改进连接主义能量最小化
Improving Connectionist Energy Minimization
摘要：
为能量最小化而设计的对称网络，如玻尔兹曼机和Hopfield网，经常被研究用于优化、约束满足和NP难问题的近似。然而，找到一个全球解决方案（即，能量函数的全局最小值）不能得到保证，并且甚至局部解也可能需要指数数量的步骤。我们提出了一个改进的标准本地激活函数用于这样的网络。改进的算法保证了树形子网络在线性时间内找到全局最小值。该算法称为activate，是统一的，并且不假设网络是树状的。它可以识别树状子网络，即使在循环拓扑结构（任意网络），并避免局部极小值沿这些树。对于无环网络，该算法保证从系统的任何初始状态收敛到全局最小值（自稳定），并在各种类型的控制器下保持正确。在消极的一面，我们表明，在存在的周期，没有统一的算法存在，保证最优性，即使在顺序异步调度。异步调度程序一次只能激活一个单元，而同步调度程序可以在单个时间步长中激活任意数量的单元。此外，没有统一的算法存在优化，甚至非循环网络时，调度器是同步的。最后，我们展示了如何使用循环割集计划的算法可以得到改进。一般的算法称为带割集激活算法，改进了激活算法，并有一定的性能保证，这与网络的环割集的大小有关。
英文摘要：
Symmetric networks designed for energy minimization such as Boltzman machines and Hopfield nets are frequently investigated for use in optimization, constraint satisfaction and approximation of NP-hard problems. Nevertheless, finding a global solution (i.e., a global minimum for the energy function) is not guaranteed and even a local solution may take an exponential number of steps. We propose an improvement to the standard local activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm, called activate, is uniform and does not assume that the network is tree-like. It can identify tree-like subnetworks even in cyclic topologies (arbitrary networks) and avoid local minima along these trees. For acyclic networks, the algorithm is guaranteed to converge to a global minimum from any initial state of the system (self-stabilization) and remains correct under various types of schedulers. On the negative side, we show that in the presence of cycles, no uniform algorithm exists that guarantees optimality even under a sequential asynchronous scheduler. An asynchronous scheduler can activate only one unit at a time while a synchronous scheduler can activate any number of units in a single time step. In addition, no uniform algorithm exists to optimize even acyclic networks when the scheduler is synchronous. Finally, we show how the algorithm can be improved using the cycle-cutset scheme. The general algorithm, called activate-with-cutset, improves over activate and has some performance guarantees that are related to the size of the network's cycle-cutset.
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