Synaptic Weight Distributions Depend on the Geometry of Plasticity

Moreover, the conclusion itself also appears highly unreasonable

Here we would like to flag this as a highly unscientific comment. We are not sure what to say other than to encourage the reviewer to channel their incredulity into an objective assessment of our theoretical and experimental evidence, as well as our work’s novelty. We encourage them to read the other reviewer’s comments to challenge their initial opinion. Reviewing scientific papers should not consist of comments along the lines of “No way, I can’t believe it”, but rather, careful assessment of the evidence provided. We are surprised we need to clarify this here, but it seems we do.

Regarding the claim that our conclusion is highly unreasonable, we’d like to point out that the data collected on synaptic weight distributions in the brain seems to support our conclusions. Notably, the distribution of synaptic spine sizes across brain regions and species [Buzsáki and Mizuseki, 2014] is highly robustly log-normal. That suggests that regardless of the architecture or function of a neural circuit, the synaptic weight distribution is log-normal, which could easily be explained if the synaptic geometry is the major determining factor, and this was preserved across species and brain regions. Thus, far from being unreasonable, our conclusion is both supported by our theory and experiments, and existing data in the neuroscience literature. We will be sure to make this point in any revisions.

And the conclusion itself lacks significance.

We strongly disagree, and must highlight again that the reviewer is making oddly subjective comments with no argument to back up their claims. Why is this conclusion insignificant? If we are correct, and changes in the weight distribution will be dominated by the synaptic geometry, then that would give us an experimental means of assessing synaptic geometry, which would be a major insight into how the brain works! We also think the reviewer's surprise regarding our conclusions are indicative of actual significance. If science were never surprising, it would be unnecessary. We do agree that future work is required for this theoretical contribution to fully leave its mark on AI and computational neuroscience, but we think this an exciting and novel first step.

In conclusion, we were very disappointed with this review. Not because it is a low score, but because it lacks any actual engagement with our paper, both in terms of the theory we develop and the evidence we present. It is very frustrating as an author to have a reviewer make various false claims about assumptions we make, then dismiss our conclusions based on the reviewer’s apparent inability to believe the conclusions without any discussion of what the theory or data presented shows.

We hope the AC will ignore this review when rendering their decision.

We are pretty disappointed with this review, as it is clear that the reviewer did not actually read our paper carefully, nor try to give constructive or insightful critiques.

First, the reviewer raises concerns about five assumptions we supposedly make in this paper, but only one of them is made, and then only partially (see item 4, below). It leads us to wonder if it is just a simple misunderstanding, (in which case that may be a result of poor communication on our part), or whether Reviewer mbP3 was simply not engaged enough with the paper to notice that we do not make the assumptions they claim we do. We agree that each of these five assumptions listed here is overly restrictive, and some are evidently incorrect, conflicting with known experimental evidence. But, as noted, we do not make these assumptions. Below, we explain this for each of the claimed assumptions we make:

“Our brain is a feedforward network with no feedback and lacks any dynamic processes.

We don’t assume that the network is feedforward, in fact, in the submission we present results with recurrent neural networks, namely CORnet (a recurrent architecture). We have also added RNN experiments to the revised paper.

During learning, only the neurons in the last layer of our brain update.

We never state this or make any assumptions this way. This comment is therefore very wide of the mark and indicates the reviewer has not engaged with the paper. Please see section 2.1 Deep networks onwards.

Our brain should exclusively perform a supervised learning task.

We don’t assume this. We only assume that the brain is engaged in a task that can be framed as regression, but we would point out that self-supervised learning and reinforcement learning also often rely on various forms of regression, thus there is nothing about the assumption of regression that forces us into supervised learning. However, we agree that verifying our theoretical statement that our analysis is loss agnostic is valuable, and we have included experiments with a self-supervised loss here to demonstrate this point.

Our brain uses gradient descent algorithms.

We do assume that an ANN trained by gradient-based methods can be used as a working model for the brain, and as a working abstraction, can be used to understand aspects of how the brain learns and works. This does not mean we are committed to the assumption that the brain explicitly calculates gradients and uses them for its updates. Rather, it means we assume that the updates are non-orthogonal to the gradient. Notably, this is a mathematical necessity if weight updates are small, because otherwise performance would not improve, see e.g. Raman, D. V., Rotondo, A. P., & O’Leary, T. (2019). Fundamental bounds on learning performance in neural circuits. Proceedings of the National Academy of Sciences, 116(21), 10537-10546. Notably, this framework is widely used in computational neuroscience literature because it allows researchers to make tractable theoretical advances that can generate biological hypotheses - as we do in this manuscript. The implications of this assumption are stated clearly and we believe this work contributes to an established literature which precisely aims at elucidating the presence of gradient-following plasticity in the brain.

The optimal values for our brain's neurons for a given problem are unique and deterministic.

This is possibly the strangest of the reviewer’s concerns, as we never state this anywhere and actually explicitly assume the opposite – that the network should be overparameterized, which would lead to there being many different possible solutions that would be arrived at depending on various stochastic factors, such as initialization and data sampling. Thus, again, the reviewer appears to have not really engaged with the paper, and possibly, not even read it beyond the abstract.

Thank you for the review!

We agree that architecture choices play a role in our experiments. However, for all architectures, we’ve observed a strong contrast between multiplicative (NE) and additive (2-norm) plasticity rules. The last two architectures, CORnet-S and CORnet-RT, had several recurrent blocks; we included them to test if recurrence can break our conclusions. In the updated pdf (see Appendix B.3), we included new RNN experiments on row-wise sequential MNIST; the new results are very consistent with our prediction. (We agree it would be valuable to try e.g. spiking networks, but we would leave it to future work.)

Section 4.4: I find the choices of the initial w_0 surprising to demonstrate the fits

Apologies for an unclear message! We wanted to provide the simplest initialization that would lead to the final weight distribution to illustrate how just knowing the final distribution is not enough. That plot could also be reproduced if we started from two Gaussian in the log space (same means as for the point initialization). A simple example would be that data itself: if we treat the recorded weight distribution as the initial conditions, and train a bit more with the negative entropy potential, the resulting weight distribution should have two slightly wider Gaussian in the log space.

Can you elaborate on the link between geometry of plasticity and the locality of the learning rule?

If $\nabla\phi(\mathbf{w})_i=f(w_i)$ (so, a function of a single weight and not others), the update becomes local since it doesn’t depend on other weights. This is true for the potentials we discuss. However, for Euclidean geometry, we additionally need $f(w_i) = w_i$. For negative entropy, we get $f(w_i) =1 + \log w_i$, so the primal-dual weight mapping stays local but is no longer Euclidean. We mention this (albeit more briefly) just before Eq. 14, but have now expanded that explanation.

Isn't the use of optimizers such as Adam a way to have a different geometry of plasticity in ANNs?

It would depend on the optimizer. A simple momentum (with scalar, but possible adaptive, parameters), wouldn’t change mirror descent solutions (Gunasekar et al. (2018)). Adam or similar optimizers might change the space of solutions, but they usually try to make gradient updates unit-variance, which intuitively should strengthen Gaussian convergence. See the added discussion after Th. 1 in Sec. 3.

minor points

Thank you, we have corrected them.

While our approach needs some further development for direct use with neural data, as per points that you raised, we’d also like to emphasize the scope of our contributions to neuroscience as we see them:

1. This work is a significant step towards aligning biological synaptic weight changes with network weight updates via gradient based methods. The results we present are both novel and technically non-trivial, and lay the foundation for such further analysis and theory. For example, determining the theoretical limits of the Gaussian approximation to the true weight change (see appendix) coupled with empirical tests with deep networks.
2. We show that, despite the common use of basic gradient descent as a plasticity model in computational neuroscience, it’s not consistent with log-normality, multiplicative updates, and Dale’s law. Our framework allows us to address this without re-evaluating various neuro-inspired models of how the brain might implement backpropagation (abstaining from whether this is the right approach or not).
3. The ANN & gradient descent approach, despite its recent popularity, is often only vaguely linked to biology. One of the problems is a lack of experimental applicability. Here we make a step in this direction: going from theory and deep learning-driven models to experiment design and predictions. Namely that we can experimentally determine the synaptic geometry of the brain.

While it might be useful for analysing deep net behaviour, again, it is not clear how this can be used to improve deep net training either.

There is recent work showing that the use of different p-norms as potentials can result in better performance than gradient descent (which is the 2-norm) [Sun et al., 2023]. Furthermore, a weight re-parameterisation that is equivalent to the negative entropy potential in certain settings, was found to facilitate training sparse networks and be beneficial for continual learning [Schwarz et al., 2019] (see also Appendix C). Finally, in RL, mirror descent with geometries has long been used for bandit tasks [Shalev-Shwartz, 2011]. Therefore an exciting future direction is to explore the differences and benefits between gradient descent and a weight update algorithm derived from estimating biological synaptic geometry. If the reviewer updates their score and the paper is accepted for publication we will add a discussion of this subject to the appendix or discussion, as we agree it is an important topic to communicate.

We hope we have clarified your questions and addressed your concerns about our neuroscientific contributions. If not, we would very much welcome further discussion.

Thank you for your review, the points you have raised have been useful for us. We address weakness 1 and question 1 together below. Regarding weakness 2, please see our response to question 3, and we also understand it to overlap with the applicability of gradient-based learning to neuroscience (shared with reviewer jXnh). Here’s a slightly shortened response:

First, our work uses gradient descent to construct a rigorous theory. However, our result ultimately relies on the large-sum behavior of update terms. If we replace the gradient $\nabla l$ wrt the loss in Eq. 5 $\nabla\phi(w^{t+1})=\nabla\phi(w^{t})-\eta\nabla l$, with anything else that drives learning, like a 3-factor Hebbian rule, the intuition would still hold: a large sum of small and mostly independent updates looks Gaussian in the dual space.

More generally, we assume that an ANN trained by gradient-based methods can be used as a working model for the brain, and as a working abstraction can be used to understand aspects of how the brain learns and works. As long as the learning algorithm the brain uses descends a gradient, even if it is never explicitly calculated, then this framework can be applied. However, this does not mean that we are committed to the brain doing gradient descent, per se. But it is a commonly applied framework that allows researchers to make tractable theoretical advances that can generate biological hypotheses - as we do in this manuscript.

How can we guarantee that the loss function / potential function separation truly represents a separation between extrinsic and intrinsic factors in the weight changes?

The motivation of the paragraph just before Sec. 2.1 was to convey a way to think about credit assignment in terms of the mirror descent framework we present in Sec. 2. We are not committed to this separation of factors per se in biology. However, we think it can be helpful to think in terms of this separation. Extrinsic credit assignment factors are those external to the synapse, such as a neuron-wide error signal multiplied by presynaptic activity (which corresponds to the gradient). These external factors are then used to update the synapse (somehow), which is the intrinsic step. These factors are not necessarily separated in real neurons, however, within the mirror descent framework they are, and the intrinsic step is captured by the potential. We hope this clarifies this point and are happy to discuss it further. We have updated our communication of this point in the manuscript.

How to evaluate whether the linearity / Gaussian assumption has been violated, in cases where the loss and potential function are unknown?

Sec. 3.1 indirectly addresses this point, in short, whether the loss and/or potential function are unknown does not impact the assessment of linearity /Gaussian assumption violation:

1. Linearity: the norm of the weight changes should be ~ an order of magnitude smaller than the weights in the dual space. This comes from applying a 1st order Taylor series approximation, and can be verified for a candidate potential.
2. Gaussian assumption: this could be broken by correlations in inputs. However, we think this is unlikely, because this is a common feature in natural sensory data [Ruderman (1994)] and neural activity [Schulz et al. (2015)] (we mention this just before Sec. 3.1). We also empirically investigate the robustness of this violation in the deep network experiments (Figs ). We will strengthen our communication of these points in the main text.

How to make the "geometry of synaptic weight changes" more interpretable for neuroscientific insight?

By “geometry of synaptic weight changes” we mean the set of properties that describe the relationship between different synaptic weight configurations. In this paper we focus on distance, i.e. which synaptic configurations are “far” or “close” to each other for a neuron. We highlight that training models with gradient descent implicitly assumes a Euclidean distance assessment of which synaptic weight configurations are close to or far from each other. However, this does not fit with neurobiology, for example synapses cannot change sign! As such, Euclidean distance does not make sense from a neuroscientific perspective because a weight change that involves a sign change is not penalized more than one that does not. We therefore focus on alternative distance functions (Bregman divergences, which are parameterized by a potential). These distance functions induce different weight change geometries, as they change which weight configurations are deemed far or close to one another.

In this sense, the true synaptic geometry of the brain is currently unknown. But our work enables a direct experimental proposal – if we (1) pick a task, (2) measure the weights before and after task learning, we can then (3) find the potential (and therefore geometry) that best fits a Gaussian change in the dual space (see Fig 1 for primal/dual space distinction).

I found the paper pretty dense, and not easy to follow what was going on where, and I always had to keep lots of things in mind at any moment.

Thank you for this comment! We agree that we had to condense a lot of things in the main text, partly because we draw on a range of prerequisites: mirror descent is not a well known topic in theoretical neuroscience.

We’d be happy to expand the intuition/re-write in some parts, and would appreciate guidance re which sections. We’re assuming that the densest part is Sec. 3 (the main result): would expanding on Lagrange multipliers from the beginning help? Additionally, would making the theorem statement more informal help?

Currently, we expanded the discussion before Th. 1 and compressed Sec. 3.1 (see the updated pdf). But we’re also hesitant to put all technical details to the appendix since we are already hiding significant theoretical contributions there to prioritize readability!

Couldn’t parse Fig 3

Left plots: with more weights (lazy regime), the weight changes become more Gaussian (CDF difference) and smaller in magnitude. Right plots: if the change is Gaussian for one potential, measuring it with the wrong potential would result in a very non-Gaussian weight change. We hope this is clear?

Where do you show this? It’s not in Fig 5.

Apologies, we should’ve made it more clear. We can rule out Euclidean geometry since it can’t produce Gaussian changes in the log space. We can estimate the geometry by finding the potential with the most Gaussian change (if the weights before and after learning are known).

I didn’t follow the explanation before eqn 14. Could do with some clarifying.

We agree there are few steps that we had to condense due to space constraints here. There is a linearization step and a local potentials discussion. We have expanded the derivation for the former in the text. Re local potentials: If $\nabla\phi(\mathbf{w})_i=f(w_i)$ (so, a function of a single weight and not others), the update becomes local since it doesn’t depend on other weights. This is true for the potentials we discuss, but not e.g. for $\phi(\mathbf{w})=\|\mathbf{w}\|_2$ (non-squared norm) since the gradient would depend on other weights.

Thank you for the review!

The big assumption that the brain is doing gradient descent…

Our result only partially relies on gradient descent. We need it, along with the result in Eq. 8, to have a rigorous theory for Gaussian behavior during learning. However, as we outline in the second paragraph after Eq. 7, our result ultimately relies on the large-sum behavior of update terms, and these updates need not come from gradient descent. So, if we replace the gradient $\nabla l$ wrt the loss in Eq. 5 (the mirror descent update), $\nabla\phi(w^{t+1})=\nabla\phi(w^{t})-\eta\nabla l$, with anything else that governs learning, like a 3-factor Hebbian rule, the intuition would still hold: a large sum of small and mostly independent updates looks Gaussian in the dual space, $\nabla\phi(w^{\infty})=\nabla\phi(w^{0}) + \sum_{t=1}^\infty g^t\approx \nabla\phi(w^{0}) +\mathcal{N}$. But such a general case would make theory much harder (unless we put some trivial assumptions like literal independence of updates). Gradient-based optimization leads to the Bregman divergence result, which in turn leads to a closed-form approximate solution, which we can show to be Gaussian. We have added this explanation to Sec. 3.

More generally, we assume that an ANN trained by gradient-based methods can be used as a working model for the brain, and as a working abstraction can be used to understand aspects of how the brain learns and works. As long as the learning algorithm the brain uses descends a gradient, even if it is never explicitly calculated, then this framework can be applied. However, this does not mean that we are committed to the brain doing gradient descent, per se. But it is a commonly applied framework that allows researchers to make tractable theoretical advances that can generate biological hypotheses - as we do in this manuscript.

We also offer another point of view: if backpropagation is taken as the working model of the brain, is it consistent with neural data? Our conclusions are that it’s not in the simplest form (gradient descent) due to the solutions it finds, but other variants such as exponentiated gradient are consistent (with log-normals, Dale’s law, and multiplicative updates). This is important and impactful given the number of computational papers that assume gradient descent as a working model of learning in the brain.

Lastly, we note that there is a sizable literature exploring learning in the brain under the assumption of gradient-guided plasticity. The question about the validity of this assumption is important – and work like ours help make progress– but this larger question is beyond the scope of our paper.

What’s the actual citation for this?? (i.e. not a review)

We referred to this: Richards and Kording. The study of plasticity has always been about gradients. The Journal of Physiology, 2023.

But the basic mathematical idea is that if you’re slowly improving your performance with small weight changes, you must be approximately following the gradient (wrt performance) since it is the mathematical direction for performance improvement. If you were not, then there would be no improvement in loss. As an additional citation (non-review) for this point, see Raman, D. V., Rotondo, A. P., & O’Leary, T. (2019). Fundamental bounds on learning performance in neural circuits. Proceedings of the National Academy of Sciences, 116(21), 10537-10546.

As the discussion period is coming to an end, we have further updated the manuscript after our discussion with reviewer DFTN. Their comments also focused on the clarity of the paper and we think our responses and changes are applicable to your comments.

Thank you again for your positive and helpful review, we hope the changes we have made to the manuscript have improved its clarity and as such we hope you would consider a score increase.