As in the title, our goal is to estimate uniformly and with high probability (and up to a constant) a potential having access to noisy pointwise estimates of the associated vector field (i.e., the gradient of the potential). Here how I tried to formalize the problem.
Suppose $d \in \mathbb{N}$ and let $\mathcal{B}_1^d$ be the unitary closed ball of $\mathbb{R}^d$ centered in the origin.
Let $\mathcal{F}$ be the class of all $C^1$ functions $F : \Omega \to \mathbb{R}$, where $\Omega$ is any open set of $\mathbb{R}^d$ containing $[-1,1]^d$, and such that $\forall x\in[-1,1]^d, \nabla F(x) \in \mathcal{B}_1^d$.
Suppose that, for each $F \in \mathcal{F}$, we have the following interaction protocol:
For each $t=1,2,\dots$
- We select a point $X_t \in [-1,1]^d$ (a selection based on past observations and possibly some randomization, but not on the knowledge of $F$)
- We observe a $\mathcal{B}_1^d$-valued random variable $Y_t^F$ such that $\mathbb{E}[Y_t^F \mid X_1,Y_1^F,\dots,X_{t-1},Y_{t-1}^F,X_t] = \mathbb{E}[Y_t^F \mid X_t] =\nabla F(X_t)$.
Basically, we query a point in the domain and we see a noisy (and bounded) reconstruction in that point of the vector field associated to the gradient of the potential $F$.
Regardless which potential $F \in \mathcal{F}$ we are interacting with, our goal is to give a uniform reconstruction of $F-F(0)$ with high-probability using the previous interaction protocol, i.e. we want to find a strategy to select the points $X_1,X_2,\dots$ and a family of estimators $(\Phi_{x,t})_{x \in (-1,1)^d,t \in \mathbb{N}}$, where $\forall x \in [-1,1]^d, \forall t \in \mathbb{N}, \Phi_{x,t} : \big([-1,1]^d\times\mathcal{B}_1^d\big)^t \to \mathbb{R}$, such that $$\forall \varepsilon >0, \forall \delta \in (0,1), \exists T \in \mathbb{N}, \forall t \ge T, \forall F \in \mathcal{F}, \\ \mathbb{P}\bigg[ \sup_{x \in [0,1]^d}\Big|\Phi_{x,t}(X_1,Y_1^F,\dots,X_t,Y_t^F) - \big(F(x)-F(0)\big)\Big| \ge \varepsilon \bigg] \le \delta.$$ Notice that estimating $F-F(0)$ instead of $F$ is just a trick to get rid of the constant we never see.
It is interesting to start with the case $d = 1$, where I'm quite confident that the problem is solvable via Monte Carlo integration in the following way. Define $$\Phi_{x,t}(x_1,y_1,\dots,x_t,y_t) = \frac{1}{t} \sum_{s=1}^t \big(y_t \cdot \mathbb{I}\{\min(x,0)\le x_t \le \max(x,0)\} \cdot \operatorname{sgn}(x)\big)$$ and select the random variables $X_1,X_2, \dots$ just as family of independent $[-1,1]$-valued uniform random variables.
But what about the general case? What are sensible estimators and strategies to solve the problem? And what are the best achievable decaying rate depending on $t, \varepsilon$ and on the dimension $d$ for the quantity $$\mathbb{P}\bigg[ \sup_{x \in [0,1]^d}\Big|\Phi_{x,t}(X_1,Y_1^F,\dots,X_t,Y_t^F) - \big(F(x)-F(0)\big)\Big| \ge \varepsilon \bigg] ?$$