Biologically-Plausible Markov Chain Monte Carlo Sampling from Vector Symbolic Algebra-encoded Distributions

P. Michael Furlong, Kathryn Simone, Nicole Sandra-Yaffa Dumont, Madeleine Bartlett, Terrence C. Stewart, Jeff Orchard, Chris Eliasmith

Abstract:

Vector symbolic algebras (VSAs) are modelling frameworks
that unify human cognition and neural network models, and some have
recently been shown to be probabilistic models akin to Kernel Mean
Embeddings. Sampling from vector-embedded distributions is an impor-
tant tool for turning distributions into decisions, in the context of cog-
nitive modelling, or actions, in the context of reinforcement learning.
However, current techniques for sampling from these distribution embed-
dings rely on knowledge of the kernel embedding or its gradient, knowl-
edge which is problematic for neural systems to access. In this paper, we
explore biologically-plausible Hamiltonian Monte Carlo Markov Chain
sampling in the space of VSA encodings, without relying on any explicit
knowledge of the encoding scheme. Specifically, we encode data using
a Holographic Reduced Representation (HRR) VSA, sample from the
encoded distributions using Langevin dynamics in the VSA vector space,
and demonstrate competitive sampling performance in a spiking-neural
network implementation. Surprisingly, while the Langevin dynamics are
not constrained to the manifold defined by the HRR encoding, the gen-
erated samples contain sufficient information to reconstruct the target
distribution, given an appropriate decoding scheme. We also demonstrate
that the HRR algebra provides a straightforward conditioning operation.
These results show that a generalized sampling model can explain how
brains turn probabilistic latent representations into concrete actions in
an encoding scheme-agnostic fashion. Moreover, sampling from vector
embeddings of distributions permits the implementation of probabilis-
tic algorithms, capturing uncertainty in cognitive models. We also note
that the ease of conditioning distributions is particularly well-suited to
reinforcement learning applications.