Deep Artificial Neural Networks exhibit critical periods
Studies of the critical period for monocular deprivation in humans rely on cohorts of patients affected by stimulus-deprivation amblyopia (reduced visual acuity in one eye), either in their infancy or childhood, caused by spontaneous or traumatic unilateral cataracts (18, 19). After surgical correction of the cataracts, the ability of the patients to regain normal acuity in the affected eye depends both on the duration of the deficit and on its age of onset. Earlier and longer deficits, as in animal studies of monocular deprivation (20, 5), lead to increasingly severe effects.
In order to replicate this experimental setup in ANNs, we train a standard convolutional network (CNN) to classify objects in smallRGB images from the CIFAR-10 dataset (21) in 10 classes. Recognition performance in the image classification task is akin to optotype symbol identification, employed in optometric acuity testing. To simulate the effect of cataracts, for the first epochs the images in the dataset are downsampled to and then upsampled back to
using bilinear interpolation, in practice blurring the image and destroying small-scale details.111We employed this method, instead of a simpler Gaussian blur, since it has a very similar effect and makes the quantification of information loss clearer. After that, the training continues for more epochs, giving the network enough time to converge and ensuring it is exposed to the same number of uncorrupted images as in the control () experiment.
In Figure 1, we graph the final performance of the network (described in Materials and Methods) as a function of the epoch at which the deficit is corrected ().
We clearly observe the existence of a critical period for this deficit in the ANN: if the blur is not removed within the first epochs, the final performance is severely decreased when compared to the baseline (from a test error of , in the absence of a deficit, to more than when the blur is present over 140 epochs, a increase). Once rescaled to account for the arbitrary time units (training epochs vs. days), the profile of the curve is also strikingly similar to the one obtained in kittens monocularly deprived from near birth and whose visual acuity upon eye-opening was tested and plotted against the length of the deficit window (23). Just like in humans and animal models (where critical periods are characteristic of early development), the critical period in the DNN also arises during the initial rapid learning phase. At this stage, the network is quickly learning a solution before the test error plateaus and the longer asymptotic convergence phase begins.
To quantify more accurately the sensitivity of the ANN to image blurring throughout its early learning phase, we introduced the deficit in a short constant window (40 epochs), starting at different epochs, and then measured the decrease in the ANN’s final performance compared to the baseline. In Figure 1, we plot the final testing error of the network against the epoch of onset of the deficit. We observe that the network’s sensitivity to blurring peaks in the central part of the early rapid learning phase (around 30 epochs), while later deficits produce little or no effect. A similar experiment was also performed on kittens by Olson and Freeman, using a window of 10-12 days during which the animals were monocularly deprived and using it to “scan” the first 4 months after birth to obtain a sensitivity profile (20).
The results obtained on the DNN are, again, in accord with the trend observed in animal models (Figure 1).222Olson and Freeman here employ a neuroanatomical readout to measure the impact of monocular deprivation on visual development. While excellent correlations between anatomy, physiology and behavior are generally observed (7, 28, 16, 29), direct comparisons with the ANN’s performance in the image classification task are less warranted in this case. We later return to this aspect and characterize the “connectivity” of ANNs during training in Information, synaptic strength, critical periods.
High-level deficits are not associated with a critical period
Critical periods of neuronal networks dealing with high-level aspects of sensory processing end later than their lower-level counterparts, with some nuclei remaining plastic even in adulthood (7). Although the inversion of the visual field has not been systematically tested on young animals (or humans), it is well-reported that adult humans can quickly adapt to it (30, 24), suggesting the absence of a critical period. In addition to the behavioral recovery from the deficit, the anatomical plasticity in the adult has also been confirmed in animal models (25).
We observe a similar behavior in ANNs: in Figure 2 we perform the same experiment as before, this time using vertical flipping of the training images as the deficit, instead of blurring. We observe that the ANN is largely unaffected by vertical flipping and that the network quickly recovers its baseline performance after deficit correction. An analogous result is also obtained when the high-level deficit is a permutation of the task labels (Figure 2). These findings suggest that the existence of a critical period depends on the structure of the input data and the nature of the deficit applied, and, as such, cannot be interpreted solely as an artifact of the optimization process or the network’s architecture. We have further explored these aspects in later experiments.
Sensory deprivation causes a less severe critical period
It has been observed in animal models, and recently indirectly confirmed in humans (26), that a period of early sensory deprivation (dark-rearing) can lengthen the critical period and thus cause less severe effects (at the same age) than those documented in light-reared animals (27). Similarly, both in humans and animal models, the effects of binocular cataracts have been characterized as being, somewhat counter-intuitively, less severe than those reported in patients and animals suffering from monocular deficits (31, 32). These findings, though, can be explained considering that, since visual experience is necessary to shape neuronal circuits, lack of experience can be, under certain circumstances, less damaging to the development of visual skills than defective experience (2).
Simulating complete visual deprivation in a neural network is not as simple as feeding a constant stimulus: a network presented with a constant blank input will rapidly become trivial and thus unable to train on new data. This is to be expected, since a blank input is a perfectly predictable stimulus and thus the network can quickly learn the (trivial) solution to the task. We instead want to model an uninformative stimulus, akin to noise. Moreover, even when the eyes are sutured or maintained in the darkness, there will be background excitation of photoreceptors that is best modeled as noise. To account for this, we simulate sensory deprivation by replacing the input images with a dataset composed of (uninformative) random Gaussian noise. This way the network is trained on solving the highly non-trivial task of memorizing the association between the finitely-many noise patterns and their corresponding labels.
Figure 2 shows that this extreme form of deficit has a much less severe effect, during its critical period, than the one obtained by only blurring images. Using Gaussian noise during the early training phase of the ANN does not provide any information on the natural images. Yet its effects are milder than those caused by a deficit (e.g., image blur), which instead conveys some information, but leads the network to (incorrectly) learn that no fine structure is present in the images. It is also interesting to note how a network trained to memorize noise patterns, while training on a completely different array of features than those needed to classify natural images, can still learn to identify images correctly with relative ease (Figure 2).
) follows a similar trend where, after a sharp increase and a peak during early development, progressive elimination of synapses (pruning) occurs and continues throughout life. ANNs, though, appear to have both a different timescale and a more abrupt decrease in the effective connectivity after peaking.(Right) Effects of deficit (image blur) on the FIM. In the presence of a critical-period-inducing deficit, the ANN employs a much larger set of connections to solve the task. This can be explained by the fact that the network, unable to find a general rule to classify the corrupted images, is forced to memorize them in order to solve the task. This reflects in increased use of the network’s resources, correctly captured by the FIM, it being a measure of the ANN’s information content.
Dependence on the optimization algorithm and architecture
Deep networks are commonly trained using stochastic gradient descent (SGD) with an annealing scheme decreasing the learning rate over the duration of the training, in order to obtain convergence despite the noise introduced by the stochastic gradients. One possible explanation for the existence of critical periods in DNNs would thus be that, once the network enters a sub-optimal local minimum because of the deficit introduced, it cannot then (when the deficit is removed) escape due to the decreased learning rate. InFigure 2 we have already observed that in the case of several deficits the network manages to escape such a hypothetical local extremum despite the decreased learning rate. In Figure 3, we confirm that a critical period exists even when the learning rate, which can be thought of as regulating the temperature of the system undergoing gradient descent (34), is kept constant.
Figure 3 also shows that deep fully-connected networks, trained on the MNIST digit classification dataset, present a critical period for the image blur deficit; therefore the convolutional structure is not necessary, nor is the use of natural images. Similarly, a ResNet-18 trained on CIFAR-10 also has a critical period, sharper than the one found in a standard convolutional network (Figure 1
). This is especially interesting, since ResNets allow for easier backpropagation of gradients to the lower layers, thus suggesting that the critical period is not caused by vanishing gradients.
Information, synaptic strength, critical periods
Deficits that result in critical periods reflect in changes of the brain architecture in the associated areas (7). This is inevitably different in artificial networks, as their connectivity is formally fixed at all times during training. However, not all the connections have the same importance for the final output of the network: This fact is captured, for example, by the Fisher Information Matrix (FIM) (35), which may be considered the artificial network equivalent of population-level measures of synaptic strength and connectivity in neuronal networks (36) (but see also (37) for alternative metrics). In this section, we briefly recall some properties of the FIM, and apply it to the study of the connections’ strength of an artificial neural network trained in the presence and absence of deficits. Using this approach we aim to determine whether the (global) response of the ANN to deficits correlates with (local) changes in connectivity, similarly to how behavioral and physiological features correlate in experience-deprived animal models (3, 28, 16, 29).
The Fisher Information Matrix
Generally, an artificial neural network encodes a distribution — parametrized by the weights — of the task variable (e.g., an image label in CIFAR-10), given an input image
. Intuitively, the importance of a specific connection can be estimated by perturbing the corresponding weight and looking at the magnitude of the change in the final distribution. Let us consider then a perturbationof the weights, which results in a perturbed distribution
. The discrepancy between the original distribution and the perturbed one can be measured by their Kullback-Leibler divergence, which, to second-order approximation, is given by:
where the expectation over is computed using the empirical data distribution given by the dataset, and is the Fisher Information Matrix (FIM). The FIM can thus be considered a local metric measuring how much the perturbation of a single weight (or a combination of weights) affects the output of the the network (38). In particular, weights with low Fisher information can be removed (pruned) without affecting the network’s performance, which suggests that the Fisher Information can be used, in an artificial network, as a measure of the effective connectivity or, more generally, of the strength of a connection (36). For these reasons, especially when drawing comparison with neuronal networks, we will sometimes use “connection strength” and Fisher Information interchangeably. As its name suggests, the FIM can also be considered as a measure of the quantity of information that the model contains about the training data (35)
. Indeed, we expect the strength of the connections to increase as we learn more information from experience. Finally, the FIM is also a semi-definite approximation of the Hessian of the loss function(39) and hence of the curvature of the loss landscape at a particular point during training, which provides a connection between the FIM and the optimization procedure (38).
We will employ the trace of the Fisher Information Matrix to measure the global or layer-wise connection strength, since it can be calculated efficiently despite the large size of our networks. We also considered computing the log-determinant of the matrix using the Kronecker-Factorized approximation (40, 41) in order to capture the behavior of off-diagonal terms, but we observed the same qualitative trend as the trace. We use ResNets to compute the FIM because, being a local measure, it is very sensitive to the irregularities of the loss landscape and ResNets have a relatively smooth landscape (42). For other network architectures we use a more stable estimator of the FIM based on the injection of noise in the weights (43) (also see Materials and Methods).
Information changes during training
In Figure 4 we estimate the global changes to the connections’ strength in a ResNet by plotting the trace of the FIM during the network training. We observe that during an initial phase the network acquires information about the data, which results in a large increase in the strength of the connections. However, once the performance in the task begins to plateau, the network starts reducing the overall strength of its connections while keeping its performance unchanged. This can be seen as a “consolidation” or “compression” phase, during which redundant connections are eliminated and non-relevant memorized information is discarded. It must be noted how a related observation on the training procedure consisting of two phases has been suggested by (17), who, however, focus on the compression of the activations of the network, rather than on the information in the weights and their connectivity. There is a fundamental difference between these two analyses: The relation between the information in the weights and the activations is non-trivial and articulated in (43). In particular, compression of the weights can imply compression of the activations, under suitable non-generic assumptions.
Interestingly, this change in the connection strength is closely related to the sensitivity to critical-period-inducing deficits as image blur, computed using the sliding window method as in Figure 1. In Figure 4 we see that the sensitivity is maximal precisely when the Fisher Information peaks. Moreover, we observe that the exponential of the sensitivity closely fits the Fisher Information, which is remarkable since the FIM is a local quantity computed at a single point during the training of a network without deficit, while the sensitivity during a critical period is computed using the test data at the end of the training of a network in the presence of a deficit. It is also interesting to see the effect of deficit introduction on the Fisher Information. From Figure 4 (Right), we see that the FIM trace grows much more in the presence of the deficit (image blur) and remains substantially higher even after it is removed. When the data are so corrupted as to be classified incorrectly, the network is forced to memorize the labels, therefore increasing the quantity of information needed to perform the same task.
Notice that a similar phenomenon is observed in the development of neuronal networks (Figure 4, Center), where the density of synaptic connections rapidly peaks during early development and slowly decreases afterwards as a results of synaptic pruning (44, 45). However, while deficits have been shown to affect the synaptic density in animal models (45), an increase in the overall number of connections is not observed. This difference can be partly attributed to the fact that in our experiments ANNs are trained using a finite dataset, making pure memorization a viable strategy to solve the task and therefore encouraging the storage of unnecessary information.
Layer-wise effects of deficits
We can gain additional insights on what is happening in an ANN affected by a deficit by looking at the layer-wise changes of the Fisher Information. In order to do so, we used the All-CNN architecture, which has a clearer subdivision in layers than a ResNet. When the network is trained without deficits (Figure 5, Left), the strongest (most important) connections are in the intermediate layers. This reflects the fact that most of the information in CIFAR-10 images is present at an “intermediate” scale, with low level information (e.g., edges) alone being insufficient for identification, and global/contextual information (e.g., background, position of objects) providing only limited benefit to the task performance. Therefore, we expect the network to focus its resources on the intermediate layers, since their receptive fields are optimal for this task.
On the other hand, if the network is initially trained on blurred data (Figure 5, Top Center and Top Right), the strength of the connections is dominated by the top layer (Layer 6). This is to be expected, since the low-level and mid-level structures of the images are destroyed and all of the remaining information is contained in the global features of the image, which can be processed by Layer 6, but not by the low-level layers with their smaller receptive fields. When the deficit is removed early in the training (Figure 5, Top Center), the network manages to “re-organize”, reducing the information contained in the last layer, and, at the same time, increasing the Fisher Information in the intermediate layers, in order to process the new features available. We refer to these changes in the Fisher Information as “Information Plasticity”. However, if the deficit is removed after the beginning of the “consolidation” phase, the quantity of information contained in the mid-level layers does not change significantly because of the network’s loss of Information Plasticity.
ANNs and neuronal networks are vastly different systems, nonetheless both exhibit critical periods during their early learning phases and these share a number of common features. In particular, we have shown that while cataract-like deficits, such as severe image blurring, present a critical period, high-level impairments, like image flipping, have little or no effect on the DNNs’ performance. These findings are in agreement with the characterizations of critical period responses to deficits in humans and animal models. In addition, we use the Fisher Information to study the development of the DNNs’ effective connections during training and recognize an early phase where the Fisher Information increases sharply, followed by a decrease. The maximum sensitivity to critical-period-inducing deficits corresponds to the maximum of the Fisher Information.
It must be noted that critical periods depend on the task explored, the deficit applied, and the readout used: Due to the fundamental differences between neuronal networks and ANNs, it is challenging to perfectly match experiments to animal models. Since visual pathways are extensively studied in the neurobiology of critical periods and DNNs are particularly suited for image classification, we chose to focus our analysis on this task.
“Behavioral” readouts upon deficit removal, both in ANNs and neuronal networks, can potentially be confounded by deficit-coping changes at different levels of the visual pathways (7, 2). Since deficits in deprived animals are mirrored by abnormalities in the circuitry of the visual pathways, we chose to also characterize their effects on network connections. While the connectivity of an ANNs is fixed by design, its “effective connectivity”, i.e., the connections that are actually employed by the network to solve the task, is not: We quantify it from the Fisher Information Matrix of the network’s weights. Sensitivity to critical periods and the trace of the Fisher Information peak at the same epoch, in accord with the evidence that skill development and critical periods in neuronal networks are modulated by changes (generally experience-dependent) in synaptic plasticity (2, 14). Our layer-wise analysis of the Fisher Information (Figure 5) also shows that, while in a non-deprived network connections are stronger in the intermediate layers, upon introduction of an image blurring deficit, the ANN reinforces higher layers to the detriment of intermediate layers, leaving low-level layers virtually untouched. If the deficit is removed after the ANN’s critical period ends, the network is not able to reverse these effect. Although the two systems are radically different, it is worth noting how similarities to such response to the introduction of visual deficits can be found in the visual pathways of animal models (and, in part, in humans). Lower levels (e.g., retina, lateral geniculate nucleus) are little (or not at all) affected by deficits (4, 46, 16). Higher-level visual areas e.g., V2 and post-V2) also show little remodeling upon deprivation (47, 48, 49), possibly because their plasticity is maintained into adulthood (7) and thus most of the neuroanatomical and physiological changes due to deprivation cluster, with varying severity, in the different layers of V1 (which we can compare to an intermediate-processing level) (3, 16).
In ANNs, the Fisher Information can be interpreted as an approximation of the local curvature of the loss landscape (39). According to this view, Figure 4 (Left) suggests that SGD encounters two different phases during the network training: At first, the network moves towards high curvature regions of the loss landscape, while in the second phase the curvature decreases and the network eventually converges to a flat minimum (as also observed in (50)). We could interpret these as the network crossing a narrow bottleneck during its training before entering a flat region of the loss surface. When combining this assumption with our deficit sensitivity analysis, we can hypothesize that the critical period occurs precisely upon crossing of this bottleneck. A different two-phase interpretation of SGD has also been proposed in (17): In the authors’ analysis, the network first learns to encode the task in the network’s activations, while in a second phase it minimizes the quantity of information, while keeping enough of it to maintain good task performance. The relationship between the biphasic behaviors we observe in the weights, and those observed by (17) in the activations, is non-trivial, as described in (43) and even defining information quantities for the activations presents a number of technical challenges.
There is evidence that convergence to flat minima (minima with low curvature) in a DNN correlates with a good generalization performance (51, 42, 52, 50). Indeed, by recalling the link between the Fisher Information and curvature, we can deduce from Figure 4 (Right) that networks impaired by the deficit converge to sharper minima than unaffected networks. However, we have found that the performance of the network is determined by an early sensitive phase, during which the network crosses high-curvature region of the loss landscape. This suggests that the final sharpness at convergence may be an epiphenomenon rather than the cause of good generalization.
Applying a deficit at the beginning of the training may be compared to the common practice of pre-training and fine-tuning an ANN, whereby the network is initially trained on a similar task, for which data are plentiful, and then fine-tuned to solve a specific task. This is generally found to improve the performance of the network. In (53) a similar practice is analyzed, but in the different setting of layer-wise unsupervised pretraining, seldom used in current practice, and suggests that the pre-training may act as a regularizer by moving the weights of the network towards an area of the loss landscape closer to the attractors for good solutions. The authors observe that the initial examples have a stronger effect in steering the network towards particular solutions. Here, we have shown that pre-training on blurred data can have the opposite effect; i.e., it can severely decrease the final performance of the network. Following the loss landscape interpretation, this would suggest that, surprisingly, a configuration of parameters able to classify blurred images is farther apart from a configuration able to correctly classify real images. On the other hand, a configuration able to memorize noise patterns can access a better solution to the original task. However, it must be remembered that the interpretation of the deficit’s effect as moving the network close to a bad attractor is difficult to reconcile with the smooth transition observed in the critical periods, since the network would either converge to this attractor, and thus have low accuracy, or not. A richer theoretical framework may therefore be needed to study critical periods purely in terms of the geometry of the loss landscape.
A peculiar interpretation of critical periods in humans and animal models is proposed in (2). Knudsen claims that the initial connectivity profile of neuronal networks is unstable and easily changed by early experiences (highly plastic). As more “samples” are observed, the connections change and reach a more stable configuration, which is not as easily modified by additional experience. Learning can, however, still happen within the newly created connectivity pattern. This is largely compatible with our findings: Sensitivity to critical-period-inducing deficits peaks when the remodeling of connections is at its maximum (Figure 4, Left) and different connectivity profiles are observed in networks trained with and without a deficit (Figure 5). This also agrees with the fact that high-level deficits such as image-flipping and label permutation, which do not require restructuring of the network’s connections in order to be corrected, do not exhibit a critical period.
Finally, we want to emphasize how our goal here is not so much to investigate the human (or animal) brain through ANNs, as to understand fundamental information processing phenomena, both in their biological or artificial implementations. It is also not our goal to suggest that, since they both exhibit critical periods, ANNs are necessarily a valid model of neurobiological information processing, although recent work has emphasized this aspect (54, 55, 15)
. We engage in an “Artificial Neuroscience” exercise in part to address a technological need to develop “explainable” artificial intelligence systems whose behavior can be understood and predicted. While traditionally well-understood mathematical models were used by neuroscientists to study biological phenomena, information processing in modern artificial networks is just as poorly understood as in biology, so we chose to exploit well-known biological phenomena as probes to study information processing in artificial networks. We have shown how our approach can help unveil how information is represented, acquired and transformed in complex networks of relatively simple components, whether biological or artificial.
In all of the experiments, unless otherwise stated, we use the following architecture, adapted from (56):
conv 96 - conv 96 - conv 192 s2 - conv 192 - conv 192 - conv 192 s2 - conv 192 - conv1 192 - conv1 10 - avg. pooling - softmax
where each conv block consists of aconv1 denotes a convolution. The network is trained with SGD, with a batch size of 128, learning rate starting from 0.05 and decaying smoothly by a factor of .97 at each epoch. We also use weight decay with coefficient 0.001. In the experiments with a fixed learning rate, we fix the learning rate to 0.001, which we find to allow convergence without excessive overfitting. For the ResNet experiments, we use the ResNet-18 architecture from (57) with initial learning rate 0.1, learning rate decay .97 per epoch, and weight decay 0.0005. When experimenting with varying network depths, we use the following architecture:
conv - [conv - conv s2] - conv - conv1 - conv1
In order to avoid interferences between the annealing scheme and the architecture, in these experiments we fix the learning rate to 0.001. The Fully Connected network used for the MNIST experiments has hidden layers of size
. All hidden layers use batch normalization followed by ReLU activations. We fix the learning rate to 0.005. Weight decay is not used. We use data augmentation with random translations up to 4 pixels and random horizontal flipping. For MNIST, we pad the images with zeros to bring them to size.
To compute the trace of the Fisher Information Matrix, we use the following expression derived directly from the definition:
where the input image is sampled from the dataset, while the label is sampled from the output posterior. Expectations are approximated by Monte-Carlo sampling. Notice, however, that this expression depends only on the local gradients of the loss with respect to the weights at a point , so it can be noisy when the loss landscape is highly irregular. This is not a problem for ResNets (42), but for other architectures we use instead a different technique, proposed in (43). More in detail, let be the standard cross-entropy loss. Given the current weights of the network, we find the diagonal matrix that minimizes:
where is a parameter that controls the smoothness of the approximation. Notice that can be minimized efficiently using the method in (58). To see how this relates to the Fisher Information Matrix, assume that can be approximated locally in as . We can then rewrite as
Taking the derivative with respect to , and setting it to zero, we obtain . We can then use to estimate the trace of the Hessian, and hence of the Fisher information.
Fitting of sensitivity curves and synaptic density profiles from the literature was performed using GraphPad Prism 7 (GraphPad Software, La Jolla, CA), using:
as the fitting equation. is the age at the time of sampling and , , and are unconstrained parameters (59).
The exponential fit of the sensitivity to the Fisher Information trace uses the expression , where , and are unconstrained parameters, is the Fisher Information trace at epoch of the training of a network without deficits and is the sensitivity computed using a window of size . That is, is the increase in the final test error over a baseline when the network is trained in the presence of a deficit between epochs and .
Supported by ONR N00014-17-1-2072, ARO W911NF-17-1-0304, AFOSR FA9550-15-1-0229 and FA8650-11-1-7156.
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