Trilevel learning (TLL) with zeroth order constraints is a fundamental problem in machine learning, arising in scenarios where gradient information is inaccessible due to data privacy or model opacity, such as in federated learning, healthcare, and financial systems. These problems are notoriously difficult to solve due to their inherent complexity and the lack of first order information. Moreover, in many practical scenarios, data may be distributed across various nodes, necessitating strategies to address trilevel learning problems without centralizing data on servers to uphold data privacy. To this end, an effective distributed trilevel zeroth order learning framework DTZO is proposed in this work to address the trilevel learning problems with level-wise zeroth order constraints in a distributed manner. The proposed DTZO is versatile and can be adapted to a wide range of (grey-box) trilevel learning problems with partial zeroth order constraints. In DTZO, the cascaded polynomial approximation can be constructed without relying on gradients or sub-gradients, leveraging a novel cut, i.e., zeroth order cut. Furthermore, we theoretically carry out the non-asymptotic convergence rate analysis for the proposed DTZO in achieving the $\epsilon$-stationary point. Extensive experiments have been conducted to demonstrate and validate the superior performance of the proposed DTZO.

Trilevel learning, also called trilevel optimization (TLO), has been recognized as a powerful modelling tool for hierarchical decision process and widely applied in many machine learning applications, such as robust neural architecture search, hyperparameter optimization, and domain adaptation. Tackling TLO problems has presented a great challenge due to their nested decision-making structure. In addition, existing works on TLO face the following key challenges: 1) they all focus on the non-distributed setting, which may lead to privacy breach; 2) they do not offer any non-asymptotic convergence analysis which characterizes how fast an algorithm converges. To address the aforementioned challenges, this paper proposes an asynchronous federated trilevel optimization method to solve TLO problems. The proposed method utilizes u-cuts to construct a hyper-polyhedral approximation for the TLO problem and solve it in an asynchronous manner. We demonstrate that the proposed u-cuts are applicable to not only convex functions but also a wide range of non-convex functions that meet the u-weakly convex assumption. Furthermore, we theoretically analyze the non-asymptotic convergence rate for the proposed method by showing its iteration complexity to obtain ϵ-stationary point is upper bounded by O(1/ϵ²). Extensive experiments on real-world datasets have been conducted to elucidate the superiority of the proposed method, e.g., it has a faster convergence rate with a maximum acceleration of approximately 80%.

Out-of-Distribution (OOD) generalization in machine learning is a burgeoning area of study. Its primary goal is to enhance the adaptability and resilience of machine learning models when faced with new, unseen, and potentially adversarial data that significantly diverges from their original training datasets. In this paper, we investigate time series OOD generalization via pre-trained Large Language Models (LLMs). We first propose a novel \textbf{T}ri-level learning framework for \textbf{T}ime \textbf{S}eries \textbf{O}OD generalization, termed TTSO, which considers both sample-level and group-level uncertainties. This formula offers a fresh theoretic perspective for formulating and analyzing OOD generalization problem. In addition, we provide a theoretical analysis to justify this method is well motivated. We then develop a stratified localization algorithm tailored for this tri-level optimization problem, theoretically demonstrating the guaranteed convergence of the proposed algorithm. Our analysis also reveals that the iteration complexity to obtain an $\epsilon$-stationary point is bounded by O($\frac{1}{\epsilon^{2}}$). Extensive experiments on real-world datasets have been conducted to elucidate the effectiveness of the proposed method.