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Control Theoretic Approach to Fine-Tuning and Transfer Learning

Erkan Bayram, Shenyu Liu, Mohamed-Ali Belabbas, Tamer Başar

TL;DR

The paper introduces a control-theoretic framework for fine-tuning and transfer learning in dynamical systems by memorizing labeled ensembles with a desire to expand the training set without forgetting prior knowledge. It replaces the computationally expensive q-folded method with a tuning without forgetting approach that updates the control u by projecting the gradient onto the kernel of the end-point mapping, thereby preserving previously learned end-points to first order while learning new samples. The authors establish a theoretical basis via bracket-generating controllability conditions for partially constrained ensembles and present a practical three-phase numerical method (kernel projection, norm minimization, and refinement) to implement the approach. A computational example demonstrates improved memory stability and learning plasticity compared to a penalty-based fine-tuning method, highlighting scalability and effectiveness for continual learning in control-based supervised tasks.

Abstract

Given a training set in the form of a paired $(\mathcal{X},\mathcal{Y})$, we say that the control system $\dot x = f(x,u)$ has learned the paired set via the control $u^*$ if the system steers each point of $\mathcal{X}$ to its corresponding target in $\mathcal{Y}$. If the training set is expanded, most existing methods for finding a new control $u^*$ require starting from scratch, resulting in a quadratic increase in complexity with the number of points. To overcome this limitation, we introduce the concept of $\textit{ tuning without forgetting}$. We develop $\textit{an iterative algorithm}$ to tune the control $u^*$ when the training set expands, whereby points already in the paired set are still matched, and new training samples are learned. At each update of our method, the control $u^*$ is projected onto the kernel of the end-point mapping generated by the controlled dynamics at the learned samples. It ensures keeping the end-points for the previously learned samples constant while iteratively learning additional samples.

Control Theoretic Approach to Fine-Tuning and Transfer Learning

TL;DR

The paper introduces a control-theoretic framework for fine-tuning and transfer learning in dynamical systems by memorizing labeled ensembles with a desire to expand the training set without forgetting prior knowledge. It replaces the computationally expensive q-folded method with a tuning without forgetting approach that updates the control u by projecting the gradient onto the kernel of the end-point mapping, thereby preserving previously learned end-points to first order while learning new samples. The authors establish a theoretical basis via bracket-generating controllability conditions for partially constrained ensembles and present a practical three-phase numerical method (kernel projection, norm minimization, and refinement) to implement the approach. A computational example demonstrates improved memory stability and learning plasticity compared to a penalty-based fine-tuning method, highlighting scalability and effectiveness for continual learning in control-based supervised tasks.

Abstract

Given a training set in the form of a paired , we say that the control system has learned the paired set via the control if the system steers each point of to its corresponding target in . If the training set is expanded, most existing methods for finding a new control require starting from scratch, resulting in a quadratic increase in complexity with the number of points. To overcome this limitation, we introduce the concept of . We develop to tune the control when the training set expands, whereby points already in the paired set are still matched, and new training samples are learned. At each update of our method, the control is projected onto the kernel of the end-point mapping generated by the controlled dynamics at the learned samples. It ensures keeping the end-points for the previously learned samples constant while iteratively learning additional samples.
Paper Structure (14 sections, 4 theorems, 31 equations, 1 figure, 4 algorithms)

This paper contains 14 sections, 4 theorems, 31 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $q$-Folded Method:
  4. Main Results
  5. Controllability on Partially Constrained Ensembles:
  6. Tuning without forgetting:
  7. A projected gradient descent method:
  8. Numerical Method for Tuning without Forgetting
  9. Approximation of $\mathcal{L}_{(u,x^i)}(\cdot)$:
  10. Phase I:
  11. Phase II
  12. Phase III
  13. A Computational Example
  14. Summary and Outlook

Key Result

lemma thmcounterlemma

Assume that the ensemble $\mathcal{X}$ consists of finite pairwise distinct points and $n > n_o$. For the readout map $R(x)=Cx$, if the set of control vector fields of $q$-folded system is bracket-generating in $E(\mathcal{M})^{(q)}(= E(\mathcal{M})^q \setminus \Delta^q)$, then there exists a contro

Figures (1)

  • Figure 1: (a) and (b) average error as a function of number of rounds for $|\mathcal{X}|=64$ for $j=16$ and $j=52$, respectively. (c) and (d) average error as a function of number of rounds for $|\mathcal{X}|=32$ for $j=8$ and $j=25$, respectively. The dark gray region is Phase I region and the light gray region is Phase III region (each round is followed by Phase II). Average error on the given set for the control functions $u^*$,$\Tilde{u}$, and $u^0$ are marked by $\bullet,\blacklozenge$, and $\times$, respectively.

Theorems & Definitions (12)

  • definition thmcounterdefinition: Memorization Property
  • definition thmcounterdefinition: Control distributions
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition
  • definition thmcounterdefinition: Linearized Controllability Property
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • ...and 2 more