Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso
本文通过二阶变分分析研究二次增长条件在机器学习与信号处理中的结构化优化问题（如泊松线性逆问题和L1正则化问题）中的作用，并给出了Lasso问题最优解唯一性的完整刻画。
Abstract In the previous paper Bello-Cruz et al. (J Optim Theory Appl 188:378–401, 2021), we showed that the quadratic growth condition plays a key role in obtaining Q-linear convergence of the widely used forward–backward splitting method with Beck–Teboulle’s line search. In this paper, we analyze the property of quadratic growth condition via second-order variational analysis for various structured optimization problems that arise in machine learning and signal processing. This includes, for example, the Poisson linear inverse problem as well as the $$\ell _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> -regularized optimization problems. As a by-product of this approach, we also obtain several full characterizations for the uniqueness of optimal solution to Lasso problem, which complements and extends recent important results in this direction.