Bridging Lottery Ticket and Grokking: Understanding Grokking from Inner Structure of Networks
Authors: Gouki Minegishi
Yusuke Iwasawa
Yutaka Matsuo
Venue: Accepted by TMLR ()

Abstract: Grokking is an intriguing phenomenon of delayed generalization, where neural networks initially memorize training data with perfect accuracy but exhibit poor generalization, subsequently transitioning to a generalizing solution with continued training. While factors
such as weight norms and sparsity have been proposed to explain this delayed generalization, the influence of network structure remains underexplored. In this work, we link the grokking phenomenon to the lottery ticket hypothesis to investigate the impact of internal
network structures. We demonstrate that utilizing lottery tickets obtained during the generalizing phase (termed grokked tickets) significantly reduces delayed generalization across various tasks, including multiple modular arithmetic operations, polynomial regression,
sparse parity, and MNIST classification. Through controlled experiments, we show that the mitigation of delayed generalization is not due solely to reduced weight norms or increased sparsity, but rather to the discovery of good subnetworks. Furthermore, we find that grokked
tickets exhibit periodic weight patterns and undergo rapid structural changes that coincide with improvements in generalization. Additionally, pruning techniques like the edge-popup algorithm can identify these effective structures without modifying the weights, thereby transforming memorizing networks into generalizing ones. These results underscore the novel insight that structural exploration plays a pivotal role in understanding grokking.

Published in Transactions on Machine Learning Research (05/2025)
Bridging Lottery Ticket and Grokking: Understanding Grokking from Inner Structure of Networks
Gouki Minegishi
Yusuke Iwasawa Yutaka Matsuo
Eminegishi, bwosowa, i @eweblab. t. t..-tokyo.cc... The University of Tokyo
Reviewed on OpenReview: https: // openreview. net/ forun? id= eQeVyuplta
Abstract
Grokking is intriguing phenomenon of delayed generalization, where neural networks initially memorize training data with perfect. accuracy but exhibit poor generalization, subsequently transitioning to a generalizing solution with continued training. While factors such as weight norms and sparsity have been proposed to explain this delayed generalization, the influence of network structure remains underexplored. this work, we link the grokking phenomenon to the lottery ticket hypothesis to investigate the impact of internal network structures. We demonstrate that utilizing lottery tickets obtained during the generalizing phase (termed grokked tickets) significantly reduces delayed generalization across various lasks, including multiple modular arithmetic operations, polynomial regression, sparse parity, and MNIST classification. Through controlled experiments, we show that the mitigation of delayed generalization is not due solely to reduced weight norms or increased sparsity, but rather to the discovery of good subnetworks. Furthermore, we find that grokked tickets exhibit periodic weight patterns, beneficial graph properties such as increased average path lengths and reduced clustering coefficients, and undergo rapid structural changes that coincide with improvements in generalization. Additionally, pruming techniques like the rddee-poppp algorithm identify these effective structures without modifying the weights, thereby transforming memorizing networks into generalizing ones. These results underscore the novel insight that structural exploration plays a pivotal role in understanding grokking,
1 Introduction
Understanding the mechanism of generalization is a central question in understanding the efficacy of neural networks. Recently, Power et al. (2022) unveiled the intriguing phenomenon of delayed generalization (grokking); neural networks initially attain memorizing network Cuem with the perfect training accuracy but poor generalization, yet further training transitions the solution to generalizing network Cgen- This phenomenon, which contradicts standard machine learning expectations, is being studied to answer the question: what underlies the transition between memorization and generalization? (Liu et al., 2022; 2023a)
Regarding the relationship between generalization and deep learning in general, it is well known that structure of networks significantly impacts generalization performance. For instance, image recognition performance has greatly improved by leveraging the structure of convolution (Krizhevsky et al., 2012). Moreover, shown in Neyshabur (2020), incorporating 3-Lasso regularization into fully connected MLPs facilitates the emergence of locality - resembling the structures in CNNs leading to improved performance in image tasks. From a slightly different perspective, Frankle & Carbin (2019) proposed the lottery licket hypothesis (LTH), which suggests that good subnetworks (good structure) help to achieve better performance with better sample efficiency (Zhang et al., 2021). Also, Ramanujan et al. (2020) shows that exploring structures alone can achieve performance comparable weight updates, suggesting that good subnetworks are enough achieve generalized performance.

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Observation (sec 3): Lottery ticket
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Figure 1: (Left) Accuracy of dense model and the lottery ticket obtained at generalizing solution (grokked ticket). When using a lottery ticket (good subnetworks), the train and test accuracy increase almost similarly, i.e., the time from memorization (4mem) to generalization (4gm) has significantly accelerated. Note that not only the subtraction (tgen - mmem) but the ratio (tgen/tment) ) is also significantly improved, meaning that it's not just matter of faster learning. (Right) Three hypotheses on why delayed gencralization is reduced with a lottery ticket. We show that it is not due to a reduction in weight norm or an increase in sparsity, but rather the discovery of good structure.
While the importance of structure is well known in general, its connection to the phenomenon of grokking has not been investigated enough. Similar to our study, several prior works connect the grokking phenomenon to the property of networks, e.g., weight norm and sparsity of networks. For example, Liu et al. (2023a) experimentally confirmed that generalizing solutions have smaller norms compared to memorizing solutions. The original paper (Power et al., 2022) showed that adding weight decay during training is necessary for triggering grokking. However, our work differs from these works by discussing the relationship between the process of discovering good structures (i.e., subnetworks) and grokking rather than merely reducing the weight norm.
To investigate the role of structure in the grokking phenomenon, we first demonstrate that when using the lottery ticket obtained at generalizing solution (referred to as grokked tickets), delayed generalization is significantly reduced. Figure 1 (left) illustrates that the train and test accuracy increase almost simultancously with grokked lickets, unlike randomly initialized dense networks where delayed generalization occurs. As will be shown later, this result is related to the pruning rate, with proper pruning rates resulting in less delay. We conducted further experiments from several perspectives to understand why delayed generalization is significantly reduced with grokked tickets. First, as illustrated in Figure 1 (right), we decompose the differences between the grokked ticket and randomly initialized dense network into three elements: (1) small weight norms, (2) sparsity and (3) good structure. We investigate which of these elements contribute to the significant reduction of delayed generalization. For (1) weight norms, we find that dense networks with the same initial weight norms as the grokked ticket do not generalize faster, indicating weight norm is not the cause. For (2) sparsity, networks with the same parameter size using well-known pruning methods (Wang et al., 2019; Lee al., 2019; Tanaka et al., 2020) also do not generalize faster, indicating sparsity is not the cause. These results suggest that (3) good structure is essential for understanding grokking.
Building on these findings, we delve deeper into understanding the nature and role of these structural properties in the grokking phenomenon. Specifically, we investigate several key aspects. First, to determine whether the acqusstitonn of the structure synchronized with improvements in generalization performance, we use the Jaccard distance (Jaccard, 1901) as metric of structural distance. Our analysis shows that the structure of the subnetwork changes rapidly during the transition from memorization to generalization. This rapid structural adaptation is synchronized with enhancements in generalization performance. Second, we
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examine both graph-theoretic properties, including increased average path lengths and reduced clustering coefficients, and the periodic nature of the Modular Addition task (Nanda et al., 2023). We find that beneficial graph properties emerge and align closely with improved generalization, and the model exploits the task's inherent periodicity to discover internally well-structured subnetworks suited for achioving better generalization. Finally, based on our results indicating that structure exploration is crucial for grokking, we demonstrate that pruning facilitates generalization. We employ the odge-popup algorithm (Ramanujan et al., 2020) to identify a good structure while keeping the weights unchanged. Our experiments show that the memorizing network can be transferred to a generalizing network through pruning without weight updates, thereby strengthening our claim that delayed generalization occurs due to the discovery of good structure.
In summary, our contributions are below:
Linking Lottery Tickets and Grokking (section 3): We investigate the role of inner structures (subnetworks) in the grokking process by connecting lottery tickets to delayed gencralization. Our results demonstrate that using lottery tickets significantly reduces the delayed generalization, highlighting the importance of subnetworks in achieving efficient generalization.
Decoupling Properties of Lottery Tickets (section 4): We systematically separate the intrinsic properties of lottery tickets into three components: (1) weight norm, (2) sparsity, and (3) good structure. Through a scrics of controlled experiments, we show that neither weight norm nor sparsity alone accounts for the reduction in delayed generalization. Instead, it is the discovery of good structural configuration within the network that is crucial.
Understanding the Characteristics of Good Structure (section 5): We delve deeper into the nature of the beneficial structures discovered by lottery tickets. Our findings reveal that grokked tickets exhibit periodic weight representations and undergo rapid structural changes synchronized with improvements in generalization performance. Furthermore, demonstrate that pruming techniques, such as the edge-popup algorithm, can identify these effective structures without altering the weights, effectively transitioning memorizing networks into generalizing ones. This highlights the pivotal role of structural exploration in mitigating delayed generalization.
2 Background
Grokking is a phenomenon where generalization happens long after overfitting the training data (as shown in Figure 1 (left)). The phenomenon was initially observed in the modular addition task ((a + b) mod for a, E (0, p 1)), and the same phenomenon has been observed in more complex datasets, encompassing modular arithmetic (Gromov, 2023; Davics et al., 2023; Rubin et al., 2023; Stander et al., 2023; Furuta et al., 2024), semantic analysis (Liu et al., 2023a), n-k parity (Merrill et al., 2023), polynomial regression (Kumar et al., 2023), hierarchical task (Murty et al., 2023) and image classification (Liu al., 2023a; Radhakrishnan et al., 2022). This paper mainly focuses on grokking in the modular arithmetic tasks commonly used in prior studies.
To understand grokking, previous works proposed possible explanations, including the slingshot mecha- nism (Thilak et al., 2022), random walk among minimizers (Millidge, 2022), formulation of good representa- tion (Liu et al., 2022), the scale of weight norm (Liu et al., 2023a; Varma et al., 2023), simplicity of Fourier features (Nanda et al., 2023) and sparsity of generalizing network (Miller et al., 2023).
Among those, one of the dominant explanations regarding how the network changes during the process of grokking are the simplicity of the generalization solution, particularly focusing the weight norms of network parameters ||0||a. For example, the original paper (Power et al., 2022) posited that weight decay plays a pivotal role in grokking, i.c., test accuracy will not increase without weight decay. Liu et al. (2023a) analyzod the loss landscapes of train and test. dalaset, verifying that grokking occurs by entering the generalization zone defined by L2 norm, with models having large initial values 6g More recently, Varma et al. (2023) demonstrated that the generalization solution could produce higher logits with smaller weight norms. In this paper, we examine the changes in the network's structure and demonstrate that the network is not simply decreasing its overall weight norms but searching for good structures within itself.
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Several studies have proposed that acquiring good representations is the key to understanding grokking. For example, Power et al. (2022); Liu et al. (2022) explained that the topology of the ideal embeddings tends to be circles or cylinders within the context of modular addition tasks. Nanda et al. (2023) identified the trigonometric algorithm by which the networks solve modular addition after grokking and showed that it grows smoothly over training. Gromov (2023) showed an analytic solution for the representations when learning modular addition with MLP. Zhong et al. (2023) show, using modular addition as a prototypical problem, that algorithm discovery in neural networks is sometimes more complex. These studies support the quality of representation as key to distinguishing memorizing and generalizing networks; however, these studies do not explain what is happening within the network's 's structure.
The lottery ticket hypothesis proposed by Frankle & Carbin (2019) has garnered attention as an explanation for why over preemaaameteri neural networks exhibit generalization capabilities (Allen- Zhu et al., 2019). Informally, the lottery ticket hypothesis states that randomly initialized over-parameterized networks include sparse subnetworks that reach good performance after train, and the existence of the subnetworks is key to achieving good generalization in deep neural networks. This claim was initially demonstrated experimentally, but theoretical foundations have also been established (Frankle et al., 2020; Sakamoto & Sato, 2022). More formally, the process involves the following steps:
1. Initialize dense network Soa and train the network for l. epochs to obtain the weights Oz
2. Perform k% pruning on the trained network based on absolute values 10cl- This process, known magnitude pruning, yields a mask m E 11.1)l
3. Reset the weights of the network to their initial values. Og and get subnetwork Seoom representing lottery ticket. Train the subnetwork for t' epochs and obtain set
After the discovery of the lottery tickets, Ramamujan et al. (2020) show that there exist strong lottery tickets, which achieve good performance without weight update. They use the edge-popup algorithm (Ramanujan et al., 2020) to selects subnetworks based on a score s (s E (R(0)). In other words, when pruning certain proportion k of weights from the given weights Oo, the model predicts using edges with the top (1 - k) scores in a forward pass. For a detailed description of edge-popup, refer to Appendix A. In section 5, we use the edge-popup algorithm to check if pruning can be worked as a force to accelerate the grokking process.
3 Lottery Tickets significantly reduces Delayed Generalization
3.1 Experiment Setup
Following Power el al. (2022) and other grokking literatures (Nakkiran et al., 2019; Liu el al., 2022; Gromov, 2023; Liu et al., 2023a), we constructed a dataset of equations of the form: (a+ + 6)%p = The task involves predicting c given a pair of a and b. Our sctup uses the following detailed configurations: p= = 67, 0 & a, b, < p. The dataset size is 2211, considering all possible pairs where a 2 b. We split it into training (40%) and test (60%) following Tiu et al. (2022).
Following Liu et al. (2022), we design the MLP as follows. Firstly, we map the one-hot encoding of a, b with the embedding weights Wemb: E. Wemba, a, E, Wembb. We then feed the embeddings E. and E, into the MT. as follows:
softmax(o((E.. + E,) W out W Wunemb)
(1)
where Wemb. Win: Wout, and W Wunemb are the trainable parameters, and a is an activation function ReLU (Nair & Hinton, 2010). The dimension of the embedding space is 500, and Wi.. projects into 48-dimensional neurons.
Following Nanda et al. (2023), we used the AdamW optimizer (Loshchilov & Hutter, 2019) with a learning rate 10 the weighting of weight decay a = 1.0, 8 = 0.9, and 32 = 0.98. We initialize weights as 8n- ~ (O, /s/ /dd)),), where dn represents the dimensionality of the layer proceding each weight. If nothing is specified, assume K = 1. Let us assume we have training datasets Surin and test datasets Stest, and train a neural network f(:; 0) where a is an input and 0 represents weight. parameters of the networks. Specifically,
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Comparing the grokking speed of dense networks and grokked tickets on various setups. (a) Modular addition with MLP, (b) Modular addition with Transformer, and (c) Other modular arithmetic tasks (represented by color) and experiments other than modular arithmetic:: (d) loss on polynomial regression, (e) accuracy on sparse parity. The dashed line represents the accuracy of the base model, and the solid line represents that of grokked tickets. In all setups, the time to generalization (Igen) is reduced by grokked tickets.
Figure 2:
the network is trained using AdamW over a cross-entropy loss and weight decay (L2 norm of weights ||0|12):
argmin Eew))~ [C([(=; (), a [1112]]
To quantitatively measure how much-dclayed generalization is reduced, we define tiem the step at which the training accuracy exceeds P%, and 'gen as the step at which the test accuracy exceeds P%. Following Kumar et al. (2024), we use 95 for modular arithmetic tasks. We use the proposition (Tgrok /gen/ (mmm) to measure the acceleration.
We compared the performance of 1) dense networks fe,, and 2) trained lottery tickets fer,om where t' is a training epoch to get the final score, t is timing of pruning, and & is pruning ratio. As a special case, when 2 tgen. denote the subnetworks grokked tickets. We tested various t and it and investigated how they change the generalization speed. By default, we used k = 0.6.
3.2 Results
Figure 2-(a) shows the test accuracy of the grokked ticket and the base model on the modular addition task. The base model refers to a dense model trained from random initial values. The grokked ticket shows an improvement in test accuracy at nearly the same time as the improvement in training accuracy of the base model. In Figure 2-(b), using experiments with transformers, the result also shows that grokked tickets result. in less delay of generalization. Following Power al. (2022), we conductod exporiments on various modular arithmetic tasks demonstrate the elimination of delayed generalization. Figure 2-(c) shows a comparison of the base model (dashed line) and grokked ticket (solid line) with various modular arithmetic tasks. Moreover, following Kumar et al. (2023); Pearce et al. (2023), we also demonstrate that delayed generalization is reduced by the grokked ticket in both the polynomial regression and sparse parity tasks in Figure 2-(d,e). The results show that grokked tickets significantly reduced delayed generalization even on various tasks. See Appendix B for detailed explanation the experimental setup.
Figure 3-(a) quantify the relationship between pruning rate and Tgrok tgen/tmen (log-scaled). When the pruning rate is 0.7, Tgrok reaches its minimum, indicating that grokked ticket significantly reduce delayed
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Figure 3: Quantitative comparison of grokking speed among different pruning rates. Note that pruning rate = 0.0 corresponds to the dense network. The definition of the Tgrok is explained in subsection 3.1
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Figure 4: (a) Comparison of the test accuracy of different epoch i in which lottery tickets are acquired. We conducted every 2k epochs. The lottery tickets obtained before 25k epochs tron-grokked tickets) do not fully generalize. Additionally, this generalization ability corresponds to the test accuracy of the base model. The lottery tickets obtained after 25k epochs (grokked tickets) reduced delayed generalization. pruning rale on grokked lickets. We conducted every 0.2 pruning rate. Most pruning ratios (0.1, 0.3, 0.5, and 0.7) accelerate the generalization, indicating that the above observation does not depend heavily on the selection of the pruning ratio.
(b) The effect of
gencralization. Note that, as shown in Figure 3-(b), the fuem is delayed when using a higher pruning rate, meaning that the grokked ticket are not simply accelerating the entire learning process but specifically speeding up the transition from memorization to generalization.
In Figure 4-(a), compare the test accuracy of different epoch t in which lottery tickets acquired. The results show that lottery tickets obtained before 25k epochs (non-grokked tickets) do not fully generalize. Additionally, this generalization ability corresponds to the test accuracy of the base model. On the other hand, lottery tickets obtained after 25k epochs (grokked tickets) get perfect generalization and reduce delayed generalization. In Figure 4-(b), we investigate the effects of pruning rate grokked ticket, indicating if it's too large (e.g., 0.9), can't generalize; if it's too small (e.g., 0.3), doesn't 't generalize quickly enough.
4 Decoupling Lottery Tickets: Norm, Sparsity, and Structure
In section 3, we cstablished that grokked tickets - subnetworks extracted at the onset of generalization significantly reduce the delayed generalization phenomenon. While these results strongly suggest that some intrinsic property of the grokked tickets is responsible for their rapid move to generalization, it not yet clear what exactly this property is. Rocall from Figure that a grokked ticket can drastically reduce delayod generalization compared to a base model (a densely trained network from scratch). To understand the key factor behind this effect, consider three hepot which illustrated in Figure 1 (right):

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Hypothesis 1 (Weight Norm): Grokked tickets have smaller weight norms than the base model, and this reduction in weight norm leads to the reduction in delayed generalization.
In Section 4.1, we test Hypothesis 1 by preparing the dense model that matches the weight norms of the grokked ticket. If these models, despite having the same weight norms, exhibit different generalization speeds, it implies that the weight norm alone does not account for the reduction in delayed generalization. Consequently, Hypothesis 1 is rejected.
Hypothesis 2 (Sparsity): Grokked tickets are sporser subnetworks compored to the base model, and this higher degree of sporsity leads to the reduction in delayed generalization.
In Section 4.2, we examine Hypothesis 2 by constructing models with the same sparsity level as the grokked ticket. If these equally sparse models do not show a reduction in delayed generalization, it indicates that sparsity alone is not responsible for the reduction in delayed generalization. Therefore, Hypothesis is rejected.
Hypothesis 3 (Good Structure): Beyond differences in weight norm sparsity, grokked tickets possess superior structural configuration that directly accelerates generalization.
If both Hypotheses and 2 are rejected, the only remaining explanation is that the good structure inherent to grokked tickets drives the reduction in delayed generalization.
4.1 Controlling Weight Norm of Initial Network
To test Hypothesis 1, we prepared two dense models with the same L2 and L1 norms the grokked ticket, named the 'controlled dense model'. Such dense models are obtained through the following process:
1. Obtain lottery tickets after full generalization m igon
10 Doomf Sgen lpp
2. Get weight Lp norm ratio rp=
|10||p
3. Create weights 0. Tp with the same Lp norm as the grokked ticket.
Figure 5-(a) shows the test accuracy of the base model, grokked ticket, and controlled dense models. Dospite having the same initial weight norms, the grokked licket arrives al generalization much faster than both controlled dense models. This result indicates that the delayed increase test accuracy is attributable not to the weight norms but to the discovery of good structure. The left of the Figure shows the dynamies of the L2 norms for each model. Similar to Liu et al. (2023b), L2 norms decrease in correspondence with the rise in test accuracy, converging towards *Generalized zone'. However, as shown on the right side of the Figure 6, the final convergence points of the L1 norms vary for each model. This phenomenon of having similar L2 norms but smaller LI norms suggests that good subnetwork (grokked ticket) weights become stronger, as indicated in Miller et al. (2023). Similar results have also been observed in Transformer (Figure 16). These results demonstrate that the weight, norm itself insufficient to explain grokking, refuting Hypothesis 1.
4.2 Controlling Sparsity
Next, we examine Hypothesis 2, which posits that higher sparsity in grokked tickets accounts for their reduced delayed generalization. To investigate this, we compared models with the same sparsity as the grokked ticket, using various pruning-at-initialiatii
(PaT) methods. Specifically, we tested three well-known Pal methods Grasp (Wang ct al., 2020), SNIP (Lee et al., 2019), and Synflow (Tanaka al., 2020)) - alongside random pruning as baseline methods. For details on each of the pruning methods, refer to the section Appendix D. Figure 5-(b) compare the transition of the test accuracy of these Pal methods and the grokked ticket. The results show that all Pal methods perform worse than the base model or, in some cases, perform worse than the random pruning. These results indicate that mere sparsity cannot account for the reduced delayed generalization, thereby rejecting Hypothesis 2.
The rejection of Hypothesis 1 and 2, this leaves only Hypothesis the key difference must lie in the good structure of grokked tickets, enabling them achieve rapid generalization and effectively shorten the delayed generalization period.
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Figure 5: (a) Test accuracy dynamics of the base model, grokked ticket, and controlled dense model (L1 norm and L2 norm). The grokked ticket reaches generalization much faster than other models. (b) Comparing test accuracy of the different pruning methods. All Pal methods perform worse than the base model or, in some cases, perform worse than the random pruming. These results indicate neither the weight norm nor the sparsity alone is the cause of delayed generalization.
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Figure 6: (Left) L2 norm dynamics of the base model, grokked ticket, and controlled dense model. (Right) L1 norm dynamics of the base model, grokked ticket, and controlled dense models. From the perspective of the L2 norm, all models appear to converge to similar solution (Gencralized zone). However, from the perspective of L1 norms, they converge different values.
5 Understanding Grokking from Inner Structure of Networks
In section 4, we demonstrated that weight norm and sparsity are insufficient explanations for delayed generalization. Instead, we suggest that good structure is essential for understanding grokking, Building on these results, this section delves deeper into understanding the nature and role of the structural properties in grokking. Specifically, we investigate the following aspects:
Q1 Is the acquisition of structure synchronized with the improvement in generalization perfor- mance? (subsection 5.1)
Q2 What exactly constitutes good structure? (subscction 5.2)
Q3 Can generalization be achieved solely through structural exploration (pruning) without weight updates? (subsection 5.3)
5.1 Progress Measure: Structural Shift Capture the Generalization Timing
In this section, we conduct a more rigorous analysis of how the good structure is acquired, and we show that the discovery of the good structure corresponds to an improvement in test accuracy.
Firstly, we propose a metric of structural changes in the network, named Jaccard Distonce (JD) using approach Paganini & Forde (2020); Jaccard (1901). We measure the jaccard distance between the mask at.
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W emb
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Figure Comparison of jaccard distance (red) and changes in test accuracy (green) on each layer. The jaccard distance is represented JD(L + ò1,(), and the test accuracy change is the difference between epoch t + öt and t. The vertical line marks the most drastic change in test accuracy. When there is significant change in test accuracy, the jaccard distance (structural change) increases rapidly.
epoch + öt and t.
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The my represents a mask obtained at t epoch via magnitude pruning and is epoch. If the two structures differ, this metric is close 1 and vice versa. Test accuracy change also represented as a difference between test accuracy at epoch t + öt and t. In Figure 7, the red line represents the results of the changes in test accuracy and jaccard distance between the mask at epoch + öt and on each layer. The results show that during significant changes in test accuracy (16k-20k), the maximum change in the mask corresponds, indicating that the discovery of good structure corresponds to an improvement in test accuracy. In the Appendix F, we demonstrate that similar results are obtained for both the polynomial regression and sparse parity tasks.
Answer to Q1
Is the acquisition of structure synchronized with the improvement in generalization performance?
Yes, the discovery of good structure, as measured by Jaccard Distance, occurs simultancously with lest accuracy improvements, indicating that structural changes drive generalization.
5.2 Analysis of Good Structures Through Periodic Structures and Graph Properties
In the previous subsection (subsection 5.1), we established that significant changes in the network's structure, as measured by Jaccard Distance, coincide precisely with improvements lest accuracy. Building on this finding, we now delve deeper into the structural properties that trigger such abrupt differences. Specifically, investigate the nature of the discovered structure from two different perspectives: (1) the periodic representations known to emerge in modular arithmetic tasks (as studied by Pearce et al. (2023); Nanda et al. (2023)); and (2) graph-theoretic properties that reveal how the network's connectivity evolves to support better generalization. By examining both viewpoints, we uncover how the network ultimately settles on a 'good structure' that drives high test accuracy.
Modular Addition and Periodicity Nanda et al. (2023) demonstrated that transformers trained on modular addition tasks rely on these periodic representations to achieve generalization. Specifically, the networks embed inputs a and b into a Fourier basis, encoding them as sine and cosine components of key frequencies wa 2rk for some & E These periodic representations are then combined using trigonometric identities within the network layers to compute the modular sum.
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Aelist
Haw 16/0
Yamn AMA
www
APp
waey Prrnee
knyy Mahel
Www
Base model, Ok
Basc model. 2k
Base model. 30k
Grokking ticket, Ok
Grokking ticket, 2k
Grokking ticket, 30k
Figure 8: (lop) Frequency entropy (FE) and test accuracy of the base model and the grokked ticket. The grokked ticket converges to a smaller frequency entropy much faster than the base model. (bottom) The transition of the input-side weights for each neuron of the base model and grokked ticket. The marks correspond to the epochs of the marks in FE dynamics. The results indicate that grokked tickets acquire good structures for generalization periodic structures.
By reverss-engineering the weights and activations of a one-layer transformer trained this task, Nanda et al. (2023) found that the model effectively computes:
cos (a + b)) = cos())a) cos(w)b) - sin(waa) sin(wwb)) sin (ww(( (a b)) sin(waa) cos(wab) cos(wya) sin(w+b).
(2)
(3)
using the embedding matrix and the attention and MLP layers. The logits for each possible output c are then computed by projecting these values via:
cos (y + b c)) = cos(wula+ + b)) cos(w,c) + sin (wx + b)) sin(w
This approach ensures that the network's out put logits exhibit constructive interference c (a b) mod p, while destructive interference suppresses other incorrect values. Given this mechanism, can be inferred that the model internally utilizes periodicity, such the addition formulas, to perform modular arithmetic.
Periodic Structure Acquisition in Grokked Tickets In this work, building this prior research, we examine the structure acquired by grokked tickets from the perspective of periodicity. First, we investigate the periodicily of each neuron by plotting the direction of the weight matrix (Wmprot); which is obtained by multiplying Wemb and Win (refer to subsection 3.1). This two-dimensional weight matrix represents the input dimensions (67) on one axis and the number of neurons (48) on the other. Figure 8 (bottom) illustrates the weights (67 dimensions) of each neuron plotted at training checkpoints (0k, 2k, 30k). In the base model at 30k steps, the weights reveal clear periodicity, whereas Ok and 2k steps, the weights exhibit random structure. Notably, grokked tickets develop this periodic structure much carlier than the base model (sec the grokked licket at 2k steps). This finding highlights that the structure of grokked lickets is well-suited for Modular Addition, as they acquire periodic structure that aligns with the task natures.
To quantify this periodicity as a good structure, we introduce Fouricr Entropy (FE) as follows. In general, the discrete Fourier transform F(w) of a function f(x) is defined as follows:
F(w) = Σ exx) (-22 2TT
N
In this case, since we want know the periodicity of each neuron's weights, f(x) is the weight of the i-th neuron on the j-th input, and is the dimension of the input. Then, the Fourier Entropy (FE) is calculated
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Weighted spectral gap
Ramanujan gap tr
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(b) Path Length, Clustering Coefficient
Figure 9: (a) Evolution of the weighted spectral gap (left) and Ramanujan gap (right) of the average across the network's layers. (b) Average path length (left) and clustering coefficient (right). Definitions and results for individual layers are provided in the Appendix and Appendix K.
as follows:
n FE = - Σ pi log
Here, p; is the normalized value of F(w), and n represents number of neurons. A low value of FE indicates that the frequency of the weights of each neuron has little variation, converging to specific frequencies, which shows that the network has acquired periodic structure.
Figure 8 (top) shows the FE of the base model (green) and grokked ticket (bluc). The results show that the grokked ticket has neurons with periodic structure an early stage (2k epochs) and exhibits a rapid decrease in FE in the early epoch. This indicates that the model discovers internally good structures (generalizing structures) suited for the task (Modular Addition). To provide more detailed analysis, have added visualizations of the weight matrices and grokked ticket masks in Appendix I.
Graph Property and Grokked Ticket The structures of the network itself exhibit various graph-theoretic properties, as analyzed in prior studies (Hoang et al., 2023b;a; You et al., 2020). Motivated by these insights, we analyze the evolution of the (1) weighted spectral gap, (2) ramanujan gap, (3) average path length, and (4) clustering coefficient of throughout, training, alongside changes in train/test accuracy. Details of cach graph property metric are provided in Appendix J.
Figure (a(a) illustrates how the weighted spectral gap (left) and Ramanujan gap (right) evolve throughout training by examining the average across the network's 's layers. Results for individual layers are provided in Appendix K. We observe that the weighted spectral gap increases sharply during the memorization phase, indicating a pronounced difference between the largest and second-largest eigenvalues while the model is overfitting This trend suggests that, in the memorization phase, the weights are dominated by a particular component (corresponding to the leading cigenvector). As the model transitions toward better generalization, however, the spectral gap narrows, reflecting more balanced utilization of the parameter space. We observe similar behavior in other layers (see Appendix K for details).
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Table 1: Test accuracy changes with different optimization methods starting from memorized solutions. WD (Weight Decay) reflects the regularization effect of weight decay, using the same optimization the base model. In EP w/o WD Podge-Popup without Weight Decay), accuracy improves solely through pruning, without weight updates. Combining pruning with weight decay in EP w/ WD results in faster gencralization than weight decay alone. Additionally, report the average values of graph property metrics on the network's s weight matrix to analyze structural changes over epochs under EP w/o WD.
Epoch
600
1000
1400
2000
Test Accuracy (%)
WD EP w/o WD EP W/ WD
0.53 + 0.31
0.95 + 0.03
1.00 + 0.00
1.00 + 0.00
0.68 = 0.19
0.80 + 0.17
0.84 + 0.16
0.92 = 0.06
0.82 + 0.04
0.96 + 0.01
0.99 + 0.00
1.00 = 0.00
Graph Property Metrics
Weighted spectral gap
0,612 + 0.05
0.810 + 0.11
0.479 = 0.04
0.358 + 0.03
Spectral gap
48.91 5.05
48.51 5.12
47.42 4.97
47.20 4.55
Ramanujan gap
-2.550 = 0.52
-4.022 0.05
-4.335 0.02
-4.555 + 0.01
Average path length
1.964 + 0.006
1.961 = 0.002
1.960 + 0.001
1.959 + 0.001
Clustering coeffieient
0.733 + 0.41
0.724 + 0.35
0.719 0.30
0.719 + 0.28
Figure 9-(b) presents metrics derived from viewing the entire network as a relational graph (You el al., 2020). The average path length (left) increases alongside the rise in test accuracy, then converges to a moderate range. This trend suggests that when the network is scarching for generalized solution (Grokkod ticket), its connectivity becomes more sparse; once that solution is found, the path length decreases and stabilizes. By contrast, the clustering coefficient (right) decreases test accuracy improves, then also settles into an intermediate value. During the phase when the network is searching for generalized solution (iee. as the model discovers the Grokked ticket), clustering diminishes but eventually stabilizes near a moderate level. Both metrics thus converge to neither extreme but remain in a balanced regime, consistent with prior studies (You et al., 2020).
Answer to Q2
What exactly constitutes a good structure?
A good structure has low Fourier Entropy, indicating periodic weight patterns aligned with the task. It also follows known graph properties, such as a rising weight spectral gap during memorization and an increasing average path length in generalization.
5.3 Pruning during Training: Pruning Promote Generalization
Based on the result that the discovery of good structure correspouds to an improvement in test accuracy, we demonstrate that pruming alone can transition from memorizing solutions to generalizing solutions without weight update, and furthermore, the combination of pruning and weight decay promotes generalization more effectively than mere regularization of weight norms.
To verify this, we introduce edge edge-popup (Ramanujan et al., 2020), a method that learns how to prune weights without weight updates. In edge-popup, each weight is assigned a score, and these scores are updated through backpropagation to determine which weights to prune. For details regarding edge refer to the Appendix A. We validate our claim by optimizing using three different methods.
WD Training from O using Weight Decay with weight update (same base model).
EP w/o WD Training from 0mem using Edge-Popup without weight update.
EP w/ WD Training from 0 using Edge-Popup and Weight Decay with weight update.
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In Table 1, the result for EP without weight decay (EP w/o WD) shows that the network achieves generalization performance of 0.92 without any changes to the weights (merely by pruning weights). This result demonstrates that generalization can be achieved solely through structural exploration without updating the weights, thereby strengthening our claim that delayed generalization occurs duc to the discovery of a good structure. Additionally, the EP w/ WD result shows the fastest improvement in test accuracy and is the most effective in promoting generalization. These insights suggest that practitioners may improve generalization by incorporating methods that directly optimize beneficial structures rather than solely relying traditional regularization techniques like weight decay. Our findings highlight the potential of grokked tickets to inform the development of new, structure-orienter regularization techniques.
In Table 1, similar to the analysis in subsection 5.2, we examined the average values of graph property metrics on the network's weight matrix under EP w/o WD. As observed in subsection 5.2, the weighted spectral gap initially increased and then declined, while the ramanujan gap continued decrease. Furthermore, both the average path length and the clustoring coofficient converged similar values.
Answer to Q3
Can generalization be achieved solely through structural exploration (pruning) without weight updates?
Yes, pruming alone significantly improves generalization, demonstrating that discovering good subnet- works is sufficient. to transition from memorization to generalization.
6 Discussion and Related Works
In this paper, we conducted set of experiments to understand the mechanism of grokking (delayed generalization). Below is a summary of observations. (1) In subsection 3.2, the use of the lottery ticket significantly reduces delayed generalization. (2) In section 4, good structure is more important factor in explaining grokking than the weight norm and sparsity by comparing it with the same weight norm and sparsity level. (3) In subsection 5.2, the structure of grokked tickets is well-suited for task. (4) In subsection 5.1, good structure gradually discovered, corresponding to improvement test accuracy. (5) subsection 5.3, pruning without updating weights from a memorizing solution increases test accuracy. In this section, we connect these results with a prior explanation of the mechanism of grokking and relevant related works.
Weight norm reduction Liu et al. (2023a) suggests that the reduction of weight norms is crucial for generalization. In Figure 5, our results go further to show that, rather than simply redueing weight norms, the network discovers good structure (subnetwork), resulting in the reduction of weight norms.
Sparsity and Efficiency Merrill et al. (2023) argued that the grokking phase transition corresponds to the emergence of a sparse subnetwork that dominates model predictions. While they empirically studied parse parily tasks where sparsity is evident, we are conducting tasks (modular arithmetic,, MNIST) commonly used in grokking research and architecture (MLP, Transformer). Furthermore, Figure 5, we demonstrate not only sparsity but also that good structure is crucial.
Representation learning Liu et al. (2022); Nanda et al. (2023); Liu et al. (2023a); Gromov (2023) showed the quality of representation as key to distinguishing memorizing and generalizing networks. Figure 8 demonstrates that good structure contributes to the acquisition good representation, suggesting the importance of inner structure (network topology) in achieving good representations.
Regularization Weight decay (Rumelhart et al., 1986) is one of the most commonly used regularization techniques and is known to be a critical factor in grokking (Power et al., 2022; Liu al., 2023a). In Appendix H. we show if the good structure discovered, the network generalizes without weight decay, indicating that weight decay works structure exploration, which suggests that weight decay contributes
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not to reducing weight but to exploring good structure. In addition, tested pruning as new force to induce generalization (Table 1). However, as shown in the results of pruning at initialization (Figure 5-(b)), improper pruning can degrade generalization performance.
Lottery ticket hypothesis The lottery ticket hypothesis (Frankle & Carbin, 2019) suggests that good structures are crucial for generalization, but it remains unclear how these structures are acquired during training and how they correspond the network's representations. To the best of my knowledge, our paper is the first to connect grokking and the lottery ticket hypothesis, demonstrating how good structures emerge (subsection 5.1) and contribute to effective representations (subscction 5.2). Building this, we further show that while faster generalization is well-documented property of winning tickets, our work goes beyond by exploring how the discovery of good structures.
Future Directions While our study focused on simple tasks, the key insight of discovering good subnetworks can extend to more complex domains like large-scale image recognition and language generation. Our findings suggest that good subnetworks, or "lottery tickets", help mitigate delayed generalization, offering a structural perspective on regularization. Traditional methods like weight decay indirectly promote structure discovery, whereas pruning and bask-based optimization provide more direct and potentially more effective approach. As shown in Table 1, combining weight decay with the edge-popup algorithm accelerates generalization, demonstrating the advantages of structure-aware techniques. Future rescarch could integrate these insights into contemporary archilectures to enhance learning efficiency and robustness. Exploring pruning-based or mask-optimization strategies within large-scale models may further validate their effectiveness real-world applications. Bridging these structural insights with mainstream training techniques could improve model generalization and efficiency in complex settings.
Additionally, based the insights regarding graph properties obtained in subsection 5.2 of this study, future work should explicitly sample neural network structures according to graph properties, demonstrated in previous research (Javaheripi et al., 2021). Incorporating the struetural insights identified here into the sampling process could facilitate the discovery of more efficient and robust network architectures.
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References
Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via over- parameterization, 2019.
Xander Davies, Lauro Langosco, and David Krueger. Unifying grokking and double descent. arXiv preprint arXiv:2303.06113, 2023.
Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks, 2019.
Jonathan Frankle, Gintare Karolina Dziugaite, Daniel M. Roy, and Michael Carbin. Stabilizing the lottery ticket hypothesis, 2020.
Hiroki Furuta, Gouki Minegishi, Yusuke Iwasawa, and Yutaka Matsuo. Interpreting grokked transformers in complex modular arithmetic, 2024.
Andrey Gromov. Grokking modular arithmetic. arXiv preprint arXiv:2301.0200 2023.
Duc N.M Hoang, Souvik Kundu, Shiwei Liu, and Zhangyang Wang. Don't just prune by magnitude! your mask topology is secret weapon. In Thirty-seventh Conference on Neural Information Processing Systems, 2023a. URL
jWV7..
Duc N.M Hoang, Shiwei Liu, Radu Marculescu, and Zhangyang Wang. REVISITING PRUNING AT INITIALIZATION THROUGH THE LENS OF RAMANUJAN GRAPH. In The Fleventh International Conference on Learning Representations, 2023b. URL https://openreviiet wwtpwwwwwwwwweeeeeeeeee.mm net//orrumidduucDsstt.
P. Jaccard. Etude comporative de la distribution florale dans une portion des Alpes et du Jura. Bulletin de la Société vaudoise des sciences naturelles. Impr. Corbaz, 1901. URL https://books. http://booos..ggge....o google. co jp/books?
id=JCHdmpEACAAJ.
Mojan Javaheripi, Bita Darvish Rouhani, and Farinaz Koushanfar. Swann: Small-world architecture for fast convergence of neural networks. TEEE Journal on Emerging and Selected Topics in Cirewils and Systems, 11(4):575 -55,5,575-55,, 2021. doi:
R...AA...02..3.1 2021.31253...
Alex Krizhevsky, Tlya Sutskever, and Geoffrey F. Hinton. Imagenet classification with deep convolutional neural networks. In F. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 25, pp. 1097-1105. Curran Associates, Inc., 2012. URL http: http://papers. pdf.
Tanishq Kumar, Blake Bordelon, Samuel J. Gershman, and Cengiz Pehlevan. Grokking as the transition from lazy to rich training dynamics. arXiv preprint arXio:2310.061 0610000611110 2023.
Tanishq Kumar, Blake Bordelon, Samuel J. Gershman, and Cengiz Pehlevan. Grokking as the transition from lazy rich training dynamics. The Twelfth International Conference on Learning Representations, 2024. URL https://openrevvew /ttp
Namhoon Lee, Thalaiyasingam Ajanthan, and Philip H. Torr. Snip: Single-shot network pruning based on connection scnsitivity, 2019.
Ziming Liu, Ouail Kitouni, Niklas Nolte, Eric J. Michaud, Max Tegmark, and Mike Williams. Towards understanding grokking: An effective theory of representation learning. arXie preprint ar 3rXw:22205.1033 2022.
Ziming Liu, Eric J Michand, and Max Tegmark. Omnigrok: Grokking beyond algorithmic data. In
International Conference on Learning Representations, 2023a.
Ziming Liu, Ziqian Zhong, and Max Tegmark. Grokking compression: A nonlinear complexity perspective. arxiv preprint ar 0rx:2231000055 0.9911 2023b.
Tya Loshchilov and Frank Hutter. Decoupled weight. decay regularization, 2019.
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William Merrill, Nikolaos Tsilivis, and Aman Shukla. A tale of two circuits: Grokking competition of sparse and dense subnetworks. arxiv preprint arXiv:2303. 110003..187. 11873, 2023.
Jack Miller, Charles O'Neill, and Thang Bui. Grokking beyond neural networks: An empirical exploration with model complexity. arXie preprint orXiv:2310.179/4 2023.
Beren Millidge.
Grokking wea---rookiinn-Grirrro
'grokking', 2022.
URL https://www.bbr www.n.been.o io/
Shikhar Murty, Pratyusha Sharma, Jacob Andreas, and Christopher D. Manning. Grokking of hierarchical structure in vanilla transformers. arXiv preprint or Xiv:2305. 2023.
Vinod Nair and Geoffrey E. Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th International Conference on International Conference on Machine Learning, ICML'10, pp. 807 814, Madison, WI, USA, 2010, Omnipress. ISBN 9781603589077.
Preetum Nakkiran, Gal Kaplun, Yamini Bansal, Tristan Yang, Boaz Barak, and llya Sutskever. Deep double descent: Where bigger models and more data hurt. arXiv preprint arXiv:1912.0220 2019.
Neel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt. Progress measures for grokking via mechanistic interpretability. In International Conference on Learning Representations, 2023.
Behnam Neyshabur. Towards learning convolutions from scratch, 2020. URL ttttps://axxxi.orr//aaa 13657.
Michela Paganini and Jessica Zosa Forde. Bespoke vs. prêt-à-porter lottery tickets: Exploiting mask similarity for trainable sub-network finding, 2020.
Adam Pearce, Asma Ghandeharioun Nada Hussein, Nithum Thain, Martin Wattenberg, and Lucas Dixo. Do machine learning models memorize generalize? https://pair http://aaarrwttthogooooo withgoogle com/ explorables/ grokking/, 2023. [Online; accessed 11-October-2023].
Alethea Power, Yuri Burda, Harri Edwards, Igor Babuschkin, and Vedant Misra. Grokking: Generalization beyond overfitting on small algorithmic datasets. ar preprind urXiv:2201.021 2022.
Adilyanarayanan Radhakrishnan, Daniel Beaglehole, Parthe Pandit, and Mikhail Belkin. Mechanism of feature learning in deep fully connected networks and kernel machines that recursively learn features. arXiv preprint ar Xiv:2221 2. 19881, 2022.
Vivek Ramanujan, Mitchell Wortsman, Aniruddha Kembhavi, Ali Farhadi, and Mohammad Rastegari. What's hidden in a randomly weighted neural network?, 2020.
Noa Rubin, Inbar Seroussi, and Zohar Ringel. Droplets good representations: Grokking as a first order phase transition in two layer networks. arXiv preprint arXiv:2310.08789, 2023.
David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. nature, (23(608)):533-5555 1986.
Keitaro Sakamoto and Issei Sato. Analyzing lottery ticket hypothesis from pac-bayesial theory perspective, 2022.
Dashiell Stander, Qinan Yu, Honglu Fan, and Stella Biderman. Grokking group multiplication with cosets. arXiv preprint 2r11::2312,06581, 2023.
Hidenori Tanaka, Daniel Kunin, Daniel L. K. Yamins, and Surya Ganguli. Pruning neural networks without any data by iteratively conserving synaptic flow, 2020.
Vimal Thilak, Etai Littwin, Shuangfei Zhai, Omid Saremi, Roni Paiss, and Joshua Susskind. The slingshot mechanism: An empirical study of adaptive optimizers and the grokking phenomenon. arv preprint ar (:Xi:2220604817 2022.
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Vikrant Varma, Rohin Shah, Zachary Kenton, János Kramár, and Ramana Kumar. Explaining grokking through circuit efficiency. arXiv preprint arXim:2309.0m 022000 2023.
Chaoqi Wang, Guodong Zhang, and Roger Grosse. Picking winning tickets before training by preserving gradient flow, 2020.
Yulong Wang, Xiaolu Zhang, Lingxi Xie, Jun Zhou, Hang Su, Bo Zhang, and Xiaolin Hu. Pruming from scratch, 2019.
Jiaxuan You, Jure Leskovec, Kaiming He, and Saining Xie. Graph structure of neural networks. In Hal Daumé III and Aarti Singh (eds.), Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 10881-10890 PMLR, 13 18 Jul 2020. URL
mlr. press/v119/you200 html.
Zhenyu Zhang, Xuxi Chen, Tianlong Chen, and Zhangyang Wang. Elficient lottery ticket finding: Less data is more, 2021.
Ziqian Zhong, Ziming Liu, Max Tegmark, and Jacob Andreas. The clock and the pizza: Two stories in mechanistic explanation of neural networks. In Neurol Information Processing Systems, 2023.
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A Edge-popup algorithm
We provide detailed explanation of the edge-popup algorithm. Edge-popup Ramanujan et al. (2020) is a method for finding effective subnetworks within randomly initialized neural networks without weight updates.
Basic Flow:
1. Random Initialization: Initialize the weights w of a large neural network randomly.
Assign Scores: Assign a score 8ij to each edge mhy randomly.
3. Select Subnetwork: Form a subnetwork g using only the edges with top 4% scores.
4. Optimize Scores: Update the scores 8ij based the performance of the subnetwork G.
Score Updates: Update the score siz of each edge wij using the following formula:
OL 7
Sij
8ij
0I,
Q is the learning rate
0L is the gradient of the loss with respect to the input of the j-th neuron öT,
Z; is the output of the i-th neuron
Actual Computation: Forward Pass: Compute using only the edges whose scores sty are the top k%.
1
Σ wiy Zih(ss))ss)
Here, A(sig) if sij is in the top k% and 0 otherwise.
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B Different configurations of the task and the architecture.
B.1 MLP for MNIST
We use 4-layer MLP for the MNIST classification. The difference from the regular classification is that using Mean Squared Error (MSF) for the loss. We adopted this setting following prior research (Liu et al., 2023a). In (Liu et al., 2023a), it was confirmed in the Appendix that grokking occurred without any problems, even when trying with cross-entropy. Figure 10 shows the test and train accuracy on various configurations. Ii is evident that grokked lickets accelerate generalization in all configuratioss, and the exploration of grokked tickets contributes to generalization.
B.2 Transformer for modular addition
Similar to Nanda et al. (2023), we use 1-layer transformer in all experiments. We use single-head attention and omit layer norm.
Hyperparameters:
dvocab = 67: Size of the input and output spaces (same as p).
demb 500: Embodding size.
dmip- 128: Width of the MLP layer.
Parameters:
Wg: Embedding layer.
poss Positional embedding.
Wo: Query matrix.
WK: Key matrix.
Wv: Value matrix.
Wo: Attention output.
Win, bm: Weights and bias of the first layer of the MLP.
Wour, bout: Weights and bias of the second layer of the MLP.
Wo: Unembedding layer.
We describe the process of obtaining the logits for the single-layer model. Note that the loss is only calculated (1) denote the token at position the number of input tokens is 2, and 2²(") is one-hot vector. We denote the attention scores A and the triangular matrix with negative infinite clements as M, which is used for causal attention.
from the logits on the final token. Let x;
layer L. Here, is 0 or 1, as
The logits are caleulated via the following equations:
(0) + W 1 (0) z WEZ ;
A softmax (1)T W Wo M),
(2) WoWr(a (1
x. (3) Wout ReL. + bya) + bout + (2)
logits softmax (Wannx)(Wz2())
Figure 10 shows the training and test accuracy of the base model (green) and the grokked ticket (blue). Across datasets (c.g., Modular Addition, MNIST) and architoctures (Transformer). the grokked ticket (bluc) consistently reaches generalization faster than the base model (green). These results underscore the importance of structural elements in grokking, regardless of the task architecture.
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MLP on MA
Transformer on MA
MLP on MNIST
1.0
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25000 30000
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Base model (train)
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Grokked tickets (train)
Figure 10: Comparison of base model (green) and grokked ticket (bluc). Each column corresponds to the different configurations of the task (Modular Addition and MNIST) and the archilecture (MLP and Transformer). The dashed line represents the results of the training data.
C Experiments with Different Seeds
To ensure reproducibility, we conduct experiments with three different seeds and present the results for experiments in Figure 2. In addition to accuracy, we plot the loss of train and test in Figure 11.
MLP MA
Transformer on MA
Base model
35
Base model test
40
model (Irain)
30
model
30
Grokked ticket (test)
25
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20
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30000
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(a) Loss of MLP on Modular Addition
(b) Loss of Transformer on Modular Addition
Figure 11: Train and Test loss of base model and grokked ticket on MLP and Transformer. In both of architectures, grokked ticket convergent faster than base model
Additionally, to ensure reproducibility, we conduct experiments with three different seeds and present the results for experiments in subsection 4.1 and Transformer architecture experiment.
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Accuracy
norm
Li
Asouric
Genera Reed zone
29
30c
Epoch
208
2000
282
3000
Epoch
1000
41000
3002
-80
1.03
Epoch
2003
2000
Base model (train)
Base model (test)
Grokking tickets
Controlled dense (L) norm)
Controlled dense (L.2 norm)
Figure 12: (Left) Test accuracy dynamics of the base model, grokked ticket, and controlled dense model (L1 norm and L2 norm) with three different seeds. The grokked ticket reaches generalization much faster than other models. (Center) L2 norm dynamics of the base model, grokked ticket, and controlled dense model. (Right) L1 norm dynamics of the base model, grokked ticket, and controlled dense models. From the perspective of the 1,2 norm, all models appear converge to similar solution (Generalized zone). However, from the perspective of L1 norms, they converge to different values.
Accurucy
1. norm
norm
Accunce
noem
-20
Epoch
Epoch
Epoch
Base model (triin)
Base model (lest)
Grokked tickets
Controlled dense (L) norm)
Controlled dense (L2 norm)
Test accuracy dynamics of the base model, grokked ticket, and controlled dense model (L1 norm and L2 norm) with three different seeds the Transformer. The grokked licket. reaches generalization much faster than other models. (Center) L2 norm dynamics of the base model, grokked ticket, and controlled dense model. (Right) L1 norm dynamics of the base model, grokked ticket, and controlled dense models. From the perspective of the L2 norm, all models appear to converge to similar solution (Generalized zone). However, from the perspective of I.1 norms, they converge to different values.
Figure 13:
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Figure 14: Comparing test accuracy of the different pruning methods. All Pal methods perform worse than the base model or, in some cases, perform worse than the random pruning with three different seeds.
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Published in Transactions on Machine Learning Research (05/2025)
D Pruning at initialization methods
Currently, the methodologies of pruning neural networks (NN) at initialization (such SNIP, GraSP, SynPlow) still exhibit a gap when compared to methods that use post-training information for pruning (like Lottery Ticket). Nonetheless, this area is experiencing a surge in research activity.
The basic flow of the pruning initialization is follows:
1. Randomly initialize a neural network f(æ; 0o)
Prune p% of the parameters in On according the scores S(0), creating mask in
3. Train the network from 0g @ m.
According to Tanaka et al. (2020), research pruning at initialization boils down to the methodology of determining the score in the above process 2, which can be uniformly described as follows:
dR ((0)
00
0L When the R is the training loss L, the resulting synaptic saliency metric is equivalent to ae @ 0 used in SNIP (Lee al., 2019). -(1 AL @ 0 use in Grasp (Wang al., 2020). Tanaka et al. (2020) proposed synflow algorithm RsF = 1s ( 10 1).. In section 4.2, all initial values were experimented with using the same weights and the same pruning rate.
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Figure 15: Jaccard distance change and Lest accuracy change polynominal regression (a) and sparse Published in Transactions on Machine Learning Research (05/2025)
E Significance of Periodicity in the Modular Addition Task
The modular addition task, defined as predicting c (a + mod p. inherently involves periodicity due to the modular arithmetic structure. Nanda et al. (2023) demonstrated that transformers trained on modular addition tasks rely on periodic representations to achieve generalization. Specifically, the networks embed inputs a and b into a Fourier basis, encoding them as sine and cosine components of key frequencies wy = 2 for some & N. These periodic representations are then combined using trigonometric identities within the network layers to compute the modular sum.
Mechanism of Periodic Representations
Nanda et al. (2023) reverse-engineered the weights and activations of a one-layer transformer trained on this task and found that the model computes:
cos(ur (a + b)) cos(waa) cos(w,b) (in sin(wx,))
sin(wx (a b)) (inaw cos(w + cos(w sin(wab),
using the embedding matrix and the attention and MLP layers. The logits for each possible output e are then computed by projecting these values using:
cos(w;(a + b- c)) = cos( (w + b)) cos(wxc) + sin(w; + b)) six(w..).
This approach ensures that the network's out put logits exhibit constructive interference at c e (a+ mod p, while destructive interference suppresses other incorrect values.
Given this mechanism, can be inferred that the model internally utilizes periodicity, such the addition formulas, to perform modular arithmetie.
F Structural Changes in Tasks Other Than the Modular Addition Task
1810
(a) Jaccard Distance on Polynominal Regression
(b) Jaccard Distance on Sparse Parity
parity (b). Structural changes (Jaccard distance) correspond to the acquisition of generalization ability.
23

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Published in Transactions on Machine Learning Research (05/2025)
G Is Weight Norm Sufficient to Explain Grokking in Transformer?
Figure 16 show the accuracy of base model, grokked ticket and Controlled dense (L1 and L2) on Transformer. The results show grokked ticket generalize faster than any other model. The results suggests that even in the case of Transformer, the discovery of grokked ticket is more important than weight norms.
Accuracy
Accuracy
Epoch
Base model (train)
Grokiedd tickets
Controlled dense (L2 norm)
Base model
Controlled dense (L.
Figure 16: Accuracy of base model, grokked ticket and Controlled dense Transformer.
Weight Decay work as Structure Explorer
Our result that the discovery of good structure happens between memorization and generalization, which indicates weight decay is essential for uncovering good structure but becomes redundant after their discovery.
In this section, we first explore the critical pruning ratio, which is the maximum pruning rate that can induce generalization without weight decay Figure 17-(a). We recognize that the critical pruning rate is between 0.8 and 0.9 because if the pruning rate increases to 0.9, the test accuracy dramatically decreases. Thus, we gradually increased the pruning rate in increments of 0.01 from 0.8 and found that the k = 0.81 is the critical pruning ratio. We then compare the behavior of the grokked ticket without weight decay and the base model. Figure 17-(b) shows the results of the experiments. The results show that if good strueture discovered, the network fully generalizes without weight decay, indicating that weight decay works as structure explorer.
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(a) Test accuracy with different pruning rate
(b) Test accuracy with the critical pruning rato (0.81)
Figure 17: The effect of pruning rate on lest accuracy without weight decay(left). Test accuracy of grokked ticket with critical pruning rate (0.81) without weight decay(right).
In this section, we show that with precise pruning ratios, the grokked ticket does not require weight decay to generalize, indicating that weight docay is essential for uncovering good subnetworks but bocomes redundant after their discovery. We first explore the critical pruning ratio, which is the maximum pruning rate that induce grokking (Figure 17-a). In this case (Figure 17-a), we recognize that the critical pruming rate is
24

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Published in Transactions on Machine Learning Research (05/2025)
between 0.8 and 0.9 because if the pruning rate increases to 0.9, the test accuracy dramatically decreases. Thus, we gradually increased the pruning rate in increments of 0.01 from 0.8 and found that the k = 0.81 is the critical pruning ratio. We then compare the behavior of the grokked ticket without weight decay (a=0.0 0.0) and the base model. Figure 17-b show the results of the experiments. As shown in the figure, the test accuracy reaches perfect generalization without weight decay. The results show that the grokked ticket. with the critical pruning ratio does not require any weight docay during the optimization.
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Published in Transactions on Machine Learning Research (05/2025)
Visualization of Weights and grokked ticket
To reveal the characteristics of the grokked ticket, we visualized the weight matrices and the corresponding masks of the grokked ticket, as shown in Figure 18. The task under consideration is Modular Addition, implemented using the following MLP architecture:
softmax(o((Ea + En))in) Wou W Wnnemb);
(4)
where E. and Es are the input embeddings, Win, Wout, and Wunemb are the respective weight matrices, and a denotes the activation function. This architecture models the relationship between inputs in the Modular Addition task.
Figure 18 visualizes the learned weight matrices WE, W Winproj, W Woutproj, and Wo (top row) after generalization, as well as the corresponding masks from the grokked ticket (bottom row). The weight matrices exhibit. periodic patterns, reflecting good structure learned during training. Furthermore, the grokked ticket masks align with these periodic characteristics, indicating that the grokked ticket has successfully acquired structures that are beneficial for the lask of Modular Addition. These results highlight the ability of the grokked Licket. to uncover meaningful patterns that contribute to the model's performance.
(Pal) methods: Random, GraSP, SNIP, and SyuFlow. Unlike the results shown in Figure 18, these methods do not exhibit periodic structures. This comparison highlights the superiority of the grokked ticket in acquiring structures that are more conducive the Modular Addition task, further emphasizing its advantage over traditional Pal methods.
For comparison, Figure 19 show visualization of masks (structures) obtained by
entedd
unimmbee eY
Outzre W
weight mask
mark
Figure 18: Visualization of weight matrices WE. Winproj. Wottpro and Wy (top), as well as the corresponding masks from the grokked ticket (bottom). Periodic patterns are observed in the weight matrices, and the masks of the grokked ticket reflect these characteristics. This indicates that the grokked ticket has acquired structures beneficial for the task (Modular Addition).
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Published in Transactions on Machine Learning Research (05/2025)
(a) Random
(b) Grasp
(c) Snip
(d) Synflow
Figure 19: Visualization of masks (structures) obtained by pruming-tt-niitallation (Pal) methods: Random,
GraSP, SNIP, and SynFlow. Compared to grokked ticket in Figure 18, it can be observed that periodic structures are not achieved.
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Published in Transactions on Machine Learning Research (05/2025)
J Graph Property Metrics
In this section, we introduce the graph-theoretic metrics used in our study; ramanujan gap, weighted spectral gap, and two measures derived from vicwing the pruned network as a relational graph the average path length and the clustering coefficient.
Ramanujan Graph. A ramanajan graph known for combining sparsity with strong connectivity. Consider k-regular graph, where every node has degree k. The eigenvalues of its adjacency matrix A are sorted
Au A1 ...
2 An-
Here, the largest eigenvalue λn equals k, and the smallest An equals k;; these trivial eigenvalues. The graph is called ramanujan graph if all non-trivid eigenvalues X (i.e., those with |vl * k) satisfy
max Ax| S 22k A.
Such graphs are sparse but exhibit ellicient information diffusson.
Ramanujan Gap. The ramanujan gap measures how closely a graph's largest non-trivial eigenvalue approaches the theoretical bound II.-I. While this was originally defined for strictly k-regular graphs, Hoang al. (2023b;a) extend irregular graphs (as often arise in unstructured pruned networks) by replacing with the average degree davgi
A. V 2 davg - A(G),
where A(G) is the largest non-trivial eigenvalue in absolute value. For bipartite graphs, the largest and smallest eigenvalues are symmetrie in magnitude, making the thire-largest eigenvalue a natural choice for ((G). Hoang et al. (2023a) reports a negative correlation between A, and performance.
Weighted Spectral Gap The spectval gap of adjacency matrix typically refers to the difference between its largest eigenvalue Ao and its sccond-largest cigenvalue A1:
Spectral Gap Ao A1-
Hoang al. (2023a) extends this to weighted spectral gap by incorporating learned weights (e.g., from training) into the adjacency matrix. This better reflects how pruming other structural modifications alter the effective connectivity, often revealing strong links to model performance.
Beyond eigenvalue- based measures, interpreting the pruned network as a relational graph (You et al., 2020) allows the computation of established network-science metrics:
Average Path Length. The average path length defined the mean distance between all pairs of nodes:
1
Σ d(i KF
= N(N- - 1)
where d(i,j) is the shortest-path distance between nodes i and j, and N the number of nodes in the graph. A smaller implies fewer hops on average to traverse between nodes, indicating a more compact network.
Clustering Coefficient. We use the average clustering coefficient C to characterize how densely nodes tend to form triangles with their neighbors. The local clustering coefficient C; of node i is given by
2e;
C (;; 11)
where k; is the degree of node i and 6, is the number of edges among its h; neighbors. The average clustering coefficient is then
Σ C
= N
1=1
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Higher values of C' indicate strong local grouping community structure.
Interestingly, prior work (You et al., 2020) has shown that when focusing on performance, the best graph structure does not necessarily push both average path length and clustering coefficient to their extremes (either too small or too large). Instead, the perffrerfoning networks often occupy a moderate or balanced regime of I and C.
K Evolution of Graph Property for each layer
Figure 20-(a) illustrates how the weighted spectral gap (left) and ramanujan gap (right) evolve throughout training across different layers, specifically analyzing Wemb, Wi. Wout, W Wunemb.
We observe that the weighted spectral gap exhibits a sharp increase during the memorization phase, indicating pronounced separation between the largest and second-larges eigenvalues while the model is overfitting. This suggests that, in the memorization phase, weights are dominated by a particular component, corresponding to the leading eigenvector. However, as training progresses and generalization improves, the spectral gap narrows, reflecting a more balanced parameter space utilization. Mcanwhile, the ramanujan gap steadily decreases from the beginning, aligning with prior studies (Hoang et al., 2023a) that report a consistent downward trend. This suggests a structural shift in the graph induced by weight matrices, possibly corresponding the network's s shift from memorization to generalization.
Figure 20-(b) presents metrics derived from viewing the entire network as relational graph (You et al., 2020). The average path length (left) increases alongside the rise in test accuracy, then stabilizes at moderate range. This implies that as the network searches for a generalizable solution (i.e., discovering the Grokked ticket), its effective connectivity becomes more sparse. Once a solution is found, the path length stabilizes, suggesting a structured and efficient parameter allocation. Conversely, the clustering coefficient. (right) decreases as test, accuracy improves, before settling at an intermediate value. This decline suggests that during the discovery of the gencralization pattern, local connectivity in the graph weakens, but does not collapse entirely. Instead, it converges to a structured regime where some degree of locality is maintained while avoiding excessive clustering, consistent with previous findings (You et al., 2020).
41.
Solla
Vole
(a) Ramanujan Gap and Weight Spectral Gap of Wemb
(b) Ramamujan Gap and Weight Spectral Gap of Win.
Tois
Ver NC
30000
21000
(c) Ramanujan Gap and Weight Spectral Gap of Wout
(d) Ramanujan Gap and Weight Spectral Gap of W Winemb
Figure 20: Evolution of the weighted spectral gap and Ramanujan gap of each layer.
29


Reviews:
[{"review_id": "X2dK8v0A2O", "reviewer": "Reviewer_dmLd", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengthes\n\n1. The paper makes a novel, interesting connection between grokking and the Lottery Ticket, which seems to me a new perspective on delayed generalization.\n\n2. Authors have perform comprehensive experiments to validate their hypothesis, that is, it is the good structures that are responsible for the fast convergence rather than reduce weight and increased sparsity. \n\n3. Authors have done some qualitative analysis to shed light on what constitute a good substructure. Finally, authors demonstrated that pruning alone can transition the network from memorizing to generalizing solutions. \n\n## Weaknesses \n\n1. Limited evaluation setting. Would the results generalize to real-world applications, such as in the large, complex vision or language models?\n\n2. For the pruning method presented in 5.3, the authors state that:\n\n> To verify this, we introduce edge-popup (Ramanujan et al., 2020), a method that learns how to prune\nweights without weight updates\n\nWhy this particular method, why not others? Would the results differ significantly if switched to other pruning methods?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates the phenomenon of grokking, where neural networks transition from memorization to generalization. The authors argue there is a connection between grokking and the Lottery Ticket, in particular, grokking has more to do with good subnetworks than weight norm and sparsity. The research demonstrates that lottery tickets obtained at the generalization phase significantly reduce delayed generalization across various tasks such as modular arithmetic, polynomial regression, sparse parity, and MNIST classification.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address weaknesses.", "broader_impact_concerns": "No", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "qx46YzEiM7", "reviewer": "Reviewer_3B6m", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "S1: I am not the biggest expert in lottery tickets and grokking, but, like many people, have read a good bit about them in the past few years, this seems like the most illuminating one I have seen. \n\nS2: Writing is good and clear. For example, outlining the questions at the top of section 5 is nice. Feel free to answer them up there too. \n\nS3: I think that the experiments have been done thoroughly. For example, I think this paper would have been viable missing one or two of the subsections of section 5. But the authors did all 3. \n\nW1: As is a general concern with lottery ticket/grokking work, it is not clear whether this is useful. Experiments focus on basically understanding lottery tickets and grokking but don't suggest any clear type of value to practitioners. This is ok, and it's still valid science. But it would be nice to start seeing the extent to which these insights help anyone. Grokking and lottery tickets are popular topics, and I'm not penalizing the paper for working on them, but research on them increasingly seems like the self-perpetuating pursuit of some curiosities. \n\nW2: Experiments are pretty small scale. But this is a problem of the lottery ticket/grokking literature at large. So I don't fault this paper for simple tasks/nets. However, it would be nice if experiments found implications for more sota models.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper works to unite the study two of machine learning's spookiest phenomena: lottery tickets and grokking. The key contribution of the paper is showing that the network structure, as studied through the lens of lottery tickets, can help to explain grokking better than sparsity or norms. The paper's central experiment is that generalization speeds up when you use lottery ticket pruning. They also do work to disentangle hypotheses about whether weight norms, sparsity, or lottery tickets explain grokking, and they show that norm and sparsity alone are insufficient (but still seem helpful).  \n\nOverall, I think that this paper is well done and interesting, though not clearly impactful. I currently lean toward acceptance with minimal revisions.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Consider adding answers to the questions asked at the top of section 5. \n\nSo what? I think it would be good to add a paragraph in the discussion to discuss future work and/or what we should do with these results.  I don't know if there's much of a \"so what,\" but feel free to discuss what there might be.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ctI6ZLFEMc", "reviewer": "Reviewer_UuyT", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n* The idea of the paper is novel and smooth; combining delayed generalization and sparse structure is novel.\n* The finding that LTH tickets can reduce delayed generalization is novel and interesting.\n* The first two findings of the paper (rejection of Hypotheses 1 and 2) are empirically supported.\n* The paper is well-written and easy to follow, and the experiments are well-supported (also, all the technical aspects of the experimental setup are included). The appendix and code provide all the details for fair reproducibility.\n\n\n## Major Weaknesses\n* From my perspective, the major limitation is the definition of a good structure throughout the whole paper. No definition is given, and then, in section 5.2, the Jaccard Distance is proposed. However, I do not understand how the JD computed over the binary masks at different epochs can be a measure to define a good structure. It is true that, as shown in Figure 8, JD looks like it’s correlated to the accuracy change; however, this JD says nothing about the structures of the network itself. It just says that it changes. There are several works that analyze the inner structure of sparse neural networks and their correlation with task performance via both basic graph properties [1,2] or expander theory [3,4,5]. In this paper, while I agree with the authors that the structure of the grokking ticket may be the reason that explains such a reduction in generalization delay, I completely miss a metric from the structure of the network that shows me that. Even in the successive experiments with edge-popup, while it’s true that the algorithm changes the binary mask of the sparse network with no weights update, the paper provides no information about the properties of the network at epoch 0 (low accuracy) vs. the last epoch (good accuracy). Understanding the different properties of these two structures (rather than how the structure changes) could lead to revealing why one structure is better than another.\n\n## Minor Weaknesses\n* The theory of the Strong Lottery Ticket Hypothesis (hence the mask found by the Edge-Popup algorithm) has been theoretically proven in [6].\n* Hypothesis 2: It would be more appropriate to use LTH rather than the PaI algorithm to reject this hypothesis since PaI algorithms are established to be difficult to train from scratch [7]. Hence, the delayed generalization could also be correlated with this phenomenon.\n* Some citations are wrongly formatted.\n* Appendix E should, in my opinion, be in the main paper.\n\n\n\n[1] Stier, Julian, and Michael Granitzer. \"Structural analysis of sparse neural networks.\" Procedia Computer Science 159 (2019): 107-116. \\\n[2] Cunegatti, Elia, et al. \"Understanding Sparse Neural Networks from their Topology via Multipartite Graph Representations.\" Transactions on Machine Learning Research \\\n[3] Pal, Bithika, et al. \"A study on the ramanujan graph property of winning lottery tickets.\" International Conference on Machine Learning. PMLR, 2022 \\\n[4] Hoang, Duc NM, and Shiwei Liu. \"Revisiting pruning at initialization through the lens of ramanujan graph.\" ICLR 2023 (2023). \\\n[5] Laenen, Steinar. \"One-shot neural network pruning via spectral graph sparsification.\" Topological, Algebraic and Geometric Learning Workshops 2023. PMLR, 2023. \\\n[6] Malach, Eran, et al. \"Proving the lottery ticket hypothesis: Pruning is all you need.\" International Conference on Machine Learning. PMLR, 2020 \\\n[7] Evci, Utku, et al. \"The difficulty of training sparse neural networks.\" arXiv preprint arXiv:1906.10732 (2019).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes an interesting study about the connection between Grokking tickets and the structure of neural networks. It first shows how sparse networks (discovered by using the Lottery Ticket Hypothesis) can reduce generalization delay with respect to a dense model. Then, the authors empirically demonstrate how both Weight Norm and Sparsity Level are not the main factors that could explain the faster generalization phenomena. The authors then motivate such reduced generalization delay in Grokking tickets based on the Fourier Entropy computed over neurons, showing how Grokking tickets exhibit earlier periodic structure during training. Then they propose a metric based on Jaccard Distance to evaluate the mask change during training and its correlation with task accuracy. The paper concludes by combining a well-known agnostic pruning algorithm (edge-popup) with weight decay to show how the sparse structure discovered by the pruning algorithm improves task performance.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would ask the author to take into consideration my Main Weakness. Even though I liked the idea of the paper and part of its execution (till section 5.2), I missed the link to what a good structure is, which should be one of the main strength points of the paper (along with Sections 4 and 3 and the rejection of Hypothesis 1 and 2).\nI think that discovering what a good structure (with some metrics) is is required to support the claim that the network structure matters in grokking tickets. It would be enough to analyze the difference in structures between the network obtained at the beginning of the training and the end with edge-popup. This step, in my opinion, would be a key to strengthening the paper's message. I'm more than willing to discuss this point with the authors during the discussion phase.\nAlso, check the minors.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper investigates the mechanism of grokking, a delayed generalization phenomenon in which neural networks transition from memorization to generalization. Specifically, the paper connects grokking to the lottery ticket hypothesis (LTH), experimentally demonstrating that the network structure has a greater impact on grokking than previously known factors such as weight norm and sparsity.

The paper has clear strengths and weaknesses. 

Strengths:

1). Paper is clear written and easy to follow.

2). The paper establishes an innovative connection between grokking and LTH, offering new insights into the mechanisms behind delayed generalization and generalization of neural networks. 

3). The paper provides strong evidence challenging previous hypotheses, i.e. that weight norm and sparsity are the main factors influencing grokking, by highlighting the importance of network structure in achieving efficient generalization.

4). During rebuttal, the authors provide detailed clarifications and additional experimental results.

Weaknesses:

1). The evidence and explanation regarding the good structure are somewhat indirect and empirical.

2). The paper lacks large-scale demonstrations and does not provide clear discussion on practical implications.  

Despite these weaknesses, the paper opens promising new directions for understanding delayed generalization and generalization of neural networks, which could be of interest to some TMLR’s audience. Therefore, I recommend acceptance.
Decision: Accept (poster)