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Speculative Sampling for Parametric Temporal Point Processes

Marin Biloš, Anderson Schneider, Yuriy Nevmyvaka

TL;DR

The paper addresses the inefficiency of sequential sampling in parametric temporal point processes by introducing a speculative sampling method that generates multiple future events in parallel while preserving exact sampling. It leverages a pretrained encoder as a proposal distribution and develops a principled framework for computing rejection constants via envelope bounds, including extensions to mixtures and common distributions. Empirical results on synthetic and real datasets show meaningful runtime speedups without sacrificing distributional fidelity, with particular impact demonstrated in high-frequency finance tasks like limit order books. The approach is model-agnostic, requiring no retraining, and bridges the gap between expressive TPP modeling and scalable, parallel generation for large-scale applications.

Abstract

Temporal point processes are powerful generative models for event sequences that capture complex dependencies in time-series data. They are commonly specified using autoregressive models that learn the distribution of the next event from the previous events. This makes sampling inherently sequential, limiting efficiency. In this paper, we propose a novel algorithm based on rejection sampling that enables exact sampling of multiple future values from existing TPP models, in parallel, and without requiring any architectural changes or retraining. Besides theoretical guarantees, our method demonstrates empirical speedups on real-world datasets, bridging the gap between expressive modeling and efficient parallel generation for large-scale TPP applications.

Speculative Sampling for Parametric Temporal Point Processes

TL;DR

The paper addresses the inefficiency of sequential sampling in parametric temporal point processes by introducing a speculative sampling method that generates multiple future events in parallel while preserving exact sampling. It leverages a pretrained encoder as a proposal distribution and develops a principled framework for computing rejection constants via envelope bounds, including extensions to mixtures and common distributions. Empirical results on synthetic and real datasets show meaningful runtime speedups without sacrificing distributional fidelity, with particular impact demonstrated in high-frequency finance tasks like limit order books. The approach is model-agnostic, requiring no retraining, and bridges the gap between expressive TPP modeling and scalable, parallel generation for large-scale applications.

Abstract

Temporal point processes are powerful generative models for event sequences that capture complex dependencies in time-series data. They are commonly specified using autoregressive models that learn the distribution of the next event from the previous events. This makes sampling inherently sequential, limiting efficiency. In this paper, we propose a novel algorithm based on rejection sampling that enables exact sampling of multiple future values from existing TPP models, in parallel, and without requiring any architectural changes or retraining. Besides theoretical guarantees, our method demonstrates empirical speedups on real-world datasets, bridging the gap between expressive modeling and efficient parallel generation for large-scale TPP applications.
Paper Structure (37 sections, 5 theorems, 43 equations, 10 figures, 12 tables, 2 algorithms)

This paper contains 37 sections, 5 theorems, 43 equations, 10 figures, 12 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Temporal point processes
  4. Rejection sampling
  5. Method
  6. Rejection constant for selected distributions
  7. General rejection constant
  8. Cataloging common distributions
  9. Efficient sampling algorithm
  10. Related work
  11. Experiments
  12. Motivating examples
  13. Real data is usually non-stationary
  14. Long-horizon sampling on benchmark data
  15. Runtime improvements
  16. ...and 22 more sections

Key Result

Theorem 3.1

Let $f_T(x)$ be a target density with piecewise linear upper bound $g_T(x)$ constructed using grid points $\{x_0, x_1, \ldots, x_n\}$ as described above, and let $f_P(x)$ be a proposal density with a lower bound $h_P(x)$, on the same grid. Then, the upper bound on the rejection sampling constant $M$

Figures (10)

  • Figure 1: Illustration of our speculative sampling method for temporal point processes.
  • Figure 2: Illustration of upper and lower linear bounds of a mixture distribution with two components. The grid points always include inflections which makes the rejection sampling exact.
  • Figure 3: Average accepted step (out of 5 proposed events) for different configurations of multivariate Hawkes process. Lower sparsity and larger adjacency values define stronger mark interactions leading to lower acceptance ratio, nevertheless, all configurations show good acceptance ratios.
  • Figure 4: Limit order book samples. (Left) Samples generated with a conventional autoregressive method. (Right) Samples generated with a speculative sampling method.
  • Figure 5: Convexity of Gamma, log-normal and Weibull distributions for different parameters. The first and second derivative, and inflection points are available in closed-form, see Table \ref{['tab:distributions']}.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Theorem 3.1: Rejection constant using linear density approximation
  • proof
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • proof
  • Lemma A.4
  • ...and 1 more