Abstract: Bilevel optimization is widely applied in many machine learning tasks such as hyper-parameter learning and meta learning. Recently, many algorithms have been proposed to solve these bilevel optimization problems, which rely on the smoothness condition of objective functions of the bilevel optimization. In fact, some machine learning tasks such as learning language model do not satisfy the smoothness condition of objective functions. More recently, some methods have begun to study generalized smooth bilevel optimization. However, these proposed methods for generalized smooth bilevel optimization only focus on the (strongly) convex lower objective function. Meanwhile, these methods only consider the generalized-smooth upper-level objective, but still require the standard smooth lower-level objective in the bilevel optimization. To fill this gap, in the paper, thus we study the generalized-smooth bilevel optimization with the nonconvex lower-level objective function, where both upper-level and lower-level objectives are generalized-smooth. We propose an efficient single-loop Hessian/Jacobian-free penalty normalized gradient (i.e., PNGBiO) method. Moreover, we prove that our PNGBiO obtains a fast convergence rate of $O(\frac{1}{T^{1/4}})$ for finding a stationary solution, where $T$ denotes the iteration number. Meanwhile, we also propose a stochastic version of our PNGBiO (i.e., S-PNGBiO) method to solve stochastic bilevel problems, and prove that our S-PNGBiO has a fast convergence rate of $O(\frac{1}{T^{1/6}})$. Some experimental results on hyper-parameter learning and meta learning demonstrate efficiency of our proposed methods.

Lay Summary:

Bilevel optimization is widely applied in many machine learning tasks such as hyper-parameter learning and meta learning. Recently, many algorithms have been proposed to solve these bilevel optimization problems, which rely on the smoothness condition of objective functions of the bilevel optimization. In fact, some machine learning tasks such as learning language model do not satisfy the smoothness condition of objective functions. More recently, some methods have begun to study generalized smooth bilevel optimization. However, these proposed methods for generalized smooth bilevel optimization only focus on the (strongly) convex lower objective function. Meanwhile, these methods only consider the generalized-smooth upper-level objective, but still require the standard smooth lower-level objective in the bilevel optimization. To fill this gap, in the paper, thus we study the generalized-smooth bilevel optimization with the nonconvex lower-level objective function, where both upper-level and lower-level objectives are generalized-smooth. We propose an efficient single-loop Hessian/Jacobian-free penalty normalized gradient (i.e., PNGBiO) method. Moreover, we prove that our PNGBiO obtains a fast convergence rate for finding a stationary solution. Meanwhile, we propose a stochastic version of our PNGBiO (i.e., S-PNGBiO) method to solve stochastic bilevel problems, and prove that our S-PNGBiO also has a fast convergence rate. Some experimental results on hyper-parameter learning and meta learning demonstrate efficiency of our proposed methods.