The Butterfly Effect: Neural Network Training Trajectories Are Highly Sensitive to Initial Conditions
Authors: Gül Sena Altıntaş
Devin Kwok
Colin Raffel
David Rolnick
Venue: ICML 2025 poster (2025)
Keywords: early training
linear mode connectivity
loss landscape
perturbation sensitivity
permutation symmetry
Primary area: deep_learning->theory
TL;DR: Tiny perturbations cause neural network training to diverge to distinct loss basins

Abstract: Neural network training is inherently sensitive to initialization and the randomness induced by stochastic gradient descent. However, it is unclear to what extent such effects lead to meaningfully different networks, either in terms of the models' weights or the underlying functions that were learned. In this work, we show that during the initial "chaotic" phase of training, even extremely small perturbations reliably causes otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time. We quantify this divergence through (i) $L^2$ distance between parameters, (ii) the loss barrier when interpolating between networks, (iii) $L^2$ and barrier between parameters after permutation alignment, and (iv) representational similarity between intermediate activations; revealing how perturbations across different hyperparameter or fine-tuning settings drive training trajectories toward distinct loss minima. Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.

The Butterfly Effect: Neural Network Training Trajectories Are Highly Sensitive to Initial Conditions
Gül Sena Altintas 12 Devin Kwok 34
Colin Raffel 12
David Rolnick 34
Abstract
8
Neural network training inherently sensitive to initialization and the randomness induced by stochastic gradient descent. However, it is unclear to what extent such effects lead to meaningfully different networks, either in terms of the mod- els' weights or the underlying functions that were learned. In this work, we show that during the initial "chaotic" phase of training, even extremely small perturbations reliably causes otherwise iden- tical training trajectories to diverge-ar effect that diminishes rapidly over training time. We quantify this divergence through (i) L distance between parameters, (ii) the loss barrier when in- terpolating between networks, (iii) L and barrier between parameters after permutation alignment. and (iv) representational similarity between inter- mediate activations; revealing how perturbations across different hyperparameter or fine-tur tne-ttning set- tings drive training trajectories toward distinct loss minima. Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.
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Figure 1. Left: illustration of the "butterfly effect": network Go is trained until time and perturbed by & early (Ex) or later (sc-) in training. Both copies are trained deterministically until T and their divergence is measured (purple loss basins). Right: barriers (training cross-entropy loss) at T versus perturbation magnitude (a = is the network's initialization scale). A perturbation of as little one weight (leftmost points) reliahly causes divergence when applied early, but not when applied at later (colors).
parameterize identical or similar functions.
Although training instability affects convergence in general (Iyer et al., 2023; Jastrzebski et al., 2020), it also prevents different runs of the same network from consistently reach- ing one particular solution, which has practical implications for model merging (Singh & Jaggi, 2020; Ainsworth et al., 2023) and ensembling. Prior work has categorized train- ing into chaotic (early) and stable (late) phases (Fort et al., 2020; Frankle et al., 2020a). but it is not clear if instability is more a product of noise (e.g. batch noise, data augmenta- tions, GPU indeterminacy). a network's current state (e.g. random vs. pre-trained), or the training procedure itself (e.g. optimizer and hyperparameter selection). A thorough understanding of instability should disentangle these fac- tors, since their influence may vary over training time and between different settings. Furthermore, depending on the task, not all of these factors can be controlled- when fine- tuning a pre- pre-trained model one can vary hyperparameters but not the initial weights, for instance, while the opposite may be true when pre-training the same model.
1. Introduction
Neural network training is known to be unstable in the sense that noise can disrupt convergence to a particular minimum (Frankle et al., 2020a; Wu et al., 2018). This is true even when considering solutions that perform equally well, since symmetries and connected minima in the loss landscape give rise to many different ways for neural network to
'Equal contribution 'University of Toronto 'Vector Institute e "cGGill University "Mila Quebec Al Institute. Correspondence to: Gül Sena Altuntay <gsaltintas @ cs. tdu>,, Devin Kwok <edein.kwok @ mail.mcciill.a>.
These limitations motivate to study training stability ab- sent the effects of noise. Drawing from the dynamical sys- tems perspective, we consider how much deterministic training map diverges when a controlled perturbation is ap- plied to its initial conditions (i.e. the starting weights of a network). By selecting initial weights from models trained
Proceedings of the 4rnd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
OOur code is available gsaltintas

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The Butterfly Effect
or fine-tuned for varying durations, we build up picture of where in the loss landscape training trajectories tend to diverge, and how sensitive trajectories are to perturbations in these regions. Crucially, our approach can quantify stabil- ity more precisely and for a wider range of models-from randomly initialized to pre-trained- was possible in prior works, which only measured instability to training noise. Our contributions are as follows:
existing solution in order improve merge-ability or model diversity, respectively.
Dynamical system stability. Neural network training has been studied as stochastic process (Smith & Le, 2018; Teh et al., 2016; Redman et al., 2024) and as dynamical system (Wu et al., 2018; Jastrzebski et al., 2020; Cohen et al., 2021). We take the latter view and analyze how small perturbations to a network's initial weights evolve over training. This has two advantages: first, can differentiate between instability due to noise vs. instability inherent to training itself (in the same way that deterministic dynamical systems exhibit chaos); and second, we can model simpler and larger (exponential) instabilities, which if present, should dominate over stochastic effects and make the stochastic perspective unnecessary (Wu et al., 2018).
1.
We show that a tiny perturbation of as little as a single weight early in training causes two otherwise identi- cally initialized and trained networks to diverge the butterfly effect (Figure 2).
2. Conversely, even networks that are stable to training noise diverge under larger perturbations. This points towards the possibility of using perturbations during training or fine-tuning to increase model diversity and ensembling performance (Figure 3).
Prior works (Wu et al., 2018; Jastrzebski et al., 2020; Cohen et al., 2021) have considered dynamical systems stability in terms of whether the L distance between a network's weights and a fixed minimum will grow over time. Our work differs in that we evaluate stability relative to a moving tra- jectory, and we also want to know if networks are diverging in function (not only in weights). To measure functional divergence, we use barriers (Frankle et al., 2020a), barri- ers accounting for neural network permutation symmetries (Entezari et al., 2022; Ainsworth et al., 2023), and represen- tational similarity via Angular CKA (Williams et al., 2021). Although these quantities are not amenable to approxima- tion by linear dynamical systems, they better capture the practical differences between networks.
3.
We find that stability improves under settings includ- ing wider/shallower networks and increased learning rate warm-up (among others), and these settings can be combined to further decrease, but not eliminate, instability near initialization (Figure 4).
4. While pre-trained networks are orders of magnitude more stable than randomly initialized networks, stabil- ity varies greatly between tasks and remarkably, more pre-training of language models can actually reduce stability in some cases (Figure 5).
5. Contrary to dynamical systems perspective, I,² and barriers (Frankle & Carbin, 2019) do not grow expo- nentially over training (Figure 6), and although 2 and barriers are strongly correlated in some cases, this is not true generally (Figure 7).
Linear mode connectivity. Barriers (Frankle et al., 2020a; Neyshabur et al., 2020) are the maximum increase in loss on linear path between two networks. Networks with barriers below some noise threshold are said to exhibit linear mode connectivity (LMC), which among other useful properties is necessary condition for the loss landscape to be locally convex (Neyshabur et al., 2020, Definition 3.1). Thus, non- zero barriers indicate that two networks belong to different convex loss basins (Goodfellow et al., 2015; Huang et al., 2017; Yunis et al., 2022).2 2
2. Related Work
Stability and optimization. Many works have studied the sta- bility of optimization by asking if neural networks converge to well-generalizingg minima (Cohen et al., 2021; Jastrzeb- ski al., 2020; Wu al., 2018) converge at all (Iyer et al., 2023; Jacot et al., 2018; Sohl-Dickstein, 2024). How- ever, these works do not consider stability relative to any particular training trajectory.
Note that we do not consider the more general notion of (non-linear) mode connectivity for several reasons. First, de- spite being non-convex, neural network training often un- derstood in terms of convex optimization. In convex regions of the loss landscape however, convex optimization behaves exactly as expected. Similarly, the dynamical systems per- spective often takes a linear or quadratic approximation of the loss landscape, which again holds precisely in convex regions. Practically, merging models by weight averaging requires linearly connected networks, which is guaranteed if
We focus on the question of whether training from an initial point tends to follow the same trajectory to specific "loss basin", i.e. linearly connected, low-loss region of the loss landscape. Although narrower in scope, this question is highly relevant for practical contexts such as when using pre-trained models or conducting fine-tuning. In order to merge or ensemble models in these contexts, one may not merely want to find a good solution, but may, for example, want to converge towards or away from a particular pre-
Followwig Entezari et al. (2022); Neyshabur et al. (2020), we make the extra assumptions needed to assume the converse, i.c. that networks with zero barrier are in the same loss basin.
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the networks are from the same convex region. Finally, both theoretical Simsek et al. (2021); Lin et al. (2024) and em- pirical (Draxler et al., 2018; Garipov et al., 2018; Sonthalia et al., 2024) works have suggested that most or all minima may be trivially connected by non-linear paths.
Representational similarity. Representational similarity compares the intermediate (hidden) outputs of two networks, and can detect functional differences even when two net- works have identical performance. Although many methods exist, we use Angular CKA (Williams et al., 2021) which we explain and justify in detail in Appendix B.2. In general, representational similarity methods are invariant to symme- tries of a network's s outputs, but not of its weights. Thus, dissimilar representations indicate greater diversity between networks (which can improve ensembling performance), but similar representations do not guarantee that two networks are in the same loss basin (due to weight symmetries), and thus is not sufficient for weight averaging to succeed.
Spawning experiments. Frankle et al. (2020a;b) and Fort et al. (2020) consider if training is stable to random batch order and data augmentations (training noise) by spawning pairs of networks from the same parent, and measuring bar- riers between them after training. They find that training becomes stable to training noise after an early period of instability. Altuntaş et al. (2023) shows that reducing early training variability by lowering learning rates, increasing batch sizes, and adding learning rate warm-up further in- crease stability to training noise, and Singh et al. (2024) relates stability in barriers to the loss landscape geometry. These findings align with observations that after an initial chaotic phase, SGD training trajectories can be approxi- mated by a linear kernel (Fort et al., 2020).
Fine- tuneet stability. Pre-trained models are generally stable to training noise and converge to the same basin during fine- tuning (Neyshabur et al., 2020). However, more recent work has found that this is not always true, as Juneja et al. (2023) discovered that training noise causes fine-tuning of language models to converge to different basins. While detrimental to model merging, this kind of instability can improve model diversity and thus ensemble performance (Lubana et al., 2023; Sadrtdinov et al., 2023), even if single basin has equivalent diversity (Lion et al., 2024).
We adopt the same parent-child spawning experiments (Frankle et al., 2020a;b; Fort et al., 2020), but we eliminate all training noise as isolate the effects of perturbation to specific training times. This not only lets us precisely identify when and how much perturbation causes instability, but also allows us to do so on both randomly initialized (chaotic) and pre-trained (stable) networks.
Our method enables us to find the threshold between sta- bility and instability for any fine-tuning setting eeeenn ones stable to training noise-since we can increase our pertur- bations to any scale necessary for inducing instability in a given network. Using our method, we identify differences in stability between language and vision models, specifically studying the fine-tuning dynamics of ResNets (He et al., 2016), ViT (Dosovitskiy et al., 2021), BERT (Devlin et al., 2019), and OLMo (Groeneveld et al., 2024).
Model averaging. Weight averaging can merge zero-barrier networks from the same training trajectory (Izmailov et al., 2018), different runs (Utans, 1996; Wortsman al., 2021), or different tasks (Mirzadeh et al., 2021) to improve infer- ence speed and even performance. Weight averaging is the basis for more sophisticated model merging strategies (II- harco et al., 2023), including those using permutations to
align diverging (Wang et al., 2020) or unrelated (Singh & Jaggi, 2020) models. Our work indicates the conditions in which models are stable with respect to barriers, and thus amenable to weight averaging.
3. Methods
In this section, we define our notion of training stability, describe the framing for our perturbation experiments, and finally, define the functional dissimilarity scores and other quantities we evaluate (barriers, barriers modulo permuta- tion, and L divergence over training time).
Permutation symmetries. Recent works have shown that independently initialized networks can converge to linearly connected basins after accounting for permutation symme- tries, or different ways to order the neurons in a network (Entezari et al., 2022; Ainsworth et al., 2023; Benzing et al., 2022). The permutations aligning two networks are most unstable early in training (Sharma et al., 2024), suggesting connection with training instability. While our work com- pares identical rather than randomly initialized networks, we apply weight and activation matching algorithms in the same manner as Ainsworth et al. (2023) to determine (1) if training instability causes permutations between networks, and (2) if undoing these permutations returns diverging tra- jectories to the same loss basin.
3.1. Training Instability
Consider training as the iterative application of a stochastic training map T : e + θ to the initial parameters 8g E θ of a neural network, so that the network's parameters after training are
Or T = T To T..... T EI, Er);
(1)
T times
where = (EI,...., .T) accounts for all of the stochastic factors influencing training, such as batch sampling, data augmentation, and nardware-induce non-determinism. Un-
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less specified, treat 0 a vector concatenation of network's parameters, and when writing T we omit the training or test data if it can be inferred from context.
or training tasks) enables comparisons between different loss landscapes.
To quantify instability, we record the rate at which a dissim- ilarity score d(Or, 07-) increases relative to the perturbation magnitude o. By sampling perturbations of different sizes and directions, we can estimate the size and shape of the local region around dg where T does not tend to cause di- vergence in terms of d.
We are interested in the degree to which training is stable, in the sense that a small perturbation to a network's ini- tial weights does not significantly change the network after training. To describe stability on a continuum, we evaluate how far OT and or have diverged after training according to various notions of similarity (Section 3.4). We choose T so that networks converge to similar level of training and test performance (see details in Appendix A).
Our experiment differs from Frankle et al. (2020a) and Fort et al. (2020) in that they use independent training noise start- ing at f, instead of a single perturbation, to induce instability. While this reflects the stability of ordinary training, our experiments have two key advantages: we can isolate our instability analysis specific parts of the training trajectory from to T,, and we can also apply much smaller or larger perturbations than training noise to quantify instability over broader scale. In Figure 10, we verify that instability in our method-i.e. to single perturbations-imppl bility in the methods of Frankle et al. (2020a); Fort et al. (2020)iie. to training noise.
If we fix a particular ç. TT describes a dynamical system whose outcome depends only on the initial parameters On- This perspective has numerous advantages. First, we can separate the effects of training noise and isolate instability to the action of T. As dynamical systems can diverge at exponential rates, dominating over stochastic effects (Wu et al., 2018), our deterministic experiments also lower bound the instability of regular stochastic graining-..e
insta-
E [d (T(O, E), T(O + e, E\)) z E (T T(O + s,))
3.3. Perturbations
for independently sampled noise & and g and a similarity measure d.
The stability of a dynamical system around a given point is direetion-dependent. We take this into consideration by sampling perturbations using two different methods, which give either a narrow or broad distribution of directions. As described in Appendix B.3, all noise samples are also nor- malized to fixed T,2 relative to the network's initialization scale.
3.2. Spawn-And-Perturl Experiment
Our experiment adapts the parent-child spawning experi- ment introduced by Frankle et al. (2020a) to the notion of stability introduced above. The procedure is follows:
1. Choose an initial state 0g for network.
Batch perturbation. Batch perturbations (Eq. 2) measure stability along the directions most likely to be explored dur- ing training, by simulating single independently sampled optimization step:
2. Train the network until the perturbation time 4, giving 0, ("o:: E.
3. Make two copies of the network, and perturb one by adding e noise with magnitude a to get 0% = 02 + oe.²
4. Train both original (0,) and perturbed (8) copies with identical training noise to get 0r = T (6,; (err and 0r = 7 (0'; C.rr).
Batch n2
Vila, 0z),
Tii Di D,
5. Measure the resulting instability via d (Or, 67), where d : θ > + is a dissimilarity score.
where VE is the gradient of the loss function 0, the net- work weights, and (z;, 94) are examples sampled from minibatch of the training dataset D. Ignoring factors like momentum, this is equivalent to taking an extra training step at time f, which is rescaled by the perturbation magnitude o instead of the learning rate.
By controlling dn and f, we can explore the stability of dif- ferent points 0 in the loss landscape. More specifically, we select between different trajectories by randomly initializ- ing O or setting it to pre-trained checkpoint from another task. We then vary the perturbation time to examine how stability evolves during training. Changing TT (by choosing different model architectures, optimizers, hyperparameters,
Gaussian perturbation. To measure stability in all di- rections generally, we use Gaussian perturbations (Eq. 3), which are scaled versions of the the network's 's distribution at random initialization. For networks initialized with a Kaiming/He normal distribution (He et al., 2015), Gaussian
For interpretability, ||=|[2 is normalized to match the expected scale of the network at initialization- see Appendix B.3 for details.
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perturbations are sampled as
Barriers modulo permutation. To consider whether training instability causes networks to converge differ- ent permuted versions of the same loss basin, we apply the weight and activation matching algorithms from Ainsworth et al. (2023) to find a permutation of neurons P that approx- imately minimizes the L distance between two networks' weights or intermediate activations, respectively. To mea- sure the degree which or and B have been permuted with respect to each other, we record both the barrier be- tween 0f and P[Or], and the fraction of identity elements (unpermuted neurons) in P.
( ]
5 ~ N (0:
2 14-1
Gausssia =
(3)
where (1) is the perturbation for the ith weight of layer 7t-1 is the number of inputs from the preceding layer (com- monly called fan-in), and N(O, s) is the normal distribution with mean O and standard deviation s.
Although this is not strictly uniform in all directions (when considering all of a network's 's weights as a single vector), we choose to match the scale of the network's initialization to ensure that perturbations do not disproportionally affect some layers more than others. Since biases and normaliza- tion weights have constant initialization, we do not perturb them in our main experiments." 4
As the two matching algorithms may not necessarily find the permutation that best minimizes barriers (i.e. the barrier modulo permatation), we follow Sharma et al. (2024) in treating the barrier between Or and P[O+-] as an upper bound, As result, if the barrier after permuting by P is significantly reduced, we can say that training instability mainly causes permutations that do not substantially change a network's function. However, if P does not reduce the barrier between 0r and 8%. this could either mean that training instability causes networks to learn different functions, or that we have merely failed find a permutation that does reduce barriers.
3.4. Evaluating Functional Similarity
We use four methods to evaluate the functional similarity of networks in our spawn-and-perturts experiment: (1) L distance in weight space ||0- - 8' 2l2. (2) the loss barrier in Equation (4), (3) the loss barrier after accounting for per- mutation symmetries using the weight matching algorithm from (Ainsworth al., 2023), and (4) the representational similarity of intermediate layers measured via Angular CKA (Williams et al., 2021).
Representational similarity. We also consider whether networks in our experiments differ in their penultimate rep- resentations using the angular version of Centered Kerel Alignment (CKA), a type of representational similarity met- ric. CKA measures the cross correlcorne between two ar- bitrary representations of the same data (Kornblith et al., 2019). As defined in Equation (5), we use Angular CKA with a linear kernel (Williams et al., 2021) to measure the distance between the outputs of the last residual or attention block. Since the resulting distance is angle, 0 indicates that two networks have perfectly similar representations, and \/2 indicates that two networks have dissimilar repre- sentations. Appendix B.2 includes full details.
L divergence. In a linear dynamical system, the L di- vergence |00-- - 8/ la can diverge exponentially over time at a rate of ||el|ze>r, where À is curvature dependent (see Appendix B.4 for derivation). To determine whether this linear approximation holds for neural network training, we measure 2 distance between parameter vectors over the course of training to look for exponential growth and to determine whether 2,2 divergences are proportional to the perturbation magnitude o.
Barriers. Barriers measure the maximum increase in loss or error along the linear path between weights (Frankle et al., 2020a; Neyshabur et al.. 2020). We measure the training loss barrier as
4. Experiments
Full training details, including hyperparameters and train/test performance, are listed in Appendix A.
sup Elabr+ (-00)+++(-00)))000)0------00--0- 1E(1)))
(4)
4.1. Early vs. Late Training Instability
We first train residual networks (He et al., 2015) on CIFAR- 10 (Krizhevsky, 2009) with SGD, using standard data aug- mentations and hyperparameter settings, including 2% warm-up period followed by linearly decaying learning rate.
where a interpolates between the networks and e is the loss function for the training data. Since our work is concerned with the shape of the loss landscape in which training oc- curs, we report the cross-entropy loss barrier for training data (Bee) and after accounting for permutations (BW M) throughout the main text. Appendix B.1 describes how we compute barriers in more detail.
Early perturbations reliably cause large barriers. Fig- ure 2 shows that training from randomly initialized networks is highly sensitive to initial conditions, as batch perturba- tions as small as 0.01% of a network's 's weights produce large barriers. We further reduce the perturbation magnitude
'Figure 9 shows results from perturbing biases and normaliza- tion weights.
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ResNet-20 Batch
ResNet-20 -Btth
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Figure 2. Stability of ResNet-22 trained on CIFAR-10 10 with SGD (details in Appendix A). Loss barriers training data the end of training (y-axis) are plotted against perturbation magnitude (x-axis) and perturbation step (color indicates fraction of total training time). Left: barriers due batch perturbation. Middle: batch perturbation barriers after accounting for permutations. Right: barriers due Gaussian perturbation. For the same plots with log-scaled y-axes, see Figure 27.
Nes-et-20 Batch
ResNet-20 Batch
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Test Ensemble Acc.
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Figure Left: same as Figure measuring representational similarity distance via Angular CKA (y-axis), defined Equation (5), between original and perturbed models after training for various perturbation times (colors) and magnitudes (x-axis). Middle: barriers versus Angular CKA. Right: test accuracy of ensemble the original and perturbed models after training (averaging logits), versus Angular CKA. See Figures 13 to 15, 21 and 23 for more hyperparameter settings and fine-tuning of VIT and BERT, respectively.
by modifying only a fraction of the weights, finding that single perturbed weight is sufficient cause instability (Figure 11).
Early instability is direction-independenn. Comparing batch versus Gaussian perturbations (Figure 2), we find that although networks are evenly distributed in direction), networks perturbed initialization have high barriers for both. This shows early instability is mainly attributable to the network's 's state, and not the direction or magnitude of perturbation. However, later instability does vary depending on the direction of perturbation, which suggests that findings that use training noise (Fort al., 2020; Frankle et al., 2020a) may not be transferrable to other kinds of perturbations.
stable to the latter (which
While prior work has shown that training noise near ini- tialization causes barriers (Fort et al., 2020; Frankle et al., 2020a), we are the first to show that barriers can occur with extremely small perturbations concentrated in the first few steps, as applying the same perturbations as early as 0.5% of the way through training results in significantly reduced barriers. We name this initial instability after the "butterfly effect" in chaotic dynamical systems.
The stability increases over the first 0.5% of training time is well within the 2% warm-up period we use, and only very large perturbations (10% initialization) result in non-zero barrier after 50% of training time. While prior works find that models become stable to training noise after the first few epochs of training (Fort et al., 2020; Frankle et al., 2020a), quantify the scale of perturbation needed to induce barriers beyond this critical point, showing that stability continues to increase throughout training.
Training divergence unlikely to be caused by permuta- tions. Comparing barriers with and without permutation alignment (Figure 2), we find that applying permutations to minimize the L distance between networks does reduce barriers." While cannot rule out the possibility that better (and more costly) alignment methods may reduce barriers, we argue that this is unlikely. Prior work aligning
We omit I distance between networks after alignment, is generally reduced greatly by weight matching (Ito al., 2025).
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differenily initialized networks finds that weight matching can reduce barriers some degree even when it is outper- formed by other methods (Peña et al., 2023; Navon et al., 2024: Ainsworth et al., 2023), whereas in our case of iden- tically initialized networks, weight matching is unable to reduce barriers at all. This suggests that training instability produces real functional differences between networks, as opposed to simply permuting weights that are otherwise equivalent.
perturbed at initialization. We find that training the shallow- wide architecture combined with 10x learning rate warm-up improves stability over each individual setting (Figure 4 right), but does not eliminate barriers at initialization.
4.4. Fine-tuning
Having explored the stability of the loss landscape along trajectories starting from random initialization, we next ex- amine stability on transfer learning trajectories. Fine-tuning is known to have greater stability since it starts from pre- trained networks that have non non-random patterns of weights (Neyshabur et al., 2020), but the relative difficulty of the pre- pre-training and transfer task can either increase decrease stability to training noise (Vlaar & Frankle, 2022).
4.2. Functional Diversity
Comparing the similarity of intermediate representations in Figures 3 and 23, we find that Angular CKA (Eq- 5) correlates with earlier and larger perturbations (left), as well as barriers after training (middle). This again indicates functional differences beyond weight symmetries. 6
We again move beyond the effects of training noise to quan- tify the exact perturbation times and scales at which transfer learning is unstable. We consider task combinations from both vision and language domains, as fine-tuning the latter known be unstable to training noise (Juneja et al., 2023).
Since model ensembling benefits from diversity, we also consider whether intentionally perturbing networks can im- prove ensembling performance. This effect is most useful for fine-tuned networks, which necessarily have reduced diversity due to being trained from the same initial state far from random initialization. Figure 3 (right) and Figure 15 show that when ensembling the original and perturbed net- works, ensemble performance indeed scales with Angular CKA dissimilarity. However, fine-tuning ViT models on CIFAR- does not share this trend (Figure 21). This contradiction may be explained by observations of similar performance between ensembling and averaging in Utans (1996).
Pre-training stability depends on the tasks involved. Starting with CIFAR-10 and CIFAR-100 (Krizhevsky, 2009), we pre-train two ResNet-50 networks with layer nor- malization on either task, and then fine-tune them on the op- posite task starting from both early (0.24% of pre-training) and late (100% of pre-training) pre- cre-trained checkpoints.
Figure 5 (left, center) and Figures 17 and 18 show that fine- tuning is generally more stable than compared to ResNet-20 (Figure 2) or training the same models from random initial- ization (Figure 19). This is especially true for later check- points (p- = 191 ep3epp1r ep319sl) 1s1)) and larger perturbations (a = 0.1, equivalent to 10% of initialization). whereas fine-tuning from earlier checkpoints is more similar in barriers with regular training (Figure 5). This shows that, as in regular training, instability is mainly function of pre-training time.
4.3. Efffec of Hyperparameter Settings
We next compare the stability of different training schemes T for ResNets trained on CIFAR-10: 10: no weight decay, 10x learning rate warm-up, 4x batch size, Adam, a shallow- wide architecture with similar numbers of parameters (exact details in Table 1).
Figures and 12 show that, in line with prior work (Altintaş et al., 2023; Vlaar& & Frankle, 2022), reducing learning rate (by increasing warm-up) and increasing batch size improve stability. Adam and weight decay reduce stability, which we speculate may be due to their effect on the loss landscape's 's sharpness, which is known to affect SGD stability (Wu et al., 2018). The shallow-wide architecture is most stable of these settings, which we speculate is due to its training dynamics being more closely aligned with the infinite-width, linearized kernel regime (Lee et al., 2019; Fort et al., 2019).
Stability is task-dependent, as transfer from CIFAR-100 to CIFAR-10 10 is more stable than in reverse. This agrees with Vlaar & Frankle (2022), who find that pre-training on related vs. random data improves or worsens (respectively) the barriers between two points along a training trajectory.
Vision Transformers (ViTs). To study different archi- tecture in the vision domain, we perturb the fine-tuning trajectories of ViTs (Dosovitskiy et al., 2021) of varying sizes on CIFAR-100 (Appendix A.2). While we were only able to consider checkpoints at the end of pre-training, Ap- pendix D.2 shows that, consistent with our previous findings, larger and earlier perturbations during fine-tuning lead to larger barriers.
Next, we consider if combination of stability- innteasiinn hyperparameters could reduce barriers to O for networks
'Note that networks with zero barrier still have non-zero Angu- lar CKA, likely because linearly connected networks can perform differently on individual examples (Yunis al., 2022).
"Our work differs that we consider two diverging trajectories.
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ResNet-50 (CIFAR-100 + CIFAR-10) Batch
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Figure 5. Stability of transfer learning on vision tasks: a ResNet-50 is pre-trained on C10AR-100 and fine-tuned on CIFAR-10 (left) or vice versa (middle). Barriers (y-axis) are plotted against perturbation magnitudes (x-axis) for various pre-training durations and perturbation times (circle marker colors). See Appendix A.2 for details, and Tables 4-5 for barriers less than 10 2 Right: fine-tuning stability of Multi-BERT on QNLI, starting from 20K, 200K, and 2000K checkpoints with early and late perturbation times (diamond marker colors). For other tasks (MRPC, RTE, SST-2, and CoLA), see Figure 22. For VIT and OLMo, see Figures 20 and 24 respectively.
Pre-training does not always increase fine-tuning stabil- ity. Having established that longer pre-training improves stability for small-scale vision models, next examine heavily pre-trained language models. This setting is par- ticularly interesting because recent work by Juneja et al. (2023) demonstrates that, unlike vision models (Neyshabur et al., 2020), training noise can cause language models to converge to distinct basins after fine-tuning.
stability does not consistently improve with pre-training time. Notably, for QNLI and RTE (Figure 22), the final pre- trained checkpoint (2000k) has the largest barriers. When evaluating the pre-trained network these tasks (Table 3), we observe that the 2000k checkpoint has worse test accu- racy, despite having lower cross-entropy when compared with the 200k and 20k checkpoints. We speculate that this may be due to overfitting on the pre-training distribution, 8 which could cause the model to become brittle to pertur- bations during fine-tuning phenomenon termed "catas- trophic overfitting" by Springer al. (2025).
To investigate this, we analyze the stability of Multi-BERT (Sellam et al., 2022), which provides intermediate check- points for every 20,000 steps during pre-training. We take checkpoints at 20k, 200k, and 2000k (100%) steps of pre- training time as starting points and fine-tune on various GLUE tasks (Wang et al., 2019): natural language infer- ence (QNLI, RTE), paraphrase and similarity assessment (MRPC), sentiment classification (SST-2), and linguistic acceptability (CoLA).
Decoder-only models. Billion- parameter decoder-only models are widely used in fine-tuning and model merging, but their training dynamics remain severely understudied. To address this gap, we fine-tune intermediate checkpoints of OLMo (Groeneveld et al., 2024) on the math problem dataset GSM8K (Cobbe et al., 2021a). Figure 24 shows that pre-training longer can again reduce stability to fine- tuning, corroborating our MultBERT findings. Moreover, we observe the same trends where earlier and larger per- turbations result in higher barriers as in previous settings.
Figures 5 and 22 show that BERT is more sensitive to the size of perturbations when compared with our vision exper- iments (Figure left, middle). For all pre-training check- points, earlier perturbations during fine-tuning consistently lead to larger barriers. However, unlike our vision settings,
MultiBERT was pre-trained for around 100 epochs.
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Barrier vs. Step
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Figure 6. Evolution of barriers (leff) and L (right) over training for standard ResNet20 trained on CIFAR-IO. Each colored line averages over all perturbation magnitudes, they are nearly indis- tinguishable.
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Figure 7. Train loss harriers vs. I,² distance after training between the original and batch-perturbed models for ResNct-20 models trained on CIFAR- -- using various hyperparameter settings (left). and BERT models fine-tuned on QNLI (right). For additional fine-tuning results, see Figures 21 and 24 to 26 for other BERT tasks, ResNet, VIT, and OLMo respectively.
5. L Divergence and Barriers
Barriers and L divergence do not evolve according to a linearized dynamical system. Figure shows the rates at which barriers and L divergence increase as training progresses." 9 Contrary the linearized dynamics derived in Appendix B.4, neither barrier nor L increase exponentially over training. More work is needed to explore other mecha- nisms that could drive these observed rates of divergence.
for distribution of perturbations, applied at any time in training, on any initial network weights, and for any training procedure. This method allows us to evaluate stability over more conditions, and at finer precision, than was possible in prior works that only consider the effects of training noise (Vlaar & Frankle, 2022; Fort et al., 2020; Frankle et al., 2020a). Our experiments show that although randomly initialized networks are extremely unstable, stability rapidly increases with training to be robust to perturbations much larger than training noise.
Barriers scale with with exponential 2,2 divergence in vision settings. Although networks can diverge in weight space without increasing barriers (Frankle et al., 2020a; Vlaar & Frankle, 2022), we find in our experiments that barriers and T,2 divergence after training exhibit a strong log-linear relationship (Figure 7 left). This finding differs from Vlaar& & Frankle (2022) in that they look the distance traveled from initialization, whereas we look at the distance between training trajectories which started from the same point. We find that the proportion of identity elements in the aligning permutations P is also related to barriers, albeit to a weaker extent (Figure 16). Since P minimizes I distance between 0r and 0z. this is likely due to the correlation between barriers and 2 divergence.
Our work is consistent with the finding in Sarnthein et al. (2023) that a student network initialized very close to a random teacher nevertheless diverges quite far after training. Further work is needed to understand why, unlike in our setting, the student in Sartthein et al. (2023) remains in the same linearly connected basin as the random teacher.
While instability near initialization is universal, many trends are inconsistent and depend on the task or model. We find that (1) the rate at which stability increases along training trajectories varies greatly, (2) more pre-training does not always improve stability during fine-tuning,
(3) L diver- gence correlates strongly with barriers in some cases but not others, and (4) the rates at which L and barriers diverge do not match that of straightforward dynamical system. While the specific counter-examples we have surfaced are sufficient evidence for these results, a detailed exploration of their underlying causes and the cireumstances in which they hold (such as in isolating the effects of task versus architecture) is left for future work. Further investigation is also needed to determine (1) if certain hyperparameter settings entirely eliminate instability at initialization, and (2) what perturbations, if any, can be used to reliably improve ensemble performance.
Interestingly however, fine-tuned language models show little or no correlation between 2 divergence and barriers (Figure 7 right, Figure 26). This suggests that the relation- ship between L and barrier may only appear in smaller- scale models, which highlights the need for large-scale and multi-modal experiments (Juneja et al., 2023).
6. Discussion& & Conclusion
We present a method for measuring whether neural network training is stable (reliably converging to the same basin),
Since barriers and I are negligible at perturbation time and grow throughout training, this indicates that our results due to instability and not just the initial perturbation.
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Acknowledgements
solve math word problems, 2021a. URL https: / arxiv.org/abs/21111
Special thanks to Gaurav Iyer, David Mickish, Ekansh Sharma, Sidak Pal Singh, and Julien Boussard for discus- sions and comments. This research was supported by an NSERC Discovery grant, the Canada CIFAR AI Chairs program, and Fonds de recherche du Québec-Nature et Technologies (FRONT doctoral research award #3528161 16). Computing resources were provided by Mila-Quebec Arti- ficial Intelligence Institute, and the NVIDIA Corporation.
Cobbe, K., Kosaraju, V., Bavarian, M., Chen, M., Jun, H., Kaiser, L., Plappert, M., Tworek, J., Hilton, J., Nakano, R., Hesse, C., and Schulman, J. Training verifiers to solve math word problems. CoRR, abs/2110.1416.. 2021b. URL httpp//////aaw...o//aa//////////
Cohen, J. M., Kaur, S., Li, Y., Kolter, J. Z., and Talwalkar, A. Gradient descent on neural networks typically occurs at the edge of stability, 2021. URL http::////arrii abs / 63.00065.
Impact Statement
This paper presents work whose goal is advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
D' Angelo, F., Andriushchenko, M., Varre, A., and Flam- marion, N. Why do we need weight decay in mod- ern deep learning?, 2024. URL http: pett:://aree http///xxvvvorrree abs/2310.04413 arXiv:2110004415 [cs] version: 2.
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Sharma. E., Kwok, D., Denton, T., Roy, D. M., Rolnick, D., and Dziugaite, G. K. Simultaneous linear connec- tivity of neural networks modulo permutation. In Bifet, A., Davis, J., Krilavičius, T., Kull, M., Ntoutsi, E., and
Strogatz, S. Nonlinear dynamics and chaos: with applica- tions to physics, biology, chemistry, and engineering. Chapman Hall book. CRC Press, Boca Raton London
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New York, second edition, first issued in hardback edition, 2019. ISBN 978-0-8133-4910-1 978-0-367-09206-1.
Yunis, D., Patel, K. K., Savarese, P. H. P., Vardi, G., Fran- kle, J., Walter, M., Livescu, K., and Maire, M. On convexity and linear mode connectivity in neural net- works. In OPT 2022: Optimization for Machine Learn- ing (NeurIPS 2022 Workshop), 2022. URL https: / //openneview.net///eeeeeeeeee//e formme? 12222KK33E....
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Zhou, Z., Yang, Y., Yang, X., Yan, J., and Hu, W. Going beyond linear mode connectivity: The layerwise linear feature connectivity. In Advances in Neural Information Processing Systems, volume 36, pp. 60853-60877, URL http:://prroceddnnggeeer.s...o paper_/lle//aeee//2/////L// bffe555s42262222880666eebbbbbbb Paper--oofference.df.
Utans, J. Weight averaging for neural networks and local resampling schemes. In Proc. AAAI-96 Workshop on Integrating Multiple Learned Models. AAAI Press, pp- 133-138, 1996.
Vlaar, T. J. and Frankle, What can linear interpolation of neural network loss landscapes tell us? Proceed- ings of the 39th International Conference on Machine Learning, volume 162, pp. 22325-22341, PMLR, 2022. URL http:::/pproreednns........o /v12/ v1 vllar22a.html.
Wang, A., Singh, A., Michael, J., Hill, F,, Levy, O., and Bowman, S. R. GLUE: multi-task benchmark and analysis platform for natural language understand- ing. In International Conference Learning Repre- sentations, 2019. URL https:ttp///cmm..tm www.reeeee.eeet........ forum?id=rJh 4m225tt7.
Wang, H., Yurochkin, M., Sun, Y., Papailiopoulos, D., and Khazaeni, Y. Federated learning with matched averag- ing. In International Conference on Learning Repre- sentations, 2020. URL forum?i i dd..iiiIFSS...
Williams, A. H., Kunz, E., Kornblith, S., and Lin- derman, S. Generalized shape metrics on neural representations. In Advances in Neural Information Processing Systems, volume 34, pp. 4750, 2021. URL http:://rroceding..eeri... paper filles/paperr222///11ee. 3522ddbeeb32277552222223333363 Paper.pdf.
Wortsman, M., Horton, M. C., Guestrin, C., Farhadi, A., and Rastegari, M. Learning neural network subspaces. Proceedings of the 38th International Conference on Ma- chine Learning, volume 139, pp. 11217-11227. PMLR, 2021. URL https: htt.:tps//pproeeenng.......o.ee v1 v1 2339/wortsmman2la.ttt....
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appet.files/ppee/////8//eee
66112000188223300011ealeeee Paper.pdf.
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The Butterfly Effect
Table 1. Hyperparameter settings for ResNet-20 trained on CIFAR-10 10, along with the accuracy and training cross-entropy loss of the perturbed model at the end of training. Each setting is averaged over batch and Gaussian perturbations applied at various time steps and scales, each configuration repeated with three seeds.
Setting
Model
Optimizer
I.R
W-up
WD
BS
Steps
Accio
CF
Standard
ResNer200-32
SGD
0.100
0.020
128
25000
0.89 + 0.00
0.01
10x Warm-up
ResNer20-32
SGD
0.100
0.200
128
25000
0.89 + 0.01 0.01
0.01
AdamW (no Weight Decay)
RcsNcr20-32
A
0.003
0.020
128
20000
0.89
0.10
Batch Size = 512
ResNei20--2
SGD
0.100
0.020
512
10000
0.88 0.00
0.15
Perturb Norm Weights
ResNer20032
SGD
0.100
0.020
128
20000
0.00
0.05
ResNet-8
ResNet8-64
SGD
0.100
0.020
128
25000
0.00
0.14
ResNet-8, 10x Warm-up
ResNet8-66 64
SGD
0.100
0.200
128
25000
0.00
0.13
SGD w. Weight Decay
ResNet20-32
SGD
0.100
0.020
V
128
20000
0.89
0.15
A. Training Details
In this section, provide details about training methodology. Unless otherwise specified, we conducted all ResNet experiments on individual NVIDIA RTX 8000 GPU with CPU cores. ViT and language model experiments were conducted on NVIDIA L40S GPUs.
A.1. R-FAR-10 Hyperparameter Experiments
We train residual convolutional models (He et al., 2015) the CIFAR-10 dataset (Krizhevsky, 2009) using the hyperparam- eter settings in Table 1. Training times are chosen so that cross-entropy loss on the training data for the perturbed model is below 0.15 after training, on average. All models have test accuracies within 2 percentage points (88-90) (88-90%). For ease of interpretation, training times are rounded up to the nearest 5000 steps. Although the models in our experiments are not fully converged, and some variations remain between different hyperparameter settings, we did not find our results correlate with different training times, or the network's final train or test performance (Figure 8).
To simplify weight and activation matching, we use layer normalization (Ba et al., 2016) instead of batch normalization, resulting in slight reduction in performance. When evaluating barriers after permutation alignment, this avoids having to do additional inference passes to correct the batch normalization statistics at each interpolation step (Jordan et al., 2023).
Barriers vs. Accuracy- Batch
Barriers vs. Accuracy
Barriers vs. CE Batch
Barriers vs. CE Gaussian
Gaussian
10
10
10
10
10
10"
10
10
0.87
0.88
0.89 Acct.
0.90
0.87
0.88
0.90
0.125
0.150
0.175 CE
0.200
0.125
0.150
0.175
Acc|,
CE
Setting
10x Wa
Perlurb Norm Weights
SGD Weight Decay
AdamW (no Weight Decay)
ResNet-8
Standard
Batch Size = 512
ResNet-8, 10x Warm-up
Figure 8. Train loss barriers against test accuracy (left) and training cross entropy loss (right) of the perturbed model at the end of training.
A.2. Finetuning Experiments
CIFAR pre-training We pre-trained two ResNet-50 models with different initializations on both CIFAR-10 and CIFAR-100 datasets, with layer normalization. Each model was trained for 75,000 steps (approximately 200 epochs) using SGD with momentum 0.9 and cosine amnealing schedule. We used a peak learning rate of 0.1 with 2.5% warm-up, with a weight decay 10 and batch size of 128. The model was trained with horizontal flips, random rotations up to 10 degrees, random translations up to pixels, and cutout augmentation with 2 pixels.
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The Butterfly Effect
CIFAR Fine-tuning Starting from our pre-trained ResNet-50 checkpoints, we fine-tuned each model CIFAR-10 (or CIFAR-100) using stochastic gradient descent with momentum 0.9 for 20,000 steps.
BERT Fine-tuning Experiments We conducted experiments on the GLUE benchmark (Wang et al., 2019) using the MultiBERT model (Sellam et al., 2022), specifically using checkpoints from google/mule ibberss Sespeerss-seed.. and 1 available on HuggingFace." All tasks share the same base hyperparameters of Adam (Loshchilov & Hutter, 2017) with a learning rate of 10 and a weight decay of 0.01, while the batch size and training duration were scaled according dataset size, as detailed in Table 2.
poogge//mlltibertss
Due computational constraints, selected QNLI from among the larger datasets with more than 100k examples (QNLI, QQP, MNLI), and followed Devlin et al. (2019) by fine-tuning for three epochs. For the medium-sized SST-2 dataset, we trained for 2,500 steps. Small datasets (RTE, MRPC) with 2.5k-3.7k examples were trained for 500 steps using batch size of 32, while the medium-small dataset CoLAA (5.7k-8.5k examples) was trained for 1,500 steps with the same batch size. These settings ensure that all models achieved a training cross-entropy loss below 0.2, although our networks appear to have overfit the fine-tuning task some cases (Table 3).
Table Task-specific hyperparameters for fine-tuning MultBERT on GLUE tasks. Training schedule transitions from step-based to epoch-based for larger datasets to ensure sufficient coverage of training data.
Dataset
Examples
Batch Size
Training Schedule
QNLI
105k
128
3 epochs
SST-2
67k
2500 steps
CoLA
8.5k
32
1500 steps
MRPC
3.7k
32
500 steps
RTE
2.5k
32
500 steps
Table 3. Multi-BERT (seed 0) test performance of pre-trained checkpoints before fine-tuning (zero-shot evaluation) and after fine-tuning.
Before Training
End of Fine-tuning
Dataset
Starting Checkpoint
Acc
CE
Acc
CE
COLA
2000k
0.61
0.68
0.84
0.58
200k
0.38
0.75
0.77
0.79
20k
0.57
0.68
0.72
0.87
MRPC
2000k
0.69
0.68
0.85
0.43
200k
0.32
0.87
0.83
0.45
20k
0.32
0.77
0.77
0.82
QNLI
2000k
0,44
0.70
0.91
0.26
200k
0.52
0.75
0.89
0.32
20k
0.54
0.70
0.84
0.38
RTE
2000k
0.47
0.70
0.66
0.86
200k
0.53
0.74
0.64
1.06
20k
0.53
0.70
0.64
1.13
SST2
2000k
0.49
0.70
0.92
0.30
200k
0.48
0.72
0.91
0.34
20k
0.51
0.70
0.88
0.48
VIT Fine-tuning
We fine-tune on CIFAR-100 starting from four Vision Transformers (ViT')
(Doso- parameters)," !I
vitskiy
et al., 2021) on HuggingFace: google/vit hoapleevvi-baeeeeee
base patchhl- (86M
http://ggggggga.ccoooooooo e/mul i i bertee seed 0-step Ok and https: htttps///hagggggafcc..////// multiberts-seed 1-step Ok. "ttpps: http///gggggaac...oooooeeaa///// weweeeevvveeeeeeeeeeeeeeeeeeeeet pasas-ppcch16--22
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The Butterfly Effect
google/vit baatttt-hss-pphhhheeee 6 222--122n2 (86M parameters), 12 google/vit-large 6-224--n222 (304M parameters), 13 and google wawagll/vit--hueepatteee 4-tprcinn142222-2111 (632M parameters). 14 All models were pre-trained on ImageNet-2Ik, with sett-baae-paeee passe-p 2erchh1662244 additionally fine-tuned on ImageNet- Ik (Russakovsky et al., 2015).
We use the same hyperparameters across all model sizes: A (Loshchilov & Hutter, 2017) optimizer with learning rate 2 x 10-4, weight decay x 10 10-4, batch size 32, and cosine annealing schedule with 10% warm-up over 5 epochs. Data augmentation consisted horizontal flips, random rotation (=10%), random translation (=16 pixels), and cutout patches (16 16). Images are resized 224 224 match the input resolution expected by the models.
OLMo Fine-tuning We fine-tune OLMo-1B'5 on GSM8K (Cobbe et al., 2021b) starting from various checkpoints provided throughout its x 740K pre-training steps (3 trillion tokens). For our setting, we select three checkpoints from different training phases: (1) first available checkpoint (4B tokens), (2) mid-way through pretraining (1.5T tokens), and (3) final checkpoint (3T tokens). We fine-tuned each checkpoint for 5,000 steps using AdamW W with learning rate of 2 x 10-6 and cosine annealing with 10% warm-up.
B. Methodological Details
In all of our experiments, we train two networks simultaneously with deterministic computations enabled, using identical random seeds for random initialization (if applicable), batch order and data augmentation. We confirm that training with no perturbations results in exactly identical networks as expected.
All evaluations of models trained from initialization are averaged over three runs, while all evaluations for fine-tuned models listed in Appendix A.2 are averaged over runs.
B.1. Computing Barriers
To compute barriers, we evaluate 11 equidistant values of a E [0, 1] along the linear path between Or and Bz--
In our definition of barriers (Equation (4)), we interpolate between the loss of the endpoints following (Sharma et al., 2024), rather than taking their average loss as in Frankle et al. (2020a). This is because the former follows from the definition of convexity and is more appropriate for describing a convex loss basin. In practice, since Or and fz have near-identical loss in our experiments, the definitions are interchangeable.
We also measure test error barriers by replacing l with the 0-1 loss over test data. In practice test barriers are slightly less than train barriers as the network reach near-zero loss on the training data but not the test data, allowing for larger barriers in the former. However, since test error barriers follow the same trends as training cross-entropy barriers, we omit them from the text.
B.2. Computing Angular CKA
There are many different representational similarity methods, and a comparison of them beyond the scope of this work. We use CKA for number of reasons: it is invariant to linear transformations (other than affine) which aligns well with the capacity of neural network layers (Korbllih al., 2019; Lange et al., 2023), it has been used to compare neural networks in other contexts (Nguyen et al., 2021), is less dependent on the weighting of principal components than SVCCA (Raghu et al., 2017), and it can be applied to networks with more intermediate outputs than the number of test inputs m (Kornblith et al., 2019). We report Angular CKA for the simple reason that it gives a distance which increases with dissimilarity, which concordance with the other measurements we make. Note that Angular CKA and CKA differ only in the application of arccosine.
17

---

The Butterfly Effect
We compute the Angular CKA between the final hidden representation follows:
OCKA(Or. (--)) = CKA (f)-
(5)
HSIC Y)
CKA(X, Y) &rccos
HSIC(X, X) HSICCY, Y)
where I. is the number of residual attention blocks, fL-1 Rmxno TR is the last block's output fixed set inputs X E gmxx, and HSIC is the Hilbert-Schmidt Independence Criterion, which measures cross correlation between centered similarity matrices.
We use the implementation by Lange et al. (2023), which includes certain modifications to speed computation. Namely, we sample m 1000 examples as Lange et al. (2023) shows CKA can be reliably estimated using reasonably few examples, and we use their reduced-biax estimator for HSIC:
HSIC(X, Y) m(m 3) (Ixx H), tr H))
where is the number of test inputs, tril extracts the lower triangular portion of matrix, = 1- /m is a centering matrix that subtracts the mean, and ( is the Frobenius norm. Effectively, this estimator ignores the diagonal of the similarity matrix HXX H.
B.3. Perturbation Scale
To ensure fair comparisons between different perturbation methods and network architectures, normalize all perturbations to have a consistent 1 magnitude, which for ease of interpretation is given relative to the network's 's size at initialization (Eq. 6). Formally, we ensure the squared norm of E matches the total variance at initialization of the perturbed weights, so that
E A.M [ M|2
VVrrl0o M]
(6)
where é is a batch or Gaussian perturbation sample, M is a 0-1 mask of the weights to perturb, is the element-wise product, Var is the expected variance (not the sample variance), and Og are the network's 's initial weights. Thus, for example, a perturbation magnitude a = 0.01 is approximately 1% the size 0g-
Finally, in order to preserve the distribution activations after each normalization layer, we do not perturb biases or normal- ization weights in experiments. While prior work has found that linear mode connectivity can vary between different layers (Vlaar & Frankle, 2022; Zhou et al., 2023; Adilova et al., 2024), we did not find that our results changed significantly depending on which layers were perturbed. Figure 9 shows results when only perturbing biases and normalization weights. In this case, we use the same scale of perturbations as we would normally assign to the weights in the layer following the biases or normalization weights.
B.4. Lincarized Approximation For 2 Divergence
A classical result of dynamical systems states that linearized system subject to a small perturbation can diverge exponentially with respect to time at a rate which depends on the largest eigenvalue (i.e. top Lyapunov exponent) of the gradient of the training map (Strogatz, 2019):
T(e; +0) =(()) + TV,O,
T ( +5) T (0o) + II 7
t=1
|00 0/ 2 S |1e
where V is the gradient of the training map with respect to the weights and À is the top eigenvalue over all the gradients at each step t.
Substituting the definition of SGD, we find that the 1.2 divergence between the original and perturbed models after
18

---

The Butterfly Effect
training depends on the curvature of the loss landscape:
T((;) = 0: - e Vrd;= = TRH,, |00 0/ V |1el|eeA
(8)
where 77, is the learning rate, H; is the Hessian of the weights, and λи is the largest eigenvalue over all I - H. Divergence results either from high positive curvature as in Wu et al. (2018), but additionally if there is any negative curvature in the perturbation direction, and AArl > implies the possibility of exponential growth in divergence over training.
C. Further Experiments
C.1. Baselines
Stability is not specific layer type. In the main text (Figure 2), excluded norm weights and biases from perturbations. However, in Figure 9, we show that perturbing only norm layers leads to similar trends. This suggests that fine-tuning stability is influenced more by overall network dynamics rather than specific layer types. While individual parameters or layers may have varying importance as noted in previous work (Adilova et al., 2024), batch perturbations already capture this effect to some extent. The increased barriers for smallest perturbations at initialization are an artifact numerical instability-inn our regular experiments, we avoid this problem by reducing the fraction of perturbed weights instead of reducing the perturbation scale beyond o 10-4
ResNet-20 Batch
ResNet-20 1e200-Batee Batch
ResNet-20 Gaussian
ResNet-20 Gaussian
1.5
1.5
1.0
0.5
0.5
0.0
0.0
10
10
10
10
10
10
10"
10"
t= 0.0%
t= 0.5%
t= 2.0%
t= 10.0%
(= 50.0%
Figure Train loss barriers before and after permutations when perturbing only normalization layers. Results are shown batch (left) and Gaussian (right) perturbations ResNet-20 trained with SGD (momentum, no weight decay), using learning rate of 0.1, 2% warm-up, and a batch size of 128 for 20,000 steps.
Frankle et al. ResNet-20 (Weight Decay) Batch
1.5
10
1.0
Noise L2
BE
-2
Bwich Parturbancs Ona stop
10
0.5
Fransle (2020) Spowning
0.0
20
40
0
20
40
Perturb step (%)
Perturb step (%)
Figure 10. Left: Comparison of training loss barriers between our butterfly setting and Frankle et al.'s spawning setting. In the spawning setting, each network is trained with different non-determinism after perturbation step t. while our method applies a single perturbation. As expected, our pertuueerturatiion approach provides lower bound on the spawning barriers. Right: L magnitude of the expected deviation between two copies of the same model, when each model takes single independent training step time f.
How do our perturbations compare to SGD noise? To establish the relative magnitude of our perturbations vs. SGD noise, we replicate the parent-child spawning experiment of Frankle et al.. Figure 10 shows that our batch perturbations are a lower
19

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The Butterfly Effect
bound on the Frankle baseline's barriers, meaning that the instability resulting from training independently for multiple steps must be at least the instability resulting from single independent training step.
Note that for this comparison, we scale batch perturbations to the expected magnitude of SGD noise at the perturbation time t. This makes batch perturbation equivalent to taking only one step at time t with different SGD noise, opposed to using different SGD noise from t onwards in the Frankle baseline.
C.2. Perturbing Only A Fraction of Weights.
We further decrease the scale of our perturbations from Figure 2 by perturbing only fraction of the weights with our smallest perturbation scale of 10-4. Strikingly, we find that perturbing as little as a single weight, which occurs when the fraction of perturbed weights is 10-6, is sufficient to create barriers at initialization (Figure 11, right), The scale this perturbation (Figure 11, left) is well below that of noise caused by hardware indeterminacy.
ResNet-20 (Weight Decay) Batch
ResNet-20 (Weight Decay)
ResNet-20 (Weight Decay) Batch
ResNet-20 (Weight Decay). Gaussian
Gaussian
Noise L2 for Model.
Noise L2 for
10
10
10
10
DGE
10
10
10
10"
10
10"
10
10"
10
10
10
10
10
Perturbed weights (%)
Perturbed weights (%)
Perturbed weights (%)
Perturbed weights (%)
0.0%
0.5%
2.0%
Figure 11. Left:
absolute L norm of the noise a function of the fraction of perturbed weights. Right: train loss barriers as function
fraction of perturbed weights.
C.3. Additional Hyperparameter Settings
ResNet-20 (Adam)- Batch
ResNet-20 (BS=512)- Batch
ResNet-20 (Weight Decay) Batch
20
3
3
BeE
2
10
2
0
0
10
10
10
10
-3
= 0.0%
t= 0.5%
t=
2.0%
t= 10.0%
t= 50.0%
Figure 12. Same as Figure 2 with AdamW without weight decay and learning rate of 0.003 (left). batch size of 512 (middle), and weight
decay (right).
In this section we present the results of training ResNet-20 on CIFAR-10 for all hyperparameter combinations listed in Table
It is well-known that neural network training is highly sensitive to optimization hyperparameters (Smith, 2018). Our experiments corraborate the complex interdependencies between key optimization hyperparameters, suggesting that further theoretical exploration is needed. We specifically focus on the impact of optimizer choice, batch size, and weight decay.
SGD enhances training stability. The choice of optimizer significantly impacts the stability of the training map. Additionally, networks trained with Adam also exhibit higher I,2 distances (Figure 7). We stipulate that this phenomenon linked the
20

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The Butterfly Effect
implicit bias of SGD. Specifically, Bradley et al. (2022) highlight that SGD's inherent noise helps it avoid high-curvature regions, which are often linked poor generalization and, in our setting, could likely contain linearly connected minima.
In our experiments, we observed that larger batch sizes lead to higher loss barriers, suggesting trade-off between batch size and stability. This phenomenon aligns with findings from Keskar et al. (2017), who show that larger batch sizes can lead to sharper minima, which might increase the loss barriers and potentially hurt generalization.
Weight decay provides slightly more stability, particularly right after initialization. D' Angelo al. (2024) highlight that weight decay modifies the optimization dynamics, enhancing the implicit regularization of SGD through loss stabilization mechanisms. They also emphasize the role of gradient clipping, which stabilizes training, particularly the edge of stability.
C.4. Functional Diversity For Additional Hyperparameter Settings
We replicate Section 4.2 on two of our additional hyperparameter settings: with weight decay, and with 10x warm-up. The results in Figures 13 to 15 closely agree with our findings in Figure 3.
ResNet-20 (Weight Decay)-
ResNet-8 (10x Warm-up) Batch
Batch
Train Angular CKA
Train Angular CKA
1.00
0.4
0.75
0.50
0.2
0.25
10
10
10
10
o
0.0%
2.0%
(= 50.0%
0.5%
10.0%
Figure 13. Representational similarity distance measured via Angular CKA for ResNet-20 with weight decay (left) 10x warm-up (right).
ResNet-8 (10x Warm-up) Batch
ResNet-20 (Weight Decay)
Ensemble Acc. vs. Angular CKA
Batch
10
10
10
BcE
BcE
10
BgE
-2
caloa
10"
10
2
w. Deort Standerd
0.2
0.25
0.50
0.75
1.00
0.5
1.0
Train Angular CKA
Train Angular CKA
Train Angular CKA
=0.0%
t=0.5%
+2.0%
=10.0%
150,0%
Figure 14. Train loss barrier versus representational similarity distance on ResNet-20 with weight decay (left), ResNet-8 with 10x warm-up (middle) with perturbation time indicated by color, and including the standard setting from Figure 3, with settings indicated by color (right).
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The Butterfly Effect
ResNet-20 Batch
ResNet-8 (10x Warm-up) Batch
Ensemble Acc. vs. Angular CKA
Test Ensemble Acc.
Test Ensemble Acc.
Test Ensemble Acc.
0.91
0.905
0.90
0.90
0.900
1V0 355 99
RecNet--, Viuggtt Daray Shandard
0.89
0.895
0.89
0.2
0.4
0.6
0.4 Train Angular CKA
0.5
1.0
Train Angular CKA
Train Angular CKA
1-00%
t=0.5
2.0%
1=100
1=500%
Figure 15. Test accuracy of an ensemble of the original and perturbed models after training, versus representational similarity distance, on ResNet-20 with weight decay (left), R with 10x warm-up (middle) with perturbation time indicated by color, nd including the standard setting from Figure 3, with settings indicated by color (right).
C.5. Fixed Points of Aligning Permutations
While weight matching is unable to reduce barriers in our case of identically initialized networks, we investigate whether the underlying mechanism proposed by (Entezari et al., 2022) that barriers arise from network permutations is still relevant to our observations. We consider if the observed barriers and I distances between the original and perturbed networks correlate with the number of fixed points in the permutations found by weight alignment (Ainsworth al., 2023). Here, fixed points refer the un-permuted elements in the aligning permutations. Since expect two networks with identical weights to be aligned by permutations consisting only of fixed points, the fraction of fixed points can be used to indicate the degree to which two diverging networks have been permuted with respect to one another.
Figure 16 suggests a weak correlation, where the number of fixed points is inversely proportional to both the barrier heights and the L distance between the networks before alignment in the ResNet settings. Unfortunately. this observation does not extend to our ViT or BERT settings, as weight matching fails identify non-trivial permutations (i.e., permutations other than the identity) for these transformer architectures.
Barriers vs. Fixed Points (WM)
Barriers vs. Fixed Points (WM)
Batch
Batch
300
20
)2.)
os' s
&20
200
BGE
L2
10
100
0
0.4
0.8 WM Fixed Points
1.0
0.4
0.6
0.8
1.0
WM Fixed Points
Suting
10x Wamm-up
Purtub Norm Weights
SGD Weight Decat
yro Weight Deray)
Botch Size 512
RosNet-8, 10x Wamm-up
Figure 16. Relationship between the fraction of tixed points (un-pentn permuted elements) in the weight matching permutation aligning the ResNet-20 models trained on CIFAR-10 from Figure 2, and L divergence (left) train barriers (right).
D. Further Fine-tuning Results
D.I. ResNet Fine-Tuning
Extended pre-training yields near-zero fine-tuning barriers in ResNet-50/CIFAR settings. Figure 5 shows that fine-tuning from later checkpoints greatly improves stability in ResNet-50 experiments. For improved readability, we zoom into the perturbations with a & 10-2 in Figure 18 and Tables and 5, revealing that transferring from CIFAR-100 CIFAR-10
22

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The Butterfly Effect
ResNet-50 (CIFAR-100 -
ResNet-50 (CIFAR-10 -
CIFAR-10) Early Phase Batch
CIFAR-100) Early Phase Batch
1.5
BCc
1.0
BEE
2
0.5
0.0
0
10
10
10
10
0
0035, Oep180st I'm 0 005%, pmm 3ep327st
t= 0.5%.pp Depl.8st
10.0%, Dep280st
- (m 0.5%%p Seps27st
tm 10.0%, p 1ep327st
Figure/ 17. Same as Figure 5 (left), but with additional pre-training times and perturbation times from early in training. Stability of transfer learning on vision tasks: a ResNet-500 is pre-trained and fine-tuned (see Appendix A.2 for details) from CIFAR- 100 to CIFAR-IO (left) or vice versa (right). Barriers (y-axis) are plotted against perturbation magnitudes (x-axis) for various combinations of initial pre-trained weights and perturbation times (colors).
ResNet-50 (CIFAR-100
ResNet-50 (CIFAR-10 +
CIFAR-10) Late Phase Batch
CIFAR-100) Late Phase Batch 0.2
0.04
BcE
BOE
0.1
0.02
0.00
0.0
10 -4
-2
1.0.00%%. p 191
35%. 191ep
1- 10.0%. 181ep/1194 Por
t= 0.000% p 57ep213st
tu p. p7e.0ie STep213st
- 1=10.0% p S7ep213st
Figure 18. Same as Figure 5 (left) and Figure 17, but with additional pre-training times and perturbation times from late in training. Barriers scales < 10-2 are near-zero (see Tables and exact values).
and vice-versa results in near-zero barriers. The CIFAR-100 to CIFAR-10 direction exhibits greater stability, which could indicate that CIFAR- 100 pre-training is better suited for optimization on CIFAR 10 than the other way around.
Random initializations are less stable than pre-trdined initializations. Figure 18, we demonstrated that pre-training improves fine-tuning stability, while Figure 17 suggested that earlier checkpoints are more brittle to perturbations. As baseline, train a ResNet-50 from random initialization CIFAR-10 and find that it exhibits even larger barriers (Figure 19), with similar magnitude to training from scratch.
ResNet-50 (Random Init.)
CIFAR-10 Batch
2
0
10 -4
10 -2
0
t- 00.0%%%
1- 0.5%
t-
10.0%
Figure 19. Same as Figures 5, 17 and 18 (left panel of each). but with randomly initialized ResNet-50 instead of networks pre- ireitraind on CIFAR-100. Train loss barriers are reported for batch perturbations during fine-tuning on CIFAR-10 using the recipe from Appendix A.2.
23

---

The Butterfly Effect
Table 4. Train loss barriers and L distance in Cifar- 100 - Cifar- 10 setting, Same as Figure 18 (left).
Starting Checkpoint
Relative Perturb Step (%)
L Distance
Train BeB
191ep319st
0.005
0.00001
39,41 = 0.23
0.00 0.00
191ep319st
0.005
0.00100
39,46 + 0.18
0.00 = 0.00
19sc
0.005
0.01000
40.41 0.82
0,00 = 0.00
191ep319st
0.5
0.00001
39.49
0.00 +
191ep319st
0.5
0.00100
39.42
0.00
191ep319st
0.5
0.01000
40.03
0.00
191ep319st
10
0.00001
38.02
0.00
191ep319st
10
0.00100
38.04
0.00
191ep319st
10
0.01000
38.19
0.00
57ep213st
0.005
0.00001
0
0.00
57ep213st
0.005
0.00100
40.93 0.15
0.00
57ep213st
0.005
0.01000
4
0.00
57ep213st
0.5
0.00001
40.90
0.00
57ep213st
0.5
0.00100
0.00 0.17
57ep213st
0.5
0.01000
4..26
0.00
57ep213st
10
0.00001
3
0.00
57ep213ss 3st
10
0.00100
39.50 # 0.13
0.00 0.00
57ep213s 3st
10
0.01000
39.63 # 0.10
0.00 # 0.00
Table 5. Train loss barriers and L distance in Cifar-10 Cifar-100 setting. Same Figure 18 (right).
Starting Checkpoint
Relative Perturb Step (%)
L Distance
Train BeF
191ep319st
0.005
0.00001
67.98 + 0.00
0.00 0.00
191ep319st
0.005
0.00100
68.05 = 0.00
0.00 # 0,00
191ep319st
0.005
0.00500
68.92
0.01 = 0.00
191ep319st
0.005
0.01000
2.27
0.01 = 0.00
191ep319st
0.5
0.00001
67.89 0.00
0.00 # 0.00
191ep319st
0.5
0.00100
67.96
0,01 =
191ep319st
0.5
0.00500
68.18
0.01 0.00
191ep319st
0.5
0.01000
68.34
0.01 0.00
191ep319st
0.00001
0.00
0.00
191ep319st
10
0.00100
64.11
0.00
191cp319st
10
0.00500
6.04
0.00
191ep319st
10
0.01000
65.54
0.00
57ep213st
0.005
0.00001
6.00
0..00
57ep213st
0.005
0.00100
62.68 0.00
0.01
57ep213st
0.005
0.00500
0.00
0.00
57ep213st
0.005
0.01000
6.39
57ep21 3st
0.5
0.00001
62.32
0.01 0.00
57ep213st
0.5
0.00100
62.34
0.00
57ep213st
0.5
0.00500
6
0.01
57cp213st
0.5
0.01000
63.21
0.01
57ep213st
10
0.00001
+
0.00
57ep213st
0.00100
99.43
0.00
57ep213st 3st
0.00500
59.50
0.00
57ep213st
0.01000 10
59.70
0.00
24

---

The Butterfly Effect
D.2. ViT Fine-Tuning
Initial fine-tuning stability experiments on ResNets and MultiBERT's suggested opposed trends between vision tasks with convolutional architectures, and language tasks with tarnsformer architectures. To disentangle whether this difference is due to the nature of the task or architecture, applied experimental procedure to Vision Transformers (ViTs), which representative of the transformer architecture and larger-scale models than ResNet-50. The sources for the pre-trained ViTs and the fine-tuning procedure we use are specified in Appendix A.2.
While we did not have access to intermediate training checkpoints for VIT models, we instead compare four ViTs of different sizes which were pre-trained on ImageNet variants: base-patch11-224-in211 Ik (86M parameters), google/vit-laggeppateee 6-224-in21k (304M parameters), and pnogle/vit-huge patch14-224-in11k (632M parameters). The size of each pre-training dataset serves as proxy for pre-training duration.
bash--atchh-62224 (86M parameters), google/vit-
Extending our findings to ViT's fine-tuned on CIFAR-100. Consistent with our findings on smaller convolutional networks, in Figure 20 we observe that earlier and larger perturbations cause more pronounced barriers. Interestingly, the google/vit-base watch16-222 variant, which underwent additional fine-tuning on ImageNet- KK after its initial ImageNet-21 K pre-training, effectively represents a longer training process, Although the exact learning rate schedule is unknown, this setting resembles the extended pre-training observed in later BERT checkpoints. Models fine-tuned from this variant (Figure 20, top left) exhibit the largest barriers among all settings, with barriers order of magnitude larger than those from google/vit-base- patch16-224-in211 (Figure 20, top right). This provides additional evidence that extended pre-trate pre-training reduces fine-tuning stability. Future work could investigate the stability of intermediate ViT checkpoints.
googleliit-base--atchh--e to
google/vit-base.
CIFAR-100 Batch
6-24416--24-in21kk to CIFAR-100
-Batch
Bce
10
10
BeE
10"
0.00
0.05
0.10
0.00
0,05
0.10
o
google/vit-aare- Datchh6-224-in21k to CIFAR-100
google/vit-huge-
patch14-224-in22k to CIFAR-100
-Bttch
-Batch
10
10%
Bc
10
10
0.000
0.005
0.010
0.000
0.005
0.010
0
0
t= 0.01%
- t = 4.80%
t 9.60%
Figure 20. Stability of various VIT architectures CIFAR-100. Training loss barriers after training (y-axis) are plotted against perturbation magnitude (X-axis) and perturbation step (color). We fine-tune four pre-trained ViTs of different sizes: ViT-Base/16-224 (86M parameters, pre-trained on ImageNet-21KK and then fine-tuned ImageNet), BTTTBas/116-22-in22 (86M parameters pre-trained ImageNet-21kk) ViT-Large//1- 1--226-n22kkk (304M parameters), and ViT-Huge/14-224-inn (632M parameters).
D.3. BERT Fine-Tuning
Figure 22 plots additional fine-tuning tasks as described in Appendix A.2.
Representational similarity for BERT models. We also provide Angular CKA plots for BERT on MRPC and QNLI datasets in Figure 23. Figure 23 shows that barriers are correlated with Angular CKA, indicating real functional differences between the networks. This consistent with our findings in vision models (Figures 3, 14 and 21 right), unlike the correlation
25

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The Butterfly Effect
googleivit-base--ptchh----
google/vit-base-patchhf--
googleswitbbase--ptchhff
Batch
Batch
Batch
Train Angular CKA
Tost Ensemble Acc.
0.7
0.9225
10
0.6
BEE
0.9200
0.5
10
0.9175
0.4
10
10 -2
0.4
0.5
0.6
0.7
0.4
0.5
0.6
0.7
Train Angular CKA
Train Angular CKA
(= 0.01%
4.80%
9.60%
Figure 21. Left: representational similarity distance measured via Angular CKA for ViT-base. Middle: test ensembling accuracy against
Angular CKA. Right: training loss barrier against Angular CKA.
between barriers and L divergence (Figure 26) which is not consistent between vision and language settings.
D.4. OLMo Fine-Tuning
Figure 24 demonstrates that the trends we observe in terms of perturbation time and magnitude extend to decoder-only large language models. Consistent with our MultBEETT findings, we find that more pre-training does not necessarily lead to improved fine-tuning stability.
D.5. 2 Divergence and Barriers
Here, we examine the relationship between the barriers and the L divergence between models at the end of training in greater detail. Figure 25 (left, middle) shows that fine-tuning on vision tasks, such as transferring from CIFAR-100 to CIFAR-10 and vice versa, follows the trends presented in Figure (left). We see that this direct relationship is weak or non-existent when transferring ViTs from ImageNet to CIFAR-100 (Figure 25 right). well as for the GLUE benchmark in study (Figure 26), with QNLI and COLA exhibiting almost correlation. Interestingly, OLMo fine-tuned on GSM8K (Figure 24, right) shows clearer correlation between barriers and L divergence, which more closely resembles our ResNet-20 results (Figure 7, left) than BERT.
26

---

The Butterfly Effect
QNLI Batch
MRPC Batch
6
2
BEE
4
2
10
10
10
10 -2
RTE Batch
SST2 Batch
COLA Batch
0.0075
BEE
BEE
BcE
0.0050
2
0.0025
0
0.0000
10
10
10
10
10
10
0
9,
0,
0,
9, 2000K, = 50.00%
8p 200K, t= 50.00%
- 6, = 20k, = 50.00%
Figure 22. Same Figure (right), but with additional pre-training times, perturbation times, and tasks. Fine-tuning stability of Multi- BERT on QNLI and MRPC, starting from 20K, 200K, and 2000K checkpoints with different perturbation times. Tasks are QNLI (top left), MRPC (top right), RTE (bottom left), SST-2 (bottom middle), and COLA (bottom right).
MRPC Batch
QNLI Batch
Train Angular OKA
Train Angular CKA
1.5
1.0
1.0
0.5
0.5
10
10
10
10
0
MRPC Batch
QNLI Batch
10¹
2
BoE
10
BeE
1
10 -2
0.5
1.0
0.5
1.0
1.5
Train Angular CKA
Train Angular CKA
6, 2000k, t- 0.2%
6, 200k, t- 0.2%
8, 20k, t 0.2%
6, 50.0%
_+- 0, = 200k,
6, = 20K,
Figure 23. Top:
representational similarity distance measured via Angular CKA for MultiBETT on MRPC (left) and QNLI (right). Bottom: barriers angular CKA on MRPC (left) and QNLI (right).
27

---

The Butterfly Effect
Olmo Finetuning Stability On
Olmo Finetuning Stability On
Gsm8K Batch
Gsm8K Batch
10
10
BeE
BeE
10
5
10
0
-3
10
10
20
L2 O2,)
0
P- 1aap--KK66 48 Tokens, t- 0%
P-Stsp-369K
1.5T 1.57 Tokens, t- 0%
p=Step-740K -3 Tokens, t- 0
P- Tokens, t- 20%
P-Stsp-38KK 1.5T Tokens. [- 20%
p=Step-740K -3T Tokens, 20%
Figure 24. Stability of fin-ttuing OLMo- IB on GSM8K mathematical reasoning tasks. We fine-tune OLMo- IB checkpoints from different pre-training stages (early, middle, and final checkpoints) on GSMSK with batch perturbations applied at various training steps. Left: Loss barriers (y-axis) plotted against perturbation magnitude (X-axis) for different checkpoint combinations and perturbation steps (colors). Right: Barriers vs. 1 distance between the original and perturbed models. Consistent with our vision experiments, earlier perturbations and later pre-training checkpoints lead higher loss barriers, demonstrating that fine-tuning stability patterns generalize from vision to language models.
Barriers vs. L Distance Batch
Barriers Vs. L Distance Batch
Barriers Vs. L2 Distance Batch
10
10
10
BoE
10'
BGE
Bce
10
10
10 -2
10
40
60
60
80
100
100
200
300
L Distance (0r, 07)
L Distance (0,, 07)
L Distance (0,, 84)
googleivit-base 16 16-224
googletvit--ug patch 14 14-224-in21 1k
Dep180st 191ep319st
3ep327st
57ep213st
googleivil buse patch 16 10a2h16-224-ir 124-im2
gonglelvillarg large patch 16 224 1k
Figure 25. Train loss barriers L distance between the perturbed and original models the end of training for fine-tuning vision models: ResNet-50 transferring from CIFAR-100 100 to CIFAR-I0 (left). ResNet-50 transferring from CIFAR-10 to CIFAR- 100 (middle). ViT-base fine-tuned on CIFAR- 100 (right). Note legend for colors different the rightmost plot.
MRPC Barriers 2 Distance Batch
RTE Barriers L.2 Distance Batch
SST2 Barriers L2 Distance Batch
COLA Barriers vs. L? Distance Batch
10
10
1
T0
1
Bee
Bit
1
10
10"
5
3.5
4.0
4.5
4.0
4.5
5.0
5,5
6.0
6.5
7.0
Distance (0r, 07)
Distance (0r, 07)
L2 Distance (0,, 05)
Distance (θy, 0()
2000k
200k
Figure 26.
Train loss barriers vs. T,2 distance between the perturhed and original models at the end of training for MRPC, RTE, SST-2,
and COLA.
28

---

The Butterfly Effect
E. Additional Log-Scale Plots
These plots are copies of main figure plots with log y-axes, and are included to display a clearer separation between smaller barriers.
ResNet-20 Batch
ResNet-20 Batch
ResNet-20 Gaussian
10
10
10
BCE
ZiWm
ICEE
BCF
10
10
10
10
10
10
10
10
10
t= 0.0%
t= 0.5%
2 2.0%
--- t= 10.0%
50.0%
Figure 27. Same data Figure with y-axis log-scale improve the readability of smaller barriers. Loss barriers on training data
the end of training (y-axis) are plotted against perturbation magnitude (x-axis) and perturbation step (color indicates fraction of total training time). Left: barriers due to batch perturbation. Middle: batch perturbation barriers after accounting permutations. Right: barriers due to Gaussian perturbation.
ResNet-20 (10x W.up) Batch
ResNet-8/64 Batch
ResNet-8 (10x Warm-up) Batch
10
10
10
BeE
BcE
BeE
1
10 -
10
10
10 -3
10 -3
10
Partubedd
a
0.0%
t- 0.5%
2 2.0%
t- 10.0%
50.0%
Figure 28. Same as Figure 4 but with y-axis in log-scalo for models trained with 20% warm-up time (left), wider/shallower ResNet8 architecture (middle), and combination of both settings (right).
29


Reviews:
[{"review_id": "a7nk8i8pG9", "reviewer": "Reviewer_sufv", "summary": "The paper studies the impact of applying isolated perturbations to model parameters during different training and fine-tuning phases. The analysis provides insights into model training stability using three quantities measuring parameter similarity or functional similarity. The findings suggest that models become more stable during training, and the effect is even more evident for pretrained models. Interestingly, analysis shows that the longer training in the case of language tasks hurts the model's stability.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful review. Please find our replies below for the quoted points.\n\n## A. Other Similarity Measures\n\n> I'm wondering how other similarity measures differ from the perspective presented by the authors, e.g., CCA or CKA index, which could be an interesting add-on to the current work.\n\nThis is an excellent idea and was also recommended by Reviewer *mmJd*.\nWe prioritized barriers in our original analysis in order to determine when networks are in the same locally convex loss basin. For our reasoning about why this is useful, see the discussion of the advantages of linear mode connectivity in our rebuttal to Reviewer *HaCY*, under **C. Non-Linear Connectivity**.\n\nRepresentational similarity measures (e.g. CKA) are a more generic way to measure functional similarity that is not sensitive to mechanistic differences between neural network weights, such as barriers that prevent effective model merging. Nevertheless, we agree that measuring similarity in a more general sense will better illuminate how perturbed training trajectories differ.\n\nAccordingly, we have now conducted the following experiment on our standard setting (figure 2):\n\n1. We compute output dissimilarity (in $L^2$ between logits, or % of disagreeing classifications) and Angular CKA [Williams et al. 2021] between pairs of networks in our experiments: [https://imgur.com/a/zOBlg5X](https://imgur.com/a/zOBlg5X).\n2. As expected, measures of functional dissimilarity increase as barriers increase.\n3. Functional similarity becomes more sensitive to batch perturbation later in training.\n\nNote that Angular CKA ranges from 0 (perfectly similar) to $\\pi$ (perfectly dissimilar), with $\\pi/2$ indicating no correlation. We plot the largest (most dissimilar) CKA value over all residual block outputs, which are computed over 10000 examples using software from [Lange et al. 2023].\n\nWe will replicate the above experiment for our fine-tuning settings for inclusion in the text.\n\n- Williams, A. H.,et al. (2021). Generalized shape metrics on neural representations.\n- Lange, R. D., et al. (2023). Deep networks as paths on the manifold of neural representations.\n\n## B. Model Diversity\n\n> How can optimally perturb a model (and at which point) increase the final model's performance of the averaged model? Would that perturbation be universal across different merging strategies or should each strategy have its own method of perturbation?\n\nThis is a very interesting and novel application which we did not previously consider.\nWe have interpreted your point as the hypothesis that *deliberately perturbing models increases model diversity, leading to improved ensembling performance*. We test this hypothesis as follows:\n\n1. We take our measurements from above, and plot Angular CKA (x-axis) against ensemble performance: [https://imgur.com/a/ah7kCfg](https://imgur.com/a/ah7kCfg).\n2. We find that the ensembles indeed perform better on more dissimilar model pairs, which is in line with the proposed hypothesis.\n3. Note that, as shown by barriers increasing with Angular CKA, model averaging performs worse for more dissimilar model pairs.\n\nBy ensembling, we mean that in each of our spawn-and-perturb experiments, we evaluate the average of the output logits for the unchanged and perturbed networks after training.\nWhile Gaussian perturbations appear to have a more consistent relationship between functional dissimilarity and ensemble performance, we leave a full exploration of which perturbations are best suited for different merging strategies to future work.", "claims_and_evidence": "All the claims are supported by clear empirical evidence.", "methods_and_evaluation_criteria": "The methods (models and benchmarks) cover various popular settings, making the study interesting for both image classification and NLP communities. \n\nThe measures used by the authors to estimate the similarities between models are well-established methods within this area. However, I'm wondering how other similarity measures differ from the perspective presented by the authors, e.g., CCA or CKA index, which could be an interesting add-on to the current work.", "theoretical_claims": "The paper does not contain any formal statements.", "experimental_designs_or_analyses": "The experiments are clearly explained to the reader and meticulously designed to precisely measure the effect of perturbations.", "relation_to_broader_scientific_literature": "The work is well positioned within the related works. My current understanding is that the work could be an important first step towards designing precise training manipulations, which in turn could be used to increase the model's performance or robustness to data shifts by model merging techniques. It would be interesting to continue this line of work and explore questions like: How can optimally perturb a model (and at which point) increase the final model's performance of the averaged model? Would that perturbation be universal across different merging strategies or should each strategy have its own method of perturbation?", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "As mentioned earlier, it would be interesting to broaden the scope of methods for computing similarity between two models. Measures such as CCA, CKA could be useful in this regard and could offer a deeper understanding of the phenomenon.", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "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": "", "extra_scores": {}}, {"review_id": "KRb68xeXva", "reviewer": "Reviewer_HaCY", "summary": "This paper studies the impact of perturbation on SGD in an empirical way. The authors focus on the condition under which the perturbation during training can lead to convergence to another basin. The authors prepared perturbation in different directions of various scales that is applied to training at different moments. The same experiment is performed for both random initialization and transfer learning. The notable findings are:\n* Stability increases as the training goes on;\n* Stability can be improved by hyperparameter tuning;\n* pre-trained models are more stable;\n* Long pre-training reduces stability when fine-tuning language models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review. We have addressed the quoted comments and questions point-by-point below.\n\n## C. Non-Linear Connectivity\n\n> Due to the continuous symmetries, there are equivalent local minima that should be considered the same basin and are not linearly connected. [...] However, these solutions are not considered to be in the same basin according to the authors. [...] I wonder if the authors know how to tell whether $\\theta_T$ and $\\theta’_T$ are connected by continuous symmetry or not.\n\nThank you for raising this crucial and subtle distinction, which is also suggested by Reviewer *tDxT*.\n\nOur definition of “loss basin” (which we refer to here as “LMC basin”) in section 2 (*Linear mode connectivity*) reflects when the loss landscape is locally convex, and is drawn from Neyshabur et al. [2020] and Frankle et al. [2020]. Although this perspective is more restrictive than a general notion of “mode connectivity” [Draxler et al. 2018, Garipov et al. 2018], ours is both theoretically and practically relevant:\n1. Despite being a non-convex problem, neural network training has many connections with convex optimization. LMC basins describe approximately convex regions of the loss landscape.\n2. Classical methods for determining the stability of dynamical systems take a linear or quadratic approximation, which holds precisely in LMC basins.\n3. Model merging applications require networks to share a LMC basin, as shown by permutation alignment research [Singh & Jaggi 2020, Entezari et al. 2022, Ainsworth et al. 2023].\n\nPrior work, both theoretical [Simsek et al. 2021, Lin et al. 2024] and empirical [Draxler et al. 2018, Garipov et al. 2018, Sonthalia et al. 2024], also points to non-linear connectivity being a trivial property of neural network minima in general. If this is the case, then knowing if $\\theta_T$ and $\\theta’_T$ are continuously connected would be rather uninformative.\n\nWe apologize for not making our definition of “loss basin” and our justifications for the choice clear, and will update the above points in the text.\n\n- Simsek, B., et al. (2021). Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances.\n- Lin, Z.,et al. (2024). Exploring neural network landscapes: Star-shaped and geodesic connectivity.\n- Sonthalia, A., et al. (2024). Do Deep Neural Network Solutions Form a Star Domain?.\n\n## D. Correspondence With SGD Noise\n\n> In this work, the authors freeze the noise in SGD, and the injected noise is the main object of study for stability analysis. In reality, the noise of SGD may be much larger than the perturbations studied in this paper and overshadow them. Thus, I don't think the stability the authors study is very relevant. [...] I wonder if the authors could quantify how important the perturbation studied in this paper is compared to the SGD noise.\n\nWe agree that we have not sufficiently established the relative magnitude of our perturbations vs SGD noise. To address this, we replicate the parent-child spawning experiment of Frankle et al. [2020] (Frankle baseline) and show that our batch perturbations are a lower bound on the Frankle baseline’s barriers [https://imgur.com/a/frankle-ThKay8T](https://imgur.com/a/frankle-ThKay8T). In this comparison, we scale batch perturbations to the expected magnitude of SGD noise at the perturbation time $t$: making batch perturbation equivalent to taking only one step at time $t$ with different SGD noise, as opposed to using different SGD noise from $t$ onwards in the Frankle baseline.\n\nWe also tested even smaller perturbations by perturbing a fraction of weights at the smallest perturb scale we used in our main experiments ($10^{-4}$).\nWe find that as little as a **single (!)** perturbed weight (at a fraction of $10^{-6}$) causes barriers at initialization [https://imgur.com/a/AvfP8Mh](https://imgur.com/a/AvfP8Mh), which is well below the scale of noise from sources such as hardware indeterminacy.\n\nThe significance of our approach is that, as per section 3.1, the exponential convergence or divergence of a deterministic dynamical system dominates the diffusion effects of SGD noise [Wu et al. 2018]. This means that, in the dynamical systems model, training trajectories will diverge as long as they are unstable to noise *once*, whereas in the diffusion model, divergence depends on noise persisting throughout training.\n\n## Other Questions\n\n> In the numerical experiments, I have the impression that the amplitude of the perturbation can be much larger than rounding error. Where could these perturbations possibly come from?\n\nSources of perturbation in regular training could include SGD noise (batch order, data augmentation, hardware indeterminism) as well as model pruning or quantization.  We do not target a specific source so as to keep our experimental findings general.", "claims_and_evidence": "All of the claims are supported by numerical evidences. The authors provide statistically significant results in controlled settings and the experiment themselves are solid. However, the interpretation of the results may be less so. I think characterizing the basin by a linearly connected low-loss region in the loss landscape is a little problematic. (I will leave the details in theoretical claims).", "methods_and_evaluation_criteria": "Thus authors use image classification and language model training problems to study the stability. These are reasonable choices and they are good enough to support the contents of this paper, but they don’t cover all possible cases for learning (prediction, RL, PINN, …).", "theoretical_claims": "This paper does not involve theoretical proofs. \n\nI find the reasoning a little problematic though. The authors characterize the basin by a linearly connected low-loss region in the loss landscape. I don’t think this is always true. There is symmetry (not limited to permutation) in the loss function (for example, as discussed in https://arxiv.org/pdf/2309.16932). Due to the continuous symmetries, there are equivalent local minima that should be considered the same basin and  are not linearly connected. A naive example would be a loss function for parameters $(u, v)$ and data point $(x, y)$: $(uvx - y)^2$. The minima are located on the curve $uv = y/x$, and these solutions provide the same training and testing performance. However, these solutions are not considered to be in the same basin according to the authors. Thus, I think the authors’ way of modeling the basin does not take into account the complexity in the landscape.", "experimental_designs_or_analyses": "I am not very familiar with fine-tuning language models, so I cannot check the experimental designs in detail in this regard. I don’t find notable problems in experiments in other cases.", "relation_to_broader_scientific_literature": "I think this paper provides an alternative point of view to existing works. As far as I know the stability works usually study the problem that at which local minimum the SGD stays. The authors instead study whether SGD converges to the same local minimum regardless of the characteristic of the local minimum. Also, I am not aware of previous works that freezes SGD noise and introduce another source of noise.", "essential_references_not_discussed": "To my knowledge there are no essential references not discussed.", "other_strengths_and_weaknesses": "The authors investigate a lot of potential causes for instability. These potential causes are well isolated in the experiments, and the experimental facts are recorded well. This paper can serve as a nice starting point for crafting theories about stability.\n\nIn this work, the authors freeze the noise in SGD, and the injected noise is the main object of study for stability analysis. In reality, the noise of SGD may be much larger than the perturbations studied in this paper and overshadow them. Thus, I don't think the stability the authors study is very relevant.", "other_comments_or_suggestions": "I have no other comments.", "questions_for_authors": "I wonder if the authors know how to tell whether $\\theta_T$ and $\\theta’_T$ are connected by continuous symmetry or not.\n\nI wonder if the authors could quantify how important the perturbation studied in this paper is compared to the SGD noise.\n\nIn the numerical experiments, I have the impression that the amplitude of the perturbation can be much larger than rounding error. Where could these perturbations possibly come from?", "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": "", "extra_scores": {}}, {"review_id": "lC7cy3ICw5", "reviewer": "Reviewer_tDxT", "summary": "In this work, the authors study how the sensitivity of training trajectories in neural network training depends changes with the distance to the initialization. The authors characterize the sensitivity through the divergene of training trajectories, which is measured in $L_2$ distance, loss barrieres, defined as $\\sup_{\\alpha \\in (0,1)} \\ell(\\alpha \\theta_T + (1-\\alpha) \\theta'_T) - \\alpha \\ell(\\theta_T) - (1-\\alpha) \\ell(\\theta'_T)$, and barriers module permutation. \n\nDifferent from previous work, the authors study the training stability by removing the noise and studying deterministic training with controlled pertubations. The authors also compare the stability of loss landscape induced by different choices of optimizers and hyperparameters. \n\nTheir main findings include:\n- a small pertubation at a single iteration, which is smaller than training noise, is sufficient to cause the divergence of two otherwise identically and trained networks.\n- stability during the early phase of training can be improved for wider or shallower networks, increasing the batch size, learning rate warm-up and weight decay.\n- pre-trained networks are multiple orders of magnitudes more stable than randomly initialized networks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the detailed review, which we address by topic below.\n\n## E. $L^2$ Divergence\n\n> Also the $L^2$ distance is an intuitive metric to quantify the divergence between two runs. [...] Can the authors therefore provide similar Figures as the ones in Figure 2 or Figure 3, where the error barrier is measured in terms of divergence? It would particular interesting for me to see, how much the permutation of neurons $P$ can reduce the $L^2$ distance between two networks.\n\nApologies for the omission. We have plotted $L^2$ divergence in place of barriers here and will add them to the appendix for all figures: [https://imgur.com/a/BKMQHQk](https://imgur.com/a/BKMQHQk).\n\nWe do not prioritize $L^2$ divergence as it is not as indicative of functional similarity as barriers. $L^2$ is sensitive to optimization choices (e.g. weight decay), and network scale symmetries. Indeed, we observe that:\n\n- The relationship between $L^2$ and barriers differs depending on hyperparameter settings for our vision experiments (section 5, figure 7 colors).\n- $L^2$ and barriers are unrelated for our language experiments (section 5, figure 8).\n\nRegarding weight matching (WM), Ito et al. [2024] finds:\n> permutations found by WM do not significantly reduce the distance between the two\nmodels [...] WM satisfies LMC by aligning the directions of singular vectors with large singular values in each layer’s weights.\n\nThis highlights another lack of correspondence between $L^2$ and barriers.\n\n- Ito, A., et al. (2024). Analysis of linear mode connectivity via permutation-based weight matching\n\n\n## C. Non-Linear Connectivity\n\n> Given the fact that there is some work, e.g. Draxler et al. [2018] or Garipov et al. [2018], which showed that the optima in neural networks from different initializations can be connected through (simple) non-linear curves, I wonder how much it makes sense to look only at the barriers along the linear path between the weights. [..] Both work show that minima reached from different initializations can be connected through simple, non-linear curves. Can the authors discuss how this paper relates to these works?\n\nThis is an important point also raised by Reviewer *HaCY*. Please see our rebuttal to Reviewer *HaCY* under **C. Non-Linear Connectivity**, where we address this issue.\n\n## A., B. Functional Similarity And Model Diversity\n\n> I think the main weakness of this paper is that it is still unclear to me in what way the diverged solutions are actually \"different\" from each other, [...] For instance, it could be interesting to understand whether all pertubations still converge to the same manifold and how different pertubations for different points in time and magnitude of pertubations are distributed to each other (in terms ofdistance).\n\n> The authors also mentioned that it is desired in ensembling methods to combine diverse solutions, [...] I think my main concern with this work is that I am not sure what a reasonable way is to connect divergence to diversity of solutions. Can the authors perhaps comment on this?\n\nThese are important observations echoed by Reviewers *sufv* and *mmJd*. Based on your feedback, we have conducted additional experiments using CKA and ensembling to measure model diversity. Please see our full rebuttal to Reviewer *sufv* for details.\n\nRegarding the first quoted passage, could you clarify what you mean by “converge to the same manifold”? This may be related to the section **C. Non-Linear Connectivity**.\n\n## Other Questions\n\n> It would be helpful to show Figures 2-6 and Figure 9 with the y-axis in log-scale to distinguish the lines even better.\n\nWe have updated these figures accordingly and will add them to the text:\n- Figures 2 and 3 [https://imgur.com/a/fig2-3-jND4s6E](https://imgur.com/a/fig2-3-jND4s6E).\n- Figures 4, 5, and 6 [https://imgur.com/a/hsJyGhZ](https://imgur.com/a/hsJyGhZ). We provide tablbes in place of Figure 4 as barriers are 0 for $\\sigma<0.01$.\n- Figure 9 [https://imgur.com/a/jUXmcZQ](https://imgur.com/a/jUXmcZQ).\n\n> What are the conclusions on network training that you draw from the results of this paper?\n\nOur main conclusion is that neural network training is unstable near initialization even without randomness, meaning that interventions to reduce SGD noise (larger batch size, reduced learning rates, longer warmup periods) cannot eliminate this instability. Similar work has found that the trainability of neural networks can be unstable to small hyperparameter changes [Sohl-Dickstein 2024].\n\nConversely, pre-trained networks can still be fine-tuned into different modes (and thus, diverse solutions) with sufficiently large perturbations. Our findings suggest ways to manage model diversity among multiple training runs, such as by adjusting pre-training length, changing SGD noise magnitude via hyperparameters, or by directly perturbing networks.\n\nWe will add this conclusion into the text.\n\n- Sohl-Dickstein, J. (2024). The boundary of neural network trainability is fractal.", "claims_and_evidence": "Given the metrics that the authors have chosen to measure divergence and stability of training runs, I think the claims of the authors are well supported by the evidence provided, for instance in Figure 2 (early pertubation has a much larger effect than later pertubation), Figure 3 (modifying training hyperparameters can reduce the butterfly effect) or Figure 4 (pre-trained networks are much more stable than randomly-initialized networks).", "methods_and_evaluation_criteria": "I think that the evaluation criteria are reasonable overall, by considering both the loss barrier with and without permutation, the authors try to exclude the fact that training instability merely causes permutations between networks (although it cannot excluded, as the authors state, that they simply failed to find barrier-reducing permutations). \n\nAlso the $L_2$ distance is an intuitive metric to quantify the divergence between two runs. In fact, it seems even more natural to me to use this metric (or after applying $L_2$-minimizing permutations). What is the reason for the authors not to show these figures and can they provide them during the rebutal period? \n\nGiven the fact that there is some work, e.g. Draxler et al. [2018] or Garipov et al. [2018], which showed that the optima in neural networks from different initializations can be connected through (simple) non-linear curves, I wonder how much it makes sense to look only at the barriers along the linear path between the weights. It would be interesting to understand whether one can observe a similar trend if one computes the barrier along a Bezier curve. See for instance Fig. 1 in Garipov et al. [2018].", "theoretical_claims": "This paper does not contain any proof. The derivation of the lineared approximation for $L_2$ divergence in Appendix C looks correct to me.", "experimental_designs_or_analyses": "I checked the training details described in Appendix A and experiment details in Appendix B, which I believe is fine. Also the computation of the loss barriers seems reasonable to me. However I am not familier with the matching algorithm from Ainsworth et al. [2023] to find permutation of neurons $P$ that approximately minimizes the $L_2$ distance between two networks' weights.", "relation_to_broader_scientific_literature": "This paper is strongly related to work on linear mode connectivity, which has been studied extensively, for instance in Frankle et al. [2020a;b], Singh et al. [2020.]. Bachmann et al. [2023] study different factors which lead to linear mode connectivity (or lack thereof), including the network architecture, training setup and the dataset, while Entezari et al. [2022] pose the conjecture that by taking into account permutation invariance, solutions trained with SGD will likely be in the same basin. \n\nThis paper follows the experimental setup as in Frankle et al. (2020a), but eliminate the effect of training noise by fixing it for different runs and isolate the effect of pertubations.", "essential_references_not_discussed": "There is some related work on mode connectivity, which have not discussed in this work, including Draxler et al. [2018] on \"Essentially No Barriers in Neural Network Energy Landscape\" and Garipov et al. [2018] on \"Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs\". \n\nBoth work show that minima reached from different initializations can be connected through simple, non-linear curves. Can the authors discuss how this paper relates to these works?", "other_strengths_and_weaknesses": "+ + I think the main strength and novelty of this work is to isolate the effect of noise in training and really just comparing (deterministic) training runs with controlled injected pertubations. This highlights the sensitivity of the network to slight variations.\n\n- - I think the main weakness of this paper is that it is still unclear to me in what way the diverged solutions are actually \"different\" from each other, since they achieve comparable training and test accuracy. For instance, it could be interesting to understand whether all pertubations still converge to the same manifold and how different pertubations for different points in time and magnitude of pertubations are distributed to each other (in terms of $L_2$ distance). The authors also mentioned that it is desired in ensembling methods to combine diverse solutions, but I am not sure what insights this work provides in this regard. I think my main concern with this work is that I am not sure what a reasonable way is to connect divergence to diversity of solutions.\nCan the authors perhaps comment on this?", "other_comments_or_suggestions": "It would be helpful to show Figures 2-6 and Figure 9 with the y-axis in log-scale to distinguish the lines even better.", "questions_for_authors": "I was first suprised that there is one point for $t=0.0 \\%$ in Figure 2 where the barrier is actually higher after permutation than without. Upon closer look I realized that the permutation is minimizing the $L_2$ distance between two networks. \nCan the authors therefore provide similar Figures as the ones in Figure 2 or Figure 3, where the error barrier is measured in terms of $L_2$ divergence? It would particular interesting for me to see, how much the permutation of neurons $P$ can reduce the $L_2$ distance between two networks. \n\nWhat are the conclusions on network training that you draw from the results of this paper?\n\nAlthough I gave a rather low score, I am open to adjust my score if the authors address my comments.", "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": "", "extra_scores": {}}, {"review_id": "RjsPYZqQgC", "reviewer": "Reviewer_mmJd", "summary": "The submission investigates the sensitivity of neural network training outcome to initial conditions. Building upon related work in optimization and training dynamics, it investigates the conditions for stability and identifies a chaotic early stage in which small perturbations cause trajectory divergence. Experimental evaluations demonstrate that effect in vision models, but find different effects in language models.\n\n### Summary after Rebuttal\nIn the rebuttal, the authors have added behavioral comparisons and removed claims with weak support, which I believe improve the submission overall. I encourage the authors to continue improving the experimental support and will keep my score as is for now.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your substantive review, to which we have replied below.\n\n## F. Comparing Dissimilar Tasks and Scales\n\n> Experiments are performed mostly on relatively small vision models with relatively simple tasks. [...] To compare different domains, one would have to compare models in the same regime varying the domain,\n\nWe agree with this assessment of our experimental scale, and analogous to figures 4-6, have added an experiment that fine-tunes ViTs from google/vit-base-patch16-224 on CIFAR-100: [https://imgur.com/a/AculLLH](https://imgur.com/a/AculLLH).\n\nWe also note that the claims below (sections 4.3 and 5) are negative statements, for which a counterexample is sufficient:\n\n1. More pre-training does not always improve fine-tuning stability (figures 4-6).\n2. Barriers do not generally scale with $L^2$ divergence (figures 7-8).\n3. Barriers and $L^2$ do not diverge exponentially proportional to perturbation scale (figure 9).\n\nIn these cases, we do not strictly need to isolate the effects of task vs architecture or scaling.\n\n## G. Fine-tuning Task Difficulty\n\n> the claim that going from hard to simple improves stability is evaluated only on CIFAR100 and CIFAR10, which are very entangled beyond ‘hard’ and ‘simple’.\n\nWe agree that the claim “Pre-training on harder tasks improves stability” is weak. We attempted to make this claim more precise by relating stability to fine-tuning performance, but preliminary evidence in this direction is inconclusive. We have therefore replaced this claim with our new findings on CKA and ensembling (below).\n\n## A., B. Functional Similarity And Model Diversity\n> Why not evaluate behavioral similarlity (CKA, SVCCA, agreement, etc)? As the authors note, the same/different basins may be relevant for merging / ensembling models, for which it is essential to know whether or not models actually learn different behavior.\n\nReviewers *sufv* and *tDxT* have also raised this excellent point. Based on your feedback, we have run additional experiments that find:\n1. Training instability (as evidenced by larger barriers) indeed correlates with functional dissimilarity (Angular CKA).\n2. Perturbations to training can be used to improve ensemble performance, with more functional dissimilarity leading to better ensembles. This shows that instability increases model diversity.\n\nPlease see our full rebuttal to Reviewer *sufv* for details and figures.\n\nFinally, to understand why we chose to focus on barriers as a measure of functional dissimilarity, please also see our rebuttal to Reviewer *HaCY* under **Non-Linear Connectivity**.\n\n## Other Questions\n\n> There are several alignment methods that improve over git re-basin, e.g. Sinkhorn based [1] or learned [2].\n\nWe note that weight matching (WM) is already able to greatly reduce barriers between *independently trained*  networks for our standard ResNet20-32 architecture [Ainsworth et al. 2023], whereas WM has little effect on the *identically spawned* networks of our experiments. The ineffectiveness of WM in our case suggests that other methods may not fare much better. Practically speaking, Peña et al. [2023] and Navon et al. [2024] are also quite costly (scaling poorly to larger models), as they must learn their permutations.\n\n> Figure 2 and 3 titles and caption could be improved. Left and middle figure have the same title. Why are Gaussian permutation barriers evaluated without accounting for permutations? Figure 3 - same as which of Figures 2?\n\nThank you for the detailed corrections. We will revise the titles/captions, and add figures for permutation-aligned barriers for Gaussian perturbations into the appendix. Note that just like in batch perturbations, permutation does not reduce Gaussian-perturbed barriers.\n\n> I would be curious to learn why the authors chose their experimental setup as they have. Are these mostly computational constraints, or based on previous work?\n\nIndeed, our experimental setup is motivated by that of previous works [Frankle et al. 2020, Vlaar & Frankle 2022, Juneja et al. 2023, Ainsworth et al. 2023]. Having said that, evaluating stability for training (as opposed to fine-tuning) would require full runs which is costly when considering many replicates, perturbation times/magnitudes, and hyperparameter settings.\n\n> Also, the submission motivates the work via two streams: optimization and work on early training phases. To me, that duality is lost in the course of the submission, and it is more of a generalization of existing work in the latter. I am curious what the authors see as the optimization perspective on their work.\n\nWe apologize for this confusion. Our intent was to disambiguate between our notion of instability (divergence of two trajectories that each independently converge to a solution) and that of work such as Wu et al. [2018] (tendency of a trajectory to fail to converge to certain solutions). We will de-emphasize the optimization perspective in the introduction to reflect its relevance to our work.", "claims_and_evidence": "Before I go into this, I generally like the paper (see strenghts and weaknesses below). However, I find the claims made way to general for the experimental support. \n\nExperiments are performed mostly on relatively small vision models with relatively simple tasks. I understand the need for tractable experiment setups, but the experimental evaluation would be strengthened significantly if showed beyond ResNet20/50 and CIFAR10 / CIFAR100. This particular combination has showed specific effects in previous work, which unfortunately doesn’t always generalize. I am therefore somewhat skeptical to what degree the results generalize and would encourage the authors to systematically assess different architectures, sizes, and tasks. As is, the insight rest on relatively narrow experimental support. For example: the claim that going from hard to simple improves stability is evaluated only on CIFAR100 and CIFAR10, which are very entangled beyond ‘hard’ and ‘simple’. A claim that general requires more experimental evaluation. \n\nThe same goes for the comparison between language and vision models. Conventional vision models (like ResNet20s on CIFAR) and even relatively small and outdated BERT models operate in very different regimes (scale, parameter-data-ratio, training task, etc). The conclusion that language and vision models behave differently are not supported by a single skewed experiment like that. If anything, there may be insights into different behavior of different regimes. To compare different domains, one would have to compare models in the same regime varying the domain, e.g. large VITs pre-trained with data augmentation to recent LLMs, considering the respective scaling laws. In fact, the authors admit that in Section 5 lines 434.", "methods_and_evaluation_criteria": "Generally yes (modulo narrow datasets+architecture combinations discussed before). The framework is very well motivated and introduced. The only question I had is why the authors chose to use only loss barriers, l2 distance or perm distance? Why not evaluate behavioral similarlity (CKA, SVCCA, agreement, etc)? As the authors note, the same/different basins may be relevant for merging / ensembling models, for which it is essential to know whether or not models actually learn different behavior. I encourage the authors to include that in their framework as well, or to explain why they chose to not include it.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "The experimental design is well done, except for the issues discussed above.", "relation_to_broader_scientific_literature": "The authors discuss the relation to previous work well throughout the paper, but explicitly in section 2.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "## Strengths\n- The paper is very well written, has a tight narrative and well motivated introduction.  \n- Relevant problem, understanding the sensitivity is fundamentally important, but also highly relevant for understanding the genesis of distinct phases, model merging, ensembling, pruning and also adversarial vulnerability of models.  \n- Well written related work, that clarified the submissions relation to existing work and identified the contribution.  \n- Extending Frankle et al. parent-child spawning experiments, the submission presents a more general framework to study the impact of some perturbation on a training trajectory.  \n- There are some very interesting results, particularly the attribution of stability to the weights (rather than direction), and suggesting the natural occurrence of permutations is not a likely phenomenon to explain trajectory divergence.\n\n## Weaknesses\nMy main issues with the paper are already discussed above: the generality of claims that are not well supported experimentally, and the obmission of behavioral similarity metrics. Beyond that:\n- There are several alignment methods that improve over git re-basin, e.g. Sinkhorn based [1] or learned [2].  \n- Figure 2 and 3 titles and caption could be improved. Left and middle figure have the same title. Why are Gaussian permutation barriers evaluated without accounting for permutations? Figure 3 - same as which of Figures 2?\n\n[1] Pena et al., Re-basin via implicit Sinkhorn differentiation, CVPR 2023.\n[2] Navon et al., Equivariant Deep Weight Space Alignment, ICML 2024", "other_comments_or_suggestions": "Typo in the paragraph heading in Section 5: \"with with\"", "questions_for_authors": "I would be curious to learn why the authors chose their experimental setup as they have. Are these mostly computational constraints, or based on previous work? \n\nAlso, the submission motivates the work via two streams: optimization and work on early training phases. To me, that duality is lost in the course of the submission, and it is more of a generalization of existing work in the latter. I am curious what the authors see as the optimization perspective on their work.", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper systematically studies the chaotic nature of neural network training i.e., little changes to initial conditions smaller than training noise, leads to highly divergent later trajectories. 

All reviewers unanimously appreciated the paper. 
- Most reviewers appreciated the breadth of models studied (`sufv`) 
- and the fact that both image and language models are studied (`sufv`, `HacY`). 
- There is agreement that the experiments are rigorous and support the claims (`HacY`: "claims are supported by numerical evidences", "statistically significant results in controlled settings", `tDXT`: "the claims of the authors are well supported by the evidence provided"). 
- The reviewers also find the results interesting and an advance on what is studied in this area (`mmJD`).

I believe that the main concerns were addressed well by the authors through additional results/arguments:
1. Adding other similarity scores (`sufv`, `mmJD`, `tDxT `) 
3. Models/tasks being outdated/small/simple (`mmJD`)
3. Why study linear mode connectivity (`HacY`, `tDXT`, ) -- the linearity of mode connectivity only makes it a more interesting phenomenon to study.
4. Noise studied is smaller than that of training noise (`HacY`) -- like above, I believe that this only makes the instabilities a more surprising object worth a study. 

_Footnote that doesn't affect the decision:_ It would be worth relating the curious findings here to the one in [1] where a student model initialized _very_ close the teacher ends up deviating far from the teacher (see their Fig 5 and discussion below). 

"Random Teachers are Good Teachers", Sarnthein et al., ICML 2023, https://arxiv.org/abs/2302.12091
Decision: Accept (poster)