SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Colin Bredenberg*
Mila-Quebec AI Institute
Montreal, Quebec H2S 3H1
Ezekiel Williams
Cristina Savin
Blake Richards†
Guillaume Lajoie†
In recent years, many researchers have proposed new models for synaptic plasticity in the brain based on principles of machine learning. The central motivation has been the development of learning algorithms that are able to learn difficult tasks while qualifying as "biologically plausible". However, the concept of a biologically plausible learning algorithm is only heuristically defined as an algorithm that is potentially implementable by biological neural networks. Further, claims that neural circuits could implement any given algorithm typically rest on an amorphous concept of "locality" (both in space and time). As a result, it is unclear what many proposed local learning algorithms actually predict biologically, and which of these are consequently good candidates for experimental investigation. Here, we address this lack of clarity by proposing formal and operational definitions of locality. Specifically, we define different classes of locality, each of which makes clear what quantities cannot be included in a learning rule if an algorithm is to qualify as local with respect to a given (biological) constraint. We subsequently use this framework to distill testable predictions from various classes of biologically plausible synaptic plasticity models that are robust to arbitrary choices about neural network architecture. Therefore, our framework can be used to guide claims of biological plausibility and to identify potential means of experimentally falsifying a proposed learning algorithm for the brain.
Over the last several decades, computational neuroscience researchers have proposed a variety of "biologically plausible" models of synaptic plasticity that seek to provide normative accounts of a variety of learning processes in the brain—these models aim to explain how modifications of cellular properties such as excitability and synaptic strengths can improve performance on a variety of important tasks for an organism, using only the information "locally" available to a real neuron in the brain. However, there is no consensus on what "biologically plausible" really means. This is due to a lack of any precise definition of locality, with individual studies adopting different, architecture-specific heuristics that are often not explicitly articulated; further, at an experimental level, the precise quantities available to neurons for computing synaptic updates are still up for debate [1, 2], and likely vary across cell types [3, 4, 5] and brain regions [6, 7, 8, 9, 10].
*Corresponding author, email at: colin.bredenberg@mila.quebec.
†Co-senior authors.
As an example, some studies postulate that global neuromodulatory signals containing reward information could be available at individual synapses, in addition to the more traditional pre- and post-synaptic information typically required for Hebbian plasticity [11, 12, 1]. Within this broad class, individual studies differ in their assumptions about this reward information (e.g. is it completely global [13] or potentially distinct for different layers or regions [14]) and in the precise details about the pre- and post-synaptic signals available (e.g. is it spike-timing [15] that is available or spike-rate [16]?). Furthermore, there are many more signals that could theoretically be available at synapses: for example, other studies propose that individual pyramidal neurons could receive detailed error (or target) feedback information at their apical dendrites [17, 18, 19, 2] which can be used for learning. Previous work has taxonomized different normative plasticity models according to these feedback signals [20]. Within this space of normative plasticity models, it can be difficult to identify which features of a given model constitute necessary, testable predictions that can be used for experimental verification—i.e., if this information were not available at a synapse, learning with the given algorithm would be impossible—and which features are due to arbitrary choices that may vary with more or less realistic neural network architectures.
As a consequence of these heterogeneous conceptions of locality and given the advancements in the field, it is arguably necessary to develop a clear and precise way to identify what variables synapses should have access to in a learning algorithm, whenever claims of locality are made. In this paper, we focus on formalizing this process. Our central contributions are as follows:
1. We develop an architecture-independent formal framework for locality that we term  $\mathbf{S}p$ -locality, which requires one to precisely specify the set of variables that synapses are assumed to have access to within a model: different choices of allowed variables for the set  $\mathbf{S}$  produce different classes of locality. Importantly, this definition can be used for any learning algorithm applied to any stochastic neural network.
2. We use our framework to group existing plasticity models into different locality classes. Intriguingly, we find that many different algorithms derive their locality from similar principles, even though they belong to different locality classes.
3. We show that our different locality classes make different experimental predictions, which makes it possible to identify which algorithms can be cleanly distinguished experimentally and which cannot.
2 Formalizing notions of locality
Our ultimate goal is a formal definition of locality that makes it easy to identify which variables are locally available to synapses in a learning algorithm within any specific model architecture. However, this can be challenging. For example, when we consider a single-compartment neuron model it seems reasonable to assume that the postsynaptic voltage is locally available to synapses. In contrast, in a multi-compartment model, voltages can be different in each compartment, and synapses likely only have access to voltages within nearby compartments. Ideally, we would avoid a definition that simply requires us to list all of the variables that must be made available to the synapse, as that would carry little conceptual weight. However, we will begin by taking this approach, as it will help us to build the groundwork for a more conceptually useful definition in what follows.
Notation. We have  $\mathbf{X}$  as a random vector in  $\mathbb{R}^{N_{\mathbf{X}}}$  denoting network variables (e.g. voltage, spike-rate, inputs, etc.), and  $\Theta$  as a random vector in  $\mathbb{R}^{N_{\Theta}}$  denoting network parameters (e.g. synaptic weights, spike threshold, etc.). We combine these as  $\mathbf{Z} = [\mathbf{X},\Theta]$  the concatenation of the two vectors. We will use  $\mathbf{Z}\sim p(\mathbf{Z})$  to denote a random vector with support  $\Omega \subseteq \mathbb{R}^{N_{\Theta} + N_{\mathbf{X}}}$ , but will also overload this notation to indicate a vector subject to either universal or existential quantification (e.g.  $\exists \mathbf{Z}$  s.t.  $\frac{\mathrm{df}(\mathbf{Z})}{\mathrm{d}\mathbf{Z}} = 0$ ) or a dummy variable in an expectation (e.g.  $\mathbb{E}_{p(\mathbf{Z})}[\mathbf{Z}] = \mu$ ). We will use the subscript " $\neq i$ " to denote removal of the  $i^{th}$  element of a vector; for example  $\mathbf{Z}_{\neq i} = [\mathbf{Z}_1,\dots ,\mathbf{Z}_{i - 1},\mathbf{Z}_{i + 1},\dots ,\mathbf{Z}_{N_{\mathbf{X}} + N_{\Theta}}]^\top$ , and will occasionally use the more general notation  $\mathbf{Z}_{\neq \nu}$ , where  $\nu$  is a set of indices, to denote the vector with the associated set of elements removed.
2.1 S-locality: graph-specific locality based on a predefined set of available variables
We begin with a definition of locality that formalizes the concept of a set of variables,  $\mathbf{S}_k\subseteq \mathbf{Z}$ , to which the  $k^{th}$  synapse can have access in order to compute synaptic updates (e.g. some neural voltages,
reward signals, etc.). We then consider the set of these sets across all synapses,  $\mathbf{S} = \{\mathbf{S}_1,\dots ,\mathbf{S}_{N_\Theta}\}$ . This inspires the following definition:
Definition 2.1. Given parameters  $\Theta \in \mathbb{R}^{N_{\Theta}}$ , random variables  $\mathbf{X} \sim p(\mathbf{X}|\Theta) \in \mathbb{R}^{N_{\mathbf{X}}}$ , and function  $f(\cdot): \mathbb{R}^{N_{\mathbf{X}} + N_{\Theta}} \to \mathbb{R}^{N_{\Theta}}$ , the update function  $f(\mathbf{Z})$  is  $\mathbf{S}$ -local with respect to some set  $\mathbf{S} = \{\mathbf{S}_k: \mathbf{S}_k \subseteq \mathbf{Z}\}$  if and only if  $\forall k, i$ :
$$
\exists \mathbf {Z} \in \Omega \text {s . t .} \frac {\partial f _ {k} (\mathbf {Z})}{\partial \mathbf {Z} _ {i}} \neq 0 \Rightarrow \mathbf {Z} _ {i} \in \mathbf {S} _ {k}. \tag {1}
$$
In short,  $\mathbf{S}$ -locality requires that if parameter update  $f_{k}(\mathbf{Z})$  has a direct dependence on  $\mathbf{Z}_{i}$ , then  $\mathbf{Z}_{i}$  must be included in the set  $\mathbf{S}_{k}$  of "allowed" variables for that parameter. For example, in the classic approach to locality in a neural network, for any synapse  $\mathbf{W}_{ij} \in \Theta$ , we might set  $\mathbf{S}_{ij} = \{\mathbf{r}_i, \mathbf{r}_j\} \subseteq \mathbf{Z}$ , allowing any  $\mathbf{S}$ -local parameter update under this definition to use pre-  $(\mathbf{r}_j)$  and post-synaptic  $(\mathbf{r}_i)$  firing rate information, and nothing else. If we wanted to allow a third factor such as a global neuromodulatory signal,  $R$ , to project to diffusely to every synapse in the network, as in reward-modulated synaptic plasticity rules, we might take  $\mathbf{S}_{ij} = \{\mathbf{r}_i, \mathbf{r}_j, R\}$ . In this manner, for any model architecture, we can manually construct a set  $\mathbf{S}_{ij}$  that defines our notion of locality for each synapse.
While our definition is intuitive and flexible, it is cumbersome: for every parameter we have to explicitly define which variables are local. This means that S-locality is architecture-specific, requiring an evaluation of which  $\mathbf{S}_k$  is an acceptable set of variables for each synapse's update rule. For example, if we switched from a single- to a multi-compartment model, we would now have to modify  $\mathbf{S}_{ij}$  to accept only the voltage of the compartment where the synapse is located. Ideally, our definition would make it obvious whether a learning rule is local in both models without having to redefine the list of which variables are local. To do that, we will use the structure of the distribution  $p(\mathbf{X},\Theta)$  to construct a definition of locality that captures intuitions across network architectures.
2.2  $p$ -locality
Despite its ability to concretely define locality for any model, S-locality is architecture-specific and cumbersome: in this section, we will develop an alternative operationalized notion of locality, called  $p$ -locality (where  $p$  stands for "probability"). Our idea is to formulate locality in terms of adjacency in the computational graph underlying network dynamics. To do this we will leverage conditional dependencies in the joint distribution  $p(\mathbf{X},\Theta)$  to generate an architecture-general concept of locality<sup>3</sup>. This is considerably more useful as both a conceptual tool and organizing principle for normative synaptic plasticity models. However, we will still call on S-locality later to formulate our final locality definition:  $\mathbf{S}p$ -locality.
Further, it is important to note that for this definition, as with S-locality, we do not restrict the functional form of parameter updates  $f_{k}(\cdot)$  — we only restrict which variables they are allowed to depend on. This is in contrast to previous, complementary work which has focused on defining allowable local computations and memory complexity constraints for 'biologically plausible' learning algorithms [21]. Our decision not to restrict  $f_{k}(\cdot)$  enables us to abstract away the details of 'allowed' subcellular computations, and will consequently enable us to make much more general statements about the locality properties of normative plasticity algorithms as a whole, without having to focus on the specifics of particular models of neural dynamics.
Definition 2.2.  $p$ -locality: Consider a probability distribution  $p(\mathbf{Z})$  over network activity variables and parameters, and an update function  $f: \mathbb{R}^{N_{\mathbf{X}} + N_{\Theta}} \to \mathbb{R}^{N_{\Theta}}$ , mapping variables and parameters to parameter updates. Assume that  $\frac{\partial p(\mathbf{Z}_i|\mathbf{Z}_{\neq i})}{\partial\Theta_k}$  and  $\frac{\partial f_k(\mathbf{Z})}{\partial\mathbf{Z}_i}$  exist for all  $\mathbf{Z} \in \Omega$  and  $i, k$  indices. The update function  $f(\mathbf{Z})$  is local with respect to  $p$  ( $p$ -local) if and only if  $\forall k, i$ :
$$
\exists \mathbf {Z} \in \Omega \text {s . t .} \frac {\partial f _ {k} (\mathbf {Z})}{\partial \mathbf {Z} _ {i}} \neq 0 \Rightarrow \mathbb {E} _ {p} \left[ \left(\frac {\partial \log p \left(\mathbf {Z} _ {i} \mid \mathbf {Z} _ {\neq i}\right)}{\partial \boldsymbol {\Theta} _ {k}}\right) ^ {2} \right] \neq 0. \tag {2}
$$
To unpack this definition and render it more intuitive, as with S-locality,  $\frac{\partial f_k(\mathbf{Z})}{\partial\mathbf{Z}_i}$  indicates a test to see whether the  $k$ th index of  $f$  has a direct dependence on a particular variable  $\mathbf{Z}_i$ —we assume
<sup>3</sup>We define dynamics in terms of probability distributions because many canonical normative plasticity models are stochastic; deterministic computation graphs are a limit case of our approach.
Figure 1: Common neural network graphical models and the permissible parameter updates for a synapse  $\mathbf{W}_{ij}$  under  $p$ -locality. a. Feedforward neural networks can be described as DAGs, so variables that are not children or coparents of  $\mathbf{W}_{ij}$  are excluded from parameter updates under  $p$ -locality (by Property 2.1). b. For fixed weights in a dynamic neural network like a recurrent neural network (RNN),  $p$ -locality excludes variables that are not children and coparents of  $\mathbf{W}_{ij}$  for all time points (also by Property 2.1). c. For an undirected graphical model like a Boltzmann machine, variables that are not neighbors of  $\mathbf{W}_{ij}$  are excluded under  $p$ -locality (by Property 2.2).
that  $f_{k}(\cdot)$  is differentiable primarily for simplicity, and differentiable approximations can always be substituted in practice if  $f_{k}(\cdot)$  happens to not be differentiable. If there is a dependency, then we want that associated random variable  $\mathbf{Z}_i$  to be considered local to  $\Theta_k$ . To measure this, we have selected the Fisher Information of  $p(\mathbf{Z}_i|\mathbf{Z}_{\neq i})$  with respect to  $\Theta_k$ . This measurement quantifies direct influence. For example, if  $\mathbf{Z}_i$  is a neuron downstream of neuron  $\mathbf{Z}_j$ , and  $\Theta_k$  is a synapse onto  $\mathbf{Z}_j$ , then by conditioning on  $\mathbf{Z}_j$ ,  $\Theta_k$  is no longer able to have any effect on the conditional probability distribution of  $\mathbf{Z}_i$  through its influence on  $\mathbf{Z}_j$ . In other words, the statistical influence of  $\Theta_k$  on  $\mathbf{Z}_i$  is mediated through the influence of  $\mathbf{Z}_j$  on  $\mathbf{Z}_i$ , and thus it is indirect. By conditioning on  $\mathbf{Z}_j$ , we seal off one pathway of indirect influence. By conditioning on all  $\mathbf{Z}_{\neq i}$ , we seal off all pathways of indirect influence, which makes  $\frac{\partial\log p(\mathbf{Z}_i|\mathbf{Z}_{\neq i})}{\partial\Theta_k} = 0$  for all  $\mathbf{Z}$ .
It is worth stressing that  $p(\mathbf{Z}) = p(\mathbf{X}|\Theta)p(\Theta)$  is a joint distribution over both  $\mathbf{X}$  and  $\Theta$ , even though most algorithms operate only on the conditional distribution  $p(\mathbf{X}|\Theta)$ . As a consequence, the marginal probability that we place on  $p(\Theta)$  can be a free choice, which is useful for clarifying the assumptions in the model. For example, we will typically assume that synapses do not have any marginalized dependence on one another, i.e.  $p(\Theta) = \prod_k p(\Theta_k)$ .
2.3 Properties of  $p$ -locality
Having defined  $p$ -locality, we can now talk about its interesting properties, which will give intuitions for how it functions and which will figure prominently in our proofs of algorithms' locality. While any  $p$ -local update could also technically be reformulated as a S-local rule, by taking the set S to be the set allowed under  $p$ -locality, the conciseness of  $p$ -locality and its convenient mathematical properties will prove to be its principal benefits over S-locality. For proofs of these properties, see Appendices A-C. To see the power and generality of  $p$ -locality as a definition, we first provide two properties that show how one can immediately make statements about the locality of a learning algorithm by simply inspecting graphs associated with the model. Assume all quantities are defined as in Definition 2.2 and that, for certain properties, the joint density satisfies mild regularity constraints (see Appendix C.1). For the first two properties we further assume that  $p(\mathbf{Z})$  is strictly positive (see Appendix B). Then we have the following:
Property 2.1. Assume  $\mathcal{G}_{\mathrm{d}}$  is a Directed Acyclic Graph (DAG) for  $p$ . If  $\frac{\partial f_k(\mathbf{Z})}{\partial \mathbf{Z}_i} \neq 0$  for  $\mathbf{Z}_i$  that is not a parent, co-parent, or child of  $\Theta_k$  in  $\mathcal{G}_{\mathrm{d}}$ , then  $f$  is not  $p$ -local.
Property 2.2. Assume that  $\mathcal{G}$  defines an Undirected Graph (UG) for  $p$ . If  $\frac{\partial f_k(\mathbf{Z})}{\partial \mathbf{Z}_i} \neq 0$  for  $\mathbf{Z}_i$  that is not a neighbour of  $\Theta_k$  in  $\mathcal{G}$ , then  $f$  is not  $p$ -local.
By these properties, as long as the conditional dependencies of  $p(\mathbf{Z})$  can be summarized by an UG or DAG (two classes which subsume many modern neural network architectures, see Figure 1 for
examples), we can identify many of the variables disallowed by  $p$ -locality. Therefore, for such a network, we can decide whether  $p$ -locality conforms to intuitions about biological plausibility as easily as we can for a hand-crafted set of allowed variables under S-locality; for most practical neural network architectures, we will see below that  $p$ -locality does behave intuitively. Thus, with  $p$ -locality we have a definition of locality that does not require an exhaustive list of the variables available to a synapse, but rather, which relies on the implicit computational structure of the model (i.e.  $p(\mathbf{Z})$ ).
Property 2.3. For any function  $b: \mathbb{R}^{N\mathbf{x} + N\Theta} \to \mathbb{R}^{N\Theta}$  defined such that  $b_{k}(\mathbf{Z}) = h_{k}(f_{k}(\mathbf{Z}), g_{k}(\mathbf{Z}))$ , where  $f(\mathbf{Z})$  and  $g(\mathbf{Z})$  are  $p$ -local and  $h_{k}$  is differentiable,  $b(\mathbf{Z})$  is also  $p$ -local.
This demonstrates that  $p$ -local functions can be arbitrarily combined without the combination losing the  $p$ -local property. It also shows that  $p$ -locality places no restrictions on the functional form of parameter updates, so long as they are exclusively functions of variables allowed under  $p$ -locality.
Property 2.4. For any function  $f(\cdot): \mathbb{R}^{N_{\mathbf{X}} + N_{\Theta}} \to \mathbb{R}^{N_{\Theta}}$ , there exists a probability distribution  $p(\mathbf{Z})$  such that the random variable  $f(\mathbf{Z})$  with  $\mathbf{Z} \sim p(\mathbf{Z})$  is  $p$ -local.
This is intended as a cautionary note: unless  $p$  defines a neural network-like probability distribution,  $p$ -locality does not necessarily conform to intuitions about biological locality (see Appendix F for examples). That  $f(\mathbf{Z})$  is  $p$ -local for some probability distribution says nothing about a function  $f(\mathbf{Z})$ ;  $p$ -locality only becomes informative when we narrow our focus to probability distributions with biological relevance.
The following properties will turn out to be extremely important in Section 3: all algorithms that we survey below with provable  $p$ -locality properties will involve the score function or the derivative of the unnormalized energy function.
Property 2.5. The derivative of the log joint distribution  $\frac{\partial\log p(\mathbf{X},\Theta)}{\partial\Theta}$  is  $p$  -local.
Property 2.6. For a probability distribution given by  $p(\mathbf{Z}) = \frac{1}{\mathcal{Z}} \exp(-E(\mathbf{Z}))$ , where  $\mathcal{Z}$  is a normalizing constant, the expression  $\frac{\partial}{\partial \Theta} E(\mathbf{Z})$  is  $p$ -local.
Property 2.7. If the parameter marginal distribution factorizes as  $p(\Theta) = \prod_{k} p(\Theta_{k})$ , i.e. the parameters are independent from one another, then the score function  $\frac{\partial \log p(\mathbf{X}|\Theta)}{\partial \Theta}$  is  $p$ -local.
Property 2.8. For a mixture distribution  $p_{12}(\mathbf{Z},\gamma) = p_1(\mathbf{Z})^\gamma p_2(\mathbf{Z})^{1 - \gamma}p(\gamma)$  for some binary variable  $\gamma \in \{0,1\}$  with nonzero probabilities, if  $f(\mathbf{Z})$  is  $p_1$ -local (or equivalently  $p_2$ -local), then  $f(\mathbf{Z})$  is  $p_12$ -local.
Intuitively, this feature holds because mixture distributions introduce more dependencies between parameters  $\Theta$  and random variables  $\mathbf{X}$ . Therefore,  $p$ -locality under a mixture of probability distributions is more permissive than  $p$ -locality under any single one of its constituent probability distributions.
2.4 A simple motivating example
To make our definition of  $p$ -locality more concrete, we will now explore how it functions for a common network architecture. Consider a simplified linear-nonlinear feedforward neuron model with  $L$  layers of neurons  $\mathbf{r}^{(l)}$  ( $\mathbf{X} = [\mathbf{r}^{(1)}, \dots, \mathbf{r}^{(L)}]$ ) connected by synaptic weights  $\mathbf{W}^{(l)}$  ( $\Theta = [\mathbf{W}^1, \dots, \mathbf{W}^{(L)}]$ ), with additive Gaussian noise at each layer. Conditioned on our feedforward weight matrices and stimuli  $s$ , the probability distribution of neural firing rates is given by:
$$
p (\mathbf {r} | \mathbf {s}, \mathbf {W}) = \prod_ {l = 1} ^ {L} \prod_ {i = 1} ^ {N _ {l}} p \left(\mathbf {r} _ {i} ^ {(l)} \mid \mathbf {r} ^ {(l - 1)}, \mathbf {W} ^ {(l)}\right) \tag {3}
$$
$$
p \left(\mathbf {r} _ {i} ^ {(l)} \mid \mathbf {r} ^ {(l - 1)}, \mathbf {W} ^ {(l)}\right) \sim \mathcal {N} \left(h \left(\mathbf {W} _ {i:} ^ {(l)} \mid \mathbf {r} ^ {(l - 1)}\right), \sigma^ {2}\right), \tag {4}
$$
where for notational simplicity we have taken  $\mathbf{r}^{(0)} = \mathbf{s}$ ,  $h(\cdot)$  is a pointwise nonlinearity, and  $\mathbf{W}_{i:}^{(l)}$  corresponds to the  $i$ th row of  $\mathbf{W}^{(l)}$ . As discussed above, most algorithms operate on the conditional distribution,  $p(\mathbf{X}|\boldsymbol{\Theta})$ , so we only define a prior over parameters for the purposes of assessing  $p$ -locality. To ensure that our joint distribution is a directed graphical model, and to ensure that we have no dependencies between parameters, we will assume that the parameters for the network are all independently distributed, i.e.  $p(\mathbf{W}) = \prod_{i,j,l} p(\mathbf{W}_{ij}^{(l)})$ , where  $\mathbf{W}_{ij}^{(l)}$  is the synapse between