SlowGuess/NeurIPS_2023_OCR huggingface dataset:
worse precision of the final optimization solution. To be more specific, given an  $\varepsilon$ -membership oracle to a convex set  $K$ , Lee et al. [42] can only guarantee the returned solution to be  $\varepsilon^{1/\gamma_0}$ -optimal and contained in  $B_2(K, \varepsilon^{1/\gamma_0})$ , where  $\gamma_0 > 6$  is a fixed constant. An understudied open direction in the optimization community is whether a variant of Lee et al. [42] can return a solution in  $B_2(K, \varepsilon + \delta)$  for  $\delta$  that is any tunable parameter, with runtime depending polynomially on  $1/\delta$ .
Open Problem F.1. Given an  $\varepsilon$ -approximate membership oracle of a convex set  $K \subseteq \mathbb{R}^m$  and an evaluation oracle of a convex function  $f$ , is there an optimization algorithm that, under sufficient regularity conditions, for any tunable  $\delta$ , finds a near-optimal solution  $\hat{x}$  such that  $f(\hat{x}) \leq \min_{x \in K} f(x) + \varepsilon^{1/\gamma_0}$  and  $\hat{x} \in B_2(K, \varepsilon + \delta)$  within  $\text{poly}(m, 1/\delta, \log(1/\varepsilon))$  calls to both oracles?
We note that resolving Open Problem F.1 will immediately lead to exponentially improved rates for our Theorem C.14 for any finite Stackelberg games. To see this, recall that the algorithms that achieve Theorem C.14 require repeated calls to the constructed APPROXMEM oracle. This oracle provides an  $\varepsilon_{1}$ -approximate response, indicating whether or not the query point belongs to the  $\varepsilon_{2}$ -conservative version of each best response polytope. The number of samples used in simulating each oracle call is exponential in  $\frac{\varepsilon_2}{\varepsilon_1}$ . The algorithm by Lee et al. [42] amplifies the inaccuracy of APPROXMEM at a polynomial rate of  $\gamma_0$ . Consequently, imposing a constraint of  $\varepsilon_{1} \lesssim \varepsilon_{2}^{\gamma_{0}}$ . However, resolving Open Problem F.1 removes this constraint and introduces a much milder one, i.e.,  $\varepsilon_{1} \lesssim \varepsilon_{2} - \delta$ , where  $\delta$  is a tunable parameter. This adjustment leads to exponential improvement in the final sample complexity.