Second-order optimization methods are computationally expensive for large-scale problems. Recently, Doikov, Chayti, and Jaggi (ICML 2023) proposed the LazyCRN method that reduces computation by studying the gradient complexity of second-order methods. Their method can achieve a gradient complexity of $\mathcal{O}( \bar d + \bar d^{1/2} \epsilon^{-3/2})$ and $\mathcal{O}( \bar d + \bar d^{1/2} \epsilon^{-1/2})$ for nonconvex and convex optimization, respectively, where $\bar d$ is the effective dimension and $\epsilon$ is the target precision. Very recently, Adil, Bullins, Sidford, and Zhang (NeurIPS 2025) improved the gradient complexity to $\mathcal{O}( \bar d + \bar d^{1/3} \epsilon^{-3/2} \ln^{18} \epsilon^{-1})$ for nonconvex optimization. However, the tightness of these methods remains open. In this work, we propose new methods that achieve an improved complexity of $\mathcal{O}( \bar d + \bar d^{1/3} \epsilon^{-3/2})$ and $\mathcal{O}( (\bar d + \bar d^{13/21} \epsilon^{-2/7}) \ln \bar d)$ for nonconvex and convex optimization, respectively, improving best-known results for both setups.

This paper studies second-order methods for convex-concave minimax optimization. Monteiro & Svaiter (2012) proposed a method to solve the problem with an optimal iteration complexity of $\mathcal{O}(\epsilon^{-3/2})$ to find an $\epsilon$-saddle point. However, it is unclear whether the computational complexity, $\mathcal{O}((N+ d^2) d \epsilon^{-2/3})$, can be improved. In the above, we follow Doikov et al. (2023) and assume the complexity of obtaining a first-order oracle as $N$ and the complexity of obtaining a second-order oracle as $dN$. In this paper, we show that the computation cost can be reduced by reusing Hessian across iterations. Our methods take the overall computational complexity of $\tilde{\mathcal{O}}( (N+d^2)(d+ d^{2/3}\epsilon^{-2/3}))$, which improves those of previous methods by a factor of $d^{1/3}$. Furthermore, we generalize our method to strongly-convex-strongly-concave minimax problems and establish the complexity of $\tilde{\mathcal{O}}((N+d^2) (d + d^{2/3} \kappa^{2/3}) )$ when the condition number of the problem is $\kappa$, enjoying a similar speedup upon the state-of-the-art method. Numerical experiments on both real and synthetic datasets also verify the efficiency of our method.

In this work, we consider bilevel optimization when the lower-level problem is strongly convex. Recent works show that with a Hessian-vector product (HVP) oracle, one can provably find an $\epsilon$-stationary point within ${O}(\epsilon^{-2})$ oracle calls. However, the HVP oracle may be inaccessible or expensive in practice. Kwon et al. (ICML 2023) addressed this issue by proposing a first-order method that can achieve the same goal at a slower rate of $\tilde{O}(\epsilon^{-3})$. In this paper, we incorporate a two-time-scale update to improve their method to achieve the near-optimal $\tilde{O}(\epsilon^{-2})$ first-order oracle complexity. Our analysis is highly extensible. In the stochastic setting, our algorithm can achieve the stochastic first-order oracle complexity of $\tilde {O}(\epsilon^{-4})$ and $\tilde {O}(\epsilon^{-6})$ when the stochastic noises are only in the upper-level objective and in both level objectives, respectively. When the objectives have higher-order smoothness conditions, our deterministic method can escape saddle points by injecting noise, and can be accelerated to achieve a faster rate of $\tilde {O}(\epsilon^{-1.75})$ using Nesterov's momentum.

This paper studies the stochastic nonconvex-strongly-concave minimax optimization over a multi-agent network. We propose an efficient algorithm, called Decentralized Recursive gradient descEnt Ascent Method (DREAM), which achieves the best-known theoretical guarantee for finding the $\epsilon$-stationary points. Concretely, it requires $\mathcal{O}(\min (\kappa^3\epsilon^{-3},\kappa^2 \sqrt{N} \epsilon^{-2} ))$ stochastic first-order oracle (SFO) calls and $\tilde \mathcal O(\kappa^2 \epsilon^{-2})$ communication rounds, where $\kappa$ is the condition number and $N$ is the total number of individual functions. Our numerical experiments also validate the superiority of DREAM over previous methods.

Bilevel optimization reveals the inner structure of otherwise oblique optimization problems, such as hyperparameter tuning, neural architecture search, and meta-learning. A common goal in bilevel optimization is to minimize a hyper-objective that implicitly depends on the solution set of the lower-level function. Although this hyper-objective approach is widely used, its theoretical properties have not been thoroughly investigated in cases where \textit{the lower-level functions lack strong convexity}. In this work, we first provide hardness results to show that the goal of finding stationary points of the hyper-objective for nonconvex-convex bilevel optimization can be intractable for zero-respecting algorithms. Then we study a class of tractable nonconvex-nonconvex bilevel problems when the lower-level function satisfies the Polyak-Ł{}ojasiewicz (PL) condition. We show a simple first-order algorithm can achieve complexity bounds of $\tilde{\mathcal{O}}(\epsilon^{-2})$, $\tilde{\mathcal{O}}(\epsilon^{-4})$ and $\tilde{\mathcal{O}}(\epsilon^{-6})$ in the deterministic, partially stochastic, and fully stochastic setting respectively.

This paper studies decentralized optimization problem, where the local objective on each node is an average of a finite set of convex functions and the global function is strongly convex. We propose an efficient stochastic variance reduced first-order method that allows the different nodes to establish their stochastic local gradient estimator with different mini-batch sizes per iteration. We prove the upper bound on the computation time of the proposed method contains the dependence on the global condition number, which is sharper than the previous results that only depend on the local condition numbers. Compared with the state-of-the-art methods, we also show that our method requires less local incremental first-order oracle calls and comparable communication cost. We further perform numerical experiments to validate the advantage of our method.

This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose a novel method by reformulating them into functionally constrained problems. Our method achieves near-optimal rates for both smooth and nonsmooth problems. To the best of our knowledge, this is the first near-optimal algorithm that works under standard assumptions of smoothness or Lipschitz continuity for the objective functions.

We study the problem of finding a near-stationary point for smooth minimax optimization. The recently proposed extra anchored gradient (EAG) methods achieve the optimal convergence rate for the convex-concave minimax problem in the deterministic setting. However, the direct extension of EAG to stochastic optimization is not efficient. In this paper, we design a novel stochastic algorithm called Recursive Anchored IteratioN (RAIN). We show that the RAIN achieves near-optimal stochastic first-order oracle (SFO) complexity for stochastic minimax optimization in both convex-concave and strongly-convex-strongly-concave cases. In addition, we extend the idea of RAIN to solve structured nonconvex-nonconcave minimax problem and it also achieves near-optimal SFO complexity.

We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}[F(x;\xi)]$ , where the component $F(x;\xi)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth.The recently proposed gradient-free method requires at most $\mathcal{O}( L^4 d^{3/2} \epsilon^{-4} + \Delta L^3 d^{3/2} \delta^{-1} \epsilon^{-4})$ stochastic zeroth-order oracle complexity to find a $(\delta,\epsilon)$-Goldstein stationary point of objective function, where $\Delta = f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$ and $x_0$ is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to $\mathcal{O}(L^3 d^{3/2} \epsilon^{-3}+ \Delta L^2 d^{3/2} \delta^{-1} \epsilon^{-3})$.

We propose a communication and computation efficient second-order method for distributed optimization. For each iteration, our method only requires O (d) communication complexity, where d is the problem dimension. We also provide theoretical analysis to show the proposed method has the similar convergence rate as the classical second-order optimization algorithms. Concretely, our method can find (∈, √dLe,)-second-order stationary points for nonconvex problem by O (√dL,∈-3/2) iterations, where L is the Lipschitz constant of Hessian. Moreover, it enjoys a local superlinear convergence under the strongly-convex assumption. Experiments on both convex and nonconvex problems show that our proposed method performs significantly better than baselines.

This paper considers stochastic first-order algorithms for minimax optimization under Polyak-{\L}ojasiewicz (PL) conditions. We propose SPIDER-GDA for solving the finite-sum problem of the form $\min_x \max_y f(x,y)\triangleq \frac{1}{n} \sum_{i=1}^n f_i(x,y)$, where the objective function $f(x,y)$ is $\mu_x$-PL in $x$ and $\mu_y$-PL in $y$; and each $f_i(x,y)$ is $L$-smooth. We prove SPIDER-GDA could find an $\epsilon$-approximate solution within ${\mathcal O}\left((n + \sqrt{n}\,\kappa_x\kappa_y^2)\log (1/\epsilon)\right)$ stochastic first-order oracle (SFO) complexity, which is better than the state-of-the-art method whose SFO upper bound is ${\mathcal O}\big((n + n^{2/3}\kappa_x\kappa_y^2)\log (1/\epsilon)\big)$, where $\kappa_x\triangleq L/\mu_x$ and $\kappa_y\triangleq L/\mu_y$.For the ill-conditioned case, we provide an accelerated algorithm to reduce the computational cost further. It achieves $\tilde{{\mathcal O}}\big((n+\sqrt{n}\,\kappa_x\kappa_y)\log^2 (1/\epsilon)\big)$ SFO upper bound when $\kappa_x\geq\sqrt{n}$. Our ideas also can be applied to the more general setting that the objective function only satisfies PL condition for one variable. Numerical experiments validate the superiority of proposed methods.