Introduction

This post is part of a sequence describing our team’s research on selection theorems for modularity, as part of this year's AI Safety Camp, under the mentorship of John Wentworth. Here, we provide some background reading for the discussion of modularity that will follow.

As we describe in more detail in our project intro, the motivating question of modularity that we started with (which is described in John’s post on the Evolution of Modularity) is why does evolution seem to have produced modular systems (e.g. organs and organ systems), but current ML systems (even genetic algorithms, which are consciously fashioned off evolutionary mechanisms) are highly non-modular? So far, most of our research has focused on the idea of modularly varying goals (MVG), but this is not the only proposed cause of modularity in the biological literature. This post serves as a literature review, with a few brief words on how we are thinking about testing some these hypotheses. Subsequent posts will discuss our team’s research agenda in more detail.

Theories of modularity

Modularity in the environment selects for modular systems

Basic idea: If we have an environment which contains a variety of subproblems and which changes rapidly in highly modular ways, this might select for a modular internal design to better adapt to these problems. In the literature, this usually takes the form of “modularly varying goals” (MVG). If we vary one aspect of the agent’s goal, but keep the rest of the goal and the environment constant, then this might produce a selection pressure for modular systems in correspondence with this modular goal structure.

Biological modularity: Many examples of modularity in biology seem to have a pretty clear correspondence with goals in the environment. For instance, type of terrain and availability of different food sources might vary somewhat independently between environments (or even within the same environment), and correspondingly we see the evolution of modular systems specifically designed to deal with one particular task (muscular system, digestive system). One can make a similar argument for reproduction, respiration, temperature regulation, etc.

Evidence: The main paper on this idea from the biological literature is Kashtan & Alon’s 2005 paper. Their methodology is to train a system (they use both neural networks and genetic algorithms) to learn a particular logic function consisting of a series of logic gates, and after a certain number of steps / generations they vary one particular part of the logic gate setup, while keeping the rest the same. Their results were that modularity (as measured by the Q-score) was selected for, and the network motifs found in the evolved networks had a visible correspondence to the modular parts of the goal. We will discuss this paper in more detail in later sections of this post.

Testing this idea: We started by trying to replicate the results of the 2005 paper. It was somewhat challenging because of the now outdated conventions used in the paper, but our tentative conclusions for now are that the paper doesn’t seem to replicate. Our plans if the replication was a success would have been to generalise to more complicated systems (one proposal was to train a CNN to recognise two digits from an MNIST set and perform an arithmetic operation on them, with the hope that it would learn a modular representation of both the individual digits and of the operator). However, this line of research has been put on hold until we get to the bottom of the null result from the Kashtan & Alon paper’s replication.

For a fuller discussion of how we’ve been testing this idea (and some of our ideas as to why the replication failed), please see this post.

Specialisation drives the evolution of modularity

Basic idea: MVG can be viewed as a subcase of this theory, because the environment fluctuating in a modular way is one possible explanation for why selection might favour specialisation. However, it isn’t the only explanation. Even in a static environment, evolution is a dynamic process, and a modular organism will be more easily able to evolve changes that improve its ability to deal with a particular aspect of its environment, without this affecting other modules (and hence having detrimental impacts on the rest of the organism). Hence specialisation (and hence modularity) might still be selected for.

Biological modularity: This theory was created partly from observation of a problem with MVG: not all environments seem to fluctuate in obviously modular ways (or even to fluctuate much at all). However, a lot of the biological intuitions of MVG carry over into this theory; namely the correspondence between different aspects of the environment that lend themselves well to specialisation, and different modular parts of the organism’s internal structure.

Evidence: The paper Specialisation Can Drive the Evolution of Modularity looked into regulatory gene networks (simulated using genetic algorithms), and examined the conditions under which they form modules. It finds that the main driver of modularity is selection for networks with multiple stable gene activity patterns that have considerable overlap. In other words, many of the genes between the patterns are identical. However, this paper only examined one particular evolutionary setting, and it’s unclear to which extent the results can be viewed as representative of a common principle of biological evolution. There don’t seem to have been any follow-up studies on this research agenda.

Testing this idea: We have no current plans to test this idea, since we think that MVG captures the core parts of it.

Direct selection for modularity, when it can perform a task / enable an adaptation that would be infeasible otherwise

Basic idea: Modularity may be selected for when it breaks a developmental constraint, and thereby makes some beneficial adaptation possible that would be virtually impossible otherwise.

Biological modularity: It’s unclear how to look for evidence for this in the biological record, because it doesn’t seem obvious how the world would be different depending on whether this was true or false. However, one form this could take that seems highly plausible in a biological context is evolving a particular architectural change at very early stages of ontogenesis, which enforces modularity (more on this in the “evidence” section).

Evidence: A well-known phenomenon in neural networks is that of “neural interference”, whereby a system designed to learn a solution to two separate tasks inevitably develops connections between the two, despite these serving no practical purpose. As an example, consider a network performing image recognition on two different images, with the aim of outputting two different classifications: clearly no connections are required between the two tasks, but these will almost certainly be created anyway during the search for a solution, and they are hard to “un-learn”. The 2000 paper Evolving Modular Architectures for Neural Networks explores the idea of both the network architecture and the parameters being genetically inherited - that way organisms can avoid this interference altogether.

However, it appears to us that this methodology may be problematic in that the researcher is deciding on the network structure before the study is even run. Looking at the above diagram, it’s fairly obvious to us that the design on the right will perform better, but this study provides no evolutionary model as to how the modular architecture on the right might be selected for by evolution in the first place. It might seem intuitive to humans to choose the modular decomposition on the right, but for evolution to find that decomposition in particular is where all the work gets done, so this theory doesn’t seem to present an actual causal explanation.

Testing this idea: If this hypothesis is correct, then there should be certain tasks that a non-modular network simply fails to solve, and so any network trained on this task will either find a modular solution or fail to converge. However, empirically this seems like it isn’t the case (the question “why aren’t the outputs of genetic algorithms modular” was the motivating one for much of this project). Whether this is because the hypothesis is false, or because ML is missing some key aspect of evolutionary dynamics, is unclear.

Modularity arises from a selection pressure to reduce connection costs in a network

Basic idea: If we apply selection pressure to reduce connection costs, this could perform a kind of regularisation that leads to a modular structure, where the only learned connections are the ones that are most important for performing a task. Furthermore, connection costs between different nodes in the network scaling with the “distance” between them might encourage a kind of “locality”, which might further necessitate modularity.

Biological modularity: Despite being intuitively appealing, this idea looks a bit weaker when put in a biological context. There are some important examples of connections being costly (e.g. the human brain), but for this to be the driving factor leading to modularity, we should expect to see connection costs playing a leading role in virtually all forms of signalling across all organisms where modularity is to be found, and this doesn’t always seem to be the case (e.g. signalling molecules in bacteria). However, it’s possible this factor contributes to modularity along with other factors like MVG, even though it’s not the driving factor (e.g. see results from the paper below).

The locality argument seems a lot more robust to biological evidence. Where there is modularity, we very often see it structured in a local way, with networks of cells performing a specific task being clustered together into a relatively small region of space (e.g. organs). Cells interact with each other in a highly localised way, e.g. through chemical and mechanical signals. This means that any group of cells performing a task which requires a lot of inter-cellular interaction is size-limited, and may have to learn to perform a specialised task.

This argument also helps explain why modularity might be observable to humans. Consider a counterfactual world in which locality didn’t apply: there would be very little visible commonality between different organisms, and so even if modularity was present we would have a hard time identifying it (Chris Olah uses a similar idea as an analogy when discussing the universality claim for circuits in neural networks).

Evidence: The 2013 paper The evolutionary origins of modularity explores this idea. Their methodology was to compare two different networks: “performance alone” (PA) and “performance and connection costs” (P&CC). The task was a form of image recognition the authors have called the “retina problem”: the networks were presented with an 8-pixel array separated into two 2x2 blocks (left and right), and tasked to decide whether a certain type of object appeared in each block. This task exhibited a natural modular decomposition. It could also be varied in a modular way: by changing the task from deciding whether an image appeared in both blocks (”L-AND-R environment”) to whether an image appeared in either block (”L-OR-R”). In this way, they could test pressure to reduce costs against the MVG hypothesis. The results were that the P&CC networks were both more modular than the PA networks, and significantly outperformed them after a finite number of iterations, but that the best PA networks outperformed the best P&CC networks (explaining why performance alone might not be sufficient to select for modularity). They also found that adding MVG further increased both the outperformance and the modularity.

Another paper exploring this hypothesis is Computational Consequences of a Bias toward Short Connections (1992). This paper focuses on architectural constraints selecting for fewer/shorter connections, rather than imposing an explicit penalty. However, it doesn’t use particularly advanced machine learning methods, and focuses more narrowly on the human brain than on general modularity across evolved systems, making it less relevant for our purposes.

Testing this idea: We plan to introduce this type of selection pressure in our replication of the Kashtan & Alon 2005 paper. That way we can test it against the MVG hypothesis, and see whether the two combined are enough to produce modularity. We will experiment with different cost measures, although to the extent that this is a strong causal factor leading to modularity, we would expect the results to be relatively robust to different ways of calculating connection costs (this was also what the 2013 paper found).

We also have plans to test the locality idea, by instantiating neurons in some kind of metric space (e.g. 2D Euclidean space, or just using some kind of indexing convention), and penalising connection costs proportionally to their distance.

For our team's most recent post, please see here .