Functional constrained optimization (FCO) has emerged as a powerful tool for solving various machine learning problems. However, with the rapid increase in applications of neural networks in recent years, it has become apparent that both the objective and constraints often involve nonconvex functions, which poses significant challenges in obtaining high-quality solutions. In this work, we focus on a class of nonconvex FCO problems with nonconvex constraints, where the two optimization variables are nonlinearly coupled in the inequality constraint. Leveraging the primal-dual optimization framework, we propose a smoothed first-order Lagrangian method (SLM) for solving this class of problems. We establish the theoretical convergence guarantees of SLM to the Karush-Kuhn-Tucker (KKT) solutions through quantifying dual error bounds. By establishing connections between this structured FCO and equilibrium-constrained nonconvex problems (also known as bilevel optimization), we apply the proposed SLM to tackle bilevel optimization oriented problems where the lower-level problem is nonconvex. Numerical results obtained from both toy examples and hyper-data cleaning problems demonstrate the superiority of SLM compared to benchmark methods.

However, a topology with a fast consensus rate, e.g., the exponential graph, generally has a large maximum degree, which incurs significant communication

Such a phenomenon is illustrated in Figure 1 where the learned predictive function from the source data may significantly differ from the true function.

The "folk wisdom" in the literature is that the focus on local optimization sidesteps the curse of dimensionality;

Sampling from a high-dimensional distribution serves as one of the key components in statistics, machine learning, and scientific computing, and constitutes the foundation of the fields including Bayesian statistics and generative models [Liu and Liu, 2001, Brooks et al., 2011, Song et al.,

Sampling from a high-dimensional distribution serves as one of the key components in statistics, machine learning, and scientific computing, and constitutes the foundation of the fields including Bayesian statistics and generative models [Liu and Liu, 2001, Brooks et al., 2011, Song et al.,