arXiv subjects

Tingkai Jia

Publications and source records attributed to Tingkai Jia.

3 recordsLinked to original sources

arXiv2026

Fully First-Order Algorithms for Online Bilevel Optimization

In this work, we study nonconvex-strongly convex online bilevel optimization (OBO) using only first-order oracle. Existing OBO algorithms are mainly based on hypergradient descent, which requires access to a Hessian-vector product (HVP) oracle and potentially incurs high computational costs. By reformulating the original OBO problem as a single-level online problem with inequality constraints and constructing a sequence of Lagrangian function, we eliminate the need for HVPs arising from implicit differentiation. Specifically, we propose a fully first-order algorithm for OBO, and provide theoretical guarantees showing that it achieves regret of $O(1 + V_T + H_{2,T})$ with a total of $O(T\log T)$ iterations, where $V_T$ measures the variation in function values and $H_{2,T}$ characterizes the drift variation of the inner-level optimal solution. We also establish a sublinear regret bound under the single-loop structure by introducing additional gradient-variation terms. Furthermore, we develop an improved variant with an adaptive inner-iteration scheme, which removes the dependence on $H_{2,T}$ and achieves regret of $O(\log T + V_T)$. Finally, under the stochastic OBO setting, we establish the regret bound for the fully first-order algorithm, i.e., $O(T^{2/3}(1 + \sigma^2) + V_T + H_{2,T})$. Numerical experiments demonstrate the feasibility of our algorithm and support our theoretical findings.

cs.LG↗

arXiv2026

Achieving Better Local Regret Bound for Online Non-Convex Bilevel Optimization

Online bilevel optimization (OBO) has emerged as a powerful framework for many machine learning problems. Prior works have developed several algorithms that minimize the standard bilevel local regret or the window-averaged bilevel local regret of the OBO problem, but the optimality of existing regret bounds remains unclear. In this work, we establish optimal regret bounds for both settings. For standard bilevel local regret, we propose an algorithm with adaptive iteration strategy that achieves the optimal regret $\Omega(1+V_T)$ with at most $O(T\log T)$ total inner-level gradient evaluations. We further develop a fully single-loop algorithm whose regret bound includes an additional gradient-variation terms. For the window-averaged bilevel local regret, we design an algorithm that captures linear environmental variation through a novel window-based analysis and achieves the optimal regret $\Omega(T/W^2)$. The algorithm also supports an efficient single-loop structure, achieving an $O(T/W)$ regret bound with $O(WT)$ total gradient evaluations. Experiments validate our theoretical findings and demonstrate the practical effectiveness of the proposed methods.

cs.LG↗

arXiv2025

Decentralized Stochastic Nonconvex Optimization under the $(L_0,L_1)$-Smoothness

This paper focuses on the decentralized stochastic optimization problem $f(\mathbf{x})=\frac{1}{m}\sum_{i=1}^m f_i(\mathbf{x})$ over a connected network of $n$ agents, where each local function has the form of $f_i(\mathbf{x}) = {\mathbb E}\left[F(\mathbf{x};{\boldsymbol \xi}_i)\right]$ which satisfies the $(L_0,L_1)$-smooth condition but possibly nonconvex and each random variable ${\boldsymbol \xi}_i$ follows distribution ${\mathcal D}_i$. We propose a novel algorithm called decentralized normalized stochastic gradient descent (DNSGD), which can achieve an $\epsilon$-stationary point at each local agent. We present a new framework for analyzing decentralized first-order methods in the $(L_0,L_1)$-smooth setting, based on the Lyapunov function related to the product of the gradient norm and the consensus error. We show that the proposed algorithm attains the upper bounds on the sample complexity of ${\mathcal O}(m^{-1}(L_f\sigma^2\Delta_f\epsilon^{-4} + \sigma^2\epsilon^{-2} + L_f^{-2}L_1^3\sigma^2\Delta_f\epsilon^{-1} + L_f^{-2}L_1^2\sigma^2))$ per agent and the communication complexity of $\tilde{\mathcal O}((L_f\epsilon^{-2} + L_1\epsilon^{-1})\gamma^{-1/2}\Delta_f)$, where $L_f=L_0 +L_1\zeta$, $\sigma^2$ is the variance of the stochastic gradient, $\Delta_f$ is the initial optimal function value gap, $\gamma$ is the spectral gap of the network, and $\zeta$ is the degree of the gradient dissimilarity. In the special case of $L_1=0$, the above results (nearly) match the lower bounds of decentralized stochastic nonconvex optimization under the standard smoothness. We also conduct numerical experiments to show the empirical superiority of our method.

math.OC↗