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 that achieves the optimal regret $Ω(1+V_T)$ 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 sublinear environmental variation through a window-based analysis and achieves the optimal regret $Ω(T/W^2)$. Experiments validate our theoretical findings and demonstrate the practical effectiveness of the proposed methods.

Decentralized Stochastic Nonconvex Optimization under the Relaxed Smoothness

This paper studies decentralized optimization problem $f(\mathbf{x})=\frac{1}{m}\sum_{i=1}^m f_i(\mathbf{x})$, where each local function has the form of $f_i(\mathbf{x}) = {\mathbb E}\left[F(\mathbf{x};{\xi}_i)\right]$ which is $(L_0,L_1)$-smooth but possibly nonconvex and the random variable ${\xi}_i$ follows distribution ${\mathcal D}_i$. We propose a novel algorithm called decentralized normalized stochastic gradient descent (DNSGD), which can achieve the $\epsilon$-stationary point on each local agent. We present a new framework for analyzing decentralized first-order methods in the relaxed smooth setting, based on the Lyapunov function related to the product of the gradient norm and the consensus error. The analysis shows upper bounds on 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 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 on decentralized nonconvex optimization in the standard smooth setting. We also conduct numerical experiments to show the empirical superiority of our method.