重塑全球环形结构以通过平滑崎岖景观加速局部优化
Reshaping Global Loop Structure to Accelerate Local Optimization by Smoothing Rugged Landscapes
AI总结：
通过重塑全局环形结构，改进优化算法以平滑能量景观，提升局部优化效率。
摘要：
具有摩擦的概率图模型表现出崎岖的能量景观，这些景观会捕获迭代优化动态。这些景观不仅由局部相互作用塑造，还关键地由图的全局环形结构决定。著名的Bethe近似将图视为树，从而忽略了全局结构，限制了其在优化中的有效性。环展开在原则上捕捉了这种全局结构，但通常由于组合爆炸而难以实现。M层构造提供了一种替代方法：制作图的M个副本，并在它们之间随机重新连接边。这提供了一个从原始图（M=1）到Bethe近似（M→∞）的受控近似序列。在这里，我们通过将均匀随机重连接替换为一个结构化的混合核Q来推广这一构造，该核设定了任意两层之间相互连接的概率。结果，全局环形结构可以在不修改局部相互作用的情况下进行塑造。我们证明，在此复制和重连接转换之后，存在一个区域，其中层间波动衰减，从而增加达到原始图能量函数全局最小值的概率。这提供了一种高度通用且实用的优化工具。通过这种方法，达到这些最优解所需的计算成本在稀疏和密集的Ising基准测试中均得到降低，包括自旋玻璃和植入实例。当与副本交换蒙特卡洛结合时，相同构造提高了最大独立集问题的多项式时间算法阈值。一个洞穴分析显示，结构化的层间耦合通过压缩构型复杂性并抑制许多亚优的亚稳态，显著平滑了崎岖的能量景观。
英文摘要：
Probabilistic graphical models with frustration exhibit rugged energy landscapes that trap iterative optimization dynamics. These landscapes are shaped not only by local interactions, but crucially also by the global loop structure of the graph. The famous Bethe approximation treats the graph as a tree, effectively ignoring global structure, thereby limiting its effectiveness for optimization. Loop expansions capture such global structure in principle, but are often impractical due to combinatorial explosion. The $M$-layer construction provides an alternative: make $M$ copies of the graph and reconnect edges between them uniformly at random. This provides a controlled sequence of approximations from the original graph at $M=1$, to the Bethe approximation as $M \rightarrow \infty$. Here we generalize this construction by replacing uniform random rewiring with a structured mixing kernel $Q$ that sets the probability that any two layers are interconnected. As a result, the global loop structure can be shaped without modifying local interactions. We show that, after this copy-and-reconnect transformation, there exists a regime in which layer-to-layer fluctuations decay, increasing the probability of reaching the global minimum of the energy function of the original graph. This yields a highly general and practical tool for optimization. Using this approach, the computational cost required to reach these optimal solutions is reduced across sparse and dense Ising benchmarks, including spin glasses and planted instances. When combined with replica-exchange Monte Carlo, the same construction increases the polynomial-time algorithmic threshold for the maximum independent set problem. A cavity analysis shows that structured inter-layer coupling significantly smooths rugged energy landscapes by collapsing configurational complexity and suppressing many suboptimal metastable states.
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