explicitly parameterise the manifold of interest. In this setting, different approaches are required. [12] introduced a constrained version of HMC for this case. Meanwhile, [102] proposed a constrained Metropolis-Hastings algorithm with a reverse projection check to ensure reversibility; this approach has since been extended in [57, 59]. Other, more recent approaches suitable for this setting include MIED [60] (see Sec. 4.3), and O-SVGD [104].
In this section, we perform an empirical evaluation of Coin MSVGD (Sec. 6.1 - 6.2) and Coin MIED (Sec. 6.3). We consider several simulated and real data experiments, including sampling from targets defined on the simplex (Sec. 6.1), confidence interval construction for post-selection inference (Sec. 6.2), and inference in a fairness Bayesian neural network (Sec. 6.3). We compare our methods to their learning-rate-dependent analogues, namely, MSVGD [82] and MIED [60]. We also include a comparison with Stein Variational Mirror Descent (SVMD) [82], and projected versions of Coin SVGD [81] and SVGD [63], which include a Euclidean projection step to ensure the iterates remain in the domain. We provide additional experimental details in App. F and additional numerical results in App. G. Code to reproduce all of the numerical results can be found at https://github.com/loissharrock/constrained-coin-sampling.
6.1 Simplex Targets
Following [82], we first test the performance of our algorithms on two 20-dimensional targets defined on the simplex: the sparse Dirichlet posterior of [73] and the quadratic simplex target of [1]. We employ the IMQ kernel and the entropic mirror map [7]; and use  $N = 50$  particles,  $T = 500$  iterations. In Fig. 1 and Fig. 6-7 (App. F.1), we plot the energy distance [87] to a set of surrogate ground truth samples, either obtained i.i.d. (sparse Dirichlet target) or using the No-U-Turn Sampler (NUTS) [42] (quadratic target). After  $T = 500$  iterations, Coin MSVGD has comparable performance to MSVGD and SVMD with optimal learning rates but significantly outperforms both algorithms for sub-optimal learning rates (Fig. 1). In both cases, the projected methods fail to converge. For the sparse Dirichlet posterior, Coin MSVGD converges more rapidly than MSVGD and SVMD, even for well-chosen values of the learning rate (Fig. 6a in App. F.1). Meanwhile, for the quadratic target, Coin MSVGD generally converges more rapidly than MSVGD but not as fast as SVMD [82], which takes advantage of the log-concavity of the target in the primal space (Fig. 6a in App. F.1).
6.2 Confidence Intervals for Post Selection Inference
We next consider a constrained sampling problem arising in post-selection inference [54, 88]. Suppose we are given data  $(X,y)\in \mathbb{R}^{n\times p}\times \mathbb{R}^n$ . We are interested in obtaining valid confidence intervals (CIs) for regression coefficients obtained via the randomised Lasso [89], defined as the solution of
$$
\hat {\beta} = \underset {\beta \in \mathbb {R} ^ {p}} {\arg \min } \left[ \frac {1}{2} | | y - X \beta | | _ {2} ^ {2} + \lambda | | \beta | | _ {1} - \omega^ {\top} \beta + \frac {\varepsilon}{2} | | \beta | | _ {2} ^ {2} \right], \tag {13}
$$
where  $\lambda, \varepsilon \in \mathbb{R}_+$  are penalty parameters and  $\omega \in \mathbb{R}^p$  is an auxiliary random vector. Let  $\hat{\beta}_E \in \mathbb{R}^q$  represent the non-zero coefficients of  $\hat{\beta}$  and  $\hat{z}_E = \mathrm{sign}(\hat{\beta}_E)$  for their signs. If the support  $E$  were known a priori, we could determine CIs for  $\beta_E$  using the asymptotic normality of  $\bar{\beta}_E = (X_E^\top X_E)^{-1}X_E^\top y$ . However, when  $E$  is based on data, these 'classical' CIs will no longer be valid. In this case, one approach is to condition on the result of the initial model selection, i.e., knowledge of  $E$  and  $\hat{z}_E$ . In practice, this means sampling from the so-called selective distribution, which has support  $\mathcal{D} = \{(\hat{\beta}_E, \hat{z}_{-E}) : \mathrm{sign}(\hat{\beta}_E) = \hat{z}_E, ||\hat{z}_{-E}||_{\infty} \leq 1\}$ , and density proportional to [e.g., 90, Sec. 4.2]
$$
\hat {g} \left(\hat {\beta} _ {E}, \hat {z} _ {- E}\right) \propto g \left(\varepsilon \left(\hat {\beta} _ {E}, 0\right) ^ {\top} - X ^ {\top} \left(y - X _ {E} \hat {\beta} _ {E}\right) + \lambda \left(\hat {z} _ {E}, \hat {z} _ {- E}\right) ^ {\top}\right). \tag {14}
$$
Synthetic Example. We first consider the model setup described in [79, Sec. 5.3]; see App. F.2 for full details. In Fig. 2, we plot the energy distance between samples obtained using Coin MSVGD and projected Coin SVGD, and a set of 1000 samples obtained using NUTS [42], for a two-dimensional selective distribution. Similar to our previous examples, the performance of MSVGD is very sensitive to the choice of learning rate. If the learning rate is too small, the particle converge rather slowly; on the other hand, if the learning rate is too big, the particles may not even converge. Meanwhile, Coin MSVGD converges rapidly to the true target, with no need to tune a learning rate. Similar to the examples considered in Sec. 6.1, the projected methods once again fail to converge to the true target, highlighting the need for the mirrored approach.
In Fig. 3, we plot the coverage of the CIs obtained using Coin MSVGD, MSVGD, SVMD, and the norejection MCMC algorithm in selective Inference [91], as we vary the nominal coverage or the total number of samples. For MSVGD and SVMD, we use RMSProp [92] to adapt the learning rate, with an initial learning rate of  $\gamma = 0.1$ , following [82]. As the nominal coverage varies, Coin MSVGD achieves similar coverage to MSVGD and SVMD; and significantly higher actual coverage than norejection (Fig. 3a). Meanwhile, as the number of samples varies, Coin MSVGD, MSVGD, and SVMD consistently all obtain a higher coverage than the fixed nominal coverage of  $90\%$  (Fig. 3b). This is important for small sample sizes, where the standard approach undercovers. The performance of MSVGD and SVMD is, of course, highly dependent on an appropriate choice of learning rate. In Fig. 11 (App. G.3), we provide additional plots illustrating how the coverage of the CIs obtained using MSVGD and SVMD can deteriorate when the learning-rate is chosen sub-optimally.
Real Data Example. We next consider a post-selection inference problem involving the HIV-1 drug resistance dataset studied in [8, 75]; see App. F.2 for full details. The goal is to identify statistically significant mutations associated with the response to the drug Lamivudine (3TC). The randomised Lasso selects a subset of mutations, for which we compute  $90\%$  CIs using 5000 samples and five methods: 'unadjusted' (the unadjusted CIs), 'standard' (the method in selectiveInference), MSVGD, SVMD, and Coin MSVGD. Our results (Fig. 4) indicate that the CIs obtained by Coin MSVGD are similar to those obtained via the standard approach, which we view as a benchmark, as well as by MSVGD and SVMD (with a well chosen learning rate). Importantly, they differ from the CIs obtained using the unadjusted approach (see, e.g., mutation P65R or P184V in Fig. 4). Meanwhile, for sub-optimal choices of the learning rate (Fig. 4b and Fig. 4c, Fig. 12 in App. G.3), the CIs obtained using MSVGD and SVMD differ substantially from those obtained using the baseline 'standard' approach and by Coin MSVGD, once more highlighting the sensitivity of these methods to the choice of an appropriate learning rate.
6.3 Fairness Bayesian Neural Network
Finally, following [60, 64, 65, 69], we train a fairness Bayesian neural network to predict whether an individual's income is greater than \(50,000\), with gender as a protected characteristic. We use the Adult Income dataset [50]. The dataset is of the form \(\mathcal{D} = \{x_i, y_i, z_i\}_{i=1}^n\), where \(x_i\) denote the feature vectors, \(y_i\) denote the labels (i.e., whether the income is greater than \)50,000\)), and \(z_i\) denote the protected attribute (i.e., the gender). We train a two-layer Bayesian neural network \(\hat{y}(x; w)\) with weights \(w\) and place a standard Gaussian prior on each weight independently. The fairness constraint is given by \(g(\theta) = \mathrm{Cov}_{(x, y, z) \sim \mathcal{D}}[z, \hat{y}(x; \theta)]^2 - t \leq 0\) for some user-specified \(t > 0\). In testing, we evaluate each method using a Calder-Verwer (CV) score [14], a standard measure of disparate impact. We run each method for \(T = 2000\) iterations and use \(N = 50\) particles.
In Fig. 5a, we plot the trade-off curve between test accuracy and CV score, for  $t \in \{10^{-6}, 10^{-5}, 10^{-4}, 10^{-3}, 2 \times 10^{-3}, 5 \times 10^{-3}, 10^{-2}\}$ . We compare the results for Coin MIED, MIED, and the methods in [65], using the default implementations. In this experiment, Coin MIED is clearly preferable to the other methods, achieving a much larger Pareto front. In Fig. 5b and Fig. 5c, we plot the constraint and the MIE versus training iterations for both coin MIED and MIED. Once again, coin MIED tends to outperform MIED, achieving both lower values of the constraint and lower values of the energy.
Summary. In this paper, we introduced several new particle-based algorithms for constrained sampling which are entirely learning rate free. Our first approach was based on the coin sampling framework introduced in [81], and a perspective of constrained sampling as a mirrored optimisation problem on the space of probability measures. Based on this perspective, we also unified several existing constrained sampling algorithms, and studied their theoretical properties in continuous time. Our second approach can be viewed as the coin sampling analogue of the recently proposed MIED algorithm [60]. Empirically, our algorithms achieved comparable or superior performance to other particle-based constrained sampling algorithms, with no need to tune a learning rate.
Limitations. We highlight three limitations. First, like any mirrored sampling algorithm, mirrored coin sampling (e.g., Coin MSVGD), necessarily depends on the availability of a mirror map that can appropriately capture the constraints of the problem at hand. Second, Coin MSVGD has a cost of  $O(N^2)$  per update. Finally, even in the population limit, establishing the convergence of mirrored coin sampling under standard conditions (e.g., strong log-concavity or a mirrored log-Sobolev inequality) remains an open problem. We leave this as an interesting direction for future work.
LS and CN were supported by the UK Research and Innovation (UKRI) Engineering and Physical Sciences Research Council (EPSRC), grant number EP/V022636/1. CN acknowledges further support from the EPSRC, grant number EP/R01860X/1.
A Calculus in the Wasserstein Space
We recall the following from [2, Chapter 10]. Suppose  $\mu \in \mathcal{P}_2(\mathbb{R}^d)$  and  $\xi \in L^{2}(\mu)$ . Let  $\mathcal{F}$  be a proper and lower semi-continuous functional on  $\mathcal{P}_2(\mathbb{R}^d)$ . We say that  $\xi \in L^{2}(\mu)$  belongs to the Fréchet subdifferential of  $\mathcal{F}$  at  $\mu$  and write  $\xi \in \partial \mathcal{F}(\mu)$  if, for any  $\nu \in \mathcal{P}_2(\mathbb{R}^d)$ ,
$$
\lim  _ {\nu \rightarrow \mu} \inf  _ {\nu \rightarrow \mu} \frac {\mathcal {F} (\nu) - \mathcal {F} (\mu) - \int_ {\mathbb {R} ^ {d}} \langle \xi (x) , t _ {\mu} ^ {\nu} (x) - x \rangle \mu (\mathrm {d} x)}{W _ {2} (\nu , \mu)} \geq 0, \tag {15}
$$
where  $t_\mu^\nu : \mathbb{R}^d \to \mathbb{R}^d$  denotes the optimal transport map from  $\mu$  to  $\nu$  [2, Chapter 7.1]. Under mild conditions [2, Lemma 10.4.1], the subdifferential  $\partial \mathcal{F}$  is single-valued,  $\partial \mathcal{F}(\mu) = \{\nabla_{W_2} \mathcal{F}(\mu)\}$ , and  $\nabla_{W_2} \mathcal{F}(\mu) \in L^2(\mu)$  is given by
$$
\nabla_ {W _ {2}} \mathcal {F} (\mu) = \nabla \frac {\partial \mathcal {F} (\mu)}{\partial \mu} (x) \quad \text {f o r} \mu \text {- a . e .} x \in \mathbb {R} ^ {d}, \tag {16}
$$
where  $\frac{\partial\mathcal{F}(\mu)}{\partial\mu}:\mathbb{R}^d\to \mathbb{R}$  denotes the first variation of  $\mathcal{F}$  at  $\mu$ , that is, the unique function such that
$$
\lim  _ {\varepsilon \rightarrow 0} \frac {1}{\varepsilon} (\mathcal {F} (\mu + \varepsilon \zeta) - \mathcal {F} (\mu)) = \int_ {\mathbb {R} ^ {d}} \frac {\partial \mathcal {F} (\mu)}{\partial \mu} (x) \zeta (\mathrm {d} x), \tag {17}
$$
where  $\zeta = \nu -\mu$  , and  $\nu \in \mathcal{P}_2(\mathbb{R}^d)$  . We will refer to  $\nabla_{W_2}\mathcal{F}(\mu)$  as the Wasserstein gradient of  $\mathcal{F}$  at  $\mu$
B Examples of Mirrored Wasserstein Gradient Flows
In this section, we outline several algorithms which arise as special cases of the MWGF introduced in Sec 3.2, namely
$$
\frac {\partial \eta_ {t}}{\partial t} + \nabla \cdot \left(w _ {t} \eta_ {t}\right) = 0, \quad \mu_ {t} = \left(\nabla \phi^ {*}\right) _ {\#} \eta_ {t}, \tag {18}
$$
where  $(w_{t})_{t\geq 0}$  are vector fields chosen to ensure the convergence of  $(\eta_t)_{t\geq 0}$  to  $\nu = (\nabla \phi)_{\#}\pi$
We will assume, throughout this section, that  $\pi \propto e^{-V}$  and that  $\nu = (\nabla \phi)_{\#} \pi \propto e^{-W}$ . The Monge-Ampere equation determines the relationship between  $W$  and  $V$ . In particular, we have [e.g., 43]
$$
e ^ {- V} = e ^ {- W \circ \nabla \phi} \det \nabla^ {2} \phi , e ^ {- W} = e ^ {- V \circ \nabla \phi^ {*}} \det \nabla^ {2} \phi^ {*}. \tag {19}
$$
Thus, the potential  $W(y)$  of the mirrored target  $\nu$  evaluated at  $y = \nabla \phi(x)$  can be expressed in terms of the potential  $V(x)$  of the original target  $\pi$  evaluated at  $x$ , as
$$
W (y) = V (x) + \log \det  \nabla^ {2} \phi (x). \tag {20}
$$
B.1 Mirrored Langevin Dynamics
Suppose that  $\mathcal{F}(\eta) = \mathrm{KL}(\eta ||\nu)$ , where, as elsewhere,  $\nu = (\nabla \phi)_{\#}\pi$  is the dual target. In addition, let  $w_{t} = -\nabla_{W_{2}}\mathcal{F}(\eta_{t})$ . In this case, it is straightforward to show that [e.g. 2, Chapter 10]  $\nabla_{W_2}\mathcal{F}(\eta) = \nabla \log \left(\frac{\eta}{\nu}\right)$ . Thus, for this choice of  $(w_{t})_{t\geq 0}$ , the MWGF in (18) reads
$$
\frac {\partial \eta_ {t}}{\partial t} - \nabla \cdot \left(\nabla \log \left(\frac {\eta_ {t}}{\nu}\right) \eta_ {t}\right) = 0, \quad \mu_ {t} = \left(\nabla \phi^ {*}\right) _ {\#} \eta_ {t}. \tag {21}
$$
Substituting  $\nabla \log \nu = -\nabla W$ , and using the fact that  $\eta_t \nabla \log (\eta_t) = \nabla \eta_t$ , this can also be written as
$$
\frac {\partial \eta_ {t}}{\partial t} - \nabla \cdot \left(\eta_ {t} \nabla W + \nabla \eta_ {t}\right) = 0, \quad \mu_ {t} = \left(\nabla \phi^ {*}\right) _ {\#} \eta_ {t}. \tag {22}
$$
This PDE is nothing more than the Fokker-Planck equation describing the evolution of the law of the overdamped Langevin SDE with respect to the mirrored target  $\nu \propto e^{-W}$ . In particular, suppose that  $x_0 \sim \mu_0$ ,  $y_0 = \nabla \phi(x_0) \sim \eta_0$ , and that  $(x_t)_{t \geq 0}$  and  $(y_t)_{t \geq 0}$  are the solutions of
$$
\mathrm {d} y _ {t} = - \nabla W (y _ {t}) \mathrm {d} t + \sqrt {2} \mathrm {d} b _ {t}, x _ {t} = \nabla \phi^ {*} (y _ {t}), \tag {23}
$$
where  $b = (b_{t})_{t\geq 0}$  is a standard Brownian motion. Then, if we let  $(\mu_t)_{t\geq 0}$  and  $(\eta_t)_{t\geq 0}$  denote the laws of  $(x_{t})_{t\geq 0}$  and  $(y_{t})_{t\geq 0}$ , it follows that  $(\mu_t)_{t\geq 0}$  and  $(\eta_t)_{t\geq 0}$  satisfy (22). This is precisely the mirrored Langevin dynamics (MLD) introduced in [43]. We remark that, using (20), the MLD in (23) can also be written as
$$
\mathrm {d} y _ {t} = - \nabla^ {2} \phi (x _ {t}) ^ {- 1} (\nabla V (x _ {t}) + \nabla \log \det  \nabla^ {2} \phi (x _ {t})) \mathrm {d} t + \sqrt {2} \mathrm {d} b _ {t}, \quad x _ {t} = \nabla \phi^ {*} (y _ {t}). \tag {24}
$$
Remark 1. It is worth distinguishing between the mirrored Langevin dynamics in (23) and the so-called mirror Langevin diffusion [e.g., 44, Equation 2], which refers to the solution of
$$
\mathrm {d} y _ {t} = - \nabla V \left(x _ {t}\right) \mathrm {d} t + \sqrt {2} \left[ \nabla^ {2} \phi \left(x _ {t}\right) \right] ^ {\frac {1}{2}} \mathrm {d} b _ {t}, \quad x _ {t} = \nabla \phi^ {*} \left(y _ {t}\right) \tag {25}
$$
These dynamics, and the corresponding Euler-Maruyama discretisation, known as the mirror Langevin algorithm or mirror Langevin Monte Carlo, were first studied in [103], and have since also been analysed in [1, 44, 61]. An alternative time discretisation has also been studied in [19].
In this case, one can show that the law  $(\mu_t)_{t\geq 0}$  of the solution of the mirror Langevin diffusion in (25) satisfies the Fokker-Planck equation [e.g., 44, App. A.2]
$$
\frac {\partial \mu_ {t}}{\partial t} + \nabla \cdot (\mu_ {t} v _ {t}) = 0, \quad v _ {t} = - [ \nabla^ {2} \phi ] ^ {- 1} \nabla \log \frac {\mu_ {t}}{\pi}. \tag {26}
$$
Recalling that  $\nabla_{W_2}\mathrm{KL}(\mu_t||\pi) = \nabla \log \frac{\mu_t}{\pi}$ , one can view (26) as a special case of the so-called Wasserstein mirror flow (WMF), defined according to [e.g. 1, App. C]