Visualizing High-Dimensional Trajectories on the Loss-Landscape of ANNs
Authors: Stefan Horoi
Jessie Huang
Guy Wolf
Smita Krishnaswamy
Venue:  (2021)

Abstract: Training artificial neural networks requires the optimization of highly non-convex loss functions. Throughout the years, the scientific community has developed an extensive set of tools and architectures that render this optimization task tractable and a general intuition has been developed for choosing hyper parameters that help the models reach minima that generalize well to unseen data. However, for the most part, the difference in trainability in between architectures, tasks and even the gap in network generalization abilities still remain unexplained. Visualization tools have played a key role in uncovering key geometric characteristics of the loss-landscape of ANNs and how they impact trainability and generalization capabilities. However, most visualizations methods proposed so far have been relatively limited in their capabilities since they are of linear nature and only capture features in a limited number of dimensions. We propose the use of the modern dimensionality reduction method PHATE which represents the SOTA in terms of capturing both global and local structures of high-dimensional data. We apply this method to visualize the loss landscape during and after training. Our visualizations reveal differences in training trajectories and generalization capabilities when used to make comparisons between optimization methods, initializations, architectures, and datasets. Given this success we anticipate this method to be used in making informed choices about these aspects of neural networks.

Under review as a conference paper at ICLR 2021
VISUALIZING HIGH-DIMENSIONAL TRAJECTORIES ON THE LOSS LANSSCAN OF ANNS
Anonymous authors
Paper under double-blind review
ABSTRACT
Training artificial neural networks requires the optimization of highly non-convex loss functions. Throughout the years, the scientific community has developed an extensive set of tools and architectures that render this optimization task tractable and a general intuition has been developed for choosing hyper parameters that help the models reach minima that generalize well to unseen data. However, for the most part, the difference in trainability in between architectures, tasks and even the gap in network generalization abilities still remain unexplained. Visualization tools have played a key role in uncovering key geometric characteristics of the loss- landscape of ANNs and how they impact trainability and generalization capabilities. However, most visualizations methods proposed so far have been relatively limited in their capabilities since they are of linear nature and only capture features in a limited number of dimensions. We propose the use of the modern dimensionality reduction method PHATE which represents the SOTA in terms of capturing both global and local structures of high- high-dimensional data. We apply this method to visualize the loss landscape during and after training. Our visualizations reveal differences in training trajectories and generalization capabilities when used to make comparisons between optimization methods, initializations, architectures, and datasets. Given this success we anticipate this method to be used in making informed choices about these aspects of neural networks.
INTRODUCTION
Artificial neural networks (ANNs) have been successfully used to solve a number of complex tasks in diverse array of domains, such as object recognition, machine translation, image generation, 3D protein structure prediction and many more. Despite being highly overparameterized for the tasks they solve, and having the capacity to memorize the entire training data, ANNs tend to generalize to unseen data. This is a spectacular feat since the highly non-convex optimization typically encountered in them should (theoretically) be significant obstacle to using these models (Blum & Rivest, 1993). Questions such as why ANNs favor generalization over memorization and why they find good minima even with intricate loss functions still remain largely unanswered. One promising research direction for answering them is to look at the loss-landscape of deep learning models. Recent work tried to approach this task by proposing various visualization methods. An emerging challenge here is how to look at such an extremely high dimensional optimization landscape (linear in the number of parameters of the network) with respect minimized loss.
In past work, loss functions and their level lines were visualized via random directions starting at a minimum, or by means of linear methods like PCA. In some case, this approach proved effective in uncovering underlying structures in the loss-landscape and link them to network characteristics, such as generalization capabilities or structural features (Keskar et al.) 2016 Li et al., 2018). However, these methods have two major key drawbacks: (1) they are linear in that they only choose directions that are linear combinations of parameter axes while the loss landscape itself is highly nonlinear, and (2) they choose only two among thousands (if not millions) of axes to visualize and ignore all others.
In this work, we utilize and adapt the PHATE dimensionality reduction method (Moon et al., 2019), which relies on diffusion-basee manifold learning, to study ANN loss landscapes by visualizing the evolution of network weights during training in low dimensions. In general, visualizations like PHATE Moon et al. (2019) are specifically designed to squeeze much variability as possible into

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two dimensions, and thus provide advantage over previous approaches. In particular our choice of using PHATE over other popular methods, such as tSNE (van der Maaten & Hinton 2008). is due to its ability to capture global and local structures of data, and in particular to keep intact the training trajectories that are traversed through during gradient descent. Indeed, during training, the high-dimensional neural networks weights change significantly while remaining on connected manifold defined by the support of viable configurations (c.g.. with sufficiently low training loss), which we refer when discussing the geometry of the loss landscape. We show that PHATE is suitable to track such continuous weight trajectories, as opposed to tSNE or UMAP that tend to shatter them. Moreover, our approach provides general view of relevant geometric patterns that emerge in the high-dimensional parameter space, providing insights regarding the properties of ANN training and reflecting on their impact on the loss landscape.
Contributions: We propose a novel loss-landscape visualization based on a variation of PHATE. implemented with cosine distance in Section 4 Our method is, to our knowledge, different from all other proposed methods for loss visualization that it is naturally nonlinear and captures data characteristics from all dimensions. In Section 5.1 we show that our method uncovers key geometric patterns characterizing loss-landscape regions surrounding good and bad generalization optima, as well as memorization optima. Finally, we establish the robustness of our method by applying it to numerous tasks, architectures, and optimizers in Sections [.3 and suggesting that our method can be used in a consistent manner to validate training and design choices.
RELATED WORK
Loss landscape visualization methods have been proposed in numerous contexts. Goodfellow et al. (2014) proposed the "linear path experiment" where the loss of ANN is evaluated at series of points 0 = (1 a)) + o0, for different values of a E [o. 1] and 0;, f corresponding to the initial parameters of the model and the found optima in parameter space respectively. This oic-dimensional linear interpolation method has allowed them to show that popular state of the art ANNs typically do not encounter significant obstacles along a straight path from initialization to convergent solution. They also used the method to visualize the loss along directions connecting two distinct minima and to show that these are linearly separated by region of higher valued loss.
This method was further developed by Im et al. (2166), who adapted it to enable the visualization of wwo-dimensional projections of the lass-land using barycentric and bilinear interpolation for groups of three or four points in parameter space. This analysis method has allowed them to establish that despite starting with the same parameter initialization, different optimization algorithms find different minima. Furthermore, they noticed that the loss-landscape around minima have characteristic shapes that are optimizer-specific and that normaaiiomaliztionns smooths the loss function.
More recently, Li et al. (2018) have addressed the scale invariance and network symmetries problems discussed in Neyshabur et al. (2017); Dinh et al. (2017). which prevented mcaningful comparisons between land-sand plots from different networks. They proposed 1D) and 2D linear interpolation plots, similar to past techniques, but where they used filter-wise normalized directions to remove the scaling effect. This method has allowed them visualize and compare the regions on the loss-landscape surrounding minima coming from multiple networks in a meaningful way and to correlate the "flatness" of the region to the generalization capabilities of the corresponding network. Furthermore, they studied the effects of the network depth, width and the presence of skip connections on the geometry of the lass-land and on network generalization.
The importance of loss landscape visualization methods like the one presented in this paper increases with the growing scientific community interest in furthering our understanding of ANNs. As our understanding of this landscape gets deeper, we start uncovering more and more high-dimensional and complex geometric and topological characteristics. For instance, Draxler et al. (2018) have found nonlinear pathways in parameter space connecting distinct minima, along which the training and test errors remain small and losses are consistently low. This suggests that minima are not situated in isolated valleys but rather connected manifolds representing low loss regions. However, such characteristics are intrinsically high-dim high-dimensional, making linear methods inadequate to visualize these structures. Even in standard applications of the linear methods one inevitably asks if the thousands (or
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even millions) of unseen directions do not hide critical features of the landscape, and if the visualized linear path is relevant to what happened during training.
In this paper, suggest the use of modern dimensionality reduction techniques to study the complex loss-landscape of ANNs. Such techniques have been extensively used in recent years to successfully study the internal learned representations of deep networks and highlight their complex geometric structures and intrinsic dimensionality (Gigante et al. [219 Horoi et al.] 2020 Recanatesi et al.) 2018; Farrell al., 2019; Maheswaranathan al. 2019). and here further advance this line of work to provide new applications and insights in the study of ANN optimization and generalization.
3 PRELIMINARIES: PHATE DIMENSIONALITY REDUCTION & VISUALIZATION
Given a n x m data matrix N. where n is the number of data points and m the number of features, PHATE computes low-dimensional embeddings of the data points (dimension d to be specified by the user) which preserve both their global and their local structure. It first computes the pairwise distance matrix D using specified distance function Ø, each element of the matrix being given by Dij = (NN Nja). To better capture local affinities between data points, an affinity matrix A is computed using an a-decaying kernel with a addativ- bandwidth Ck corresponding to the k-NN distance of the i-th data point. The elements of A are given by:
1
(-(2)) Dan Enei
))
exp (-(2z) Ek.j
Aij KR (k,a(i.)) = exp
The decaying factor a regulates the decay rate of the kernel (smaller a = kernel with lighter tails), corresponding the Gaussian.
The affinity matrix is then row-normalized to obtain the diffusion operator P, row-stochastic Markov transition matrix with element Pt, giving the probability of jumping from the i-th to the j-th data point in one time step. One of the reasons PHATE excels at capturing global structures in data, especially high-dimensiinn
trajectories and branches, is that leverages the diffusion operator (also used, Coifman & Lafon 2006 to construct diffusion maps) by running the implicit Markov chain forward in time. This is accomplished by raising the matrix P to the power I, effectively taking t random walk steps and revealing intrinsic geometric structure through the affinity of the data at scale that grows with the number of steps. The optimal value for t is automatically chosen to be the knee point of the Neumann entropy of P.
To enable dimensionality reduction while retaining diffusion geometry information from the operator, PHATE leverages information geometry define a pairwise potential distance an M-divergence ID., = || log P... log Pj=:le between corresponding 1-step diffusion probability distributions of the points, which provides global context to each data point. The resulting information distance matrix ID is finally embedded into a visualizable low dimensional (2D or 3D) space by metric multidimensional scaling (MDS), thereby squeezing the intrinsic geometric information into two dimensions (as is also done in (SNE), rather than producing a high dimensional alternative coordinate system like PCA or diffusion maps.
We reason that since PHATE learns a data manifold via diffusion, and places points in the global context of other points, it can keep trajectories connected from beginning to end without shattering them. Indeed, this property was established quantitatively in Moon et al. (2019) via Denoised Manifold Affinity Preservation (DeMAP) metric. Here, it leads the preservation trajectory flow directions and relations between them, demonstrated in Figure in comparison with other methods, on trajectory-like data and ANN training trajectories, as well as in our results presented in the remainder of the paper.
4 METHODS
The loss-landscape is a high dimensional function f(O) that assigns a loss value every possible vector / in the high- digh-dimensional parameter space of an ANN denoted e. The dimensionality of the space renders the task of completely visualizing the loss-landscape virtually impossible. By gathering locally connected loss-landscape patches throughout training and around minima found via gradient descent, we hope to reconstruct manifolds on which these trajectories lie. Despite
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the thousands, if not millions, of dimensions of the parameter space Q hypothesize that these trajectories and manifolds are intrinsically of low dimensional and thus PHATE, which reduced dimensionality via manifold learning and information geometry. will help facilitate their visualization in two specific settings: 1. To characterize the region of the loss-landscape surrounding minima; and 2. To simultaneously visualize multiple training trajectories corresponding to different parameter initializations and optimizers in order to make training choices.
In setting (1) we use the following "jump and retrain" experiment. Given a trained ANN and minimum found during training with an optimizer:
Let 0, represent the vector of network parameters at the minimum;
For seed E (0, 3, 4]:
For step_sie E {0.25, 0.5, 0.75, 1}:
1. Choose the random vector Useed in the parameter space of the network O corre- sponding to the random seed seed;
2. Filter normalize the vector proposed in Li et al. (2018) to obtain Useedi
Set the network parameters at each jump initialization to be Ojump-init = 0, slep_size &ueeedi
4. Retrain for 50 epochs using optimizer with 0.9 momentum on the original training set;
5. Record network parameters 0 for each seed and step size combination and at the end of each epoch.
* Apply PHATE to the data matrix [Oseed,step step.tt..pesseppohh and visualize the embeddings.
In setting (2), we simultaneously visualize multiple training trajectories corresponding to different parameter initializations and optimizers. Here, we simply follow the optimizer from initialization to convergence, and record the parameter vector at each step.
We note that methods that pick random directions or surfaces around the minima projected to random directions are not able visualize entire trajectories as the spaces and planes change. This is distinct advantage of using all of the dimensions of the parameter space. In both settings, PHATE has allowed us to bypass the key drawbacks of previously proposed linear interpolation methods by: 1. Capturing variance in data from all relevant dimensions and embed it in a low-dimensions space; and 2. preserving high-dimensional trajectories and global structures of data in parameter space, All modern dimeeniionlity--rductiio techniques would have achieved (1) in one way or another and with various degrees of success. The reason our method is based on cosine-distance PHATE, is to better accomplishes (2) by preserving trajectories and global relationships in high-dimensional data to a greater extent than any other oftte--f-thh-ane art dimensionality-reduee reduucton technique. Figure demonstrates this by showing comparison of multiple such techniques, namely PHATE, PCA, t-SNE (van der Maaten & Hinton 2008) and UMAP (McInnes & Healy] 2018). and how they each embed the results of jump and retrain experiment (Figure IA and B) and an artificial data set having a tree-like structure (Figure [C) in 2D space.
While some trajectory- structure is visible in all low-dimensional embeddings, only PHATE properly captures intra-trajectory variance. PHATE is also the only technique that captures the global relationships in between trajectories while t-SNE and UMAP have a tendency to cluster points that are close in parameter space and disregard the global structure of the data. On the artificial data set, what we observe that the embeddings of the linear method PCA are highly affected by the noise in the data while t-SNE and UMAP have tendency of shattering trajectories that should be connected.
5 CASE STUDIES AND APPLICATIONS
5.1 GENERALIZATION VS. MEMORIZATION
Making sure that machine learning models can generalize new data is one of the most important steps in solving any data analysis task, and is core problem in the field. It is so important that oftentimes large fractions of the scarcely available data are not used for training but rather for testing, to ensure that models indeed learn to solve the task in relevant manner without overfitting their training data, Arguably one of the most important and general questions in machine learning at the moment is: by looking only given trained model and its training data, can we predict how well the model will perform on unseen data? There is even an entire NeurIPS 2020 competition entirely focused on answering this question (Tiang al. 2020
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Figure 1: 2D embeddings of the jump and retrain experiment results for a WResNer28-2 network using the modern dimensionality reduction techniques PHATE (1), t-SNE (2), UMAP (3) using euclidean (A) and cosine (B) distances and the linear methods PCA (A4) and random plane projection (B4). C: embeddings found using the same techniques of artificial data set having fully connected tree- like structure. We see that PHATE consistently retains continuous trajectory structures while other embeddings (ISNE/UMAP) shatter the structure, show chaotic trends because of uninformative projections to low dimensions (PCA, random directions). See Appendix A for higher resolution.
Previous work offers partial and vague answer to this question by linking the "flatness" of neural network optima to the extent to which the network is able to generalize. This idea was initially presented in Hochreiter & Schmidhuber (1997) and has recently been the focus of numerous papers (e.g.. Chaudhari et al. 2016 Keskar et al. 2016 Swirszez et al.) 2016 Dinh et al.] 2017; Li et al.) 2018). Flat minima in parameter space, around which the value of the loss function is fairly constant, are believed to correspond to configurations where the network generalizes well to the unseen data. Sharp minima, on the other hand, correspond to ones where the network is expected to generalize poorly. Following the underlying idea of these propositions, we believe that studying the geometry of the loss landscape around minima has the potential of uncovering key characteristics about the network's i ability to generalize. Here, we show that jump-and-retrann experiments combined with our visualization method reveal stark differences between neural network configurations that memorize (or overfit) versus ones that generalize.
For our empirical results we trained multiple WResNet28-2 networks (wide ResNet of depth 28 and 2 times as wide standard ResNet (He et al.) 2016)) presented in ZZaoruyko & Komodakis (2016) on the CIFARIO 10 dataset (Krizhevsky et al. 2009) using batches of size 128 for 200 epochs with the SGD optimizer with 0.9 momentum and learning rate of 0.1 which decays by factor of 10 at epoch 150. In some cases we did not use data augmentation and weight decay and in others we used le-4 weight decay and random crops and horizontal flips for the training data. Adding the data augmentation and the weight decay has allowed us to find optima that generalize better, increasing the accuracy on the test data from around 85% to around 95%. We will be referring the optima with -95% test accuracy "good optima" since they generalize better, and that with lower test accuracy as "bad optima". Using the same training procedure as the one for the bad optima, we also trained WResNer28-2 network for memorization by completely randomizing the labels of the training data. This modification was used Zhang et al. (017): Arpit et al. (2017): KKueger et al. (2017) to show that neural networks have the ability to completely memorize the training data sets. We then the jump-and- -nt--rrrinnrr experiment for these three types minima and visualized the results using PHATE in Figure 2 This allows to analyze and compare the regions in parameter space around the good, bad and memorization minima, which all achieve -0 loss on their respective training sets.
Despite only -7% difference in their accuracy the test data set, the PHATE embedding of the trajectories surrounding each generalization minimum is significantly different as seen in Fig. 2l. The minimum that shows good generalization has distinctive star flower shaped pattern. This
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Figure 2: 2D PHATE embeddings of the jump and retrain experiment conducted on good (a,d) and one bad (b,e) generalization minima that achieve 94.39% and 87.29% accuracy on the test data respectively, as well one memorization minimum (c,f). The initial points of the jumps (before retraining) are marked by diamonds of different colors based on slep_size in all plots. All trajectories are colored by their respective direction of the initial jump. Plots a,b,e have ascending huc, meaning the color gets darker as retraining progresses, Plots d,e,f have descending huc, meaning the color gets whiter as retraining progresses. In and e the points are colored by the direction (seed) of the jumps in parameter space and the color grows darker as retraining progresses. In contrast to more continuous trajectories returning near the optimum the good generalization case, memorization displays a more random pattern where weights move out before moving back, often switching direction during retraining. Bad generalization appears to be more similar to memorization, where no continuous trajectories were observed. See Appendix B for higher resolution version.
pattern a jump-and-retrain experiment indicates that even when points are thrown far from the minimum, the landscape is wide enough that they return to it. Moreover, they tend return in the same direction they were pushed out, leading to a flower-pettr convergence pattern. Such a structure is the high-dimensional equivalent to "wide valley" described in past work studying flatness of optima. While the starting points are all set on five distinct directions from the minimum, the plotted restricted any subspace of the parameter space.
retraining trajectories
In the bad minimum on the other hand, the trajectories start off near the middle of the plot (darker points in the middle of Fig. 2b) but during the retraining, the trajectories diverge toward the edges before coming back towards the middle of the plot. This outward movement is stark contrast with the consistent retraining trajectories surrounding the good generalization minimum, which immediately return to the valley. This indicates that the region around the oud-generalization minimum is most probably not wide (or flat) high-dimensional valley and contains nonnegligible amount of nonconvexities, preventing training trajectories to systematically converge towards the minimum. The "memorization" minimum plot Fig. D looks similar to the bad generalization plot, with trajectories that go outward at small step sizes of the "jump." " However, curiously at larger step sizes of the "jump," the trajectories seem to return without going outward first, but they do not return immediately, ie., they show some lateral movement, potentially indicating bumpiness in the landscape that they avoiding. Thus we see a clear pattern in the good generalization minima of inward movement upon the jump and retrain, and more outward behavior in the bad generalization and memorizations. See supplemental video for an animation of these trajectories, and Appendix B for trajectories colored by loss.
5.2 OPTIMIZATION
We analyzed three stochastic gradient-descent optimization methods commonly used for training ANNs: SGD (vanilla), SGD with momentum (SGD_M) and Adam. The major difference that Adam uses adaptive step sizes based off previous iterations while SGD and SGD_M .M use fixed step

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sizes. We are interested in answering questions including how the network evolves during training with different optimizer and whether different optima are reached at the end of training.
We trained Wese-ResNet networks of depth 28 and twice wide, which is the same in the previous section. The same initial learning rate of 0.1 and learning rate decay schedule employed with each optimizer, data augmentation is used to assist training while no weight decay used. All experiments achieve close to 100% training accuracy and 90% test accuracy. A fixed set of five seeds is used across optimizers.
First we examine individual trajectories of network weights from the same seed using different optimizers (Fig. B) Compared the smooth trajectory from using Adam, SGD with or without momentum seems to exhibit scattered patterns. It should be noted that the magnitudes of network weights learned by Adam are much larger than by SGD or SGD M (Appendix D) The evolution and resulting network weights from the Adam optimizer appears to differ more significantly from SGD or SGD SGD_M, where the resulting parameters appear to be further away from the initialization.
Figure 3: Comparing PHATE embedding of network weights during training using different optimiz- ers. The same seed was used across optimizers, Adam produces a more smooth trajectory.
Figure 4: PHATE embedding (cosine distance) of network parameters during training using dif- ferent optimizers (SGD, SGD_M __M, and Adam). The five initialization marked by golden stars. (Diffe)nn optimization algorithms reach distinct minima that all lie in a disc-shaped region. ADAM seems to search an orthogonal space of the loss landscape from SGD with momentum, as seen by the "disappearance" of the trajectory, (c) The Adam trajectories, plotted by themselves are continuous, however distances between final weights reached by Adam are much larger than other optimizers.See Appendix B for higher resolution images
In Fig. 4 the distinctiveness between optima reached from different initializations and optimizers are clearly shown. We see in the Fig. 4[) a) that when starting from the same seed, SGD and SGD) with momentum travel in similar directions. However, with momentum. Without momentum, a similar minima is reached. However, the path towards the minima longer with more moves sideways, indicating that the space searched less effectively, In Fig. 4bb) the fact that Adam travels so far in the weight space, causes the initial points look very close together. From the initial points, the SGD with momentum travels in shorter paths towards different minima. However, Adam seems to exhibit disconnected jump. potentially flying off into an orthogonal space what is visualized, but interestingly returning to same region of minima as SGD_M. The seemingly discontinuity of the trajectories in the middle plots indicates that Adam searches space very differently than SGD. Fig. 4[c) shows that the Adam trajectorics are indeed connected in continuous line, so the trajectory itself is not discontinuous, but rather in different subspace than the SGD M trajectory.
shorter and straighter training path
5.3 ROBUSTNESS
One of the main problems encountered in previous attempts to visualize ANN loss landscapes resides in the diffieulty of comparing them across tasks and initializations. Here, show that
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visualization approach, applied to the results of jump-and-retrain experiments, alleviates this concern. To this end, we trained multiple WResNer28-2 networks by following the procedure described in found one
Sec. [ and starting from multiple parameter initializations. For each initialization, good (~94% test acc.) and one bad optima (-86% test acc.), giving the jump-and-retrain results shown in Fig. Further,
also trained similar network the CIFAR 100 dataset, following the same training procedure, except here the data augmentation was comprised not only of random horizontal flips and crops, but also of random rotations of up to 45 in each direction and random modifications to the brightness, contrast, saturation and hue of the images. Since CIFARION 100 poses significantly harder classification task than CIFARIO, our networks only reaches -60% test accuracy for the bad optimum trained without data augmentation and weight decay, and ~70% test accuracy for the good optimum. The fact that the networks only reached 70% accuracy is just a reflection of the complexity of the task, with better accuracy requiring better fine-tuning of the architecture and the training procedure. The results of these experiments are shown in Fig.
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Figure 5: 2D PHATE embeddings of the jump and retrain experiment ran on different initialization for the WResNer28-2 2 network on CIFARIO (a, and d) and on CIFAR 100 (e and f), colored by epoch of retraining. Patterns that resemble bad generalization are exhibited the first plot for all three settings in (a), (c) and (e), while good generalization patterns shown in the second plots (b), (d) and (f). demonstrates that our approach is robust across tasks. See Appendix forr a higher resolution version.
The plots for different initializations the WResNet28-2 are consistently showing the same star- shaped patterns, characterizing the retraining trajectories surrounding minima that generalize well. The plots for the bad minima also share a similar pattern, not only across initialization but also across tasks. Strikingly, the star-shaped pattern is better defined in Fig. 5T. corresponding to the good optimum WResNet28-2 trained on CIFAR100. This offers a key insight into the effect of regularization the geometric characteristies of the found minima. It seems that regularization techniques enhance generalization by forcing the optimizer into wider valleys.
6 DISCUSSION AND CONCLUSION
We proposed novel approach to visualize ANN loss landscapes based on the -tate-of-the-art PHATE dimensionality reduction, which is able to capture branch-like structures in high-dimensional data in two dimensional representations. Our approach enables geometric exploration retrained trajectories surrounding generalization and memorization optima, found via ANN training, provide insight into generalization capabilities the network. Further, it enables meaningful comparison training trajectories across optimizers. Finally, our proposed paradigm is demonstrably consistent and robust across initializations and supervised learning tasks. We expect in future work this visualization approach to enable more methodical paradigms for the development of deep models that generalize better, train faster, and provide fundamental understanding of their capabilities.
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Kevin R. Moon, David van Dijk, Zheng Wang, Scott Gigante, Daniel B. Burkhardt, William S. Chen, Kristina Yim, Antonia van den Blzen, Matthew J. Hirn, Ronald R. Coifman, Natalia B. Ivanova, Guy Wolf, and Smita Krishnaswamy. Visualizing structure and transitions in high-dimensional biological data. Nature Biotechnology. 37(12): 1482-149222 December 2019.
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Stefano Recanatesi, Matthew Farrell, Guillaume Lajoie, Sophie Deneve, Mattia Rigotti, and Eric Shea-Brown. Signatures and mechanisms of low-dimensional neural predictive manifolds. bioRxiv, pp. 471987, 2018.
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Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. In Edwin R. Hancock Richard C. Wilson and William A. P. Smith (eds.), Proceedings of the British Machine Vision Conference (BMVC), pp. 87.1-87.12. BMVA Press, September 2016. ISBN 1-901725-59-6. doi: 10.5244/C. 30.87.
C. Zhang. S. Bengio, M. Hardt, B. Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ArXiv, abs/161 66 1.03530, 2017.
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Under review as a conference paper at ICLR 2021
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Under review as a conference paper at ICLR 2021
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Under review as a conference paper at ICLR 2021
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Reviews:
[{"review_id": "wKmxdqiRWDE", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\nThe authors propose to visualize training trajectories with the dimension reduction and visualization tool - PHATE. The authors try to reconstruct manifolds on which the training trajectories lie. It argues that the method captures data characteristics from all dimensions and shows consistent geometric patterns for loss landscape surrounding good and bad generalization optima.\n\n\nStrength\n- The approach embeds multiple training trajectories in one visualization by adopting dimension reduction approaches, which is not explored before. The comparison of multiple dimension reduction approaches is also lacking in previous literatures.\n- The paper has a nice review and summarization of previous approaches on loss surface visualization.\n\nWeakness\n\nMy major concern is that the insights observed from the methods are not quite straightforward for interpretation and some conclusions are not clear or novel. \n\n- In page 4, the authors claimed “We note that methods that pick random directions...are not able to visualize entire trajectories as the spaces and planes change” this observation is also made in previous work[1] and PCA is used for visualizing the training trajectories. However, this is not necessarily a disadvantage for visualizing the flatness of the minima as discussed follows.\n\n- To characterize the region of loss landscape, the authors propose to adopt “jump and retrain” (setting 1) and this involves visualizing multiple trajectories together. However, the rationale to do this is not quite clear and the interpretation of the visualization is hard. It is not quite clear about the advantage of proposed visualization (Figure 2) on explaining the “flatness” in comparison with previous 1D or 2D visualization with randomized directions. On the other hand, the computation cost (multiple initialization, multiple steps and retraining) for generating those figures could be significant.\n\n- The proposed visualization should be valuable for comparing different trajectories produced by different optimizers as all trajectories can be plotted together, however, the visualization (figure 4) does not convey a clear message about which optimizer (trajectory) is better in performance (generalization). \n\n- The individual trajectory visualization (Figure 3) in sec 5.2 seems not to differ too much with the PCA based visualization [1][2]. What is the takeaway message from Figure 3? It is not clear how the observation “the resulting parameters appear to be further away from the initialization” is made from the figure.\n\nQuestions\n\nIn section 5.3, the authors claim there are consistent and distinct patterns for “good” and “bad” minima. Is there a smooth transition between the “good” and “bad”? Will the pattern differ with a much better or worse initialization? e.g., what is the pattern for random initialization (which is the “worst”)? \n\n\n[1] Li et al, Visualizing the Loss Landscape of Neural Nets, NIPS 2018.\n[2] Eliana Lorch. Visualizing deep network training trajectories with pca. ICML Workshop on Visualization for Deep Learning, 2016.", "extra_scores": {}, "extra_text": {}}, {"review_id": "UnFjCpVKAXK", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper suggests using PHATE, a modern dimensionality reduction method, for visualizing the training trajectories of deep networks. It argues that PHATE visualizations can bring to light interesting aspects of the training dynamics that are missed by other dimensionality reduction algorithms because PHATE does a better job at preserving both local and global structure in the data. This is not the first application of PHATE in the context of deep learning, but to my knowledge it is the first application to deep learning trajectories.\n\nThe paper shows that PHATE visualizations of train-and-jump trajectories (where the model is repeatedly pushed away from a minimum and then allowed to retrain) can have features that are correlated with how well a model generalizes. It also shows that different optimization algorithms can lead to distinguishable visual features.\n\nWhile the use of better visualization techniques may lead to better understanding of neural network dynamics, I am not convinced that the paper succeeds in making this case:\n\n1. The generalization results (Figure 2) are interpreted using known results about the generalization properties of flat vs. sharp minima. These results are nice, and are potentially useful. However, they were only demonstrated in a few settings. Do these results hold more generally, when using other architectures and different data sets?\n\n2. I felt that Section 5.2 on visualizing the trajectories of different optimizers did not have a clear take away message.\n\n3. Except for Figure 1, PHATE is not compared against more commonly used reduction algorithms such as tSNE and UMAP. For example, regarding Figure 2: Could we draw similar conclusions about generalization using other techniques?\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "U7FiNHxjUpi", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "### 1. Brief summary:\nThe authors use a new dimensionality reduction technique called PHATE (that was first introduced in a different paper) to study the weight-space positions and training trajectories of several DNN architectures (ResNet, WideResNet) on vision classification tasks (CIFAR-10, CIFAR-100). They use different optimizers (SGD, SGD+Momentum, Adam), study very good and decent (they call them bad) optima, and the solutions to memorization of random labels. They perform two kinds of experiments: 1) perturbing a single optimum and retraining from there, visualizing those trajectories, and 2) visualizing several random init -> optimum trajectories from different inits. They then draw some conclusions from the PHATE projections for the two kind of experiments and different optimizers and kinds of optima.\n\n### 2. Strengths\n* I like the introduction of the PHATE algorithm and the authors explanation of its advantages and why it might be a good fit for the weight space trajectory visualization of DNNs. \n* The paper is well motivated. Understanding the loss landscape of DNNs is likely very important and still very underexplored.\n* I like the comparison to other algorithms (PCA, t-SNE) and the synthetic tree-like structure they visualize in addition to real DNNs to demonstrate the advantageous properties of PHATE.\n* I like that the others study the stability of their visualization to retraining and seed change -- that is very good and should be the norm in all papers that make claims that depend on stochasticity.\n\n### 3. Weaker points\n1. The scope of the paper is, in my opinion, a bit too limited. A lot of space is used to introduce the PHATE algorithm that is, as I understand it, /not/ a novel contribution of this work.\n\n2. While the visualizations look compelling, the conclusions that are drawn from them are often too strong. I don't see how the visualization can be connected to flatness, dimensionality etc directly. They do serve as a good visual guide, but I feel the paper doesn't establish the connection to those quantities very well. I see that let's say for the random labels and real labels, the projections look different, but how do I link this to e.g. the size of the low-loss basin around those two?\n\n3. The authors flip between a language suggesting that there are some specific dimensions that are being visualized (e.g. \"potentially flying off to an orthogonal subspace\" and appreciating that the PHATE technique doesn't preserve dimensions in any meaningful way. I would advise to soften the claims to reflect the non-linear, adaptive nature of PHATE and the difficulty of connecting the embeddings to any particular directions.\n\n4. The experiment type 1 = going in random directions off an optimum and retraining back doesn't seem to push far enough. What would happen if I perturbed more? Would the solutions glide back as they do? How does this compare between the real labels and random labels? I'd say this would be a nice validation at least -- if you go to far, you should not go back. If you do, the PHATE projections do not project the way you believe. If you don't, that's interesting on its own and can be used for comparison between cases (Adam, SGD, random labels ...).\n\n5. The Figure 3 results for a single trajectory look uninformative -- what would happen if you did this multiple times from different seeds? Are the Adam paths smoother generically, or just this particular random seed? You should either make the experiment statistically meaningful in some way, or explain what the reader is supposed to take away from it. As is, it doesn't do much.\n\n6. The experiments of type 2: random inits -> training -> optima, look promising. However, they really break my intuition for what PHATE does and if it is at all relevant. My main worry is this: why do the inits end up coinciding? From [1] and [4] it is quite certain that a) the inits are mutually orthogonal to a high degree and b) so are their optimized endpoints. How come your inits get mapped to the same point sometimes, but other times they do not? This really puts your results into question for me, especially because you use the visualizations to make pretty strong claims about the loss landscape structure.\n\n7. (correct me if I am wrong) Only WideResNet and ResNet on CIFAR-10/100 were used in this paper. Those are two pretty similar architectures and two very similar vision tasks. I would appreciate a much broader set of experiments to make sure the claims here hold. For example, \na. what do skip connections do? both architectures here have them, and\nb. How would MNIST, FASHION MNIST, SVHN do? Those are cheap to train and easy to use datasets and I would expect the authors do use them\nc. How would a simple feed-forward CNN do?\nd. What about fully-connected nets? Would they behave the same way?\nEmpathically, I am /not/ asking for ImageNet -- I know that it is very hard to run and while it would be nice to have, there is no need to use it in every paper. But I would want to see a much broader set of experiments  on other CNN-based architectures as a well as other, potentially weaker datasets. As is, the paper doesn't have the sufficient experimental support to see the generality of the interpretation claims. \n\n### 4. Relevant papers that might be worth adding/exploring\nThere are some papers that I think might be relevant here and that the authors might want to wish to add / read / explore. \n\n[1] /Large Scale Structure of Neural Network Loss Landscapes/ by Stanislav Fort, Stanislaw Jastrzebski (https://arxiv.org/abs/1906.04724) at NeurIPS 2019 build a geometric model of the low-loss manifolds of DNNs incorporating the observations of 1. connectivity of init to optimum, the high-dim nature of the manifolds and their connectedness. They also visualize the loss landscape on sections that might be relevant here. They also show that they can find N-dimensional surfaces connected N+1 independent optima together, going beyond the 2 optima on a 1-d path.\n\n[2] You might be missing the second paper that established the connectivity between different modes: Garipov, T., Izmailov, P., Podoprikhin, D., Vetrov, D. P., and Wilson,  A. G.  /Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs/ at http://arxiv.org/abs/1802.10026\n\n[3] /Measuring the Intrinsic Dimension of Objective Landscapes/ by Chunyuan Li, Heerad Farkhoor, Rosanne Liu, Jason Yosinski (https://arxiv.org/abs/1804.08838) establishes that the loss landscape low-loss manifolds have a low /intrinsic dimension/ that can be interpreted as high d of the manifolds (done in https://arxiv.org/abs/1906.04724)\n\n[4] /Deep Ensembles: A Loss Landscape Perspective/ by Stanislav Fort, Huiyi Hu, Balaji Lakshminarayanan (https://arxiv.org/abs/1912.02757) look at the cosine similarity of the weight space positions of multiple runs as well as a single run. They also visualize the loss landscape between optima on different affine and non-affine sections.\n\nIn your justification for why PHASE might be good at visualizing the trajectories you mention that they are low dimensional. There is a number of works showing that it might or might not be the case:\n[5] Gradient Descent Happens in a Tiny Subspace by Guy Gur-Ari, Daniel A. Roberts, Ethan Dyer (https://arxiv.org/abs/1812.04754) shows the gradients in a mostly low-D subspace, but the actual learning happens orthogonal to that.\n[6] Emergent properties of the local geometry of neural loss landscapes by Stanislav Fort, Surya Ganguli (https://arxiv.org/abs/1910.05929) dissects the Hessian structure and finds low-D signal + high-D noise, where the signal is not in the learning directions but rather forms constraints.\n[7] Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra by V. Papyan (arXiv:2008.11865) has a on overview of the low-D structures in the DNN Hessians that might be relevant.\n\n \n### 5. Summary\nI like that you introduced a new dimensionality reduction technique to visualizing loss landscape trajectories and positions for DNNs. The reasons why it might be better than others are compelling. However, I think that the scope of the experiments you provide is limited, the claims you make a bit stronger than would be justified by the results shown, and there are some worrisome features of some of the embedded trajectories that make my question the validity of your overall conclusions. I think the paper has a promise, but it needs a bit more work. **I am open to revising my score** if you address my questions well.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "RLHLYMYJlx5", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "### Paper Summary\n\nThis paper uses PHATE to visualize the progression of neural net parameters during learning in neural networks to provide insight into generalizable vs. non-generalizable minima, and the behaviors of different optimization algorithms.  PHATE is an improvement over previous visualization techniques due to its approach to finding a manifold, allowing it to plot in two dimensions multiple trajectories that do not otherwise share a plane.  This is then used to plot trajectories in \"jump and retrain\" experiments, in which a minimum is found, and then perturbations made to the network parameters, before restarting training.  It is shown that in the networks experimented on, minima with good test set performance reliably funnel the new learning trajectories into the same minimum, while minima with poor test set performance see the perturbed initializations find other minima, thus demonstrating the \"flatness vs sharpness\" of minima.  Trajectories produced by SGD, SGD with momentum, and Adam are also compared.  Adam is shown to travel further, but along a smoother trajectory.\n\n### Originality\n\nThe visualization approach is not new, but is new to neural network trajectories.  The visualizations effectively confirm suspected properties of neural networks, but do not offer anything particularly new.  The behaviors of learning algorithms are demonstrated, but not much discussed, so little new is learned.\n\n### Significance\n\nVisualization is important to understanding the behaviors of high-dimensional learning trajectories, and this is better for plotting multiple trajectories than previous approaches.  It is an effective approach, and is superior at demonstrating known neural network behaviors than others.  Insights are not significant, but the visualization of them is improved.\n\n### Clarity\n\nVery clearly written, and easy to understand.\n\n### Quality\n\nEnjoyable paper, which offers a new, effective tool, but no real new insights into neural networks.  The jump-and-retrain experiments are effective and well-chosen.  The demonstrations of learning algorithm are not as effective, and I'm not sure what to take away from them.  I invite the authors to help me find the significance of these experiments.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers and I agree that the paper is well motivated and that there are good comparisons to prior work. However, the scope of the paper is rather limited, and there were some doubts about the overall conclusions and whether the current results fully support them. As such, I cannot recommend the paper for publication.
Decision: Reject