A Blocked Linear Method for Optimizing Large Parameter Sets in Variational Monte Carlo

Abstract

We present a modification to variational Monte Carlo's linear method optimization scheme that addresses a critical memory bottleneck while maintaining compatibility with both the traditional ground state variational principle and our recently-introduced variational principle for excited states. For wave function ansatzes with tens of thousands of variables, our modification reduces the required memory per parallel process from tens of gigabytes to hundreds of megabytes, making the methodology a much better fit for modern supercomputer architectures in which data communication and per-process memory consumption are primary concerns. We verify the efficacy of the new optimization scheme in small molecule tests involving both the Hilbert space Jastrow antisymmetric geminal power ansatz and real space multi-Slater Jastrow expansions. Satisfied with its performance, we have added the optimizer to the QMCPACK software package, with which we demonstrate on a hydrogen ring a prototype approach for making systematically convergent, non-perturbative predictions of Mott-insulators' optical band gaps.

QMCPACK: an open sourceab initioquantum Monte Carlo package for the electronic structure of atoms, molecules and solids

“…One important limitation of the LM comes when the number of variational parameters rises to 10,000 or more, at which point the contributions to H and S made by each Markov chain become cumbersome to store in memory, especially when running one Markov chain per core on a large parallel system in which per-core memory is limited. QMCPACK currently addresses this memory bottleneck using the blocked LM [35], a recent algorithm that separates the variable space into blocks, estimates the most important variable-change directions within each block, and then uses these directions to construct a reduced and vastly more memory efficient LM eigenvalue problem to generate an update direction in the overall variable space. Like excited state targeting, this is a new feature that can be expected to evolve in time, and has been made openly available to the community in the spirit of rapid dissemination.…”

Complementary first and second derivative methods for ansatz optimization in variational Monte Carlo

“…This enables the expected strong coupling between them to be handled more accurately by the LMstyle diagonalization within that block. While the BLM has been successfully applied up to about 25,000 parameters and found to closely reproduce the results of the standard LM, [6] it remains a relatively new method, and the present study will provide additional data on its efficacy.…”

“…A more extensive description of the BLM and its precise memory usage can be found in its original paper. [6] We divide parameters evenly among blocks, but one could implement the use of tailored blocks of varying sizes. It is advisable to choose the block size to be large enough to keep important parameters of the same type, such as all of those for a Jastrow factor, within the same block.…”

“…The cost is given only approximately because while it is normally dominated by the cost of the N b blocks in the BLM, there are additional contributions related to how many directions are retained from the first BLM diagonalization and how many old directions are used. [6]…”

Size Consistent Excited States via Algorithmic Transformations between Variational Principles

“…Although we have chosen to test this strategy using Ω as the variational principle and the VMC linear method [18,[30][31][32][33][34] as the wave function update method, we expect it to be effective for other variational principles and updated methods as well.…”