Thus a gap has arisen between theory and practice.

Nonsmooth nonconvex optimization problems broadly emerge in machine learning and business decision making, whereas two core challenges impede the development of efficient solution methods with finite-time convergence guarantee: the lack of computationally tractable optimality criterion and the lack of computationally powerful oracles. The contributions of this paper are two-fold.

Nonsmooth nonconvex optimization problems broadly emerge in machine learning and business decision making, whereas two core challenges impede the development of efficient solution methods with finite-time convergence guarantee: the lack of computationally tractable optimality criterion and the lack of computationally powerful oracles. The contributions of this paper are two-fold. Second, we propose the gradient-free method (GFM) and stochastic GFM for solving a class of nonsmooth nonconvex optimization problems and prove that both of them can return a (\delta,\epsilon) -Goldstein stationary point of a Lipschitz function f at an expected convergence rate at O(d {3/2}\delta {-1}\epsilon {-4}) where d is the problem dimension. Two-phase versions of GFM and SGFM are also proposed and proven to achieve improved large-deviation results. Finally, we demonstrate the effectiveness of 2-SGFM on training ReLU neural networks with the \textsc{Minst} dataset.

The stochastic compositional optimization (SCO) is popular in many real-world applications, including risk management, reinforcement learning, and meta-learning. However, most of the previous methods for SCO require the smoothness assumption on both the outer and inner functions, which limits their applications to a wider range of problems. In this paper, we study the SCO problem in that both the outer and inner functions are Lipschitz continuous but possibly nonconvex and nonsmooth. In particular, we propose gradient-free stochastic methods for finding the (\delta, \epsilon) -Goldstein stationary points of such problems with non-asymptotic convergence rates. Our results also lead to an improved convergence rate for the convex nonsmooth SCO problem. Furthermore, we conduct numerical experiments to demonstrate the effectiveness of the proposed methods.

Nonsmooth nonconvex optimization problems broadly emerge in machine learning and business decision making, whereas two core challenges impede the development of efficient solution methods with finite-time convergence guarantee: the lack of computationally tractable optimality criterion and the lack of computationally powerful oracles. The contributions of this paper are two-fold. Second, we propose the gradient-free method (GFM) and stochastic GFM for solving a class of nonsmooth nonconvex optimization problems and prove that both of them can return a (\delta,\epsilon) -Goldstein stationary point of a Lipschitz function f at an expected convergence rate at O(d {3/2}\delta {-1}\epsilon {-4}) where d is the problem dimension. Two-phase versions of GFM and SGFM are also proposed and proven to achieve improved large-deviation results. Finally, we demonstrate the effectiveness of 2-SGFM on training ReLU neural networks with the \textsc{Minst} dataset.

Zeroth-order (ZO) method has been shown to be a powerful method for solving the optimization problem where explicit expression of the gradients is difficult or infeasible to obtain. Recently, due to the practical value of the constrained problems, a lot of ZO Frank-Wolfe or projected ZO methods have been proposed. However, in many applications, we may have a very large number of nonconvex white/black-box constraints, which makes the existing zeroth-order methods extremely inefficient (or even not working) since they need to inquire function value of all the constraints and project the solution to the complicated feasible set. In this paper, to solve the nonconvex problem with a large number of white/black-box constraints, we proposed a doubly stochastic zeroth-order gradient method (DSZOG) with momentum method and adaptive step size. Theoretically, we prove DSZOG can converge to the $\epsilon$-stationary point of the constrained problem. Experimental results in two applications demonstrate the superiority of our method in terms of training time and accuracy compared with other ZO methods for the constrained problem.

As Large Language Models (LLMs) grow dramatically in size, there is an increasing trend in compressing and speeding up these models. Previous studies have highlighted the usefulness of gradients for importance scoring in neural network compressing, especially in pruning medium-size networks. However, the substantial memory requirements involved in calculating gradients with backpropagation impede the utilization of gradients in guiding LLM pruning. As a result, most pruning strategies for LLMs rely on gradient-free criteria, such as weight magnitudes or a mix of magnitudes and activations. In this paper, we devise a hybrid pruning criterion, which appropriately integrates magnitude, activation, and gradient to capitalize on feature map sensitivity for pruning LLMs. To overcome memory requirement barriers, we estimate gradients using only forward passes. Based on this, we propose a Memory-effIcieNt structured prunIng procedure for LLMs (MINI-LLM) to remove no-critical channels and multi-attention heads. Experimental results demonstrate the superior performance of MINI-LLM over existing gradient-free methods on three LLMs: LLaMA, BLOOM, and OPT across various downstream tasks (classification, multiple-choice, and generation), while MINI-LLM maintains a GPU memory footprint akin to gradient-free methods.

Parameter-efficient tuning methods such as LoRA could achieve comparable performance to model tuning by tuning a small portion of the parameters. However, substantial computational resources are still required, as this process involves calculating gradients and performing back-propagation throughout the model. Much effort has recently been devoted to utilizing the derivative-free optimization method to eschew the computation of gradients and showcase an augmented level of robustness in few-shot settings. In this paper, we prepend the low-rank modules into each self-attention layer of the model and employ two derivative-free optimization methods to optimize these low-rank modules at each layer alternately. Extensive results on various tasks and language models demonstrate that our proposed method achieves substantial improvement and exhibits clear advantages in memory usage and convergence speed compared to existing gradient-based parameter-efficient tuning and derivative-free optimization methods in few-shot settings.

Linearized Graph Neural Networks (GNNs) have attracted great attention in recent years for graph representation learning. Compared with nonlinear Graph Neural Network (GNN) models, linearized GNNs are much more time-efficient and can achieve comparable performances on typical downstream tasks such as node classification. Although some linearized GNN variants are purposely crafted to mitigate ``over-smoothing", empirical studies demonstrate that they still somehow suffer from this issue. In this paper, we instead relate over-smoothing with the vanishing gradient phenomenon and craft a gradient-free training framework to achieve more efficient and effective linearized GNNs which can significantly overcome over-smoothing and enhance the generalization of the model. The experimental results demonstrate that our methods achieve better and more stable performances on node classification tasks with varying depths and cost much less training time.