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The constant-error curves connecting two given networks discovered by the proposed method are not unique. We trained two different polygonal chains with the same endpoints and different random seeds using VGG-16 on CIFAR-10. We then measured the Euclidean distance between the turning points of these curves. For VGG-16 ... |
Although high accuracy connecting curves can often be very simple, such as a polygonal chain with only one bend, we note that line segments directly connecting two modes generally incur high error. For VGG-16 on CIFAR-10 the test error goes up to [MATH] in the center of the segment. For ResNet-158 and Wide ResNet-28-10... |
In this paper we focus on connecting pairs of networks trained using the same hyper-parameters, but from different random initializations. Building upon our work, Gotmare et al. have recently shown that our mode connectivity approach applies to pairs of networks trained with different batch sizes, optimizers, data augm... |
To motivate the ensembling procedure proposed in the next section, we now examine how far we need to move along a connecting curve to find a point that produces substantially different, but still useful, predictions. Let [MATH] and [MATH] be two distinct sets of weights corresponding to optima obtained by independently... |
and for [MATH] the error of an ensemble is already as low as the error of an ensemble of the two independently trained networks used as the endpoints of the curve. Thus even by moving away from the endpoint by a relatively small distance along the curve we can find a network that produces meaningfully different predict... |
Fast Geometric Ensembling In this section, we introduce a practical ensembling procedure, Fast Geometric Ensembling (FGE), motivated by our observations about mode connectivity. |
In the previous section, we considered ensembling along mode connecting curves. Suppose now we instead only have one set of weights [MATH] |
corresponding to a mode of the loss. We cannot explicitly construct a path [MATH] as before, but we know that multiple paths passing through [MATH] exist, and it is thus possible to move away from [MATH] in the weight space without increasing the loss. Further, we know that we can find diverse networks providing meanin... |
Inspired by these observations, we propose the Fast Geometric Ensembling (FGE) method that aims to find diverse networks with relatively small steps in the weight space, without leaving a region that corresponds to low test error. |
While inspired by mode connectivity, FGE does not rely on explicitly finding a connecting curve, and thus does not require pre-trained endpoints, and so can be trained in the time required to train a single network. |
Let us describe Fast Geometric Ensembling. First, we initialize a copy of the network with weights [MATH] set equal to the weights of the trained network [MATH] . Now, to force [MATH] to move away from |
[MATH] without substantially decreasing the prediction accuracy we adopt a cyclical learning rate schedule [MATH] (see Figure , left), with the learning rate at iteration [MATH] defined as |
[EQUATION] where [MATH] , the learning rates are [MATH] , and the number of iterations in one cycle is given by even number [MATH] . Here by iteration we mean processing one mini-batch of data. We can train the network [MATH] using the standard [MATH] -regularized cross-entropy loss function (or any other loss that can... |
Figure (left) illustrates the adopted learning rate schedule. During the periods when the learning rate is large (close to [MATH] ), [MATH] is exploring the weight space doing larger steps but sacrificing the test error. When the learning rate is small (close to [MATH] ), [MATH] is in the exploitation phase in which th... |
To find a good initialization [MATH] for the proposed procedure, we first train the network with the standard learning rate schedule (the schedule used to train single DNN models) for about [MATH] of the time required to train a single model. After this pre-training is finished we initialize FGE with [MATH] and run the... |
[MATH] , and then ensemble all of the models gathered across these runs. Cyclical learning rates have also recently been considered in Smith and Topin and Huang et al. Our proposed method is perhaps most closely related to Snapshot Ensembles |
, but has several distinctive features, inspired by our geometric insights. In particular, Snapshot Ensembles adopt cyclical learning rates with cycle length on the scale of [MATH] to [MATH] epochs from the beginning of the training as they are trying to do large steps in the weight space. However, according to our ana... |
[MATH] to [MATH] epochs in the last stage of the training. As illustrated in Figure (left), the step sizes made by FGE between saving two models (that is the euclidean distance between sets of weights of corresponding models in the weight space) are on the scale of [MATH] for Preactivation-ResNet-164 on CIFAR-100. For ... |
as opposed to the cosine schedule in Snapshot Ensembles. Fast Geometric Ensembling Experiments In this section we compare the proposed Fast Geometric Ensembling ( FGE ) technique against ensembles of independently trained networks ( Ind ), and SnapShot Ensembles ( SSE |
a recent state-of-the-art fast ensembling approach. For the ensembling experiments we use a [MATH] -layer Preactivation-ResNet in addition to the VGG-16 and Wide ResNet-28-10 models. Links for implementations to these models can be found in the supplement. |
We compare the accuracy of each method as a function of computational budget. For each network architecture and dataset we denote the number of epochs required to train a single model as [MATH] For a [MATH] budget, we run each of Ind, FGE and SSE |
[MATH] times from random initializations and ensemble the models gathered from the [MATH] runs. In our experiments we set [MATH] for VGG-16 and Wide ResNet-28-10 (WRN-28-10) models, and |
[MATH] for ResNet-164, since [MATH] epochs is typically sufficient to train this model. We note the runtime per epoch for FGE, SSE, and Ind is the same, and so the total computation associated with [MATH] budgets is the same for all ensembling approaches. |
For Ind, we use an initial learning rate of 0.1 for ResNet and Wide ResNet, and 0.05 for VGG. For FGE, with VGG we use cycle length [MATH] epochs, and a total of [MATH] |
models in the final ensemble. With ResNet and Wide ResNet we use [MATH] epochs, and the total number of models in the final ensemble is [MATH] for Wide ResNets and [MATH] for ResNets. For VGG we set the learning rates to [MATH] [MATH] ; for ResNet and Wide ResNet models we set [MATH] |
[MATH] . . For SSE, we followed Huang et al. and varied the initial learning rate [MATH] and number of snapshots per run [MATH] We report the best results we achieved, which corresponded to |
[MATH] for ResNet, [MATH] for Wide ResNet, and [MATH] for VGG. The total number of models in the FGE ensemble is constrained by network choice and computational budget. Further experimental details are in the supplement. |
Table summarizes the results of the experiments. In all conducted experiments FGE outperforms SSE, particularly as we increase the computational budget. The performance improvement against Ind is most noticeable for CIFAR-100. With a large number of classes, any two models are less likely to make the same predictions. ... |
Figure (right) illustrates the results for Preactivation-ResNet-164 on CIFAR-100 for one and two training budgets. The training budget [MATH] is [MATH] epochs. Snapshot Ensembles use a cyclical learning rate from the beginning of the training and they gather the models for the ensemble throughout training. To find a go... |
Diversity of predictions of the individual networks is crucial for the ensembling performance [e.g., 19 We note that the diversity of the networks averaged by FGE is lower than that of completely independently trained networks. Specifically, two independently trained ResNet-164 on CIFAR-100 make different predictions o... |
on CIFAR100 for ResNet-164). However, for a given computational budget FGE can propose many more high-performing networks than independent training, leading to better ensembling performance (see Table ). |
6.1 ImageNet ImageNet ILSVRC-2012 is a large-scale dataset containing [MATH] million training images and [MATH] validation images divided into [MATH] classes. |
CIFAR-100 is the primary focus of our ensemble experiments. However, we also include ImageNet results for the proposed FGE procedure, using a ResNet-50 architecture. We used a pretrained model with top- [MATH] test error of [MATH] to initialize the FGE procedure. We then ran FGE for [MATH] epochs with a cycle length of... |
[MATH] . The top- [MATH] test error-rate of the final ensemble was [MATH] . Thus, in just [MATH] epochs we could improve the accuracy of the model by [MATH] using FGE. The final ensemble contains [MATH] models (including the pretrained one). Despite the harder setting of only [MATH] epochs to construct an ensemble, FGE... |
Huang et al. on ImageNet, [MATH] error, which was also achieved using a ResNet-50. Discussion and Future Work We have shown that the optima of deep neural networks are connected by simple pathways, such as a polygonal chain with a single bend, with near constant accuracy. We introduced a training procedure to find thes... |
There are so many exciting future directions for this research. At a high level we have shown that even though the loss surfaces of deep neural networks are very complex, there is relatively simple structure connecting different optima. Indeed, we can now move towards thinking about valleys of low loss, rather than iso... |
These valleys could inspire new directions for approximate Bayesian inference, such as stochastic MCMC approaches which could now jump along these bridges between modes, rather than getting stuck exploring a single mode. One could similarly derive new proposal distributions for variational inference, exploiting the fla... |
Acknowledgements. Timur Garipov was supported by Ministry of Education and Science of the Russian Federation (grant 14.756.31.0001). Timur Garipov and Dmitrii Podoprikhin were supported by Samsung Research, Samsung Electronics. Andrew Gordon Wilson and Pavel Izmailov were supported by Facebook Research and NSF IIS-1563... |
Appendix A Supplementary Material We organize the supplementary material as follows. Section A.1 discusses the computational complexity of the proposed curve finding method. Section A.2 describes how to apply batch normalization at test time to points on curves connecting pairs of local optima. Section A.3 provides for... |
discusses a trivial construction of curves connecting two modes, where points on the curve represent reparameterization of the endpoints, unlike the curves in the main text. Section A.9 provides details of experiments on FGE. Finally, Section A.10 describes pathways traversed by FGE. |
A.1 Computational complexity of curve finding The forward pass of the proposed method consists of two steps: computing the point |
[MATH] and then passing a mini-batch of data through the DNN corresponding to this point. Similarly, the backward pass consists of first computing the gradient of the loss with respect to [MATH] , and then multiplying the result by the Jacobian [MATH] . The second step of the forward pass and the first step of the back... |
In our experiments we use curve parametrizations of a specific form. The general formula for a curve with one bend is given by [EQUATION] |
Here the parameters of the curve are given by [MATH] and coefficients [MATH] For this family of curves the computational complexity of the first step of the method is [MATH] , as we only need to compute a weighted sum of [MATH] |
[MATH] and [MATH] . The Jacobian matrix [EQUATION] thus the additional computational complexity of the backward pass is also [MATH] , as we only need to multiply the gradient with respect to |
[MATH] by a scalar. Thus, the total additional computational complexity is [MATH] . In practice we observe that the gap in time-complexity between one epoch of training a single model and one epoch of the proposed method with the same network architecture is usually below [MATH] |
A.2 Batch Normalization Batch normalization ( Ioffe and Szegedy ) is essential to modern deep learning architectures. Batch normalization re-parametrizes the output of each layer as |
[EQUATION] where [MATH] and [MATH] are the mean and standard deviation of the output [MATH] [MATH] is a constant for numerical stability and [MATH] and |
[MATH] are free parameters. During training, [MATH] and [MATH] are computed separately for each mini-batch and at test time statistics aggregated during training are used. |
When connecting two DNNs that use batch normalization, along a curve [MATH] , we compute [MATH] and [MATH] for any given [MATH] over mini-batches during training, as usual. In order to apply batch-normalization to a network on the curve at the test stage we compute these statistics with one additional pass over the dat... |
A.3 Formulas for curves with [MATH] bends For [MATH] bends [MATH] , the parametrization of a polygonal chain connecting points [MATH] is given by |
[EQUATION] for [MATH] and [MATH] For [MATH] bends [MATH] , the parametrization of a Bezier curve connecting points [MATH] and [MATH] is given by |
[EQUATION] A.4 Curve Finding Experiments All experiments on curve finding were conducted with TensorFlow ( Abadi et al. and as baseline models we used the following implementations: |
ResNet-bottleneck- [MATH] and Wide ResNet-28-10 ( ); ResNet-158 ( ); A reimplementation of VGG-16 without batch-normalization from ( |
); Table summarizes the results of the curve finding experiments with all datasets and architectures. For each of the models we report the properties of loss and the error on the train and test datasets. For each of these metrics we report 3 values: “Max” is the maximum values of the metric along the curve, “Int” is a ... |
[MATH] where [MATH] metric [MATH] represents the train loss or the error on the train or test dataset and “Min” is the minimum value of the error on the curve. “Int” represents a mean over a uniform distribution on the curve, and for the train loss it coincides with the loss ( [MATH] in the paper. We use an equally-spa... |
We also examined the quantity “Mean” defined as [MATH] , which coincides with the loss [MATH] from the paper, but in all our experiments it is nearly equal to “Int”. |
Besides convolutional and fully-connected architectures we also apply our approach to RNN architecture on next word prediction task, PTB dataset Marcus et al. ). As a base model we used the implementation available at |
As the main loss we consider perplexity. The results are presented in Table A.5 Train loss and test accuracy surfaces In this section we provide additional visualizations. Fig. and Fig. |
show visualizations of the train loss and test accuracy for ResNet- [MATH] on CIFAR-100 and VGG- [MATH] on CIFAR-10. A.6 Curve Ensembling |
Here we explore ensembles constructed from points sampled from these high accuracy curves. In particular, we train a polygonal chain with one bend connecting two independently trained ResNet-164 networks on CIFAR-100 and construct an ensemble of networks corresponding to [MATH] points placed on an equally-spaced grid o... |
Furthermore, we can improve the ensemble on the chain without adding additional parameters or computational expense, by accounting for the pattern of increased training and test loss towards the centres of the linear paths shown in Figure While the training and test accuracy are relatively constant, the pattern of loss... |
, which is inversely proportional to the loss. Figure , bottom right, illustrates the test loss of ResNet-164 on CIFAR-100 before and after temperature scaling. After rescaling the predictions of the networks, the test loss along the curve decreases and flattens. Further, the test error-rate of the ensemble constructed... |
[MATH] independently trained networks. However, directly ensembling on the curves requires manual intervention for temperature scaling, and an additional pass over the training data for each of the networks ( [MATH] in this case) at test time to perform batch normalization as described in section A.2 . Moreover, we als... |
A.7 The Effects of Increasing Parametrization One possible factor that influences the connectedness of a local minima set is the overparameterization of neural networks. In this section, we investigate the relation between the observed connectedness of the local optima and the number of parameters (weights) in the neur... |
For each value of [MATH] , Figure shows the worst training loss along the curve, maximum of losses of the endpoints, and the ratio of the length of the curve and the line segment connecting the two modes. Increasing the number of parameters we are able to reduce the difference between the worst value of the loss along ... |
A.8 Trivial connecting curves For convolutional networks with ReLU activations and without batch normalization we can construct a path connecting two points in weight space such that the accuracy of each point on the curve (excluding the origin of the weight space) is at least as good as the minimum of the accuracies o... |
Let [MATH] and [MATH] be two sets of weights. This path of interest consists of two parts. The first part connects the point [MATH] with [MATH] and the second one connects the point [MATH] with [MATH] We describe only the first part [MATH] of the path, such that |
[MATH] , as the second part is completely analogous. Let the weights of the network [MATH] be [MATH] where [MATH] are the weights and biases of the [MATH] -th layer, and [MATH] is the total number of layers. Throughout the derivation we consider the inputs of the network fixed. The output of the [MATH] -th layer [MATH]... |
and [MATH] It is easy to see that logits of the network with weights [MATH] are equal to [MATH] for all [MATH] Note that the predicted labels corresponding to the logits [MATH] and |
[MATH] are the same, so the accuracy of all networks corresponding to [MATH] is the same. Algorithm 1 Fast Geometric Ensembling 0: weights [MATH] , LR bounds [MATH] ,cycle length [MATH] (even), number of iterations [MATH] |
0: ensemble [MATH] {Initialize weight with [MATH] [MATH] for [MATH] do [MATH] {Calculate LR for the iteration} [MATH] {Stochastic gradient update} |
if [MATH] then [MATH] {Collect weights} end if end for A.9 Fast geometric ensembling experiments Alg. provides an outline of the algorithm. As baseline models we used the following implementations: |
VGG-16 ( ); Preactivation-ResNet-164 ( ); ResNet-50 ImageNet ( ); Wide ResNet-28-10 ( ); For the FGE (Fast Geometric Ensembling) strategy on ResNet we run the FGE routine summarized in Alg. [MATH] after epoch [MATH] of the usual (same as Ind) training for [MATH] epochs. The total training time is thus [MATH] epochs. Fo... |
A.10 Polygonal chain connecting FGE proposals In order to better understand the trajectories followed by FGE we construct a polygonal chain connecting the points that FGE ensembles. Suppose we run FGE for [MATH] learning rate cycles obtaining [MATH] points [MATH] |
in the weight space that correspond to the lowest values of the learning rate. We then consider the polygonal chain consisting of the line segments connecting |
[MATH] to [MATH] for [MATH] . We plot test accuracy and train error along this polygonal chain in Figure . We observe that along this curve both train loss and test error remain low, agreeing with our intuition that FGE follows the paths of low loss and error. Surprisingly, we find that the points on the line segments ... |
# Source: arxiv 1804.01144 # Title: Self-Organization and Artificial Life: A Review # Sections: all # Downloaded: 2026-03-03T01:58:37.596528+00:00 |
Self-Organization and Artificial Life: A Review Abstract Self-organization has been an important concept within a number of disciplines, which Artificial Life (ALife) also has heavily utilized since its inception. The term and its implications, however, are often confusing or misinterpreted. In this work, we provide a ... |
What is self-organization? The term “self-organizing system” was coined by Ashby, ( 1947 to describe phenomena where local interactions between independent elements lead to global behaviors or patterns. The phrase is used when an external observer perceives a pattern in a system with many components, and this pattern i... |
A formal definition of the term runs into difficulties in agreeing on what is a system , what is organization , and what is self |
(Gershenson and Heylighen,, 2003 , none of which are perfectly straightforward. However, a pragmatic approach focuses on when it is useful to describe a system as self-organizing (Gershenson,, 2007 . This utility typically comes when an observer identifies a pattern at a higher scale but is also interested in phenomena... |
Self-organization has been an important concept within a number of disciplines, such as statistical mechanics (Wolfram,, 1983 ; Crutchfield,, 2012 , supramolecular chemistry (Lehn,, 2017 , and computer science (Mamei et al.,, 2006 . Artificial Life (ALife) frequently draws heavily on self-organizing systems in differen... |
Usage Ashby coined the term “self-organizing system” to show that a machine could be strictly deterministic and yet exhibit a self-induced change of organization (Ashby,, 1947 . This notion was further developed within cybernetics (von Foerster,, 1960 ; Ashby,, 1962 . In many contexts, a thermodynamical perspective has... |
The recent subfield of guided self-organization explores mechanisms by which self-organization can be regulated for specific purposes—that is, how to find or design dynamics for a system such that it will have particular attractors or outcomes (Prokopenko,, 2009 ; Ay et al.,, 2012 ; Polani et al.,, 2013 ; Prokopenko,, ... |
While there may be no single agreed-on definition of self-organization, this lack need not be an insurmountable obstacle for its study, any more than a lack of a unanimous formal definition of “life” has been an obstacle for progress in the fields of biology or ALife. In what follows, we provide a concice review of how... |
Domains One way to classify ALife research is to divide it into soft hard , and wet domains, roughly referring to computer simulations, physical robots, and chemical/biological research (including living technology as the application of ALife (Bedau et al.,, 2009 ), respectively. Self-organization has played a central ... |
Soft ALife Soft ALife, or mathematical and computational modeling and simulation of life-like behaviors, has been linked to self-organization in many sub-domains. Cellular automata (CAs) (Ilachinski,, 2001 , one of the most popular modeling frameworks used in earlier forms of soft ALife, are well-explored, illustrative... |
(Langton,, 1984 1986 ; Reggia et al.,, 1993 ; Sipper,, 1998 , and evolution by variation and natural selection (Sayama,, 1999 2004 ; Salzberg and Sayama,, 2004 ; Suzuki and Ikegami,, 2006 ; Oros and Nehaniv,, 2007 2009 . Similarly, partial differential equations (PDEs), a continuous counterpart of CAs, have an even lon... |
Another representative class of soft ALife that shows self-organization comprises models of collective behavior of self-driven agents (Vicsek and Zafeiris,, 2012 . Reynolds’ Boids model (Reynolds,, 1987 is probably the best known in this category. In this work, self-propelled agents (“boids”) move in a continuous space... |
Such collective behavior models have been brought to artificial chemistry studies (Dittrich et al.,, 2001 ; Banzhaf and Yamamoto,, 2015 as well, such as swarm chemistry and its variants (Sayama,, 2008 ; Kreyssig and Dittrich,, 2011 ; Sayama,, 2011 2012 ; Erskine and Herrmann,, 2015 , in which kinetically and chemically... |
(Doursat,, 2011 ; Doursat et al.,, 2012 , in which researchers attempt to achieve a successful merger of self-organization and programmable architectural design, by discovering or designing agent rules that result in specific desired high-level patterns. |
Other examples of self-organization in soft ALife are found in simulation models of artificial societies. Their roots can be traced back to the famous segregation models developed by Sakoda and Schelling back in the early 1970s (Sakoda,, 1971 ; Schelling,, 1971 ; Hegselmann,, 2017 , in which simple, independent decisio... |
Hard ALife Robots can be considered to be life-like artifacts in their ability to sense their physical environment and take action in response. Physical agents, even very simple ones, can evoke in the observer a particularly strong sense of being animate. From W. Grey Walter’s tortoises (Walter,, 1950 1951 , to simple ... |
Physical hardware has the strong advantage that the physical characteristics of the system (dynamics, sensor performance, actuator noise profiles, etc.) are by definition realistic, whereas simulations are necessarily simplified and typically fail to capture phenomena that only become evident through material experimen... |
Physical experiments have been used to explore self-organizing phenomena in a variety of areas. Aggregation of objects has been studied from a physics perspective (Giomi et al.,, 2013 ; in ways inspired by behavior observed in living systems, such as cockroaches or bees (Garnier et al.,, 2008 ; Kernbach et al.,, 2009 ;... |
Wet ALife Wet ALife, or physico-chemical synthesis of life-like behaviors, extensively utilizes self-organization as its core principle. A classic example is the spatial pattern formation in experimentally realized reaction-diffusion systems, such as the Belousov-Zhabotinsky reaction (Vanag and Epstein,, 2001 ; Adamatz... |
More recently, dynamic behaviors of macroscopically visible chemical droplets, a.k.a. liquid robots (Čejková et al.,, 2017 , has become a focus of active research in ALife. In this line of research, interactions among chemical reactions, physical micro-fluid dynamics and possibly other not-yet-fully-understood microsco... |
Perspectives As already mentioned above, we can understand a self-organizing system as one in which organization increases in time. However, it can be shown that, depending on how the variables of a system are chosen, the same system can be said to be either organizing or disorganizing (Gershenson and Heylighen,, 2003 ... |
Looking at the perspectives of ALife, it can be useful to think of self-organization as the common language that unifies the soft, hard and wet domains. The ALife community can progress owing to shared concepts and definitions, and despite the mentioned difficulties, self-organization stands as a common ground on which... |
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