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Convexity and strict convexity for compositional neural networks in high-dimensional optimal control

Lars Grüne, Konrad Kleinberg, Thomas Kruse, Mario Sperl

TL;DR

This work investigates how compositional structure in high-dimensional optimal control enables neural-network-based representations that avoid the curse of dimensionality. It extends the CAD-free approximation results for strictly convex terminal costs to two directions: enforcing strict convexity via stage costs using an extended state-space reformulation, and achieving weak approximations when the cost is only convex. The authors derive explicit neural-network size bounds that depend on compositional features, demonstrate how to reformulate problems to preserve compositional structure, and provide gradient-descent-based lemmas to support weak optimization guarantees. Collectively, the results show that, under structured dynamics and cost components, neural networks can efficiently approximate optimal controls in high dimensions, with clear theoretical guarantees and constructive complexity estimates. This lays groundwork for practical, scalable NN-based feedback synthesis in large-scale control systems, including linear-quadratic-like settings and beyond.

Abstract

Neural networks (NNs) have emerged as powerful tools for solving high-dimensional optimal control problems. In particular, their compositional structure has been shown to enable efficient approximation of high-dimensional functions, helping to mitigate the curse of dimensionality in optimal control problems. In this work, we build upon the theoretical framework developed by Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022), particularly their results on NN approximations for compositional functions in optimal control. Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) establishes that, under suitable assumptions on the compositional structure and its associated features, optimal control problems with strictly convex cost functionals admit a curse-of-dimensionality-free approximation of the optimal control by NNs. We extend this result in two directions. First, we analyze the strict convexity requirement on the cost functional and demonstrate that reformulating a discrete-time optimal control problem with linear transitions and stage costs as a terminal cost problem ensures the necessary strict convexity. Second, we establish a generalization of Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) which provides weak error bounds for optimal control approximations by NNs when the cost functional is only convex rather than strictly convex.

Convexity and strict convexity for compositional neural networks in high-dimensional optimal control

TL;DR

This work investigates how compositional structure in high-dimensional optimal control enables neural-network-based representations that avoid the curse of dimensionality. It extends the CAD-free approximation results for strictly convex terminal costs to two directions: enforcing strict convexity via stage costs using an extended state-space reformulation, and achieving weak approximations when the cost is only convex. The authors derive explicit neural-network size bounds that depend on compositional features, demonstrate how to reformulate problems to preserve compositional structure, and provide gradient-descent-based lemmas to support weak optimization guarantees. Collectively, the results show that, under structured dynamics and cost components, neural networks can efficiently approximate optimal controls in high dimensions, with clear theoretical guarantees and constructive complexity estimates. This lays groundwork for practical, scalable NN-based feedback synthesis in large-scale control systems, including linear-quadratic-like settings and beyond.

Abstract

Neural networks (NNs) have emerged as powerful tools for solving high-dimensional optimal control problems. In particular, their compositional structure has been shown to enable efficient approximation of high-dimensional functions, helping to mitigate the curse of dimensionality in optimal control problems. In this work, we build upon the theoretical framework developed by Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022), particularly their results on NN approximations for compositional functions in optimal control. Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) establishes that, under suitable assumptions on the compositional structure and its associated features, optimal control problems with strictly convex cost functionals admit a curse-of-dimensionality-free approximation of the optimal control by NNs. We extend this result in two directions. First, we analyze the strict convexity requirement on the cost functional and demonstrate that reformulating a discrete-time optimal control problem with linear transitions and stage costs as a terminal cost problem ensures the necessary strict convexity. Second, we establish a generalization of Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) which provides weak error bounds for optimal control approximations by NNs when the cost functional is only convex rather than strictly convex.

Paper Structure

This paper contains 13 sections, 9 theorems, 67 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The compositional framework and applications in optimal control
  3. Preliminaries on compositional functions
  4. Optimal control with compositionality
  5. Enforcing strict convexity by introducing stage costs
  6. Strict convexity for problems with linear dynamics and separated stage costs
  7. The extended state space problem and its compositional structure
  8. Neural network approximations of optimal controls with stage costs
  9. Weak approximations for non-strictly convex problems
  10. Weak error estimates for gradient descent
  11. Neural networks approximations for optimizers in non-strictly convex minimization problems
  12. Neural network approximations of optimal controls in non-strictly convex problems
  13. Conclusion

Key Result

Theorem 2.5

Let $(f, \mathcal{G}^f, \mathcal{L}^f)$ satisfy Assumption asm:compositional_functions and let $([)]{ r_{\max}^f, \Lambda^f, L_{\max}^f, \lvert\mathcal{V}_G^f\rvert}$ be features of $(f, \mathcal{G}^f, \mathcal{L}^f)$. Then for any $n_w \in \mathbb{N}$ there exists a $\mathrm{NN}$$f^\mathrm{NN}$ wit where $C > 0$ is a constant determined by $\{ ( d_{i,j}, m_{i, j} ) \in \mathbb{N}^2 : f_{i,j} \in

Figures (4)

  • Figure 1: Illustration of a compositional function.
  • Figure 2: Extended dynamic $\mathbf{f}$
  • Figure 3: Extended terminal cost $\mathbf{g}$
  • Figure 4: Compositional structure of $f$

Theorems & Definitions (18)

  • Definition 2.1: Definition 3.1 in kang2022feedforward
  • Definition 2.3: Definition 4.3 in kang2022feedforward
  • Definition 2.4: Definition 4.1 in kang2022feedforward
  • Theorem 2.5: Theorem 4.10 in kang2022feedforward
  • Example 2.6
  • Remark 2.7
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • Lemma 3.5
  • ...and 8 more