SlowGuess/NeurIPS_2023_OCR huggingface dataset:
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A Existence and uniqueness of global minimiser
In this section, we discuss assumptions under which the global minimiser of the optimisation problem
$$
L (Q) = \int \ell (\theta) d Q (\theta) + \lambda D (Q, P) \tag {7}
$$
over  $\mathcal{P}(\mathbb{R}^J)$  exists and is unique. We assume throughout that the optimisation problem is not pathological, in the sense that there exists a measure  $\widehat{Q}\in \mathcal{P}(\mathbb{R}^{J})$  such that  $L(\widehat{Q}) < \infty$ . This is in applications often trivial to verify. A good candidate for  $\widehat{Q}$  is typically the reference measure  $P$ .
Loss assumptions Let  $\ell : \mathbb{R}^J \to \mathbb{R}$  be a loss satisfying the following assumptions:
(L1) The loss  $\ell$  is bounded from below which means that
$$
c := \inf  \left\{\ell (\theta): \theta \in \mathbb {R} ^ {J} \right\} > - \infty . \tag {8}
$$
(L2) The loss is norm-coercive which means that
$$
\ell (\theta) \rightarrow \infty \tag {9}
$$
$$
i f \left\| \theta \right\|\rightarrow \infty .
$$
(L3) The loss  $\ell$  is lower semi-continuous which means that
$$
\lim  _ {\theta \rightarrow \theta_ {0}} \inf  _ {\theta} \ell (\theta) \geq \ell \left(\theta_ {0}\right) \tag {10}
$$
for all  $\theta_0\in \mathbb{R}^J$
Regulariser assumptions Let  $D: \mathcal{P}(\mathbb{R}^J) \times \mathcal{P}(\mathbb{R}^J) \to [0, \infty]$  be a regulariser and  $P \in \mathcal{P}(\mathbb{R}^J)$  a reference measure. We define  $D_P(\cdot) \coloneqq D(\cdot, P)$  for notational convenience. We assume the following for  $D_P$ :
(D1) The function  $D_P$  is lower semi-continuous w.r.t. to the topology of weak-convergence, i.e. for all sequences  $(Q_n)_{n \in \mathbb{N}} \subset \mathcal{P}(\mathbb{R}^J)$  and all  $Q$  with  $D_P(Q) < \infty$ , it holds that  $Q_n \xrightarrow{\mathcal{D}} Q$  implies
$$
\lim  _ {n \rightarrow \infty} \inf  _ {n \rightarrow \infty} D _ {P} (Q _ {n}) \geq D _ {P} (Q). \tag {11}
$$
Here,  $\xrightarrow{\mathcal{D}}$  denotes convergence in distribution.
(D2)  $D_P$  is strictly convex, i.e. for all  $Q_1 \neq Q_2 \in \mathcal{P}(\mathbb{R}^J)$  with  $D_P(Q_1) < \infty$  and  $D_P(Q_2) < \infty$ , it holds that
$$
D _ {P} (\alpha Q _ {1} + (1 - \alpha) Q _ {2}) <   \alpha D _ {P} (Q _ {1}) + (1 - \alpha) D _ {P} (Q _ {2}) \tag {12}
$$
with  $\alpha \in (0,1)$
The next theorem provides an existence result for the optimisation problem (7). The result is similar in spirit to Lemma 2.1 in Knoblauch (2021) with the important difference that our assumptions are easier to verify, since they are formulated in terms of  $\ell$  and  $D_P$ .
Theorem 3 (Existence of global minimiser). Under the assumptions (L1)-(L3) and (D1) there exists a probability measure  $Q^{*} \in \mathcal{P}(\mathbb{R}^{J})$  with
$$
L \left(Q ^ {*}\right) = \inf  \left\{L (Q): Q \in \mathcal {P} \left(\mathbb {R} ^ {J}\right) \right\}. \tag {13}
$$
Proof. Let  $c > -\infty$  be the lower bound for  $\ell$ . It follows immediately that  $L(Q) \geq c$  for all  $Q \in \mathcal{P}(\mathbb{R}^J)$  since  $D(P, Q) \geq 0$ . As a consequence we know that
$$
\infty > L ^ {*} := \inf  \left\{L (Q): Q \in \mathcal {P} \left(\mathbb {R} ^ {J}\right) \right\} \geq c > - \infty . \tag {14}
$$
By definition of the infimum we can construct a sequence  $l_{n} = L(Q_{n}) \in \mathbb{R}$  in the image of  $L$  such
$$
l _ {n} \rightarrow L ^ {*} \tag {15}
$$
for  $n\to \infty$ . We now show by contradiction that the corresponding sequence  $\left(Q_n\right)\subset \mathcal{P}(\mathbb{R}^J)$  is tight. Assume that  $\left(Q_{n}\right)$  is not tight. By definition we can then find an  $\epsilon >0$  such that for each  $k\in \mathbb{N}$  there exists  $n = n_k\in \mathbb{N}$  with  $Q_{n_k}([-k,k]^J)\leq 1 - \epsilon$ . We set  $A_{k}\coloneqq [-k,k]^{J}\subset \mathbb{R}^{J}$  and obtain
$$
\begin{array}{l} l _ {n _ {k}} = L \left(Q _ {n _ {k}}\right) (16) \\ = \int_ {A _ {k}} \ell (\theta) d Q _ {n _ {k}} (\theta) + \int_ {\mathbb {R} ^ {J} \backslash A _ {k}} \ell (\theta) d Q _ {n _ {k}} (\theta) + \lambda D (Q, P) (17) \\ \geq \int_ {A _ {k}} \ell (\theta) d Q _ {n _ {k}} (\theta) + \int_ {\mathbb {R} ^ {J} \backslash A _ {k}} \ell (\theta) d Q _ {n _ {k}} (\theta) (18) \\ \geq c Q _ {n _ {k}} \left(A _ {k}\right) + \inf  \left\{\ell (\theta): \theta \in \mathbb {R} ^ {J} \backslash A _ {k} \right\} Q _ {n _ {k}} \left(\mathbb {R} ^ {J} \backslash A _ {k}\right) (19) \\ \geq c Q _ {n _ {k}} \left(A _ {k}\right) + \epsilon \inf  \left\{\ell (\theta): \theta \in \mathbb {R} ^ {J} \backslash A _ {k} \right\}. (20) \\ \end{array}
$$
Due to the coerciveness of  $\ell$ , we know that  $\inf \left\{\ell(\theta) : \theta \in \mathbb{R}^J \backslash A_k\right\} \to \infty$  for  $k \to \infty$  and therefore  $l_{n_k} \to \infty$  for  $k \to \infty$ . However, this is a contradiction: The sequence  $(l_n)$  is convergent and therefore in particular bounded. As a consequence, it cannot contain the unbounded sub-sequence  $(l_{n_k})$ . It follows that the sequence  $(Q_n)$  is tight. By Prokhorov's theorem we can now extract a sub sequence  $(Q_{n_k})$  of  $(Q_n)$  and a measure  $Q^* \in \mathcal{P}(\mathbb{R}^J)$  such that
$$
Q _ {n _ {k}} \xrightarrow {\mathcal {D}} Q ^ {*} \tag {21}
$$
for  $k \to \infty$ . Due to Lemma 5.1.7 in Ambrosio et al. (2005) the lower semi-continuity of  $\ell$  implies that  $Q \mapsto \int \ell(\theta) dQ(\theta)$  is lower semi-continuous. This combined with the lower semi-continuity of  $D_P$  gives
$$
\lim  _ {k \rightarrow \infty} \inf  L \left(Q _ {n _ {k}}\right) \geq L \left(Q ^ {*}\right). \tag {22}
$$
From this it immediately follows that
$$
L \left(Q ^ {*}\right) \leq \lim  _ {k \rightarrow \infty} \inf  L \left(Q _ {n _ {k}}\right) = L ^ {*}, \tag {23}
$$
but by definition  $L^{*}$  is the global minimum of  $L$  which implies  $L^{*} \leq L(Q^{*})$ . We therefore conclude that  $L(Q^{*}) = L^{*}$ .
Theorem 3 only shows the existence of a global minimiser. In order to show uniqueness we use the convexity assumption (D2). The proof is the same as in finite dimensions and only included for completeness.
Theorem 4 (Uniqueness of global minimiser). Assume that (D2) holds. Then, the global minimiser of  $L$  is unique (whenever it exists).
Proof. Assume there exists two probability measures  $Q_{1}, Q_{2} \in \mathcal{P}(\mathbb{R}^{J})$  such that
$$
L \left(Q _ {1}\right) = L ^ {*} = L \left(Q _ {2}\right). \tag {24}
$$
where  $\infty > L^{*} \coloneqq \inf \left\{L(Q) : Q \in \mathcal{P}(\mathbb{R}^{J})\right\} > -\infty$ . We define the probability measure  $Q_{3} \coloneqq \frac{1}{2} Q_{1} + \frac{1}{2} Q_{3}$ . By strict convexity we obtain
$$
L \left(Q _ {3}\right) <   \frac {1}{2} L \left(Q _ {1}\right) + \frac {1}{2} L \left(Q _ {3}\right) = L ^ {*}, \tag {25}
$$
which is a contradiction to  $Q_{1}$  and  $Q_{2}$  being global minimisers.
Note that in the literature on GVI (Knblauch et al., 2022) it is common to assume that the regulariser is definite, i.e.
$$
D (P, Q) = 0 \Longleftrightarrow P = Q \tag {26}
$$
for all  $P, Q \in \mathcal{P}(\mathbb{R}^J)$ . We did not use this assumption in neither Theorem 3 nor Theorem 4. However, the next lemma shows that it is basically implied by strict convexity.
<sup>4</sup>A sequence of probability measures  $(Q_{n})$  is called tight if and only if for every  $\epsilon > 0$  there exists a compact set  $K \in \mathbb{R}^{J}$  such that for all  $n \in \mathbb{N}$  holds:  $Q_{n}(K) > 1 - \epsilon$ .
Lemma 1. Let  $D_P: \mathcal{P}(\mathbb{R}^J) \to [0, \infty]$  be strictly convex and assume further  $D(Q, Q) = 0$  for all  $Q \in \mathcal{P}(\mathbb{R}^J)$ . Then it follows that  $D(Q, P) = 0$  implies  $P = Q$ .
Proof. We prove the claim by contradiction. Assume that there exists  $P \neq Q$  such that  $D(P, Q) = 0$ . The strict convexity and  $D(P, P) = 0$  imply combined that
$$
\begin{array}{l} D \left(\frac {1}{2} P + \frac {1}{2} Q, P\right) <   \frac {1}{2} D (P, P) + \frac {1}{2} D (Q, P) (27) \\ = 0. (28) \\ \end{array}
$$
However, we know that  $D\left(\frac{1}{2} P + \frac{1}{2} Q, P\right) \geq 0$  by assumption. This is a contradiction.
Discussion on loss assumptions The assumptions on the loss  $\ell$  in (L1) and (L3) are rather weak. Typically loss functions in machine learning are bounded from below and continuous (and therefore in particular lower semi-continuous). However, norm-coercivity can be violated. Consider for example the squared loss
$$
\ell (\theta) := \sum_ {n = 1} ^ {N} \left(y _ {n} - f _ {\theta} \left(x _ {n}\right)\right) ^ {2}, \tag {29}
$$
where  $f_{\theta}$  is the parametrisation of a neural network with one hidden layer, i.e.  $\theta = (w,A)$  and
$$
f _ {\theta} (x) = w ^ {T} \sigma (A x), \tag {30}
$$
where  $\sigma : \mathbb{R} \to \mathbb{R}$  is an activation function which is applied pointwise to the vector  $Ax$  and has the property that  $\sigma(0) = 0$ . It is now possible to find a sequence of parameters  $(\theta_k)_{k \in \mathbb{N}} \subset \mathbb{R}^J$  with  $\|\theta_k\| \to \infty$  such that  $\ell(\theta_k)$  does not converge to infinity. Define  $w_k := k(1 \ldots 1)$ ,  $A_k := 0$  and  $\theta_k = (w_k A_k)$  for  $k \in \mathbb{N}$ . Then we obviously have that
$$
\left\| \theta_ {k} \right\| = \left\| w _ {k} \right\|\rightarrow \infty \tag {31}
$$
for  $k\to \infty$  but
$$
\begin{array}{l} \ell \left(\theta_ {k}\right) = \sum_ {n = 1} ^ {N} \left(y _ {n} - f _ {\theta_ {k}} \left(x _ {n}\right)\right) ^ {2} (32) \\ = \sum_ {n = 1} ^ {N} \left(y _ {n} - w ^ {T} \sigma (0)\right) ^ {2} (33) \\ = \sum_ {n = 1} ^ {N} y _ {n} ^ {2}, (34) \\ \end{array}
$$
which is constant and therefore does not converge to  $\infty$ . A similar, but notationally more involved, construction can be made for neural networks with more than one hidden layer. However, this is an issue that can be easily resolved by adding what is known as weight decay to the loss. For example, consider for  $\gamma > 0$  the loss
$$
\ell (\theta) := \sum_ {n = 1} ^ {N} \left(y _ {n} - f _ {\theta} \left(x _ {n}\right)\right) ^ {2} + \gamma \| \theta \| ^ {2} \tag {35}
$$
with weight decay. This loss is by construction norm-coercive and therefore the previous existence proof applies.
Discussion on regulariser assumptions The assumptions (D1) and (D2) are quite weak. The KL-divergence for example is known to be lower semi-continuous (Polyanskiy and Wu, 2014, Theorem 3.7) and strictly convex (Polyanskiy and Wu, 2014, Theorem 4.1). This immediately implies lower semi-continuity and convexity of  $\mathrm{KL}(\cdot ,P)$  for any fixed  $P$ . The MMD is also known to be strictly convex (Arbel et al., 2019, Lemma 25), whenever it is well-defined, which can be guaranteed under weak assumptions on  $\kappa$  (Muandet et al., 2017, Lemma 3.1). The lower semi-continuity properties also depend on the kernel  $\kappa$ . However, for bounded kernels it is trivial to verify. We include the proof for completeness, but assume this has been shown before elsewhere.
Lemma 2. Let the kernel  $\kappa : \mathbb{R}^J \times \mathbb{R}^J$  be continuous and bounded:  $\| \kappa \|_{\infty} := \sup_{\theta, \theta' \in \mathbb{R}^J} |k(\theta, \theta')| < \infty$  and  $P$  be fixed. Then  $\mathrm{MMD}(\cdot, P)$  is continuous and therefore, in particular, lower semi-continuous.
Proof. Let  $(Q_{n})_{n\in \mathbb{N}}$  and  $Q^{*}$  be such that
$$
Q _ {n} \xrightarrow {\mathcal {D}} Q ^ {*} \tag {36}
$$
for  $n\to \infty$  . This immediately implies that
$$
Q _ {n} \otimes Q _ {n} \xrightarrow {\mathcal {D}} Q ^ {*} \otimes Q ^ {*} \tag {37}
$$
for  $n\to \infty$  , where  $Q^{*}\otimes Q^{*}$  denotes the product measure of  $Q^{*}$  with itself. Further, note that the kernel mean embedding  $\mu_P$  is continuous as integral with respect to the second component of a continuous function and bounded since
$$
\begin{array}{l} | \mu_ {P} (\theta) | = \left| \int \kappa \left(\theta , \theta^ {\prime}\right) d P \left(\theta^ {\prime}\right) \right| (38) \\ \leq \int | \kappa (\theta , \theta^ {\prime}) | d P \left(\theta^ {\prime}\right) (39) \\ \leq \| \kappa \| _ {\infty}. (40) \\ \end{array}
$$
By the definition of weak convergence for measures, we therefore have
$$
\begin{array}{l} \iint \kappa (\theta , \theta^ {\prime}) d \left(Q _ {n} \otimes Q _ {n}\right) (\theta , \theta^ {\prime}) \longrightarrow \iint \kappa (\theta , \theta^ {\prime}) d \left(Q ^ {*} \otimes Q ^ {*}\right) (\theta , \theta^ {\prime}) (41) \\ \int \mu_ {P} (\theta) d Q _ {n} (\theta) \longrightarrow \int \mu_ {P} (\theta) d Q ^ {*} (\theta) (42) \\ \end{array}
$$
for  $n\to \infty$  . This immediately implies continuity of  $\mathrm{MMD}(\cdot ,P)$  with respect to the topology of weak convergence.
Notice that most kernels common in machine learning, such as the squared exponential or the Matérn kernel, are continuous and bounded and therefore Lemma 2 applies.
Remark 1. The astute reader may have noticed that our existence proof only guarantees the existence of measure  $Q^{*} \in \mathcal{P}(\mathbb{R}^{J})$ . However, the Wasserstein gradient flow is by definition only formulated in the space of probability measures with finite second moment, denoted  $\mathcal{P}_2(\mathbb{R}^J)$ . Assumptions which guarantee that  $Q^{*} \in \mathcal{P}_{2}(\mathbb{R}^{J})$  are easy to formulate. For example, we can require that there exists  $C > 0$  and  $R > 0$  such that the loss  $\ell$  satisfies
$$
| \ell (\theta) | > C \| \theta \| ^ {2} \tag {43}
$$
for all  $\| \theta \| > R$ . This immediately implies that  $Q^{*} \in \mathcal{P}_{2}(\mathbb{R}^{J})$  since otherwise
$$
\int | \ell (\theta) | d Q ^ {*} (\theta) = \infty \tag {44}
$$
gives a contradiction to the finiteness of  $L(Q^{*})$ . However, even if (43) is violated, the reference measure  $P$  may still guarantee that  $Q^{*} \in \mathcal{P}_{2}(\mathbb{R}^{J})$ . For example, if  $P \in \mathcal{P}_{2}(\mathbb{R}^{J})$ , then  $D_{P}(Q^{*})$  will typically be large if  $Q^{*} \notin \mathcal{P}_{2}(\mathbb{R}^{J})$  and the global minimiser is therefore in a sense unlikely to have fat tails. We therefore assume  $Q^{*} \in \mathcal{P}_{2}(\mathbb{R}^{J})$  throughout the paper and consider it to be a minor practical concern.
B Realising the Wasserstein gradient flow
In this section, we identify a suitable stochastic process that allows us to follow the WGF.
Let  $L^{\mathrm{fe}}: \mathcal{P}(\mathbb{R}^J) \to (-\infty, \infty]$  be the free energy discussed in Section 3.2 given as
$$
L ^ {\mathrm {f e}} (Q) := \int V (\theta) d Q (\theta) + \frac {\lambda_ {1}}{2} \int \kappa (\theta , \theta^ {\prime}) d Q (\theta) d Q \left(\theta^ {\prime}\right) + \lambda_ {2} \int \log (q (\theta)) q (\theta) d \theta , \tag {45}
$$
where  $\lambda_1, \lambda_2 \geq 0$  are constants,  $V: \mathbb{R}^J \to \mathbb{R}$  is the potential,  $\kappa: \mathbb{R}^J \times \mathbb{R}^J \to \mathbb{R}$  is symmetric. We will write  $L$  for  $L^{\mathrm{fe}}$  from now on to simplify notation. The Wasserstein gradient of  $L$  is given as (cf. Chapter 9.1 Villani, 2003, Equation 9.4)
$$
\nabla_ {W} L [ Q ] (\theta) = \nabla V (\theta) + \lambda_ {1} \left(\nabla_ {1} \kappa * Q\right) (\theta) + \lambda_ {2} \nabla \log (q (\theta)), \tag {46}
$$
where  $\nabla_{1}\kappa :\mathbb{R}^{J}\times \mathbb{R}^{J}\to \mathbb{R}^{J}$  is the (vector-valued) derivative of  $\kappa$  with respect to the first component,  $\nabla$  denotes the euclidean gradient with respect to  $\theta$  and  $(\nabla_{1}\kappa *Q)(\theta)\coloneqq \int \nabla_{1}\kappa (\theta ,\theta^{\prime})dQ(\theta^{\prime})$  for  $\theta \in \mathbb{R}^J$ . The corresponding Wasserstein gradient flow is therefore given as (cf. Chapter 9.1 Villani, 2003, Equation 9.3)
$$
\partial_ {t} q (t, \theta) = \nabla \cdot \left(q (t, \theta) (\nabla V (\theta) + \lambda_ {1} (\nabla_ {1} \kappa * Q) (\theta) + \lambda_ {2} \nabla \log (q _ {t} (\theta)))\right). \tag {47}
$$
In general the probability density evolution of a stochastic process is—via the Fokker-Planck equation—associated with the adjoint of the (infinitesimal) generator of the stochastic process. We will therefore try to identify the generator associated to the density evolution in (47). To this end let  $h \in C_c^2(\mathbb{R}^J, \mathbb{R})$  where  $C_c^2(\mathbb{R}^J, \mathbb{R})$  denotes the space of twice continuously differentiable functions with compact support. We multiply both sides of (47) with  $h$ , integrate, and apply the partial integration rule to obtain
$$
\begin{array}{l} \frac {d}{d t} \int h (\theta) q (t, \theta) d \theta = - \int \nabla_ {W} L [ Q (t) ] (\theta) \cdot \nabla h (\theta) q (t, \theta) d \theta . (48) \\ = - \int (\nabla V (\theta) + \lambda_ {1} (\nabla_ {1} \kappa * Q _ {t}) (\theta)) \cdot \nabla h (\theta) d Q _ {t} (\theta) (49) \\ - \lambda_ {2} \int \nabla \log (q _ {t} (\theta)) \cdot \nabla h (\theta) d Q _ {t} (\theta). (50) \\ \end{array}
$$
By chain-rule and partial integration, (50) can be rewritten as
$$
\begin{array}{l} - \lambda_ {2} \int \nabla \log (q _ {t} (\theta)) \cdot \nabla h (\theta) d Q _ {t} (\theta) = - \lambda_ {2} \int \nabla q _ {t} (\theta) \cdot \nabla h (\theta) d \theta (51) \\ = \lambda_ {2} \int \Delta h (\theta) d Q _ {t} (\theta). (52) \\ \end{array}
$$
Putting everything together, we obtain
$$
\frac {d}{d t} \int h (\theta) q (t, \theta) d \theta = \int (A [ Q (t) ] h) (\theta) d Q _ {t} (\theta), \tag {53}
$$
where  $\{A[Q]\}_{Q\in \mathcal{P}(\mathbb{R}^J)}$  is a family of operators defined as
$$
\left(A [ Q ] h\right) (\theta) := - \left(\nabla V (\theta) + \lambda_ {1} \left(\nabla_ {1} \kappa * Q\right) (\theta)\right) \cdot \nabla h (\theta) + \lambda_ {2} \Delta h. \tag {54}
$$
for  $h \in C_c^2(\mathbb{R}^J, \mathbb{R})$ . The reader may recognize this operator family as the generator of a so-called nonlinear Markov processes (Kolokoltsov, 2010, Chapter 1.4). The nonlinearity in this case refers to the dependency on the measure  $Q$ . Linear Markov processes have no measure-dependency. This family of generators corresponds to a McKean-Vlasov process of the form
$$
d \theta (t) = - \left(\nabla V (\theta (t)) + \lambda_ {1} \left(\nabla_ {1} \kappa * Q _ {t}\right) (\theta (t))\right) d t + \sqrt {2 \lambda_ {2}} d B (t), \tag {55}
$$
where  $\left(B(t)\right)_{t > 0}$  is a Brownian motion and  $Q_{t}$  the law of  $\theta (t)$ . In other words: The solution to (55) has the time marginals  $Q(t)$  such that (53) holds for every  $h\in C_c^2 (\mathbb{R}^J,\mathbb{R})$ . Furthermore, the corresponding pdfs  $\left(q(t)\right)$  satisfy the nonlinear Fokker-Planck equation given as
$$
\partial_ {t} q _ {t} = A ^ {*} \left[ Q _ {t} \right] q _ {t}, \tag {56}
$$
where  $A^{*}[Q]$  denotes the  $L^2$ -adjoint of the operator  $A[Q]$  and is given as
$$
\left(A ^ {*} [ Q ] h\right) (\theta) = \nabla \cdot \left(h (\theta) \left(\nabla V (\theta) + \lambda_ {1} \left(\nabla_ {1} \kappa * Q\right) (\theta) + \lambda_ {2} \nabla \log (h (\theta))\right)\right) \tag {57}
$$