For many computational problems involving randomness, intricate geometric features of the solution space have been used to rigorously rule out powerful classes of algorithms. This is often accomplished through the lens of the multi Overlap Gap Property ($m$-OGP), a rigorous barrier against algorithms exhibiting input stability. In this paper, we focus on the algorithmic tractability of two models: (i) discrepancy minimization, and (ii) the symmetric binary perceptron (\texttt{SBP}), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. Our first focus is on the limits of online algorithms. By establishing and leveraging a novel geometrical barrier, we obtain sharp hardness guarantees against online algorithms for both the \texttt{SBP} and discrepancy minimization. Our results match the best known algorithmic guarantees, up to constant factors. Our second focus is on efficiently finding a constant discrepancy solution, given a random matrix $\mathcal{M}\in\mathbb{R}^{M\times n}$. In a smooth setting, where the entries of $\mathcal{M}$ are i.i.d. standard normal, we establish the presence of $m$-OGP for $n=\Theta(M\log M)$. Consequently, we rule out the class of stable algorithms at this value. These results give the first rigorous evidence towards a conjecture of Altschuler and Niles-Weed~\cite[Conjecture~1]{altschuler2021discrepancy}. Our methods use the intricate geometry of the solution space to prove tight hardness results for online algorithms. The barrier we establish is a novel variant of the $m$-OGP. Furthermore, it regards $m$-tuples of solutions with respect to correlated instances, with growing values of $m$, $m=\omega(1)$. Importantly, our results rule out online algorithms succeeding even with an exponentially small probability.

翻译：对于许多涉及随机性的计算问题，解空间的复杂几何特征已被用于严格排除强大类别的算法。这通常通过多重重叠间隙性质（$m$-OGP）这一视角实现，该性质构成了针对输入稳定性算法的严格障碍。本文聚焦于两类模型的算法可解性：（i）差异最小化，以及（ii）对称二元感知器（\texttt{SBP}）——一个随机约束满足问题，同时也是单层神经网络的玩具模型。我们首先关注在线算法的局限性。通过建立并利用一种新颖的几何障碍，我们为\texttt{SBP}和差异最小化问题中的在线算法获得了尖锐的困难性保证。我们的结果与已知的最佳算法保证相匹配（仅相差常数因子）。其次，我们关注在给定随机矩阵$\mathcal{M}\in\mathbb{R}^{M\times n}$时高效寻找常数差异解的问题。在平滑设定下（其中$\mathcal{M}$的条目为独立同分布的标准正态分布），我们证明了当$n=\Theta(M\log M)$时$m$-OGP的存在性。因此，在该参数值下我们排除了稳定算法类。这些结果为Altschuler与Niles-Weed的猜想~\cite[Conjecture~1]{altschuler2021discrepancy}提供了首个严格证据。我们的方法利用解空间的精细几何结构，为在线算法证明了紧致的困难性结果。所建立的障碍是$m$-OGP的一种新颖变体。此外，它涉及针对相关实例的$m$元组解，且$m$取增长值（$m=\omega(1)$）。重要的是，我们的结果排除了在线算法甚至以指数小概率成功的可能性。