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Lattice Random Walk Discretisations of Stochastic Differential Equations

Samuel Duffield, Maxwell Aifer, Denis Melanson, Zach Belateche, Patrick J. Coles

Abstract

We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random values. This approach is a significant departure from traditional floating point discretisations and offers several advantages; including compatibility with stochastic computing architectures that avoid floating-point arithmetic in place of directly manipulating the underlying probability distribution of a bitstream, elimination of Gaussian sampling requirements, robustness to quantisation errors, and handling of non-Lipschitz drifts. We prove weak convergence and demonstrate the advantages through experiments on various SDEs, including state-of-the-art diffusion models.

Lattice Random Walk Discretisations of Stochastic Differential Equations

Abstract

We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random values. This approach is a significant departure from traditional floating point discretisations and offers several advantages; including compatibility with stochastic computing architectures that avoid floating-point arithmetic in place of directly manipulating the underlying probability distribution of a bitstream, elimination of Gaussian sampling requirements, robustness to quantisation errors, and handling of non-Lipschitz drifts. We prove weak convergence and demonstrate the advantages through experiments on various SDEs, including state-of-the-art diffusion models.

Paper Structure

This paper contains 31 sections, 5 theorems, 47 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lattice Random Walk
  3. Weak Convergence
  4. Selecting $\delta_x$
  5. Diagonal Diffusion Assumption
  6. Lattice Random Walk for Ordinary Differential Equations
  7. Related Work
  8. Advantages of Lattice Random Walk
  9. Stochastic Computing
  10. No Gaussian Sampling
  11. Robustness to (Quantisation) Error
  12. Non-Lipschitz Drifts
  13. Experiments
  14. Ornstein-Uhlenbeck Process
  15. Poisson Random Effects Model (Non-globally Lipshitz)
  16. ...and 16 more sections

Key Result

Theorem 1

Consider the SDE eq:sde with drift function $f(x,t)$ and diagonal diffusion matrix $\sigma(x,t)$ that are sufficiently smooth. Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a test function with bounded derivatives. Then the LRW discretisation eq:lrw with spatial stepsize $\delta_{x, i} = \Theta(\sqr where $x_N$ is the discretized solution at time $T = N\delta_t$ and $X(T)$ is the true SDE solution

Figures (7)

  • Figure 1: Visualisation of exact SDE, Euler-Maruyama and LRW (with equal stepsize).
  • Figure 2: Comparison of stochastic computing pipelines. LRW enables stochastic computing for SDE simulation without bitstream accumulation.
  • Figure 3: Sensitivity to $\delta_x$ for an OU process. KL divergence between true stationary and LRW distributions, as a function of $\delta_t$ and $\delta_x$. Averaged over 10 different seeds. The dotted line corresponds to the binary update condition \ref{['eq:rot']}.
  • Figure 4: Robustness to quantisation for an OU process. KL divergence between true stationary distribution and the discretisations. Averaged over 50 different seeds and randomly sampled OU parameters.
  • Figure 5: Robustness to stepsize for a non-Lipschitz Poisson random effects model. Error from the ergodic mean of the generated samples and the true parameter, averaged over 50 different seeds.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1: Weak convergence of the LRW discretisation
  • proof
  • Theorem 2: Allowable range for $\delta_x$
  • proof
  • Theorem 3: Feasibility condition for $\delta_x$
  • proof
  • Theorem 4: Reduction from ternary to binary
  • proof
  • Theorem 1: Weak convergence of the LRW discretisation
  • proof