Kronecker-Factored Approximation

Data-free preconditioning builds the Kronecker factors of KFAC from synthetic probes instead of private data. DP-KFC validates a probe by matching the eigenvalue decay of the private factors; we turn this criterion into a random-matrix account of what a probe can reconstruct. A single dichotomy organizes it: in the factorization $A \otimes G$, the error-signal factor $G$ and the architecture-determined bulk of the activation factor $A$ are reconstructible without data, while only the data-induced spikes of $A$ are not. We prove the $G$ half is data-insensitive up to an explicit error-energy bound, with zero violations over a full training trajectory across four modalities and a natural-image benchmark. We locate the failure boundary at a Baik-Ben Arous-Péché threshold: a bulk-only probe suffers an exact damage $s^2/(1-s^2)$ from a data-induced spike, which becomes invisible once it exceeds $(d/n)^{1/4}$, confirmed on an injected-spike surrogate. Holding the probe spectrum fixed and varying only its eigenbasis, we show that spectral-shape matching is neither sufficient nor the operative object: a probe with a near-exact matched spectrum (log-ratio dispersion $0.02$) still inflates the residual $2 \times$-$90 \times$ when its eigenvectors are random. A closed-form whitened-volume law explains this and predicts the within-spectrum ordering of the critical volume, though not its cross-dataset magnitude. Downstream, per-layer tuning does not transfer to accuracy and the headroom is a small oracle-minus-white gap. Under a privacy mechanism the source ordering survives while learning does, then inverts at a noise cross-over where the flat probe overtakes the matched ones.

Earnings call transcription presents a realistic continual learning scenario where Automatic Speech Recognition (ASR) systems are updated with new recordings but risk catastrophic forgetting of previously learned financial speech patterns. Adapting Whisper-small from a general-domain speech corpus to the Earnings22 benchmark reveals the opposite: all evaluated methods exhibit improvement on held-out financial speech from the same domain during adaptation rather than forgetting. This task complementarity makes regularization counterproductive. Nevertheless, when regularization is warranted, a layer-wise mini-Kronecker-Factored Approximate Curvature (mini-KFAC) approximation seems to achieve lower word error rates (WERs) and more stable optimization than standard diagonal Elastic Weight Consolidation (EWC). The results suggest that task complementarity should be assessed before applying continual learning constraints, and prefer mini-KFAC EWC over diagonal EWC when needed.

Training data attribution for diffusion models aims to identify the training samples that influence a generated instance, but existing methods either require costly per-sample gradient computation or query-specific model optimization. Moreover, most methods attribute changes in a proxy loss rather than changes in the actual model's generative behavior. We address these limitations by formulating attribution directly with a local score discrepancy measure, which applies to any diffusion variant (including DDPM, EDM, and flow matching), and by showing that such measure can be estimated without retraining, as a preconditioned gradient similarity. We instantiate this estimator as Training-data Influence via score Discrepancy (TID), which uses Kronecker-factored curvature to avoid random projections and per-sample gradient storage. We then distill TID into TIDE, a forward-only student trained online to reproduce the teacher's rankings from the diffusion model's internal activations. Under counterfactual evaluation on CIFAR-10, ArtBench-10, and MS-COCO, TID matches or outperforms state-of-the-art approaches, while TIDE retains most of TID's accuracy at four to five orders of magnitude lower per-query cost, attributing generated samples in milliseconds and faster than the generation itself.

Data attribution tools used in practice, TRAK, LoGRA/LogIX, and EK-FAC, run after training: they recompute per-sample gradients in a separate pass over the training set, and curvature-aware variants need a further pass to estimate covariance. Traceprop records projected per-sample gradients and K-FAC covariance statistics inside the training backward pass itself. On Pythia-1B LoRA fine-tunes on an A100, this adds 2.6% to training time, against 10.7% for LogIX's best inline configuration, random-init (4.1x lower, one-sided Mann-Whitney p = 9e-5, n = 10 per arm); at Pythia-6.9B the numbers are 11.3% and 27.0% (p = 0.004). LogIX's default configuration, PCA-init, is both slower and lower quality than random-init, so we compare against random-init throughout. On a small transformer where LDS is measurable, Traceprop matches LogIX's best configuration: pooled LDS difference +0.0022, 95% CI [-0.0038, +0.0075]. Two things do not work: at Pythia-160M/SST-2, every method is indistinguishable from a low noise ceiling, and on planted-backdoor and mislabel detection, gradient attribution does not beat gradient norm or representation similarity. The contribution here is systems, not a new estimator: LogIX's attribution quality, in one pass instead of two.

Spiking Neural Networks (SNNs) offer an energy-efficient alternative to conventional neural networks by exploiting sparse, binary spikes, and event-driven computation. However, the training of SNNs remains challenging, as spiking activations create a sharp loss landscape that hinders training, and diagonal-curvature optimizers such as the Adam family may fail to capture this geometry. The extension of curvature-based optimization methods to SNNs is further complicated by the sparse, discrete, and temporally recurrent nature of their underlying dynamics. To address these limitations, we propose SpiKFAX, a second-order optimization method that formulates a computationally tractable, Kronecker-factored approximation of the Fisher information matrix specifically adapted to the structure of SNNs. Empirical evaluation across five architectures and seven datasets demonstrates that SpiKFAX consistently yields improvements in test accuracy and training stability relative to other popular optimizers.

How should a Kronecker-factored Hessian approximation be chosen for post-training quantization? We address this question through WaterKron, which combines two-sided GPTQ with row- and column-dependent waterfilling scales and entropy coding. We derive its high-rate distortion with respect to the full Hessian using an explicit Kronecker-Hessian mismatch factor $Φ$. This factor quantifies the asymptotic distortion penalty due to the Kronecker Hessian approximation and provides a criterion for selecting the factors optimally. Minimizing $Φ$ leads to a Gaussian covariance-fitting problem with classical ``flip-flop'' updates. We thus give a rate-distortion justification for using the resulting FlipFlop Hessian in quantization. We evaluate it empirically, finding that the FlipFlop Hessian consistently improves KL divergence and perplexity over input-only, marginal, and Frobenius-based Hessian choices.

We accelerate a family of algorithms for neural network quantization whose geometry is informed by any Kronecker-factored approximation of the Hessian. GPTQ-style adaptive rounding typically uses one-sided information derived from input activations. Two-sided Kronecker-factored Hessian approximations can additionally capture correlations across output coordinates, but applying GPTQ directly in the vectorized weight domain is computationally expensive. Building on the two-sided adaptive-rounding formulation used by BoA and YAQA, we introduce BaKron, an efficient solver that combines anti-diagonal parallelism with a recursive divide-and-conquer construction. For an $m\times n$ weight matrix, BaKron uses $O(m+n)$ sequential steps while reducing the total work from $O(m^2n^2)$ to $O(mn(m+n))$. Thus, it matches the cubic scaling of GPTQ while exploiting richer curvature information. Moreover, BaKron is modular with respect to both the base quantizer and the Hessian estimator. We also provide practical benchmarks, consider a range of Hessians that BaKron can be called with, find an efficient technique to compute these Hessians, and evaluate the algorithm experimentally.

Shampoo-style optimizers approximate gradient covariance matrices using Kronecker-factored structures. Recent work~\cite{lin2026understanding} showed that such approximations can be viewed as projections under Bregman matrix divergences, leading to different Kronecker-factored preconditioners. However, it remains unclear what role the choice of divergence plays when the covariance is not exactly Kronecker-factored. We study this question through the spectrum of the covariance matrix. We show that Frobenius, von Neumann, and LogDet divergences distribute the unavoidable Kronecker approximation error differently across the covariance spectrum. We further show that their Kronecker factors are governed by divergence-weighted residuals rather than the raw approximation error, explaining how these spectral preferences are realized in the resulting preconditioners. Empirically, we observe that the top covariance eigenspace is substantially better aligned with the Hessian matrix, while the tail spectrum is much noisier and unreliable. Motivated by these findings, we propose a subspace-aware Kronecker optimizer that applies eigenvalue-based preconditioning in the top subspace and uses an adaptive isotropic acceleration constant in the bottom subspace.

Differentially private optimization suffers from a fundamental geometric mismatch: deep networks have highly anisotropic loss landscapes, yet DP-SGD injects isotropic noise. Second-order preconditioning can resolve this, but estimating curvature typically requires private data (consuming privacy budget) or public data (introducing distribution shift). We show that the Fisher Information Matrix decouples into architectural sensitivity, recoverable via synthetic noise, and input correlations, approximable from modality-specific frequency statistics. We propose DP-KFC, which constructs KFAC preconditioners by probing networks with structured synthetic noise, requiring neither private nor public data. Empirically, DP-KFC consistently outperforms DP-SGD and adaptive baselines across diverse modalities in strong privacy regimes ($\varepsilon \leq 3$). DP-KFC matches private-data preconditioners while public-data variants degrade by up to $4.8\%$, showing that curvature can be estimated without consuming privacy budget or introducing distribution shift. This enables privacy-preserving learning in specialized domains (e.g., medical applications) where regulatory constraints make data scarce.

Bilevel optimization (BO) is widely applicable to many machine learning problems. Scaling BO, however, requires repeatedly computing hypergradients, which involves solving inverse Hessian-vector products (IHVPs). In practice, these operations are often approximated using crude surrogates such as one-step gradient unrolling or identity/short Neumann expansions, which discard curvature information. We build on implicit function theorem-based algorithms and propose to incorporate Kronecker-factored approximate curvature (KFAC), yielding curvature-aware hypergradients with a better performance efficiency trade-off than Conjugate Gradient (CG) or Neumann methods and consistently outperforming unrolling. We evaluate this approach across diverse tasks, including meta-learning and AI safety problems. On models up to BERT, we show that curvature information is valuable at scale, and KFAC can provide it with only modest memory and runtime overhead. Our implementation is available at https://github.com/liaodisen/NeuralBo.

Online Continual Learning (OCL) for image classification represents a challenging subset of Continual Learning, focusing on classifying images from a stream without assuming data independence and identical distribution (i.i.d). The primary challenge in this context is to prevent catastrophic forgetting, where the model's performance on previous tasks deteriorates as it learns new ones. Although various strategies have been proposed to address this issue, achieving rapid convergence remains a significant challenge in the online setting. In this work, we introduce a novel approach to training OCL models that utilizes the Natural Gradient Descent optimizer, incorporating an approximation of the Fisher Information Matrix (FIM) through Kronecker Factored Approximate Curvature (KFAC). This method demonstrates substantial improvements in performance across all OCL methods, particularly when combined with existing OCL tricks, on datasets such as Split CIFAR-100, CORE50, and Split miniImageNet.

Task Arithmetic yields a modular, scalable way to adapt foundation models. Combining multiple task vectors, however, can lead to cross-task interference, causing representation drift and degraded performance. Representation drift regularization provides a natural remedy to disentangle task vectors; however, existing approaches typically require external task data, conflicting with modularity and data availability constraints (e.g., privacy requirements). We propose a dataless approach by framing regularization against representation drift as a curvature matrix approximation problem. This allows us to leverage well-established techniques; in particular, we adopt Kronecker-Factored Approximate Curvature and obtain a practical regularizer that achieves state-of-the-art results in task addition and negation. Our method has constant complexity in the number of tasks and promotes robustness to task vector rescaling, eliminating the need for held-out tuning.

Low-Rank Adaptation (LoRA) fine-tunes large models by learning low-rank updates on top of frozen weights, dramatically reducing trainable parameters and memory. In this work, we address the gap between training with full steps with low-rank projections (SVDLoRA) and LoRA fine-tuning. We propose LoRSum, a memory-efficient subroutine that closes this gap for gradient descent by casting LoRA optimization as a proximal sub-problem and solving it efficiently with alternating least squares updates, which we prove to be an implicit block power method. We recover several recently proposed preconditioning methods for LoRA as special cases, and show that LoRSum can also be used for updating a low-rank momentum. In order to address full steps with preconditioned gradient descent, we propose a scaled variant of LoRSum that uses structured metrics such as K-FAC and Shampoo, and we show that storing the diagonal of these metrics still allows them to perform well while remaining memory-efficient. Experiments on a synthetic task, CIFAR-100, and language-model fine-tuning on GLUE, SQuAD v2, and WikiText-103, show that our method can match or improve LoRA baselines given modest compute overhead, while avoiding full-matrix SVD projections and retaining LoRA-style parameter efficiency.

The Fisher information is a fundamental concept for characterizing the sensitivity of parameters in neural networks. However, leveraging the full observed Fisher information is too expensive for large models, so most methods rely on simple diagonal approximations. While efficient, this approach ignores parameter correlations, often resulting in reduced performance on downstream tasks. In this work, we mitigate these limitations and propose Generalized Fisher-Weighted SVD (GFWSVD), a post-training LLM compression technique that accounts for both diagonal and off-diagonal elements of the Fisher information matrix, providing a more accurate reflection of parameter importance. To make the method tractable, we introduce a scalable adaptation of the Kronecker-factored approximation algorithm for the observed Fisher information. We demonstrate the effectiveness of our method on LLM compression, showing improvements over existing compression baselines. For example, at a 20 compression rate on the MMLU benchmark, our method outperforms FWSVD, which is based on a diagonal approximation of the Fisher information, by 5 percent, SVD-LLM by 3 percent, and ASVD by 6 percent compression rate.