Restricted Boltzmann Machines are Hard to Approximately Evaluate or Simulate.
P. Long and R. Servedio.
27th International Conference on Machine Learning (ICML), 2010, pp. 703-710.


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 $\neq$ 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 $e^{Kn}$. This hardness result holds even if the parameters of $M$ are constrained to be at most $\psi(n)$ for any function $\psi(n) = \omega(n)$, and if the number of hidden nodes of $M$ is at most $n$.

We then show that assuming RP $\neq$ NP, there is no polynomial-time randomized algorithm for the following problem: given the parameters of an RBM $M$, generate a random example from a probability distribution whose total variation distance from the distribution defined by $M$ is at most $1/12.$


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