Stochastic Compositional Gradient Descent Under Compositional Constraints

This work studies constrained stochastic optimization problems where the objective and constraint functions are convex expressed as compositions of functions. The problem arises in context fair classification, regression, design queuing systems. Of particular interest is large-scale setting an oracle provides gradients constituent functions, goal to solve with a minimal number calls oracle. Owing compositional form, provided by do not yield unbiased estimates or gradients. Instead, we construct approximate tracking inner function evaluations, resulting quasi-gradient saddle point algorithm. We prove that proposed algorithm guaranteed find optimal feasible solution almost surely. further establish requires $\mathcal {O}(1/\epsilon ^{4})$ data samples order obtain notation="LaTeX">$\epsilon$ -approximate while also ensuring zero violation. result matches sample complexity gradient descent method for unconstrained improves upon best-known results settings. efficacy tested on both classification regression problems. numerical show outperforms state-of-the-art algorithms terms convergence rate.