Luu Duc Trung

Truong Xuan Khanh, Truong Quynh Hoa, Luu Duc Trung et al.

We give the first quantitative prediction of grokking delay under AdamW. Treating the delay as a first-passage time, we derive a closed-form law T_grok - T_mem = (1 / 2 kappa_LL eta lambda) log(V_mem / V_star), where V_t = ||theta_t||^2 is the squared parameter norm, V_star is an architecture-dependent threshold, and kappa_LL absorbs the AdamW correction to the clean-SGD contraction rate 2 eta lambda. Calibrating (kappa_LL, V_star) on a single hyperparameter cell predicts grokking delays on 26 held-out runs with MAPE 17.7% over a 41x delay range; the law generalises to MLPs (MAPE 18.0%, N=34) and degrades to 23.3% on cross-task extension (N=46, 43.5x range), with a structured residual in which V_star / V_mem stays comparatively stable within architecture (CV about 14% on the 1L transformer). First-passage of V_t is necessary but not sufficient. A quantile-margin theorem establishes that positive delay requires both norm separation V_mem > V_post and angular reachability of a threshold alpha_star = arcsin(C / V_T_mem^(1/2)), where C is computable from the empirical NTK feature map and the validation-margin quantile. Calibrating C on modulus p=89 predicts alpha_star = 47.2 degrees at p=97 (observed 47.8 degrees, error 1.3%) as a prior cross-cell prediction. Causal interventions that freeze the norm or remove weight decay at memorisation eliminate grokking (0/6 vs. 3/3 baseline), trapping the angular displacement near 12 degrees. kappa_LL is empirically measured per architecture rather than derived from (beta_1, beta_2, epsilon); within-architecture CV stays at most 15% across four architectures, but values differ by about 2x between architectural variants beyond depth alone. Empirical scope is algorithmic tasks (modular arithmetic, sparse parity) under AdamW; whether the law transfers to natural-language scale models is open.

Truong Xuan Khanh, Truong Quynh Hoa, Luu Duc Trung et al.

Grokking is the sudden generalization that appears long after a model has perfectly memorized its training data. Although this phenomenon has been widely observed, there is still no quantitative theory explaining the length of the delay between memorization and generalization. Prior work has noted that weight decay plays an important role, but no result derives tight bounds for the delay or explains its scaling behavior. We present a first-principles theory showing that grokking arises from a norm-driven representational phase transition in regularized training dynamics. Training first converges to a high-norm memorization solution and only later contracts toward a lower-norm structured representation that generalizes. Our main result establishes a scaling law for the delay: T_grok - T_mem = Theta((1 / gamma_eff) * log(||theta_mem||^2 / ||theta_post||^2)), where gamma_eff is the effective contraction rate of the optimizer (gamma_eff = eta * lambda for SGD and gamma_eff >= eta * lambda for AdamW). The upper bound follows from a discrete Lyapunov contraction argument, and the matching lower bound arises from dynamical constraints of regularized first-order optimization. Across 293 training runs spanning modular addition, modular multiplication, and sparse parity tasks, we confirm three predictions: inverse scaling with weight decay, inverse scaling with learning rate, and logarithmic dependence on the norm ratio (R^2 > 0.97). We further find that grokking requires an optimizer that can decouple memorization from contraction: SGD fails under hyperparameters where AdamW reliably groks. These results show that grokking is a predictable consequence of norm separation between competing interpolating representations and provide the first quantitative scaling law for the delay of grokking.

Truong Xuan Khanh, Truong Quynh Hoa, Luu Duc Trung et al.

Grokking -- delayed generalisation long after memorisation -- lacks a predictive mechanistic explanation. We identify the normalised spectral entropy $\tilde{H}(t)$ of the representation covariance as a scalar order parameter for this transition, validated on 1-layer Transformers on group-theoretic tasks. Five contributions: (i) Grokking follows a two-phase pattern: norm expansion then entropy collapse. (ii) $\tilde{H}$ crosses a stable threshold $\tilde{H}^* \approx 0.61$ before generalisation in 100% of runs (mean lead: 1,020 steps). (iii) A causal intervention preventing collapse delays grokking by +5,020 steps ($p=0.044$); a norm-matched control ($n=30$, $p=5\times10^{-5}$) confirms entropy -- not norm -- drives the transition. (iv) A power-law $ΔT = C_1(\tilde{H}-\tilde{H}^*)^γ+C_2$ ($R^2=0.543$) predicts grokking onset with 4.1% error. (v) The mechanism holds across abelian ($\mathbb{Z}/97\mathbb{Z}$) and non-abelian ($S_5$) groups. Crucially, MLPs show entropy collapse without grokking, proving collapse is necessary but not sufficient -- architecture matters. Code: https://anonymous.4open.science/r/grokking-entropy