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Based on the conclusion mentioned above, the proximity distances of each individual with different ranks in the population are calculated. For convenience, it is assumed that there are [MATH] solutions in [MATH] , and [MATH] individuals in the current population. Consequently, there will be [MATH] proximity distances. ... |
[EQUATION] For an intuitive understanding, an example is illustrated in Fig. to present the motivation of the proximity distance assignment for individuals who are ranked as [MATH] . In Fig. , the black solid circles refer to the reference points, the black triangles marked by [MATH] [MATH] , and [MATH] refer to the in... |
Noted here that, the proximity distance assignment for the individuals with rank [MATH] is also employed in the IGD indicator In summary, smaller proximity distance reveals the better proximity when the exact PF of the problem to be optimized is unknown. Furthermore, the algorithm of the proximity distance assignment i... |
Input: Current population [MATH] with size [MATH] ; reference points [MATH] with size [MATH] Output: Proximity distances matrix [MATH] |
for [MATH] to [MATH] do [MATH] [MATH] [MATH] Calculate the rank of [MATH] for [MATH] to [MATH] do if [MATH] then [MATH] [MATH] else if |
[MATH] then [MATH] [MATH] 10 11 else 12 [MATH] [MATH] 13 14 end if 15 16 end for 17 18 end for Return [MATH] Algorithm 4 Assign Proximity Distance |
III-D Generating Offspring The process of generating offspring in the proposed algorithm is similar to that in genetic algorithms, in addition to the selection of individuals for filling up the gene pool from which the parent solutions are selected to generate offspring. In this subsection, the processes of generating ... |
Step 1: Select solutions from the current population to fill up the gene pool, until it is full. Step 2: Select two parent solutions from the gene pool and remove them from the gene pool. |
Step 3: Employ the Simulated binary crossover (SBX) operator to generate offspring with the selected parent solutions. Step 4: Employ the polynomial mutation operator to mutate the generated offspring. |
Step 5: Repeat Steps until the gene pool is empty. Input: Current population [MATH] ; Gene pool size [MATH] Output: Gene pool [MATH] |
[MATH] [MATH] while the size of [MATH] is less than [MATH] do [MATH] Randomly select two individuals from [MATH] [MATH] Obtain the rank of [MATH] |
[MATH] Obtain the rank of [MATH] [MATH] Obtain the proximity distances of [MATH] [MATH] Obtain the proximity distances of [MATH] |
if [MATH] then [MATH] [MATH] 10 11 else if [MATH] then 12 [MATH] [MATH] 13 14 else 15 if [MATH] then 16 [MATH] [MATH] 17 18 else if |
[MATH] then 19 [MATH] [MATH] 20 21 else 22 [MATH] Randomly select one individual from [MATH] 23 [MATH] [MATH] 24 25 end if 26 27 |
end if 28 29 end while Return [MATH] Algorithm 5 Filling Up the Gene Pool The binary tournament selection approach is employed in Algorithm to select individuals from the current population to fill up the gene pool. In other words, the binary tournament selection |
approach is employed to select individuals from the current population. Specifically, two individuals which are denoted by [MATH] and [MATH] are randomly selected from the current population first (line ). Then, their ranks and the proximity distances are obtained (lines ). Next, the individual with smaller rank value ... |
and the polynomial mutation operators are employed for the corresponding crossover and mutation operations in the proposed algorithm. It has been reported that two solutions selected in a large search space is not necessary to generate promising offspring |
. Generally, two ways can be employed to solve this problem. One is the mating restriction method to limit the offspring to be . The other one is to use SBX with a large distribution index |
. In the proposed algorithm, the latter one is utilized due to its simplicity. III-E Environmental Selection When the offspring have been generated, the size of the current population is greater than that of the available slots. As a consequence, the environmental selection takes effects to select a set of representati... |
is employed to solve this LAP. Input: [MATH] [MATH] , and [MATH] ; Available slots size [MATH] Output: [MATH] [MATH] [MATH] [MATH] |
[MATH] while [MATH] do [MATH] [MATH] [MATH] [MATH] end while if [MATH] then [MATH] [MATH] 10 11 else 12 [MATH] Uniformly select [MATH] reference points from [MATH] |
13 [MATH] Select [MATH] individuals from [MATH] 14 [MATH] [MATH] 15 end if 16 Return [MATH] Algorithm 6 Environmental selection III-F |
Computational Complexity In this subsection, the computational complexity of the proposed algorithm is analyzed. For convenience, it is assumed that the problem to be optimized is with [MATH] objectives, [MATH] decision variables, [MATH] desired solutions for decision-makers, and the computational complexity is analyze... |
III-G Discussions Loss of selection pressure is a major issue for traditional MOEAs in effectively solving MaOPs because of the traditional domination comparisons between individuals giving a large proportion of non-dominated solutions. In the proposed algorithm, the dominance relation of all the individuals are compar... |
When the number of solutions to be selected is larger than the available slots, the representatives are chosen from a global view in the proposed algorithm. For convenience of understanding, it is first assumed that [MATH] representatives need to be selected from [MATH] solutions where [MATH] . Then the selection of [M... |
. If the individuals are selected by finding the individual who has the least distance to the reference points, the diversity is not necessarily guaranteed. |
IV Experiments To evaluate the performance of the proposed algorithm in solving MaOPs, a series of experiments is performed. Particularly, NSGA-III |
, MOEA/D , HypE , RVEA , and KnEA are selected as the state-of-the-art peer competitors. Although the IGD -EMOA can be viewed as the peer algorithm based on IGD indicator, it is merely capable of solving MaOPs with no more than [MATH] objectives. As a consequence, IGD -EMOA is excluded from the list of peer competitors... |
The remaining of this section is organized as follows. At first, the selected benchmark problems used in this experiment are introduced. Then, the chosen performance metric is given to measure the quality of the approximate Pareto-optimal solutions |
IV-A Benchmark Test Problems The widely used scalable test problems DTLZ1-DTLZ7 from the DTLZ benchmark test suite and WFG1-WFG9 from the WFG benchmark test suite |
are employed in our experiments. Specifically, each objective function in one given [MATH] -objective test problem of DTLZ has [MATH] decision variables, and [MATH] is set to be [MATH] for DTLZ1, [MATH] for DTLZ2-DTLZ6, and [MATH] for DTLZ7 problems. Moreover, each objective function of a given problem in WFG test suit... |
IV-B Performance Metric The widely used Hypervolume (HV) which simultaneously measures the convergence and diversity of the MaOEAs is selected as the performance metric in these experiments. Specifically, the reference points for the calculation of HV are set to be [MATH] for DTLZ1, [MATH] for DTLZ2-DTLZ6, and [MATH] f... |
is applied for the calculation when [MATH] , otherwise the exact approach proposed in is utilized . In our experiments, all the HV values are normalized to [MATH] by dividing the HV value of the origin with the corresponding reference point. Moreover, higher HV values indicate a better performance of the corresponding ... |
IV-C Parameter Settings In this subsection, the baseline parameter settings which are adopted by all the compared MaOEAs are declared first. Then the special parameter settings required by each MaOEA are provided. |
IV-C Number of Objectives Test problems with [MATH] [MATH] , and [MATH] objectives are considered in the experiments because the proposed algorithm aims specifically at effectively solving MaOPs. |
IV-C Number of Function Evaluations and Stop Criterion All compared algorithms are individually executed [MATH] independent times. The maximum number of function evaluations for each compared MaOEA in one independent run is set to be [MATH] [MATH] , and [MATH] for [MATH] -, [MATH] -, and [MATH] -objective, respectively... |
IV-C Statistical Approach Because of the heuristic characteristic of the peer evolutionary algorithms, all the results, which are measured by the performance metric over [MATH] independent runs for each competing algorithm, are statistically evaluated. In this experiment, the Mann-Whitney-Wilcoxon rank-sum test |
with a [MATH] significance level is employed for this purpose. IV-C Population Size In principle, the population size can be arbitrarily assigned. However, the population size of the proposed algorithm, NSGA-III, MOEA/D, and RVEA depends on the number of the associated reference points or reference vectors. For a fair ... |
and the configurations are listed in Table IV-C Genetic Operators The SBX and polynomial mutation are employed as the genetic operators. Moreover, the probabilities of the crossover and mutation are set to be [MATH] and [MATH] , respectively. The distribution indexes of mutation and crossover are set to be [MATH] , in ... |
In solving the proposed nadir point estimation method for constructing the Utopian PF, evolutionary algorithm is employed. To be specific, the SBX and polynomial mutation, both of whose distribution index are set to be [MATH] , and probabilities for crossover and mutation are set to be [MATH] and [MATH] , respectively,... |
IV-D Experimental Results and Analysis In this subsection, the results, which are II and III , respectively. Furthermore, the numbers with bold face imply the best mean values over the corresponding test problem with a given objective number (the second and third columns in Tables II and III ) against all compared algo... |
From the results measured by HV on DTLZ1-DTLZ7 test problems (Table II ), it is clearly shown that MaOEA/IGD achieves the best performance among its peer competitors upon [MATH] - and [MATH] -objective DTLZ1 and DTLZ7, while performs slightly worse upon [MATH] -objective DTLZ1 by KnEA and DTLZ7 by RVEA. Furthermore, Ma... |
The HV results from WFG1-WFG9 test problems III . For [MATH] -objective WFG test problems, MaOEA/IGD shows a better performance on WFG1, WFG2, WFG7, and WFG9 than its peer competitors, and performs a little worse than that of KnEA on WFG5, RVEA on WFG6, and NSGA-III on WFG8 test problems. Although MaOEA/IGD does not sh... |
Briefly, MaOEA/IGD wins 9 times out of the 12 comparisons upon the test problems whose PF shapes are linear (i.e., DTLZ1, DTLZ7, WFG1, and WFG3), which can be interpreted that the sampled reference points from the Utopian PF for the proposed algorithm are the Pareto-optimal solutions due to the linear feature of the PF... |
Theoretically, the major shortcoming of HV indicator against IGD is its much higher computational complexity. However, noted that from Tables II and III , the proposed algorithm, which is designed based on the IGD indicator, outperforms HypE, which is motivated by the HV indicator, upon all test problems with the selec... |
that the HV result is largely affected by the nadir points of the problem to be optimized. In HypE, the nadir points are determined as the evolution continues. In this way, the obtained nadir point would be inaccurate during the early evolution process (the reasons have been discussed in reviewing the nadir point estim... |
. In practice, that number is unknown and unavailable of such may lead to a poor performance. IV-E Investigation on Nadir Point Estimation |
In this subsection, we will investigate the performance of the proposed DNPE on estimating the nadir point. To be specific, two peer competitors including WC-NSGA-II and PCSEA which have been discussed in Section II are utilized to perform comparisons on selected test problems. In these comparisons, the numbers of func... |
[EQUATION] where [MATH] denotes the [MATH] -th element of the estimated nadir point derived from the extreme points [MATH] is met. The experimental results for DTLZ1, DTLZ2, and WFG2 with [MATH] -, [MATH] -, [MATH] -, and [MATH] -objective are plotted in Fig. . Please note that the reason of choosing these three test p... |
The results performed by compared nadir point estimation methods on [MATH] -, [MATH] -, [MATH] -, and [MATH] -objective DTLZ1, DTLZ2, and WFG2 are illustrated in Figs. 5a 5b , and 5c , respectively. It is clearly shown in Fig. 5a that these compared algorithms find the satisfactory nadir points of the DTLZ1 which is wi... |
Conclusion and Future Works In this paper, an IGD indicator-based evolutionary algorithm is proposed for solving many-objective optimization problems. In order to obtain a set of uniformly distributed reference points for the calculation of the IGD indicator, a decomposition-based nadir point estimation method is desig... |
Moreover, experiments are performed by the proposed decomposition-based nadir point estimation method against a couple of competitors over three representative test problems (DTLZ1, DTLZ2, and WFG2) with challenging features in PF shapes and objective value scales, the experimental results reveal the satisfactory resul... |
Yanan Sun (S’15-M’18) received a Ph.D. degree in engineering from the Sichuan University, Chengdu, China, in 2017. From 2015.08-2017.02, he is a jointly Ph.D. student financed by the China Scholarship Council in the School of Electrical and Computer Engineering, Oklahoma State University (OSU), USA. He is currently a P... |
Gary G. Yen (S’87-M’88-SM’97-F’09) received a Ph.D. degree in electrical and computer engineering from the University of Notre Dame in 1992. Currently he is a Regents Professor in the School of Electrical and Computer Engineering, Oklahoma State University (OSU). Before joined OSU in 1997, he was with the Structure Con... |
Dr. Yen was an associate editor of the IEEE Control Systems Magazine, IEEE Transactions on Control Systems Technology Automatica Mechantronics IEEE Transactions on Systems, Man and Cybernetics, Parts A and B and I EEE Transactions on Neural Networks . He is currently serving as an associate editor for the IEEE Transact... |
Zhang Yi (F’16) received a Ph.D. degree in mathematics from the Institute of Mathematics, The Chinese Academy of Science, Beijing, China, in 1994. Currently, he is a Professor at the Machine Intelligence Laboratory, College of Computer Science, Sichuan University, Chengdu, China. He is the co-author of three books: Con... |
# Source: arxiv 1802.10026 # Title: Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs # Sections: all # Downloaded: 2026-03-03T04:14:18.518203+00:00 |
Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs Abstract The loss functions of deep neural networks are complex and their geometric properties are not well understood. We show that the optima of these complex loss functions are in fact connected by simple curves over which training and test accuracy are n... |
Introduction The loss surfaces of deep neural networks (DNNs) are highly non-convex and can depend on millions of parameters. The geometric properties of these loss surfaces are not well understood. Even for simple networks, the number of local optima and saddle points is large and can grow exponentially in the number ... |
. Moreover, the loss is high along a line segment connecting two optima [e.g., 17 These two observations suggest that the local optima are isolated. |
In this paper, we provide a new training procedure which can in fact find paths of near-constant accuracy between the modes of large deep neural networks. Furthermore, we show that for a wide range of architectures we can find these paths in the form of a simple polygonal chain of two line segments. Consider, for examp... |
The left panel shows a plane defined by three independently trained networks. In this plane, all optima are isolated, which corresponds to the standard intuition. However, the middle and right panels show two different paths of near-constant loss between the modes in weight space, discovered by our proposed training pr... |
We believe that this geometric discovery has major implications for research into multilayer networks, including (1) improving the efficiency, reliability, and accuracy of training, (2) creating better ensembles, and (3) deriving more effective posterior approximation families in Bayesian deep learning. Indeed, in this... |
In particular, our contributions include: The discovery that the local optima for modern deep neural networks are connected by very simple curves, such as a polygonal chain with only one bend. |
A new method that finds such paths between two local optima, such that the train loss and test error remain low along these paths. |
Using the proposed method we demonstrate that such mode connectivity holds for a wide range of modern deep neural networks, on key benchmarks such as CIFAR-100. We show that these paths correspond to meaningfully different representations that can be efficiently ensembled for increased accuracy. |
Inspired by these observations, we propose Fast Geometric Ensembling (FGE), which outperforms the recent state-of-the-art Snapshot Ensembles |
, on CIFAR-10 and CIFAR-100, using powerful deep neural networks such as VGG-16, Wide ResNet-28-10, and ResNet-164. On ImageNet we achieve |
[MATH] top- [MATH] error-rate improvement for a pretrained ResNet-50 model by running FGE for only [MATH] epochs. We release the code for reproducing the results in this paper at |
The rest of the paper is organized as follows. Section discusses existing literature on DNN loss geometry and ensembling techniques. Section introduces the proposed method to find the curves with low train loss and test error between local optima, which we investigate empirically in Section . Section |
then introduces our proposed ensembling technique, FGE, which we empirically compare to the alternatives in Section Finally, in Section we discuss connections to other fields and directions for future work. |
Note that we interleave two sections where we make methodological proposals (Sections ), with two sections where we perform experiments (Sections ). Our key methodological proposal for ensembling, FGE, is in Section |
Related Work Despite the success of deep learning across many application domains, the loss surfaces of deep neural networks are not well understood. These loss surfaces are an active area of research, which falls into two distinct categories. |
The first category explores the local structure of minima found by SGD and its modifications. Researchers typically distinguish sharp and wide local minima, which are respectively found by using large and small mini-batch sizes during training. Hochreiter and Schmidhuber and Keskar et al. for example, claim that flat m... |
Li et al. proposed a new visualization method for the loss surface near the minima found by SGD. Applying the method for a variety of different architectures, they showed that the loss surfaces of modern residual networks are seemingly smoother than those of VGG-like models. |
The other major category of research considers global loss structure. One of the main questions in this area is how neural networks are able to overcome poor local optima. Choromanska et al. investigated the link between the loss function of a simple fully-connected network and the Hamiltonian of the spherical spin-gla... |
Lee et al. showed that under mild conditions gradient descent almost surely converges to a local minimizer and not a saddle point, starting from a random initialization. |
In recent work Freeman and Bruna theoretically show that local minima of a neural network with one hidden layer and ReLU activations can be connected with a curve along which the loss is upper-bounded by a constant that depends on the number of parameters of the network and the “smoothness of the data”. Their theoretic... |
By contrast, we propose a much simpler training procedure that can find near-constant accuracy polygonal chains with only one bend between optima, even on a range of modern state-of-the-art architectures. Inspired by properties of the loss function discovered by our procedure, we also propose a new state-of-the-art ens... |
Xie et al. proposed a related ensembling approach that gathers outputs of neural networks from different epochs at the end of training to stabilize final predictions. More recently, Huang et al. proposed |
snapshot ensembles , which use a cosine cyclical learning rate to save “snapshots” of the model during training at times when the learning rate achieves its minimum. In our experiments, we compare our geometrically inspired approach to Huang et al. , showing improved performance. |
Draxler et al. simultaneously and independently discovered the existence of curves connecting local optima in DNN loss landscapes. To find these curves they used a different approach inspired by the |
Nudged Elastic Band method from quantum chemistry. Finding Paths between Modes We describe a new method to minimize the training error along a path that connects two points in the space of DNN weights. Section 3.1 |
introduces this general procedure for arbitrary parametric curves, and Section 3.2 describes polygonal chains and Bezier curves as two example parametrizations of such curves. In the supplementary material, we discuss the computational complexity of the proposed approach and how to apply batch normalization at test tim... |
3.1 Connection Procedure Let [MATH] and [MATH] in [MATH] be two sets of weights corresponding to two neural networks independently trained by minimizing any user-specified loss [MATH] , such as the cross-entropy loss. Here, [MATH] is the number of weights of the DNN. Moreover, let |
[MATH] be a continuous piecewise smooth parametric curve, with parameters [MATH] , such that [MATH] To find a path of high accuracy between [MATH] and [MATH] , we propose to find the parameters |
[MATH] that minimize the expectation over a uniform distribution on the curve, [MATH] [EQUATION] where the distribution [MATH] on [MATH] is defined as: |
[MATH] The numerator of ( ) is the line integral of the loss [MATH] on the curve, and the denominator [MATH] is the normalizing constant of the uniform distribution on the curve defined by [MATH] Stochastic gradients of [MATH] in Eq. ( ) are generally intractable since [MATH] depends on [MATH] Therefore we also propose... |
[EQUATION] where [MATH] is the uniform distribution on [MATH] . The difference between ( ) and ( ) is that the latter is an expectation of the loss [MATH] with respect to a uniform distribution on [MATH] , while ( is an expectation with respect to a uniform distribution on the curve. The two losses coincide, for exampl... |
To minimize ( ), at each iteration we sample [MATH] from the uniform distribution [MATH] and make a gradient step for [MATH] with respect to the loss [MATH] . This way we obtain unbiased estimates of the gradients of [MATH] , as |
[EQUATION] We repeat these updates until convergence. 3.2 Example Parametrizations Polygonal chain The simplest parametric curve we consider is the polygonal chain (see Figure , right). The trained networks |
[MATH] and [MATH] serve as the endpoints of the chain and the bends of the chain are the parameters [MATH] of the curve parametrization. Consider the simplest case of a chain with one bend [MATH] . Then |
[EQUATION] Bezier curve A Bezier curve (see Figure , middle) provides a convenient parametrization of smooth paths with given endpoints. A quadratic Bezier curve [MATH] with endpoints [MATH] and [MATH] is given by |
[EQUATION] These formulas naturally generalize for [MATH] bends [MATH] (see supplement). Curve Finding Experiments We show that the proposed training procedure in Section |
does indeed find high accuracy paths connecting different modes, across a range of architectures and datasets. Moreover, we further investigate the properties of these curves, showing that they correspond to meaningfully different representations that can be ensembled for improved accuracy. We use these insights to pro... |
In particular, we test VGG- [MATH] , a [MATH] -layer Wide ResNet with widening factor [MATH] and a [MATH] -layer ResNet on CIFAR-10, and VGG- [MATH] [MATH] -layer ResNet-bottleneck |
on CIFAR-100. For CIFAR-10 and CIFAR-100 we use the same standard data augmentation as Huang et al. We provide additional results, including detailed experiments for fully connected and recurrent networks, in the supplement. |
For each model and dataset we train two networks with different random initializations to find two modes. Then we use the proposed algorithm of Section |
to find a path connecting these two modes in the weight space with a quadratic Bezier curve and a polygonal chain with one bend. We also connect the two modes with a line segment for comparison. In all experiments we optimize the loss ), as for Bezier curves the gradient of loss ) is intractable, and for polygonal chai... |
Figures and show the results of the proposed mode connecting procedure for ResNet-164 on CIFAR-100. Here loss refers to [MATH] -regularized cross-entropy loss. For both the Bezier curve and polygonal chain, train loss (Figure , left) and test error (Figure , middle) are indeed nearly constant. In addition, we provide p... |
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