Do Deep Neural Network Solutions Form a Star Domain?

## Abstract

Abstract It has recently been conjectured that neural network solution sets reachable via stochastic gradient descent (SGD) are convex, considering permutation invariances [ entezari_permutation_invariances_2022 ] . This means that a linear path can connect two independent solutions with low loss, given the weights of one of the models are appropriately permuted. However, current methods to test this theory often require very wide networks to succeed [ ainsworth_git_re_basin_2022 , benzing_random_2022 ] . In this work, we conjecture that more generally, the SGD solution set is a star domain that contains a star model that is linearly connected to all the other solutions via paths with low loss values, modulo permutations. We propose the Starlight algorithm that finds a star model of a given learning task. We validate our claim by showing that this star model is linearly connected with other independently found solutions. As an additional benefit of our study, we demonstrate better uncertainty estimates on the Bayesian Model Averaging over the obtained star domain. Further, we demonstrate star models as potential substitutes for model ensembles. Our code is available at https://github.com/aktsonthalia/starlight .

## 1 Introduction

The learning problem for a neural network is inherently
characterized by a non-convex loss landscape, leading to multiple possible solutions rather than a singular one. Efforts to comprehend this landscape and the set of solutions have been ongoing.

A significant early discovery in this area [garipov_mode_connectivity_2018] demonstrated that almost any two independent solutions could be connected through a simple low-loss curve. While this finding highlighted the vastness of the solution set, other research has focused on its complexity.

For instance, permutation symmetries allow neuron positions in different layers to be jointly swapped without changing the function represented by the neural network [brea2019weight, singh_jaggi_otfusion_2021, ainsworth_git_re_basin_2022, sinkhorn_re_basin_2023]. [entezari_permutation_invariances_2022] proposed that when accounting for these symmetries, the solution set found by stochastic gradient descent (SGD) essentially becomes convex, i.e., any pair of independent solutions can be connected through a low-loss line segment after an appropriate permutation is applied to one of the models. Notably, [sharma2024simultaneous] investigate the stronger property of *simultaneous* linear connectivity, wherein permuting a given model linearly connects it to several other models. However, recent works [ainsworth_git_re_basin_2022, xiao2023compact] study convexity in the context of the formulation in [entezari_permutation_invariances_2022]. Our work therefore refers to their conjecture as the “convexity conjecture” (Conjecture 1) while acknowledging that other, stronger forms of convexity can be formulated.

The convexity conjecture has faced challenges.
Subsequent studies [juneja2022linearreveals, benzing_random_2022, ainsworth_git_re_basin_2022, altintas_disentangling_2023, sinkhorn_re_basin_2023] revealed that even after the application of permutation-finding algorithms,
two distinct solutions in the parameter space might still be separated by a high loss barrier [frankle2020lottery, entezari_permutation_invariances_2022] upon performing linear interpolation.
These studies attribute this discrepancy to various factors, including network depth and width, dataset complexity [ainsworth_git_re_basin_2022] and high learning rates [altintas_disentangling_2023]. Theoretical investigation [entezari_permutation_invariances_2022, ferbach2024proving] suggests that in general, the conjecture needs wide networks to hold, and that deeper networks might need to be even wider than their shallower counterparts to satisfy the conjecture.

[Uncaptioned image]: x1.png

In response to these findings, our research introduces the star domain conjecture.
We propose that solutions in deep neural networks (DNNs) form a star domain rather than a convex set, modulo permutation symmetries.
A star domain is a set $A$ with at least one special element, known as a star point, $a_{0}\in A$ that is connected to every other element in $A$.
A convex set is a specific instance of a star domain.
The star domain conjecture thus proposes that in cases where convexity [entezari_permutation_invariances_2022] does not hold, a weaker form of convexity (i.e., star-shaped connectivity) still exists.

The star domain conjecture is still a stronger assertion than mode connectivity [garipov_mode_connectivity_2018] which states that any two models $\theta_{A}$ and $\theta_{B}$ can be connected through a possibly non-linear path in the solution space. As a special case, this path could be as simple as a piece-wise linear path comprising a third point $\theta_{C}$ such that $(\theta_{A},\theta_{C})$ and $(\theta_{B},\theta_{C})$ are linearly connected. In contrast, our conjecture implies that all pairs of solutions are interconnected via a shared third solution, the star point, which is common to all solution pairs: $\exists\theta_{C}$ such that $\forall\theta_{A},\theta_{B}\in S$, $(\theta_{A},\theta_{C})$ and $(\theta_{B},\theta_{C})$ are linearly connected, where $S$ is the solution set.

We substantiate our star domain conjecture with empirical evidence by introducing the Starlight algorithm to identify a candidate star model for a given learning task. Starlight finds a model that is linearly connected with a finite set of independent solutions. We demonstrate that these star model candidates have low loss barriers with an arbitrary set of solutions that were not used in constructing the star model candidates. This provides strong evidence that there exist star models that are linearly connected with other solutions.

In addition to validating the conjecture, our research delves into the distinctive characteristics of star models. We find that sampling from the star domain for Bayesian Model Averaging (BMA) leads to better uncertainty estimates than ensembles. Additionally, we demonstrate star models as a possible substitute to model ensembles, with lower inference time and memory footprint. These differences highlight the potential advantages of star models in various neural network applications.

We summarise our contributions:

- 1.
The star domain conjecture for characterizing connectivity in neural network solution sets.
- 2.
The Starlight algorithm for identifying a star model for a gradient-based learning task.
- 3.
Analysis of practical benefits shown by the star models.

## 2 Related work

We introduce the relevant development of findings toward the understanding of DNN solution sets.

Mode Connectivity.
[garipov_mode_connectivity_2018] and [draxler_essentially_no_barriers2018] concurrently discovered mode connectivity.
[gotmare2018using_mode_connectivity] soon followed, showing non-linear connectivity even between networks obtained using different training schemes. [kuditipudi_explaining_connectivity_dropout_noise_2019] explained mode connectivity via dropout stability and noise stability.
[benton_loss_simplexes_2021] went on to show that there exist not only simple paths, but also *volumes* of low loss, connecting several DNN solutions. These works focus on general, *non-linear* connectivity, while we study a stricter condition, viz., linear connectivity.

Linear Mode Connectivity (LMC). [frankle2020lottery] were the first to study LMC. Later, [entezari_permutation_invariances_2022] proposed that SGD solution sets are convex modulo permutations, while [singh_jaggi_otfusion_2021, ainsworth_git_re_basin_2022, sinkhorn_re_basin_2023] introduced "re-basin" methods, i.e., methods for bringing different solutions into the same basin. Recent work [ainsworth_git_re_basin_2022, altintas_disentangling_2023, benzing_random_2022] also noted failure cases for LMC, while
[ferbach2024proving] theoretically investigated convexity for sufficiently wide nets.
Our analysis builds upon these findings and reveals evidence for a weaker property, viz., star-shaped connectivity, in cases where convexity does not hold.

Star-shaped connectivity (SSC). [annesi2023star] provide valuable insights for SSC in the loss landscape for the simple case of the negative spherical perceptron. In contrast, we consider more complex models and learning tasks, and propose a novel verification method for SSC.
Concurrently to us, [lin2024exploring] obtain star models linearly connected with a finite number of solutions to larger nets like the VGG16 [simonyan2014vgg].
In contrast, our work additionally considers permutation invariances [ainsworth_git_re_basin_2022, entezari_permutation_invariances_2022] and provides evidence that star models trained this way might be connected to infinitely many other solutions.

Practical Applications. Mode connectivity has found applications in
model fusion ([garipov_mode_connectivity_2018, singh_jaggi_otfusion_2021]), adversarial robustness ([zhao_adversarial_bridging_mode_connect_2020, wang_robust_mode_connectivity_2023]),
continual learning ([mirzadeh_lmc_multitask_contiunal_2020, wen_opc_connectivity_2023]), and federated learning ([wang2020federated, ainsworth_git_re_basin_2022]). In contrast, our work focuses on understanding the surface of the loss landscape. However, we also explore potential applications, e.g., Bayesian Model Averaging.

## 3 The star domain conjecture

### 3.1 Background: the convexity conjecture

Here, we formally state the convexity conjecture, starting with basic notations. A neural network is a function $f_{\theta}(\cdot)$ parameterized by $\theta\in\Theta$, where $\Theta$ is the parameter space.
Given a dataset $\mathcal{D}$, we formulate a non-negative loss $\mathcal{L}(\theta)=\mathcal{L}\left(\theta;\mathcal{D}\right)\geq 0$ and minimize $\mathcal{L}\left(\theta\right)$ to find a solution in $\Theta$.

The solution set is $S:=\{\theta\mid\mathcal{L}\left(\theta\right)\approx 0\}$.

The loss barrier was first defined by [frankle2020lottery]. We use the formulation in [entezari_permutation_invariances_2022], i.e., the barrier between $\theta_{A},\theta_{B}\in\Theta$ is $B\left(\theta_{A},\theta_{B}\right):=\max_{t\in[0,1]}\widetilde{\mathcal{L}}_{
t}(\theta_{A},\theta_{B})$, where

$$ $\displaystyle\widetilde{\mathcal{L}}_{t}(\theta_{A},\theta_{B}):=$ $\displaystyle\,\mathcal{L}\left((1-t)\cdot\theta_{A}+t\cdot\theta_{B}\right)- \left((1-t)\cdot\mathcal{L}\left(\theta_{A}\right)+t\cdot\mathcal{L}\left( \theta_{B}\right)\right)\vspace{-1em}$ (1) $$

is the difference between the loss value at $t$, and the linear interpolation of the losses at the end-points.

[Uncaptioned image]: x2.png

Two solutions $\theta_{A},\theta_{B}\in\Theta$ are said to be linearly mode-connected, or LMC [frankle2020lottery], when their loss barrier is
approximately zero: $B\left(\theta_{B},\theta_{A}\right)\approx 0$.

The convexity conjecture is constructed upon a parameter space where the permutation symmetries are factored out.
A permutation invariance [brea2019weight] can be formulated as an equivalence relation $\sim$ between two points $\theta_{A},\theta_{B}$ in the parameter space such that $\theta_{A}\sim\theta_{B}$ if and only if there exists a permutation $\pi$ of the parameters such that $\pi(\theta_{A})=\theta_{B}$ *and* the functions represented by them are identical: $f_{\theta_{A}}(x)=f_{\theta_{B}}(x)$ for all $x$. Given two points $\theta_{A}$ and $\theta_{B}$, we look for the permutation of $\theta_{B}$ that connects it to $\theta_{A}$ (or vice versa) with as low a loss barrier as possible [ainsworth_git_re_basin_2022, entezari_permutation_invariances_2022, sinkhorn_re_basin_2023]. A winning permutation [entezari_permutation_invariances_2022] for models $\theta_{A}$ and $\theta_{B}$ is defined as

$$ $\displaystyle\underset{\theta_{A}\rightarrow\theta_{B}}{\pi}:=\operatorname*{ arg\,min}_{\pi\in\mathcal{P}_{\theta_{A}}}B\left(\pi(\theta_{A}),\theta_{B}\right)$ (2) $$

where $\mathcal{P}_{\theta}:=\{\pi\,|\,\pi(\theta)\sim\theta\}$ is the set of all function-preserving permutations of $\theta$.

###### Conjecture 1 .

Convexity Conjecture [entezari_permutation_invariances_2022].
Let $S{}$ be the set of SGD-reachable solutions for a deep neural network $f(\theta)$ trained for a certain task.
Let $\theta_{A},\theta_{B}\in S{}$ be two solutions. Then, there exists a minimum width $h$ such that if $f(\theta)$ is wider than $h$, then with high probability, $\theta_{B}$ can be permuted to obtain $\tilde{\theta}_{B}=\underset{\theta_{B}\rightarrow\theta_{A}}{\pi}(\theta_{B})$ such that $\theta_{A}$ and $\tilde{\theta}_{B}$ are highly likely to be linearly mode-connected, i.e., $B\left(\tilde{\theta}_{B},\theta_{A}\right)\approx 0$.

We refer to this as the (quasi-) convexity [ainsworth_git_re_basin_2022, xiao2023compact] conjecture, because, by definition, a convex set is precisely a set where the line segment between any two elements is included in the set. The conjecture provides a geometric intuition that the solution set is generally convex, modulo permutations.

Theoretical results only validate the conjecture in limited settings, given sufficiently wide networks [entezari_permutation_invariances_2022, ferbach2024proving]. Empirical validations exhibit mixed accounts. [ainsworth_git_re_basin_2022] notably achieve zero barrier between two ResNet-20-32 models trained on CIFAR-10, but there remains a loss barrier between narrower models, even after weight matching. They further report network depth and dataset complexity as aggravating factors. [benzing_random_2022] provide interesting insights using their activation-matching permutation algorithm. While fully connected networks (FCNs) live in the same loss valley even at initialization, convolutional nets (CNNs) are usually not connected even after considering permutation invariances. [sinkhorn_re_basin_2023] introduce Sinkhorn re-basin, a differentiable permutation-finding approach; however, even with two-layer NNs, the barrier between CIFAR-10 models, albeit low, remains non-zero. For CNN architectures like VGG, the barrier is substantially high. [altintas_disentangling_2023] show that aggravating factors for LMC include the Adam optimizer [kingma2014adam], absence of warmup, and task complexity.

Hence, in cases where strong evidence for the convexity conjecture is absent, it is important to consider other possible topologies for general DNN solution sets.
To this end, we propose the star domain conjecture for characterizing DNN solution sets that do not enjoy convexity [entezari_permutation_invariances_2022] modulo permutations.

###### Conjecture 1 .

### 3.2 The star domain conjecture

We propose a weaker form of convexity for characterizing DNN solution sets.
We argue that DNN solution sets are generally star domains, modulo function-preserving permutations. While [annesi2023star] demonstrates this property for simple spherical negative perceptrons (without permutations), we argue that it holds for even deeper, more complex nets after considering permutation invariances.

We start with the necessary definitions to make a formal description of the conjecture. A set $A\subset\mathbb{R}^{n}$ is a star domain if there exists an element $a_{0}\in A$ such that for any other element $a\in A$ and $0\leq t\leq 1$, $(1-t)\cdot a_{0}+t\cdot a\in A$, i.e., all points on the line segment between $a_{0}$ and $a$ lie in $A$. We call such $a_{0}$ a star point. In the context of the parameter space, we refer to the star point of a star-domain-shaped solution set as a star model.

###### Conjecture 2 .

Star Domain Conjecture.
Consider the set $S{}$ of SGD-reachable solutions to a deep neural network $f(\cdot)$ trained to execute a certain machine learning task. Let $h$ be the minimum width for which $S{}$ becomes convex modulo permutations. Then there exists a constant $0<\alpha<1$ such that if $f(\cdot)$ is wider than $\alpha h$, then $S{}$ is highly likely to be a star domain modulo permutation symmetries, i.e., there exists a *star model* $\theta^{\star}\in S{}$ such that for any other solution $\theta\in S{}$, it is possible to obtain $\tilde{\theta}=\underset{\theta\rightarrow\theta^{\star}}{\pi}(\theta)$ such that $B\left(\tilde{\theta},\theta^{\star}\right)\approx 0$.

A convex set is a special case of a star domain, where all the elements are star points. The star domain conjecture is thus a relaxation of the convexity conjecture.

###### Conjecture 2 .

### 3.3 Finding a star model

We provide empirical evidence for the star domain conjecture via two steps. First, we present a method for finding a star model. Second, we verify that the model found is indeed a star model: it has a low loss barrier with an arbitrary solution in $S{}$. Here, we focus on the first step.

We consider a necessary condition for a star model $\theta^{\star}$: given an arbitrary set of models $Z=\{\theta_{1},\theta_{2},\dots,\theta_{N}\}\subset S{}$, $\theta^{\star}$ has to be connected to all of them, modulo permutation invariances.

We present a recipe for finding such a $\theta^{\star}$.

We first obtain a finite set $Z=\{\theta_{1},\theta_{2},\dots,\theta_{N}\}$ of models, independently trained with different random seeds controlling the initialization, batch composition, and augmentation. We then formulate a loss function that, for fixed $Z$, encourages low loss barriers between $\theta$ and some permuted versions of $\{\theta_{1},\theta_{2},\dots,\theta_{N}\}$. The objective may be expressed as

$$ $\displaystyle\theta^{\star}_{Z}=\operatorname*{arg\,min}_{\theta}\frac{1}{N} \sum\limits_{\theta_{n}\in Z}B\left(\theta,\underset{\theta_{n}\rightarrow \theta}{\pi}(\theta_{n})\right)$ (3) $$

where $\underset{\theta_{n}\rightarrow\theta}{\pi}$ is the winning permutation defined in Section 3.1 that permutes $\theta_{n}$ without changing the represented function while minimizing the loss barrier against $\theta$. To solve this optimization problem, we propose to minimize the expected loss on the linear interpolation between the model in question $\theta$ and each source model $\theta_{n}$, after permutations. We modify the training objective as $\theta^{\star}_{Z}=\operatorname*{arg\,min}_{\theta}\widetilde{\mathcal{L}}_{Z
}(\theta)$
where

$$ $\displaystyle\widetilde{\mathcal{L}}_{Z}(\theta):=\frac{1}{N}\sum_{n=1}^{N} \int_{0}^{1}\mathcal{L}\left((1-t)\cdot\theta+t\cdot\underset{\theta_{n} \rightarrow\theta}{\pi}(\theta_{n})\right)dt$ (4) $$

This expresses the expected loss on the set of line segments between $\theta$ and $\underset{\theta_{n}\rightarrow\theta}{\pi}(\theta_{n})$, where each source model $\theta_{n}\sim\text{Unif}(Z)$ is chosen at random and then each point on the line segment is sampled as $t\in\text{Unif}[0,1]$. The optimization problem in eq. 4
involves computational challenges.
Resolving the continuous integral over $t$ is non-trivial for complex learning problems.
Furthermore, $\underset{\theta_{n}\rightarrow\theta}{\pi}$ assumes access to the winning permutation. However, the winning permutation depends on $\theta$, which constantly changes during optimization.
We introduce the following solutions.

Figure: Algorithm 1 Starlight: Training a Star Model.

#### Monte-Carlo optimization scheme.

Instead of estimating $\widetilde{\mathcal{L}}_{Z}(\theta)$ precisely at every iteration, we rely on a Monte-Carlo estimation scheme, inspired by the parameter-curve fitting method by [garipov_mode_connectivity_2018]. At iteration $k\geq 1$,
we sample $\theta_{n^{(k)}}$ uniformly from $Z$
and $t^{(k)}$ from $\text{Unif}[0,1]$ (see LABEL:appendix:different_sampling_schemes for ablations using alternative sampling schemes). Hence, we obtain a single point on the manifold, calculate the cross-entropy loss at this point, and subsequently the gradients for updating $\theta$.

#### Finding optimal permutations.

We perform weight matching [ainsworth_git_re_basin_2022], i.e., we seek a permutation $\pi_{n}$ that maximizes the dot product $\theta{}\cdot\pi_{n}(\theta_{n})$, for each $\theta_{n}\in Z$. This procedure aligns each source model $\theta_{n}$ with the candidate star model $\theta$. This operation is performed at the beginning of every epoch instead of every *iteration*, significantly speeding up the optimization process.

Algorithm 1 describes the detailed procedure. Once we find a $\theta$ that has a low expected loss $\widetilde{\mathcal{L}}_{Z}(\theta)$ on the linear paths to a finite set of source models $Z$, we may verify if this $\theta$ is likewise linearly connected with an arbitrary solution $\theta_{N+1}\notin Z$.

### 3.4 Empirical evidence

We introduced Starlight to find a candidate star model. Now, we propose a method to verify if the model found in Section 3.3 is a star model by checking its linear connection to an arbitrary solution $\theta_{N+1}\notin Z$, i.e., not part of the set of source models used for finding the star model. We refer to such models as held-out solutions $H$ that are disjoint from the source models: $H\cap Z=\emptyset$.

**Table 1: Empirically verifying the star domain conjecture. “Regular loss” and “Star loss” indicate training losses for regular models in $Z{}$ and star models $\theta^{\star}$, respectively. “Star-regular” refers to the barrier $B\left(\theta^{\star},\theta_{h}\right)$ between a star model and one of the heldout models in $H$. For comparison, “Regular-regular” is the loss barrier $B\left(\theta_{A},\theta_{B}\right)$ between two arbitrary models. We report values up to one standard deviation over several runs, except for ImageNet. In each case, star models exhibit significantly lower loss barriers with other models, than the corresponding average loss barrier between two regular models.**
| Dataset | Architecture | Regular loss | Star loss | Regular-regular | Star-regular |  |
| --- | --- | --- | --- | --- | --- | --- |
| CIFAR-10 | ResNet-18 | $0.001\pm 0.000$ | $0.001\pm 0.000$ | $0.383\pm 0.056$ | $0.078\pm 0.007$ |  |
| CIFAR-10 | ResNet-18 (Adam) | $0.001\pm 0.000$ | $0.015\pm 0.000$ | $1.368\pm 0.551$ | $0.335\pm 0.022$ |  |
| CIFAR-10 | VGG11 | $0.003\pm 0.000$ | $0.022\pm 0.000$ | $0.515\pm 0.034$ | $0.131\pm 0.005$ |  |
| CIFAR-10 | VGG19 | $0.001\pm 0.000$ | $0.059\pm 0.000$ | $1.281\pm 0.153$ | $0.336\pm 0.078$ |  |
| CIFAR-10 | DenseNet | $0.001\pm 0.000$ | $0.157\pm 0.000$ | $4.634\pm 0.727$ | $1.729\pm 0.409$ |  |
| CIFAR-100 | ResNet-18 | $0.004\pm 0.001$ | $0.005\pm 0.000$ | $2.905\pm 0.047$ | $0.756\pm 0.049$ |  |
| CIFAR-100 | DenseNet | $0.006\pm 0.000$ | $0.635\pm 0.000$ | $6.920\pm 0.216$ | $3.735\pm 0.180$ |  |
| ImageNet-1k | ResNet-18 | $0.711$ | $1.380$ | $5.948$ | $2.794$ |  |

Figure: Figure 1: Starness of a star model vs source models. We plot the loss barriers $B(\theta^{\star},\theta_{h})$ between star models $\theta^{\star}$ and heldout models $\theta_{h}\in H$ at different numbers of source models $Z$ used for learning the star model $\theta^{\star}$ (orange points). The heldout set is disjoint with the source models: $H\cap Z=\emptyset$. We provide a reference point given by the loss barrier between two regular solutions $B(\theta_{A},\theta_{B})$ for $\theta_{A},\theta_{B}\in S{}$ (blue plot). The error bars indicate one standard deviation across five held-out models $|H|=5$. Incorporating more source models $|Z|$ enables finding a better star model with a lower loss barrier against an arbitrary solution.
Refer to caption: x3.png

Figure: Figure 2: “Starness” vs. model width and depth. For starness vs. model width (left), we vary the width of a WideResNet (depth $22$) from $1\times$ to $8\times$. For starness vs. model depth, we vary the depth of a WideResNet (width $1\times$) from $22$ to $40$ layers. For each depth-width combination, we plot the loss barriers $B(\theta^{\star},\theta_{h})$ between star models $\theta^{\star}$ and heldout models $\theta_{h}\in H$ on the y-axis. As a reference point, we plot the barrier between two regular solutions $B(\theta_{A},\theta_{B})$, on the x-axis. The points are annotated with the corresponding widths or depths. Star models consistently enjoy better linear connections with regular models, than do the regular models amongst each other.
Refer to caption: x5.png

Figure: Figure 3: Loss barriers for star models. We interpolate between a star model $\theta^{\star}$ and regular models that are trained with SGD. There are two types of regular models, depending on whether they are used for finding the star model: source models $Z$ are used, and heldout models $H$ are not. Along the interpolation, we visualize the loss barrier by plotting the loss and accuracy values (orange curves). For these curves, $t=0$ corresponds to the star model $\theta^{\star}$. For reference, we plot the interpolation between two arbitrary regular models (blue curves). The error bands correspond to one standard deviation.
Refer to caption: x7.png

We describe our main findings with reference to ResNet-18 [resnet_he_2016] models trained on CIFAR [krizhevsky2009learning] using SGD, using $50$ source models and $5$ held-out models. We present results for additional architectures (e.g., VGG [simonyan_very_2015] and DenseNet [huang_densely_2018]), a large-scale dataset (ImageNet-1k [deng2009imagenet]) and settings (for instance, Adam [kingma2014adam]) in Table 1 and LABEL:appendix:additional_results.
Likewise, our empirical findings are built upon the training loss and accuracy, but we confirm that they also transfer over to test loss and accuracy in LABEL:appendix:test_metrics. We largely use standard recipes to train the models in our experiments, with the exception of star models where we additionally incorporate the steps in Algorithm 1. We further describe our experimental setup in LABEL:appendix:implementation_details. We summarize our observations below.

Convexity conjecture does not hold.
In Figure 3, we show loss barriers between two independently trained solutions (blue “regular-regular” curves).
We observe that the loss increases and accuracy drops significantly at around $t=0.5$, even after applying the algorithm [ainsworth_git_re_basin_2022] to find the winning permutation.
We present another piece of evidence that the convexity conjecture does not hold for thin ResNets, reconfirming the findings of [ainsworth_git_re_basin_2022].

Star model has low loss barriers with other solutions.
In Figure 3,
we show the training losses and accuracies along linear paths between the candidate star model $\theta^{\star}$ and other types of solutions (either source models $Z$ or held-out models $H$). They are indicated with red curves. As a reference, we always plot the confidence interval of loss and accuracy values along the line segments between two regular solutions (blue curves).
For the source models in $Z$, star-to-regular connections enjoy essentially zero loss barriers, in contrast with regular-to-regular connections, which remain significantly higher at $0.381$, for CIFAR-10.
This demonstrates that it is possible to find a model $\theta^{\star}$ simultaneously connected to $|Z|=50$ models.
The same is true for the line segments between the star model $\theta^{\star}$ and a held-out model picked from $|H|=5$ models; even though the barrier between the star model and the heldout model is non-zero, it remains as low as $0.077$ compared to $0.381$ for the regular-to-regular case.

A greater number of source models enhances “starness”.
Our star model is constructed from the set of source models $Z$.
We question whether greater $|Z|$ induces greater “starness” of the solution found by Starlight.
In Figure 1, we plot the loss barrier against the number of source models $2\leq|Z|\leq 50$ used to construct the star model.
For statistical significance, we include loss barrier statistics between two regular, independently trained models in $S$ with error bars indicating one standard deviation.
We observe that the loss barriers between these star models and the held-out models decrease as $|Z|$ increases. The decreasing trend has not saturated after $|Z|=50$. We stopped there because of computational limits. However, including more source models is likely to enhance connectivity between the obtained star model and the other solutions even further.

Effect of model width and depth.
Prior work stresses the importance of model width and depth [ainsworth_git_re_basin_2022, entezari_permutation_invariances_2022] in determining loss barriers between two solutions. We investigate the effect of model width and depth for residual nets. Specifically, we consider WideResNets [zagoruyko_wide_resnets_2017] of widths $1\times$, $2\times$, $4\times$, and $8\times$ that of a normal ResNet (depth $22$). We also consider ResNets of depths $22$, $28$, $34$, and $40$. We compare the barriers achieved by “regular-regular” and “star-regular” pairs for each case Figure 2. Our investigation confirms existing reports of decreasing loss barriers as model width increases. We observe significantly lower star-regular barriers than regular-regular barriers for models of identical widths (e.g., roughly $0.004$ compared to $0.012$ at width $8x$). In fact, it is possible to fit a linear regression line to the observed barrier values, wherein the star-regular barriers are about a third of the regular-regular barriers at any given width (Figure 2, left). We draw similar conclusions from varying depth (Figure 2, right), although the change in barriers as we change the model depth is not quite as pronounced as it is for the varying width case. This observation also motivates the statement of our conjecture, as we explain in LABEL:appendix:barrier_relationships.

Effect of optimizer.
While both the convexity conjecture and the star domain conjecture involve solution sets obtained through SGD, we also investigate the impact of using the Adam optimizer [kingma2014adam]. Specifically, we train $15$ regular models, with $|H|=5$ and ($|Z|=10$). We then train a star model and evaluate its barriers with the models in the held-out set. Results can be found in Table 1 (second row). We observe that Adam-trained regular solutions have a higher loss barrier between them ($1.368$) compared to SGD-trained regular solutions ($0.383$). Likewise, the barrier between the star model and regular models also increases from $0.078$ for SGD solutions to $0.335$ for Adam solutions. While both “regular-regular” and “star-regular” connections suffer with this change of optimizer, “star-regular” connections still fare significantly better than “regular-regular” connections. This finding suggests that Adam solutions are also highly likely to enjoy star-shaped connectivity.

Caveats. Despite the promising observations above, our star domain conjecture is not theoretically validated and thus remains a conjecture.
From the empirical perspective, loss barriers between the star model and other solutions often yield values that are significantly greater than zero.
However, we emphasize that this paper is focused on providing the lower bound in evidence supporting the star domain conjecture.
Considering a larger number of source models for the star model construction, improving Starlight, and developing a better algorithm for finding the winning permutations will potentially contribute to the discovery of better star models in the solution set.

Conclusion.
Our experimental results confirm existing reports that the convexity conjecture requires very wide networks to hold, and has otherwise several failure cases for which we propose a relaxed version, viz., the star domain conjecture.
We obtain strong empirical evidence that the star model found through Starlight is likely to be a true star model.
Our analysis thus sheds further light on solution set geometry for narrower and deeper networks, as well as for complex learning tasks where the convexity conjecture struggles.
We invite the community to expand upon our findings and converge toward a more accurate understanding of the loss landscape.

## 4 Practical applications

The star domain conjecture introduces a novel dichotomy of solution types: “star” and “non-star” models.
Most solutions are non-star and lack linear connections with other solutions.
However, in Section 3.2, we have presented strong evidence for the existence of star models.
In this section, we examine the properties and potential benefits of star models in practice.
Section 4.1 explores whether star models and the surrounding star domain provide a better posterior for Bayesian Model Averaging. In Section 4.2, we propose star models as a practical alternative to model ensembling.

**Table 2: Model fusion performances. “Regular” indicates single models; Ensemble indicates a vanilla average of the probability vectors across the member models; “Star” indicates a model found using Starlight using the regular models as the source set $Z$. ResNet18 has been used throughout. We show one standard deviation for the error bars. In addition, we report the accuracy of the best member in the ensemble (“Best of $n$”) and the accuracy of the best star model (“Best of $3$”). Star models perform better than single, regular models but use only a fraction of the compute required by the ensemble at test time.**
| Dataset      #Models | Regular | Best of $n$ | Ensemble | Star | Best of $3$ |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
|  | $2$ | $95.2\pm 0.03$ | $95.24$ | $95.8$ | $95.3\pm 0.16$ | $95.43$ |  |
| CIFAR-10 | $5$ | $95.1\pm 0.14$ | $95.24$ | $96.0$ | $95.2\pm 0.15$ | $95.27$ |  |
|  | $50$ | $95.1\pm 0.16$ | $95.44$ | $96.3$ | $95.3\pm 0.20$ | $95.54$ |  |
|  | $2$ | $77.3\pm 0.16$ | $77.49$ | $79.6$ | $78.0\pm 0.24$ | $78.14$ |  |
| CIFAR-100 | $5$ | $77.4\pm 0.21$ | $77.68$ | $80.4$ | $78.1\pm 0.02$ | $78.15$ |  |
|  | $50$ | $77.3\pm 0.28$ | $77.94$ | $81.3$ | $78.4\pm 0.10$ | $78.48$ |  |
| Train / test complexity | $\mathcal{O}(1)/\mathcal{O}(1)$ | $\mathcal{O}(n)/\mathcal{O}(n)$ | $\mathcal{O}(n)/\mathcal{O}(1)$ |  |  |  |  |

Figure: (a) AUROC (Max. Probability)
Refer to caption: x15.png

### 4.1 Bayesian model averaging

Bayesian model averaging (BMA) enhances uncertainty estimation by averaging predictions from the posterior of models in the parameter space.
Posterior families in the literature range from simple Gaussian [bbb] and Bernoulli [dropout] distributions to more complex geometries like splines [garipov_mode_connectivity_2018] and simplices [benton_loss_simplexes_2021].
Here, we examine if the star domain provides a good posterior family for BMA-based uncertainty estimation.

Setup.
The posterior of interest is the collection of line segments between the star model $\theta^{\star}$ and other solutions $\{\theta_{1},\cdots,\theta_{N}\}$ that are independently found.
Similarly to Starlight, we sample first from the model index of $\{1,\cdots,N\}$ uniformly and then sample from the line segment $\text{Unif}[0,1]$.
As in standard BMA, we consider a set of models sampled from the posterior and the post-softmax average of these models.
We use ResNet-18 models trained on CIFAR-10. As a baseline, we present the BMA for the independent solutions $\{\theta_{1},\cdots,\theta_{N}\}$.

Evaluation.
We assess the predictive uncertainty of the BMA-based confidence estimates. For the ranking metric, we use the area under ROC curve (AUROC). We consider both max-probability and entropy-based confidence measures. We also show results based on the expected calibration error (ECE).

Results.
Figure 4 shows uncertainty quantification at different numbers of posterior samples from $2$ to $19$.
BMA using the star domain posterior consistently exhibits better AUROC values than baseline deep-ensemble estimates.
However, ECE is worse than that of the deep ensemble.
The star domain posterior provides avenues for more precisely ranked uncertainty estimates, albeit absolute-value uncertainty quantification may not be precise.

Conclusion.
Our proposed star domain posterior offers better uncertainty estimates than the deep ensemble baseline in rank-based predictive uncertainty evaluation.

### 4.2 Potential usage in model fusion

Given a fixed amount of training data, a popular approach to maximize model generalizability is ensembling, i.e., fusing predictions from multiple independent models. This basic approach suffers from computational complexities during both training and inference. Every input has to be processed by individual member models at test time. Storing multiple models also leads to a higher memory footprint, scaling linearly with the number of ensemble members.

Starlight can also be understood as a method for aggregating multiple source models $Z=\{\theta_{1},\cdots,\theta_{N}\}$ into a single model $\theta^{\star}$.
From a computational perspective, star models reduce the necessary time and storage complexity during inference.
We investigate whether the star models provide an enhanced generalization compared to the individual models.

Setup.
We slightly modify the training objective of Starlight to align it with a better generalization capability of the star model. We add a cross-entropy term $\mathcal{L}\left(\theta\right)$ so that $\mathcal{L}_{\text{total}}(\theta,Z)=\widetilde{\mathcal{L}}_{Z}(\theta)+
\mathcal{L}\left(\theta\right)$, where $\widetilde{\mathcal{L}}_{Z}(\theta)$ is the original optimization objective for the star model discovery in eq. 4.

Evaluation.
We evaluate test accuracies for star models trained with varying numbers of source models ($|Z|$) and compare them to ensembles using the same source models.

Results.
Results in Table 2 show that star models consistently outperform regular models ($78.4\%$ vs. $77.3\%$) for CIFAR-100 with $|Z|=50$). While less accurate than ensembles over $Z$, star models require only a fraction of the compute during inference.

Conclusion.
“Starness” of a solution may enhance generalization.
In scenarios where test-time inference costs are critical, star models could be a promising alternative to vanilla ensembles.

## 5 Conclusion

This paper proposes a novel understanding of SGD loss landscapes. The traditional picture before [garipov_mode_connectivity_2018] was one of extreme non-convexity, in contrast with the current picture of near-perfect convexity in a canonical, modulo-permutations space [entezari_permutation_invariances_2022] for extremely wide nets. Our claim becomes relevant when narrower and deeper nets, complex datasets, and different optimization schemes are considered. We propose a weaker form of convexity in these cases, i.e., the solution set is a star domain modulo permutations. Our empirical findings support this hypothesis. We propose the Starlight algorithm to find candidate “star models” and verify that they are indeed linearly connected to other solutions. In addition to the empirical evidence for the star domain conjecture, we present potential use cases for star models in practice, including uncertainty estimation through Bayesian model averaging, and model fusion.

## Peer Reviews

confidence: 4
contribution: 3
initial_rating: 6
presentation: 4
questions: See weaknesses.
soundness: 3
strengths: The paper is well written, and the conjecture makes sense and is easy to understand. The experiments are quite convincing, though one could obviously alway try larger/deeper networks.
summary: The paper proposes a weaker version of the conjecture that the minima recovered by SGD are convex up to permutations, proposing instead that this set is star-shaped up to permutations. A small theoretical motivation is given for shallow linear networks, but the main evidence is empirical, where the authors propose an algorithm to find the star model at the center of the start shape from a ensemble of trained networks. They then observe that the star-shaped conjecture is approximately true for networks where the convex conjecture is not. 

Finally two alternatives are proposed to ensembling: sampling from the star shape, or taking the star model. Both seem to be promising.
weaknesses: The authors do not propose any intuition why star-shape is the right choice to fix the convex up to permutation conjecture. In the theoretical results, can star-shapedness be guaranteed for significantly smaller widths than convexity?

Star-shape is significantly weaker than convexity (especially in high dimension, I would guess), so it is not completely surprising that one can reach a lower barrier. On the other hand switching from convexity to star shape makes this conjecture much harder to check and compute, since one has to find the star model at the center, and the experiments suggest that one needs a very large amount of models to find this star model accurately.

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 3
questions: I might be missing it but are there any experiments where Starlight algorithm finds a model where permutations fail to reduce the error barrier?
soundness: 2
strengths: - Paper is easy to follow and give sufficient background.
- Introduced the star-conjecture that is stronger than mode connectivity as in [1] but weaker than weak linear connectivity at defined in [2]
summary: # 

The paper raises a conjecture if the trained neural networks, after taking into account for the permutation symmetries, form a star domain, i.e. there exists a neural network that is linearly connected to all the other networks. 

Authors propose an algorithm to find the model that is star model by utilizing permutation matching algorithms and by defining a loss that minimizes the error barrier between the star model and other networks in the set.

Authors empirically validate the conjecture by showing that one can find a star model that is linearly connected to all the networks. 

Authors show that model found by Starlight leads to mild improvement in generalization than individual models but still is a bit worse than model ensembling
weaknesses: - Relationships with other notions of linear connectivity is not very well discussed.
- If I understand correctly, the experiments only conducted when there is weak linear connectivity modulo permutation holds since weak linear connectivity implies the star conjecture:
    - If weak linear connectivity modulo permutation holds, one can pick any model $\theta$ from the set $Z = \{\theta_1, \dots, \theta_N\}$ and find permutation $P_1, \dots, P_N$ that linearly connects the model $\theta_i$ to $theta$. This should lead to star connectivity
- Results for Starlight over model ensembling are not very extensive,

Minor comments:

- the bibliography needs many fixes. Specially because authors cite the arxiv version of the paper but should cite the publications

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 2
questions: 1- This questions is related to the title and the whole paper `DO DEEP NEURAL NETWORK SOLUTIONS FORM A STAR DOMAIN?'
Do authors by 'solution' mean the global minima of objective function? If yes, how can authors make sure that global minima are reached in practice by SGD (objective functions are highly non-convex), let say in practical settings like limited width of the neural network? This question also concerns your empirical observations. 
Also `The learning problem for a neural network is inherently characterized by a non-convex loss landscape, leading to multiple possible solutions rather than a singular one.' As far as I am aware, the first and main concern in optimization of NN is non-global optima solutions. 
And could you please elaborate what is the role of SGD here? Why not other optimization methods? 

2-Theorem 1 is targeting a toy model of neural networks (shallow, linear, ...), could you please elaborate how this theory under restrictive Assumption 1 can support your Conjecture? Could authors also elaborate on how Assumption 1 makes sense in practice? 

3-Line 236: How can you reach all those solutions, while it is computationally too expensive? In fact, there are infinitely many solutions for neural networks, it kinda seems too far to extract a set of solutions. How large that set need to be to leave you a star point? 

4-Line 267: It is not clear for me how authors handle finding optimal permutations? Could you please elaborate more?  

5-Theorem 1 is valid for $m\to \infty$. How authors can handle this in practice? 

6-Line 502, it is stated that this approach is useful to decrease computational expenses, but I am not sure how it can happened following your proposed setting in 236 and solving (3). 

7-Better to clarify about importance of start shaped models sooner in text regarding your comments in section 4.2.
soundness: 2
strengths: The paper is written clearly at some parts but I believe the writing can still be improved. 
For example, it was not clear at the beginning of paper why such a star model property is important in NN, but authors explain this later 
in their section 4. 
The main contribution of the paper seems to be interesting, but I am still not convinced about their details.
summary: This paper conjectures that SGD solution set of neural networks is a star domain. 
They have proved their conjecture for a toy model of neural networks, namely shallow linear neural networks with just one-dimensional input data . They also propose a Starlight algorithm to find a star model of a given task. It is mentioned that the benefit of having star domain for network solution is in Bayesian Model Averaging. Authors also supported their claims by proving empirical studies.
weaknesses: I think beyond their Theorem 1 that is proved for a really toy model of neural networks under strict assumptions, there is no theoretical support over their claimed conjecture. 
I also believe there are confusing points in the paper that need to be clarified, please see my question list below.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: -  Theorem 1 is established for an oversimplified two-layer linear network. Can the author at least show results with two-layer network with nonlinear activations? Otherwise, the theory is too weak to support the star domain conjecture.

-  Follow-up questions : How do the authors envision extending the theoretical framework of the star domain conjecture to encompass a broader range of neural network architectures and training methods? What specific conditions or assumptions must hold for the star domain conjecture to be valid in practical applications?

-  About Empirical Validation: how do the authors plan to validate the star domain conjecture across different types of neural network architectures (e.g., transformers, recurrent networks)? Can the authors provide additional insights or results regarding the performance of the Starlight algorithm in high-dimensional parameter spaces?

- About algorithm complexity: Given the complexity of the Starlight algorithm, what steps have the authors taken to ensure its computational efficiency and ease of implementation in practical settings? Are there potential optimizations or modifications to the Starlight algorithm that could enhance its performance, especially in large-scale applications?
soundness: 3
strengths: The introduction of the star domain conjecture adds a novel perspective to existing theories regarding the connectivity of neural network solutions. The Starlight algorithm is well-defined and offers practical means to identify star models, contributing to the field of model ensemble methods. The paper includes thorough empirical validation across various architectures and datasets, strengthening the claims made.
summary: The paper presents a conjecture regarding the nature of solution sets for deep neural networks (DNNs) trained via stochastic gradient descent (SGD). It introduces the concept of a "star domain," proposing that these solution sets can be characterized by a special "star model" that connects to all other solutions in a low-loss path, modulo permutation invariances. The paper proposes the ``Starlight" algorithm that finds a star model of a given learning task and confirms existing reports that the convexity conjecture requires very wide networks to hold based on experimental results.
weaknesses: While the empirical results are promising, the theoretical underpinnings of the star domain conjecture could be elaborated further. The authors acknowledge challenges in validating the conjecture theoretically but could provide more insights into potential avenues for future theoretical exploration.

The experiment results primarily focus on specific settings (e.g., ResNet, CIFAR-10). Broader validation across different tasks and architectures could enhance the generalizability of the findings.

The algorithm's complexity may pose challenges for reproducibility. Detailed discussions on implementation and computational overhead would be beneficial.

Meta-review: This paper proposes and studies the “star domain conjecture,” which considers the shape of the global solutions of neural networks (up to permutations). The authors conjecture that there exist certain “star points” in the solution set such that the line segment between any element in the solution set and a star point entirely lies in the solution set. The conjecture is a stronger one than mode connectivity and a weaker one than the convexity (up to permutations) conjecture. The authors propose an algorithm named Starlight, aimed at finding a star solution given many “source solutions.” Using the algorithm, the paper verifies the conjecture by extensive experiments on different datasets, models, optimizers, etc. Lastly, the paper draws connections to Bayesian model averaging and model fusion, suggesting that the star solution can potentially be useful as a “summary model” of multiple different models.

The paper presents a bold and interesting conjecture and offers empirical/theoretical support of the claim. Numerical evaluations are quite thorough, in support of the proposed conjecture. The new conjecture sheds light on our understanding of the neural loss landscape.

The main weakness of the paper, pointed out by the reviewers, is that the paper does not provide a good insight or theory on why this star domain conjecture should be true in neural networks. The paper does have a theorem on a 2-layer linear neural network with univariate inputs, but the theorem does not quantify the effect of finite width. However, given the complex nature of the nonconvex training losses, it is deemed that the lack of formal theoretical explanations does not significantly harm the merit of the paper.

Overall, I believe that the paper can contribute to a deeper understanding of neural network training. I hence recommend acceptance.

Decision: Accept (Poster)