SlowGuess/NeurIPS_2023_OCR huggingface dataset:
As  $K(x,x) = h(0)$  and  $K(x,y) > 0$  for all  $x,y$ , and that  $\mu$  is an empirical measure  $\mu = \frac{1}{N}\sum_{i=1}^{N}\delta_{x_i}$ , the last quantity is equal to
$$
\frac {1}{N ^ {2}} \sum_ {i} | x _ {i} | ^ {2} \sum_ {j} K (x _ {i}, x _ {j}) \geq \frac {1}{N ^ {2}} \sum_ {i} | x _ {i} | ^ {2} K (x _ {i}, x _ {i}) \geq \frac {h (0)}{N} \int | x | ^ {2} \mu (d x).
$$
In total, we derive that  $b$  is dissipative on average.
Smoothness of the drift: We have, for  $\mu$  in a compact set of  $\mathcal{P}_2(\mathbb{R}^d)$
$$
\begin{array}{l} \left| b (x, \mu) - b \left(x ^ {\prime}, \mu\right) \right| \\ \leq \left| \int \nabla_ {2} K (x, y) - \nabla_ {2} K \left(x ^ {\prime}, y\right) \mu (d y) \right| + \left| \int \left(K (x, y) - K \left(x ^ {\prime}, y\right)\right) \nabla V (y) \mu (d y) \right| \\ \leq L \left| x - x ^ {\prime} \right| \left(1 + \int \left| \nabla V (y) \right| \mu (d y)\right) \\ \leq L \left| x - x ^ {\prime} \right| \left(1 + C \int (1 + | y | ^ {2}) \mu (d y)\right) <   L ^ {\prime} \left| x - x ^ {\prime} \right|. \\ \end{array}
$$
Moreover, take  $\mu, \nu$  in the same compact set, and let  $\pi$  be the optimal coupling (in  $\mathcal{W}_2$  sense). Then,
$$
\begin{array}{l} | b (x, \mu) - b (x, \nu) | ^ {2} \\ \leq 2 \left| \int \nabla_ {2} K (x, y) - \nabla_ {2} K (x, z) \pi (d y, d z) \right| ^ {2} \\ + 2 \left| \int K (x, y) \nabla V (y) - K (x, z) \nabla V (z) \pi (d y, d z) \right| ^ {2} \\ \leq 2 L ^ {2} \mathcal {W} _ {2} ^ {2} (\mu , \nu) \\ + 2 \left| \int K (x, y) \nabla V (y) - K (x, z) \nabla V (y) + K (x, z) \nabla V (y) - K (x, z) \nabla V (z) \pi (d y, d z) \right| ^ {2} \\ \leq 2 L ^ {2} \mathcal {W} _ {2} ^ {2} (\mu , v) + 4 \left(L ^ {2} + L ^ {\prime 2}\right) \mathcal {W} _ {2} ^ {2} (\mu , v). \\ \end{array}
$$
C.3 Two-player Zero-sum Continuous Games
Proof of Corollary 3. Recall that
$$
b (q, \mu) := \int \left( \begin{array}{c} - \nabla_ {x} K (q _ {1}, q _ {2} ^ {\prime}) \\ \alpha \nabla_ {y} K (q _ {1} ^ {\prime}, q _ {2}) \end{array} \right) \mu (d q ^ {\prime}), \quad q = (q _ {1}, q _ {2}).
$$
Smoothness of drift: For a fixed  $\mu \in \mathcal{P}_2(\mathbb{R}^d)\mathbb{R}^{2d}$  and  $q,r\in \mathbb{R}^{2d}$  we have
$$
\begin{array}{l} | b (q, \mu) - b (r, \mu) | ^ {2} \leq \int \left| \left(\frac {- \nabla_ {x} K (q _ {1} , q _ {2} ^ {\prime})}{\alpha \nabla_ {y} K (q _ {1} ^ {\prime} , q _ {2})}\right) - \left(\frac {- \nabla_ {x} K (r _ {1} , q _ {2} ^ {\prime})}{\alpha \nabla_ {y} K (q _ {1} ^ {\prime} , r _ {2})}\right) \right| ^ {2} \mu (d q ^ {\prime}) \\ \leq L ^ {2} \left| q _ {1} - r _ {1} \right| ^ {2} + \alpha^ {2} L ^ {2} \left| q _ {2} - r _ {2} \right| ^ {2} \\ \leq L ^ {2} | q - r | ^ {2}, \\ \end{array}
$$
where we used  $L$ -Lipschitzness of  $\nabla_x K$  and  $\nabla_y K$ .
Now, for a fixed  $q \in \mathbb{R}^{2d}$ , and  $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d) \mathbb{R}^{2d}$  with optimal coupling  $\pi$ , we have
$$
\begin{array}{l} | b (q, \mu) - b (q, \nu) | ^ {2} \leq \iint \left| \left(\frac {- \nabla_ {x} K (q _ {1} , r _ {2})}{\alpha \nabla_ {y} K (r _ {1} , q _ {2})}\right) - \left(\frac {- \nabla_ {x} K (q _ {1} , r _ {2} ^ {\prime})}{\alpha \nabla_ {y} K (r _ {1} ^ {\prime} , q _ {2})}\right) \right| ^ {2} \pi (d r, d r ^ {\prime}) \\ \leq L ^ {2} \iint | r _ {2} - r _ {2} ^ {\prime} | ^ {2} + | r _ {1} - r _ {1} ^ {\prime} | ^ {2} \pi (d r, d r ^ {\prime}) \\ = L ^ {2} \mathcal {W} _ {2} ^ {2} (\mu , v). \\ \end{array}
$$
Average dissipativity of drift: Suppose  $\nabla_{x}K$  and  $-\nabla_{y}K$  are  $(a,\beta)$ -dissipative. Then
$$
\begin{array}{l} \int \langle q, b (q, \mu) \rangle \mu (d q) = \iint \langle q _ {1}, - \nabla_ {x} K (q _ {1}, q _ {2} ^ {\prime}) \rangle + \alpha \langle q _ {2}, \nabla_ {y} K (q _ {1} ^ {\prime}, q _ {2}) \rangle \mu (d q ^ {\prime}) \mu (d q) \\ \leq \int - a \alpha \left(\left| q _ {1} \right| ^ {2} + \left| q _ {2} \right| ^ {2}\right) \mu (d q) + 2 \beta , \\ \end{array}
$$
implying that  $b(\cdot ,\cdot)$  is  $(a\alpha ,2\beta)$ -dissipative on average.
If, on the other hand, the domains  $\mathcal{X}$  and  $\mathcal{Y}$  are bounded, observe that by Cauchy-Schwarz
$$
\left| \int \langle q, b (q, \mu) \rangle \mu (d q) \right| \leq \iint | q _ {1} | | \nabla_ {x} K (q _ {1}, q _ {2} ^ {\prime}) | + \alpha | q _ {2} | | \nabla_ {y} K (q _ {1} ^ {\prime}, q _ {2}) | \mu (d q ^ {\prime}) \mu (d q) \leq M,
$$
where  $M = \sup_{q_1 \in \mathcal{X}, q_2 \in \mathcal{Y}} |q_1| |\nabla_x K(q_1, q_2')| + \alpha |q_2| |\nabla_y K(q_1', q_2)|$ . Also denoting by  $R = \sup_{q \in \mathcal{X} \times \mathcal{Y}} |q|^2$ , we see that for any  $\alpha > 0$ ,  $b(\cdot, \cdot)$  is  $(\alpha, M + \alpha N)$ -dissipative on average, as
$$
\int \langle q, b (q, \mu) \rangle \mu (d q) + \alpha \int | q | ^ {2} \mu (d q) \leq M + \alpha N.
$$
Optimistic algorithm fits Assumption 4: Recall the iterates
$$
q _ {k + 1} ^ {i} = q _ {k} ^ {i} + \gamma_ {k + 1} \left(2 b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) - b \left(q _ {k - 1} ^ {i}, \widehat {\mu} _ {k - 1}\right)\right) + \sqrt {2 \gamma_ {k + 1}} \sigma \Xi_ {k + 1} ^ {i},
$$
where  $\Xi_{k + 1}^{i} = (\xi_{k + 1}^{i},\zeta_{k + 1}^{i})$ . Notice that the bias of this iteration is
$$
\varepsilon_ {k + 1} ^ {i} = b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) - b \left(q _ {k - 1} ^ {i}, \widehat {\mu} _ {k - 1}\right).
$$
For brevity, let us write  $\mathcal{F}_k\coloneqq \mathcal{F}_{\tau_k}$ . We have
$$
\begin{array}{l} \mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] = \mathbb {E} \left[ \left| q _ {k + 1} ^ {i} - q _ {k} ^ {i} - \gamma_ {k + 1} b (q _ {k} ^ {i}, \widehat {\mu} _ {k}) - \sqrt {2 \gamma_ {k + 1}} \sigma \Xi_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] \\ \leq 3 \mathbb {E} \left[ \left| q _ {k + 1} ^ {i} - q _ {k} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 3 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 6 \gamma_ {k + 1} \tau (1 + \alpha) d. \\ \end{array}
$$
$$
\begin{array}{l} \mathbb {E} \left[ \left| q _ {k + 1} ^ {i} - q _ {k} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] \leq 2 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| 2 b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) - b \left(q _ {k - 1} ^ {i}, \widehat {\mu} _ {k - 1}\right) \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 2 \gamma_ {k + 1} \tau (1 + \alpha) d \\ = 2 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) + \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 2 \gamma_ {k + 1} \tau (1 + \alpha) d \\ \leq 4 \gamma_ {k + 1} ^ {2} \mathbb {E} \big [ | b (q _ {k} ^ {i}, \widehat {\mu} _ {k}) | ^ {2} \mid \mathcal {F} _ {k} \big ] + 4 \gamma_ {k + 1} ^ {2} \mathbb {E} \big [ | \varepsilon_ {k + 1} ^ {i} | ^ {2} \mid \mathcal {F} _ {k} \big ] + 2 \gamma_ {k + 1} \tau (1 + \alpha) d \\ \end{array}
$$
Combining the last two inequalities, we have
$$
\begin{array}{l} \mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] \leq 3 \left(4 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 4 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 2 \gamma_ {k + 1} \tau (1 + \alpha) d\right) \\ + 3 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ | b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) | ^ {2} \mid \mathcal {F} _ {k} \right] + 6 \gamma_ {k + 1} \tau (1 + \alpha) d \\ = 1 2 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 1 5 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| b \left(q _ {k} ^ {i}, \widehat {\mu} _ {k}\right) \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 1 2 \gamma_ {k + 1} \tau (1 + \alpha) d \\ \end{array}
$$
Since  $\gamma_{k + 1}\to 0$ , we can assume that  $12\gamma_{k + 1}^2\leq 1 / 2$ , which implies
$$
\mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] \leq 3 0 \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| b (q _ {k} ^ {i}, \widehat {\mu} _ {k}) \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 2 4 \gamma_ {k + 1} \tau (1 + \alpha) d,
$$
which is exactly what we are after.
C.4 Kinetic Equations
Proof of Corollary 4. Smoothness of the drift: Let  $x, y \in \mathbb{R}^d$  and  $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d)$  and set  $\pi$  be an optimal coupling between  $\mu$  and  $\nu$  (in  $W_1$  sense). Then
$$
\left| b (x, \mu) - b (y, \nu) \right| \leq \left| \nabla V (x) - \nabla V (y) \right| + \left| \int \nabla W (x - z) \mu (d z) - \int \nabla W (y - z) \nu (d z) \right|.
$$
By  $L$ -Lipschitzness of  $\nabla V$ , the first term is bounded by  $L|x - y|$ . For the second term, using the coupling, we can write it as
$$
\begin{array}{l} \left| \iint \nabla W (x - z _ {1}) - \nabla W (y - z _ {2}) \pi (d z _ {1}, d z _ {2}) \right| \leq \iint | \nabla W (x - z _ {1}) - \nabla W (y - z _ {2}) | \pi (d z _ {1}, d z _ {2}) \\ \leq L \iint | x - y + z _ {2} - z _ {1} | \pi (d z _ {1}, d z _ {2}) \\ \leq L \iint | x - y | + | z _ {2} - z _ {1} | \pi (d z _ {1}, d z _ {2}) \\ \leq L \left| x - y \right| + L W _ {1} (\mu , \nu) \\ \leq L | x - y | + L W _ {2} (\mu , \nu). \\ \end{array}
$$
Putting these together we get
$$
\left| b (x, \mu) - b (y, v) \right| \leq 2 L \left(\left| x - y \right| + \mathcal {W} _ {2} (\mu , v)\right).
$$
Average dissipativity of the drift: First we show that for  $x \in \mathbb{R}^d$  and a probability measure  $\mu$ , we have
$$
\iint \langle x, \nabla W (x - y) \rangle \mu (d x) \mu (d y) \geq - M _ {W} / 2. \tag {C.1}
$$
This holds, since
$$
\begin{array}{l} \iint \langle x, \nabla W (x - y) \rangle \mu (d x) \mu (d y) \\ = \iint \langle x - y + y, \nabla W (x - y) \rangle \mu (d x) \mu (d y) \\ = \iint \langle x - y, \nabla W (x - y) \rangle \mu (d x) \mu (d y) + \iint \langle y, \nabla W (x - y) \rangle \mu (d x) \mu (d y) \\ \geq - M _ {W} + \iint \langle y, \nabla W (x - y) \rangle \mu (d x) \mu (d y) \\ \geq - M _ {W} - \iint \langle x, \nabla W (x - y) \rangle \mu (d x) \mu (d y), \\ \end{array}
$$
where in the penultimate inequality we used the assumption (which implies  $\langle \nabla W(x),x\rangle \geq -M_W$ ), and in the last one, we used the that  $W$  is symmetric (which implies  $\nabla W(-z) = -\nabla W(z)$ ), and used Fubini's theorem to exchange integrals. Bringing the last term to the left and dividing by 2 shows (C.1).
To show average dissipativity, it suffices to observe
$$
\begin{array}{l} - \int \langle x, b (x, \mu) \rangle \mu (d x) = \int \langle x, \nabla V (x) \rangle \mu (d x) + \iint \langle x, \nabla W (x - y) \rangle \mu (d y) \mu (d x) \\ \geq \alpha \int | x | ^ {2} \mu (d x) - \beta - M _ {W} / 2. \\ \end{array}
$$
Proximal algorithm fits Assumption 4: Note that this implicit algorithm corresponds to the following proximal step
$$
x _ {k + 1} ^ {i} = \arg \min  _ {x} \left\{V (x) + \frac {1}{N} \sum_ {j = 1} ^ {N} W (x - x _ {k} ^ {j}) + \frac {1}{2 \gamma_ {k + 1}} \left| x - \left(x _ {k} ^ {i} + \sqrt {2 \gamma_ {k + 1}} \xi_ {k + 1} ^ {i}\right) \right| ^ {2} \right\}.
$$
By defining the perturbation as
$$
P _ {k + 1} ^ {i} = \varepsilon_ {k + 1} ^ {i} = \nabla V (x _ {k + 1} ^ {i}) - \nabla V (x _ {k} ^ {i}) + \frac {1}{N} \sum_ {j = 1} ^ {N} (\nabla W (x _ {k + 1} ^ {i} - x _ {k} ^ {j}) - \nabla W (x _ {k} ^ {i} - x _ {k} ^ {j})),
$$
we see that the algorithm (Kin-Prox) fits the template (SAA). For brevity, let us write  $\mathcal{F}_k\coloneqq \mathcal{F}_{\tau_k}$ . We only have to show that
$$
\mathbb {E} \big [ | \varepsilon_ {k + 1} | ^ {2} \mid \mathcal {F} _ {k} \big ] = \sum_ {i = 1} ^ {N} \mathbb {E} \big [ | \varepsilon_ {k + 1} ^ {i} | ^ {2} \mid \mathcal {F} _ {k} \big ] = \mathcal {O} (\gamma_ {k + 1} ^ {2} | \boldsymbol {b} (x _ {k}) | ^ {2} + \gamma_ {k + 1}).
$$
$$
\begin{array}{l} \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} = \left| \nabla V (x _ {k + 1} ^ {i}) - \nabla V (x _ {k} ^ {i}) + \frac {1}{N} \sum_ {j = 1} ^ {N} (\nabla W (x _ {k + 1} ^ {i} - x _ {k} ^ {j}) - \nabla W (x _ {k} ^ {i} - x _ {k} ^ {j})) \right| ^ {2} \\ \leq 2 \left| \nabla V \left(x _ {k + 1} ^ {i}\right) - \nabla V \left(x _ {k} ^ {i}\right) \right| ^ {2} + \frac {2}{N} \sum_ {j = 1} ^ {N} \left| \nabla W \left(x _ {k + 1} ^ {i} - x _ {k} ^ {j}\right) - \nabla W \left(x _ {k} ^ {i} - x _ {k} ^ {j}\right) \right| ^ {2} \\ \leq 2 L ^ {2} \left| x _ {k + 1} ^ {i} - x _ {k} ^ {i} \right| ^ {2} + \frac {2 L ^ {2}}{N} \sum_ {j = 1} ^ {N} \left| x _ {k + 1} ^ {i} - x _ {k} ^ {i} \right| ^ {2} \\ = 4 L ^ {2} \left| x _ {k + 1} ^ {i} - x _ {k} ^ {i} \right| ^ {2}. \\ \end{array}
$$
For brevity, let
$$
f (x) = \nabla V (x) + \frac {1}{N} \sum_ {j = 1} ^ {N} \nabla W (x - x _ {k} ^ {j}),
$$
noticing that  $\varepsilon_{k + 1}^{i} = f(x_{k + 1}^{i}) - f(x_{k}^{i})$ . By the update rule (Kin-Prox)
$$
\mathbb {E} \big [ | x _ {k + 1} ^ {i} - x _ {k} ^ {i} | ^ {2} \mid \mathcal {F} _ {k} \big ] \leq 2 \gamma_ {k + 1} ^ {2} \mathbb {E} \big [ | f (x _ {k + 1} ^ {i}) | ^ {2} \mid \mathcal {F} _ {k} \big ] + 4 \gamma_ {k + 1} d.
$$
Moreover, we have that  $|f(x_{k + 1}^i)|^2 \leq 2|f(x_{k + 1}^i) - f(x_k^i)|^2 + 2|f(x_k^i)|^2$ . Since  $\gamma_{k + 1} \to 0$ , we can assume that  $16L^{2}\gamma_{k + 1}^{2} < \frac{1}{2}$ . All in all, this gives
$$
\begin{array}{l} \mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] \leq 4 L ^ {2} \mathbb {E} \left[ \left| x _ {k + 1} ^ {i} - x _ {k} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] \\ \leq 8 L ^ {2} \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ | f (x _ {k + 1} ^ {i}) | ^ {2} \mid \mathcal {F} _ {k} \right] + 1 6 L ^ {2} \gamma_ {k + 1} d \\ \leq 1 6 L ^ {2} \gamma_ {k + 1} ^ {2} \mathbb {E} \left[ \left| f \left(x _ {k + 1} ^ {i}\right) - f \left(x _ {k} ^ {i}\right) \right| ^ {2} \mid \mathscr {F} _ {k} \right] + 1 6 L ^ {2} \gamma_ {k + 1} ^ {2} \left| f \left(x _ {k} ^ {i}\right) \right| ^ {2} + 1 6 L ^ {2} \gamma_ {k + 1} d \\ \leq 1 6 L ^ {2} \gamma_ {k + 1} ^ {2} \mathbb {E} [ | \varepsilon_ {k + 1} ^ {i} | ^ {2} \mid \mathcal {F} _ {k} ] + 1 6 L ^ {2} \gamma_ {k + 1} ^ {2} | f (x _ {k} ^ {i}) | ^ {2} + 1 6 L ^ {2} \gamma_ {k + 1} d \\ \leq \frac {1}{2} \mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathcal {F} _ {k} \right] + 1 6 L ^ {2} \gamma_ {k + 1} ^ {2} \left| f \left(x _ {k} ^ {i}\right) \right| ^ {2} + 1 6 L ^ {2} \gamma_ {k + 1} d. \\ \end{array}
$$
$$
\mathbb {E} \left[ \left| \varepsilon_ {k + 1} ^ {i} \right| ^ {2} \mid \mathscr {F} _ {k} \right] \leq 3 2 L ^ {2} \gamma_ {k + 1} ^ {2} \left| f \left(x _ {k} ^ {i}\right) \right| ^ {2} + 3 2 L ^ {2} \gamma_ {k + 1} d.
$$
Summing over  $i$  and observing that  $\sum |f(x_k^i)|^2 = |\pmb {b}(x_k)|^2$  concludes the proof.