Original: Mechanistic Lens on Mode Connectivity
Revised: Mechanistic Lens on Mode Connectivity

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: With the rise of pretrained models, fine-tuning has become increasingly important.
Revised: With the rise of pretrained models, fine-tuning has become increasingly important.

Original: However, naive fine-tuning often does not eliminate a model’s sensitivity to spurious cues.
Revised: However, naive fine-tuning often does not eliminate a model’s sensitivity to spurious cues.

Original: To understand and address this limitation, we study the geometry ofneural network loss landscapes through the lens of mode-connectivity.
Revised: To understand and address this limitation, we study the geometry of neural network loss landscapes through the lens of mode-connectivity.

Original: We tackletwo questions: 1) Are models trained on different distributions mode-connected? 2)
Revised: We tackle two questions: 1) Are models trained on different distributions mode-connected? 2)

Original: Can we fine tune a pre-trained model to switch modes?
Revised: Can we fine tune a pre-trained model to switch modes?

Original: We define a notion of mechanistic similarity based on shared invariances and show linearly-connected modesare mechanistically similar.
Revised: We define a notion of mechanistic similarity based on shared invariances and show linearly-connected modes are mechanistically similar.

Original: We find naive fine-tuning yields linearly connectedsolutions and hence is unable to induce relevant invariances.
Revised: We find naive fine-tuning yields linearly connected solutions and hence is unable to induce relevant invariances.

Original: We also propose andvalidate a method of “mechanistic fine-tuning” based on our gained insights.
Revised: We also propose and validate a method of “mechanistic fine-tuning” based on our gained insights

Original:  Introduction
Revised:  Introduction

Original: Consider modes that rely on different features (highlighted yellow) to make their predictions.
Revised: Consider modes that rely on different features (highlighted yellow) to make their predictions.

Original: Are such mechanistically dissimilar modes connected via paths of high accuracy?
Revised: Are such mechanistically dissimilar modes connected via paths of high accuracy?

Original: Does difference in mechanisms affect the simplicity of their paths?
Revised: Does difference in mechanisms affect the simplicity of their paths?

Original: And, can we exploit this connectivity to switch between modes?
Revised: And, can we exploit this connectivity to switch between modes?

Original: Deep neural networks (DNNs) suffer from various shortcomingsof robustness [1, 2, 3], often relying on spurious or shortcut cuesthat do not generalize robustly but are “simpler” to learn [4, 5].
Revised: Deep neural networks (DNNs) suffer from various shortcomings of robustness [1, 2, 3], often relying on spurious or shortcut cues that do not generalize robustly but are “simpler” to learn [4, 5].

Original: For ex., in vision tasks, models can exploit features such as background or texture to identify object categories; however, shaperelated features are likely to be more robust in practice [6, 7, 8].
Revised: For ex., in vision tasks, models can exploit features such as background or texture to identify object categories; however, shaperelated features are likely to be more robust in practice [6, 7, 8].

Original: Invariant prediction and related approaches [9] aim to producesuch robust models by accounting for the causal mechanismsunderlying the data generating process, hence inducing invariance to such “spurious” features and learning ones that generalizestrongly [10].
Revised: Invariant prediction and related approaches [9] aim to produce such robust models by accounting for the causal mechanisms underlying the data generating process, hence inducing invariance to such “spurious” features and learning ones that generalize strongly [10].

Original: Meanwhile, when trained on different data distributions, a model may end up learning different invariances.
Revised: Meanwhile, when trained on different data distributions, a model may end up learning different invariances.

Original: In thiswork, we introduce the notion of mechanistic similarity to describe models that share invariances, but may otherwise differ intheir predictions.
Revised: In this work, we introduce the notion of mechanistic similarity to describe models that share invariances, but may otherwise differ in their predictions.

Original: Our motivating question is whether fine-tuningcan alter a model’s learned invariances.
Revised: Our motivating question is whether fine-tuning can alter a model’s learned invariances.

Original: Specifically, if a modelhas learned to rely on spurious features in its training data, can weget it to break that “bad habit” by fine-tuning it on some “clean”data that does not contain such spurious features?
Revised: Specifically, if a model has learned to rely on spurious features in its training data, can we get it to break that “bad habit” by fine-tuning it on some “clean” data that does not contain such spurious features?

Original: We considerthis question through the lens of mode-connectivity [11, 12],which argues relatively simple paths connect DNN minimizers via paths of high accuracy or low loss.
Revised: We consider this question through the lens of mode-connectivity [11, 12], which argues relatively simple paths connect DNN minimizers via paths of high accuracy or low loss.

Original: 1.1 Preliminaries / Notations
Revised: 1.1 Preliminaries / Notations

Original: Model: Consider a neural network f : X × R d → R n .
Revised: Model: Consider a neural network f : X × R d → R n .

Original: The model’s output decision is denotedˆ y ( f ( x, θ ))
Revised: The model’s output decision is denoted ˆ y ( f ( x, θ ))

Original: ∈ Y = { 1 , ..., K } for input x ∈ X ⊂ R n and parameters θ ∈ R d .
Revised: ∈ Y = { 1 , ..., K } for input x ∈ X ⊂ R n and parameters θ ∈ R d .

Original: The ground truth targetis denoted y ∈ Y .
Revised: The ground truth target is denoted y ∈ Y .

Original: The model’s loss on a dataset D ∈ X × Y for parameter setting θ is denoted as L ( f ( D ; θ )) .
Revised: The model’s loss on a dataset D ∈ X × Y for parameter setting θ is denoted as L ( f ( D ; θ )) .

Original: We denote a continuous path connecting two set of parameters θ 1 , θ 2 as γ θ 1 → θ 2 ( t ) ,where γ θ 1 → θ 2 (0) = θ 1 , and γ θ 1 → θ 2 (1) =
Revised: We denote a continuous path connecting two set of parameters θ 1 , θ 2 as γ θ 1 → θ 2 ( t ) , where γ θ 1 → θ 2 (0) = θ 1 , and γ θ 1 → θ 2 (1) =

Original: θ 2 .
Revised: θ 2 .

Original: Mode-connectivity is formalized as follows.
Revised: Mode-connectivity is formalized as follows.

Original: Definition 1.
Revised: Definition 1.

Original: Mode-Connectivity (along a Path.)
Revised: Mode-Connectivity (along a Path.)

Original: Assume modes θ 1 , θ 2 achieve L ( D ; θ ) <
Revised: Assume modes θ 1 , θ 2 achieve L ( D ; θ ) <

Original: ϵ , forsome small value ϵ .
Revised: ϵ , for some small value ϵ .

Original: We call θ 1 , θ 2 mode-connected along γ θ 1 → θ 2 ( t ) if moving along the path neveryields increase in loss, i.e., ∀ t ∈ [0 , 1] , L ( f ( D , γ θ 1 → θ 2 ( t ))) <
Revised: We call θ 1 , θ 2 mode-connected along γ θ 1 → θ 2 ( t ) if moving along the path never yields increase in loss, i.e., ∀ t ∈ [0 , 1] , L ( f ( D , γ θ 1 → θ 2 ( t ))) <

Original: ϵ .
Revised: ϵ .

Original: Prior works have found modes in modern neural networks’ landscapes to be connected viarelatively simple paths [11, 12, 13, 14].
Revised: Prior works have found modes in modern neural networks’ landscapes to be connected via relatively simple paths [11, 12, 13, 14].

Original: We thus restrict our experiments to the following:
Revised: We thus restrict our experiments to the following:

Original: [Equation]
Revised: [Equation]

Original:  Here, θ m denotes a set of parameters that is explicitly optimized to identify a quadratic path connecting the two modes θ 1 , θ 2 (see App. B.4).
Revised: i [Equation] Here, θ m denotes a set of parameters that is explicitly optimized to identify a quadratic path connecting the two modes θ 1 , θ 2 (see App. B.4).

Original: We note that modes which do not appear to be linearlyconnected can often be linearly connected via an appropriate permutation of neurons that preservesthe model’s functional nature (i.e., produces the same outputs) (see App. B.5) [15, 16, 17].
Revised: We note that modes which do not appear to be linearly connected can often be linearly connected via an appropriate permutation of neurons that preserves the model’s functional nature (i.e., produces the same outputs) (see App. B.5) [15, 16, 17].

Original: Data: We assume there is a latent space Z ⊂ R m that instantiates a data-generating process (DGP)
Revised: Data: We assume there is a latent space Z ⊂ R m that instantiates a data-generating process (DGP)

Original: G : Z → X × Y and induces the datapoint ( x, y ) as follows: ( x, y ) :
Revised: G : Z → X × Y and induces the datapoint ( x, y ) as follows: ( x, y ) :

Original: = G ( z ) .
Revised: = G ( z ) .

Original: X and Y are assumedconditionally independent given Z , and define G X , G Y as the components of G producing X and Y .
Revised: X and Y are assumed conditionally independent given Z , and define G X , G Y as the components of G producing X and Y .

Original: Similar to prior work on disentanglement [18, 19, 20, 21, 22] and nonlinear ICA [23, 24, 25, 26, 27],we assume G X ( . ) has a valid inverse G − 1 X : X → Z and the latent dimensions are statisticallyindependent, i.e., z i ∈
Revised: Similar to prior work on disentanglement [18, 19, 20, 21, 22] and nonlinear ICA [23, 24, 25, 26, 27], we assume G X ( . ) has a valid inverse G − 1 X : X → Z and the latent dimensions are statistically independent , i.e., z i ∈

Original: Z i ⊥ z j ∈ Z j , ∀
Revised: Z i ⊥ z j ∈ Z j , ∀

Original: ( i, j ) , i ̸
Revised: ( i, j ) , i ̸

Original: =
Revised: =

Original: j .
Revised: j .

Original: To empirically validate our claims, we need theability to intervene on spurious cues in the data and generate counterfactuals (see Def. 2).
Revised: To empirically validate our claims, we need the ability to intervene on spurious cues in the data and generate counterfactuals (see Def. 2).

Original: We thuspropose to use synthetic datasets with known spurious cues–e.g., CIFAR-10 with a located box cue;see App.
Revised: We thus propose to use synthetic datasets with known spurious cues–e.g., CIFAR-10 with a located box cue; see App.

Original: B.3 for details and visualizations.
Revised: B.3 for details and visualizations.

Original:  Towards a Definition of Mechanistic Similarity
Revised:  Towards a Definition of Mechanistic Similarity

Original: Figure 2: Summarizing Mechanistic Similarity: We define mechanistic similarity of two modes by assessing their response to unit interventions on the data-generating process, i.e., interventions on specific dimensions of the latent vector z (e.g., background and shape in the figure).

Original: Modes invariant to the same set of interventions (denoted ∼ ) are termed mechanistically similar.
Revised: interventions specific shape set of interventions (denoted ∼ ) are termed mechanistically similar

Original: While a surprising result, it is unclear if connectivity emerges in modes that rely on different mechanisms (e.g., shape vs. background; see Fig. 1).
Revised: While a surprising result, it is unclear if connectivity emerges in modes that rely on different mechanisms (e.g., shape vs. background; see Fig. 1).

Original: Toanswer this, we must first design a notion of mechanistic (dis)similarity.
Revised: To answer this, we must first design a notion of mechanistic (dis)similarity.

Original: The intuition behind our definitionwill be the following question: do models that succeedin a similar manner, fail in a similar manner?
Revised: The intuition behind our definition will be the following question: do models that succeed in a similar manner, fail in a similar manner?

Original: We propose to assess failures by measuring a model’s responseto relevant data transformations: if two models usesimilar mechanisms, they should respond similarly totransformed inputs; by choosing transforms that encodetask-relevant vulnerabilities, we can make this definition operationally well-motivated.
Revised: We propose to assess failures by measuring a model’s response to relevant data transformations: if two models use similar mechanisms, they should respond similarly to transformed inputs; by choosing transforms that encode task-relevant vulnerabilities, we can make this definition operationally well-motivated.

Original: For ex., randomizing the synthetic cue in Fig.
Revised: For ex., randomizing the synthetic cue in Fig.

Original: 5, we can assess whethera model relies on the cue.
Revised: 5, we can assess whether a model relies on the cue.

Original: This is analogous to theuse of visual illusions (invalid percepts) for designingmodels of early visual processing in neuro-/cognitivescience [28, 29].
Revised: This is analogous to the use of visual illusions (invalid percepts) for designing models of early visual processing in neuro-/cognitivescience [28, 29].

Original: We first define the following.
Revised: We first define the following.

Original: Definition 2.
Revised: Definition 2.

Original: (Unit Interventions and Counterfactuals.)
Revised: (Unit Interventions and Counterfactuals.)

Original: An isomorphism
Revised: An isomorphism

Original: A α i
Revised: A α i

Original: i : Z i × Z i
Revised: i : Z i × Z i

Original: → Z idefines a unit intervention on the i th dimension of the state z if it alters its value by adding apredefined scalar α i .
Revised: → Z i defines a unit intervention on the i th dimension of the state z if it alters its value by adding a predefined scalar α i .

Original: The isomorphism E : X × R m → X defines a counterfactual if it altersa datapoint x by changing its corresponding latent z =
Revised: The isomorphism E : X × R m → X defines a counterfactual if it alters a datapoint x by changing its corresponding latent z =

Original: G − 1 X ( x ) via a set of unit interventions{A} :
Revised: G − 1 X ( x ) via a set of unit interventions {A} :=

Original: = {A α i i } mi =1 .
Revised: {A α i i } mi =1 .

Original: Specifically, we have, E ( x ; (cid:98) A ) =
Revised: Specifically, we have, E ( x ; (cid:98) A ) =

Original: G X ◦
Revised: G X ◦

Original: A α m m ◦ · · · ◦ A α 1 1 ◦ G − 1 X ( x ) .
Revised: A α m m ◦ · · · ◦ A α 1 1 ◦ G − 1 X ( x ) .

Original: Unit interventions on the data-generating process allow precise manipulation of a state z , while acounterfactual maps the changed state into the observable data space.
Revised: Unit interventions on the data-generating process allow precise manipulation of a state z , while a counterfactual maps the changed state into the observable data space.

Original: Due to independence of latentdimensions, our definition of unit interventions easily composes and can model broader notions ofinterventions [30]; combined with counterfactuals, unit interventions are thus sufficient to assess amodel’s response to general data transformations, as we show below.
Revised: Due to independence of latent dimensions, our definition of unit interventions easily composes and can model broader notions of interventions [30]; combined with counterfactuals, unit interventions are thus sufficient to assess a model’s response to general data transformations, as we show below.

Original: Definition 3.
Revised: Definition 3.

Original: (Invariance.)
Revised: (Invariance.)

Original: The model f ( . ; θ ) is termed invariant to unit intervention A
Revised: The model f ( . ; θ ) is termed invariant to unit intervention A

Original: i if counterfactuals generated by A i do not yield increase in loss, i.e., L ( f ( D ; θ ))
Revised: i if counterfactuals generated by A i do not yield increase in loss, i.e., L ( f ( D ; θ ))

Original: =
Revised: =

Original: E α ∈Z i L ( f ( E ( D ; A αi ); θ )) .
Revised: E α ∈Z i L ( f ( E ( D ; A αi ); θ )) .

Original: Lemma 1.
Revised: Lemma 1.

Revised: (Exhaustiveness of Unit Interventions.)

Original: If f ( . ; θ ) is invariant to unit interventions
Revised: If f ( . ; θ ) is invariant to unit interventions

Original: A i and A j , it must be invariant to theircomposition; if it is not invariant to A i or A j , it cannot be invariant to their composition.
Revised: A i and A j , it must be invariant to their composition; if it is not invariant to A i or A j , it cannot be invariant to their composition.

Revised: That is, unit interventions exhaustively characterize a model.

Original: [Figure]
Revised: [Figure]

Original: Dissimilar Modes.
Revised: Dissimilar Modes.

Original: We train ResNet18 models on our synthetic CIFAR-10 with box-cues and the original dataset (denoted θ C , θ NC , respectively). synthetic cues.
Revised: We train ResNet18 models on our synthetic CIFAR-10 with box cues and the original dataset (denoted θ C , θ NC , respectively)

Original: denote train/test counterfactuals datasets, where either the cue is present (w/ Cue), absent (w/o Cue), randomized (Rand. Cue), or the underlying image is randomized (Rand.
Revised: /test counterfactuals datasets, where either the cue is present (w/ Cue), absent (w/o Cue), randomized (Rand. Cue), or the underlying image is randomized (Rand.

Original: Img).
Revised: Img).

Original: We find the two modes can be connected via quadratic, but not linear, paths.
Revised: We find the two modes can be connected via quadratic, but not linear, paths.

Original: Moreover, minimizers near θ NC yield the same performance upon randomization of the cue, while ones near θ C lose performance substantially, showing lack of shared invariances and mechanistic dissimilarity.
Revised: Moreover, minimizers near θ NC yield the same performance upon randomization of the cue, while ones near θ C lose performance substantially, showing lack of shared invariances and mechanistic dissimilarity.

Original: See App.
Revised: See App.

Original: D for further results.
Revised: D for further results.

Original: Essentially, the above lemma states that studying a model’s response to unit interventions is sufficientto characterize it: if a model is invariant to a set of unit interventions, it must be invariant to theircomposition; similarly, lack of invariance to a single unit intervention is sufficient to precludeinvariance to the composition of those interventions.
Revised: Essentially, the above lemma states that studying a model’s response to unit interventions is sufficient to characterize it: if a model is invariant to a set of unit interventions, it must be invariant to their composition; similarly, lack of invariance to a single unit intervention is sufficient to preclude invariance to the composition of those interventions.

Original: The lemma thus allows us to define mechanisticsimilarity of two modes as sharing of invariances to the same unit interventions.
Revised: The lemma thus allows us to define mechanistic similarity of two modes as sharing of invariances to the same unit interventions.

Original: Definition 4.
Revised: Definition 4.

Original: (Mechanistic Similarity.)
Revised: (Mechanistic Similarity.)

Original: Consider a set of unit interventions {A} :
Revised: Consider a set of unit interventions {A} :

Original: =
Revised: =

Original: {A α i i } , wherei ∈ [ m ] .
Revised: {A α i i } , where i ∈ [ m ] .

Original: For parameters θ , denote the subset of interventions the model is invariant to as I ( θ ) ⊂ {A} .
Revised: For parameters θ , denote the subset of interventions the model is invariant to as I ( θ ) ⊂ {A} .

Original: Then, f ( . ; θ 1 ) and f ( . ; θ 2 ) are said to be mechanistically similar if I ( θ 1 )
Revised: Then, f ( . ; θ 1 ) and f ( . ; θ 2 ) are said to be mechanistically similar if I ( θ 1 )

Original: =
Revised: =

Original: I ( θ 2 ) .
Revised: I ( θ 2 ) .

Original:  Relating Mechanistic Similarity and Mode Connectivity
Revised:  Relating Mechanistic Similarity and Mode Connectivity

Original: We first present the following proposition which follows directly from the results by [31, 32, 33] andanswers the question: is it even possible for mechanistically dissimilar modes to be connected?
Revised: We first present the following proposition which follows directly from the results by [31, 32, 33] and answers the question: is it even possible for mechanistically dissimilar modes to be connected?

Original: Proposition 1.
Revised: Proposition 1.

Original: (Mechanistically Dissimilar Modes are Connected.)
Revised: (Mechanistically Dissimilar Modes are Connected.)

Original: Assume θ 1 , θ 2 are two mechanistically dissimilar modes of loss L ( f ( D ; θ )) on a given dataset D .
Revised: Assume θ 1 , θ 2 are two mechanistically dissimilar modes of loss L ( f ( D ; θ )) on a given dataset D .

Original: Given sufficient overparamterization,there exists a continuous path that connects the two modes (in the sense of Def. 1).
Revised: Given sufficient overparamterization, there exists a continuous path that connects the two modes (in the sense of Def. 1).

Original: That is, even if two modes use completely different mechanisms to fit a dataset D , as long asthey achieve zero loss on it, there will exist a continuous path that connects them.
Revised: That is, even if two modes use completely different mechanisms to fit a dataset D , as long as they achieve zero loss on it, there will exist a continuous path that connects them.

Original: However, aswe mentioned before, beyond the surprising fact that modes in the landscape are connected at all,the further intriguing result is that they are connected via relatively simple paths.
Revised: However, as we mentioned before, beyond the surprising fact that modes in the landscape are connected at all, the further intriguing result is that they are connected via relatively simple paths.

Original: We empiricallydemonstrate that this property continues to hold for mechanistically dissimilar modes, provided oneuses non-linear connectivity paths .
Revised: We empirically demonstrate that this property continues to hold for mechanistically dissimilar modes, provided one uses non-linear connectivity paths .

Original: Specifically, we train VGG-13 and ResNet-18 models on syntheticdatasets and plot accuracy/loss on counterfactual datasets (e.g., see Fig.
Revised: Specifically, we train VGG-13 and ResNet-18 models on synthetic datasets and plot accuracy on counterfactual datasets (e.g., see Fig.

Original: 5 & App.
Revised: 5 & App.

Original: D).
Revised: D).

Original: We analyzequadratic paths identified using without cue data, with cue data, linear path, and linear path afterpermuting neurons to match in activations (see App. B.5).
Revised: We analyze quadratic paths identified using without cue data, with cue data, linear path, and linear path after permuting neurons to match in activations (see App. B.5).

Original: We see that we can identify quadratic,but not linear , paths that connect mechanistically dissimilar modes in the sense of Def. 1.
Revised: We see that we can identify quadratic, but not linear , paths that connect mechanistically dissimilar modes in the sense of Def. 1.

Original: Thus,mechanistic similarity affects the functional form of connectivity paths between modes.
Revised: Thus, mechanistic similarity affects the functional form of connectivity paths between modes.

Original: Moreover, wesee different points along the connectivity paths respond differently to counterfactuals, indicating lackof mechanistic similarity along the path .
Revised: Moreover, we see different points along the connectivity paths respond differently to counterfactuals, indicating lack of mechanistic similarity along the path .

Original: Building on these results, we next argue that mechanisticallysimilar modes must be connected via linear paths and vice versa.
Revised: Building on these results, we next argue that mechanistically similar modes must be connected via linear paths and vice versa.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: CBFT 74.1 71.
Revised: CBFT 74.1 71.

Original: [Figure]
Revised: [Figure]

Original: Conjecture 1.
Revised: Conjecture 1.

Original: (Mechanistic Similarity Enforces Linear Connectivity.)
Revised: (Mechanistic Similarity Enforces Linear Connectivity.)

Original: If, up to permutations ofneurons, θ 1 , θ 2 show linear connectivity on a dataset D , then they must be mechanistically similar.
Revised: If, up to permutations of neurons, θ 1 , θ 2 show linear connectivity on a dataset D , then they must be mechanistically similar.

Original: Ifthey cannot be connected linearly, the modes must be mechanistically dissimilar.
Revised: If they cannot be connected linearly, the modes must be mechanistically dissimilar.

Original: [Figure]
Revised: [Figure]

Original: E for further results.
Revised: results.

Original: In App.
Revised: In App.

Original: F, we show the above conjecture holds locally by analyzing the landscape up to a second-orderapproximation.
Revised: F, we show the above conjecture holds locally by analyzing the landscape up to a second-order approximation.

Original: We provide extensive experiments todemonstrate the conjecture holds true in real settings(see Fig.
Revised: We provide extensive experiments to demonstrate the conjecture holds true in real settings (see Fig.

Original: 4 & App.
Revised: 4 & App.

Original: E).
Revised: E).

Original: Specifically, we train VGG-13and ResNet-18 models on our synthetic datasets andfine-tune these trained models on the original data without cues for 100 epochs, different initial learning rates,and a step-decay schedule.
Revised: Specifically, we train VGGand ResNet-18 models on our synthetic datasets and fine-tune these trained models on the original data without cues for 100 epochs, different initial learning rates, and a step-decay schedule.

Original: We see that whenever linearconnectivity is exhibited, the modes respond similarlyto counterfactual datasets (i.e., they are mechanistically similar).
Revised: We see that whenever linear connectivity is exhibited, the modes respond similarly to counterfactual datasets (i.e., they are mechanistically similar).

Original: Meanwhile, when linear connectivitydoes not emerge, the fine-tuned mode responds differently on counterfactuals: e.g., fine-tuning using a largelearning rate exhibits clear invariance to the spuriouscue, while the original mode does not.
Revised: Meanwhile, when linear connectivity does not emerge, the fine-tuned mode responds differently on counterfactuals: e.g., fine-tuning using a large learning rate exhibits clear invariance to the spurious cue, while the original mode does not.

Original: Our resultsthus provide nuance to prior claims that all modes arelinearly connected up to permutations [15, 16]: wefind the landscape is a collection of basins of linearlyconnected modes that follows similar mechanisms toproduce their outputs.
Revised: Our results thus provide nuance to prior claims that all modes are linearly connected up to permutations [15, 16]: we find the landscape is a collection of basins of linearly connected modes that follows similar mechanisms to produce their outputs.

Original: The training pipeline’s inherentbiases (e.g., simplicity bias in SGD [4, 5, 36]) showspreferential behavior for certain basins, due to whichlinear connectivity may emerge (up to permutations)for the same pipeline but slightly different settings thatdo not shift bias towards another basin (e.g., changedinitializations).
Revised: The training pipeline’s inherent biases (e.g., simplicity bias in SGD [4, 5, 36]) shows preferential behavior for certain basins, due to which linear connectivity may emerge (up to permutations) for the same pipeline but slightly different settings that do not shift bias towards another basin (e.g., changed initializations).

Original: Meanwhile, two mechanistically dissimilar modes cannot be connected linearly, exhibitingan increase in loss along the linear path between them.
Revised: Meanwhile, two mechanistically dissimilar modes cannot be connected linearly, exhibiting an increase in loss along the linear path between them.

Original: Mechanistic Fine-Tuning: Consider a mode θ C that we want to fine-tune on some minimal “clean”data to create a mode θ FT that does not use some undesirable mechanism.
Revised: Mechanistic Fine-Tuning: Consider a mode θ C that we want to fine-tune on some minimal “clean” data to create a mode θ FT that does not use some undesirable mechanism.

Original: This premise follows fromrecent work on removing reliance on spurious cues in DNNs [34, 35].
Revised: This premise follows from recent work on removing reliance on spurious cues in DNNs [34, 35].

Original: We now show our developedinsights can be used to address this problem (see App.
Revised: We now show our developed insights can be used to address this problem (see App.

Original: C for details).
Revised: C for details).

Original: Specifically, we propose toregularize the fine-tuning process to induce a high loss barrier between linear interpolation of thecurrent state of θ FT and θ C .
Revised: Specifically, we propose to regularize the fine-tuning process to induce a high loss barrier between linear interpolation of the current state of θ FT and θ C .

Original: As per our analysis, the existence of this barrier will imply lack of sharedinvariances and hence mechanistic dissimilarity.
Revised: As per our analysis, the existence of this barrier will imply lack of shared invariances and hence mechanistic dissimilarity.

Original: To ensure the unshared invariance correspondsto ignoring vs. using the spurious cue, we further add an invariance loss that asks class-centroidsproduced by θ FT on data with and without cue to be the same.
Revised: To ensure the unshared invariance corresponds to ignoring vs. using the spurious cue, we further add an invariance loss that asks class-centroids produced by θ FT on data with and without cue to be the same.

Original: We find this approach, termed
Revised: We find this approach, termed

Original: Connectivity-Based Fine-Tuning (CBFT), outperforms recent baselines [34, 35] and naive fine-tuningon clean and counterfactual data, indicating emergence of desired invariances (see Tab. 1 & App. 3).
Revised: Connectivity-Based Fine-Tuning (CBFT), outperforms recent baselines [34, 35] and naive fine-tuning on clean and counterfactual data, indicating emergence of desired invariances (see Tab. 1 & App. 3).

Original: A Related Work
Revised: A Related Work

Original: Model Similarity: Prior works have tried to assess similarity of two models trained under differentsettings via the notion of prediction mismatch [37, 38, 39], which involves finding which samplestwo models produce different outputs on.
Revised: Model Similarity: Prior works have tried to assess similarity of two models trained under different settings via the notion of prediction mismatch [37, 38, 39], which involves finding which samples two models produce different outputs on.

Original: In contrast to such works, we propose to assess modelsimlarity via their response to relevant data transformations, where relevance depends on the task anduser choices.
Revised: In contrast to such works, we propose to assess model simlarity via their response to relevant data transformations, where relevance depends on the task and user choices.

Original: If two models use similar mechanisms, they should respond similarly to transformedinputs; by choosing transforms that encode task-relevant vulnerabilities, we can make this definitionoperationally well-motivated.
Revised: If two models use similar mechanisms, they should respond similarly to transformed inputs; by choosing transforms that encode task-relevant vulnerabilities, we can make this definition operationally well-motivated.

Original: For example, by randomizing the synthetic cue in Figure 5, wecan assess whether the model relies on the cue or not.
Revised: For example, by randomizing the synthetic cue in Figure 5, we can assess whether the model relies on the cue or not.

Original: This is analogous to the use of visualillusions (inaccurate perceptual inferences) for determining valid computational models of early visualprocessing in cognitive science [29] and neuroscience [28], and relates to the idea of Goldstone’stheorem in physics [40, 41], which claims moving in the direction of an operator’s symmetry does notyield increase in system’s energy.
Revised: This is analogous to the use of visual illusions (inaccurate perceptual inferences) for determining valid computational models of early visual processing in cognitive science [29] and neuroscience [28], and relates to the idea of Goldstone’s theorem in physics [40, 41], which claims moving in the direction of an operator’s symmetry does not yield increase in system’s energy.

Original: Intuitively, then, we can expect the number of shared symmetriesto be a valid proxy for how benign a path must be for two mechanistically dissimilar modes to beconnected.
Revised: Intuitively, then, we can expect the number of shared symmetries to be a valid proxy for how benign a path must be for two mechanistically dissimilar modes to be connected.

Original: We also note our work is among the first works to examine the mechanistic similarityof different networks.
Revised: We also note our work is among the first works to examine the mechanistic similarity of different networks.

Original: While many previous works have asked whether a neural network learns thetrue causal mechanisms generating the data, ours is the second work to examine the mechanisticsimilarity of different networks, after Nanda et al.
Revised: While many previous works have asked whether a neural network learns the true causal mechanisms generating the data, ours is the second work to examine the mechanistic similarity of different networks, after Nanda et al.

Original: [42], who introduce a method (“STIR”) formeasuring how similar the mechanisms learned by two networks are.
Revised: [42], who introduce a method (“STIR”) for measuring how similar the mechanisms learned by two networks are.

Original: We also note the idea ofidentifying easily manipulable latents to infer interpretable modules is also popular in generativemodels [43, 44, 45, 46].
Revised: We also note the idea of identifying easily manipulable latents to infer interpretable modules is also popular in generative models [43, 44, 45, 46].

Original: Mode connectivity.
Revised: Mode connectivity.

Original: Existence of a single, continuous manifold connecting global minimizers was firstidentified theoretically by [33] and empirically discovered in concurrent works under the title of “modeconnectivity” by [11] and [12].
Revised: Existence of a single, continuous manifold connecting global minimizers was first identified theoretically by [33] and empirically discovered in concurrent works under the title of “mode connectivity” by [11] and [12].

Original: A geometrical characterization of the manifold connecting minimizerswas provided by [32], who showed the manifold is primarily composed of affine subspaces.
Revised: A geometrical characterization of the manifold connecting minimizers was provided by [32], who showed the manifold is primarily composed of affine subspaces.

Original: Modeconnectivity results have been used for designing and analyzing algorithms for several practicallyrelevant applications, such as ensembling [14, 47, 48, 49], network pruning [13, 15], fine-tuning [50],adversarial robustness [51], and multi-task/continual learning [52].
Revised: Mode connectivity results have been used for designing and analyzing algorithms for several practically relevant applications, such as ensembling [14, 47, 48, 49], network pruning [13, 15], fine-tuning [50], adversarial robustness [51], and multi-task/continual learning [52].

Original: This last work is closely relatedto ours: they find that they can train a model which is linearly mode connected to models trained onindividual tasks.
Revised: This last work is closely related to ours: they find that they can train a model which is linearly mode connected to models trained on individual tasks.

Original: During the course of this work, we became aware of the contemporary paper by[53], who empirically investigate if minimizers connected via linear paths follow similar strategiesto produce their decisions.
Revised: During the course of this work, we became aware of the contemporary paper by [53], who empirically investigate if minimizers connected via linear paths follow similar strategies to produce their decisions.

Original: Their analysis focuses on NLP tasks and follows an alternative analysis;hence, their results can be regarded as further verification of our claims about linear mode connectivityon a different modality.
Revised: Their analysis focuses on NLP tasks and follows an alternative analysis; hence, their results can be regarded as further verification of our claims about linear mode connectivity on a different modality.

Original: Fine-tuning.
Revised: Fine-tuning.

Original: Fine-tuning is a well-established practice in deep learning.
Revised: Fine-tuning is a well-established practice in deep learning.

Original: The most basic fine-tuningmethod is to treat the pre-trained model as an initialization, and continue training with new data.
Revised: The most basic fine-tuning method is to treat the pre-trained model as an initialization, and continue training with new data.

Original: A variant is to train only a subset of parameters, such as the final classification layer [34].While[34] argue that “last layer re-training is sufficient for robustness to spurious correlations”, our morein-depth evaluation shows that this method actually does not eliminate sensitivity to spurious features.
Revised: A variant is to train only a subset of parameters, such as the final classification layer [34].While [34] argue that “last layer re-training is sufficient for robustness to spurious correlations”, our more in-depth evaluation shows that this method actually does not eliminate sensitivity to spurious features.

Original: Our findings are more congruent with those of [50], who find that fine-tuned models tend to remainlinearly connected to the pretraining mode, suggesting that fine-tuning may fail to make fundamentalchanges to a model’s behavior.
Revised: Our findings are more congruent with those of [50], who find that fine-tuned models tend to remain linearly connected to the pretraining mode, suggesting that fine-tuning may fail to make fundamental changes to a model’s behavior.

Original: Model editing.
Revised: Model editing.

Original: Model editing refers to fine-tuning approaches that aim to make a targeted change toa particular aspect of model’s behavior without incidentally affecting other aspects.
Revised: Model editing refers to fine-tuning approaches that aim to make a targeted change to a particular aspect of model’s behavior without incidentally affecting other aspects.

Original: For instance, [54]give the example of correcting a prediction errors on a particular example without changing a model’sprediction on other examples.
Revised: For instance, [54] give the example of correcting a prediction errors on a particular example without changing a model’s prediction on other examples.

Original: Most work to date on model editing aims to make such changesthat are “local” in input space, i.e. only affecting the model’s “understanding” of who the currentprime minister of the UK is [55].
Revised: Most work to date on model editing aims to make such changes that are “local” in input space, i.e. only affecting the model’s “understanding” of who the current prime minister of the UK is [55].

Original: Mechanistic fine-tuning shares this high-level goal of “targeted”fine-tuning, however, we aim to make edits that are local in a different sense: we want to make atargeted change to the causal mechanism the model implements to make predictions.
Revised: Mechanistic fine-tuning shares this high-level goal of “targeted” fine-tuning, however, we aim to make edits that are local in a different sense: we want to make a targeted change to the causal mechanism the model implements to make predictions.

Original: Specifically, weaim to make a model invariant to (e.g. spurious/nuisance) features that is was not already invariantto (or vice versa), without changing its learned representations of the features themselves, or itsinvariance to other features.
Revised: Specifically, we aim to make a model invariant to (e.g. spurious/nuisance) features that is was not already invariant to (or vice versa), without changing its learned representations of the features themselves, or its invariance to other features.

Original: Such a change would tend to influence many of a model’s predictions,making approaches such as MEND [55] inappropriate for our setting.
Revised: Such a change would tend to influence many of a model’s predictions, making approaches such as MEND [55] inappropriate for our setting.

Original: We believe [56] is the workmost similar to ours in this respect; three significant differences are: (i) their method only works
Revised: We believe [56] is the work most similar to ours in this respect; three significant differences are: (i) their method only works

Original: Linearly
Revised: Linearly

Original: Nonlinearly
Revised: Nonlinearly

Original: Linearly
Revised: Linearly

Original: Nonlinearly
Revised: Nonlinearly

Original: Mode Mode
Revised: Mode Mode

Original: Mech.
Revised: Mech.

Original: Mech.
Revised: [Table]

Original: Connected Connected
Revised: Connected Connected

Original: Connected Connected
Revised: Connected Connected

Original: Mechanistically Similar ✓ ✓ ✓ ✓
Revised: Mechanistically Similar ✓ ✓ ✓ ✓

Original: Mechanistically Dissimilar ✗ ✓ ✗ ✗ given a large model pre-trained on lots of data, (ii)they rely on style transfer, (iii) they only tune some ofthe layers.
Revised: Mechanistically Dissimilar ✗ ✓ ✗ ✗ given a large model pre-trained on lots , (ii) tune some of

Original: B Training Setup and Dataset Visualizations 
Revised: B Training Setup and Dataset Visualizations B.1 Training details

Original: B.1 Training details

Original: When training from scratch (e.g., in Fig. 3), we train models for 100 epochs with a batch-size of256.
Revised: When training from scratch (e.g., in Fig. 3), we train models for 100 epochs with a batch-size of 256.

Original: Learning rate starts at 0 .
Revised: Learning rate starts at 0 .

Original: and is dropped by a factor of 10 at the 40 th and 80 th epochs.
Revised: and is dropped by a factor of 10 at the 40 th and 80 th epochs.

Original: No dataaugmentations are used.
Revised: No data augmentations are used.

Original: When fine-tuning to assess linear connectivity (e.g., in Fig. 4), we trainmodels for a further 100 epochs on data without cues using different initial learning rates, but thesame step-decay schedule (decay factor of 0.1 at decay epochs 40 and 80).
Revised: When fine-tuning to assess linear connectivity (e.g., in Fig. 4), we train models for a further 100 epochs on data without cues using different initial learning rates, but the same step-decay schedule (decay factor of 0.1 at decay epochs 40 and 80).

Original: When using syntheticdatasets, if a proportion c is to be assigned the cue feature, we use the first c % samples of all classes toassign them the respective cues.
Revised: When using synthetic datasets, if a proportion c is to be assigned the cue feature, we use the first c % samples of all classes to assign them the respective cues.

Original: We do not store the samples beforehand and use manually designeddataloaders that allow for easy manipulation of samples in an online manner, enabling straightforwardcounterfactual evaluations.
Revised: We do not store the samples beforehand and use manually designed dataloaders that allow for easy manipulation of samples in an online manner, enabling straightforward counterfactual evaluations.

Original: We note the code is currently under review, but will be released soon.
Revised: We note the code is currently under review, but will be released soon.

Original: B.2 Fine-Tuning details
Revised: B.2 Fine-Tuning details

Original: We train models on the synthetic data with cue features, reserving 2500 training samples as “clean”data for further fine-tuning to remove reliance on the spurious cue.
Revised: We train models on the synthetic data with cue features, reserving 2500 training samples as “clean” data for further fine-tuning to remove reliance on the spurious cue.

Original: Depending on the method, thefine-tuning setup involves different hyperparameters.
Revised: Depending on the method, the fine-tuning setup involves different hyperparameters.

Original: For consistency, we follow the use of a cosineschedule for fine-tuning on clean data, as done by Kirichenko et al.
Revised: For consistency, we follow the use of a cosine schedule for fine-tuning on clean data, as done by Kirichenko et al.

Original: [34] and Kumar et al.
Revised: [34] and Kumar et al.

Original: Naive Fine-Tuning .
Revised: Naive Fine-Tuning .

Original: We use different initial learning rates, including medium ( 0 . 01 ) and small( 0 . 001 ).
Revised: We use different initial learning rates, including medium ( 0 . 01 ) and small ( 0 . 001 ).

Original: For a large learning rate, we note that while fine-tuning on a minimal set does induce goodinvariance properties, the performance on the original, without cue data (called NC in tables) is oftenrather poor.
Revised: For a large learning rate, we note that while fine-tuning on a minimal set does induce good invariance properties, the performance on the original, without cue data (called NC in tables) is often rather poor.

Original: Hence, we disinclude those results.
Revised: Hence, we disinclude those results.

Original: LLRT (Kirichenko et al.
Revised: LLRT (Kirichenko et al.

Original: We freeze the model parameters at their current state, remove thefinal linear layer, and replace it with a randomly initialized one.
Revised: We freeze the model parameters at their current state, remove the final linear layer, and replace it with a randomly initialized one.

Original: The layer is fine-tuned on clean datafor 100 epochs with a cosine decay schedule that starts at a LR of 30.
Revised: The layer is fine-tuned on clean data for 100 epochs with a cosine decay schedule that starts at a LR of 30.

Original: LPFT (Kumar et al.
Revised: LPFT (Kumar et al.

Original: First, we follow the protocol above for LLRT and get a new linear layer.
Revised: First, we follow the protocol above for LLRT and get a new linear layer.

Original: Thereafter, the entire model is fine-tuned on clean data for 20 epochs with initial learning rates of0 . 01 , 0 . 001 , and 0 . 0001 .
Revised: Thereafter, the entire model is fine-tuned on clean data for 20 epochs with initial learning rates of 0 . 01 , 0 . 001 , and 0 . 0001 .

Original: The best retrieved results on validation data are reported.
Revised: The best retrieved results on validation data are reported.

Original: CBFT .
Revised: CBFT .

Original: We initialize the model at the spurious mode θ C and first run a warmup epoch without theinvariance loss (i.e., only the barrier loss and clean data loss are used).
Revised: We initialize the model at the spurious mode θ C and first run a warmup epoch without the invariance loss (i.e., only the barrier loss and clean data loss are used).

Original: This is similar to the LLRTstep in LPFT and helps move the model away from θ C .
Revised: This is similar to the LLRT step in LPFT and helps move the model away from θ C .

Original: Thereafter, the model is fine-tuned for 20epochs with an initial learning rate of 0 . 01 .
Revised: Thereafter, the model is fine-tuned forepochs with an initial learning rate of 0 . 01 .

Original: B.3 Data Generating Process and Visualizations
Revised: B.3 Data Generating Process and Visualizations

Original: Since our goal is to assess the role of mechanistic similarity on connectivity of modes, we needmodels that we know for a fact rely on different mechanisms for making their decisions.
Revised: Since our goal is to assess the role of mechanistic similarity on connectivity of modes, we need models that we know for a fact rely on different mechanisms for making their decisions.

Original: To thisend, we propose to use easily manipulable synthetic datasets.
Revised: To this end, we propose to use easily manipulable synthetic datasets.

Original: Such datasets have been used byprior works for better understanding several important topics, such as transfer learning [57], domaingeneralization [58, 59], disentanglement [60, 61], self- and semi-supervised learning [22, 62, 19],and inductive biases of neural networks [6, 7, 63].
Revised: Such datasets have been used by prior works for better understanding several important topics, such as transfer learning [57], domain generalization [58, 59], disentanglement [60, 61], self- and semi-supervised learning [22, 62, 19], and inductive biases of neural networks [6, 7, 63].

Original: Our data generation process (DGP) is illustrated in
Revised: Our data generation process (DGP) is illustrated in

Original: Figure 5 and involves augmenting the natural DGP with synthetic cue features that are conditioned on the sample category.
Revised: Figure 5 and involves augmenting the natural DGP with synthetic cue features that are conditioned on the sample category.

Original: By design, the dataset is easy to intervene on for creating counterfactual samples(see Def. 2).
Revised: By design, the dataset is easy to intervene on for creating counterfactual samples (see Def. 2).

Original: We specifically focus on interventions which break the cue’s correlation with respect totarget category by randomizing it (e.g., uniformly changing location of the box cue in the “located CIFAR-10” dataset).
Revised: We specifically focus on interventions which break the cue’s correlation with respect to target category by randomizing it (e.g., uniformly changing location of the box cue in the “located CIFAR-10” dataset).

Original: Vice-versa, we also analyze the case where the cue remains intact, but theunderlying image is randomized (e.g., putting a cat instead of a dog).
Revised: Vice-versa, we also analyze the case where the cue remains intact, but the underlying image is randomized (e.g., putting a cat instead of a dog).

Original: This helps analyze how much agiven mode relies on possibly semantically relevant features that originate from the source image,especially under partial correlation.
Revised: This helps analyze how much a given mode relies on possibly semantically relevant features that originate from the source image, especially under partial correlation.

Original: We highlight that such low-complexity cues can be viewed asstand-ins for spurious or shortcut features that are commonplace in realistic settings [2, 1], allowingus to analyze if modes that use spurious versus non-spurious features are connected.
Revised: We highlight that such low-complexity cues can be viewed as stand-ins for spurious or shortcut features that are commonplace in realistic settings [2, 1], allowing us to analyze if modes that use spurious versus non-spurious features are connected.

Original: We provideseveral visualizations of randomly sampled datapoints from different classes and their counterfactualsin Figure 6 (CIFAR-10 with box cue), Figure 7 (CIFAR-100 with box/color cue), and Figure 8(Dominoes: CIFAR-10 with concatenated FashionMNIST images).

Original: Learning curves with train/testaccuracies for models trained on different proportions of samples with cue features for these datasetsare reported in Figure 9a, Figure 10a (VGG /

Original: -18 on CIFAR-10 with box cue), Figure 9b,

Original: , and networks [5] will ensure the model preferentially uses them for making its decisions.

Original: Datasets (right).

Original: We use three datasets: (1) CIFAR-10 with a 3 × 3 box whose location depends on the sample category; (2) CIFAR-100 with 3 × 3 boxes colored according to the first digit of the object label, and located according to the second digit; (3) and Dominoes [5], wherein CIFAR-10 images are concatenated same class FashionMNIST images.

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: B.4 Identifying Quadratic Paths
Revised: B.4 Identifying Quadratic Paths

Original: The qudaratic path is defined as follows.
Revised: The qudaratic path is defined as follows.

Original: [Equation]
Revised: [Equation]

Original: The set of parameters θ m can be thought of as vertex of a parabola that helps anchor the curve.
Revised: The set of parameters θ m can be thought of as vertex of a parabola that helps anchor the curve.

Original: Toidentify this set of parameters, we follow [11] and train points randomly sampled from the quadraticpath to achieve zero loss on a given dataset D .
Revised: To identify this set of parameters, we follow [11] and train points randomly sampled from the quadratic path to achieve zero loss on a given dataset D .

Original: That is,
Revised: That is,

Original: [Equation]
Revised: [Equation]

Original: Note that consequently, a quadratic connectivity path necessarily depends on the dataset used forits identification and it is not mandatory that it will generalize across datasets/distributions.
Revised: Note that consequently, a quadratic connectivity path necessarily depends on the dataset used for its identification and it is not mandatory that it will generalize across datasets/distributions.

Original: Thisis precisely what we see in our results in Figure 3, where we are able to identify quadratic modeconnectivty between two sets of parameters on a given dataset, but those paths do not generalize tocounterfactual datasets.
Revised: This is precisely what we see in our results in Figure 3, where we are able to identify quadratic modeconnectivty between two sets of parameters on a given dataset, but those paths do not generalize to counterfactual datasets.

Original: B.5 Finding Permutations for Linear Connectivity
Revised: B.5 Finding Permutations for Linear Connectivity

Original: Given two sets of parameters θ 1 , θ 1 , identifying the linear path between them involves merelyinterpolating the parameters.
Revised: Given two sets of parameters θ 1 , θ 1 , identifying the linear path between them involves merely interpolating the parameters.

Original: [15] argue that parameters discovered using SGD can always be linearlymode connected up to permutations, i.e., there exists a permutation π that connects the π ( θ 1 ) , θ 2 inthe sense of Def. 1.
Revised: [15] argue that parameters discovered using SGD can always be linearly mode connected up to permutations, i.e., there exists a permutation π that connects the π ( θ 1 ) , θ 2 in the sense of Def. 1.

Original: To empirically analyze this claim in our work, we identify π by maximizing thesimilarity of activations produced by model with parameters θ 1 and θ 2 .
Revised: To empirically analyze this claim in our work, we identify π by maximizing the similarity of activations produced by model with parameters θ 1 and θ 2 .

Original: That is,
Revised: That is,

Original: [Equation]
Revised: [Equation]

Original: [Equation] Given the above problem involves discrete optimization, we propose to solve it greedily by computingrepresentations at each layer of the two models, finding a permutation that matches the representationsmaximally, and then repeating the process for the next layer.
Revised: Given the above problem involves discrete optimization, we propose to solve it greedily by computing representations at each layer of the two models, finding a permutation that matches the representations maximally, and then repeating the process for the next layer.

Original: For finding the permutation, we use
Revised: For finding the permutation, we use

Original: SciPy’s linear assignment solver [64].
Revised: SciPy’s linear assignment solver [64].

Original: We use representations over a batch-size of 512 and run thematching process over the entire dataset.
Revised: We use representations over a batch-size of 512 and run the matching process over the entire dataset.

Original: Let π <l denotes permutations before layer l .
Revised: Let π <l denotes permutations before layer l .

Original: Initialize: θ 1 , θ 2 , dataset D , model f , π lL 

Revised: Initialize: θ 1 , θ 2 , dataset D , model f , π lL

Original: =
Revised: =

Original: I , where I is identity permutation.l ←[-
-]
Revised: I , where I is identity permutation. l ←[-
-]

Original: L ← # of Layersfor x ∈ D dol ← 1for l ≤ L doπ l = arg minπ
Revised: L ← # of Layers for x ∈ D do l ←for l ≤ L do π l = arg min

Original:  || f l ( x ; π <l ( θ 1 ))
Revised: π || f l ( x ; π <l ( θ 1 ))

Original: − f l ( x ; θ 2 ) ||
Revised: − f l ( x ; θ 2 ) ||

Original: ▷ f l denotes representation at layer ll ← l + 1end forend for
Revised: ▷ f l denotes representation at layer l l ← l +end for end for

Original: Our algorithm is able to recover the ground-truth permutation if a given model’s neurons are intentionally randomized (while maintaining functional connectivity).
Revised: Our algorithm is able to recover the ground-truth permutation if a given model’s neurons are intentionally randomized (while maintaining functional connectivity).

Original: This serves as a sanity check thatconfirms the validity of our technique.
Revised: This serves as a sanity check that confirms the validity of our technique.

Original: C Mechanistic Fine Tuning: Overriding Decision-Making Rules by Driving aModel Beyond Mechanistic Barrier
Revised: C Mechanistic Fine Tuning: Overriding Decision-Making Rules by Driving a Model Beyond Mechanistic Barrier

Original: Fine tuning is commonly used as a strategy to improve sample efficiency by taking a pre-trainedmodel as an initialization and training it further on a new small data set.
Revised: Fine tuning is commonly used as a strategy to improve sample efficiency by taking a pre-trained model as an initialization and training it further on a new small data set.

Original: This section explores theidea of mechanistic fine tuning , where we specifically aim to override existing mechanisms of apretrained model by fine tuning it on a small out-of-distribution dataset.
Revised: This section explores the idea of mechanistic fine tuning , where we specifically aim to override existing mechanisms of a pretrained model by fine tuning it on a small out-of-distribution dataset.

Original: Note that here we are nottrying to teach the model how to perform new extra tasks (e.g., classifying new image categories,etc.), but rather attempting to override its existing mechanisms, e.g., for how it classifies images witha fixed set of classes.
Revised: Note that here we are not trying to teach the model how to perform new extra tasks (e.g., classifying new image categories, etc.), but rather attempting to override its existing mechanisms, e.g., for how it classifies images with a fixed set of classes.

Original: Mechanistic fine-tuning with connectivity-based fine-tuning (CBFT) Consider a set of parametersθ C found by training a model f ( . ) on dataset D C that has some spurious or shortcut cue that a modelcan use to achieve zero loss.
Revised: Mechanistic fine-tuning with connectivity-based fine-tuning (CBFT) Consider a set of parameters θ C found by training a model f ( . ) on dataset D C that has some spurious or shortcut cue that a model can use to achieve zero loss.

Original: Practically, this situation is commonplace because spurious or shortcutcues are often highly correlated with the actual target category [2].
Revised: Practically, this situation is commonplace because spurious or shortcut cues are often highly correlated with the actual target category [2].

Original: Recent work has aimed to useminimal data with “clean” samples, i.e., samples which lack the spurious cue to remove the model’sreliance on such cues [34, 35], while trying to maintain any other useful features it has learned.
Revised: Recent work has aimed to use minimal data with “clean” samples, i.e., samples which lack the spurious cue to remove the model’s reliance on such cues [34, 35], while trying to maintain any other useful features it has learned.

Original: Wenow show that our newfound understanding of neural network loss landscapes from the perspectiveof mechanistic similarity can be used to address this problem.
Revised: We now show that our newfound understanding of neural network loss landscapes from the perspective of mechanistic similarity can be used to address this problem.

Original: To this end, we recall from Figure 4that when fine-tuning models with different learning rates on original datasets, the only situationθ FT demonstrates invariance to cues embedded in the dataset is when it is not linearly connected tothe original mode θ C .
Revised: To this end, we recall from Figurethat when fine-tuning models with different learning rates on original datasets, the only situation θ FT demonstrates invariance to cues embedded in the dataset is when it is not linearly connected to the original mode θ C .

Original: As per Conjecture 1, this lack of linear connectivity indicates θ FT and θ C aremechanistically dissimilar due to lack of a shared invariance.
Revised: As per Conjecture 1, this lack of linear connectivity indicates θ FT and θ C are mechanistically dissimilar due to lack of a shared invariance.

Original: We thus argue that a valid strategy toremove a model’s reliance on spurious cues is by forcing it to move to a different region in thelandscape that does not exhibit linear connectivity to the θ C .
Revised: We thus argue that a valid strategy to remove a model’s reliance on spurious cues is by forcing it to move to a different region in the landscape that does not exhibit linear connectivity to the θ C .

Original: Operationalizing this idea has a challenge however: there can be multiple regions that do not linearlyconnect with θ C
Revised: Operationalizing this idea has a challenge however: there can be multiple regions that do not linearly connect with θ C

Original: and we speficially want ones that boast our desired invariances–i.e., regions that areinvariant to the spurious cue.
Revised: and we speficially want ones that boast our desired invariances–i.e., regions that are invariant to the spurious cue.

Original: We propose to circumvent this issue by assuming the existence of aminimal “clean” dataset D NC , similar to prior work, that can be used to enforce desired invarianceson the model’s representations.
Revised: We propose to circumvent this issue by assuming the existence of a minimal “clean” dataset D NC , similar to prior work, that can be used to enforce desired invariances on the model’s representations.

Original: Specifically, assume L ( ., . )
Revised: Specifically, assume L ( ., . )

Original: denotes a classification loss, such ascross-entropy and D i denotes the subset of a dataset D corresponding to samples that belong to thei th class in an K − class classification problem.
Revised: denotes a classification loss, such as cross-entropy and D i denotes the subset of a dataset D corresponding to samples that belong to the i th class in an K − class classification problem.

Original: Then, we can use the following two-step procedurefor connectivity-based fine tuning (CBFT):
Revised: Then, we can use the following two-step procedure for connectivity-based fine tuning (CBFT):

Original: [Equation]
Revised: [Equation]

Original: In the above, γ θ → θ C ( t ) denotes the linear path between θ and θ C .
Revised: In the above, γ θ → θ C ( t ) denotes the linear path between θ and θ C .

Original: The first step aims to maximize themodel loss L CE along the linear path up to an upper bound λ 1 ; meanwhile, the second loss aims toensure the model predicts correct labels on the clean dataset D NC , while simultaneously promotinginvariance to spurious cues by producing the same average representation across all classes on both
Revised: The first step aims to maximize the model loss L CE along the linear path up to an upper bound λ 1 ; meanwhile, the second loss aims to ensure the model predicts correct labels on the clean dataset D NC , while simultaneously promoting invariance to spurious cues by producing the same average representation across all classes on both

Original: D NC and D C .
Revised: D NC and D C .

Original: We note the set of parameters θ is initialized to θ C ; the method turns out to be fairlyrobust to the values of λ 1 and λ 2 , so we set both to 1 and never tune them.
Revised: We note the set of parameters θ is initialized to θ C ; the method turns out to be fairly robust to the values of λ 1 and λ 2 , so we set both to 1 and never tune them.

Original: In Table [Table] we empirically validate CBFT against recent baselines used for removing a model’sreliance on spurious cues.
Revised: In Table 3, we empirically validate CBFT against recent baselines used for removing a model’s reliance on spurious cues.

Original: As we show, while these methods are performant, they do not work wellon counterfactuals datasets, e.g., they continue to perform well even if we randomize the image!
Revised: As we show, while these methods are performant, they do not work well on counterfactuals datasets, e.g., they continue to perform well even if we randomize the image!

Original: Incontrast, we see that not only CBFT performs better on clean data, but it in fact shows the desiredbehaviors: sensitivity to randomization of the image and invariance to spurious cues.
Revised: In contrast, we see that not only CBFT performs better on clean data, but it in fact shows the desired behaviors: sensitivity to randomization of the image and invariance to spurious cues.

Revised: We compare our method, Connectivity-Based Fine-Tuning (CBFT), against Fine-tuning with a medium/small learning rate (FT M/S ), LLR [34], and LPFT [35].

Revised: ∼ denotes invariance is desirable, i.e., accuracy should be similar to that on NC; ↓ indicates lower accuracy is desirable; best results are in bold.

Revised: We generally see that all baselines yield large degradations in its absence of cues; even achieving very high accuracy when the underlying image is randomized.

Revised: Meanwhile, CBFT is able to break reliance on cues, inducing representations that are often completely invariant to its presence.

Original:  [Table]
Revised: [Figure] [Table] [Figure] [Table]

Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: D Non-Linear Connectivity of Mechanistically Dissimilar Modes
Revised: D Non-Linear Connectivity of Mechanistically Dissimilar Modes

Original: We train VGG-13 and ResNet-18 models on our synthetic CIFAR-10 / CIFAR-100 / Dominoesdatasets with cue features (see Figure 6, Figure [Table] and Figure 8) and the original datasets themselves.
Revised: We train VGG-13 and ResNet-18 models on our synthetic CIFAR-10 / CIFAR-100 / Dominoes datasets with cue features (see subsection B.3) and the original datasets themselves.

Original: Parameters of the corresponding models are denoted θ C , θ NC .
Revised: Parameters of the corresponding models are denoted θ C , θ NC .

Original: We identify connectivity paths alongpairs of parameters, specifically evaluating quadratic paths identified using the data without cue(denoted Quadratic w/o Cue), quadratic path identified using data with cue (denoted Quadratic w/ 
Revised: We identify connectivity paths along pairs of parameters, specifically evaluating quadratic paths identified using the data without cue (denoted Quadratic w/o Cue), quadratic path identified using data with cue (denoted Quadratic w/ Cue), linear path (denoted

Original: Cue), linear path (denoted Linear), and linear path after permuting θ C to maximally match θ NC ’sactivations (denoted Linear Permuted).
Revised: Linear), and linear path after permuting θ C to maximally match θ NC ’s activations (denoted Linear Permuted).

Original: In the following, plot titles denote evaluation dataset, includingdatasets where either the cue is present (denoted w/ Cue), absent (denoted w/o Cue), randomized(denoted Rand. Cue), or the underlying image is randomized but the cue remains the same (denoted Rand.
Revised: In the following, plot titles denote evaluation dataset, including datasets where either the cue is present (denoted w/ Cue), absent (denoted w/o Cue), randomized (denoted Rand. Cue), or the underlying image is randomized but the cue remains the same (denoted Rand.

Original: Img).
Revised: Img).

Original: Line colors denote the proportion of dataset that has synthetic cues.
Revised: Line colors denote the proportion of dataset that has synthetic cues.

Original: Across all ourresults, we see the set of parameters θ NC yields the same performance upon randomization of the cue,while θ C loses its performance substantially–i.e., the two modes are mechanistically dissimilar due tolack of shared invariances (see Def. 4).
Revised: Across all our results, we see the set of parameters θ NC yields the same performance upon randomization of the cue, while θ C loses its performance substantially–i.e., the two modes are mechanistically dissimilar due to lack of shared invariances (see Def. 4).

Original: Nonetheless, we can identify quadratic (but not linear) pathsthat connect these mechanistically dissimilar modes in the sense of Def. 1, hence corroborating Prop.1across several datasets and model architectures, and showing mechanistically dissimilar modes canbe connected via relatively simple paths as well .
Revised: Nonetheless, we can identify quadratic (but not linear) paths that connect these mechanistically dissimilar modes in the sense of Def. 1, hence corroborating Prop.1 across several datasets and model architectures, and showing mechanistically dissimilar modes can be connected via relatively simple paths as well .

Original: However, different points on the connectivity pathsrespond differently to counterfactuals, indicating mechanistic dissimilarity despite connectivity vialow-loss paths.
Revised: However, different points on the connectivity paths respond differently to counterfactuals, indicating mechanistic dissimilarity despite connectivity via low-loss paths.

Original: [Figure]
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Original: Cue
Revised: Cue

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Original: [Table]
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Original: Rand
Revised: Rand

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Original: Img
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Original: C
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Original: [Table]
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Original: w/ Cue
Revised: w/ Cue

Original: C
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Revised: w/o Cue

Original: C
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Original: [Table] [Table]
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Original: Rand.
Revised: Rand.

Original: Cue
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Original: C
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Original: Img
Revised: Img

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Original: Accuracy.

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Original: Cue

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Original: Rand

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Original: C

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Original: C [Table] [Table] Rand.
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Original: C

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Original: Img

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Original: [Figure][Table] [Figure] Accuracy.
Revised: [Figure]

Original: [Figure]
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Original: -18 on CIFAR-10 with Box

Original: We plot test/train accuracy/loss curves along different connectivity paths and see thorough corroboration of our claims in the main text: Mechanistically dissimilar minimizers can be connected via nonlinear paths on a given dataset, but behave different on counterfactuals, indicating lack of mechanistic similarity.

Original: [Figure]
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Original: Rand
Revised: Rand

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Original: Cue
Revised: Cue

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Original: [Table]
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Original: Rand
Revised: Rand

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Original: Img
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Original: C
Revised: C

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Revised: w/ Cue

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Revised: C

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Revised: w/o Cue

Original: C
Revised: C

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Original: NC
Revised: NC

Original: Rand.
Revised: Rand.

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Original: C
Revised: C

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Original: Rand.
Revised: Rand.

Original: Img
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Original: [Table][Figure] [Figure] Rand
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Original: Cue

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Original: Rand

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Original: w/ Cue

Original: C

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Original: C [Table] [Figure] [Table]
Revised: [Table]

Original: NC

Original: Rand.

Original: Cue

Original: C

Original: [Table] [Figure][Table] Rand.
Revised: [Figure]

Original: Img

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Original: Rand

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Original: Cue

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Original: Rand

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Original: Img

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Original: C

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Original: [Figure][Table] [Figure] [Table]
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Original: w/o Cue

Original: C [Table] [Figure] [Table]
Revised: [Table]

Original: NC

Original: Rand.

Original: Cue

Original: C

Original: [Table] [Figure][Table] Rand.
Revised: [Figure]

Original: Img

Original: [Figure] [Table] [Figure] [Table]
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Original: [Figure][Table] [Figure] [Figure]
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Original: Rand
Revised: Rand

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Original: Cue
Revised: Cue

Original: [Figure]
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Original: [Table]
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Original: Rand
Revised: Rand

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Original: Img
Revised: Img

Original: C
Revised: C

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Revised: w/ Cue

Original: C
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Original: C
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Original: [Table]
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Original: NC
Revised: NC

Original: Rand.
Revised: Rand.

Original: Cue
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Original: C
Revised: C

Original: [Table]
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Original: [Figure]
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Original: [Table]
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Original: Rand.
Revised: Rand.

Original: Img
Revised: Img

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Original: E Linear Connectivity and Mechanistic Similarity
Revised: E Linear Connectivity and Mechanistic Similarity

Original: We train VGG-13 and ResNet-18 models on our synthetic CIFAR-10 / CIFAR-100 / Dominoesdatasets with cue features (see Figure 6, Figure 7, and Figure 8).
Revised: We train VGG-13 and ResNet-18 models on our synthetic CIFAR-10 / CIFAR-100 / Dominoes datasets with cue features (see App. B.3).

Original: Corresponding models are denoted θ C .
Revised: Corresponding models are denoted θ C .

Original: These models are then fine-tuned on the original CIFAR-10/CIFAR-100 datasets that do not haveany cue features.
Revised: These models are then fine-tuned on the original CIFAR-10/CIFAR-100 datasets that do not have any cue features.

Original: We use different learning rates (LR) and train for 100 epochs with a step-decayschedule (decay at epoch 40, 80 by a factor of 0 . 1 ).
Revised: We use different learning rates (LR) and train for 100 epochs with a step-decay schedule (decay at epoch 40, 80 by a factor of 0 . 1 ).

Original: Corresponding models are denoted θ FT .
Revised: Corresponding models are denoted θ FT .

Original: In thefollowing, plot titles denote evaluation dataset, including datasets where either the cue is present(denoted w/ Cue), absent (denoted w/o Cue), randomized (denoted Rand.
Revised: In the following, plot titles denote evaluation dataset, including datasets where either the cue is present (denoted w/ Cue), absent (denoted w/o Cue), randomized (denoted Rand.

Original: Cue), or the underlyingimage is randomized but the cue remains the same (denoted Rand.
Revised: Cue), or the underlying image is randomized but the cue remains the same (denoted Rand.

Original: Img).
Revised: Img).

Original: Line colors denote theproportion of dataset that has synthetic cues.
Revised: Line colors denote the proportion of dataset that has synthetic cues.

Original: Across all our results, we see that for a small enough learning rate, θ FT exhibits linear connectivitywith θ C on the synthetic dataset (in the sense of Def.1); correspondingly, counterfactual evaluationsillustrate linear connectivity as well.
Revised: Across all our results, we see that for a small enough learning rate, θ FT exhibits linear connectivity with θ C on the synthetic dataset (in the sense of Def.1); correspondingly, counterfactual evaluations illustrate linear connectivity as well.

Original: This indicates the models respond similarly to interventions onthe dataset and are hence mechanistically similar and connected (see Def. 4).
Revised: This indicates the models respond similarly to interventions on the dataset and are hence mechanistically similar and connected (see Def. 4).

Original: Meanwhile, increasingthe learning rate induces barriers along the linear path.
Revised: Meanwhile, increasing the learning rate induces barriers along the linear path.

Original: Correspondingly, we find linear connectivitydoes not hold on the synthetic dataset and models respond differently to counterfactual evaluations.
Revised: Correspondingly, we find linear connectivity does not hold on the synthetic dataset and models respond differently to counterfactual evaluations.

Original: That is, they are mechanistically dissimilar and not connected.[Figure] [Figure] [Figure]
Revised: That is, they are mechanistically dissimilar and not connected.

Original: [Figure]

Original: [Figure]

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Original: w/ Cue

Original: C 0.

Original: [Figure]

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Original: FT w/o Cue

Original: C 0.

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Original: FT

Original: Rand.

Original: Cue

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Original: FT

Original: Rand.

Original: Img

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Original: FT w/

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Original: Cue

Original: C 0. [Figure][Figure] FT
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Original: Rand

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Original: Cue

Original: C 0.

Original: [Figure][Figure] FT Rand.
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Original: Img

Original: C 0.

Original: [Figure][Figure] FT [Figure]
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Original: C

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Original: F Lemma / Claims
Revised: F Lemma / Claims

Original: F.1 Lemma 1.
Revised: F.1 Lemma 1.

Original: Lemma 1.
Revised: Lemma 1.

Original: If f ( . ; θ ) is invariant to unit interventions
Revised: If f ( . ; θ ) is invariant to unit interventions

Original: A i and A j , it must be invariant to theircomposition; if it is not invariant to A i or A j , it cannot be invariant to their composition.
Revised: A i and A j , it must be invariant to their composition; if it is not invariant to A i or A j , it cannot be invariant to their composition.

Original: Proof.
Revised: Proof.

Original: Assume the parameterization θ exhibits invariance to the intervention A i .
Revised: Assume the parameterization θ exhibits invariance to the intervention A i .

Original: Independently,consider another intervention A j .
Revised: Independently, consider another intervention A j .

Original: Then, f ( E ( x ; {A i , A j } ); θ ) =
Revised: Then, f ( E ( x ; {A i , A j } ); θ ) =

Original: f ( G X ◦
Revised: f ( G X ◦

Original: A i ◦ A j ◦ G − 1 X ( x ); θ ) =
Revised: A i ◦ A j ◦ G − 1 X ( x ); θ ) =

Original: f
Revised: f

Original: ( G X ◦ A i ◦ G − 1 X
Revised: ( G X ◦ A i ◦ G − 1 X

Original: ( E ( x ; A j )); θ )
Revised: ( E ( x ; A j )); θ )

Original: =
Revised: =

Original: f ( E ( E ( x ; A j ); A i ); θ )
Revised: f ( E ( E ( x ; A j ); A i ); θ )

Original: =
Revised: =

Original: f ( E ( x ; A j ); θ ) , where the last equalityhappens due to the assumed invariance of A i .
Revised: f ( E ( x ; A j ); θ ) , where the last equality happens due to the assumed invariance of A i .

Original: Now, if θ exhibits invariance to A j as well, we havef ( E ( x ; {A i , A j } ); θ ) =
Revised: Now, if θ exhibits invariance to A j as well, we have f ( E ( x ; {A i , A j } ); θ ) =

Original: f ( E ( x ; A j ); θ )
Revised: f ( E ( x ; A j ); θ )

Original: =
Revised: =

Original: f
Revised: f

Original: ( x ; θ ) , i.e., the parameterization θ is invariant to thesimultaneous operation (i.e., composition) of A i and A j .
Revised: ( x ; θ ) , i.e., the parameterization θ is invariant to the simultaneous operation (i.e., composition) of A i and A j .

Original: Meanwhile, if θ is invariant A i , but notto A j , we have f ( E ( x ; {A i , A j } ); θ ) =
Revised: Meanwhile, if θ is invariant A i , but not to A j , we have f ( E ( x ; {A i , A j } ); θ ) =

Original: f ( E ( x ; A j ); θ ) ̸
Revised: f ( E ( x ; A j ); θ ) ̸

Original: = f ( x ; θ ) , i.e., the parameterization θ is notinvariant to the simultaneous operation (i.e., composition) of A i and A j .
Revised: = f ( x ; θ ) , i.e., the parameterization θ is not invariant to the simultaneous operation (i.e., composition) of A i and A j .

Original: Note that the derivation above did not rely on the fact that the interventions are “unit”, in the sensethat they act on independent dimensions.
Revised: Note that the derivation above did not rely on the fact that the interventions are “unit”, in the sense that they act on independent dimensions.

Original: However, if one considers general interventions that canact on multiple dimensions of the latent space simultaneously, then a given intervention can undothe effects of another one.
Revised: However, if one considers general interventions that can act on multiple dimensions of the latent space simultaneously, then a given intervention can undo the effects of another one.

Original: For example, assume a parameterization is not invariant to interventionsthat yield rotations, but nothing else.
Revised: For example, assume a parameterization is not invariant to interventions that yield rotations, but nothing else.

Original: Then, two invariant interventions can make an object rotateby equal and opposite angles, while changing some other dimensions of the latent state that themodel is invariant to.
Revised: Then, two invariant interventions can make an object rotate by equal and opposite angles, while changing some other dimensions of the latent state that the model is invariant to.

Original: In this case, the interventions end up undoing their effect, and the overall statechange does not yield any influence on the model output.
Revised: In this case, the interventions end up undoing their effect, and the overall state change does not yield any influence on the model output.

Original: By assuming unit interventions that enforcetransformations on specific dimensions, we can circumvent this failure mode.
Revised: By assuming unit interventions that enforce transformations on specific dimensions, we can circumvent this failure mode.

Original: F.2 Proposition 1.
Revised: F.2 Proposition 1.

Original: Proposition 1.
Revised: Proposition 1.

Original: (Mechanistically Dissimilar Modes are Connected.)
Revised: (Mechanistically Dissimilar Modes are Connected.)

Original: Assume θ 1 , θ 2 are two mechanistically dissimilar modes of loss L ( f ( D ; θ )) on a given dataset D .
Revised: Assume θ 1 , θ 2 are two mechanistically dissimilar modes of loss L ( f ( D ; θ )) on a given dataset D .

Original: Given sufficient overparamterization,there exists a continuous path that connects the two modes (in the sense of Def. 1).
Revised: Given sufficient overparamterization, there exists a continuous path that connects the two modes (in the sense of Def. 1).

Original: Proof.
Revised: Proof.

Original: The proof follows trivially from the results of [32, 31].
Revised: The proof follows trivially from the results of [32, 31].

Original: Therein, it is shown all loss minimizerslie on a single continuous manifold given sufficient overparameterization.
Revised: Therein, it is shown all loss minimizers lie on a single continuous manifold given sufficient overparameterization.

Original: That is, regardless of theunderlying mechanism leading to zero loss, the minimizer will necessarily lie on the manifold ofparameterizations achieving zero loss.
Revised: That is, regardless of the underlying mechanism leading to zero loss, the minimizer will necessarily lie on the manifold of parameterizations achieving zero loss.

Original: F.3 Conjecture 1.
Revised: F.3 Conjecture 1.

Original: Conjecture 1.
Revised: Conjecture 1.

Original: (Mechanistic Similarity Enforces Linear Connectivity.)
Revised: (Mechanistic Similarity Enforces Linear Connectivity.)

Original: If, up to permutations ofneurons, θ 1 , θ 2 show linear connectivity on a dataset D , then they must be mechanistically similar.
Revised: If, up to permutations of neurons, θ 1 , θ 2 show linear connectivity on a dataset D , then they must be mechanistically similar.

Original: Ifthey cannot be connected linearly, the modes must be mechanistically dissimilar.
Revised: If they cannot be connected linearly, the modes must be mechanistically dissimilar.

Original: Neural networks boast the well-known permutation symmetry phenomenon in their structure: permuting neurons, while accounting for the fan-in and fan-out weights, yields a model that is functionallythe same [65, 15, 32].
Revised: Neural networks boast the well-known permutation symmetry phenomenon in their structure: permuting neurons, while accounting for the fan-in and fan-out weights, yields a model that is functionally the same [65, 15, 32].

Original: That is, after permutation, the model encodes the exact same function as theoriginal model; in the language of this paper, we can say the model uses the exact same mechanismsproducing its outputs before and after the permutation.
Revised: That is, after permutation, the model encodes the exact same function as the original model; in the language of this paper, we can say the model uses the exact same mechanisms producing its outputs before and after the permutation.

Original: To avoid this degeneracy, we will assumethat we are analyzing two minimizers θ 1 , θ 2 that necessarily are not permutations of each other.
Revised: To avoid this degeneracy, we will assume that we are analyzing two minimizers θ 1 , θ 2 that necessarily are not permutations of each other.

Original: In practice, one can run recent methodologies on “neural alignment” to ensure this assumption isvalid [16, 17, 66].
Revised: In practice, one can run recent methodologies on “neural alignment” to ensure this assumption is valid [16, 17, 66].

Original: Proof.
Revised: Proof.

Original: As per Lemma 1, we only need to establish invariance / covariance to unit interventions forcharacterizing the mechanisms underlying a model’s decision rules and, correspondingly, ascertainmechanistic similarity between two model parameterizations.
Revised: As per Lemma 1, we only need to establish invariance / covariance to unit interventions for characterizing the mechanisms underlying a model’s decision rules and, correspondingly, ascertain mechanistic similarity between two model parameterizations.

Original: To that end, we consider a unitintervention A i that we assume the minimizer θ 1 is invariant to.
Revised: To that end, we consider a unit intervention A i that we assume the minimizer θ 1 is invariant to.

Original: We will analyze the loss of the modelparameterized with linear interpolation of θ 1 , θ 2 on a counterfactual sample E ( x ; A i ) generated usingintervention A i .
Revised: We will analyze the loss of the model parameterized with linear interpolation of θ 1 , θ 2 on a counterfactual sample E ( x ; A i ) generated using intervention A i .

Original: For brevity, we denote the latent state of z as z =
Revised: For brevity, we denote the latent state of z as z =

Original: G − 1 X ( x ) ; correspondingly, wedenote the intervened latent state as A α
Revised: G − 1 X ( x ) ; correspondingly, we denote the intervened latent state as A α

Original: i i ( z ) =
Revised: i i ( z ) =

Original: z + ∆ z , where ∆ z is 0 in all but the i th dimension,where it is equal to ∆ z i = α i .
Revised: z + ∆ z , where ∆ z is 0 in all but the i th dimension, where it is equal to ∆ z i = α i .

Original: We can thus write: E ( x ; A α i i ) =
Revised: We can thus write: E ( x ; A α i i ) =

Original: G X ◦
Revised: G X ◦

Original: A α
Revised: A α

Original: i i ◦ G − 1 X ( x ) =
Revised: i i ◦ G − 1 X ( x ) =

Original: G X (˜ z ) =
Revised: G X (˜ z ) =

Original: G X ( z + ∆ z ) .
Revised: G X ( z + ∆ z ) .

Original: We now consider the parameterization along a general path γ θ 1 → θ 2 ( t ) such that γ θ 1 → θ 2 (0)
Revised: We now consider the parameterization along a general path γ θ 1 → θ 2 ( t ) such that γ θ 1 → θ 2 (0)

Original: = θ 1 and617 γ θ 1 → θ 2 (1) =
Revised: = θ 1 and γ θ 1 → θ 2 (1) =

Original: θ 2 .
Revised: θ 2 .

Original: We assess its loss on the counterfactual data via a second-order expansion along thedata-generating process:
Revised: We assess its loss on the counterfactual data via a second-order expansion along the data-generating process:

Original: [Equation]
Revised: [Equation]

Original: The parameterization along a general path connecting the two minimizers θ 1 , θ 2 can be written in thefollowing form: γ θ 1 → θ 2 ( t ) = θ 1 + ∆ θ ( t, 1) , where ∆ θ ( t, 1) =
Revised: The parameterization along a general path connecting the two minimizers θ 1 , θ 2 can be written in the following form: γ θ 1 → θ 2 ( t ) = θ 1 + ∆ θ ( t, 1) , where ∆ θ ( t, 1) =

Original: γ θ 1 → θ 2 ( t ) − θ 1 .
Revised: γ θ 1 → θ 2 ( t ) − θ 1 .

Original: Then, expandingthe loss achieved by the model with this parameterization on the original data up to second-orderalong the change in parameters, we get the following.
Revised: Then, expanding the loss achieved by the model with this parameterization on the original data up to second-order along the change in parameters, we get the following.

Original: [Equation]
Revised: [Equation]

Original: where the loss and the gradient term can be ignored because θ 1 is a minimizer of the loss on dataset
Revised: where the loss and the gradient term can be ignored because θ 1 is a minimizer of the loss on dataset

Original: D .
Revised: D .

Original: Now, substituting Equation 6 into Equation 5, we get the following.
Revised: Now, substituting Equation 6 into Equation 5, we get the following.

Original: [Equation]
Revised: [Equation]

Original: where the last equality follows because of the assumed invariance of θ 1 to the intervention
Revised: where the last equality follows because of the assumed invariance of θ 1 to the intervention

Original: A α
Revised: A α

Original: i
Revised: i

Original: i .
Revised: i .

Original: We break the argument into two cases:
Revised: We break the argument into two cases:

Original: Linear case: If the connectivity path γ θ 1 → θ 2 ( t ) is linear, the change in loss movingfrom θ 1 to θ 2 , along the displacement vector θ 2 − θ 1 , is zero.
Revised: Linear case: If the connectivity path γ θ 1 → θ 2 ( t ) is linear, the change in loss moving from θ 1 to θ 2 , along the displacement vector θ 2 − θ 1 , is zero.

Original: Since θ 1 is a minimizer, this implies the displacement vector lies in the null-space of the Hessian, i.e.,∆ θ (2 , 1) T ∇ 2 θ [ L ( f ( x ; θ 1 ))]
Revised: Since θ 1 is a minimizer, this implies the displacement vector lies in the null-space of the Hessian, i.e., ∆ θ (2 , 1) T ∇ 2 θ [ L ( f ( x ; θ 1 ))]

Original: ∆ θ (2 , 1) = 0 .
Revised: ∆ θ (2 , 1) = 0 .

Original: Correspondingly, for any point in this linear path,we have, ∆ θ ( t, 1) T ∇ 2 θ
Revised: Correspondingly, for any point in this linear path, we have, ∆ θ ( t, 1) T ∇ 2 θ

Original: [ L ( f ( x ; θ 1 ))]
Revised: [ L ( f ( x ; θ 1 ))]

Original: ∆ θ ( t, 1) = 

Revised: ∆ θ ( t, 1) =

Original: ∀ t ∈ [0 , 1] .
Revised: ∀ t ∈ [0 , 1] .

Original: Substituting this relation into Equation 7, we get L ( f ( E ( x ; A α i i ) ; γ θ 1 → θ 2 ( t )))
Revised: Substituting this relation into Equation 7, we get L ( f ( E ( x ; A α i i ) ; γ θ 1 → θ 2 ( t )))

Original: = 

Revised: =

Original: and all parameterizations along the linearpath share invariances with the parameterization θ 1 .2.
Revised: and all parameterizations along the linear path share invariances with the parameterization θ 1 . 2.

Original: Non-Linear case: If the connectivity path γ θ 1 → θ 2 ( t ) is not linear, then there exists an interpolationalong the linear path connecting minimizers θ 1 , θ 2 that has a loss higher than the two minimizers.
Revised: Non-Linear case: If the connectivity path γ θ 1 → θ 2 ( t ) is not linear, then there exists an interpolation along the linear path connecting minimizers θ 1 , θ 2 that has a loss higher than the two minimizers.

Original: That is, the displacement vector ∆ θ ( t, 1) does not lie in the null-space of the Hessian and∆ θ ( t, 1) T ∇ 2 θ [ L ( f ( x ; θ 1 ))]
Revised: That is, the displacement vector ∆ θ ( t, 1) does not lie in the null-space of the Hessian and ∆ θ ( t, 1) T ∇ 2 θ [ L ( f ( x ; θ 1 ))]

Original: ∆ θ ( t, 1) ̸ = 0 .
Revised: ∆ θ ( t, 1) ̸ = 0 .

Original: Substituting this relation into Equation 7, we getL ( f ( E ( x ; A [Equation] [Table] [Equation]
Revised: Substituting this relation into Equation 7, we get L ( f ( E ( x ; A α i i ) ; γ θ 1 → θ 2 ( t )))

Original: [Table]

Original: [Equation] [Table] [Equation] [Table]
Revised: ̸ = 0 .

Original: [Equation]