Stochastic Quadrature

The partition function is fundamental for probabilistic graphical models—it is required for inference, parameter estimation, and model selection. Evaluating this function corresponds to discrete integration, namely a weighted sum over an exponentially large set. This task quickly becomes intractable as the dimensionality of the problem increases. We propose an approximation scheme that, for any discrete graphical model whose parameter vector has bounded norm, estimates the partition function with arbitrarily small error. Our algorithm relies on a near minimax optimal polynomial approximation to the potential function and a Clenshaw-Curtis style quadrature. Furthermore, we show that this algorithm can be randomized to split the computation into a high-complexity part and a low-complexity part, where the latter may be carried out on small computational devices. Experiments confirm that the new randomized algorithm is highly accurate if the parameter norm is small, and is otherwise comparable to methods with unbounded error.

While our initial approach provides desirable statistical guarantees, typical constructions of such approximations are themselves not amenable to efficient inference. Here, we develop a class of Monte Carlo sampling algorithms for efficiently approximating the value of the partition function, as well as the associated pseudo-marginals. More precisely, for pairwise models with n vertices and m edges, the complexity can be reduced from O(d^k) to O(k^4+kn+m), where d >= 4m is the parameter dimension. We also consider the uses of stochastic quadrature for the problem of maximum-likelihood (ML) parameter estimation. For completely observed data, our analysis gives rise to a probabilistic bound on the log-likelihood of the model. Maximizing this bound yields an approximate ML estimate which, in analogy to the moment-matching of exact ML estimation, can be interpreted in terms of pseudo-moment-matching. We present experimental results illustrating the behavior of this approximate ML estimator.

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