arXiv subjects

Lesi Chen

Publications and source records attributed to Lesi Chen.

18 recordsLinked to original sources

arXiv2026

Halpern Iteration Achieves $\tilde{\mathcal{O}}(\epsilon^{-1/p})$ $p$th-Order Oracle Complexity for Monotone Variational Inequalities

We study second- and higher-order methods for solving smooth monotone variational inequalities (MVI). Monteiro and Svaiter (SIAM J. Optim., 2012) showed that a second-order method, NPE, converges at the rate of $\mathcal{O}(T^{-1.5})$. For convex-concave minimax optimization, a subset of MVI problems, Chen, Liu, Luo, and Zhang (COLT 2025) recently improved the complexity to $\tilde{\mathcal{O}}( T^{-1.75})$ . However, it is open whether the conjectured complexity for MVI can be improved. In this paper, by using a large-step inexact Halpern iteration, we propose a novel Halpern-NPE method that achieves an even faster rate of $\tilde{\mathcal{O}}(T^{-2})$ for solving MVIs. We also provide the $p$th-order generalization of our method. We first introduce an Anchored Tensor Method (ATM) that achieves the rate of $\mathcal{O}(T^{-(p-1)})$, and then combine it with the Halpern iteration to achieve a faster convergence rate of $\tilde{\mathcal{O}}(T^{-p})$. This improves all prior results for $p \ge 2$ and matches the classical extragradient method for $p=1$.

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arXiv2026

Optimal Convex Optimization with Inexact Second-Order Oracles

In this paper, we present a novel second-order method called Accelerated Inexact Newton Extragradient (AINE) for convex optimization using $\delta$-inexact Hessians. We show that AINE can find an $\epsilon$-solution in the inexact second-order oracle (ISO) complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_2/\epsilon)^{2/7} )$ when the Hessian is $L_2$-Lipschitz continuous, and a better complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_3/\epsilon)^{1/5} )$ when the third-order derivative is $L_3$-Lipschitz continuous. Notably, each iteration of our method can be conducted in the same running time as matrix multiplication up to logarithmic factors. In addition, we also establish matching oracle complexity lower bounds for both setups, demonstrating the optimality of our methods.

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arXiv2026

Second-Order Bilevel Optimization with Accelerated Convergence Rates

This paper studies second-order methods for nonconvex-strongly-convex bilevel optimization. We propose a novel fully second-order bilevel approximation method (FSBA) that achieves an iteration complexity of $\tilde{\mathcal{O}}(\epsilon^{-1.5})$ for finding the $(\epsilon, \mathcal{O}(\sqrt{\epsilon}))$ second-order stationary point of the hyper-objective function. Our results demonstrate that second-order methods can achieve an accelerated convergence rate than first-order methods in bilevel optimization. To address the heavy computational cost associated with the second-order oracle, we introduce a lazy variant of FSBA, called LFSBA, which reuses second-order information across several iterations. We prove that LFSBA exhibits better computational complexity than FSBA by a factor of $\sqrt{d}$, where $d$ is the dimension of the problem. We also apply a similar idea to nonconvex strongly-concave minimax optimization and propose the lazy minimax cubic-regularized Newton (LMCN) method with better computational complexity compared to existing second-order methods.

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arXiv2026

Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(\epsilon^{-4/(3p+1)})$ $p$th-Order Oracle Complexity

When the objective has Lipschitz continuous $p$th-order derivatives, it is known that convex-concave minimax problems can be solved with $\mathcal{O}(\epsilon^{-2/(p+1)})$ $p$th-order oracle calls. This complexity upper bound was speculated to be optimal as it is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(\epsilon^{-4/(3p+1)})$ by applying the Monteiro-Svaiter acceleration. We also establish a lower complexity bound of $\Omega(\epsilon^{-2/(3p-1)})$, suggesting a gap still exists for $p \ge 2$.

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arXiv2025

On the Condition Number Dependency in Bilevel Optimization

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 $\epsilon$-stationary point with first-order methods when the upper-level problem is nonconvex, and the lower-level problem is strongly convex. Recent works achieve a $\tilde{\mathcal{O}}(\bar \kappa_y^{7/2} \epsilon^{-2})$ upper bound that is near-optimal in $\epsilon$. In this work, we establish a new $\Omega(\kappa_y^{5/2} \epsilon^{-2})$ lower bound, where $\kappa_y \le \bar \kappa_y$ is the lower-level condition number. Our lower bound establishes the first provable gap {in terms of condition number dependency} between bilevel problems and minimax problems in this setup, and \textit{is tight up to logarithmic factors when the lower-level function is quadratic.} Our lower bounds can be extended to various settings. (1) For second-order and arbitrarily smooth problems, we show lower bounds of $\Omega(\kappa_y^{9/4} \epsilon^{-7/4})$ and $\Omega(\kappa_y^{13/6} \epsilon^{-5/3})$, respectively. (2) For convex--strongly-convex problems, we improve the previously best lower bound (Ji and Liang, JMLR 2022) from $\Omega(\kappa_y /\sqrt{\epsilon})$ to $\Omega(\kappa_y^{3/2} / \sqrt{\epsilon})$. (3) For stochastic nonconvex--strongly-convex problems, we also show the lower bounds of $\Omega(\kappa_y^4 \epsilon^{-4})$ and $\Omega(\kappa_y^{9/2} \epsilon^{-4})$ for stochastic Hessian-vector-product and stochastic first-order methods, respectively.

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arXiv2025

Faster Gradient Methods for Highly-Smooth Stochastic Bilevel Optimization

This paper studies the complexity of finding an $\epsilon$-stationary point for stochastic bilevel optimization when the upper-level problem is nonconvex and the lower-level problem is strongly convex. Recent work proposed the first-order method, F${}^2$SA, achieving the $\tilde{\mathcal{O}}(\epsilon^{-6})$ upper complexity bound for first-order smooth problems. This is slower than the optimal $\Omega(\epsilon^{-4})$ complexity lower bound in its single-level counterpart. In this work, we show that faster rates are achievable for higher-order smooth problems. We first reformulate F$^2$SA as approximating the hyper-gradient with a forward difference. Based on this observation, we propose a class of methods F${}^2$SA-$p$ that uses $p$th-order finite difference for hyper-gradient approximation and improves the upper bound to $\tilde{\mathcal{O}}(p \epsilon^{-4-p/2})$ for $p$th-order smooth problems. Finally, we demonstrate that the $\Omega(\epsilon^{-4})$ lower bound also holds for stochastic bilevel problems when the high-order smoothness holds for the lower-level variable, indicating that the upper bound of F${}^2$SA-$p$ is nearly optimal in the highly smooth region $p = \Omega( \log \epsilon^{-1} / \log \log \epsilon^{-1})$.

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arXiv2025

Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ Second-Order Oracle Complexity

Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(\epsilon^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been speculated to be optimal because the upper bound is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ by generalizing the optimal second-order method for convex optimization to solve the convex-concave minimax problem. We further apply a similar technique to lazy Hessian algorithms and show that our proposed algorithm can also be seen as a second-order ``Catalyst'' framework (Lin et al., JMLR 2018) that could accelerate any globally convergent algorithms for solving minimax problems.

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arXiv2025

Faster Newton Methods for Convex and Nonconvex Optimization in Gradient Complexity

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.

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arXiv2024

Second-Order Min-Max Optimization with Lazy Hessians

This paper studies second-order methods for convex-concave minimax optimization. Monteiro and 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.

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arXiv2024

Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems

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.

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arXiv2023

Faster Stochastic Algorithms for Minimax Optimization under Polyak--{\L}ojasiewicz Conditions

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 $\epsilon$-optimal 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 $\tilde{{\mathcal O}}\big((n+\sqrt{n}\,\kappa_x\kappa_y)\log (\kappa_y/\epsilon) \log(1/\epsilon)\big)$ SFO upper bound when $\kappa_y \gtrsim \sqrt{n}$. Our ideas can also be applied to a more general setting where the objective function only satisfies the PL condition for one variable. Numerical experiments validate the superiority of proposed methods.

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arXiv2023

Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles

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 ${\mathcal{O}}(\epsilon^{-2})$ 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 $\tilde{\mathcal{O}}(\epsilon^{-3})$. In this paper, we incorporate a two-time-scale update to improve their method to achieve the near-optimal $\tilde {\mathcal{O}}(\epsilon^{-2})$ first-order oracle complexity. Our analysis is highly extensible. In the stochastic setting, our algorithm can achieve the stochastic first-order oracle complexity of $\tilde {\mathcal{O}}(\epsilon^{-4})$ and $\tilde {\mathcal{O}}(\epsilon^{-6})$ 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 $\tilde {\mathcal{O}}(\epsilon^{-1.75})$ using Nesterov's momentum.

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arXiv2023

Communication Efficient Distributed Newton Method with Fast Convergence Rates

We propose a communication and computation efficient second-order method for distributed optimization. For each iteration, our method only requires $\mathcal{O}(d)$ 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~$\big(ε, \sqrt{dLε}\,\big)$-second-order stationary points for nonconvex problem by $\mathcal{O}\big(\sqrt{dL}\,ε^{-3/2}\big)$ 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.

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arXiv2023

Faster Gradient-Free Algorithms for Nonsmooth Nonconvex Stochastic Optimization

We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}_{\xi} [F(x; \xi)]$, 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 $\mathcal{O}(L^3 d^{3/2} \epsilon^{-3}+ \Delta L^2 d^{3/2} \delta^{-1} \epsilon^{-3})$.

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arXiv2023

On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis

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 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-{\L}ojasiewicz (PL) condition. We show a simple first-order algorithm can achieve better 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. The complexities in the first two cases are optimal up to logarithmic factors.

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arXiv2022

An Efficient Stochastic Algorithm for Decentralized Nonconvex-Strongly-Concave Minimax Optimization

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.

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arXiv2022

On the Complexity of Decentralized Smooth Nonconvex Finite-Sum Optimization

We study the decentralized optimization problem $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{m}\sum_{i=1}^m f_i({\bf x})$, where the local function on the $i$-th agent has the form of $f_i({\bf x})\triangleq \frac{1}{n}\sum_{j=1}^n f_{i,j}({\bf x})$ and every individual $f_{i,j}$ is smooth but possibly nonconvex. We propose a stochastic algorithm called DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST) method, which achieves an $\epsilon$-stationary point at each agent with the communication rounds of $\tilde{\mathcal O}(L\epsilon^{-2}/\sqrt{\gamma}\,)$, the computation rounds of $\tilde{\mathcal O}(n+(L+\min\{nL, \sqrt{n/m}\bar L\})\epsilon^{-2})$, and the local incremental first-oracle calls of ${\mathcal O}(mn + {\min\{mnL, \sqrt{mn}\bar L\}}{\epsilon^{-2}})$, where $L$ is the smoothness parameter of the objective function, $\bar L$ is the mean-squared smoothness parameter of all individual functions, and $\gamma$ is the spectral gap of the mixing matrix associated with the network. We then establish the lower bounds to show that the proposed method is near-optimal. Notice that the smoothness parameters $L$ and $\bar L$ used in our algorithm design and analysis are global, leading to sharper complexity bounds than existing results that depend on the local smoothness. We further extend DEAREST to solve the decentralized finite-sum optimization problem under the Polyak-{\L}ojasiewicz condition, also achieving the near-optimal complexity bounds.

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arXiv2022

Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization

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.

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