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Separable approximations of optimal value functions under a decaying sensitivity assumption

Mario Sperl, Luca Saluzzi, Lars Grüne, Dante Kalise

TL;DR

This work addresses the curse of dimensionality in computing the optimal value function $V(x)$ for interconnected optimal control problems by introducing a decaying-sensitivity mechanism over a graph. It constructs local, low-dimensional components $\\Psi_l^j$ defined on neighborhoods $\\mathcal{B}_l(j)$ and proves a global approximation $V(x) \approx \sum_j \\Psi_l^j(H_l^{j} x) + V(0)$ with error bounded by $(s-1)\\gamma(l+1)$, yielding a $d$-separable representation. The approach is instantiated in the discrete-time LQR setting, where the exact $V(x) = x^\top P x$ can be recovered via a spatially exponential decaying $P$, and in nonlinear examples using SDRE for the Allen-Cahn equation, with numerical experiments illustrating how neighborhood size and graph structure influence accuracy. The results suggest a practical pathway to curse-of-dimensionality-free neural-network approximations of $V$ in large-scale networked systems and offer guidance on selecting neighborhood radii to achieve desired accuracy.

Abstract

An efficient approach for the construction of separable approximations of optimal value functions from interconnected optimal control problems is presented. The approach is based on assuming decaying sensitivities between subsystems, enabling a curse-of-dimensionality free approximation, for instance by deep neural networks.

Separable approximations of optimal value functions under a decaying sensitivity assumption

TL;DR

This work addresses the curse of dimensionality in computing the optimal value function for interconnected optimal control problems by introducing a decaying-sensitivity mechanism over a graph. It constructs local, low-dimensional components defined on neighborhoods and proves a global approximation with error bounded by , yielding a -separable representation. The approach is instantiated in the discrete-time LQR setting, where the exact can be recovered via a spatially exponential decaying , and in nonlinear examples using SDRE for the Allen-Cahn equation, with numerical experiments illustrating how neighborhood size and graph structure influence accuracy. The results suggest a practical pathway to curse-of-dimensionality-free neural-network approximations of in large-scale networked systems and offer guidance on selecting neighborhood radii to achieve desired accuracy.

Abstract

An efficient approach for the construction of separable approximations of optimal value functions from interconnected optimal control problems is presented. The approach is based on assuming decaying sensitivities between subsystems, enabling a curse-of-dimensionality free approximation, for instance by deep neural networks.
Paper Structure (10 sections, 4 theorems, 54 equations, 4 figures, 1 table)

This paper contains 10 sections, 4 theorems, 54 equations, 4 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Optimal Control Problem
  4. Problem Formulation
  5. Separable Approximation of the Optimal Value Function
  6. Discrete-Time LQR
  7. Numerical Examples
  8. Random Linear Quadratic Regulator
  9. Nonlinear example: Allen-Cahn equation
  10. Conclusion

Key Result

Lemma 3

Consider an OCP of the form PF.system with a $\mathcal{C}^1$-optimal value function $V$. Define $\delta_{ij}$ as in PF.sensitivity and assume that there exists $\tilde{\gamma} \in \mathcal{L}$ such that for all $i,j \in [s]$, $i \neq j$, and $x \in \Omega$ it holds that Then Assumption ass.Decay holds with $\gamma(\cdot) = s \max_{x \in \Omega} \lVert x \rVert \, \tilde{\gamma}(\cdot)$.

Figures (4)

  • Figure 1: Neighborhoods in the graph of the semi-discrete heat equation.
  • Figure 2: Stepwise approximation of a point $x \in \mathbb{R}^3$ with the functions $\Psi_l^j$.
  • Figure 3: Decay in logarithmic scale of the first column of the absolute value of solution of the Riccati equation varying the parameter $k$.
  • Figure 4: Decay in logarithmic scale of the first column of $|P(y_0)|$ varying the parameter $r$.

Theorems & Definitions (14)

  • Definition 1
  • Example 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Remark 6
  • Remark 7
  • Definition 8: cf. Def. 1 in zhang2023optimal
  • ...and 4 more