Meta-Learning the Inductive Biases of Simple Neural Circuits
Authors: Will Dorrell
Maria Yuffa
Peter E. Latham
Venue: Submitted to ICLR 2023 (2023)
Keywords: Inductive Bias
Generalisation
Meta-Learning
Spiking Neural Network
Neuroscience

Abstract: Animals receive noisy and incomplete information, from which we must learn how to react in novel situations. A fundamental problem is that training data is always finite, making it unclear how to generalise to unseen data. But, animals do react appropriately to unseen data, wielding Occam's razor to select a parsimonious explanation of the observations. How they do this is called their inductive bias, and it is implicitly built into the operation of animals' neural circuits. This relationship between an observed circuit and its inductive bias is a useful explanatory window for neuroscience, allowing design choices to be understood normatively. However, it is generally very difficult to map circuit structure to inductive bias. In this work we present a neural network tool to bridge this gap. The tool allows us to meta-learn the inductive bias of neural circuits by learning functions that a neural circuit finds easy to generalise, since easy-to-generalise functions are exactly those the circuit chooses to explain incomplete data. We show that in systems where the inductive bias is known analytically, i.e. linear and kernel regression, our tool recovers it. Then, we show it is able to flexibly extract inductive biases from differentiable circuits, including spiking neural networks. This illustrates the intended use case of our tool: understanding the role of otherwise opaque pieces of neural functionality, such as non-linearities, learning rules, or connectomic data, through the inductive bias they induce.

META- LEARNING THE INDUCTIVE BIASES OF SIMPLE NEURAL CIRCUITS
Anonymous authors Paper under double-blind review
ABSTRACT
Animals receive noisy and incomplete information, from which we must learn how to react in novel situations. A fundamental problem is that training data is always finite, making it unclear how to generalise to unseen data. But, animals do react appropriately to unseen data, wielding Occam's razor to select parsimo- nious explanation of the observations. How they do this is called their inductive bias, and is implicitly built into the operation of animals' neural circuits. This relationship between an observed circuit and its inductive bias is useful explana- tory window for neuroscience, allowing design choices to be understood norma- tively. However, it is generally very difficult to map circuit structure to inductive bias. In this work we present a neural network tool to bridge this gap. The tool allows to meta-learn the inductive bias of neural circuits by learning functions that a neural circuit finds easy to generalise, since -oneeea-geee functions are exactly those the circuit chooses explain incomplete data. We show that in systems where the inductive bias is known analytically, i.c. linear and ker- nel regression, our tool recovers it. Then, we show it is able to flexibly extract inductive biases from differentiable circuits, including spiking neural networks. This illustrates the intended use of our tool: understanding the role of otherwise opaque pieces of neural functionality, such as non-linearities, learning rules, connectomic data, through the inductive bias they induce.
INTRODUCTION
Generalising to unseen data is fundamental problem for animals and machines: you receive set of noisy training data, say an assignment of valence to the activity of a sensory neuron, and must fill in the gaps to predict valence from activity, Fig. IA. This is hard since, without prior assumptions, is completely underconstrained. Many explanations or hypotheses perfectly fit any dataset (Hume) 1748), but different choices will lead wildly different outcomes. Further, the training data is likely noisy; how you choose to sift the signal from the noise can heavily influence generalisation, Fig. IB.
Generalising requires prior assumptions about likely explanations of the data. For example, prior belief that small changes in activity lead to correspondingly small changes in valence would bias you towards smoother explanations, breaking the tie between options and 2 in Fig. 1A. is a learner's inductive bias that chooses certain, otherwise similarly well- well-fitting, explanations over others.
A) VALENCE
B) VALENCE
VALENCE
OPTION 2
OPTION 2
SHALLOW NETWORK
TRAINING DATA
X
OPTION
DEEP NETWORK
OPTION
SENSORY NEURON ACTIVITY
SENSORY NEURON ACTIITYY
SENSORY NEURON ACTIVITY
Figure 1: Generalisation Requires Prior Assumptions. A: The same dataset is perfectly fit by many functions. B: Different assumptions about signal quality lead to different fittings. C: Training a (shallow) or 8 (deep) layer ReLU network on the same dataset leads to different generalisations.

---

The inductive bias of learning algorithm, such neural network, can be powerful route to understanding in both Machine Learning and Neuroscience. Classically, the success of convolu- tional neural networks can be attributed to their explicit inductive bias towards consslationinvariinn classifications (LeCun et al. 1998), and these ideas have since been very successfully extended to networks with range of structural biases (Bronstein et al.] (02T). Further, many network features have been linked to implicit regularisation of the network, such as the stochasticity of SGD (Mandt ct al., 2017), parameter initialisation (Glorot & Bengio 2010), early stopping (Hardt et al. 2016),
or low rank biases of gradient descent Gunasekar et al. (0177).
In neuroscience, the inductive bias has been used to assign normative roles to representational or structural choices via their effect on generalisation. For example, the non non-linearity in neural net- work models of the cerebellum has been shown to have a strong effect on the network's ability to generalise functions with different frequency content (Xie et al., 2022). Experimentally, these network properties vary across the cerehellum, hence this work suggests that each part of the cere- bellum may be tuned to tasks with particular smoothness properties. This is exemplary of a spate of recent papers applying similar techniques to visual representations (Bordelon et al., 2020; Pandey et al. 202I), mechanosensory representations (Pandey et al., 2021p. and olfaction (Harris) 2019))
Despite the potential of using inductive bias to understand neural circuits, the approach is limited, since mapping from learning algorithms to their inductive bias is highly non-trivial. Numerous circuit features (learning rules, architecture, non-linearities, etc.) influence generalisation. For ex- ample, training two simple ReLU networks of different depth to classify three data points leads to different generalisations for non-obvious reasons, Fig. IC. In constrained cases analytic bridges have mapped learning algorithms to their inductive bias. In particular, the study of kernel regres- sion, an algorithm that maps data points to a feature space in which linear regression to labels is then performed (Sollich) 1998; Bordelon al., 2020; [Smon et al., 202T), has been influential: all the cited examples of understanding in neuroscience via inductive bias have used this bridge. However, it severely limits the approach: most biological circuits cannot be well approximated as performing a fixed feature map then linearly regressing to labels!
Here, we develop a flexible neural network approach that is able to meta-learn the inductive bias of essentially any differentiable supervised learning algorithm. It follows a meta-learning framework [Vanschoren] 2019): an outer neural network (the meta-learner) assigns labels to a dataset, this labelled dataset is then used in the inner optimisation to train the inner neural network (the learner). The meta-learner is then trained on a meta-loss which measures the generalisation error of the learner to unseen data. Through gradient descent on the meta-loss, the meta-learner meta-learns to label data in a way that the leamer finds easy to generalise. These easy-to- -ooeeer-gee functions form description of the inductive bias. In other words, if the network receives few training points from this function it will generalise appropriately, and generally the network will regularly use this function to explain finite datasets.
To our knowledge, the most related work is Li et al. (2021). Li et al. view sets of neural networks, trained or untrained, as a distribution over the mapping from imput to labels. They fit this distribution by meta-learning the parameters of a gaussian process which assigns a label distribution to each input. This provides an interpretable summary of fixed sets of network. In our work we do something very different: rather than focusing on a fixed, static set of networks, we find the inductive biases of learning algorithms via meta-learning casily learnt functions.
In the following sections we describe our scheme, and validate it by comparing to the known induc- tive biases of linear and kernel regression. We then extend in several ways. First, networks are inductively biased towards areas of function space, not single functions. Therefore we learn a set of orthogonal functions that a learner finds easy to generalise, providing a richer characterisation of the inductive bias. Second, we introduce a framework that asks how a given design choice (architecture, learning rule, non-linearity) effects the inductive bias. To do that, we assemble two networks that differ only by the design choice in question, then meta-learn function that network finds much easier to generalise than the other. This can be used to explain why a particular circuit feature present. We again validate both schemes against linear and kernel regression. Finally we show our tool's 's flexibility in series of more adventurous examples: we validate it on a challenging differ- entiable learner (a spiking neural network); we show works in high- diigh-iinsssons by meta-learnin MNIST labels; and we highlight its explanatory power for neuroscience by using it to normatively explain patterns in recent connectomic data via their inductive bias.
2

---

2
A NEURAL NETWORK TO META-LEARN INDUCTIVE BIASES
Our main contribution is meta-learning framework for extracting the inductive bias of differen- tiable learning algorithms, Fig. 2A, that we describe in this section. In the outer-loop neural network, the meta-leamer, assigns labels to input sampled from some distribution, hence creating the real-world function that our circuit of interest will try learn. The inner-loop learning algo- rithm, the learner, is the circuit whose inductive bias we want to extract; for example, biological sensory processing circuit that assigns valences to inputs. When provided with training dataset of inputs and labels the learner adjusts its parameters according to its internal learning rules. Then the generalisation eπror of the trained learner measured on a held-out test set, and this is used as the meta-loss train the meta-learner. This process repeats, retraining the learner at every iteration and iteratively developing the meta-learner's weights, until the meta-learner is labelling the data in way that the learner finds easy to generalise after training a few datapoints (we used around 30). Thus, the meta-learner has extracted a function towards which the learner is inductively biased.
As just outlined, the meta-learner will find the easiest-to-generalis function, usually the one that assigns all inputs the same label. To avoid this trivial function, we introduce a term in the meta-loss that forces the distribution of labels to take a particular (non-constant) form. Specifically, it measures the Sinkhorn divergence between the meta-learner's ' label distribution and a uniform distribution from (other divergences also work, Appendix B). The full pseudocode is Algorithm
A)
L
Lg
B) LINEAR LEARNER:
0.75
C) KERNEL LEARNER:
META-LEARNER
LEARNER
off
fo(x)
(xx)
LATITUDE
p(x)
LONGITUDE
Figure 2: Meta-Learning the Inductive Bias. A: The meta-learner labels a dataset which used to train the learner. Gradient descent is performed loss made of the learner's 's generalisation error to unseen data (Cy), and the Sinkhorn divergence between the meta-learner's label distribution and target distribution (Ca), here choosen to be uniform from -1 1 to 1. B: The meta-learner learns linearly separable labelling of data sampled from a circle for a ridge regression learner. C: For kernel regression learner and data sampled from the surface of sphere, the meta-learner's labelling is very close to the predicted spherical harmonic (99% of norm within first order harmonics).
Algorithm 1: Pseudocode for Meta-Learning the Learner's Inductive Bias
Initialise meta-learner: fa(x)
2 while Step count < Total do
Generate dataset from input distribution: p(z)
Label using metalearner: & = fx
Split inputs and labels into test and train datasets: Drr & Dr
Train leaner using Dr, giving trained learner network: g(æ)
Label Dre using trained learner:
(xx)
Compute the generalisation error of the leaner: Lg E,, 22
Compute the Sinkhorn Divergence of metalearne''s labels from uniform [-1, Ca
10
Take & gradient step on meta-loss: Lg La
end
3

---

Our meta-learner must fit function that the learner generalise. To enable the meta- meta-learner to learn all functions the learner might plausibly generalise well, its function class could usefully be a superset of the learner's. s. Therefore, we choose the meta-leamer's architecture to be a slightly larger version of the learner's (though, beyond this, our findings appear robust, Appendix D).
We validate our scheme by meta-learning sensible functions for linear and kernel learners, whose inductive biases are known. First, for ridge regression on data sampled from a 2D circle the meta- learner assigns linearly separable labels, Fig. 2B; exactly the labels linear circuits easily generalise.
Next, we meta-learn kernel ridge regression's inductive bias. Kemel regression involves projecting the input data through fixed mapping to a feature space (e.g. the last hidden layer of fixed neural network) and performing linear regression from feature space to labels. Bordelon et al. (2020) show that the inductive bias of kernel regression can be understood through the kernel cigenfunctions (z)( with eigenvalue {),}). These are defined on input distribution p(æ:) via a kernel k(æ, x') that measures the similarity of two inputs in feature space:
f A(zz, x' (z))u((")dp(a²) ddp(=))= = (x).
(I)
The algorithm is inductively biased towards higher eigenvalue eigenfunctions; i.e., fewer train- ing points are needed to reach a given generalisation error when fitting high vs. low eigenvalue eigenfunctions. General functions can be understood by projecting onto the eigenbasis. Hence our meta-learner, in searching for kernel regression's 's sosiest-to-generalis non-constant function, should choose the highest eigenvalue eigenfunction.
To test this, we meta-learn the inductive bias a two-layer neural network with fixed first layer weights. We sample data uniformly from the sphere and randomly connect large hidden layer of ReI.U neurons to the three input neurons. The elements of this random weight matrix are drawn iid from standard normal, and the learning algorithm performs ridge regression the hidden layer activities. Previous work has analytically derived the kernel for this network, and computed it's eigenfunctions (Cho & Saul 2009; Mairal & Vert 2018), which are spherical harmonics. The higher the frequency of the spherical harmonics the lower its eigenvalue. Matching this, our network meta-learns one of the set of lowest frequency spherical harmonics, Fig. 2C.
3 META-LEARNING AREAS OF FUNCTION SPACE
Having validated our tool on some simple test cases, we now extend it find richer characterisation of the inductive bias. A given learning algorithm is inductively biased towards areas of function space, not just one particular function. To gain access to this larger space, we learn a series of meta- learners. The first of these is exactly as described above, then we iteratively introduce additional meta-learners. To ensure each meta-learne learns to the meta-loss that penalises the square of the dot product between the current labelling and that of all the previously trained meta-leamers, Fig. 3A. On a dataset {x.))
aspect of the inductive bias we add term
s
2
Lorthog = Σ (E (Z (X)) (w
for each previous meta-learners fe, (xx) and the current meta-learner for (x). From the learner's perspective nothing has changed, at each meta-step it simply learns to fit the meta-learner that is currently being trained. But each additional meta-learner must discover an easy-to- --gnrerllee func- tion that is orthogonal to all previous meta-learners.
We again validate this scheme against linear and kernel regression. When tested on linear regression of 2D data the meta-learners learn two orthogonal linearly separable labellings, then third orthog- onal function that the learner struggles to generalise, as expected, Fig. 3B. We then test on the same kernel regression network we described previously. Theory predicts that the meta-learners should learn the eigenfunctions in decreasing order of their eigenvalue, and we find that this is true to a good approximation, Fig. 3C, learning approximations the three first order spherical harmonies, and then three approximations to second order spherical harmonics.
For linear classifiers (e.g. linear and kernel regression), the full set of orthogonal functions explains the entire inductive bias. This won't 't be true in general. Nonetheless, we expect the set of orthogonal functions will still be helpful guide to network's inductive bias, even for non-linear classifiers.
4

---

A)
B) LINEAR LEARNER:
Lp
Ly
MSE: 0.003
0.75
MSE: 0.003
MSE: 0.1
12
ld
X3 22 Z1
0
META-LEARNER 3
0.4
META-LEARNER 2
C) KERNEL LEARNER:
META-LEARNER
MSE: 0,00004
MSE: 0.0007
MSE: 0.001
LEARNER
(xx)
LATITUDE
MSE: 0,007
MSE: 0,004
MSE: 0.004
X ~ p(x)
LONGITUDE
Figure 3: Meta-Learning Many Functions. A: We learn many meta-learners, each of which has to label orthogonally to all previous meta-leamnrs. B: For a linear learner the meta-learners learn two orthogonal linear functions and an orthogonal but hard learn third function. C: For kernel learner we learn 6 meta-learners, the first 3 approximate well first order spherical harmonics (96% norm overlap), and the next second order spherical harmonics (91% % norm overlap), as predicted.
FINDING THE EFFECT OF DESIGN CHOICES ON THE INDUCTIVE BIAS
Our work is motivated by the desire to understand how design choices in learning algorithms such architecture, learning rule, and non-linearities lead to downstream generalisation effects, particularly in biological networks. One additional setting which we have found useful is to compare two networks with some architectural difference between them, and learn functions that one of the networks finds much easier to generalise than the other. In this way, can build intuition for the impact design features on the inductive bias. To illustrate this we again create a meta-learner that labels data, but this time the labels used train two learners. We then train the meta- beaanee so that one learner (the chosen student) much better at generalising than the other (neglected student). This is done by minimising the generalisation errors of the chosen student minus the neglected student, Fig. 4A. Validating this approach on well understood algorithms, we show that it can find functions that kernel regression algorithm is able to learn better than linear regression, Fig. 4B, i.c. non-linearly separable function.
A)
LC
L
B)
META-LEARNER TARGET
Lg
In
0.75
LATTTUDE
CHOSEN STUDENT: KERNEL LEARNER
NEGLECTED STUDENT: LINEAR LEARNER
CHOSEN'S FIT MSE: 0.02
NEGLECTED'S 's FIT MSE: 0,2
META-LEARNER
0.08
0.08
LONGITUDE
x (X))
Figure 4: Meta-learning design choice impact. A: Labellings are learnt such that a chosen student generalises much better than neglected one. B: The meta-learner finds non-linear labelling for which kerel regression generalises an order of magnitude better than linear regression.
5

---

This illustrates some of the games that can be played in this setting. For example, you could play a co-operative game, in which you meta-learn a function that a set of learners all find easy to gen- eralise, and each learner could have different connectivity matrices to match the distribution in real animals, ensuring the tool does not over-fit to some specifie details. However as the losses become more complex training becomes harder, for example this adversarial setting between chosen and neglected student is hard to make robust if the two learners are relatively similar.
META-LEARNING APPLIED TO MORE COMPLEX LEARNING ALGORITHMS
So far we have developed and tested a suite of tools for extracting the inductive bias of learning algo- rithms. We now apply tools to networks whose inductive bias cannot be understood analytically. Specifically: we show our method works on challenging differentiable learner, a spiking neural network; we validate method our tool can give normative explanations for biological circuit features, by meta-learning the impact of connectivity structures the generalisation of model of the fly mushroom body. Our tool is flexible: by taking gradients through the training procedure we can meta-learn inductive biases for networks trained using PyTorch. We will provide code on github that produces our figures, including a basic ReLU network (Appendix A which should be easily adapted to networks of interest.
high-dimensional MNIST example; and illustrate how
5.1 SPIKING NEURAL NETWORK
The brain, unlike artificial neural networks, computes using spikes. How is open question. A recent exciting advance in this area is the surrogate gradient method, which permits gradient based training of spiking neural networks by smoothing the discontinuous gradient (Neftci 2019 Zenke & Vogels 202I). We make use of this development to meta-learn the inductive bias of spiking network, providing a challenging test case for our method.
META-LEARNER TARGETS
LEARNER FITS
A)
4
Lg
FIRST FUNCTION
m(t)
READOUT u(t)
SECOND FUNCTION 0.75
HIDDEN LIF LAYER
VALENCE,
O O
META-LEARNER
(x(
SPIKE TRAIN (AΦ)
-0.75
THIRD FUNCTON
NEURON
0.75
TIME,
p(A@)
-0.75
-0.5m
0
0.5 -0.5 0
0.5
PHASE DIFFERENCE, A$
Figure 5: Meta-learning through Spiking Network. A: Labellings are learnt that the spik- ing network, with weights trained via surrogate gradient descent, finds easy to generalise. Phase differences, Áé, are sampled uniformly and used to generate spike train by sampling from pois- son process with the following rates: for half the neurons rn Yau (1 + sin(t + B,))), where is neuron index and f,, are uniformly sampled offsets; for the other half we add phase shift: To = "mo + sin(L 0,, + 2 These populations represent sensory neurons the two ears, and Aø is the interaural phase difference. This activity feeds into population of linear-integrate- and-fire neurons, then onwards to a readout lincar-integrate neuron. The valence assigned the sum of the readout's activity over time. B: We learn three orthogonal meta-learners (as in section 33) and find the spiking network finds easiest to learn low frequency functions. Left: the meta-learner's target function. Right: the spiking network's labelling. As can be seen, the spiking network captures the main behaviour, but increasingly poorly at higher frequencies.

---

We study modification of model developed for recent tutorial (Goodman al. 2022 Zenke 2019), which is trained to assign a label to an incoming spike train. The network is a model of an interaural phase difference detection circuit. The input spike train is parameterised by phase difference, Ań, that generates two sets of spike trains, one in each ear, Fig. Fig. These spikes are processed through hidden layer of iinea-integrrae-and--ii neurons (III),, before reaching clas- sification layer. A real-valued valence is assigned by summing the output neuron's 's activity over the trial. The meta-learning framework is before: the meta-learner assigns valences to input phase differences, these labels are used to train the spiking network by surrogate gradient descent, then the meta-learner trained to minimise the learner's 's generalisation error and distribution loss. Our method works well, finding simple smoothness prior, Fig. B..
5.2 A HIGII-DIMENSIONLL MNIST EXAMPLE
Next, we test out method on a high- digeediiensional input dataset. Thus far, to visualise our results, we have only considered low dimensional input data. We demonstrate that our method continues to work high-dimensions by applying to a dataset made of the 0 and I MNIST digits (LeCun) 1998.. We meta-learn labelling of this dataset that simple convolutional neural network finds easy to generalise. Our meta-learner's 's architecture is also a convolutional neural network whose outputs are bounded between 0 and I, and the meta-learner must learn eusy-to-gencralis labelling with high variance. We find that the meta-learner consistently rediscovers the MNIST digits within the dataset, separating each digit into its class, figure 6 We return to the important question of understanding high-dimensional inductive biases in the discussion.
A)
La Z
Lg
B)
META-LEARNER LABELLED MNIST DIGITS
1.00
1.00
0.00
0.00
t
META-LEARNER
LEARNER
1.00
0.02
0.11
0,00
fx)
Figure 6: Meta-Learning on MNIST A: A meta-learner receives MNIST Os and 1s, and assigns labels, bounded between 0 and 1, that have high variance and can be easily generalised by the learner. B: 99% of digits are assigned a label, shown in title, consistent with MNIST class.
5.3 INTERPRETING CONNECTIVITY PATTERNS THROUGH THEIR INDUCED INDUCTIVE BIAS
A large maturing source of neuroscience data is (a list of which neurons connect to one another). However, there is currently a dearth of methods for interpreting this data (Litwin-Kumar & Turaga 2019). In this section, we show our tool can be used to give normative roles to connectomic patterns through their induced inductive bias. We study model of the fly mushroom body, beautiful circuit that fruit flies use to assign valence to odours (Aso et all 2014 Hige 2018). for which connectomic 2022)
data has recently become available (Zheng
2018
Odorants trigger a subset of the fly's olfactory receptors. These activations are represented in a small glomerular population (input neurons), projected large layer of Kenyon cells (hidden neurons), then onwards to output neurons that signal various dimensions of the odour's valence, Fig. 7 An error signal is provided if the fly misclassifies a good odour as bad, or vice versa, allowing the fly to update its weights and learn appropriate responses. Classically, the input-to-hidden hidddneen connectivity was assumed random; i.e., each hidden neuron connects to a few randomly selected input neurons. However, connectomic data has shown that hidden neurons preferentially connect to some inputs, and there are input groupings if a hidden neuron connects to one member of group it likely connects to many, Fig 7 (Zheng et al. 2018 2022). Zavitz al. (202T) tested networks with this connectivity on a battery of tasks and found that, compared to random, (1) they were better at identifying odours that activated over-connected inputs, and (2) they generalised assigned valence across a group (i.c. if you assign high valence to the activation of one neuron, you do the same for other neurons in the same group).
7

---

A)
SPARSELY CONNECTED LEARNER
B)
RANDOM 2-SPARSE LEARNER PREFERS NEURON BASIS
OUTPUT VALENCE:
MUSHHOOOM BODY OUTPUT
NEURONS (MBONS)
HCDEN NEURONS: KENYON CELLS
C)
Overconnected
OVERCONNECTION
PREFERRED NEURON
INPUT NEURONS:
GLOMERULI
x Unit Norm
-1
-1
1
-1
NEURON ACTIVITY
D) GROUPED CONNECTIVITY
E)
UNCONSTRAINED GROUPING WITHIN GROUP CLASSIFICATION
POSITIVE GROUPING
PROSS-GROUP GENERALISATION
NEURON
AVERAGE GROUP
GROUP
GROUP
NEURON 1
AVERAGE GROUP
Figure 7: Understanding Connectivity via Inductive Bias. A: We model the fly mushroom body as ReLU network with one large hidden layer. Each hidden neuron is connected to two of the five input neurons. B: The meta-leamer finds the labelling the learner generalises most easily. We show this labelling projected against each of the input neuron activities. As can be seen, the labelling depends on only one neuron's s activity, second from left. C: In the overconnected setting each hidden neuron still connects to inputs, but there is a strong bias towards connecting to the first, highlighted neuron. As a result, the meta-learner settles on a labelling that depends only on this neuron's ss activity. D: We explore the impacts of group connectivity, in which the input neurons are divided into two groups, and hidden neurons tend to be connected to two neurons from the same group. E: We train the meta-learner, and find that it's labelling depends only on neurons within the same group. The plot shows the projection of the datapoints into a subspace defined by the two neurons in the red group. The labelling depends linearly on position within this subspace. F: However, if the input-hidden connections are constrained to be positive, the meta-learner's ' labelling depends only on the average activity within each group, i.c. if one member of a group increases the output, so do all members; hence, the function generalises across group members.
We used our tool to verify and develop these findings by examining the effect of different connectiv- ity patterns on the inductive bias of a sparsely-connected model of the mushroom body, Fig. 77. A.AAs As baseline, fully connected networks are biased towards smooth functions, appendix A the simplest being those that assign valence based on one direction in the input space: high at one end, low at the other, like in Fig. - C. However, which direction is unimportant; they 're all equally easy to learn. Sparsity breaks this degeneracy, aligning the casiest to learn functions with the input neuron basis, figure 7 As such, sparse connectivity, which is ubiquitous in neuroscience, ensures the fly is best at assigning labels based on the activity of small collections of neurons. Next, we intro- duced the observed connectomic structure. Biasing the connectivity broke the degeneracy amongst neuron axes. The networks were, fairly intuitively, best at generalising functions that depended on the activity of overconnected inputs, figure 7 matching Zavitz et al. 2202T) Finally we introduce connectivity groups, figure 7Pp Without additional changes this does little, the neuron basis is still preferred and, unlike Zavitz et al. (02TT) generalisation across inputs not observed, figure Z Only when we additionally constrain the input-to-hidder connections to be excitatory (i.e. positive) do we see that the circuit becomes inductively biased towards functions that generalise across groups of inputs, figure F In retrospect this can be understood intuitively: positive weights and grouped connectivity ensure that a hidden neuron that is activated by one input will also be activated by other group members, encouraging generalisation. This effect is removed by permitting negative weights, which let members of the same group excite or inhibit the same hidden neuron.
Thus, verify the findings of Zavitz et al. 2021 without needing presuppose battery of tasks. In doing so we highlight how our tool can be used to gain insight into the role of circuit design choices, in particular, the importance of the neuron basis for sparsely connected networks.
8

---

6 DISCUSSION & CONCLUSIONS
We presented a meta-lea approach to extract the inductive bias of differentiable supervised learning algorithms, which we hope will be useful in normatively interpreting the role of features of biological networks. This approach required few assumptions beyond those that make the inductive bias an interesting way to conceptualise a circuit in the first place. We required, first, the circuit must be interpretable as performing supervised learning. Second, the input data must be specified. And, third, you must specify the way the circuit learns, and be able to take gradients through this learning process. We will discuss each of these requirements and ways they could be relaxed; regardless, is heartening that any circuit satisfying these will, in principle, suffice. The analytic bridge between kernel regression and its inductive bias (Bordelon et al., 2020; (Simon et al. 202T) has already found multiple uses in biology in just a few years (Bordelon & Pehlevan | et al. [2021] Harris
2019
Xie al., 2022). , despite its stringent assumptions. We hope that relaxing those assumptions will offer a route to allow these ideas to be applied more broadly.
The first requirement is that the learner performs supervised learning. This is often reasonable. Some circuits contain explicit supervision or eπror signals, like the fly mushroom body or the cerebellum (Shadmehr 2020), and generally brain areas that make predictions (i.e., all internal models), can use their prediction errors as a learning signal. A Alternatively, some circuits are well modelled as one area providing a supervisory signal for another, as in classic systems consolidation (McClelland et al. 1995 or receiving supervision from past version of themselves through replay (van de Ven al, 2020.. Nevertheless, modelling circuits as performing supervised learning will always be an ap- proximation, most simply due to unmodelled effects such as neuromodulation. As illustration of how our framework could be extended beyond supervised learning. we consider how neuromodula- tion could be incorporated. There exist models of how neuromodulation influences circuit function, for example by sharpening neural non-linearities (Ferguson & Cardin 2020; Aston-Jones & Cohen 2005), and these could be included in the learner model. The meta-learner could then meta-learn two outputs, one label and one neuromodulator. It's goal would be to meta-learn functions that the learner finds easy to generalise with limited quantities of neuromodulatory attention applied to specific, exemplar, training points.
Next, we could relax our second assumption, access to an input distribution, which is often lacking- This can be avoided by using real neural data as the input. Or, if neural data is limited, generative modelling could be used fit the neural data distribution and new samples drawn from that distri- bution. Finally, one could imagine single meta-learner that creates not only the label, but also the data. That is, the meta-learner could generate the entire dataset by transforming a noise sample into an input-output pair. This would have to be carefully regularised to avoid trivial input distributions, but could in principle learn the input statistics that particular networks are tuned to process.
Finally, in slightly kooky way, we could avoid specifyying the learning algorithm by interfacing with an animal directly! Animals' inductive biases are objects of interest in their own right, but can also give insight into the underlying neural processing. These biases could be studied by replacing the inner learner with a real animal that is trained on a labelled dataset from the meta-learner, then tested new datapoints. Since we cannot compute gradients through the computations of living animal, the meta-learner could be optimised using black- box optimisation procedures that rely only on meta-loss evaluations, like the Nelder-Mead method (Singer & Nelder 2009).
Despite our optimism for this approach, there remain challenges. Most fundamentally, sets of func- tions that learner easily generalises are still hard to interpret. We have shown how our tool can provide insight for low- low-dimensional inputs (Fig. 2 5), by comparing to ground truth labels (Fig. 6), or by projecting the learnt functions onto an appropriate basis (Fig. 7). However, to make the concept of inductive bias more powerful, more tools are needed to interpret the resulting functions.
To conclude, the inductive bias a promising angle from which to understand learning algorithms. Analytic bridges between circuit design and inductive bias have already 'explained' the presence of aspects of the circuit through their effect on the network's s generalisation properties in both artificial
(Canatar et al. 202 Bahri et al. 2021) and biological (Bordelon & Pehlevan| 2021 Pandey et al.) 2021 Harris 2019 Xie et al. (022) networks. However, these techniques require very constrain- ing assumptions. We have dramatically loosened these assumptions and shown meta-learning approach can flexibly extract the inductive bias of neural circuits. We have shown its utility in interpreting connectomic data, and we believe it will prove useful on other datasets and problems.

---

REFERENCES
Yoshinori Aso, Daisuke Hattori, Yang Yu, Rebecca M Johnston, Nirmala A Iyer, Teri-TB Ngo, Heather Dionne, LF Abbott, Richard Axel, Hiromu Tanimoto, al. The neuronal architecture of the mushroom body provides a logic for associative learning. elife, 3:c04577, 2014.
Gary Aston-Jones and Jonathan D Cohen. An integrative theory of locus coeruleus-norepinehhrie function: adaptive gain and optimal performance. Annu. Rev. Neurosi., 28:403 2005.
Yasaman Bahri, Ethan Dyer, Jared Kaplan, Jachoon Lee, and Utkarsh Sharma. Explaining neural scaling laws. arXiv preprint arXi::2102.06701, 2021.
Blake Bordelon and Cengiz Pehlevan. Population codes enable learning from few examples by shaping inductive bias. BioRxiv, 2021.
Blake Bordelon, Abdulkadir Canatar, and Cengiz Pehlevan. Spectrum dependent learning curves in kernel regression and wide neural networks. In International Conference on Machine Learning, pp. 1024-00244 1034. PMLR, 2020.
Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv preprint arXiv:2104. 1..004.1344 13478, 2021.
Abdulkadir Canatar, Blake Bordelon, and Cengiz Pehlevan. Spectral bias and task muskl alignment explain generalization in kernel regression and infinitely wide neural networks. Nature communi- cations, 12(1):1-12, 2021.
Youngmin Cho and Lawrence Saul. Kernel methods for deep learning. Advances in neural informa- fion processing systems, 22, 2009.
Josip Djolonga. torch-too-sample. httpss//gghabb.ccmmmoospspoooerr samp
2020.
Katie A Ferguson and Jessica A Cardin. Mechanisms underlying gain modulation in the cortex. Nature Reviews Neuroscience, 21(2):80-92, 2020.
Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference artificial intelligence and statistics, pp. 249-256. JMLR Workshop and Conference Proceedings, 2010.
Dan Goodman, Tomas Fiers, Richard Gao, Marcus Ghosh, and Perez. Spiking neural network models in neuroscience cosyne tutorial 2022. Y77cre2ML9s, 2022.
Nicholas
Arthur Gretton, Karsten M Borgwardt, Malte Rasch, Bernhard Schölkopf, and Alexander Smola.
A kernel two-sample test. The Journal of Machine Learning Research. 13(1):723-773, 2012.
Suriya Gunasekar, Blake E Woodworth, Srinadh Bhojanapalli, Behnam Neyshabur, and Nati Sre- bro. Implicit regularization in matrix factorization. Advances in Neural Information Processing Systems, 30, 2017.
Moritz Hardt, Ben Recht, and Yoram Singer. Train faster, generalize better: Stability stochastic gradient descent. In International conference machine learning, pp. 1225-1233 PMLR, 2016.
Kameron Decker Harris. arXiv:1909.02603 2019.
Additive function approximation in the brain. arXiv preprint
Toshihide Hige. What can tiny mushrooms in fruit flies tell us about learning and memory? Neuro- science Research, 129:8-16, 2018.
David Hume. An enquiry concerning human understanding and other writings. 1748.
Yann LeCun. The mnist database of handwritten digits. http://yann. lecun. com/exdb/nmist// 1998.
Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEFF, 86(1) 86(11):222 1):2782-2324, 1998.
10

---

Michael Y Li, Erin Grant, and Thomas L Griffiths. Meta-learning inductive biases of learning systems with gaussian processes. In Fifth Workshop on Leamnin at the Conference on Neural Information Processing Systems, 2021.
Ashok Litwin-Kumar and Srinivas C Turaga. Constraining computational models using electron microscopy wiring diagrams. Current opinion in neurobiology, 58:94-100, 2019.
Julien Mairal and Jean-Philippe Vert. Machine learning with kernel methods. Lecture Notes, Jan- uary, 10, 2018.
Stephan Mandt, Matthew Hoffman, and David M Blei. Stochastic gradient descent as approximate bayesian inference. arXiv preprint TeXiv:/704.000 04289, 2017.
James I McClelland, Bruce I. McNaughton, and Randall C O'Reellly. Why there complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory. Psychological review, 102(3):419, 1995.
Emre Neftci, Hesham Mostafa, and Friedemann Zenke. Surrogate gradient learning spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks. IEEE Signal Processing Magazine, 36(6):51-6) 2019.
Biraj Pandey, Marius Pachitariu, Bingni W Brumton, and Kameron Decker Harris. Structured ran- dom receptive fields enable informative sensory encodings. bioRxiv, 2021.
Reza Shadmehr. Population coding in the cerebellum: a machine learning perspective. Journal of neurophysiology, 124(6):0022-2551, 2020.
James B Simon, Madeline Dickens, and Michael R DeWeese. Neural tangent kernel eigenvalues accurately predict generalization. arXiv preprint arXiv:2110.03922, 2021.
Saša Singer and John Nelder. Nelder-mead algorithm. Scholarpedia, 4(7):2928, 2009.
Peter Sollich. Learning curves for gaussian processes. Advances in neural information processing systems, 11, 1998.
Gábor Székely and Maria L Rizzo. Energy statistics: A class of statistics based on distances. Journal of statistical planning and inference, 12398): 1272, 2013.
Gido M van de Ven, Hava T Siegelmann, and Andreas S Tolias. Brain-inspired replay for continual learning with artificial neural networks. Nature communications, 11(1):1-14, 2020.
Joaquin Vanschoren. Meta-learning. In Automated machine learning, pp. 35-61. Springer, Cham, 2019.
Marjorie Xie, Samuel Muscinelli, Kameron Decker Harris, and Ashok Litwin Kumaar Task- dependent optimal representations for cerebellar learning, bioRxiv, 2022.
D Zavitz, EA Amematsro, A Borisyuk, and SJ Caron. Connectivity patterns that shape olfactory representation in a mushroom body network model. 2021.
Freidemann Zenke. Spytorch. https:///eendooorrgheeooo//21220 oY0.a// 2019.
Friedemann Zenke and Tim P Vogels. The remarkable robustness of surrogate gradient learning for instilling complex function spiking neural networks. Neural computation, 33(4):899-925, 2021.
Zhihao Zheng, J Scott Lauritzen, Eric Perlman, Camenzind G Robinson, Matthew Nichols, Daniel Milkie, Omar Torrens, John Price, Corey B Fisher, Nadiya Sharifi, et al. A complete electron microscopy volume of the brain of adult drosophila melanogaster. Cell, 174(3):730-74) 2018.
Zhihao Zheng, Feng Li, Corey Fisher, Iqbal Ali, Nadiya Sharifi, Steven
Joseph Hsu, Najla Masoodpanah, Lucia Kmecova, Tom Kazimiers, et al. Structured sampling of olfactory input by the fly mushroom body. Current Biology, 32(15):3334-334, 2022.
11

---

A META-LEARNING A SIMPLE BACKPROP TRAINED RELU NETWORK
A simple test of our framework is a feedfoward ReLU network with 2 hidden layers, learnt using gradient descent. While the functions this network finds easy generalise cannot be extracted analytically, our tool finds that, unsurprisingly, these networks are biased towards smooth explana- tions of the data, learning six smooth orthogonal classifications that increase in frequency and mean squared error [83, We include this example as the simplest PyTorch implementation of learning the inductive bias of network trained by gradient descent, the hope that the code can be easily adapted for future use.
A)
B)
BACK-PROP TRAINED RELU LEARNER
Ld
L
A
MSE: 0.07
MSE: 0.03
MSE: 0.04
LATITUDE
META-LEARNER
LEARNER TRAINED GRADIENT DESCENT
MSE: 0.19
MSE: 0.16
MSE: 0.19
fo(x)
pX))
LONGITUDE
Figure 8: Meta-learning a Backprop Trained Network A: Schematic We meta-learn the functions
2-layer ReLU network trained using backprop finds easy generalise. B: Low Frequency Bias
B USING DIFFERENT DIVERGENCE MEASURES
To persuade the meta-learner find non-trivial functions, include divergence loss that forces the meta-learner's 's label distribution to take a particular form: uniform between -1 and 1. In this section we show that the particular divergence that we use has little impact on the solutions we find for learning the casiest-to-generalis function of the kernel learner, figure 2 Figure shows that variety of divergence metrics can used, sinkhorn, an energy statistic from (zékely & Rizzo (2013). and the maximum mean discrepancy from Gretton et al. (2012) (implementations from Djolonga (2020)). In each case the learnt function has over 99% norm in the space first order spherical harmonics, demonstrating that the meta-learner has learnt appropriately.
SINKHORN: 99%
ENERGY STATISTIC: 99.5%
MMD: 99.6%
LATITUDE
0.75
0
-0.75
LONGITUDE
Figure 9: We three different divergence metrics to penalise the meta-learner's label distribution for deviating from uniform between -I and 1. We add these to the meta-loss, along with the gener- alisation error of the simple kemnel learner introduced in section 2. For all three divergence metrics the meta-learner learns very close approximation to first order spherical harmonic, as predicted.
12

---

C RANDOM RE-SEEDING CANNOT REPLACE ORTHOGONALIISATINN
We introduced the orthogonalisation procedure in section so that successive meta-leamners have explore larger areas of function space. It is legitimate to wonder whether simply re-running the optimisation would have had the same effect. Here we show it does not, and orthogonalisation is needed explore the learner's inductive bias fully.
We re-run the meta-leamee optimisation without any orthogonality terms for the kernel learner de- scribed in section In figure I0 we find that the meta-learners find different functions, but only approximations to the first order spherical harmonics. This makes sense, the meta-learner is tasked with finding the sosiest-to-genraliss non-constant function. For this particular learner there is degenerate space of such functions and so re-running the meta-learner simply draws another sample from this space functions. However, to access the second order spherical harmonics that this learner still learns, just worse, we need something like the orthogonality constraint.
Figure 10: Re-running the meta-learner 10 times on the kernel regression algorithm from section produces approximations to the first order spherical harmonies, but second order functions.
D IMPACT OF META-LEARNER ARCHITECTURE ON EXTRACTED FUNCTIONS
We chose the meta-learner 's architecture to be slightly larger version of the learner's. Our motiva- tion for this that we want the function class of the meta-learner be a super-set the learner's. s, so that it can learn all the functions the learner could plausibly generalise well.
We tested how robust our results were architectural changes in the meta-learner. We used the simple feedforward 2-hidden layer ReLU network as in Appendix A, trained by backpropagation, and learnt 6 orthogonal functions that the learner finds easy to generalise. We took our meta-learner to be similar feedforward network of different depth from 4 to O hidden layers. Figure II shows that the specific choice meta-learner didn't 't matter for meta-learners with between 2 to 4 layers. Each learnt 3 approximations to first order spherical harmonics, and 3 to second order harmonics. But a linear learner can't 't learn more than three orthogonal functions, so failed to find more than three orthogonal generalisable functions.
13

---

A)
ZERO LAYER METAVLEARNER (LE., LNEAR)
THREE LAYER META-LEARNEN
MSE: 0.05
MSE: 0,016
MSE: 0.03
MSE: 0,008
MSE: 0.03
MSE: 0.007
LAnTLOE
MIE:: 0.19
MSE: 0.16
MSE: 0.19
MSE: 0.2
MSE: 0.2
MSE: 0.11
0.03
a.
B)
Two LAYER META-LEAINER
FOUR LAYER META-LENRNER
MSE: 0.019
MSE: 0.026
0.03
MSE:
LinrURE
0.11
MSE: 0.19
MSE: 0.21
MSE: 0.016
MSEI
LONGITUDE
LONGITUDE
Figure 11:
We meta-learn susy-to-generalise functions for a simple ReLU network using meta- learnrrs of different depths. This process fails find more than orthogonal functions when the meta-learner is linear (A.) (note the small label spread, symptom failure to learn), but meta- learners with (B.) 2, (C.) 3, (D.) all find qualitatively similar results
14


Reviews:
[{"review_id": "gwZGyWq82GZ", "reviewer": "Reviewer_VQde", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "This paper proposes a simple method for visualizing the inductive bias of a supervised learning algorithm. The method involves meta-learning the labels for a dataset, such that the learning algorithm is able to easily generalize when trained on a subset of those labels. For example, if the inputs are 1D, the meta-learned labels will map out a function over the 1D space that corresponds to the \"easiest\" function for the learning algorithm to learn, hence one can interpret that function as representative of the inductive bias of the algorithm. The paper applies their method to a couple of toy examples (linear and kernel regression in 2D) to build intuition, as well as to a feedforward ReLU network and a spiking neural network.", "summary_of_the_review": "Overall, I found this approach pretty interesting as a way to understand the function of a network. I think I would need to see more examples of being able to visualize or understand the inductive bias of larger-scale networks, or at the very least more toy examples where the inductive bias was something deeper beyond \"smooth functions are easier to model\". If this concern is addressed I would consider increasing my score.", "strength_and_weaknesses": "Strengths:\n- Novel idea, well motivated\n- Clear explanations and plenty of simple examples to build intuition\n- The approach behaves reasonably for simple problems\n\nWeaknesses\n- The method is really only applied to simple networks. Why not apply the method to visualize or understand the inductive biases of more complex networks (even stuff like networks trained on MNIST—which is still a relatively easy task). Is it because the meta-learning fails for these bigger networks, or are the resulting learned labels hard to interpret?\n- Most (all?) of the examples in the paper are cases where the learning algorithm has a bias towards smooth functions. It would be nice to see an example where the learning algorithm had some other kind of bias, and to show that the method recovers that specific inductive bias. You could even use simple 1D regression problems (instead of 2D) for simplicity. For example, the original MAML paper has a nice toy problem where they use MAML to meta-learn a network that has an inductive bias towards sinusoidal functions (by meta-training on sinusoids). If one were to take that network, and try to recover it's inductive bias using this method, could you recover what kind of function class was used to meta-train the network in the first place? I think that would be another powerful example of the technique (to demonstrate that the meta-learner can uncover structure besides \"smooth\" data).\n- Instead of the orthogonalization procedure, what what happen if you just initialized using multiple random seeds and re-ran the meta-learning procedure? Would different random seeds solve the problem that the orthogonalization is trying to address?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "dMWFI5uRwb", "reviewer": "Reviewer_TZjT", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "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": "The goal of the paper is to introduce a numerical procedure for extracting inductive biases, defined as ease of generalization,  for a differentiable trained recognition model. This procedure involves two layers of optimization: an inner learning model optimized on a training set of input-output pairs, and a meta-learner which is optimized so as to pick the training set that makes it easier for the inner learner to generalize to unseen data. ", "summary_of_the_review": "I like this a lot, found the approach clever and elegant. I imagine it would make a fun contributed talk/spotlight", "strength_and_weaknesses": "+ conceptually simple procedure, easy to apply to a range of models\n+ sequential introduction of multiple meta-learners, intriguing multi-learner scenarios for comparative across models analysis\n+ code provided for all demos\n\n- there is a little bit of circularity in that the set of functional hypotheses being considered itself comes from a functional class with its own inductive biases. \n- would have liked a little more details about the multi meta-learner scenario from the perspective of the inner loop learner", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "Q0RpN7Ne9c", "reviewer": "Reviewer_wT2V", "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": "## Updated Score. Changed from 3 to 5 \n\nThis paper proposes to address the critical question of understanding the inductive bias of machine learning models, i.e., what kind of function/problem models are most suited based on their design choices or other parameters like learning rules, etc. \n\nThe authors choose to design a model that can be used to characterize the inductive biases of various models systematically. The authors show that the meta-learner for given tasks and simple, relatively well-understood models can learn the inductive bias functions that the learning model itself has. They extend their study to more complex \"standard\" neural networks trained with backprop and spiking neural networks. \n\nAlthough the paper offers an interesting approach, one important objection is that, as opposed to the paper from Li et al. 2021, the meta-learners would themselves require a meta-learner to understand their own biases. \n\n", "summary_of_the_review": "The paper addresses an interesting question for both the ML and Neuroscience community. Given an observed model, how can we determine its inductive bias when its theoretical study is hard or impossible? \n\nThe paper, nonetheless, does not address critical questions and make enough predictions to make this paper even a proof of concept. It appears novel because it uses neural networks instead of GPs as meta-learners. Still, the resulting interpretability is lost because the meta-learners inductive bias is hard to understand. \n\nI look forward to hearing from the authors and clarifying points I might have missed. ", "strength_and_weaknesses": "The paper is well-written, and the breadth of models studied appears sufficient to touch a broad audience of ML and Neuroscience enthusiasts. The authors offer an alternative to the GP (Li et al.) and theoretical (Bordelon et al.) approaches. \n\n\nThere are many weaknesses with the current approach, which I believe makes it a bit premature for publication. \n\n1) The choice of the Sinkhorn distance seems somewhat arbitrary. How does the selection of the different objectives affect learning? \n2) The paper offers primarily qualitative comparisons (i.e., visual) rather than quantitative ones. Are there ways to properly understand the quality of the approximation of the implicit by different types of meta-learners? \n3) How does the meta-learner's choice affect the implicit bias (given the meta-learners have their own inductive biases)?\n4) What type of novel, non-trivial, insights can we obtain from the current paper? Various future directions and discussions are nicely laid out. However, as of now without them, this paper does not offer enough to be even a proof of concept. \n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "CdpDiq8mPcg", "reviewer": "Reviewer_QUHj", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "The authors present a meta-learning framework to infer a system's or circuit's inductive bias. In their work, the authors claim that their method connects architectural design choices to function space features. This work demonstrates the usability of the method in inferring the inductive bias for relatively simple models, i.e. linear and kernel regression as well as for shallow neural networks and spiking neural networks. The authors also leverage their framework and coupled with an adversarial training strategy, they demonstrate a potential application of learning functions that distinguish the inductive bias of two models with different architectures. Although current experimental evidence is limited to low-dimensional toy problems, the proposed method seems promising in understanding the inductive bias enforced by different architectural choices in high-dimensional end-to-end learning.", "summary_of_the_review": "Although the general idea seems exciting and promising, I have my doubts over the scalability of this method. Unless authors can provide more evidence from higher dimensional problems (maybe something with MNIST where you try to visualize canonical functions it learns over the pixel space), the utility of the method seems to be limited. Moreover, it is not very clear from the discussions how this method can be used in computational neuroscience. Therefore, my current assessment is marginally below the threshold but I am happy to revisit my rating if the authors address some of my concerns raised above. ", "strength_and_weaknesses": "Strengths:\n1. The paper is very well-written (barring a few typos) and it is easy to follow and understand the core tenets of the proposed framework.\n2. The authors present incremental evidence to test out different hypothesis pertaining to their proposal. In doing so, they enable the reader to develop the intuitions behind what the expect from the proposed framework, when put in practice.\n3. The idea seems simple and elegant, yet powerful and highly applicable to systems identification problems in both the fields of machine learning and neuroscience.\n\nWeakenesses:\n1. A key weakness of the work is its limited empirical validation or experimental evidence, specifically for non-toy problems. Although the intuitions presented in toy tasks are clear and illustrate the usefulness of the method, it remains to be seen how well the bootstrapped learning framework performs in high dimensions. \n2. A considerable failure mode in high-dimensional bootstrapped learning, something that this work proposes with the use of a learner that uses outputs of the meta-learner and vice versa, is dimension collapse. Specifically, it has been shown (for self-supervised bootstrapped learning) that often the functions learned are low-rank and therefore, the outputs do not span the entire space. Owing to this possible failure mode, it is reasonable to wonder whether this framework would be as applicable in high-dimensions as it is for low dimensionality problems.\n3. I was a bit confused about the utility of the proposed method for biological circuits, specifically how does the proposal fit in when attempting to understand the inductive bias of biological circuits. Firstly, the framework expects the learner to be given some output. How would such a training protocol look like in a particular circuit in a behaving animal? It would be nice to clarify this in the main text as a major motivation revolves around using the framework for biological circuits. Secondly, is it necessary that the system requires gradients to learn? In my understanding, the framework would work if there is some learning happening in the learner. Or could the framework still work without any plasticity in the learner? Have the authors tried testing the limits of their proposal by varying the learning rate for the learner. \n4. The authors present their framework and demonstrate that it works for low-dimensional toy settings but it is not clear why or how it does so. It would be nice to have some intuitive explanation or insights from learning theory or dynamical systems that conveys how this meta-learning setup converges to the inductive bias functions of a learning system. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject