Bilevel optimization minimizes an objective function, defined by an upper-level problem whose feasible region is the solution of a lower-level problem. We study the oracle complexity of finding an $ε$-stationary point with first-order methods when the upper-level problem is nonconvex and the lower-level problem is strongly convex. Recent works (Ji et al., ICML 2021; Arbel and Mairal, ICLR 2022; Chen el al., JMLR 2025) achieve a $\tilde{\mathcal{O}}(κ^4 ε^{-2})$ upper bound that is near-optimal in $ε$. However, the optimal dependency on the condition number $κ$ is unknown. In this work, we establish a new $Ω(κ^2 ε^{-2})$ lower bound and $\tilde{\mathcal{O}}(κ^{7/2} ε^{-2})$ upper bound for this problem, establishing the first provable gap between bilevel problems and minimax problems in this setup. Our lower bounds can be extended to various settings, including high-order smooth functions, stochastic oracles, and convex hyper-objectives: (1) For second-order and arbitrarily smooth problems, we show $Ω(κ_y^{13/4} ε^{-12/7})$ and $Ω(κ^{17/10} ε^{-8/5})$ lower bounds, respectively. (2) For convex-strongly-convex problems, we improve the previously best lower bound (Ji and Liang, JMLR 2022) from $Ω(κ/\sqrtε)$ to $Ω(κ^{5/4} / \sqrtε)$. (3) For smooth stochastic problems, we show an $Ω(κ^4 ε^{-4})$ lower bound.

Although upper bound guarantees for bilevel optimization have been widely studied, progress on lower bounds has been limited due to the complexity of the bilevel structure. In this work, we focus on the smooth nonconvex-strongly-convex setting and develop new hard instances that yield nontrivial lower bounds under deterministic and stochastic first-order oracle models. In the deterministic case, we prove that any first-order zero-respecting algorithm requires at least $Ω(κ^{3/2}ε^{-2})$ oracle calls to find an $ε$-accurate stationary point, improving the optimal lower bounds known for single-level nonconvex optimization and for nonconvex-strongly-convex min-max problems. In the stochastic case, we show that at least $Ω(κ^{5/2}ε^{-4})$ stochastic oracle calls are necessary, again strengthening the best known bounds in related settings. Our results expose substantial gaps between current upper and lower bounds for bilevel optimization and suggest that even simplified regimes, such as those with quadratic lower-level objectives, warrant further investigation toward understanding the optimal complexity of bilevel optimization under standard first-order oracles.

We propose a reduction for non-convex optimization that can (1) turn an stationary-point finding algorithm into an local-minimum finding one, and (2) replace the Hessian-vector product computations with only gradient computations. It works both in the stochastic and the deterministic settings, without hurting the algorithm's performance. As applications, our reduction turns Natasha2 into a first-order method without hurting its theoretical performance. It also converts SGD, GD, SCSG, and SVRG into algorithms finding approximate local minima, outperforming some best known results.

Going beyond the assumption of convex-concavity poses a big challenge.

This work investigates the performance limits of projected stochastic first-order methods for minimizing functions under the $(\alpha,\tau,\mathcal{X})$-projected-gradient-dominance property, that asserts the sub-optimality gap $F(\mathbf{x})-\min_{\mathbf{x}'\in \mathcal{X}}F(\mathbf{x}')$ is upper-bounded by $\tau\cdot\|\mathcal{G}_{\eta,\mathcal{X}}(\mathbf{x})\|^{\alpha}$ for some $\alpha\in[1,2)$ and $\tau>0$ and $\mathcal{G}_{\eta,\mathcal{X}}(\mathbf{x})$ is the projected-gradient mapping with $\eta>0$ as a parameter. For non-convex functions, we show that the complexity lower bound of querying a batch smooth first-order stochastic oracle to obtain an $\epsilon$-global-optimum point is $\Omega(\epsilon^{-{2}/{\alpha}})$. Furthermore, we show that a projected variance-reduced first-order algorithm can obtain the upper complexity bound of $\mathcal{O}(\epsilon^{-{2}/{\alpha}})$, matching the lower bound. For convex functions, we establish a complexity lower bound of $\Omega(\log(1/\epsilon)\cdot\epsilon^{-{2}/{\alpha}})$ for minimizing functions under a local version of gradient-dominance property, which also matches the upper complexity bound of accelerated stochastic subgradient methods.

We consider the problem of finding stationary points in Bilevel optimization when the lower-level problem is unconstrained and strongly convex. The problem has been extensively studied in recent years; the main technical challenge is to keep track of lower-level solutions $y^*(x)$ in response to the changes in the upper-level variables $x$. Subsequently, all existing approaches tie their analyses to a genie algorithm that knows lower-level solutions and, therefore, need not query any points far from them. We consider a dual question to such approaches: suppose we have an oracle, which we call $y^*$-aware, that returns an $O(\epsilon)$-estimate of the lower-level solution, in addition to first-order gradient estimators {\it locally unbiased} within the $\Theta(\epsilon)$-ball around $y^*(x)$. We study the complexity of finding stationary points with such an $y^*$-aware oracle: we propose a simple first-order method that converges to an $\epsilon$ stationary point using $O(\epsilon^{-6}), O(\epsilon^{-4})$ access to first-order $y^*$-aware oracles. Our upper bounds also apply to standard unbiased first-order oracles, improving the best-known complexity of first-order methods by $O(\epsilon)$ with minimal assumptions. We then provide the matching $\Omega(\epsilon^{-6})$, $\Omega(\epsilon^{-4})$ lower bounds without and with an additional smoothness assumption on $y^*$-aware oracles, respectively. Our results imply that any approach that simulates an algorithm with an $y^*$-aware oracle must suffer the same lower bounds.

This paper considers the optimization problem of the form $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{n}\sum_{i=1}^n f_i({\bf x})$, where $f(\cdot)$ satisfies the Polyak--{\L}ojasiewicz (PL) condition with parameter $\mu$ and $\{f_i(\cdot)\}_{i=1}^n$ is $L$-mean-squared smooth. We show that any gradient method requires at least $\Omega(n+\kappa\sqrt{n}\log(1/\epsilon))$ incremental first-order oracle (IFO) calls to find an $\epsilon$-suboptimal solution, where $\kappa\triangleq L/\mu$ is the condition number of the problem. This result nearly matches upper bounds of IFO complexity for best-known first-order methods. We also study the problem of minimizing the PL function in the distributed setting such that the individuals $f_1(\cdot),\dots,f_n(\cdot)$ are located on a connected network of $n$ agents. We provide lower bounds of $\Omega(\kappa/\sqrt{\gamma}\,\log(1/\epsilon))$, $\Omega((\kappa+\tau\kappa/\sqrt{\gamma}\,)\log(1/\epsilon))$ and $\Omega\big(n+\kappa\sqrt{n}\log(1/\epsilon)\big)$ for communication rounds, time cost and local first-order oracle calls respectively, where $\gamma\in(0,1]$ is the spectral gap of the mixing matrix associated with the network and~$\tau>0$ is the time cost of per communication round. Furthermore, we propose a decentralized first-order method that nearly matches above lower bounds in expectation.

We provide a first-order oracle complexity lower bound for finding stationary points of min-max optimization problems where the objective function is smooth, nonconvex in the minimization variable, and strongly concave in the maximization variable. We establish a lower bound of $\Omega\left(\sqrt{\kappa}\epsilon^{-2}\right)$ for deterministic oracles, where $\epsilon$ defines the level of approximate stationarity and $\kappa$ is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the $\epsilon$ and $\kappa$ dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of $\Omega\left(\sqrt{\kappa}\epsilon^{-2} + \kappa^{1/3}\epsilon^{-4}\right)$. It suggests that there is a significant gap between the upper bound $\mathcal{O}(\kappa^3 \epsilon^{-4})$ in (Lin et al., 2020a) and our lower bound in the condition number dependence.

Recent advances in randomized incremental methods for minimizing $L$-smooth $\mu$-strongly convex finite sums have culminated in tight complexity of $\tilde{O}((n+\sqrt{n L/\mu})\log(1/\epsilon))$ and $O(n+\sqrt{nL/\epsilon})$, where $\mu>0$ and $\mu=0$, respectively, and $n$ denotes the number of individual functions. Unlike incremental methods, stochastic methods for finite sums do not rely on an explicit knowledge of which individual function is being addressed at each iteration, and as such, must perform at least $\Omega(n^2)$ iterations to obtain $O(1/n^2)$-optimal solutions. In this work, we exploit the finite noise structure of finite sums to derive a matching $O(n^2)$-upper bound under the global oracle model, showing that this lower bound is indeed tight. Following a similar approach, we propose a novel adaptation of SVRG which is both \emph{compatible with stochastic oracles}, and achieves complexity bounds of $\tilde{O}((n^2+n\sqrt{L/\mu})\log(1/\epsilon))$ and $O(n\sqrt{L/\epsilon})$, for $\mu>0$ and $\mu=0$, respectively. Our bounds hold w.h.p. and match in part existing lower bounds of $\tilde{\Omega}(n^2+\sqrt{nL/\mu}\log(1/\epsilon))$ and $\tilde{\Omega}(n^2+\sqrt{nL/\epsilon})$, for $\mu>0$ and $\mu=0$, respectively.

Imitation learning (IL) consists of a set of tools that leverage expert demonstrations to quickly learn policies. However, if the expert is suboptimal, IL can yield policies with inferior performance compared to reinforcement learning (RL). In this paper, we aim to provide an algorithm that combines the best aspects of RL and IL. We accomplish this by formulating several popular RL and IL algorithms in a common mirror descent framework, showing that these algorithms can be viewed as a variation on a single approach. We then propose LOKI, a strategy for policy learning that first performs a small but random number of IL iterations before switching to a policy gradient RL method. We show that if the switching time is properly randomized, LOKI can learn to outperform a suboptimal expert and converge faster than running policy gradient from scratch. Finally, we evaluate the performance of LOKI experimentally in several simulated environments.