Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?

TL;DR

This paper investigates whether the covariance matrix in the Adaptive Metropolis algorithm can collapse to zero without a lower bound, finding that eigenvalues generally do not tend to zero, thus challenging the necessity of the lower bound.

Contribution

The study analyzes variants of the Adaptive Metropolis algorithm without explicit covariance lower bounds, showing eigenvalues typically remain positive.

Findings

Eigenvalues of the covariance matrix do not tend to collapse to zero.

The lower bound parameter psilon is not always necessary.

Variants without explicit bounds behave well in practice.

Abstract

The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk Metropolis algorithm. The proposal distribution has the following time-dependent covariance matrix at step $n+1$ \[ S_n = Cov(X_1,...,X_n) + \epsilon I, \] that is, the sample covariance matrix of the history of the chain plus a (small) constant $\epsilon>0$ multiple of the identity matrix $I$. The lower bound on the eigenvalues of $S_n$ induced by the factor $\epsilon I$ is theoretically convenient, but practically cumbersome, as a good value for the parameter $\epsilon$ may not always be easy to choose. This article considers variants of the AM algorithm that do not explicitly bound the eigenvalues of $S_n$ away from zero. The behaviour of $S_n$ is studied in detail, indicating that the eigenvalues of $S_n$ do not tend to collapse to zero in general.