ADOPT: Modified Adam Can Converge with Any $β_2$ with the Optimal Rate

Adam is one of the most popular optimization algorithms in deep learning. However, it is known that Adam does not converge in theory unless choosing a hyperparameter, i.e., $\beta_2$, in a problem-dependent manner. There have been many attempts to fix the non-convergence (e.g., AMSGrad), but they require an impractical assumption that the gradient noise is uniformly bounded. In this paper, we propose a new adaptive gradient method named ADOPT, which achieves the optimal convergence rate of $\mathcal{O} ( 1 / \sqrt{T} )$ with any choice of $\beta_2$ without depending on the bounded noise assumption. ADOPT addresses the non-convergence issue of Adam by removing the current gradient from the second moment estimate and changing the order of the momentum update and the normalization by the second moment estimate. We also conduct intensive numerical experiments, and verify that our ADOPT achieves superior results compared to Adam and its variants across a wide range of tasks, including image classification, generative modeling, natural language processing, and deep reinforcement learning. The implementation is available at https://github.com/iShohei220/adopt.

Interpreting Grokked Transformers in Complex Modular Arithmetic

Grokking has been actively explored to reveal the mystery of delayed generalization. Identifying interpretable algorithms inside the grokked models is a suggestive hint to understanding its mechanism. In this work, beyond the simplest and well-studied modular addition, we observe the internal circuits learned through grokking in complex modular arithmetic via interpretable reverse engineering, which highlights the significant difference in their dynamics: subtraction poses a strong asymmetry on Transformer; multiplication requires cosine-biased components at all the frequencies in a Fourier domain; polynomials often result in the superposition of the patterns from elementary arithmetic, but clear patterns do not emerge in challenging cases; grokking can easily occur even in higher-degree formulas with basic symmetric and alternating expressions. We also introduce the novel progress measure for modular arithmetic; Fourier Frequency Sparsity and Fourier Coefficient Ratio, which not only indicate the late generalization but also characterize distinctive internal representations of grokked models per modular operation. Our empirical analysis emphasizes the importance of holistic evaluation among various combinations.

Grokking Tickets: Lottery Tickets Accelerate Grokking

Grokking is one of the most surprising puzzles in neural network generalization: a network first reaches a memorization solution with perfect training accuracy and poor generalization, but with further training, it reaches a perfectly generalized solution. We aim to analyze the mechanism of grokking from the lottery ticket hypothesis, identifying the process to find the lottery tickets (good sparse subnetworks) as the key to describing the transitional phase between memorization and generalization. We refer to these subnetworks as ''Grokking tickets'', which is identified via magnitude pruning after perfect generalization. First, using ''Grokking tickets'', we show that the lottery tickets drastically accelerate grokking compared to the dense networks on various configurations (MLP and Transformer, and an arithmetic and image classification tasks). Additionally, to verify that ''Grokking ticket'' are a more critical factor than weight norms, we compared the ''good'' subnetworks with a dense network having the same L1 and L2 norms. Results show that the subnetworks generalize faster than the controlled dense model. In further investigations, we discovered that at an appropriate pruning rate, grokking can be achieved even without weight decay. We also show that speedup does not happen when using tickets identified at the memorization solution or transition between memorization and generalization or when pruning networks at the initialization (Random pruning, Grasp, SNIP, and Synflow). The results indicate that the weight norm of network parameters is not enough to explain the process of grokking, but the importance of finding good subnetworks to describe the transition from memorization to generalization. The implementation code can be accessed via this link: \url{https://github.com/gouki510/Grokking-Tickets}.