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From Few-Shot Optimal Control to Few-Shot Learning

Roman Chertovskih, Nikolay Pogodaev, Maxim Staritsyn, A. Pedro Aguiar

TL;DR

The paper tackles unconstrained nonlinear optimal control problems under a few-shot setting by lifting them to a linear super-problem on the dual Banach space and performing an exact variational analysis. This yields a monotone descent method with self-adjusting steps, enabling rapid convergence with few control updates. The framework is extended to mean-field control and linked to ML contexts, offering a unified perspective on information propagation in self-interacting neural networks. The proposed approach promises computational efficiency in high-dimensional or distributed systems and suggests PDE-based strategies for few-shot learning applications.

Abstract

We present an approach to solving unconstrained nonlinear optimal control problems for a broad class of dynamical systems. This approach involves lifting the nonlinear problem to a linear ``super-problem'' on a dual Banach space, followed by a non-standard ``exact'' variational analysis, -- culminating in a descent method that achieves rapid convergence with minimal iterations. We investigate the applicability of this framework to mean-field control and discuss its perspectives for the analysis of information propagation in self-interacting neural networks.

From Few-Shot Optimal Control to Few-Shot Learning

TL;DR

The paper tackles unconstrained nonlinear optimal control problems under a few-shot setting by lifting them to a linear super-problem on the dual Banach space and performing an exact variational analysis. This yields a monotone descent method with self-adjusting steps, enabling rapid convergence with few control updates. The framework is extended to mean-field control and linked to ML contexts, offering a unified perspective on information propagation in self-interacting neural networks. The proposed approach promises computational efficiency in high-dimensional or distributed systems and suggests PDE-based strategies for few-shot learning applications.

Abstract

We present an approach to solving unconstrained nonlinear optimal control problems for a broad class of dynamical systems. This approach involves lifting the nonlinear problem to a linear ``super-problem'' on a dual Banach space, followed by a non-standard ``exact'' variational analysis, -- culminating in a descent method that achieves rapid convergence with minimal iterations. We investigate the applicability of this framework to mean-field control and discuss its perspectives for the analysis of information propagation in self-interacting neural networks.

Paper Structure

This paper contains 13 sections, 1 theorem, 26 equations, 2 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Contribution and Novelty. Organization of the Paper
  3. THE INTUITION: GEOMETRIC CONTROL
  4. Challenges
  5. Summary of the Approach
  6. Lifting
  7. Duality argument
  8. Exact variational analysis
  9. Descent method
  10. Generalizations
  11. EXTENSION: MEAN-FIELD CONTROL
  12. ML PERSPECTIVES
  13. CONCLUSION: FEW-SHOT LEARNING

Key Result

Proposition 1

Let $\bar{u}, u \in \mathcal{U}$ be arbitrary controls with the corresponding primal and dual states $\mu = \mu[u]$, $\bar{\mu} = \mu[\bar{u}]$ and $\bar{p} = p[\bar{u}]$. Then, the following representation holds:

Figures (2)

  • Figure 1: Example \ref{['KuramE']}: Baseline, $\rho_0$, and optimized, $\rho_T$, densities together with its target profile $\hat{\rho}$ on the flat representation of $\mathbb S^1$, $\mathrm{x} \in [0, 2\pi]$.
  • Figure 2: Example \ref{['AtteE']}: Optimized density snapshots as heatmaps over the torus' fundamental domain (rendered on a relative color scale).

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Example 1: Kuramoto model
  • Example 2: Attention-inspired model