W IRING U P V ISION : M INIMIZING S UPERVISED S YNAPTIC U PDATES N EEDED TO P RODUCE A P RIMATE V ENTRAL S TREAM
Anonymous authors
Paper under double-blind review
A BSTRACT
After training on large datasets, certain deep neural networks are surprisingly good models of the neural mechanisms of adult primate visual object recognition.
Nevertheless, these models are considered poor models of the development of the visual system because they posit millions of sequential, precisely coordinated synaptic updates, each based on a labeled image.
While ongoing research is pursuing the use of unsupervised proxies for labels, we here explore a complementary strategy of reducing the required number of supervised synaptic updates to produce an adult-like ventral visual stream (as judged by the match to V1, V2, V4,
IT, and behavior).
Such models might require less precise machinery and energy expenditure to coordinate these updates and would thus move us closer to viable neuroscientiﬁc hypotheses about how the visual system wires itself up.
Relative to standard model training on labeled images in ImageNet, we here demonstrate that the total number of supervised weight updates can be substantially reduced using three complementary strategies:
First, we ﬁnd that only 2% of supervised updates (epochs and images) are needed to achieve ∼ 80% of a fully trained model’s match to adult ventral stream.
Speciﬁcally, training beneﬁts predictions of higher visual cortex the most whereas predictions of earlier areas improve only marginally over the course of training.
Second, by improving the random distribution of synaptic connectivity, we ﬁnd that 54% of the brain match can already be achieved “at birth” (i.e. no training at all).
Third, we ﬁnd that, by training only ∼ 5% of model synapses, we can still achieve nearly 80% of the match to the ventral stream.
This approach further improves on ImageNet performance over previous attempts in computer vision of minimizing trained components without substantially increasing the number of trained parameters.
These results reﬂect ﬁrst steps in modeling not just primate adult visual processing during inference, but also how the ventral visual stream might be “wired up” by evolution (a model’s “birth” state) and by developmental learning (a model’s updates based on visual experience).
I NTRODUCTION
Particular artiﬁcial neural networks (ANNs) are the leading mechanistic models of visual processing in the primate visual ventral stream (Schrimpf et al., 2020;
Kubilius et al., 2019; Dapello et al., 2020).
After training on large-scale datasets such as ImageNet (Deng et al., 2009) by updating weights based on labeled images, internal representations of these ANNs partly match neural representations in the primate visual system from early visual cortex V1 through V2 and V4 to high-level
IT (Yamins et al., 2014; Khaligh-Razavi & Kriegeskorte, 2014; Cadena et al., 2017; Tang et al., 2018; Schrimpf et al., 2018; Kubilius et al., 2019), and model object recognition behavior can partly account for primate object recognition behavior (Rajalingham et al., 2018; Schrimpf et al., 2018).
Recently, such models have been criticized due to how their learning departs from brain development because they require many more labeled examples than is reasonable for biological systems’ limited waking (visual) experience (Seibert, 2018; Zador, 2019).
For example, all the current top models of the primate ventral stream rely on trillions of supervised synaptic updates, i.e. the training of millions of parameters with millions of labeled examples over dozens of epochs.
In biological systems, on the other hand, the at-birth synaptic wiring as encoded by the genome already provides structure that is sufﬁcient for macaques to exhibit adult-like visual representations after a few months (Movshon & Kiorpes, 1988; Kiorpes & Movshon, 2004; Seibert, 2018), which restricts the amount of experience dependent learning.
Furthermore, different neuronal populations in cortical circuits undergo different plasticity mechanisms: neurons in supragranular and infragranular layers adapt more rapidly than those in layer 4 which receives inputs from lower areas (Diamond et al., 1994; Schoups et al., 2001), while current artiﬁcial synapses, on the other hand, all change under the same plasticity mechanism.
While current models provide a basic understanding of the neural mechanisms of adult ventral stream inference, can we start to build models that provide an understanding of how the ventral stream “wires itself up” – models of the initial state at birth and how it develops during postnatal life?
Related Work.
Several papers have addressed related questions in machine learning: Distilled student networks can be trained on the outputs of a teacher network (Hinton et al., 2015; Cho &
Hariharan, 2019; Tian et al., 2019), and, in pruning studies, networks with knocked out synapses perform reasonably well (Cheney et al., 2017; Morcos et al., 2018), demonstrating that models with many trained parameters can be compressed (Wu et al., 2018) which is further supported by the convergence of training gradients onto a small subspace (Gur-Ari et al., 2018).
Tian et al.
(2020) show that a pre-trained encoder’s ﬁxed features can be used to train a thin decoder with performance close to full ﬁne-tuning and recent theoretically-driven work has found that training only BatchNorm layers (Frankle et al., 2021) or determining the right parameters from a large pool of weights (Frankle et al., 2019; Ramanujan et al., 2019) can already achieve high classiﬁcation accuracy.
Unsupervised approaches are also starting to develop useful representations without requiring many labels by inferring internal labels such as clusters or representational similarity (Caron et al., 2018; Wu et al., 2018; Zhuang et al., 2019; H ´ enaff et al., 2019; Konkle & Alvarez, 2020; Zhuang et al., 2020).
Many attempts are also being made to make the learning algorithms themselves more biologically plausible (e.g. Lillicrap et al., 2016; Scellier & Bengio, 2017; Pozzi et al., 2020).
Nevertheless, all of these approaches require many synaptic updates in the form of labeled samples or precise machinery to determine the right set of weights.
In this work, we take ﬁrst steps of relating ﬁndings in machine learning to neuroscience and using such models to explore hypotheses about the product of evolution (a model’s “birth state”) while simultaneously reducing the number of supervised synaptic updates (a model’s visual experience dependent development) without sacriﬁcing high brain predictivity.
Our contributions follow from a framework in which evolution endows the visual system with a well-chosen, yet still largely random “birth” pattern of synaptic connectivity (architecture + initialization), and developmental learning corresponds to training a fraction of the synaptic weights using very few supervised labels.
We do not view the proposed changes as fully biological models of post-natal development, only that they more concretely correspond to biology than current models.
Solving the entire problem of development all at once is too much for one study, but even partial improvements in this direction will likely be informative to further work.
Speciﬁcally,
we build models with a fraction of supervised updates (training epochs and labeled images) that retain high similarity to the primate ventral visual stream (quantiﬁed by a brain predictivity score from benchmarks on Brain-Score (Schrimpf et al., 2018)) and ﬁnd that layers corresponding to higher visual regions such as IT are most dependent on training, 2.
we improve the “at-birth” synaptic connectivity to show that even low-capacity evolutionarily encoded information might lead to reasonable initial representations with no training at all, 3.
we propose a thin, “critical training” technique which reduces the number of trained synapses while maintaining high brain predictivity and improves over previous computer vision attempts to minimize trained components, 4.
we combine these three techniques to build models with two orders of magnitude fewer supervised synaptic updates but high brain predictivity relative to a fully trained model
Code and pre-trained models are available through GitHub: https://anonymous.4open.
science/r/anonymous-
M ODELING P RIMATE V ISION
We evaluate all models on a suite of ventral stream benchmarks in Brain-Score (Schrimpf et al., 2018; 2020), and we base the new models presented here on the CORnet-S architecture, one of the most accurate models of adult primate visual processing (Kubilius et al., 2019).
Brain-Score benchmarks.
To obtain quantiﬁed scores for brain-likeness, we use a thorough set of benchmarks from Brain-Score (Schrimpf et al., 2018).
To keep scores comparable, we only included those neural benchmarks from Brain-Score (Schrimpf et al., 2018) with the same predictivity metric.
All benchmarks feed the same images to a candidate model that were used for primate experiments while “recording” activations or measuring behavioral outputs.
Speciﬁcally, the V1 and
V2 benchmarks present 315 images of naturalistic textures and compare model representations to primate single-unit recordings from Freeman et al.
(2013) (102 V1 and 103 V2 neurons); the V and IT benchmarks present 2,560 naturalistic images and compare models to primate Utah array recordings from Majaj et al.
(2015) (88 V4 and 168 IT electrodes).
A linear regression is ﬁt from model to primate representations in response to 90% of the images and its prediction score on the held-out 10% of images is evaluated with Pearson correlation, cross-validated 10 times.
The behavioral benchmark presents 240 images and compares model to primate behavioral responses from
Rajalingham et al.
A logistic classiﬁer is ﬁt on models’ penultimate representations on 2, separate labeled images.
The classiﬁer is then used to estimate probabilities for 240 held-out images.
Per-image confusion patterns between model and primate are compared with a Pearson correlation.
All benchmark scores are normalized by the respective ceiling.
We primarily report the average brain predictivity score as the mean of V1, V2, V4, IT, and behavioral scores.
We note that the Brain-Score benchmarks in this study are based on limited data and thus present a possible limitation.
Nonetheless, they are the most extensive set of primate ventral stream neuronal and behavioral benchmarks that is currently available and the scores generalize to new experiments (Kubilius et al., 2019).
Brain-Score provides separate sets of data as public benchmarks which we use to determine the type of distribution in Section 4, and the layer-to-region commitments of reference models.
CORnet-S. One of the current best model architectures on the Brain-Score benchmarks is
CORnet-S (Kubilius et al., 2019), a shallow recurrent model which anatomically commits to ventral stream regions.
CORnet-S has four computational areas, analogous to the ventral visual areas V1,
V2, V4, and IT, and a linear decoder that maps from neurons in the model’s last visual area to its behavioral choices.
The recurrent circuitry (Figure 3B) uses up- and down-sampling convolutions to process features and is identical in each of the models visual areas (except for V1 COR ), but varies by the total number of neurons in each area.
We base all models developed here on the CORnet-S architecture and use the same hyper-parameters as proposed in (Kubilius et al., 2019).
Representations are read out at the end of anatomically corresponding areas.
H IGH SCORES IN BRAIN PREDICTIVITY CAN BE ACHIEVED WITH FEW SUPERVISED UPDATES
We evaluated the brain predictivity scores of CORnet-S variants that were trained with a combination of fewer epochs and images.
Models were trained with an initial learning rate of 0.1, divided bywhen loss did not improve over 3 epochs, and stopping after three decrements.
Figure 1 shows model scores on neural and behavioral Brain-Score measures, relative to a model trained for 43 epochs on all 1.28M labeled ImageNet images.
In Panel A, we compare the average score over the ﬁve brain measures of various models to the number of supervised updates that each model was trained with, deﬁned as the number of labeled images times the number of epochs.
While a fully trained model reaches an average score of .42 after 55,040,000 supervised updates ( epochs × 1.28M images), a model with only 100,000 updates already achieves 50% of that score, and 1,000,000 updates increase brain predictivity scores to 76%.
Models are close to convergence score after 10,000,000 supervised updates with performance nearly equal to full training (97%).
Scores grow logarithmically with an approximate 5% score increase for every order of magnitude more supervised updates.
[Figure]
Figures 1B and C show individual neural and behavioral scores of models trained with fewer training epochs or labeled images independently.
Early to mid visual representations (V1, V2, and V4 scores) are especially closely met with only few supervised updates, reaching 50% of the ﬁnal trained model in fractions of the ﬁrst epoch (Figure 1B).
After only one full iteration over the training set, V1, V2, and V4 scores are close to their ﬁnal score (all > 80%) while IT requires two epochs to reach a comparable level.
Behavioral scores take slightly longer to converge ( > 80% after 7 epochs).
Similarly, when training until convergence with fractions of the 1.28M total images
50,000 images are sufﬁcient to obtain high neural scores (80% of full training in V1, V2, V4, IT).
Behavioral scores again require more training: half the standard number of labeled images is needed to surpass 80%.
Concretely relating supervised updates to primate ventral stream development, Seibert (2018) establishes that no more than ∼ 4 months – or 10 million seconds – of waking visual experience is needed to reach adult-level primate IT cortex (as assessed by its capability to support adult level object recognition).
From this estimate, we can compute how many supervised updates per second different models in Figure 1A would require (assuming those updates are evenly distributed over themillion seconds).
For instance, the fully trained model’s 55 million supervised updates translate to 5.5 updates every second, whereas the model with 1 million updates and 76% relative brain predictivity translates to one labeled image update every 10 seconds which appears more plausible given the upper limit of 2-3 saccades per second in humans (Yarbus, 1967; Gibaldi & Sabatini, 2020).
“A T - BIRTH ” SYNAPTIC CONNECTIVITY YIELDS REASONABLE BRAIN PREDICTIVITY WITH NO TRAINING AT ALL
If few supervised updates can get model representations fairly close to a fully trained model (Figure 1), how close are the initial representations without any training?
In relation to biology and following the introduced framework of treating all consecutive training as developmental learning, these “at-birth” synaptic connections would result from information encoded in the genome as a product of evolution.
Due to the genome’s capacity bottleneck, it is infeasible to precisely encode every synapse.
Primary visual cortex alone contains ∼ 1.4E8 neurons per hemisphere (Leuba & Kraftsik, 1994), ∼ 1E synapses per neuron, each requiring ∼ 37 bits per synapse (Zador, 2019).
Thus, without any clever rules, specifying the connections in only one hemisphere of V1 could require up to ∼ 5.2E12 bits – orders of magnitude more than the entire genome’s 1GB = 8E9 bits (Zador, 2019).
Sampling synaptic weights from reasonably compressed distributions on the other hand places little memory requirements on genetic encoding while potentially yielding useful initial weights.
Current machine learning techniques for initializing weights, such as Kaiming Normal (He et al., 2015), sample from a Gaussian distribution centered around zero
To test the hypothesis that the genome might already encode more powerful initial representations with synaptic wiring sampled from distributions speciﬁed by only few bits, we explored multidimensional distributions as a more expressive alternative.
These distributions only require a small number of parameters, but unlike current generic initializers, we explicitly specify them for each layer.
To determine the right parameterization, we compress a trained model’s weights into clusters which we then sample from (“ Weight Compression, WC ”).
More speciﬁcally, for all convolutional layers except the ﬁrst, we cluster the kernel weights and later sample from the clusters.
We determine the number of clusters with the elbow method (Thorndike, 1953): 11 for V1, 13 for V2, 16 for V4, and 15 for IT.
To capture the relative importance of clusters we ﬁt a normal distribution to the cluster frequency over kernels.
In batch normalization layers, we ﬁt one normal distribution each to the weights and biases.
For the ﬁrst convolutional layer only, we employ a Gabor prior on the weights following studies in V1 (Hubel & Wiesel, 1962; Jones & Palmer, 1987)
(Appendix B).
This results in 33 KB (4,166 parameters) to specify network initialization, compared to 423 MB for a trained model’s weights (assuming 8 byte per parameter).
Model interpretability studies (Zeiler & Fergus, 2013; Olah et al., 2020;
Cammarata et al., 2020) classify model weights, comparable to WC ’s representation.
Visualizing the weight compressions from trained CORnet-S weights (Figure 2B), we ﬁnd that the ﬁrst layer’s Gabor ﬁlters qualitatively align with an analysis by Cammarata et al.
Cluster centers seem to represent an intuitive division of channel types with opposite types in every layer.
Applying WC to CORnet-S, we ﬁrst obtain a compressed and clustered set of parameters, from which we sample entirely new weights to yield a new model CORnet-S WC .
This model is not trained at all and we only evaluate the goodness of its initial wiring on the suite of Brain-Score benchmarks.
Strikingly, we ﬁnd that even without any training, CORnet-S WC achieves 54 ± 1 .
5% of the brain predictivity score relative to a fully-trained model (Figure 2), representing a 12 percent point improvement ( n = 10 seeds; permutation test p <
1 E − 5 ) over the Kaiming Normal initialized model with a score of 43 ± 1 .
Early ventral stream regions V1 and V2 are predicted especially well with no loss in score but we note that these two benchmarks are less well predicted by the trained model to begin with.
V4 scores also approximate those of a trained model relatively well
The major drop occurs in the IT and especially behavioral scores where CORnet-S WC only reaches 39% and 6% of the trained model’s score respectively.
Similarly, a trained linear decoder on
CORnet-S WC ’s IT representations only reaches 5% of a trained model’s ImageNet top-1 accuracy.
While intuitively sampling weights from a compression of trained weights should somewhat recover reasonable brain predictivity scores, we note that not every implementation satisﬁes both high brain predictivity and sufﬁcient compression for the genome bottleneck (Appendix B.3).
Weight Compression explores the hypothesis that evolution may have discovered an initialization strategy with improved at-birth representations (relative to current initializations).
WC is most likely not how evolution found the at-birth synaptic connections, but shows that with nearly identical capacity, an alternative initialization distribution leads to networks that are more brain-like in their adult state – revealing a new space of possibilities (hypotheses) that should be considered (see Appendix B.4 for more details on biological plausibility).
These ﬁndings further suggests that matching representations in higher visual regions might be especially dependent on visual experience whereas early visual regions might already be reasonably well speciﬁed without experience.
T RAINING THIN DOWN - SAMPLING LAYERS REDUCES THE NUMBER OF UPDATED SYNAPSES WHILE MAINTAINING HIGH BRAIN PREDICTIVITY
While improved “at-birth” connectivity can reach 54% of a fully-trained model’s score, additional experience dependent updates appear necessary to reach higher predictivities.
With standard train- ing, each iteration simultaneously updates all of the millions of synaptic weights in the neural network, which may be difﬁcult to implement biologically.
Alternatively, learning could take place preferentially in speciﬁc components.
Cortical circuits are heterogeneous and different neuronal populations undergo distinct plasticity mechanisms.
For example, neurons in supra- and infragranular layers adapt more rapidly than those in layer 4, where inputs from lower areas arrive, as observed in rat somatosensory cortex (Diamond et al., 1994) and primate V1 (Schoups et al., 2001).
As a proof-of-principle that training a reduced set of layers can retain high performance, we propose a novel thin training technique, which we term Critical Training (CT; Figure 3A) .
CT updates only the weights in critical layers, instead of updating every single model synapse.
In CORnet-S, each of the blocks has one down-sampling layer to produce an area’s ﬁnal representation (Figure 3B).
We explore successive variants of applying CT up to a block in the architecture and then training the following blocks, e.g. freezing V1, V2, V4 with critical training of the respective down-sampling layers and additional IT training.
The ﬁnal CT ventral stream model is almost completely frozen and only the synapses generating each cortical area’s output are trained.
We compared Critical Training against two alternative approaches: 1) reducing the trained parameters by freezing entire model blocks, for instance keeping V1 and V2 blocks ﬁxed while training V and IT blocks.
We term this block-wise freezing and training approach Downstream Training (DT) .
And 2) an approach proposed by Frankle et al.
(2021) where only the BatchNorm parameters in a network are trained while all other parameters are kept at their initial values (“ BatchNorm, BN ”).
Compared to standard back-propagation training all the weights, all three approaches ( CT, DT, BN ) reduce the number of trained parameters (Figure 3C).
However, while the average score with DT (gray) already drops below 65% with over a quarter of trained parameters remaining and BN drops to 62% with very few parameters, CT (blue) maintains over 75% with only 1.4 out of 52.8 million parameters trained.
The choice of which critical layers to train also matters: training the connecting layers between regions – i.e. the last (down-sampling, default) or ﬁrst (up-sampling) layer – retains most of the performance whereas training layers such as BatchNorm performs worse.
By reducing the number of trained parameters, Critical Training also yields engineering beneﬁts with more than 40% of the ImageNet score maintained at < 3% of parameters trained – a signiﬁcant improvement over 22% accuracy with BatchNorm (Frankle et al., 2021) while adding only a small number of additional weights.
CT further reduces training time by 30% per epoch.
H IGH BRAIN PREDICTIVITY CAN BE ACHIEVED WITH A RELATIVELY SMALL NUMBER OF SUPERVISED SYNAPTIC UPDATES
All three training reduction methods independently minimize the number of supervised synaptic updates required to reach a reasonably high brain predictivity score.
Reducing the number of supervised updates minimizes the number of epochs and images (Section 3); Weight Compression (WC) improves the at-birth synaptic connectivity for high initial scores with no training at all (Section 4); and Critical Training (CT) reduces the number of synapses that are updated during training (Section 5).
Testing synergies between these strategies, we combined all three methods to build novel models that only require a small number of supervised synaptic updates to reasonably capture the mechanisms of adult ventral visual stream processing and object recognition behavior.
Figure 4A shows the average brain predictivity of a range of models with varying numbers of supervised synaptic updates relative to a standard trained CORnet-S (black dot, 3,000 trillion supervised synaptic updates).
With a reduced number of supervised updates (training epochs and labeled images)
but standard initialization and training all weights (light blue dots), models require 5.2 trillion updates to achieve > 50% of the score of a fully trained model and about 100 trillion updates to reach 80% brain score.
Adding WC+CT (dark blue dots), the corresponding model already reaches 53% at birth with 0 supervised synaptic updates.
At 0.5% the updates of a fully trained model (14 trillion vs. 3,000 trillion), models then reach 79% of the score ( (cid:21) model with modeling choices marked in Figures 1 to 3).
Continuing standard training from this 79% model, we can achieve 100% of the score with 15 additional epochs of 1,028 trillion supervised synaptic updates (one third of the fully trained model’s 3,000 trillion updates).
Reference models (gray dots) MobileNet (Howard et al., 2017) and ResNet (He et al., 2016) obtain high scores, but also require many supervised synaptic updates.
HMAX (Riesenhuber & Poggio, 1999) is fully speciﬁed with no updates but lacks in score.
[Figure]
C T
[Figure]
e tt e r
W C
+
C
[Figure]
o r
[Figure]
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[Figure]
We next examined interactions between methods by comparing models initialized with WC and trained with CT to models with standard initialization and training all weights, when both are trained with fewer epochs and images.
Figure 4B shows the percent point difference between the two model families.
WC+CT yield strong beneﬁts (green numbers) in a regime with few supervised updates, improving by up to 27 percent points when training for only 1 epoch on 1,000 images.
With many updates on the other hand, WC+CT is less advantageous than standard training (red numbers): with all 43 epochs and 1.28M images, the score reduces by 17 percent points.
WC+CT therefore most positively interact with a small budget of supervised updates (which is the focus of this work).
D ISSECTING TRAINING REDUCTIONS
We asked whether the developed techniques would generalize to architectures other than the
CORnet-S architecture they were based on.
Beyond establishing the methods as more general, this can be seen as a novel way to construct model taxonomies.
We therefore applied Weight Compression (WC) and Critical Training (CT) to ResNet-50 (He et al., 2016) and MobileNet-V1 (Howard et al., 2017) architectures, both high-performing models on Brain-Score.
We used WC distributions determined on CORnet-S, i.e. we tested transfer without re-ﬁtting.
WC+CT maintain 91% of the score in ResNet despite an almost 80% reduction in parameters.
When applied to MobileNet, the average score drops by 22% and parameters are reduced less strongly (43%).
This difference in performance could be due to MobileNet already being very compressed, or having a less similar architecture.
With most analyses so far comparing an average score, we dissected the relative contributions of
WC and CT to individual benchmarks (Figure 5B).
We compared KN to WC initialization, as well as resulting models after critical training ( KN+CT and WC+CT ).
WC initialization improves most over KN in early visual regions V1 and V2, while additional training with CT is most beneﬁcial in mid- to high-level visual cortex V4 and IT, as well as the behavioral benchmark.
[Figure]
D ISCUSSION
We developed a range of models with neural and behavioral scores approaching those of the current leading model of the adult ventral visual stream as quantiﬁed in Brain-Score, while requiring only a fraction of supervised synaptic updates.
These models were built by complementarily 1) reducing the number of supervised updates, i.e. training epochs and labeled images; 2) improving the “at birth” distribution of synaptic connectivity; and 3) training only critical synapses at the end of each model area.
The techniques and resulting models proposed here are ﬁrst steps to more closely modeling not just adult primate visual processing, but also exploring the underlying mechanisms of evolution and developmental learning.
These proof-of-principle demonstrations are far from accounting for the rich information encoded in the genome or the developmental learning that together result in adult mechanisms of visual processing, and require further experimental validation.
We here started from CORnet-S, one of the leading models on Brain-Score, but does not fully predict all brain measurements (0.42 absolute score).
We veriﬁed favorable transfer to models with similar architectures such as ResNet, but generalization to an already compressed MobileNet was limited (Figure 5A).
Relating to genomic mechanisms, the proposed techniques should generalize to other domains such as auditory processing.
With the capacity bottleneck in the genome, mechanisms for wiring up would likely be shared between similar systems.
The fact that early visual areas converge earlier during training (Figure 1) and are better predicted than higher areas by WC initialization is consistent with developmental studies of the primate ventral stream.
In humans, behaviors that rely on lowlevel spatial and temporal processing of visual inputs reach adult-like performance considerably earlier than complex visual behaviors that rely on higher cortical regions, such as face perception (Ellemberg et al., 1999; Grill-Spector et al., 2008).
A critical component in more closely modeling primate development is to reduce the dependence on labels altogether.
Recent unsupervised approaches are starting to rival the classiﬁcation performance of supervised models (Caron et al., 2018; H´enaff et al., 2019; Zhuang et al., 2020) and combining them with the advances presented here could further reduce the number of synaptic updates.
More precise biological measurements are required to quantify the number of (parallel) experiencedependent updates.
Current unsupervised techniques however still require back-propagation which is routinely criticized as non-biological, among others due to the propagation of gradients (Grossberg, 1987; Whittington & Bogacz, 2019; Hunsberger, 2017).
Local learning rules (L¨owe et al., 2019; Xiong et al., 2020) might alleviate these concerns and with critical training (Figure 3), it could be sufﬁcient to learn in only a subset of layers.
The changes to model initialization and training presented here serve as a proof-of-principle that models can be changed to more closely align with primate development by reducing training steps with labeled images and improving initialization.
It is also possible to achieve high brain predictivity when training only a fraction of weights, but all these models are still far from the actual biological mechanisms.
We expect future work in this direction to further close the gap with improved evolutionarily encoded wiring mechanisms and developmental learning rules.
A B ENCHMARK DETAILS
We use the benchmarks as implemented in www.github.com/brain-score/ brain-score at commit 96b0711, and convert base models to brain models with www.github.com/brain-score/model-tools at commit 2f778c6.
Images were presented at 4 degrees without aperture for the V1 and V2 benchmarks and at 8 degrees for the V4,
IT, and behavior benchmarks.
Models committed to an input size of 8 degrees visual angle.
B W EIGHT COMPRESSION DETAILS
For all convolutional layers except the ﬁrst, we cluster kernel weights in a layer using the k-means algorithm (Fix & Hodges, 1951).
The number of clusters is determined using elbow (Thorndike, 1953) (see Table 1).
To capture the relative importance of clusters we ﬁt a normal distribution N f for each cluster with µ f as the cluster frequency over kernels and σ f as the frequency standard deviation.
To sample weights for a kernel, we ﬁrst sample a cluster distribution i ∼ N f per kernel and then obtain channel weights by sampling from a Gaussian with (cid:126)µ i as the cluster center and the standard deviation (cid:126)σ i of clustered weights.
In batch normalization layers, we ﬁt one normal distribution each to the weights and biases.
B. C OMPRESSING THE FIRST LAYER WITH A G ABOR PRIOR
The weight compression approach we use in Section 4 is based on different initialization techniques, applied to different layers.
For the very ﬁrst layer of size 7 × 7 we found a Gabor ﬁlter most effective following studies in V1 (Hubel & Wiesel, 1962; Jones & Palmer, 1987).
To generate the Gabor kernels we ﬁt trained channel weights to a Gabor function
[Equation]
where
[Equation]
LAYER
NUM CLUSTERS
V1.conv
V1.conv
V2.
[Table]
input V2.conv V2.conv V2.
[Table]
V2.skip
V4.conv input V4.conv V4.conv V4.conv
V2.skip
IT.conv IT.conv IT.conv IT.skip x rot and y rot are the orthogonal and parallel orientations relative to the grating, θ is the angle of the grating orientation, f is the spatial frequency of the grating, φ is the phase of the grating relative to the Gaussian envelope, σ x and σ y are the standard deviations of the Gaussian envelope orthogonal and parallel to the grating, which can be deﬁned as multiples ( n x and n y ) of the inverse of the grating frequency and C is a scaling factor.
The function is ﬁt per channel, which leads to a set of Gabor parameter for each of the 3 RGB channels.
We then ﬁt a multidimensional mixture of Gaussians to the combination of all ﬁlter parameter per kernel, resulting in a kernel parameter set.
For the three RGB input channels in the ﬁrst layer and the 8 Gabor parameters we therefore ﬁt to 3 × 8 = 27 parameters.
We evaluate the best number of components (number of distinct Gaussian distributions) based on the Bayesian Information Criterion (Schwarz, 1978) and use 4 components for the ﬁrst layer of CORnet-S. To generate new kernels we sample a kernel parameter set from this mixture distribution and apply them to the described
Gabor function that spans the weight values.
B.
[Equation]
In addition to convolutional layers, models consist of several Batchnorm layers, which contain a learnable bias and weight term.
To initialize these terms, we ﬁt a normal distribution per weight and bias vector of the trained values and sample from this distribution.
Note that BatchNorm layers contain running average means and standard deviations for normalization purposes, which are applied at validation time.
Those terms are set to zero when no training has happened, but cause score changes once the model has processed the dataset.
During training the mean and standard deviation of the current batch are used instead.
B. A LTERNATIVE APPROACHES
We have explored a variety of weight compression methods applied to different layers and evaluate their performance “at birth” without training and when trained with critical training.
Figure 7 shows brain predictivities of several alternative compression methods implemented as follows:
• WC Weight compression approach with clustering as described in Section 4, using a Gabor prior approach for the ﬁrst layer, noisy cluster sampling for convolutional layers and ﬁtted normal distributions for Batchnorm layers (4,166 parameters for CORnet-S). compress weights “at without any and after training critical Our best clustering-based approach WC achieved similar re
- sults as the Mixture Gaussian approach ( ∼ 3 percent points mean difference) but leads to more diverse clusters. Performance drops when solely sampling weights from kernel based normal distributions ( Kernel normal ) and additionally disabling the Gabor prior ( No Gabor prior ) • Mixture Gaussian Instead of sampling weights from cluster centers, this approach uses multidimensional distributions for convolutional layers with kernel size > 1 .
We ﬁt a mixture Gaussian distribution per layer to the weights of a channel over all kernels.
To sample a new kernel, we sample individual channels from this distribution.
For convolutional layers with a kernel size of 1 we draw weights from a normal distribution adjusted per kernel as described in the next item (428,114 parameters for CORnet-S).
• Kernel normal
All weights are sampled based on normal distributions.
We ﬁt mean and standard deviation to the weights of one trained kernel and resample a new kernel from this distribution.
We do this separately for every kernel to generate a whole layer.
This approach is similar to the BatchNorm sampling method where we compress BatchNorm weight and bias terms instead of kernels (433,735 parameters for CORnet-S).
• No Gabor prior To evaluate the importance of the Gabor prior we use the Kernel normal model and apply the same normal distribution approach to layer one instead of Gabor sampling.
Performance drops by 13 percent points without training, and by 9 percent points after critical training.
(20,026 parameters for CORnet-S)
We speculate that increasing the number of clusters would follow logarithmic growth with respect to the performance of the model: i.e. a small number of clusters can already yield useful wiring (as explored here), while adding more clusters will have a positive effect but with decreasing effect size.
B. B IOLOGICAL P LAUSIBILITY
Weight Compression (WC)’s cluster-based initialization generates an “at-birth” network that already captures some useful aspects of the visual inputs in a compressed manner and allows the system to learn faster. The way that these clusters are determined (from a previously trained network) is not biologically plausible and does not correspond to any evolutionary mechanism we are aware of. However, independent of how these clusters are determined, WC shows that it is possible to encode certain priors about a system’s wiring diagram in a very compressed manner.
Evolution also acquired aspects of the visual inputs (with a different strategy) and encoded them in the genome in a lower information regime.
For example, we know that there are multiple neuro-developmental mechanisms, such as spontaneous retinal waves and axon guidance cues, that signiﬁcantly shape the architecture and function of the visual system requiring no (or very little) visual experience ((Huberman et al., 2008)).
These mechanisms depend on a relatively small number of proteins encoded in a genome and give rise to a highly complex pattern of at-birth synaptic connectivity representing a very large compression of information.
C WC INITIALIZED AND CT TRAINED MODEL ANALYSIS
Our best model WC+CT beneﬁts from a combination of improved initialization through weight compression, and critical training.
Figure 8A shows models with standard initialization and training all weights, but with fewer supervised updates (cf.
Figure 1), models that only train down-sampling layers ( CT ), and models that combine critical training with weight compression ( WC+CT ).
A model initialized with weight compression achieves (only WC ) 54% brain predictivity score with 0 supervised synaptic updates.
Figure 8B and C show detailed brain predicitivity scores, relative to a fully trained model, for models initialized and trained with WC+CT (B) and models initialized with standard Kaiming Normal and training all weights (C) when trained with a range of epochs and labeled images.
The speciﬁc benchmark scores when either training with all labeled images for a varying number of epochs (Figure 9A) or when training with fewer labeled images until convergence (Figure 9B) show the benchmarks of early visual achieve the best results, relative to a fully trained model.
The V1 score is identical over all training states, since we do not train the V1 area.
Notably, ImageNet performance of these networks seems to not be predictive of their brain predictivity, since even untrained networks with at-chance ImageNet performance correspond reasonably well to e.g. V1 (Figures 1 and 2).
New normative tasks might be required to explain these results, such as model robustness to image corruptions (Dapello et al., 2020).
[Equation] TRAINING REDUCTIONS – DETAILS D. T RANSFER TO R ES N ET AND M OBILE N ET
To show the generalization of our approach we applied the weight compression methods to a ResNet50 (He et al., 2016) and a MobileNet (Howard et al., 2017) (version 1, multiplier 1.0, image size 224) architecture.
We do not regenerate sampling distributions or clusters based on the new architectures trained weights, but used the CORnet-S based distributions to sample new weights for the different architectures.
Since CORnet-S is inspired by ResNet modules, we applied our critical training approach by training all conv3 layers (equivalent down sampling layers) of ResNet50.
For MobileNet we explored various layer mappings.
When training only the very few layers that result in reduced feature size, which are implemented as depthwise separable convolutional layers and appear three times overall, performance performance dropped close to random.
Those layers however are mapped to CORnet-S’ conv2 layers due to their 3 × 3 kernels whereas critical training in CORnet-S trains conv3 down-sampling layers with a kernel size of 1 × 1 .
To transfer our critical training approach, we therefore additionally train the 1 × 1 MobileNet layers corresponding to conv3.
This training version allows for more training but still reduces the amount of trained parameters by 43% while maintaining 78% of the original score.
For both transfer methods we initialize the ﬁrst layer using the Gabor method based on CORnet-S’s mixture-of-Gaussian distribution.
Since the
Gabor function is scalable we can produce Gabor kernels of varying size.
Furthermore we disable
BatchNorm biases and weights in all transfer models by freezing them to default values.
We found that transferring those distributions on new architectures harms brain predictivity scores.
Nevertheless, the BatchNorm layers still normalize activations by applying the running average and standard deviation.
D. C OMPARISON OF TECHNIQUES TO REDUCE SUPERVISED SYNAPTIC UPDATES (F IG . 5B)
To analyse the relative contributions of Weight Compression and Critical Training we compare brain predictivitiy scores of different models in Figure 5B:
• KN
A model initialized by standard Kaiming Normal initialization without training.
• WC A model initialized by our Weight Compression initialization, described in Section 4, without training.
• KN+CT
The KN -initialized model trained with Critical Training until convergence,
i
e. three downstream layers and the decoder are trained and all other layers remain unchanged.
• WC+CT The WC -initialized model with Critical Training .
V1 scores do not change because weights in the V1 model area are all frozen.
E T RAINING DETAILS
We used PyTorch 0.4.1 and trained the model using the ImageNet 2012 training set Deng et al.
We used a batch size of 256 images and trained on a QuadroRTX6000 GPU until convergence.
We start with a learning rate of 0.1 and decrease it four times by a factor of ten when training loss does not decrease over a period of three epochs.
For optimization, we use Stochastic Gradient
Descent with a weight decay 0.0001, momentum 0.9, and a cross-entropy loss between image labels and model logits.
We trained all models with these settings except the standard Mobilenet, where we used the pretrained tensorﬂow model.
Since the number of epochs for this model are not clearly stated, we use the published value of 100 training epochs Howard et al.
The training time of a full CORnet-S with standard Imagenet dataset for 43 epochs is ∼ 2.5 days.
All variations with less weights/images/epochs trained in shorter time.
Reference models trained for 4 days at most under the described settings.
If not further speciﬁed, we show results of one training run.
When showing error bars we used seeds 0, 42 and 94.
Code to reproduce our analyses from scratch, including the framework for weight compression and critical training, as well as pre-trained models, will be made available through GitHub.