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Stochastic Learning of Computational Resource Usage as Graph Structured Multimarginal Schrödinger Bridge

Georgiy A. Bondar, Robert Gifford, Linh Thi Xuan Phan, Abhishek Halder

TL;DR

The paper addresses learning the time-varying, jointly distributed availability of heterogeneous hardware resources (e.g., CPU, LLC, memory bandwidth) by formulating it as a graph-structured multimarginal Schrödinger bridge problem. It introduces path, barycentric, and series-parallel MSBP formulations, with maximum-likelihood interpretations, and develops efficient Sinkhorn-based projection algorithms that scale linearly with the number of snapshots or cores. Theoretical guarantees of convexity and duality are paired with practical algorithms that exploit graph structure to reduce computational complexity, allowing nonparametric learning directly from scattered profile data without gridding. Empirical validation on single-core NMPC and multi-core Canneal benchmarks demonstrates accurate distributional predictions and fast convergence, highlighting the approach’s potential for fine-grained dynamic resource scheduling in resource-constrained cyber-physical systems.

Abstract

We propose to learn the time-varying stochastic computational resource usage of software as a graph structured Schrödinger bridge problem. In general, learning the computational resource usage from data is challenging because resources such as the number of CPU instructions and the number of last level cache requests are both time-varying and statistically correlated. Our proposed method enables learning the joint time-varying stochasticity in computational resource usage from the measured profile snapshots in a nonparametric manner. The method can be used to predict the most-likely time-varying distribution of computational resource availability at a desired time. We provide detailed algorithms for stochastic learning in both single and multi-core cases, discuss the convergence guarantees, computational complexities, and demonstrate their practical use in two case studies: a single-core nonlinear model predictive controller, and a synthetic multi-core software.

Stochastic Learning of Computational Resource Usage as Graph Structured Multimarginal Schrödinger Bridge

TL;DR

The paper addresses learning the time-varying, jointly distributed availability of heterogeneous hardware resources (e.g., CPU, LLC, memory bandwidth) by formulating it as a graph-structured multimarginal Schrödinger bridge problem. It introduces path, barycentric, and series-parallel MSBP formulations, with maximum-likelihood interpretations, and develops efficient Sinkhorn-based projection algorithms that scale linearly with the number of snapshots or cores. Theoretical guarantees of convexity and duality are paired with practical algorithms that exploit graph structure to reduce computational complexity, allowing nonparametric learning directly from scattered profile data without gridding. Empirical validation on single-core NMPC and multi-core Canneal benchmarks demonstrates accurate distributional predictions and fast convergence, highlighting the approach’s potential for fine-grained dynamic resource scheduling in resource-constrained cyber-physical systems.

Abstract

We propose to learn the time-varying stochastic computational resource usage of software as a graph structured Schrödinger bridge problem. In general, learning the computational resource usage from data is challenging because resources such as the number of CPU instructions and the number of last level cache requests are both time-varying and statistically correlated. Our proposed method enables learning the joint time-varying stochasticity in computational resource usage from the measured profile snapshots in a nonparametric manner. The method can be used to predict the most-likely time-varying distribution of computational resource availability at a desired time. We provide detailed algorithms for stochastic learning in both single and multi-core cases, discuss the convergence guarantees, computational complexities, and demonstrate their practical use in two case studies: a single-core nonlinear model predictive controller, and a synthetic multi-core software.
Paper Structure (24 sections, 8 theorems, 73 equations, 11 figures, 3 tables)

This paper contains 24 sections, 8 theorems, 73 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic Learning of Computational Resource
  4. MSBP and Graph Structures
  5. Classical SBP and its Maximum Likelihood Iterpretation
  6. MSBP Formulation
  7. Path Structured MSBP
  8. Barycentric MSBP
  9. Series-Parallel Graph Structured MSBP
  10. Algorithms
  11. Projections for Path Structured MSBP
  12. Projections for Barycentric MSBP
  13. Projections for Series-Parallel Graph-structured MSBP
  14. Overall Algorithm
  15. Experiments
  16. ...and 9 more sections

Key Result

Lemma 1

warner1990modern Given finite sets $\mathcal{S}_{1}\subseteq\mathcal{S}_{2}$ with $\vert\mathcal{S}_1\vert=\nu_1$ and $\vert\mathcal{S}_2\vert=\nu_2$, we have $\vert\mathcal{S}_{2}\setminus\mathcal{S}_{1}\vert = \nu_2 - \nu_1$.

Figures (11)

  • Figure 1: (a) The classical (bimarginal) SBP computes the most likely measure-valued curve connecting two given probability measures $\mu_1,\mu_2$ at times $\tau_1,\tau_2$ respectively. (b) The MSBP computes the most likely measure-valued curve connecting multiple, here three given probability measures $\mu_1,\mu_2, \mu_3$ at times $\tau_1,\tau_2,\tau_3$ respectively. In both subfigures, the feasible measured-valued curves are shown in the top with darker (resp. lighter) hues for higher (resp. lower) probability. The given measures are shown in the bottom as colored scatter plots (red = high probability, blue = low probability) in the ground coordinates $(\xi_1,\xi_2,\xi_3)^{\top}\in\mathbb{R}^3$.
  • Figure 2: The path tree for sequentially observed $\{\mu_{\sigma}\}_{\sigma\in\llbracket s\rrbracket}$.
  • Figure 3: The tree graph for barycentric MSBP (Sec. \ref{['subsec:BaryStructuredMSBP']}) with measures $\{\mu_{\sigma}^j\}_{(j,\sigma)\in\left(\{0\}\cup\llbracket J\rrbracket\right)\times\llbracket s\rrbracket}$, where $J=2$.
  • Figure 4: The graph for series-parallel graph-structured MSBP (Sec. \ref{['subsec:SeriesParallelGraphStructuredMSBP']}) with $J$ parallel paths of length $s$.
  • Figure 5: Components of the measured feature vector $\bm{\xi}$ in \ref{['defFeatureVec']} for the single core experiment in Sec. \ref{['subsec:SingleCoreExperiment']}, for all of the five control cycles for 500 executions of the NMPC software, for a fixed cyber-physical context.
  • ...and 6 more figures

Theorems & Definitions (13)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • Proposition 3
  • Remark 3
  • Lemma 2
  • Proposition 4
  • Remark 4
  • ...and 3 more