SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In other words, the population  $\rho$  in (7a) is subject to an isotropic diffusive force with diffusion coefficient  $\alpha > 0$ , a drift force due to the time-varying confining potential  $\int f_1 \mathrm{d}\mu(t) - \alpha \log \tilde{\rho}$ , and a self-interaction force via the interaction potential  $W$ . If we consider the measure  $\mu$  to be given and fixed in time, this corresponds exactly to the type of parabolic equation studied in [CMV03]. Here however the dynamics are more complex due to the coupling of the confining potential with the dynamics (7b) for  $\mu(t)$  via the coupling potential  $f_1$ . Before presenting the analysis of this model, let us give a bit more intuition on the meaning and the structure of these dynamics.
In the setting where  $\mu$  represents a binary classifier, we can think of the distribution  $\bar{\rho}$  as modelling all those individuals carrying the true label 1, say, and the distribution  $\rho(t)$  as modelling all those individuals carrying a true label 0, say, where 0 and 1 denote the labels of two classes of interest. The term  $\int f_1(z,x)\mu(t,\mathrm{d}x)$  represents a penalty for incorrectly classifying an individual at  $z$  with true label 0 when using the classifier  $\mu(t,x)$ . In other words,  $\int f_1(z,x)\mu(t,\mathrm{d}x) \in [0,\infty)$  is increasingly large the more  $z$  digresses from the correct classification 0. Similarly,  $\int f_1(z,x)\rho(t,\mathrm{d}z) \in [0,\infty)$  is increasingly large if the population  $\rho$  shifts mass to locations in  $z$  where the classification is incorrect. The terminology aligned objectives refers to the fact that in (7) both the population and the classifier are trying to evolve in a way as to maximize correct classification. Analogously, the term  $\int f_2(z,x)\bar{\rho}(\mathrm{d}z)$  is large if  $x$  would incorrectly classify the population  $\bar{\rho}$  that carries the label 1. A natural extension of the model (7) would be a setting where also the population carrying labels 1 evolves over time, which is simulated in Section E.2. Most elements of the framework presented here would likely carry over the setting of three coupled PDEs: one for the evolution of  $\rho(t)$ , one for the evolution of  $\bar{\rho}(t)$  and one for the classifier  $\mu(t)$ .
$$
\alpha \Delta \rho - \alpha \operatorname {d i v} (\rho \nabla \log \tilde {\rho}) = \alpha \operatorname {d i v} (\rho \nabla \delta_ {\rho} K L (\rho | \tilde {\rho}))
$$
forces the evolution of  $\rho(t)$  to approach  $\tilde{\rho}$ . In other words, it penalizes (in energy) deviations from a given reference measure  $\tilde{\rho}$ . In the context of the application at hand, we take  $\tilde{\rho}$  to be the initial distribution  $\rho(t = 0)$ . The solution  $\rho(t)$  then evolves away from  $\tilde{\rho}$  over time due to the other forces that are present. Therefore, the term  $KL(\rho|\tilde{\rho})$  in the energy both provides smoothing of the flow and a penalization for deviations away from the reference measure  $\tilde{\rho}$ .
The self-interaction term  $W * \rho$  introduces non-locality into the dynamics, as the decision for any given individual to move in a certain direction is influenced by the behavior of all other individuals in the population. The choice of  $W$  is application dependent. Very often, the interaction between two individuals only depends on the distance between them. This suggests a choice of  $W$  as a radial function, i.e.  $W(z) = \omega(|z|)$ . A choice of  $\omega : \mathbb{R} \rightarrow \mathbb{R}$  such that  $\omega'(r) > 0$  corresponds to an attractive force between individuals, whereas  $\omega'(r) < 0$  corresponds to a repulsive force. The statement  $|z| \omega'(|z|) = z \cdot \nabla_z W(z) \geq -D$  in Assumption 3 therefore corresponds to a requirement that the self-interaction force is not too repulsive. Neglecting all other forces in (7a), we obtain the non-local interaction equation
$$
\partial_ {t} \rho = \operatorname {d i v} \left(\rho \nabla W * \rho\right)
$$
which appears in many instances in mathematical biology, mathematical physics, and material science, and it is an equation that has been extensively studied over the past few decades, see for example [Car+11; BCY12; CCH14; BCL09; BLL12; CMV06; CFG23] and references therein. Using the results from these works, our assumptions on the interaction potential  $W$  can be relaxed in many ways, for example by allowing discontinuous derivatives at zero for  $W$ , or by allowing  $W$  to be negative.
The dynamics (7b) for the algorithm  $\mu$  is a non-autonomous transport equation,
$$
\partial_ {t} \mu = \operatorname {d i v} (\mu v),
$$
where the time-dependence in the velocity field
$$
v (t, x) := \nabla_ {x} \left(\int f _ {1} (z, x) \mathrm {d} \rho (t, z) + \int f _ {2} (z, x) \mathrm {d} \tilde {\rho} (z) + \frac {\beta}{2} \| x - x _ {0} \| ^ {2}\right),
$$
comes through the evolving population  $\rho(t)$ . This structure allows to obtain an explicit solution for  $\mu(t)$  in terms of the initial condition  $\mu_0$  and the solution  $\rho(t)$  to (7a) using the method of characteristics.
Proposition 5. Assume that there exists a constant  $c > 0$  such that
$$
\left\| \int \nabla_ {x} f _ {1} (z, x) \mathrm {d} \rho (z) + \int \nabla_ {x} f _ {2} (z, x) \mathrm {d} \bar {\rho} (z) \right\| \leq c (1 + \| x \|) \quad \forall \rho \in \mathcal {P} _ {2} (\mathbb {R} ^ {d}) a n d \forall x \in \mathbb {R} ^ {d}. \tag {8}
$$
Then the unique distributional solution  $\mu(t)$  to (7b) is given by
$$
\mu (t) = \Phi (t, 0, \cdot) _ {\#} \mu_ {0}, \tag {9}
$$
where  $\Phi (t,s,x)$  solves the characteristic equation
$$
\partial_ {s} \Phi (s, t, x) + v (s, \Phi (s, t, x)) = 0, \quad \Phi (t, t, x) = x. \tag {10}
$$
Proof. Thanks to Assumption 1, we have that  $v \in C^{1}(\mathbb{R} \times \mathbb{R}^{d}; \mathbb{R}^{d})$ , and by (8), we have
$$
\| v (t, x) \| \leq c (1 + \| x \|) \quad \text {f o r a l l} t \geq 0, x \in \mathbb {R} ^ {d}.
$$
By classical Cauchy-Lipschitz theory for ODEs, this guarantees the existence of a unique global solution  $\Phi(t,s,x)$  solving (10). Then it can be checked directly that  $\mu(t)$  as defined in (9) is a distributional solution to (7b).
In the characteristic equation (10),  $\Phi(s, t, x)$  is a parametrization of all trajectories: if a particle was at location  $x$  at time  $t$ , then it is at location  $\Phi(s, t, x)$  at time  $s$ . Our assumptions on  $f_1, f_2$  and  $\bar{\rho}$  also ensure that  $\Phi(s, t, \cdot): \mathbb{R}^d \to \mathbb{R}^d$  is a  $C^1$ -diffeomorphism for all  $s, t \in \mathbb{R}$ . For more details on transport equations, see for example [De 07].
Remark 2. Consider the special case where  $\mu_0 = \delta_{x_0}$  for some initial position  $x_0\in \mathbb{R}^d$ . Then by Proposition 5, the solution to (7b) is given by  $\mu (t) = \delta_{x(t)}$ , where  $x(t)\coloneqq \Phi (t,0,x_0)$  solves the ODE
$$
\dot {x} (t) = - v (t, x (t)), \qquad x (0) = x _ {0},
$$
which is precisely of type (3).
For the case of competing objectives, the two models we consider can be written as
$$
\partial_ {t} \rho = - \operatorname {d i v} (\rho [ \nabla (f _ {1} (z, b (\rho)) - \alpha \log (\rho / \tilde {\rho}) - W * \rho ]),
$$
$$
b (\rho) := \operatorname * {a r g m i n} _ {\bar {x}} \int f _ {1} (z, \bar {x}) \mathrm {d} \rho (z) + \int f _ {2} (\bar {x}, z ^ {\prime}) \mathrm {d} \bar {\rho} \left(z ^ {\prime}\right) + \frac {\beta}{2} \| \bar {x} - x _ {0} \| ^ {2}
$$
for (5), and
$$
\frac {\mathrm {d}}{\mathrm {d} t} x = - \nabla_ {x} \left(\int f _ {1} (z, x) r (x) (\mathrm {d} z) + \int f _ {2} (x, z ^ {\prime}) \mathrm {d} \bar {\rho} (z ^ {\prime}) + \frac {\beta}{2} \| x - x _ {0} \| ^ {2}\right),
$$
$$
r (x) := \underset {\hat {\rho} \in \mathcal {P}} {\operatorname {a r g m a x}} \int f _ {1} (z, x) \mathrm {d} \hat {\rho} (z) - \alpha K L (\hat {\rho} | \tilde {\rho}) - \frac {1}{2} \int \hat {\rho} W * \hat {\rho}.
$$
for (6).
A.2 Definitions and notation
Here, and in what follows,  $\mathrm{I}_d$  denotes the  $d \times d$  identity matrix, and  $\mathrm{id}$  denotes the identity map. The energy functionals we are considering are usually defined on the set of probability measures on  $\mathbb{R}^d$ , denoted by  $\mathcal{P}(\mathbb{R}^d)$ . If we consider the subset  $\mathcal{P}_2(\mathbb{R}^d)$  of probability measures with bounded second moment,
$$
\mathcal {P} _ {2} (\mathbb {R} ^ {d}) := \left\{\rho \in \mathcal {P} (\mathbb {R} ^ {d}): \int_ {\mathbb {R} ^ {d}} \| z \| ^ {2} \mathrm {d} \rho (z) <   \infty \right\},
$$
then we can endow this space with the Wasserstein-2 metric.
Definition 4 (Wasserstein-2 Metric). The Wasserstein-2 metric between two probability measures  $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d)$  is given by
$$
W _ {2} (\mu , \nu) ^ {2} = \inf  _ {\gamma \in \Gamma (\mu , \nu)} \int \| z - z ^ {\prime} \| _ {2} ^ {2} \mathrm {d} \gamma (z, z ^ {\prime})
$$
where  $\Gamma$  is the set of all joint probability distributions with marginals  $\mu$  and  $\nu$ , i.e.  $\mu(\mathrm{d}z) = \int \gamma(\mathrm{d}z, z') \mathrm{d}z'$  and  $\nu(\mathrm{d}z') = \int \gamma(z, \mathrm{d}z') \mathrm{d}z$ .
The restriction to  $\mathcal{P}_2(\mathbb{R}^d)$  ensures that  $W_{2}$  is always finite. Then the space  $(\mathcal{P}_2(\mathbb{R}^d), W_2)$  is indeed a metric space. We will make use of the fact that  $W_{2}$  metrizes narrow convergence of probability measures. To make this statement precise, let us introduce two common notions of convergence for probability measures, which are a subset of the finite signed Radon measures  $\mathcal{M}(\mathbb{R}^d)$ .
Definition 5. Consider a sequence  $(\mu_n) \in \mathcal{M}(\mathbb{R}^d)$  and a limit  $\mu \in \mathcal{M}(\mathbb{R}^d)$ .
- (Narrow topology) The sequence  $(\mu_n)$  converges narrowly to  $\mu$ , denoted by  $\mu_n \rightharpoonup \mu$ , if for all continuous bounded functions  $f: \mathbb{R}^d \to \mathbb{R}$ ,
$$
\int_ {\mathbb {R} ^ {d}} f (z) \mathrm {d} \mu_ {n} (z) \rightarrow \int_ {\mathbb {R} ^ {d}} f (z) \mathrm {d} \mu (z).
$$
- (Weak-\* topology) The sequence  $(\mu_n)$  converges weakly-\* to  $\mu$ , denoted by  $\mu_n \stackrel{*}{\rightharpoonup} \mu$ , if for all continuous functions vanishing at infinity (i.e.  $f: \mathbb{R}^d \to \mathbb{R}$  such that for all  $\epsilon > 0$  there exists a compact set  $K_\epsilon \subset \mathbb{R}^d$  such that  $|f(z)| < \epsilon$  on  $\mathbb{R}^d \setminus K_\epsilon$ ), we have
Let us denote the set of continuous functions on  $\mathbb{R}^d$  vanishing at infinity by  $C_0(\mathbb{R}^d)$ , and the set of continuous bounded functions by  $C_b(\mathbb{R}^d)$ . Note that narrow convergence immediately implies that  $\mu_n(\mathbb{R}^d) \to \mu(\mathbb{R}^d)$  as the constant function is in  $C_b(\mathbb{R}^d)$ . This is not necessarily true for weak- $*$  convergence. We will later make use of the Banach-Alaoglu theorem [Ala40], which gives weak- $*$  compactness of the unit ball in a dual space. Note that  $\mathcal{M}(\mathbb{R}^d)$  is indeed the dual of  $C_0(\mathbb{R}^d)$  endowed with the sup-norm, and  $\mathcal{P}(\mathbb{R}^d)$  is the unit ball in  $\mathcal{M}(\mathbb{R}^d)$  using the dual norm. Moreover, if we can ensure that mass does not escape to infinity, the two notions of convergence in Definition 5 are in fact equivalent.
Lemma 6. Consider a sequence  $(\mu_n) \in \mathcal{M}(\mathbb{R}^d)$  and a measure  $\mu \in \mathcal{M}(\mathbb{R}^d)$ . Then  $\mu_n \rightharpoonup \mu$  if and only if  $\mu_n \stackrel{*}{\rightharpoonup} \mu$  and  $\mu_n(\mathbb{R}^d) \to \mu(\mathbb{R}^d)$ .
This follows directly from Definition 5. Here, the condition  $\mu_n(\mathbb{R}^d) \to \mu(\mathbb{R}^d)$  is equivalent to tightness of  $(\mu_n)$ , and follows from Markov's inequality [Gho02] if we can establish uniform bounds on the second moments, i.e. we want to show that there exists a constant  $C > 0$  independent of  $n$  such that
$$
\int \| z \| ^ {2} \mathrm {d} \mu_ {n} (z) <   C \quad \forall n \in \mathrm {N}. \tag {11}
$$
Definition 6 (Tightness of probability measures). A collection of measures  $(\mu_n) \in \mathcal{M}(\mathbb{R}^d)$  is tight if for all  $\epsilon > 0$  there exists a compact set  $K_{\epsilon} \subset \mathbb{R}^{d}$  such that  $|\mu_n| (\mathbb{R}^d \setminus K_{\epsilon}) < \epsilon$  for all  $n \in \mathbb{N}$ , where  $|\mu|$  denotes the total variation of  $\mu$ .
Another classical result is that the Wasserstein-2 metric metrizes narrow convergence and weak- $*$  convergence of probability measures, see for example [San15, Theorem 5.11] or [Vil03, Theorem 7.12].
Lemma 7. Let  $\mu_n, \mu \in \mathcal{P}_2(\mathbb{R}^d)$ . Then  $W_2(\mu_n, \mu) \to 0$  if and only if
$$
\mu_ {n} \rightharpoonup \mu \quad a n d \quad \int_ {\mathbb {R} ^ {d}} \| z \| ^ {2} \mathrm {d} \mu_ {n} (z) \rightarrow \int_ {\mathbb {R} ^ {d}} \| z \| ^ {2} \mathrm {d} \mu (z).
$$
Remark 3. Note that  $\mu_n \rightharpoonup \mu$  can be replaced by  $\mu_n \xrightarrow{*} \mu$  in the above statement thanks to the fact that the limit  $\mu$  is a probability measure with mass 1, see Lemma 6.
Next, we consider two measures  $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$  that are atomless, i.e.  $\mu(\{z\}) = 0$  for all  $z \in \mathbb{R}^d$ . By Brenier's theorem [BB00] (also see [Vil03, Theorem 2.32]) there exists a unique measurable map  $T: \mathbb{R}^d \to \mathbb{R}^d$  such that  $T_{\#}\mu = \nu$ , and  $T = \nabla \psi$  for some convex function  $\psi: \mathbb{R}^d \to \mathbb{R}$ . Here, the push-forward operator  $\nabla \psi_{\#}$  is defined as
$$
\int_ {\mathbb {R} ^ {d}} f (z) \mathrm {d} \nabla \psi_ {\#} \rho_ {0} (z) = \int_ {\mathbb {R} ^ {d}} f (\nabla \psi (z)) \mathrm {d} \rho_ {0} (z)
$$
for all Borel-measurable functions  $f: \mathbb{R}^d \mapsto \mathbb{R}_+$ . If  $\rho_1 = \nabla \psi_{\#} \rho_0$ , we denote by  $\rho_s = [(1 - s) \mathrm{id} + s\nabla \psi]_{\#} \rho_0$  the displacement interpolant between  $\rho_0$  and  $\rho_1$ . We are now ready to introduce the notion of displacement convexity, which is the same as geodesic convexity in the geodesic space  $(\mathcal{P}_2(\mathbb{R}^d), W_2)$ . We will state the definition here for atomless measures, but it can be relaxed to any pair of measures in  $\mathcal{P}_2$  using optimal transport plans instead of transport maps. In what follows, we will use  $s$  to denote the interpolation parameter for geodesics, and  $t$  to denote time related to solutions of (4), (5) and (6).
Definition 7 (Displacement Convexity). A functional  $G: \mathcal{P} \mapsto \mathbb{R}$  is displacement convex if for all  $\rho_0, \rho_1$  that are atomless we have
$$
G (\rho_ {s}) \leq (1 - s) G (\rho_ {0}) + s G (\rho_ {1}),
$$
where  $\rho_s = [(1 - s)\mathrm{id} + s\nabla \psi]_{\#}\rho_0$  is the displacement interpolant between  $\rho_0$  and  $\rho_1$ . Further,  $G:\mathcal{P}\mapsto \mathbb{R}$  is uniformly displacement convex with constant  $\eta >0$  if
$$
G (\rho_ {s}) \leq (1 - s) G (\rho_ {0}) + s G (\rho_ {1}) - s (1 - s) \frac {\eta}{2} W _ {2} (\rho_ {0}, \rho_ {1}) ^ {2},
$$
where  $\rho_{s} = [(1 - s)\mathrm{id} + s\nabla \psi ]_{\#}\rho_{0}$  is the displacement interpolant between  $\rho_0$  and  $\rho_{1}$ .
Remark 4. In other words,  $G$  is displacement convex (concave) if the function  $G(\rho_s)$  is convex (concave) with  $\rho_s = [(1 - s\mathrm{id} + s\nabla \psi ]_{\#}\rho_0$  being the displacement interpolant between  $\rho_0$  and  $\rho_{1}$ . Contrast this with the classical notion of convexity (concavity) for  $G$ , where we require that the function  $G((1 - s)\rho_0 + s\rho_1)$  is convex (concave).
In fact, if the energy  $G$  is twice differentiable along geodesics, then the condition  $\frac{\mathrm{d}^2}{\mathrm{d}s^2} G(\gamma_s) \geq 0$  along any geodesic  $(\rho_s)_{s \in [0,1]}$  between  $\rho_0$  and  $\rho_1$  is sufficient to obtain displacement convexity. Similarly, when  $\frac{\mathrm{d}^2}{\mathrm{d}s^2} G(\rho_s) \geq \eta W_2(\rho_0, \rho_1)^2$ , then  $G$  is uniformly displacement convex with constant  $\eta > 0$ . For more details, see [McC97] and [Vil03, Chapter 5.2].
A.3 Steady states
The main goal in our theoretical analysis is to characterize the asymptotic behavior for the models (4), (5) and (6) as time goes to infinity. The steady states of these equations are the natural candidates to be asymptotic profiles for the corresponding equations. Thanks to the gradient flow structure, we expect to be able to make a connection between ground states of the energy functionals, and the steady states of the corresponding gradient flow dynamics. More precisely, any minimizer or maximizer is in particular a critical point of the energy, and therefore satisfies that the first variation is constant on disconnected components of its support. If this ground state also has enough regularity (weak differentiability) to be a solution to the equation, it immediately follows that it is in fact a steady state.
To make this connection precise, we first introduce what exactly we mean by a steady state.
Definition 8 (Steady states for (4)). Given  $\rho_{\infty} \in L_{+}^{1}(\mathbb{R}^{d}) \cap L_{loc}^{\infty}(\mathbb{R}^{d})$  with  $\| \rho_{\infty}\|_{1} = 1$  and  $\mu_{\infty} \in \mathcal{P}_{2}(\mathbb{R}^{d})$ , then  $(\rho_{\infty},\mu_{\infty})$  is a steady state for the system (4) if  $\rho_{\infty} \in W_{loc}^{1,2}(\mathbb{R}^{d})$ ,  $\nabla W*\rho_{\infty} \in L_{loc}^{1}(\mathbb{R}^{d})$ ,  $\rho_{\infty}$  is absolutely continuous with respect to  $\tilde{\rho}$ , and  $(\rho_{\infty},\mu_{\infty})$  satisfy
$$
\nabla_ {z} \left(\int f _ {1} (z, x) \mathrm {d} \mu_ {\infty} (x) + \alpha \log \left(\frac {\rho_ {\infty} (z)}{\bar {\rho} (z)}\right) + W * \rho_ {\infty} (z)\right) = 0 \quad \forall z \in \operatorname {s u p p} \left(\rho_ {\infty}\right), \tag {12a}
$$
$$
\nabla_ {x} \left(\int f _ {1} (z, x) \mathrm {d} \rho_ {\infty} (z) + \int f _ {2} (z, x) \mathrm {d} \tilde {\rho} (z) + \frac {\beta}{2} \| x - x _ {0} \| ^ {2}\right) = 0 \quad \forall x \in \operatorname {s u p p} \left(\mu_ {\infty}\right) \tag {12b}
$$
in the sense of distributions.
Here,  $L_{+}^{1}(\mathbb{R}^{d})\coloneqq \{\rho \in L^{1}(\mathbb{R}^{d}):\rho \geq 0\}$
Definition 9 (Steady states for (5)). Let  $\rho_{\infty} \in L_{+}^{1}(\mathbb{R}^{d}) \cap L_{loc}^{\infty}(\mathbb{R}^{d})$  with  $\| \rho_{\infty}\|_{1} = 1$ . Then  $\rho_{\infty}$  is a steady state for the system (5) if  $\rho_{\infty} \in W_{loc}^{1,2}(\mathbb{R}^{d})$ ,  $\nabla W * \rho_{\infty} \in L_{loc}^{1}(\mathbb{R}^{d})$ ,  $\rho_{\infty}$  is absolutely continuous with respect to  $\tilde{\rho}$ , and  $\rho_{\infty}$  satisfies
$$
\nabla_ {z} \left(f _ {1} (z, b (\rho_ {\infty})) - \alpha \log \left(\frac {\rho_ {\infty} (z)}{\tilde {\rho} (z)}\right) - W * \rho_ {\infty} (z)\right) = 0 \quad \forall z \in \mathbb {R} ^ {d}, \tag {13}
$$
in the sense of distributions, where  $b(\rho_{\infty}) \coloneqq \operatorname{argmin}_{x} G_{c}(\rho_{\infty}, x)$ .
Definition 10 (Steady states for (6)). The vector  $x_{\infty}\in \mathbb{R}^{d}$  is a steady state for the system (6) if it satisfies
$$
\nabla_ {x} G _ {d} \left(x _ {\infty}\right) = 0.
$$
In fact, with the above notions of steady state, we can obtain improved regularity for  $\rho_{\infty}$ .
Lemma 8. Let Assumptions 1-3 hold. Then the steady states  $\rho_{\infty}$  for (4) and (5) are continuous.
Proof. We present here the argument for equation (5) only. The result for (4) follows in exactly the same way by replacing  $f_{1}(z,b(\rho_{\infty}))$  with  $-\int f_1(z,x)\mathrm{d}\mu_\infty (x)$ .
Thanks to our assumptions, we have  $f_{1}(\cdot ,b(\rho_{\infty})) + \alpha \log \tilde{\rho} (\cdot)\in C^{1}$ , which implies that  $\nabla (f_1(\cdot ,b(\rho_\infty)) + \alpha \log \tilde{\rho} (\cdot))\in L_{loc}^{\infty}$ . By the definition of a steady state,  $\rho_{\infty}\in L^{1}\cap L_{loc}^{\infty}$  and thanks to Assumption 3 we have  $W\in C^2$ , which implies that  $\nabla W*\rho_{\infty}\in L_{loc}^{\infty}$ . Let
$$
h (z) := \rho_ {\infty} (z) \nabla \left[ f _ {1} (z, b \left(\rho_ {\infty}\right)) + \alpha \log \tilde {\rho} (z) - \left(W * \rho_ {\infty}\right) (z) \right].
$$
Then by the aforementioned regularity, we obtain  $h \in L_{loc}^{1} \cap L_{loc}^{\infty}$ . By interpolation, it follows that  $h \in L_{loc}^{p}$  for all  $1 < p < \infty$ . This implies that  $\operatorname{div}(h) \in W_{loc}^{-1,p}$ . Since  $\rho_{\infty}$  is a weak  $W_{loc}^{1,2}$ -solution of (13), we have
$$
\Delta \rho_ {\infty} = \operatorname {d i v} (h),
$$
and so by classic elliptic regularity theory we conclude  $\rho_{\infty} \in W_{loc}^{1,p}$ . Finally, applying Morrey's inequality, we have  $\rho_{\infty} \in C^{0,k}$  where  $k = \frac{p - d}{p}$  for any  $d < p < \infty$ . Therefore  $\rho_{\infty} \in C(\mathbb{R}^d)$  (after possibly being redefined on a set of measure zero).
For ease of notation, we write  $G_{a}:\mathcal{P}(\mathbb{R}^{d})\times \mathcal{P}(\mathbb{R}^{d})\mapsto [0,\infty ]$  as
$$
G _ {a} ((\rho , \mu)) = \alpha K L (\rho | \tilde {\rho}) + \mathcal {V} (\rho , \mu) + \mathcal {W} (\rho),
$$
$$
\mathcal {V} (\rho , \mu) = \iint f _ {1} (z, x) \mathrm {d} \rho (z) \mathrm {d} \mu (x) + \int V (x) \mathrm {d} \mu (x),
$$
$$
\mathcal {W} (\rho) = \frac {1}{2} \iint W (z _ {1} - z _ {2}) \mathrm {d} \rho (z _ {1}) \mathrm {d} \rho (z _ {2}),
$$
with potential given by  $V(x)\coloneqq \int f_2(z,x)\mathrm{d}\bar{\rho} (z) + \frac{\beta}{2}\| x - x_0\| ^2$
In order to prove the existence of a unique ground state for  $G_{a}$ , a natural approach is to consider the corresponding Euler-Lagrange equations
$$
\alpha \log \frac {\rho (z)}{\tilde {\rho} (z)} + \int f _ {1} (z, x) \mathrm {d} \mu (x) + (W * \rho) (z) = c _ {1} [ \rho , \mu ] \quad \text {f o r a l l} z \in \operatorname {s u p p} (\rho), \tag {14a}
$$
$$
\int f _ {1} (z, x) \mathrm {d} \rho (z) + V (x) = c _ {2} [ \rho , \mu ] \quad \text {f o r a l l} x \in \operatorname {s u p p} (\mu), \tag {14b}
$$
where  $c_{1}, c_{2}$  are constants that may differ on different connected components of  $\mathrm{supp}(\rho)$  and  $\mathrm{supp}(\mu)$ . These equations are not easy to solve explicitly, and we are therefore using general non-constructive techniques from calculus of variations. We first show continuity and convexity properties for the functional  $G_{a}$  (Lemma 9 and Proposition 10), essential properties that will allow us to deduce existence and uniqueness of ground states using the direct method in the calculus of variations (Proposition 11). Using the Euler-Lagrange equation 14, we then prove properties on the support of the ground state (Corollary 12). To obtain convergence results, we apply the HWI method: we first show a general 'interpolation' inequality between the energy, the energy dissipation and the metric (Proposition 13); this fundamental inequality will then imply a generalized logarithmic Sobolev inequality (Corollary 14) relating the energy to the energy dissipation, and a generalized Talagrand inequality (Corollary 15) that allows to translate convergence in energy into convergence in metric. Putting all these ingredients together will then allow us to conclude for the statements in Theorem 2.
Lemma 9 (Lower semi-continuity). Let Assumptions 1-3 hold. Then the functional  $G_{a}:\mathcal{P}\times \mathcal{P}\to \mathbb{R}$  is lower semi-continuous with respect to the weak-\* topology.
Proof. We split the energy  $G_{a}$  into three parts: (i)  $KL(\rho |\tilde{\rho})$ , (ii) the interaction energy  $\mathcal{W}$ , and (iii) the potential energy  $\mathcal{V}$ . For (i), lower semi-continuity has been shown in [Pos75]. For (ii), we can directly apply [San15, Proposition 7.2] using Assumption 3. For (iii), note that  $V$  and  $f_{1}$  are lower semi-continuous and bounded below thanks to Assumption 1, and so the result follows from [San15, Proposition 7.1].