Restricted Boltzmann Machines are Hard to Approximately Evaluate or Simulate

Restricted Boltzmann Machines (RBMs) are a type of probability model over the Boolean cube {-1, 1}n that have recently received much attention. We establish the intractability of two basic computational tasks involving RBMs, even if only a coarse approximation to the correct output is required. We first show that assuming P = NP, for any fixed positive constant K (which may be arbitrarily large) there is no polynomial-time algorithm for the following problem: given an n-bit input string x and the parameters of a RBM M, output an estimate of the probability assigned to x by M that is accurate to within a multiplicative factor of eKn . This hardness result holds even if the parameters of M are constrained to be at most (n) for any function (n) that grows faster than linearly, and if the number of hidden nodes of M is at most n. We then show that assuming RP = NP, there is no polynomial-time randomized algorithm for the following problem: given the parameters of an RBM M, generate a rand...

• Type: Conference
• Year: 2010
• Where: ICML
• Authors: Philip M. Long, Rocco A. Servedio
• Keywords: ICML 2010, Machine Learning, Polynomial-time Randomized Algorithm, RBM M, Tasks Involving Rbms