Jincheng Cao

Jincheng Cao, Ruichen Jiang, Nazanin Abolfazli et al.

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires $\tilde{\mathcal{O}}(\max\{1/ε_f^{2},1/ε_g^{2}\}) $ stochastic oracle queries to obtain a solution that is $ε_f$-optimal for the upper-level and $ε_g$-optimal for the lower-level. This guarantee improves the previous best-known complexity of $\mathcal{O}(\max\{1/ε_f^{4},1/ε_g^{4}\})$. Moreover, for the case that the upper-level function is non-convex, our method requires at most $\tilde{\mathcal{O}}(\max\{1/ε_f^{3},1/ε_g^{3}\}) $ stochastic oracle queries to find an $(ε_f, ε_g)$-stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are $\tilde{\mathcal{O}}(\sqrt{n}/ε)$ and $\tilde{\mathcal{O}}(\sqrt{n}/ε^{2})$ for the convex and non-convex settings, respectively, where $ε=\min \{ε_f,ε_g\}$.

Jincheng Cao, Ruichen Jiang, Erfan Yazdandoost Hamedani et al.

In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max\{1/\sqrt{ε_{f}}, 1/ε_g\})$ iterations to find a solution that is $ε_f$-suboptimal and $ε_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th Hölderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max\{ε_{f}^{-\frac{2r-1}{2r}},ε_{g}^{-\frac{2r-1}{2r}}\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.

Jincheng Cao, Ruichen Jiang, Erfan Yazdandoost Hamedani et al.

In this paper, we study the problem of solving a simple bilevel optimization problem, where the upper-level objective is minimized over the solution set of the lower-level problem. We focus on the general setting in which both the upper- and lower-level objectives are smooth but potentially nonconvex. Due to the absence of additional structural assumptions for the lower-level objective-such as convexity or the Polyak-Łojasiewicz (PL) condition-guaranteeing global optimality is generally intractable. Instead, we introduce a suitable notion of stationarity for this class of problems and aim to design a first-order algorithm that finds such stationary points in polynomial time. Intuitively, stationarity in this setting means the upper-level objective cannot be substantially improved locally without causing a larger deterioration in the lower-level objective. To this end, we show that a simple and implementable variant of the dynamic barrier gradient descent (DBGD) framework can effectively solve the considered nonconvex simple bilevel problems up to stationarity. Specifically, to reach an $(ε_f, ε_g)$-stationary point-where $ε_f$ and $ε_g$ denote the target stationarity accuracies for the upper- and lower-level objectives, respectively-the considered method achieves a complexity of $\mathcal{O}\left(\max\left(ε_f^{-\frac{3+p}{1+p}}, ε_g^{-\frac{3+p}{2}}\right)\right)$, where $p \geq 0$ is an arbitrary constant balancing the terms. To the best of our knowledge, this is the first complexity result for a discrete-time algorithm that guarantees joint stationarity for both levels in general nonconvex simple bilevel problems.