GrokAlign: Geometric Characterisation and Acceleration of Grokking

A key challenge for the machine learning community is to understand and accelerate the training dynamics of deep networks that lead to delayed generalisation and emergent robustness to input perturbations, also known as grokking. Prior work has associated phenomena like delayed generalisation with the transition of a deep network from a linear to a feature learning regime, and emergent robustness with changes to the network's functional geometry, in particular the arrangement of the so-called linear regions in deep networks employing continuous piecewise affine nonlinearities. Here, we explain how grokking is realised in the Jacobian of a deep network and demonstrate that aligning a network's Jacobians with the training data (in the sense of cosine similarity) ensures grokking under a low-rank Jacobian assumption. Our results provide a strong theoretical motivation for the use of Jacobian regularisation in optimizing deep networks -- a method we introduce as GrokAlign -- which we show empirically to induce grokking much sooner than more conventional regularizers like weight decay. Moreover, we introduce centroid alignment as a tractable and interpretable simplification of Jacobian alignment that effectively identifies and tracks the stages of deep network training dynamics. Accompanying \href{https://thomaswalker1.github.io/blog/grokalign.html}{webpage} and \href{https://github.com/ThomasWalker1/grokalign}{code}.

LoRA Users Beware: A Few Spurious Tokens Can Manipulate Your Finetuned Model

Parameter Efficient FineTuning (PEFT), such as Low-Rank Adaptation (LoRA), aligns pre-trained Large Language Models (LLMs) to particular downstream tasks in a resource-efficient manner. Because efficiency has been the main metric of progress, very little attention has been put in understanding possible catastrophic failures. We uncover one such failure: PEFT encourages a model to search for shortcut solutions to solve its fine-tuning tasks. When very small amount of tokens, e.g., one token per prompt, are correlated with downstream task classes, PEFT makes any pretrained model rely predominantly on that token for decision making. While such spurious tokens may emerge accidentally from incorrect data cleaning, it also opens opportunities for malevolent parties to control a model's behavior from Seamless Spurious Token Injection (SSTI). In SSTI, a small amount of tokens correlated with downstream classes are injected by the dataset creators. At test time, the finetuned LLM's behavior can be controlled solely by injecting those few tokens. We apply SSTI across models from three families (Snowflake Arctic, Apple OpenELM, and Meta LLaMA-3) and four diverse datasets (IMDB, Financial Classification, CommonSense QA, and Bias in Bios). Our findings reveal three astonishing behaviors. First, as few as a single token of SSTI is sufficient to steer a model's decision making. Second, for light SSTI, the reliance on spurious tokens is proportional to the LoRA rank. Lastly, with aggressive SSTI, larger LoRA rank values become preferable to small rank values as it makes the model attend to non-spurious tokens, hence improving robustness.

Next Token Perception Score: Analytical Assessment of your LLM Perception Skills

Autoregressive pretraining has become the de facto paradigm for learning general-purpose representations in large language models (LLMs). However, linear probe performance across downstream perception tasks shows substantial variability, suggesting that features optimized for next-token prediction do not consistently transfer well to downstream perception tasks. We demonstrate that representations learned via autoregression capture features that may lie outside the subspaces most informative for perception. To quantify the (mis)alignment between autoregressive pretraining and downstream perception, we introduce the Next Token Perception Score (NTPS)-a score derived under a linear setting that measures the overlap between autoregressive and perception feature subspaces. This metric can be easily computed in closed form from pretrained representations and labeled data, and is proven to both upper- and lower-bound the excess loss. Empirically, we show that NTPS correlates strongly with linear probe accuracy across 12 diverse NLP datasets and eight pretrained models ranging from 270M to 8B parameters, confirming its utility as a measure of alignment. Furthermore, we show that NTPS increases following low-rank adaptation (LoRA) fine-tuning, especially in large models, suggesting that LoRA aligning representations to perception tasks enhances subspace overlap and thus improves downstream performance. More importantly, we find that NTPS reliably predicts the additional accuracy gains attained by LoRA finetuning thereby providing a lightweight prescreening tool for LoRA adaptation. Our results offer both theoretical insights and practical tools for analytically assessing LLM perception skills.

Ditch the Denoiser: Emergence of Noise Robustness in Self-Supervised Learning from Data Curriculum

Self-Supervised Learning (SSL) has become a powerful solution to extract rich representations from unlabeled data. Yet, SSL research is mostly focused on clean, curated and high-quality datasets. As a result, applying SSL on noisy data remains a challenge, despite being crucial to applications such as astrophysics, medical imaging, geophysics or finance. In this work, we present a fully self-supervised framework that enables noise-robust representation learning without requiring a denoiser at inference or downstream fine-tuning. Our method first trains an SSL denoiser on noisy data, then uses it to construct a denoised-to-noisy data curriculum (i.e., training first on denoised, then noisy samples) for pretraining a SSL backbone (e.g., DINOv2), combined with a teacher-guided regularization that anchors noisy embeddings to their denoised counterparts. This process encourages the model to internalize noise robustness. Notably, the denoiser can be discarded after pretraining, simplifying deployment. On ImageNet-1k with ViT-B under extreme Gaussian noise ($\sigma=255$, SNR = 0.72 dB), our method improves linear probing accuracy by 4.8% over DINOv2, demonstrating that denoiser-free robustness can emerge from noise-aware pretraining. The code is available at https://github.com/wenquanlu/noisy_dinov2.

Joint Embedding vs Reconstruction: Provable Benefits of Latent Space Prediction for Self Supervised Learning

Reconstruction and joint embedding have emerged as two leading paradigms in Self Supervised Learning (SSL). Reconstruction methods focus on recovering the original sample from a different view in input space. On the other hand, joint embedding methods align the representations of different views in latent space. Both approaches offer compelling advantages, yet practitioners lack clear guidelines for choosing between them. In this work, we unveil the core mechanisms that distinguish each paradigm. By leveraging closed form solutions for both approaches, we precisely characterize how the view generation process, e.g. data augmentation, impacts the learned representations. We then demonstrate that, unlike supervised learning, both SSL paradigms require a minimal alignment between augmentations and irrelevant features to achieve asymptotic optimality with increasing sample size. Our findings indicate that in scenarios where these irrelevant features have a large magnitude, joint embedding methods are preferable because they impose a strictly weaker alignment condition compared to reconstruction based methods. These results not only clarify the trade offs between the two paradigms but also substantiate the empirical success of joint embedding approaches on real world challenging datasets.

SplInterp: Improving our Understanding and Training of Sparse Autoencoders

Sparse autoencoders (SAEs) have received considerable recent attention as tools for mechanistic interpretability, showing success at extracting interpretable features even from very large LLMs. However, this research has been largely empirical, and there have been recent doubts about the true utility of SAEs. In this work, we seek to enhance the theoretical understanding of SAEs, using the spline theory of deep learning. By situating SAEs in this framework: we discover that SAEs generalise ``$k$-means autoencoders'' to be piecewise affine, but sacrifice accuracy for interpretability vs. the optimal ``$k$-means-esque plus local principal component analysis (PCA)'' piecewise affine autoencoder. We characterise the underlying geometry of (TopK) SAEs using power diagrams. And we develop a novel proximal alternating method SGD (PAM-SGD) algorithm for training SAEs, with both solid theoretical foundations and promising empirical results in MNIST and LLM experiments, particularly in sample efficiency and (in the LLM setting) improved sparsity of codes. All code is available at: https://github.com/splInterp2025/splInterp

An Empirically Grounded Identifiability Theory Will Accelerate Self-Supervised Learning Research

Self-Supervised Learning (SSL) powers many current AI systems. As research interest and investment grow, the SSL design space continues to expand. The Platonic view of SSL, following the Platonic Representation Hypothesis (PRH), suggests that despite different methods and engineering approaches, all representations converge to the same Platonic ideal. However, this phenomenon lacks precise theoretical explanation. By synthesizing evidence from Identifiability Theory (IT), we show that the PRH can emerge in SSL. However, current IT cannot explain SSL's empirical success. To bridge the gap between theory and practice, we propose expanding IT into what we term Singular Identifiability Theory (SITh), a broader theoretical framework encompassing the entire SSL pipeline. SITh would allow deeper insights into the implicit data assumptions in SSL and advance the field towards learning more interpretable and generalizable representations. We highlight three critical directions for future research: 1) training dynamics and convergence properties of SSL; 2) the impact of finite samples, batch size, and data diversity; and 3) the role of inductive biases in architecture, augmentations, initialization schemes, and optimizers.

Learning from Reward-Free Offline Data: A Case for Planning with Latent Dynamics Models

A long-standing goal in AI is to build agents that can solve a variety of tasks across different environments, including previously unseen ones. Two dominant approaches tackle this challenge: (i) reinforcement learning (RL), which learns policies through trial and error, and (ii) optimal control, which plans actions using a learned or known dynamics model. However, their relative strengths and weaknesses remain underexplored in the setting where agents must learn from offline trajectories without reward annotations. In this work, we systematically analyze the performance of different RL and control-based methods under datasets of varying quality. On the RL side, we consider goal-conditioned and zero-shot approaches. On the control side, we train a latent dynamics model using the Joint Embedding Predictive Architecture (JEPA) and use it for planning. We study how dataset properties-such as data diversity, trajectory quality, and environment variability-affect the performance of these approaches. Our results show that model-free RL excels when abundant, high-quality data is available, while model-based planning excels in generalization to novel environment layouts, trajectory stitching, and data-efficiency. Notably, planning with a latent dynamics model emerges as a promising approach for zero-shot generalization from suboptimal data.

Eidetic Learning: an Efficient and Provable Solution to Catastrophic Forgetting

Catastrophic forgetting -- the phenomenon of a neural network learning a task t1 and losing the ability to perform it after being trained on some other task t2 -- is a long-standing problem for neural networks [McCloskey and Cohen, 1989]. We present a method, Eidetic Learning, that provably solves catastrophic forgetting. A network trained with Eidetic Learning -- here, an EideticNet -- requires no rehearsal or replay. We consider successive discrete tasks and show how at inference time an EideticNet automatically routes new instances without auxiliary task information. An EideticNet bears a family resemblance to the sparsely-gated Mixture-of-Experts layer Shazeer et al. [2016] in that network capacity is partitioned across tasks and the network itself performs data-conditional routing. An EideticNet is easy to implement and train, is efficient, and has time and space complexity linear in the number of parameters. The guarantee of our method holds for normalization layers of modern neural networks during both pre-training and fine-tuning. We show with a variety of network architectures and sets of tasks that EideticNets are immune to forgetting. While the practical benefits of EideticNets are substantial, we believe they can be benefit practitioners and theorists alike. The code for training EideticNets is available at \href{https://github.com/amazon-science/eideticnet-training}{this https URL}.

Curvature Tuning: Provable Training-free Model Steering From a Single Parameter

The scaling of model size and data size has reshaped the paradigm of AI. As a result, the common protocol to leverage the latest models is to steer them towards a specific downstream task of interest through {\em fine-tuning}. Despite its importance, the main methods for fine-tuning remain limited to full or low-rank adapters--containing countless hyper-parameters and lacking interpretability. In this paper, we take a step back and demonstrate how novel and explainable post-training steering solutions can be derived theoretically from {\em spline operators}, a rich mathematical framing of Deep Networks that was recently developed. Our method--coined \textbf{Curvature Tuning (CT)}--has a single parameter that provably modulates the curvature of the model's decision boundary henceforth allowing training-free steering. This makes CT both more efficient and interpretable than conventional fine-tuning methods. We empirically validate its effectiveness in improving generalization and robustness of pretrained models. For example, CT improves out-of-distribution transfer performances of ResNet-18/50 by 2.57\%/1.74\% across seventeen downstream datasets, and improves RobustBench robust accuracy by 11.76\%/348.44\%. Additionally, we apply CT to ReLU-based Swin-T/S, improving their generalization on nine downstream datasets by 2.43\%/3.33\%. Our code is available at \href{https://github.com/Leon-Leyang/curvature-tuning}{https://github.com/Leon-Leyang/curvature-tuning}.