In the high-dimensional limit the spherical Boltzmann machine admits exact equations for training dynamics, Bayesian evidence, and cascades of phase transitions tied to mode alignment with data, which connect to generative phenomena including double descent and out-of-equilibrium biases.
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Progress measures for grokking via mechanistic interpretability
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Neural networks often exhibit emergent behavior, where qualitatively new capabilities arise from scaling up the amount of parameters, training data, or training steps. One approach to understanding emergence is to find continuous \textit{progress measures} that underlie the seemingly discontinuous qualitative changes. We argue that progress measures can be found via mechanistic interpretability: reverse-engineering learned behaviors into their individual components. As a case study, we investigate the recently-discovered phenomenon of ``grokking'' exhibited by small transformers trained on modular addition tasks. We fully reverse engineer the algorithm learned by these networks, which uses discrete Fourier transforms and trigonometric identities to convert addition to rotation about a circle. We confirm the algorithm by analyzing the activations and weights and by performing ablations in Fourier space. Based on this understanding, we define progress measures that allow us to study the dynamics of training and split training into three continuous phases: memorization, circuit formation, and cleanup. Our results show that grokking, rather than being a sudden shift, arises from the gradual amplification of structured mechanisms encoded in the weights, followed by the later removal of memorizing components.
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LOCOS scores attention heads via OV-circuit output projection onto answer-token unembedding directions and identifies non-literal retrieval heads whose ablation collapses performance on non-literal benchmarks more than prior literal-copy detectors.
A descent-free method recovers the singularity order k of dead directions in neural networks from the directional-Fisher rate, classifies them, and assembles global learning coefficients matching closed forms.
Observable Matrix Dynamics (OMD) is a new diagnostic framework that uses random matrix theory on distance matrices to distinguish diffusive relaxations from phase-transition-like reorganizations during neural network training.
During pretraining, language models exhibit natural ungrokking where learned rules are forgotten based on their support frequency in the corpus, with asymmetric editability of rule survival.
Dead-Direction Signatures provide closed-form spectral readings of dead directions in network activations and gradients that track rank deficits at singular minima, offering a cheap directional alternative to SGLD-based LLC.
The normalized inverse-scale direction of LayerNorm's affine parameters is an exact algebraic kernel of the post-final-norm centred activation covariance for any input distribution in LayerNorm transformers.
Grokking delay under cross-entropy is mediated primarily by logit scale and resulting softmax saturation, with weight norm acting only as an upstream handle that adds 1-2% beyond the scale.
Fragility, the activation noise level causing probe accuracy collapse, reveals evolving lexical-to-compositional moral encoding, layer robustness gradients, and fine-tuning differences invisible to saturated probing accuracy.
AVLLMs route audio-visual information sequentially in video tasks and via parallel streams for interleaved items, allowing early token discard with little performance loss across models and scales.
Dead directions recover Watanabe's RLCT contribution and triple (λ, m, ν) from directional Fisher curvature decay rates in original parameter space for singular models, extended via K-FAC to networks and gauge-equivariant optimizers.
CEDAR learns an invertible rotation of vision-language embeddings to concentrate semantics into sparse, axis-aligned coordinates for improved interpretability.
This paper presents Markovian Circuit Tracing (MCT) as a benchmark and pipeline to extract and test state-transition structures in transformer activations using synthetic HMM tasks, demonstrating that state patching improves counterfactual predictions.
Tensor similarity is a symmetry-invariant metric that measures functional equivalence between tensor-based networks using a recursive algorithm for cross-layer mechanisms.
The Divergent Remote Association Test (DRAT) is the first creativity test that significantly predicts LLMs' scientific ideation ability, unlike prior tests such as DAT or RAT.
A first-passage time model produces the law T_grok - T_mem = (1 / 2 kappa_LL eta lambda) log(V_mem / V_star) that predicts grokking delays with 17.7% MAPE on held-out AdamW runs after calibrating two parameters on one cell.
Susceptibilities applied to regret in deep RL agents reveal stagewise internal development in parameter space of a gridworld model that policy inspection alone cannot detect, validated via activation steering.
Transformers store count information internally but cannot read it out as digits due to near-orthogonal alignment with output-head rows; updating digit rows or applying LoRA to attention layers improves constrained and unconstrained counting respectively.
ILDR detects the geometric reorganization preceding grokking by measuring when inter-class centroid separation exceeds intra-class scatter by 2.5 times its baseline in penultimate-layer representations.
Diffusion models show grokking on modular addition by composing periodic operand representations in simple data regimes or by separating arithmetic computation from visual denoising across timesteps in varied regimes.
Effective cascade dimension D(t) crosses D=1 at the grokking transition in MLPs and Transformers, with opposite directions for modular addition versus XOR, consistent with attraction to a shared critical manifold.
Transformer weights at early training stages are closed-form compositions of bigram, token-interchangeability, and context mappings that directly reflect text-corpus statistics and explain the emergence of semantic associations.
Small transformers reproduce known Bayesian posteriors with 10^{-3} to 10^{-4} bit accuracy in verifiable wind-tunnel tasks via residual belief states, FFN updates, and attention routing, while MLPs do not.
EGD equalizes gradient speeds across singular directions, eliminating or shortening grokking plateaus on modular addition and sparse parity problems.
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Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity
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Algebraic Dead Directions in LayerNorm Transformers: A Forward-Pass-Only Diagnostic at LLM Scale
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What Does the Weight Norm Control in Grokking? Logit-Scale Mediation under Cross-Entropy
Grokking delay under cross-entropy is mediated primarily by logit scale and resulting softmax saturation, with weight norm acting only as an upstream handle that adds 1-2% beyond the scale.
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When Probing Accuracy Saturates, Fragility Resolves: A Complementary Metric for LLM Pre-Training Analysis
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A first-passage time model produces the law T_grok - T_mem = (1 / 2 kappa_LL eta lambda) log(V_mem / V_star) that predicts grokking delays with 17.7% MAPE on held-out AdamW runs after calibrating two parameters on one cell.
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Diffusion models show grokking on modular addition by composing periodic operand representations in simple data regimes or by separating arithmetic computation from visual denoising across timesteps in varied regimes.
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