Srujan Teja Thomdapu

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

Bi-level optimization, where the objective function depends on the solution of an inner optimization problem, provides a flexible framework for solving a rich class of problems such as hyper-parameter optimization and model-agnostic meta-learning. This work puts forth the first S tochastic B i-level F rank- W olfe (SBFW) algorithm to solve the stochastic bi-level optimization problems in a projection-free manner. We utilize a momentum-based gradient tracker that results in a sample complexity of $\mathcal {O}(/eps^{-3})$ for convex outer objectives and $\mathcal {O}(/eps^{-4})$ for non-convex outer objectives with strongly convex inner objectives. The stochastic compositional optimization problems (a special case of bi-level optimization problems entailing the minimization of a composition of two expected-value functions) are also considered within the same rubric. The proposed S tochastic C ompositional F rank- W olfe (SCFW) algorithm is shown to achieve a sample complexity of $\mathcal {O}(/eps^{-2})$ for convex objectives and $\mathcal {O}(/eps^{-3})$ for non-convex objectives, at par with the state-of-the-art rates of projection-free algorithms for single-level problems. The usefulness and flexibility of SBFW and SCFW algorithms are demonstrated via extensive numerical tests. We show that SBFW outperforms the state-of-the-art methods for the problem of matrix completion with denoising, and achieve improvements of up to $82\%$ in terms of the wall-clock time required to achieve a particular level of accuracy. Furthermore, we demonstrate the improved performance of SCFW over the competing projection-free variants on the policy evaluation problem in reinforcement learning.

Well-designed queuing systems form the backbone of modern communications, distributed computing, and content delivery architectures. Designs balancing infrastructure costs and user experience indices require tools from teletraffic theory and operations research. A standard approach to designing such systems involves formulating optimization problems that strive to maximize the pertinent utility functions while adhering to quality-of-service and other physical constraints. In many cases, formulating such problems necessitates making simplistic assumptions on arrival and departure processes to keep the problem simple. This work puts forth a stochastic optimization framework for designing queuing systems where the exogenous processes may have arbitrary and unknown distributions. We show that many such queuing design problems can generally be formulated as stochastic optimization problems where the objective and constraint are non-linear functions of expectations. The compositional structure obviates the use of classical stochastic approximation approaches where the stochastic gradients are often required to be unbiased. To this end, a constrained stochastic compositional gradient descent algorithm is proposed that utilizes a tracking step for the expected value functions. The non-asymptotic performance of the proposed algorithm is characterized via its iteration complexity. Numerical tests allow us to validate the theoretical results and demonstrate the efficacy of the proposed algorithm.