High-Dimensional Scaling of Piecewise Deterministic Monte Carlo
Efficiency Comparison & Asymptotic Variance Estimation
6/16/2026
1 What is PDMP?
A PDMP: Forward Event-Chain Monte Carlo
A Blog Entry on Bayesian Computation by an Applied Mathematician
$$
$$
1.1 Evolution of Markov Chain Monte Carlo
(1953–)
(1978–)
(2008–)
1.2 PDMP: Mathematical Definition
1.3 Classical MCMC methods turn out to be PDMPs in disguise
Many PDMPs behave similarly in high dimensions.
discretized by symplectic integrator
O(d^{1.25}) complexity
approximated by (random) piecewise linear interpolation
O(d^{1.5}) complexity
1.4 Digression: Killer applications of PDMP
Sparse Markov Random Field with d=10
Local Implementation O(d^1)
Exploiting sparsity, BPS atteins better scaling than HMC (MSE/sec.)
Stochastic Gradient O(n^0)
x: sample size n, y: log ESS
Logistic regression with d=16
Using an appropriate control variate, the efficiency of Zig-Zag is O(1)
in the limit of n\to\infty, it outperforms Langevin.
2 Towards Better Jump Strategy: A Scaling Analysis
We compare two famous PDMP methods under the following condition:
- ODE: Fixed
- Jump: Reflection + Refreshment vs. Combination
- Target: High dimensional standard Gaussian
2.1 FECMC vs. BPS
2.2 Jump Strategy in FECMC vs. BPS
2.3 Empirical Comparison: BPS vs. FECMC
Effective Sample Size of U(x)=\|x\|^2, the negative log-density of 100 dimensional standard Gaussian, estimated over 1000 runs
2.4 Scaling Analysis = Deriving a Diffusion Limit
\text{Plotting } \textcolor{#0096FF}{Y_t^{(d)}}=\frac{\lvert\textcolor{#0096FF}{X}_{d\textcolor{#0096FF}{t}}^{\textcolor{#0096FF}{(d)}}\rvert^2-d}{\sqrt{d}} \text{ with } d=10^2,10^3,10^4:
2.5 Theorem 1： Diffusion Limits of FECMC & BPS
dY_t^{\textcolor{#0096FF}{\text{B}}}=-\frac{\sigma^2_{\textcolor{#0096FF}{\text{B}}}(\rho)}{4}Y_t^{\textcolor{#0096FF}{\text{B}}}\,dt+\sigma_{\textcolor{#0096FF}{\text{B}}}(\rho)\,dB_t \sigma^2_{\textcolor{#0096FF}{\text{B}}}(\rho)=8\int^\infty_0e^{-\rho s}\operatorname{E}[R_0^{\textcolor{#0096FF}{\text{B}}}R_s^{\textcolor{#0096FF}{\text{B}}}]\,ds
dY_t^{\textcolor{#E95420}{\text{F}}}=-\frac{\sigma^2_{\textcolor{#E95420}{\text{F}}}(\rho)}{4}Y_t^{\textcolor{#E95420}{\text{F}}}\,dt+\sigma_{\textcolor{#E95420}{\text{F}}}(\rho)\,dB_t \sigma^2_{\textcolor{#E95420}{\text{F}}}(\rho)=8\int^\infty_0e^{-\rho s}\operatorname{E}[R_0^{\textcolor{#E95420}{\text{F}}}R_s^{\textcolor{#E95420}{\text{F}}}]\,ds
2.6 Theorem 2: Analytic Expression of \sigma^2’s
While BPS atteins maximum at non-trivial value of \rho, FECMC achieves maximum at \rho=0
