We introduce LOREN, a curvature-aware zeroth-order (ZO) optimization method for fine-tuning large language models (LLMs). Existing ZO methods, which estimate gradients via finite differences using random perturbations, often suffer from high variance and suboptimal search directions. Our approach addresses these challenges by: (i) reformulating the problem of gradient preconditioning as that of adaptively estimating an anisotropic perturbation distribution for gradient estimation, (ii) capturing curvature through a low-rank block diagonal preconditioner using the framework of natural evolution strategies, and (iii) applying a REINFORCE leave-one-out (RLOO) gradient estimator to reduce variance. Experiments on standard LLM benchmarks show that our method outperforms state-of-the-art ZO methods by achieving higher accuracy and faster convergence, while cutting peak memory usage by up to 27.3% compared with MeZO-Adam.

Abstract Second-order methods can accelerate deep neural network training, but their adoption is limited by the cost and instability of estimating and inverting curvature matrices. We revisit Kronecker-factored Fisher approximations via an empirical structural analysis of activation and gradient statistics in modern architectures. Across a range of models, we find that activation statistics capture most of the effective curvature directions, while gradient statistics mainly act as a global or diagonal rescaling. Based on this observation, we propose mean activation curvature , a scalable curvature surrogate that yields two optimizers, MAC and SMAC , offering different trade-offs between expressiveness and efficiency. We further extend the construction to self-attention layers with a structured approximation that retains the role of attention scores, and provide a convergence analysis under standard assumptions. Experiments on vision and language models show that MAC / SMAC matches or improves the accuracy of existing second-order baselines while reducing training time and memory usage. Code: github.com/hseung88/mac .

Second-order optimization methods for training neural networks, such as KFAC, exhibit superior convergence by utilizing curvature information of loss landscape. However, it comes at the expense of high computational burden. In this work, we analyze the two components that constitute the layer-wise Fisher information matrix (FIM) used in KFAC: the Kronecker factors related to activations and pre-activation gradients. Based on empirical observations on their eigenspectra, we propose efficient approximations for them, resulting in a computationally efficient optimization method called MAC. To the best of our knowledge, MAC is the first algorithm to apply the Kronecker factorization to the FIM of attention layers used in transformers and explicitly integrate attention scores into the preconditioning. We also study the convergence property of MAC on nonlinear neural networks and provide two conditions under which it converges to global minima. Our extensive evaluations on various network architectures and datasets show that the proposed method outperforms KFAC and other state-of-the-art methods in terms of accuracy, end-to-end training time, and memory usage.

Adaptive gradient methods are computationally efficient and converge quickly, but they often suffer from poor generalization. In contrast, second-order methods enhance convergence and generalization but typically incur high computational and memory costs. In this work, we introduce NYSACT, a scalable first-order gradient preconditioning method that strikes a balance between state-of-the-art first-order and second-order optimization methods. NYSACT leverages an eigenvalue-shifted Nyström method to approximate the activation covariance matrix, which is used as a preconditioning matrix, significantly reducing time and memory complexities with minimal impact on test accuracy. Our experiments show that NYSACT not only achieves improved test accuracy compared to both first-order and second-order methods but also demands considerably less computational resources than existing second-order methods.