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Fixed-time-stable ODE Representation of Lasso

Liang Wu, Yunhong Che, Wallace Gian Yion Tan, Efstathios Iliakis, Richard D. Braatz, Ján Drgoňa

Abstract

Lasso problems arise in many areas, including signal processing, machine learning, and control, and are closely connected to sparse coding mechanisms observed in neuroscience. A continuous-time ordinary differential equation (ODE) representation of the Lasso problem not only enables its solution on analog computers but also provides a framework for interpreting neurophysiological phenomena. This article proposes a fixed-time-stable ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based fixed-time-stable ODE system for solving the corresponding Karush-Kuhn-Tucker (KKT) conditions. Moreover, the settling time of the ODE is independent of the problem data and can be arbitrarily prescribed. Numerical experiments verify that the trajectory reaches the optimal solution within the prescribed time.

Fixed-time-stable ODE Representation of Lasso

Abstract

Lasso problems arise in many areas, including signal processing, machine learning, and control, and are closely connected to sparse coding mechanisms observed in neuroscience. A continuous-time ordinary differential equation (ODE) representation of the Lasso problem not only enables its solution on analog computers but also provides a framework for interpreting neurophysiological phenomena. This article proposes a fixed-time-stable ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based fixed-time-stable ODE system for solving the corresponding Karush-Kuhn-Tucker (KKT) conditions. Moreover, the settling time of the ODE is independent of the problem data and can be arbitrarily prescribed. Numerical experiments verify that the trajectory reaches the optimal solution within the prescribed time.

Paper Structure

This paper contains 10 sections, 8 theorems, 44 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Fixed-time-stable ODE for Unconstrained Smooth Optimization
  5. Method
  6. Transforming non-smooth Lasso into smooth nonnegative QP
  7. Transforming non-negative QP into fixed-time-stable Newton ODE
  8. Discussion about Implementation
  9. Numerical Examples
  10. Conclusion

Key Result

Lemma II.2

If there exists a positive-definite Lyapunov function $V\in\mathcal{C}^1(\mathcal{D},{\mathbb R})$ (where $\mathcal{D} \subset \mathbb{R}^{n_x}$ is a neighborhood of the equilibrium $x^*$) satisfying where the parameters $k>0$ and $\alpha\in(0,1)$, then the settling time $T(x_0)$ for the system eqn_continuous_dynamic can be bounded by $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Left: the trajectory $\|x(t)-x^*\|$ of Experiment (1) for one Lasso example; Right: the trajectory $\|x(t)-x^*\|$ of Experiment (2) for one Lasso example.

Theorems & Definitions (19)

  • Definition II.1: Finite-time-stable polyakov2011nonlinear
  • Lemma II.2: Garg2021fixedpolyakov2011nonlinear
  • Definition II.3: Fixed-time-stable polyakov2011nonlinear
  • Lemma II.4: polyakov2011nonlinearGarg2021fixed
  • Proposition II.5
  • proof
  • Remark II.6
  • Proposition II.7
  • proof
  • Remark III.1
  • ...and 9 more