Ordering-based approaches to causal discovery identify topological orders of causal graphs, providing scalable alternatives to combinatorial search methods. Under the Additive Noise Model (ANM) assumption, recent causal ordering methods based on score matching require an accurate estimation of the Hessian diagonal of the log-densities. In this paper, we aim to improve the approximation of the Hessian diagonal of the log-densities, thereby enhancing the performance of ordering-based causal discovery algorithms. Existing approaches that rely on Stein gradient estimators are computationally expensive and memory-intensive, while diffusion-model-based methods remain unstable due to the second-order derivatives of score models. To alleviate these problems, we propose Score-informed Neural Operator (SciNO), a probabilistic generative model in smooth function spaces designed to stably approximate the Hessian diagonal and to preserve structural information during the score modeling. Empirical results show that SciNO reduces order divergence by 42.7% on synthetic graphs and by 31.5% on real-world datasets on average compared to DiffAN, while maintaining memory efficiency and scalability. Furthermore, we propose a probabilistic control algorithm for causal reasoning with autoregressive models that integrates SciNO's probability estimates with autoregressive model priors, enabling reliable data-driven causal ordering informed by semantic information. Consequently, the proposed method enhances causal reasoning abilities of LLMs without additional fine-tuning or prompt engineering.

This paper explores second-order optimization methods in Federated Learning (FL), addressing the critical challenges of slow convergence and the excessive communication rounds required to achieve optimal performance from the global model. While existing surveys in FL primarily focus on challenges related to statistical and device label heterogeneity, as well as privacy and security concerns in first-order FL methods, less attention has been given to the issue of slow model training. This slow training often leads to the need for excessive communication rounds or increased communication costs, particularly when data across clients are highly heterogeneous. In this paper, we examine various FL methods that leverage second-order optimization to accelerate the training process. We provide a comprehensive categorization of state-of-the-art second-order FL methods and compare their performance based on convergence speed, computational cost, memory usage, transmission overhead, and generalization of the global model. Our findings show the potential of incorporating Hessian curvature through second-order optimization into FL and highlight key challenges, such as the efficient utilization of Hessian and its inverse in FL. This work lays the groundwork for future research aimed at developing scalable and efficient federated optimization methods for improving the training of the global model in FL.

It is good that the authors study reasonably modern and well-known (AlexNet and VGG) deep nets, so that the compression ratios seem directly relevant to recent CNN work. However, the authors do not seem to compare their work to ANY of the now-large literative on related recent methods for compressing CNN and other deep nets. They dismiss OBD in Section 2 as unsuitable for "today's large scale neural networks", but that seems unfair since OBD's saliencies only require the Hessian diagonals, and several modern methods to approximate the required Hessian diagonals could be used for baselines. But even without using ODB-related methods, there are many other baselines they could of and should of reported for comparison, including methods that reduce weights to fewer bits or approximate the CNN filter matrices using cheap SVD-based compression, or ones which approximate the fully connected layers using randomized projection (e.g. The authors claim in Section 2 that such work is often "orthogonal to network pruning", but that seems to miss the point: once those methods are used, is there any real advantage to the proposed method?

Second-order information is valuable for many applications but challenging to compute. Several works focus on computing or approximating Hessian diagonals, but even this simplification introduces significant additional costs compared to computing a gradient. In the absence of efficient exact computation schemes for Hessian diagonals, we revisit an early approximation scheme proposed by Becker and LeCun (1989, BL89), which has a cost similar to gradients and appears to have been overlooked by the community. We introduce HesScale, an improvement over BL89, which adds negligible extra computation. On small networks, we find that this improvement is of higher quality than all alternatives, even those with theoretical guarantees, such as unbiasedness, while being much cheaper to compute. We use this insight in reinforcement learning problems where small networks are used and demonstrate HesScale in second-order optimization and scaling the step-size parameter. In our experiments, HesScale optimizes faster than existing methods and improves stability through step-size scaling. These findings are promising for scaling second-order methods in larger models in the future.

Catastrophic forgetting remains a challenge for neural networks, especially in lifelong learning scenarios. In this study, we introduce MEtaplasticity from Synaptic Uncertainty (MESU), inspired by metaplasticity and Bayesian inference principles. MESU harnesses synaptic uncertainty to retain information over time, with its update rule closely approximating the diagonal Newton's method for synaptic updates. Through continual learning experiments on permuted MNIST tasks, we demonstrate MESU's remarkable capability to maintain learning performance across 100 tasks without the need of explicit task boundaries.

Second-order optimization uses curvature information about the objective function, which can help in faster convergence. However, such methods typically require expensive computation of the Hessian matrix, preventing their usage in a scalable way. The absence of efficient ways of computation drove the most widely used methods to focus on first-order approximations that do not capture the curvature information. In this paper, we develop HesScale, a scalable approach to approximating the diagonal of the Hessian matrix, to incorporate second-order information in a computationally efficient manner. We show that HesScale has the same computational complexity as backpropagation. Our results on supervised classification show that HesScale achieves high approximation accuracy, allowing for scalable and efficient second-order optimization. First-order optimization offers a cheap and efficient way of performing local progress in optimization problems by using gradient information. However, their performance suffers from instability or slow progress when used in ill-conditioned landscapes. Such a problem is present because firstorder methods do not capture curvature information which causes two interrelated issues. First, the updates in first-order have incorrect units (Duchi et al. 2011), which creates a scaling issue.

We introduce AdaHessian, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the Hessian. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and ADAM. The main disadvantage of traditional second order methods is their heavier per-iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in AdaHessian, including: (i) a new variance reduction estimate of the Hessian diagonal with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that AdaHessian achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of ADAM. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that AdaHessian: (i) achieves 1.80\%/1.45\% higher accuracy on ResNets20/32 on Cifar10, and 5.55\% higher accuracy on ImageNet as compared to ADAM; (ii) outperforms ADAMW for transformers by 0.27/0.33 BLEU score on IWSLT14/WMT14 and 1.8/1.0 PPL on PTB/Wikitext-103; and (iii) achieves 0.032\% better score than AdaGrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of AdaHessian is comparable to first-order methods, and that it exhibits robustness towards its hyperparameters. The code for AdaHessian is open-sourced and publicly available.