Evolution through large models

Evolution through Large Models
Joel Lehman
OpenAI
joel@openai.comJonathan Gordon
OpenAI
gordonjo@openai.com
Shawn Jain
OpenAI
jains@openai.comKamal Ndousse
Anthropic∗
kamal.ndousse@gmail.com
Cathy Yeh
OpenAI
cathy@openai.comKenneth O. Stanley
OpenAI
kennethostanley@gmail.com
June 20, 2022
Abstract
This paper pursues the insight that large language models (LLMs)
trained to generate code can vastly improve the eectiveness of mutation
operators applied to programs in genetic programming (GP). Because such
LLMs benet from training data that includes sequential changes and
modications, they can approximate likely changes that humans would
make. To highlight the breadth of implications of such evolution through
large models (ELM), in the main experiment ELM combined with MAP-
Elites generates hundreds of thousands of functional examples of Python
programs that output working ambulating robots in the Sodarace domain,
which the original LLM had never seen in pre-training. These examples
then help to bootstrap training a new conditional language model that can
output the right walker for a particular terrain. The ability to bootstrap
new models that can output appropriate artifacts for a given context in
a domain where zero training data was previously available carries im-
plications for open-endedness, deep learning, and reinforcement learning.
These implications are explored here in depth in the hope of inspiring new
directions of research now opened up by ELM.
1 Introduction
For many in the evolutionary computation (EC) community, the rise of deep
learning (DL) has raised questions on its implications for EC. Both approaches
∗Work done at OpenAI.
1arXiv:2206.08896v1  [cs.NE]  17 Jun 2022
scale well with compute and both can yield useful discoveries and meaningful
surprises. Yet are they ultimately competing paradigms, or rather are they
complementary? In this paper we explore the latter possibility, of consider-
able synergy, by highlighting an untapped implication of large language models
(LLMs; [1, 2]) for both genetic programming (GP; [3, 4]) and open-endedness
[5{7].
In particular, in this new Evolution through Large Models (ELM) approach,
an LLM trained on code can suggest mutations that are intelligent, thereby fa-
cilitating a dramatically more eective mutation operator that sidesteps many
of the challenges that previously existed for evolving programs [8]. Interest-
ingly, the benets of ELM are also reciprocal back to deep learning: the set of
samples generated through the LLM can eventually constitute a new training
set in a novel domain that can then ne-tune the LLM to perform well in the
new domain, a novel data-generation procedure. Furthermore, this approach
ultimately opens up new opportunities in the pursuit of open-endedness by in-
creasing the generative capabilities of the LLM solely through its own generated
data.
LLMs have recently yielded impressive results in automated code genera-
tion [9, 10]. These models bootstrap from human knowledge by learning from
very large datasets to achieve general coding competency. The fact that such
bootstrapping is possible is clearly relevant to GP. After all, GP is in eect a
generative approach to programming. While it might seem at rst glance that
LLMs therefore might out-compete or subsume GP, in fact GP does still oer
an advantage in situations where the particular class of programs targeted by
the search is far (or even completely lacking) from the training distribution of
the LLM. In such cases, the LLM oers limited recourse (prompt engineering to
learn an entirely new domain would be prohibitive), while GP can in principle
evolve in any space (though in practice some spaces may be intractable due to
the amount of mutation necessary to get consistent signal on tness).
Interestingly (and perhaps surprisingly), the best of both worlds is easily at-
tainable by combining them: simply by prompting the LLM to generate changes
the LLM can serve as a highly sophisticated mutation operator embedded within
an overarching evolutionary algorithm. This way, the LLM in concert with evo-
lution can steer each other towards the right region of the solution space even
though neither evolution with a conventional mutation operator nor the LLM
on its own could generate anything close. In eect, program evolution using
LLM-based perturbation begins to bridge the divide between evolutionary algo-
rithms and those that operate on the level of human ideas. That is, LLMs can
be trained to approximate how humans intentionally change programs, while
staying on the manifold of functionality. Furthermore, such LLMs can be fur-
ther ne-tuned on successful perturbations for the purposes of self-improvement,
culminating in a novel technique for iteratively enhancing the performance of
ELM.
To highlight the potential of this approach, in this paper an entire dataset
in a novel domain is generated from only a single mediocre starting example
designed by hand by humans. In particular, the domain is called Sodarace
2
[11, 12], where two-dimensional ambulating robots of arbitrary morphology are
constructed for diverse terrains. Sodarace is cheap to simulate, allowing fast
iteration, and also makes it easy to appreciate the sophistication of designs in-
tuitively by simply watching the robot walk. In this way, it facilitates quick
assessment of whether a design is successful both quantitatively and qualita-
tively.
To make the contribution of ELM explicit in the experiments in this paper,
the Sodaracers are encoded as raw Python programs that output an enumeration
of the ambulating robots' components. That way, it is possible to demonstrate
that ELM is a form of GP that can operate on a modern programming language
directly, with no special provisions needed beyond the generic (i.e. notpreviously
trained in Sodarace) existing code-generating LLM.
A nal important insight unlocked by this approach is that the ability to
generate diverse solutions in a domain or part of the search space where there
was little to no training data is foundational to bootstrapping an open-ended
process [6, 13, 14]. After all, open-endedness is fundamentally about searching
outside the distribution of previous experience, which is exactly what ELM
helps the LLM to do. Because this novel capability has potentially far-reaching
implications, we have chosen in this work to focus on the implications of the
generated data that can be produced by ELM. Of course, ELM is applicable in
many other contexts that will undoubtedly be explored in the future.
More specically, experiments that follow show that generated data is su-
ciently rich that it can serve as training data for ne-tuning LLMs to generate
code for viable new Sodaracers consistently, and furthermore that reinforcement
learning (RL) can even ne-tune an augmented LLM to output Sodaracers con-
ditionally , depending on the terrain. In the future, such conditional invention
has the potential to unlock entirely new kinds of open-ended processes, just
as humans have open-endedly built civilization over centuries by conditionally
inventing its constituents.
In short, the main contributions of this paper are (1) the ELM method
for eciently evolving programs through LLMs, (2) a technique for improving
ELM's ability to search over time by ne-tuning its LLM-based mutation oper-
ator, (3) a demonstration of ELM in a domain not included in the training data
of the LLM, and (4) validation that data generated through ELM can bootstrap
enhanced LLMs that oer a novel path towards open-endedness.
2 Background
This section reviews previous work in genetic programming, large language mod-
els, and open-endedness.
2.1 Genetic Programming
The eld of genetic programming (GP) applies evolutionary algorithms to evolve
computer programs to solve problems [3, 4, 15]. The promise of GP is that com-
3
puter code is a computationally universal representation that underlies much
modern technology, including articial intelligence itself. Therefore it is con-
ceivable for GP to automatically evolve programs that achieve human-level (or
beyond) performance across diverse application domains [16]. However, there
are obstacles in practice to its successful and widespread application to chal-
lenging problems.
One obstacle is that scaling GP to evolve increasingly complicated pro-
grams can be challenging [8], and that eectively applying GP to a new do-
main can require signicant domain expertise. A researcher often must explic-
itly specify what functions, variables, and control structures are available to
evolution [3, 17], which limits what ultimately can be evolved. In contrast,
a human programmer can open-endedly decide what libraries to import and
how to write many interdependent subroutines or classes. Research aims to lift
these constraints, often through enabling modular reuse of code: e.g. through
automatically dened functions [3], data-mining populations to nd common
sub-components [18], or attempts to use solutions to previous problems when
solving new ones [19]. However, no method yet enables GP to scalably operate
on human-designed programming languages with a minimum of domain-specic
tweaking.
A second obstacle is that nearly all GP methods explore through random per-
turbations of code, unlike humans, who through active practice improve their
prociency in making deliberate, complex, and coupled modications to pro-
grams [20, 21]. Unlike perturbing e.g. neural network weights, wherein continu-
ous parameters subject to small enough perturbations can predictably generate
small changes in functionality [22, 23], perturbing code requires discrete changes
that often dramatically shift functionality [24], thereby complicating search.
While there exist approaches towards more directed generation of ospring (e.g.
building probabilistic models of high-performing programs [25], evolving repro-
duction operators [26], or applying less-impactful mutation operators [24]), the
problem remains at core unsolved.
In contrast to GP, humans learn to reason about code in its full complexity
through experimentation and learning. This iterative eort leaves a permanent
signature in repositories of code, such as GitHub. The next section describes
progress in training large language models upon such repositories as a potential
way to bypass the above obstacles.
2.2 Large Language Models
Large language models (LLMs; [1, 2, 27]), trained on internet-scale data, have
progressed at an impressive pace in recent years. The main idea (in auto-
regressive models such as GPT-3 [2]) is to train increasingly-large neural net-
works (built on the popular transformer architecture [28], sometimes with bil-
lions of parameters) on the seemingly simple task of next-token prediction (i.e.
given a sequence of tokens seen so far, predict the proceeding token). Scaling
such LLMs (and formulating problems of interest as natural language processing
tasks) has resulted in groundbreaking performance across a wide range of tasks
4
[2, 29], including program synthesis [9, 10, 30].
In particular, by training LLMs on large-scale coding data, e.g. from GitHub,
it is possible to produce models with impressive function-synthesis capabilities
[9, 10], highlighting the possibility to bootstrap the ability to
uently code from
large-scale data. A further development are dimodels that are trained on
dis from GitHub [31]. A di is an incremental change to a le that is com-
mitted to a version control system such as GitHub, accompanied by a commit
message describing the intent of the change. In this way, di models are trained
how, given a piece of code and any potential commit message, to suggest an
informed change . Through the lens of evolutionary algorithms, such di mod-
els can be viewed as intelligent perturbation operators, providing a way to walk
over the manifold of code (in a controllable way) through mimicking human pro-
grammers. An interesting further possibility is that such models are amenable
to further training through gradient descent, implying a potentially-powerful
mechanism for self-adaptation (e.g. through reinforcing successful dis during
evolution). Both di models and their capacity for self-adaptation are explored
in this work as a way to improve GP. However, it is also important to note that
general language models not trained directly on dis can also act in eect like
di models when given the right kinds of prompts (see Section 3.1).
2.3 Open-endedness
With origins in the open-ended evolution community [6, 13, 32, 33] within ar-
ticial life, the eld of open-endedness seeks to create algorithmic systems that
produce never-ending innovation [5]. Given the primacy of search with ML,
research within open-endedness naturally has focused on rening algorithms for
open-ended search, such as those driven by novelty [34, 35] or curiosity [36, 37].
While such focus has indeed lead to algorithmic progress, there is a growing
awareness of the criticality of the environment in which open-ended algorithms
are applied [38{41].
That is, the environment limits what can arise within the system and for how
long its products can remain interesting. As a result, some have argued for more
complex environments for open-endedness, such as video games [38, 39], and
others have argued that features of the environment should co-evolve with agents
[40, 42]. Yet a theory for what specic forms of additional such complexity is
needed for enduring open-endedness has been lacking. This paper contributes a
possible theory, arguing that agents outputting inventions into the environment
in response to previous inventions may be a principled route to such continuing
open-endedness.
One challenge in evolving aspects of the environment (such as inventions),
is how they are encoded. Most research applies encodings that are specically
t to describe some xed part of a larger environment, e.g. a xed way of
describing edges within a maze [43], or the shape of a 2-D landscape [40]. While
sometimes the encodings of these parts are universal (e.g. the CPPN encoding of
landscapes in [40] can describe any landscape, and the RNN encoding of Dennis
et al. [42] can describe any maze), it is unclear how to expand the representation
5
to include more of the environment without relying upon ad-hoc principles. This
paper argues that computer programs are a general and powerful encoding for
continually expanding the richness of an existing environment.
3 Approach: Evolution through Large Models
Three distinct components facilitate ELM. First is the novel mutation operator
driven by an LLM. Second is an evolutionary outer loop that calls this mutation
operator. Finally, the third component is a method for updating the LLM to
improve based on its preceding performance. Each of these is detailed in this
section.
3.1 Mutation through Di
The main idea behind ELM centers on rethinking the mutation operator for
code by exploiting the capabilities of LLMs. In conventional GP, the language
of the code and the types of changes allowed through mutation are both chosen
intentionally to yield a reasonable chance that perturbations can lead to useful
functional changes [3]. In contrast, LLMs unlock an entirely dierent basis
for mutation: it would be more ideal if the mutation operator understood the
code and how it can be changed in interesting ways, more like a human than a
stochastic event.
LLMs can indeed be trained to output code in an autoregressive manner by
exposing them to extensive programming examples [9, 10]. A di model [31]
can similarly be autoregressively trained on a collection of code dis (e.g. from
GitHub). Each di targets a single le, where the le and di are short enough
to t into the context of the LLM. The model is trained to predict the di
(formatted, for example, in unied di format [44]) from the concatenation of
the le and the commit message, where the loss includes only the tokens that
make up the di, thereby encouraging the model to predict the di but not to
memorize the le and commit message. In other words, the model learns to
predict plausible changes to code from examples of changes made to code by
human programmers. It is important to note that the idea of di models (or
their initial training) [31] is not a contribution of this paper, but di models are
rather a tool applied here in a new context (to produce mutations).
To achieve meaningful mutations, ELM can choose among a set of commit
messages , which convey to the LLM the details of the operation it should per-
form in lieu of mutation. These messages oer signicant power and nuance
for calibrating mutation operators that is likely highly novel to anyone familiar
with implementing mutation in GP or evolutionary algorithms in general. In
the experiment in this paper, the three commit messages and their respective
probabilities of being chosen are:
•Changed make walker function. (40% chance)
•Changed parameters in make walker function. (30% chance)
6
•Small change to make walker function. (30% chance)
Of course, any commit message is conceivable. The LLM's ability to interpret
general natural language means that the scope for research exploration (and
domain-specicity) here is vast.
As a simple experiment to highlight di models' ability to intelligently mod-
ify code, an implementation of a function with an adjustable amount of bugs is
perturbed with either a simple GP mutation operator or with a 300M parameter
di model. The hypothesis is that an intelligent perturbation operator will be
better able to make multiple correlated changes to code (in this case to correct
the bugs). The 4-Parity task (which is inspired by a standard GP benchmark
[3]) serves as a representative test-bed. Note that a correct implementation
of 4-Parity returns the sum of the four input bits, modulo two. Up to ve
bugs are introduced to 4-Parity, rst by incrementally misnaming each of the
variables in the sum calculation; and for the fth bug, the modulo is changed
from two to three. Then, perturbation operators are tested for their ability to
(in one perturbation step) change the buggy version of the code to one that
successfully passes unit tests. Results in gure 1 highlight how with increasing
bugs GP mutation becomes exponentially more unlikely to produce a successful
solution (note that nomutation from GP solves all ve, given 100 ;000 trials).
In contrast, the di operator is able to x all ve bugs, and its performance
is impacted more by the number of dierent types of bugs (i.e. the fth bug
aects the modulo calculation rather than renaming variables) than by the raw
number of bugs itself. Further details (including a supporting experiment with
another task with similar results) are given in Appendix A.
Because the tools involved in an ELM implementation are unconventional,
we nally wanted to highlight here several alternatives for implementing such
systems in practice today. One option is to use models available on the OpenAI
API that can edit through following instructions [45, 46]. A second option is
to create an intelligent mutation operator through few-shot prompting instead
of through explicit training (as in the di model). That is, one could design
prompts for a model trained on code (like Codex [9] or GPT-6-J [47]). To
show the potential to replicate (or improve upon) the results in this paper, we
conducted a simple experiment comparing (on the 4-Parity problem) prompt
engineering and edit mode to the di model. Figure 2 shows how models from
the API outperform the di model used in the paper. Further experimental
details can be found in Appendix A.
3.2 The Evolutionary Algorithm and Implications for Open-
Endedness
Because the mutation operator is eectively a modular component for many
evolutionary algorithms [48, 49], ELM can be implemented within a diversity of
contexts. Of course, the approach is most applicable to a case where the genetic
encoding is through a known programming language, because that is how the
benets of the LLM will be realized. Genetic encodings in natural language
7
Figure 1: Comparing di mutation to GP mutation in xing 4-Parity
bugs. The gure shows how the ability of a single mutation to produce correct
solutions changes as bugs are incrementally added to a working 4-Parity imple-
mentation. Note that success percentage is shown in log scale , i.e. success for
GP mutation decreases exponentially in the number of mutations (and produces
no solutions when there are ve bugs). In contrast, di mutation degrades only
with the fth bug. The conclusion is that LLM-based mutation can indeed make
multiple sensible coupled changes to code.
or any other language at which LLMs excel are also conceivable, but of course
the utility of such encodings would depend on how they are applied and their
mapping to a potentially useful phenotype. The experiments in this paper focus
on Python 3 genotypes, which are also by their nature variable in length. The
ability to use modern programming languages as genotypes without the need
for any special accommodation is a key benet of ELM.
While there are many options for the evolutionary algorithm in the outer
loop, we have chosen in this paper to implement ELM within a quality diversity
(QD) algorithm [50, 51]. An important motivation for this choice is that the
emergence of the ability to search intelligently for arbitrarily complex programs
is tantalizingly close to overcoming some of the key obstacles to open-endedness
[14], and ELM is an opportunity to highlight this opportunity.
Recall that we do not yet know how to make an algorithm that exhibits
genuinely open-ended divergence. While there has been progress towards open-
endedness in recent years, the state of the art remains weak open-endedness ,
wherein novel and interesting discovery continues only for a brief time, even-
tually ending in a plateau as the possibilities are exhausted [5, 40, 43, 52{54].
In contrast, in strong open-endedness , the process would never plateau{if we
left and returned a year later, or even a million years later, its products would
continue to become more interesting over time. No algorithm comes close to
such an achievement, though it is evidently possible in nature.
The question then is what stands between today's algorithms and tractable
strong open-endedness. This gap remains despite that recent work in open-
8
Figure 2: Comparing alternate LLM-based mutations in xing 4-Parity
bugs. The performance of dierent mutation operators in xing bugs is shown
as bugs are incrementally added to a working 4-Parity implementation. Both
edit mode and prompt-engineering approaches outperform the di model applied
in this paper's experiments. The conclusion is that both prompt-engineering on
LLMs trained on code and using edit mode models from the OpenAI API are
viable options to build upon the work in this paper.
endedness appears to make progress. For example, the Enhanced POET algo-
rithm continues to generate diverse and increasingly complex terrains for bipedal
robots to solve [40]. In their hide-and-seek experiment, Baker et al. [54] show
agents discovering increasingly complex strategies like assembling blocks into
a hideout. Yet despite such algorithms clearly demonstrating the capability
to continue to invent new solutions, all such demonstrations share a singular
downfall: they slow down and eventually end. Formalizing ELM within a QD
framework in eect oers a novel opportunity to address this challenge.
This opportunity connects to the diculty of formulating an articial en-
vironment that imposes no limit on what even the most capable open-ended
algorithm can achieve, as noted in the Background. The challenge of devising
articial environments with unbounded potential raises the intriguing question
of what property our universe and planet possess that is lacking in current ar-
ticial environments. This question is critically important for open-endedness
because in the absence of that property, open-ended algorithms cannot demon-
strate their full potential. If the problem indeed stems from the fact that ar-
ticial environments to date oer only nite possible experiences until their
potential is exhausted, then to overcome this bottleneck the environment itself
needs to possess the potential to change forever.
Since the emergence of intelligence in nature, much environmental change
has been driven by the intelligent agents themselves. Eventually, humans ac-
quired the ability to leave detached artifacts in the environment that radically
alter its potential for themselves and other agents, like a house, a vehicle, or
even a program . Unlike new organisms that are evolved over generations, such
detached conditional things (DCTs) are generated intentionally as a condition
9
of the observations of the agent. Once DCTs enter the world, open-endedness
accelerates because the environment is rapidly updating even within the course
of a single lifetime.
Each DCT creates an opportunity for further DCTs. For example, the in-
vention of the door creates the opportunity for keys to be invented, which then
sets the stage for lock picks, and so on. And because they are detached, DCTs
can leave a permanent legacy in the environment well beyond the lifetime of
their inventor. In this way, invention in the era of DCTs is open-ended, and
accordingly has continued for thousands of years, from re and wheels to space
stations and computers.
This theory of DCTs supplies an abstract answer to the problem of a limited
environment: Agents must be able to imprint the environment with DCTs in
response to those already present within it. However, realizing DCTs in practice
requires addressing a separate question: how can agents be enabled to eciently
invent DCTs of limitless complexity in a new domain?
Interestingly, computer programs are universal representations, meaning
that the procedure of assembling new artifacts can naturally be described algo-
rithmically. For example, programmers have leveraged code to help create enor-
mously complex artifacts (like the layouts of computer chips or instructions for
3-D printers to produce complex physical objects). Of course, programs them-
selves can function as DCTs. In this way, a procedure that can search through
modern program space and ultimately generate such programs conditionally is
a candidate for creating open-ended environments of unlimited capacity. The
experiment in this paper will demonstrate in more detail how ELM makes such
a construct conceivable; the signicance of QD is that its ability to generate a
diverse space of artifacts can serve as the bootstrap to obtaining agents capable
of generating DCTs. In short, the QD algorithm is generating training data
that can transform the LLM into a kind of DCT generator.
While any QD algorithm can work with ELM, the specic algorithm in the
experiment in this paper is MAP-Elites [51, 55] (Figure 3). The core of MAP-
Elites is a uniformly-spaced grid of niches (called the map), that spans user-
specied dimensions of solution diversity, called the behavior characterization .
Upon initialization, a single pre-existing (hand-designed in this paper) solution
is evaluated and placed into the map. In each iteration thereafter, an inhabited
niche is randomly chosen and the solution within that niche is perturbed by the
di model and evaluated. The new candidate solution is assigned its niche from
its behavior characterization, and if that niche is unlled or the new solution out-
performs the niche's current inhabitant, it becomes the champion of that niche;
otherwise, the candidate is discarded. In this way, over iterations of search, the
map gradually lls with increasingly diverse and high-quality solutions.
3.3 Fine-tuning the Di Operator
Interestingly, because the mutation (di) operator is itself an LLM, it has the
potential to be improved with respect to the domain. While self-adaptation [56{
58] has a long tradition in evolutionary computation, including algorithms such
10
Map of Diverse Champions
Python Program Diff Model Python Program
Width of Sodaracer Height of
Sodaracer Figure 3: MAP-Elites with ELM. In each iteration, an existing Python
solution is sampled from the map of solutions for each independent replica of
a di model. Each replica generates a batch of dis that are applied to the
sampled solution to generate modied candidate solutions. These candidates are
evaluated and are then inserted into the map if they either establish a new niche
or outperform the niche's current champion. Over iterations, a single initial seed
program gives rise to a diversity of high-performing Python programs.
as CMA-ES [58] and natural evolution strategies [59], the kinds of improvements
possible in ELM are unique by oering the possibility of the LLM learning how
to think about change . That is, ideas for changes that are most promising in one
domain might be dierent than in another, and the richness of the LLM oers
the potential to capture such nuance through experience. In particular, the
pre-trained di model can be trained further (which is called ne-tuning ) with
accepted dis (by MAP-Elites) from initial iterations or runs of ELM. That way,
the di operator updates to understand better the kinds of modications that
lead to either higher quality, more novelty, or both. This ne-tuning technique
can cause ELM itself to improve over iterations. Of course, over a long run,
the ideal kinds of changes might change; continually ne-tuning based on recent
experience can potentially track such drifting opportunities. In this paper, the
potential of ne-tuning is demonstrated through a single ne-tuning iteration,
but the investigation of such continual renement is an open research opportu-
11
nity. Note that the prompt-engineering approach to LLM mutation described
at the end of Section 3.1 can also benet from ne-tuning in this way.
4 Experiment and Results
The primary motivation for the experiment that follows is to give a taste of the
breadth of implications of ELM, to evolutionary computation, to deep learning,
and to open-endedness. For this purpose, this experiment focuses on the prob-
lem of the invention of complex artifacts (which could eventually serve as DCTs
in a future more ambitious experiment). While the potential scope of applica-
tions for ELM is broad, the opportunity to learn to invent complex artifacts
in an arbitrary domain extends directly from the augmented ability to search
through programs; seeing this inventive capability realized thereby highlights
novel opportunities opening up.
The experiment will aim to bootstrap from a few hand-written (and barely
functional) examples of an invention into an LLM-based inventor that can
u-
idly output appropriate inventions conditioned on its environment. This concept
is demonstrated in the domain of Sodarace [11, 12], a physics-based invention
domain that serves as a cheap-to-simulate microcosm of invention. The goal in
Sodarace is to construct from collections of masses and oscillating springs two-
dimensional robots that can locomote competently. A wide range of interesting
Sodaracer robots are possible, as highlighted by previous ML research [12] and
the origins of the domain: Sodarace began as a web application called Sodacon-
structor, wherein the human design of Sodaracers was suciently compelling
for an online community to form around the endeavor [11].
An individual Sodaracer (Figure 4) is composed of a variable-sized collection
of point masses (each fully-described by its initial 2-D position) and oscillating
springs that connect masses together. The motion of the robot is driven by
the oscillation of its springs, and each spring has parameters specifying the
amplitude and phase of its oscillation (by convention all springs have the same
period). To evaluate a particular Sodaracer, it is simulated in a specic terrain
for a xed amount of time and its ability to traverse that terrain is measured
(i.e. how far the Sodaracer's center of mass moves along the x-axis); additionally,
to measure the diversity of solutions for MAP-Elites, features capturing gross
aspects of the robot's morphology (i.e. its initial height, width, and total mass)
are recorded. While a search algorithm could operate directly in the space of
masses and springs, here instead LLMs output Python code that describes the
morphology of the Sodaracer. For examples of such source code, see Appendix B
and G. In this way, the programs evolved by ELM are in eect indirect encodings
[60{63] for Sodaracers. That is, in principle any indirect encoding expressible
through code could be evolved from scratch or modied by ELM.
More ambitiously than only generating a repertoire of Sodaracer designs,
the experiment will attempt to implement an entire invention pipeline that
ultimately yields a novel kind of conditional LLM that can input a terrain and
output an appropriate Sodaracer for that terrain. ELM thereby serves as the
12
Figure 4: An Example Sodaracer. The objective in the Sodarace domain
is to design a Sodaracer that locomotes eectively across the ground terrain.
Labeled in the image are examples of a mass and a spring that connects two
masses together. A Sodaracer design consists of a variable number of masses
and springs, where springs also have oscillatory parameters that determine the
Sodaracer's motion.
initial data generation phase of this pipeline, showing in this way how ELM
can serve in general as a way of generating domain data for downstream deep
learning where it did not previously exist. Furthermore, in the future the ability
to train such conditional inventors could serve as a foundation for an open-ended
world of DCT-generating agents.
In practice, the aim of the invention pipeline is to create an agent that can
output complex artifacts conditionally, based on its observation of the environ-
ment. If invention is conceived as an action, then learning to invent condition-
ally can be viewed as a reinforcement learning (RL) problem [64]. That is, for
any given observation, the agent can be rewarded depending on the success of
the resultant invention. For example, in Sodarace, the agent might observe a
specic terrain such as a hill and then output a design for a Sodaracer arti-
fact. The reward then depends upon the performance of the Sodaracer in the
observed terrain.
This approach sounds straightforward{it is simply RL with complex outputs{
but there is a problem. If the agent has no prior experience in the domain (e.g. in
Sodarace), then outputting even a valid (let alone working) artifact is eectively
impossible. As a result, there is no gradient for RL and it cannot bootstrap into
the new domain.
Therefore, to get RL started, some form of pretraining is necessary. In
eect, the RL ne-tuning described above is actually the last step in a pipeline ,
where the preceding step is to teach the agent something preliminary about its
domain. For that purpose, an LLM can be trained on a large set of examples
from the target domain. For example, these examples could be Sodarace walker
designs. After exposure to enough such designs, in principle the LLM knows
something about the domain and can output sample artifacts from the training
distribution. With such knowledge later passed on to RL, it should now be
13
possible to bootstrap into conditional ne-tuning.
However, there is still a problem: where did all the examples come from
for training the LLM? If the hope is for the conditional inventor eventually to
invent in a novel domain like Sodarace where a generic LLM is unlikely to have
any exposure, then the source for all the examples needed to train the LLM
is itself elusive. As a consequence, the pipeline needs yet one more preceding
step{which is where ELM comes in{to generate a set of example artifacts from
scratch, which could then train the LLM that will eventually bootstrap RL.
Generating a diverse and large set of initial training examples is a search
problem. However, because no LLM yet has any exposure to the right kind of
data, it is a search problem within the invention space rather than within the
weight space of neural networks. Searching for diverse functional examples (to
get a wide pre-training distribution) within the space of artifacts is the natural
role of QD (i.e. MAP-Elites in this paper). Combined with the di function, the
result is ELM, which yields a novel approach to generating training examples,
thereby bootstrapping the entire process.
To recap, what emerges is a three-stage invention pipeline for training con-
ditional inventors (Figure 5):
1.ELM. Search for a diverse set of example artifacts (e.g. Sodaracers on
at
ground).
2.Pre-train the LLM with examples from ELM. The LLM accordingly
learns to output example inventions from the training distribution.
3.Learn to invent conditionally. Splice new conditional inputs onto the
LLM and ne tune it through RL to produce appropriate inventions for
the conditions it observes.
4.1 Encoding Sodaracers with Python
Previous experiments targeting Sodarace have leveraged specialized evolution-
ary encodings [12]. Instead, in this work plain-text Python code acts as a
generic representation for inventions. By showing how Python can be used to
represent artifacts in an arbitrary domain, it opens up the possibility of using
it as a generic encoding in diverse future domains. More specically, in the ex-
periments an individual is evaluated by executing its code through the Python
interpreter. The product of the interpreter (for a viable individual) is a data
structure containing the description of a Sodaracer (i.e. a Python dictionary
containing lists of both point masses and springs), which can then be passed
to the Sodarace simulator to evaluate the encoded Sodaracer's behavior. Note
that Sodaracers are encoded in Python throughout the invention pipeline, i.e.
ELM evolves Python programs and the language models in both latter stages
of the pipeline are trained to output Python programs.
Preliminary experiments showed that the di model's initial performance
(i.e. before ne-tuning) in creating useful perturbations depended upon the de-
sign of the \interface" through which Sodaracers were procedurally constructed.
14
To review:
Conditional
Inventors
LMInvention Spec
Conditional RL on
LM
Inventon Pipeline
Pre-train LM on many
examples in simple
starting context Run QD algorithm to
generate initial dataset
(ELM) Open-ended
Process
Figure 5: The Invention Pipeline. (left) A three-staged training pipeline
bootstraps from a single example of an invention to an LLM that can output an
invention tailored to its current condition. The hope for the future is for such
a conditional inventor agent to help seed an open-ended process, wherein inter-
actions between agents and their inventions spur continual innovation. (right)
In the Sodarace domain, the conditional inventor observes the terrain, which
conditions the LLM to output the specication of the desired invention.
That is, while a Sodaracer can be constructed in Python by directly adding el-
ements to a Python dictionary with keys such as \joints" and \muscles," a
more Pythonic interface (which was more eective and is what is used in the
experiments) is to create a simple class with two methods: \add joint" (to add
a spring) and \add muscle" (to add a point mass.) The idea is that such an
interface (here encapsulated in a class called \walker creator") is closer to the
training distribution of Python code (although still no Sodarace examples in this
format exist). For example, below is the encoding of a simple hand-designed
square Sodaracer (that is also used in the experiments as a seed), as well as its
translation after being executed into a dictionary of joints and muscles. The
interface also includes logic for ensuring that the Sodaracer will not break the
underlying Box2D physics engine, e.g. that each joint is connected only to so
many muscles, that the strength of muscles is limited, and that there is a mini-
mum distance between joints. Note that the benet of evolving a program that
produces a data structure rather than directly evolving the data structure itself
relates to the benets of indirect encoding (i.e. a program can leverage regu-
larities through loops, conditionals, and functions, to eciently encode large
complex structures) [60]. Figure 6 shows an image of this walker when instan-
tiated.
15
1from walk_creator import walker_creator
2
3def make_square(wc, x0, y0, x1, y1):
4 """ Make a square with top left x0,y0 and top right x1,y1 """
5 j0 = wc.add_joint(x0, y0)
6 j1 = wc.add_joint(x0, y1)
7 j2 = wc.add_joint(x1, y1)
8 j3 = wc.add_joint(x1, y0)
9
10 return j0, j1, j2, j3
11
12
13def make_walker():
14 wc = walker_creator()
15
16 # the main body is a square
17 sides = make_square(wc, 0, 0, 10, 10)
18 center = wc.add_joint(5, 5)
19
20 # connect the square with distance muscles
21 for k in range(len(sides)-1):
22 wc.add_muscle(sides[k], sides[k+1])
23 wc.add_muscle(sides[3], sides[0])
24
25 # one prong of the square is a distance muscle
26 wc.add_muscle(sides[3], center)
27
28 # the other prongs from the center of the square are active
29 wc.add_muscle(sides[0], center, False, 5.0, 0.0)
30 wc.add_muscle(sides[1], center, False, 10.0, 0.0)
31 wc.add_muscle(sides[2], center, False, 2.0, 0.0)
32
33 return wc.get_walker()
Listing 1: Example Sodaracer-generating program.
5 Pipeline Stage 1: Data Generation through
ELM
Recall that the aim in this rst stage is to generate a large variety of high-quality
examples from a single example starter program through ELM. In this stage of
the pipeline, the Sodarace environment is a simple
at terrain.
Recall that ELM in this experiment will evolve through MAP-Elites (Fig-
16
1{
2 "joints": [(0, 0), (0, 10), (10, 10), (10, 0), (5, 5)],
3 "muscles": [
4 [0, 1, {"type": "distance"}],
5 [1, 2, {"type": "distance"}],
6 [2, 3, {"type": "distance"}],
7 [3, 0, {"type": "distance"}],
8 [3, 4, {"type": "distance"}],
9 [0, 4, {"type": "muscle", "amplitude": 2.12, "phase": 0.0}],
10 [1, 4, {"type": "muscle", "amplitude": 2.12, "phase": 0.0}],
11 [2, 4, {"type": "muscle", "amplitude": 2.12, "phase": 0.0}],
12 ],
13}
Listing 2: Intermediate Sodaracer representation from running the above
Python seed program.
Figure 6: Instantiation of a hand-designed square Sodaracer. A video
of this walker can be viewed at https://y2u.be/jeP8Nsulu48
ure 3) [51]. The core of MAP-Elites is a uniformly-spaced grid of niches (called
themap), that spans user-specied dimensions of solution diversity, called the
behavior characterization . In experiments here, the behavior characterization
consists of the height, width, and mass of Sodaracers, and the map is a 12
1212 grid into which any behavioral characterization can be mapped. Upon
initialization, a single hand-designed solution is evaluated and placed into the
map. In each iteration thereafter, an inhabited niche is randomly chosen and
the solution within that niche is perturbed by the di model and evaluated. The
new candidate solution is assigned its niche from its behavior characterization,
and if that niche is unlled or the new solution outperforms the niche's current
inhabitant, it becomes the champion of that niche; otherwise, the candidate is
discarded. In this way, over iterations of search, the map gradually lls with
increasingly diverse and high-quality solutions.
To address the need for seed solutions, four simple seeds were written that
explore dierent architectural motifs: the Square seed, the Radial seed, and two
CPPN-like seeds (CPPN stands for compositional pattern-producing network
[61]); note that source code for these seeds is provided in Appendix B. The
Square seed instantiates a polygon-like bias, by including a function that creates
17
Figure 7: The three seed solutions. From top to bottom: CPPN seed,
radial seed, and square seed. A video of these walkers is at https://y2u.be/
jeP8Nsulu48 (same video as for Figure 6).
a square composed of four masses from two coordinates, and code that calls that
function and connects the masses together with a for-loop. The Radial seed
instead implements a radial bias by replacing the square-generating function
with a function that places a given number of masses in a circular shape. Finally,
the CPPN-like seeds roughly implement the CPPN-based encoding used by
previous work in Sodarace [12], i.e. it places masses and connects springs between
them as a mathematical function of their coordinates. The CPPN-based seed's
code can be neatly divided into (1) implementing the core functionality of a
CPPN, and (2) describing a particular instantiation of a CPPN, and thus enables
exploring the consequences of letting core functionality of the encoding itself
evolve. To this end, there are two CPPN seeds, one in which the CPPN encoding
is xed, called the CPPN-Fixed seed, and one where it is mutable, called the
CPPN-Mutable seed. Note that these seed programs were not highly-tuned as
the videos in Figure 7 highlight.
5.1 Experimental Details and Results
Three independent runs of ELM were conducted with each seed, running for
1;024;000 evaluations each (composed of 2,000 iterations of 512 dis per itera-
tion). A 300M parameter pretrained di model [31] served as the perturbation
operator in these experiments.
One metric of success for ELM is the number of niches lled, which represents
the diversity of data generated by ELM, under the hypothesis that such diverse
data will benet later pipeline stages. Figure 8 shows that runs of ELM tend to
discover a large proportion of niches, highlighting how the system can bootstrap
from a single user-provided example to ll the space of desired possibilities.
However, the speed of spreading through niches varies across seeds; in particular,
introducing loops and/or function composition is required for the Square seed
18
Figure 8: Amount of niches lled across seeds. The gure shows the
percentage of all niches (1,728 in total) that are lled by the end of ELM runs
across dierent seeds. Results are averaged across three independent runs for
each seed. In general, nearly all seeds ll the map, although the Square seed
proceeds more slowly than other seeds.
to spread into high-mass niches (e.g. to connect many squares together), which
emerges slowly in some runs.
Beyond diversity, the quality of solutions is also important. A gross measure
of quality is the maximum tness discovered by runs, shown in Figure 9. A
more nuanced metric that takes both quality and diversity into account is the
QD score [50], calculated as the sum of the performance of all champions in the
nal map. This metric, shown averaged over runs in Figure 10, rewards both
quality (having higher scores in each niche) and diversity (having discovered
more niches), and thus serves as a succinct measure of ELM's goal of accumu-
lating diverse, high-quality solutions (and in later stages in the pipeline, of how
well an LLM has modeled the distribution of solutions that ELM has uncov-
ered). Attainment of QD diers across seeds; while the CPPN seed uncovers
diversity most quickly, the Radial seed generates higher-quality solutions on av-
erage. The relationship between the seed and the products of search is complex
and deserves further future study (see also Appendix D for further analysis of
seed robustness).
Fine-tuning the di model on accepted dis from an initial series of runs
greatly increased performance (Figure 11); while Sodarace-generating programs
are out-of-distribution for the pretrained di model (applying a Python encoding
to this domain is a novel enterprise), ne-tuning eectively aligns the di model
with the domain, an interesting result. Figure 11c shows how the ne-tuned di
model produces a signicantly higher percentage of dis that are valid (i.e. able
to be applied) and runnable (i.e. the patched program is executable). Because of
their higher performance, the output of runs applying the ne-tuned di model
are the ones passed to later stages in the pipeline.
Note that further rounds of ne-tuning are possible (e.g. ne-tuning the di
19
Figure 9: Maximum tness across seeds. The maximum performance at-
tained on average by dierent seeds is shown. The results suggest that ELM's
capacity to nd high-tness solutions is somewhat robust to seed design.
model again from the improved products of the rst round); however preliminary
experiments showed diminishing returns. Future work could explore how to
continually improve such models, such as by identifying and encouraging more
impactful perturbations instead of including and weighting equally all accepted
dis.
The seeds and ne-tuned di model also qualitatively impact the kinds of
solutions discovered by ELM. While the Radial seed performs well quantita-
tively (in terms of quality and diversity), it turns out that its products tend to
exploit chaotic dynamics that seem overt to the
at terrain (this hypothesis is
tentatively validated in the Stage 3 experiments). The Square and CPPN seeds
in contrast are more likely to output inventions that leverage more predictable
dynamics. For these reasons, the Radial seed runs were not ultimately used in
future stages.
A video selection of the highest-quality Sodaracers from these initial runs
that showcases the considerable diversity uncovered can be viewed at https:
//y2u.be/QNyNtvwA9FI. An example of a lineage of Sodaracers progressing
from the Square seed to a high-quality nal Sodaracer can be seen at https:
//y2u.be/M9pAJuX6dyM. In short, ELM shows that by combining the an
intelligent LLM-based mutation operator with a QD algorithm it is possible to
generate hundreds of thousands of working training examples in a completely
novel domain where no data was previously available.
6 Pipeline Stage 2: Language Model Training
The product of Stage 1 is a collection of programs, whereas Stage 3 RL requires
an initial model that can output valid Sodaracer-generating programs. Thus,
the second stage of the invention pipeline ne-tunes an LLM on the products
20
Figure 10: Quality diversity score across seeds. Shown is the average nal
QD score attained by runs initialized from dierent seeds. The conclusion is
that ne-tuning the di model has a signicant impact on attained QD score,
as does the choice of seed.
of ELM, which serves as the initialization for an RL-based conditional inventor.
To do so rst requires compiling the results of Stage 1 into a ne-tuning dataset.
While there are many ways to distill a dataset of programs from runs of
ELM, a simple thresholded approach is adopted here (although see Appendix
E for another simple approach that did not work in practice). The main idea is
to append all reasonably-capable solutions for each niche.
In more detail, from each run all solutions ever admitted to the map are
included, subject to meeting a minimal bar for performance. Some parts of
the behavior space oer more stringent challenges (i.e. it is more dicult to
locomote when required to be tall but not wide and to have low mass), and yet
in some terrains encountered in Stage 3, those kinds of solutions may yet be
most eective despite their low level of absolute performance. Thus, for each
niche, the maximum performance across all runs is calculated, and the minimal
bar for inclusion is set as a percentage of that per-niche score. With a higher
percentage threshold, less data is included, but the quality of that data will be
higher.
As noted in the previous section, solutions from the Radial seed were qualita-
tively chaotic. Furthermore, preliminary experiments suggest that such chaotic
behavior signicantly harms downstream Stage 3 performance. For these rea-
sons Radial runs of ELM were excluded from the LLM datasets. Datasets for
each of the remaining treatments were compiled from 9 runs from ELM with the
ne-tuned di model (3 runs for each of the Square, CPPN-Fixed, and CPPN-
Mutable seeds). In total, the 50% cut-o threshold dataset consisted of 280K
examples, and the 80% cut-o threshold dataset contained a subset of 95K of
those examples.
A variety of pretrained code-generating models were then ne-tuned with
these examples (using the standard LLM log-probability loss), leaving out 5%
21
(a) Niches Reached
(b) QD Score
(c) Di Quality
Figure 11: The impact of ne-tuning the di model on the performance
of ELM. For both the pretrained di model and the ne-tuned one, shown are
(a) the number of niches reached, (b) QD score of the produced map, and (c)
percentage of valid/runnable dis proposed. The experiments demonstrate that
ne-tuning the di model improves performance of the evolutionary process
across all three metrics.
22
Figure 12: Test loss across model sizes. The minimum test loss achieved
by training runs on the 80% Percentage Threshold dataset across model sizes
is shown. Model sizes above 85M may not better-t the data, and random
initialization hurts performance.
of the data to serve as a test set. Models ranging from 0.1M to 680M parameters
were trained (architectural details for these models can be seen in Appendix C).
Also, as a control to support the hypothesis that Sodarace models benet from
code-generation pretraining, a 300M model was also trained instead from a
random initialization (signied with \RI" in charts that follow).
Minimum test-losses (i.e. loss on generated Sodaracers held-out from the ne-
tuning dataset) of the 80% Percentage Threshold models are shown in Figure
12. The 50% Percentage Threshold models exhibit qualitatively similar results
across model size (but as both thresholds represent dierent datasets, loss values
are not directly comparable between them). The conclusions are that model
sizes above 85M may not better t the data, and that random initialization
does hurt performance relative to ne-tuning from a model pretrained on code.
However, loss is not the whole story. The interesting question for Stage 2
is whether the LLMs trained from the data generated in Stage 1 can generate
the same diversity and quality of data. Therefore, the QD score metric and
number of niches discovered (both of which were also reported for Stage 1) are
calculated for samples taken from trained LLMs. Because these metrics can be
maximized by a model that memorizes the data, and because empirically QD
score was more correlated with loss on the training set rather than the test set,
the LLM checkpoint for each model is selected on the basis of lowest training
loss. In particular, 1,024 samples are taken from each model, which are then
evaluated and inserted into a new MAP-Elites map. For comparison, the same
metrics are calculated using the Stage 1 dataset, by taking the same number
of samples from it and evaluating them in the same way. These results are
shown in Figure 13, highlighting that the model samples achieve a similar level
of performance as dataset samples, suggesting that they have modeled the data
well. Also, there is a slight but consistent QD benet from models trained on
the 80% cuto dataset, re
ecting the higher average QD of that dataset.
23
(a) Number of Niches Filled
(b) QD Score
Figure 13: Measuring the quality and diversity of model samples. Two
metrics evaluating samples from trained LLMs are shown (across model size and
training dataset): (a) the percentage of niches discovered and (b) the QD score
achieved. The 80% threshold dataset is on average less diverse but of higher
quality than the 50% threshold dataset, and induces the same properties in
models trained upon it. There is not a trend in increasing quality or diversity as
model size increases beyond 85M, and random initialization hurts performance.
24
Figure 14: Generalization tests. In this test, the model is asked to complete
samples, taken from the rst half of the dataset from the unseen runs. These
unseen originals are shown in the videos at https://y2u.be/8C2K5fk28HI. From
top to bottom: Wheel, from radial seed; Galloper, from square seed; Runner,
from CPPN seed.
A natural further question is how well the model will do when taken out of
distribution, i.e. how well has it really internalized the dynamics of Sodarace?
That is, the training and test set for ne-tuning are taken from the same runs,
and thus the model will likely have encountered all of the motifs in the test set,
and so it may not be a representative test of how well the model will generalize
in the future. A preliminary test in this spirit is to take the rst half of the
Python programs describing several inventions from unseen runs, and explore
the capacity of dierent models to generate functional completions. Though the
Radial seed usually produced chaotic Sodaracers, in one preliminary run of ELM
with the Radial seed, a functional wheel was discovered. As noted previously
data from this run (or any other radial runs) was not used to train the models
in Stage 2, nor was it used to ne-tune the di model in Stage 1; thus the ability
to complete the wheel can serve as a proxy for generalization. Similarly, two
other high-performing individuals were taken from other preliminary runs of
the CPPN seed and the Square seed, to create a set of three out-of-distribution
completion tests. See Figure 14 for visualizations of these walkers, including
videos; source code for these generalization examples can be found in Appendix
F). Note that further tests of generalization are documented in Appendix H.
For each of the three completion tasks, 1,024 completion samples are taken
from each model and then evaluated in simulation. In contrast to the in-
distribution metrics, in this generalization-focused test, performance was more
correlated with the model's test loss rather than training loss, and thus what
checkpoint to evaluate for each model was selected on the basis of lowest test
25
Figure 15: Out of distribution completion performance. Shown is the
percentage of the original solutions' performance that is attained by completions
from trained LLMs. The percentage shown is the maximum attained over 1,024
independent completion samples from each model. The results are averaged
over three out-of-distribution solutions (taken from runs not included in LLM
training). The conclusion is that the 80% threshold models perform better than
the 50% threshold, and that there is no obvious trend in performance once model
size reaches 85M parameters.
loss. Results are shown in Figure 15, highlighting that larger models, and those
trained on the 80% threshold, generally perform better at this task. Note that
the randomly-initialized (RI) 300M model signicantly underperforms, provid-
ing more evidence that pretraining on code provides a valuable prior.
Videos of the best-performing sample for the Wheel completion from each
model are at https://y2u.be/-LW2cCwSdRU (for the 80% threshold dataset;
the random-initialized 300M model is not shown because it generated no valid
samples for this completion). For the Galloper and Runner completions, the
structure and/or behavior of completions often does not match the original
sample (especially for the Galloper). In the following linked video, a higher-
performing completion is shown for both of the Galloper and the Runner: https:
//y2u.be/XR3L4cZ83xU.
Overall, these results show that an LLM can eectively integrate synthetic
data generated through ELM in a novel domain.
7 Pipeline Stage 3: Conditional RL
In the nal stage, reinforcement learning (RL) is invoked to ne-tune the pre-
trained LLM output by Stage 2 of the pipeline. The goal is to produce a model
that outputs Python programs representing Sodaracers in response to particular
terrains . Importantly, the output of Stage 2 is an unconditional model, in the
sense that it samples Sodaracers from a distribution dened by the output of
Stage 1, without considering the terrain in which the samples will be deployed.
The rst step in Stage 3 is thus to convert the model to a conditional one, i.e. a
26
model that accepts terrains as inputs, and produces samples of Sodaracers in
response.
To achieve this functional form, we rst introduce the notion of a terrain
embedding network (TEN). The role of the TEN is to map a representation
of the terrain to a representation that can be used by the model to sample
conditionally. In particular, the output of TENs is a vector (or sequence of
vectors) in d, the dimension in which the model embeds tokens. That way, the
output of the TEN can be treated as the activation from a given prex, and
the model can proceed in eect now sampling conditioned on the output of the
TEN.
Concretely, an unconditional autoregressive LLM denes a sampling distri-
bution over a sequence of tokens x= (x1; : : : ; x n) asp(x) =Qn
i=1p(xijx<i).
In this stage, we introduce the additional module fTEN, which represents ter-
rains tinRd. AsfTEN(t)2Rd, we can consider the resulting conditional model
without further modication:
p(xjt) =nY
i=1p(xijx<i; fTEN(t)): (1)
This approach is similar to the controllable transformer proposed by Keskar
et al. [65], but with the conditional codes being the output of a TEN, rather
than particular tokens from the existing vocabulary.
Given a distribution over terrains p(t), an RL setting is constructed to train
the parameters of the TEN and further netune the LLM parameters to the
conditional setting. In particular, an episode now consists of sampling tp(t),
and sampling a program from the conditional distribution dened in Equa-
tion (1). The program is converted to a Sodaracer, evaluated in simulation with
the terrain t, and the reward is dened as the absolute distance traversed by
the Sodaracer in a given period of time.
7.1 Terrain Distributions
In this experiment, the distribution over terrains that the model is exposed to
is chosen to explore the viability of producing conditional inventors with the
Invention Pipeline. The future vision is to lay the groundwork for the ability to
deploy agents capable of conditional invention in rich, potentially multi-agent
environments that support the development of open-ended processes. In such
settings, it stands to reason that learning to output complex artifacts condi-
tioned on observations of the environment would be a prerequisite to ongoing
open-ended innovation.
However, in preliminary experiments in the Sodarace domain, learning tended
to \gravitate" towards collapsed solutions, wherein a single program is produced
that achieves reasonable performance on a subset of the terrains in the distri-
bution support. To reduce the viability of such an outcome and simulate a
scenario where conditionality is essential, a small and discrete set of terrains for
which a single program cannot achieve good performance provides a test where
conditional solutions should be signicantly more advantageous.
27
(a)left-wall
 (b)right-wall
(c)bumpy
 (d)tunnel
Figure 16: Terrains used in experiments . A small set of terrains from which
distributions that force conditionality can be constructed. The terrains are (a)
left-wall , (b) right-wall , (c) bumpy , and (d) tunnel . The Sodaracers
produced by the models are incapable of scaling the walls in left-wall and
right-wall , and therefore must produce dierent Sodaracers for these two
terrains. Similarly, achieving good performance in the tunnel terrain can only
be achieved with short Sodaracers, which struggle to locomote as quickly as
taller ones, encouraging the model to distinguish between these terrains. Finally,
Sodaracers that are procient in locomotion on
at terrains tend to perform
poorly on bumpy , encouraging the model to produce yet another Sodaracer for
this terrain. In contrast to tunnel , which requires Sodaracers with a particular
morphology, achieving good performance on bumpy requires modifying the way
Sodaracers locomote . Example Sodaracers are added to the gures to illustrate
the scale of the terrains.
In the experiments, uniform distributions are considered over sets of terrains
as illustrated in Figure 16. Two subsets are considered, both of which contain
left-wall andright-wall . One set additionally contains tunnel , and the
other includes bumpy . These sets were specically chosen such that the models
are incapable of producing a single Sodaracer that achieves good performance
on all terrains; to maximize the learning objective, the model must leverage the
TEN to incorporate conditionality.
7.2 Parametrizing TENs
Two parametrizations for the TEN are explored.
Discrete Codes. The terrain distribution has a discrete and nite support.
As such, a simple parametrization wherein the terrains are treated as additional
tokens in the existing vocabulary, and the embedding for each terrain is learned
separately may be used. The advantage of such a parametrization is that it
28
introduces a relatively small number of new parameters to be optimized with
RL, and it is conceptually simple to understand and debug. However, the main
disadvantages of such a parameterization are that (i) the number of parameters
scales with the size of the terrain set, and (ii) it does not allow the model to
naturally generalize to unseen terrains at test-time, which may be an important
constraint for downstream open-ended processes.
ResNets. An alternative parametrization is visual representations of the ter-
rains, which can then be processed by visual recognition models. In particular,
a ResNet50 [66] embeds images into Rdas a TEN when experimenting with
visual representations of terrains. The main advantages of this parametrization
are that it is quite general, could conceivably be used in multiple settings (e.g.
teaching a code-generating LLM to write programs in response to visual input,
and in theory can generalize to unseen terrains. The main drawback of this
approach is that it introduces a large number of new parameters that must be
optimized using a sparse RL signal. Conversely, for large terrain distributions,
this approach makes it possible to amortize the number of additional parameters
necessary for designing conditional inventors.
7.3 Experimental Details and Results
Each RL episode consists of sampling a batch of terrains from the distribution,
producing samples from the conditional LLM, and evaluating them in simulation
to produce the reward.
Proximal policy optimization [PPO; 67] is the RL algorithm, in conjunc-
tion with generalized advantage estimation [GAE; 68], with default hyper-
parameters. In preliminary experiments, we found it important to add a KL
term (between the policy network and the pre-trained LLM from Stage 2) to the
reward function , as proposed by Christiano et al. [69] and Stiennon et al. [70].
The value network is parametrized as a scalar-function version of the policy
network, i.e. a separate LLM with a separate prepended TEN initialized from
the Stage 2 models. Figure 17 illustrates the architectures and pipelines for the
policy and value-function networks. Each iteration consists of batches of 1,024
samples (distributed over 32 GPUs), and training runs consist of 100 iterations.
RL is run on pretrained, 300M-parameter LLMs trained with datasets having
cuto thresholds in f50%, 80%g. Recall that we use the cuto threshold to con-
trol the tradeo between data quality and quantity, such that higher thresholds
result in smaller pretraining datasets with a higher density of quality instances.
For each dataset and terrain distribution combination, three runs are performed
using dierent seeds, and the performance is averaged over samples from the
resulting model for each terrain, from over all runs, though we exclude a small
number of runs that diverged during training. To compute a measure of the per-
formance of datasets and pretrained LLMs, we invoke test-time compute: 1,024
Sodaracers are sampled uniformly and evaluated from each dataset/model (re-
call that there is one model for both cuto thresholds), and the best-performing
29
(a) Policy network
(b) Value-function network
Figure 17: Illustration of the RL architecture. The conditional policy (a)
and value-function (b) are depicted, both augmented with a separate TEN for
terrain embeddings. The policy is conditioned on a particular terrain (via the
TEN) and prompt, and produces a sample, which is interpreted as Python code.
The code is then used to compile a Sodaracer, which is evaluated in a simulation
to procuce a reward R. The value function is conditioned on the same terrain
(via its own TEN) and prompt, and outputs an estimation of the value ( V) of
every token output by the policy sample. During learning, the value-function is
trained to predict advantages estimated using GAE [68].
30
Sodaracer is considered for each terrain. Figures 18 and 19 detail the results of
these experiments with the tunnel andbumpy distributions, respectively.
In short, Figures 18 and 19 help us understand whether RL is able to discover
conditional solutions, which we interpret as conditional inventors of Sodaracers
that are capable of locomoting on particular terrains. Moreover, Figures 18
and 19 enable us to compare the performance of Sodaracers produced at dif-
ferent stages of the pipeline, and how performance is aected by the choice
of cuto threshold. A particularly interesting question is whether RL is able
to consistently improve upon the performance of test-time compute with the
pretrained models produced in Stage 2.
The RL procedure was at times brittle: training sometimes diverged, and
some results were inconsistent. Divergence tended to be more frequent using the
ResNet TENs, which is unsurprising considering the ResNets introduce many
more parameters to the model, which are in turn trained with an extremely
impoverished distribution of images (one for each terrain in the distribution).
Despite the fragility, RL ne-tuning is successful in producing conditional
inventors in this domain: the models tend to produce a single Sodaracer for
each terrain, which dier across terrains in the distribution. Importantly, the
produced Sodaracers achieve good performance for the conditioned terrain,
while failing to locomote on the other terrains. Videos showcase Sodaracers
invented for the Tunnel distribution { https://y2u.be/e53NwdT4RdM { and for
the bumpy distribution { https://y2u.be/WEM1dBtLLTw. In short, the main
result is the outputs of Stage 3, and thus the complete pipeline, are conditional
inventors of the desired form.
Moreover, in most cases, the RL models are comparable to or better than the
best-performing Sodaracers sampled from the dataset or the pretrained LLM.
This consistency implies that Stage 3 enables the models to learn to use the
TENs in conjunction with the LLMs, and further can ne-tune the models'
outputs to improve performance, though not always by signicant margins.
Models trained with a cuto of 80% tend to achieve slightly better per-
formance, and proved more stable during training, though the dierences are
not signicant. This result implies that the tradeo between data quality and
quantity may play a role in downstream tasks (such as RL ne-tuning), a point
that warrants further investigation in future work. One interesting avenue for
research in this direction is to consider pretraining procedures that include in-
formation regarding the quality of the instances (where such information is
available), e.g. as proposed by Chen et al. [71].
Finally, we note that \collapsed" solutions in which the same Sodaracer is
produced every time a particular terrain is observed (as opposed to signicantly
dierent samples each time the same terrain is seen) are sensible in this set-
ting, as there should exist a dominant Sodaracer for each terrain. However,
interestingly, in true open-ended systems this property may not hold: if the
environment is constantly shifting, that excludes the existence of single, dom-
inant inventions. In such a setting, the stochasticity of the model is expected
to be benecial, enabling the model to adapt and produce a diversity of useful
solutions.
31
left-wall right-wall tunnel average
Terrain05001000150020002500Max FitnessStage1
Stage2
Stage3-discrete
Stage3-vision(a) Cuto 50%
left-wall right-wall tunnel average
Terrain05001000150020002500Max FitnessStage1
Stage2
Stage3-discrete
Stage3-vision
(b) Cuto 80%
Figure 18: Comparing performance of models and datasets across the
stages of the pipeline on the terrain distribution including the tun-
nel. Results are detailed when training the LM using a dataset with a cuto
of (a) 50%, and (b) 80%. Stage 3 models are able to discover conditional solu-
tions in both cases, consistently perform comparably to test-time compute on
the dataset, and better than the Stage 2 pretrained LMs. For the 80% cuto
threshold, while performance is better than with 50% at all stages, the pipeline
struggles to improve performance over that of the dataset. Conversely, for the
50% cuto threshold, the Stage 3 (discrete) model improves upon all stages,
demonstrating the ability of the pipeline to improve the performance of the
models.
32
left-wall right-wall bumpy average
Terrain05001000150020002500Max FitnessStage1
Stage2
Stage3-discrete
Stage3-vision(a) Cuto 50%
left-wall right-wall bumpy average
Terrain05001000150020002500Max FitnessStage1
Stage2
Stage3-discrete
Stage3-vision
(b) Cuto 80%
Figure 19: Comparing performance of models and datasets across the
stages of the pipeline on the terrain distribution including bumpy.
Results are detailed when training the LM using a dataset with a cuto of (a)
50%, and (b) 80%. Similar trends can be seen to those in Figure 18: Stage 3
models are able to discover conditional solutions and consistently perform com-
parably to test-time compute on the dataset, improving upon Stage 2 models.
For the 80% cuto threshold, the pipeline achieves comparable performance to
that of the dataset, while improving performance for the 50% cuto threshold.
33
7.4 Qualitative Observations
Several interesting structures and solution classes were qualitatively observed
throughout the experiments, which provide additional insight into the pipeline's
ability to conditionally invent solutions to dierent terrains. One such example
is the emergence of very short Sodaracers, which arose in response to the tun-
nelterrain. The video visualizations at https://y2u.be/P9A1ruI3 tU highlight
examples of such Sodaracers produced in response to tunnel .
Another interesting class of Sodaracers appeared in earlier experiments with
ELM; a wheel-like structure emerged during the evolutionary process, and per-
severed throughout the pipeline. During Stage 3, the wheel proved particularly
adept at locomoting in the bumpy terrain, and consistently emerged as the so-
lution to bumpy produced by the Stage 3 models for that terrain across RL
runs. Unfortunately, the wheel did not re-emerge in the ELM runs used in the
nal experiments in this paper. The video at https://y2u.be/l5PVSLDknWM
demonstrates several solutions of this form discovered by RL when trained with
thebumpy terrain distribution as well as the tunnel distribution. For con-
trast, this video (https://y2u.be/Mo-rXnFq6vQ) show failure modes on bumpy
for several Sodaracers eective in locomoting on
at terrains.
Such qualitative observations provide further evidence that the pipeline is
capable of producing interesting inventors and creative solutions to problems,
even in a simplied domain that is not open-ended. We hypothesize that when
unleashed in more complex domains, this capability of conditional invention
will contribute to the open-endedness of the induced process by continually
introducing new objects to the environment, and thus changing its properties
for other agents.
8 Discussion and Conclusion
An important dierence between natural evolution and most of EC is the very
beginning{nature began with a single \example" or seed, the rst cell on Earth,
that was already bestowed with critical initial functionality and information. In
contrast, runs in EC usually begin with randomized congurations with little or
no useful information. Because programming languages like Python for humans
are natural modalities for formalizing complicated ideas and relationships, such
a program could serve as a seed more in the spirit of nature. However, the
problem then is that arbitrary mutations to an already-formulated program are
very unlikely to be useful.
A few years ago, the idea that the mutation operator could \know" how to
perturb such programs in reasonable and promising ways would be fanciful, but,
as shown in this paper. the emergence of LLMs has now made such capabilities
a reality. The MAP-Elites algorithm combined with ELM easily bootstraps
datasets of hundreds of thousands of examples in a completely foreign domain
(to the initial LLM) from initial human-written seeds. The validity of this
generated data is conrmed by the invention pipeline that follows{conditional
34
LLMs were ultimately trained starting from this data that cannot be trained
from scratch.
More broadly, the main idea introduced here is that LLMs trained on code
open up a signicant new kind of intelligent GP enabled by ELM that is no
longer at the mercy of the raw search landscape induced by code. While the
experiment in this paper points to a set of implications for open-endedness, deep
learning, and RL, the potential applications are numerous and many previous
challenges in the GP eld could be revisited with this new tool.
The experiment in this paper shows that intelligent LLM-based mutation op-
erators can successfully drive exploration by being combined with other search
algorithms (e.g. MAP-Elites in this work). Furthermore, optimizing such mu-
tation operators based on the quality of their output during the search itself
appears to make them work even better for exploration. Not only are the dis-
coveries of such search potentially useful in their own right (like wheels in the
Sodarace domain), but they oer an entirely new option for generating exam-
ple data or optimizing existing solutions in domains where data is sparse or
non-existent. For example, such search through LLM-based perturbation could
feasibly be applied to optimize the MAP-Elites search algorithm itself, or for
LLM architecture and hyperparameter search.
From the perspective of open-endedness, the challenge in principle is that
the search is by denition continually and even intentionally shifting out of
distribution. As soon as a new invention or DCT is achieved, open-endedness
demands that its now-familiar comfort zone be at least partially abandoned for
new frontiers. The experiment here wherein LLMs trained from simple
at-
ground walkers were able to leverage that knowledge to appropriately generate
specialized walkers for dierent terrains shows just this kind of informed leap
to a new frontier. If such a process of leaps upon leaps can be made to con-
tinue indenitely, then an unbounded explosion of emergent complexity could
be within reach.
One important question for future work is the extent to which the resultant
model can interpolate or extrapolate to examples (i.e. environments) outside its
training distribution. While RL can harness existing knowledge in the LLM to
bootstrap into new tasks, extrapolating principles from such knowledge is much
harder and likely to require further weight updates through additional learning.
It is possible that a sophisticated future open-ended system would entangle both
continual evolution and RL for DCTs together.
Overall, the hope is that the simple core insight that the ecacy of mutation
in GP can now dramatically improve through ELM will inspire a broad array of
novel applications and research directions. The observation that EC can benet
directly and dramatically from advances in deep learning (and deep learning
from EC further down the invention pipeline) can also help to motivate the
further pursuit of synergies between the elds.
35
Acknowledgments
Thank you to Je Clune for substantive insightful feedback and thoughtful
discussion about the project and this paper. Thanks also to Glenn Powell
for consistent useful input and ideas during team meetings and discussions.
We also thank the Supercomputing team for their work, which enabled our
experimentation.
References
[1] Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran
Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine
Bosselut, Emma Brunskill, et al. On the opportunities and risks of foun-
dation models. arXiv preprint arXiv:2108.07258 , 2021.
[2] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Ka-
plan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry,
Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger,
Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jerey
Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz
Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam
McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language
models are few-shot learners. arXiv preprint arXiv:2005.14165 , 2020.
[3] J. R. Koza. Genetic Programming: On the Programming of Computers by
Means of Natural Selection . MIT Press, Cambridge, MA, 1992.
[4] Wolfgang Banzhaf, Peter Nordin, Robert E Keller, and Frank D Fran-
cone. Genetic programming: an introduction: on the automatic evolution
of computer programs and its applications . Morgan Kaufmann Publishers
Inc., 1998.
[5] K.O. Stanley, J Lehman, and L Soros. Open-endedness: The last grand
challenge you've never heard of. O'Reilly Online , December 19, 2017.
[6] Russell K Standish. Open-ended articial evolution. International Journal
of Computational Intelligence and Applications , 3(02):167{175, 2003.
[7] Mark A. Bedau, John S. McCaskill, Norman H. Packard, Steen Rasmussen,
Chris Adami, David G. Green, Takashi Ikegami, Kunihiko Kaneko, and
Thomas S. Ray. Open problems in articial life. Articial Life , 6:363{376,
2000.
[8] Michael O'Neill, Leonardo Vanneschi, Steven Gustafson, and Wolfgang
Banzhaf. Open issues in genetic programming. Genetic Programming and
Evolvable Machines , 11(3):339{363, 2010.
36
[9] Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde
de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas
Joseph, Greg Brockman, et al. Evaluating large language models trained
on code. arXiv preprint arXiv:2107.03374 , 2021.
[10] Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrit-
twieser, R emi Leblond, Tom Eccles, James Keeling, Felix Gimeno,
Agustin Dal Lago, et al. Competition-level code generation with alpha-
code. arXiv preprint arXiv:2203.07814 , 2022.
[11] P. McOwan and E. Burton. Sodarace website. URL http://sodarace.net ,
2000{2013.
[12] Paul Szerlip and Kenneth O. Stanley. Indirectly Encoding Running and
Jumping Sodarace Creatures for Articial Life. Articial Life , 21(4):432{
444, 11 2015. ISSN 1064-5462. doi: 10.1162/ARTL-a-00185. URL https:
//doi.org/10.1162/ARTL a00185.
[13] Mark A Bedau, John S McCaskill, Norman H Packard, Steen Rasmussen,
Chris Adami, David G Green, Takashi Ikegami, Kunihiko Kaneko, and
Thomas S Ray. Open problems in articial life. Articial life , 6(4):363{
376, 2000.
[14] Kenneth O Stanley, Joel Lehman, and Lisa Soros. Open-endedness: The
last grand challenge you've never heard of. O'Reilly Radar Online Article ,
December 2017.
[15] William B Langdon and Riccardo Poli. Foundations of genetic program-
ming . Springer Science & Business Media, 2013.
[16] John R Koza, Martin A Keane, Matthew J Streeter, William Mydlowec,
Jessen Yu, and Guido Lanza. Genetic programming IV: Routine human-
competitive machine intelligence , volume 5. Springer Science & Business
Media, 2006.
[17] Markus Brameier and Wolfgang Banzhaf. A comparison of linear genetic
programming and neural networks in medical data mining. IEEE Transac-
tions on Evolutionary Computation , 5(1):17{26, 2001.
[18] Istvan Jonyer and Akiko Himes. Improving modularity in genetic pro-
gramming using graph-based data mining. In FLAIRS Conference , pages
556{561, 2006.
[19] Gregory Seront. External concepts reuse in genetic programming. In work-
ing notes for the AAAI Symposium on Genetic programming , pages 94{98.
MIT/AAAI Cambridge, 1995.
[20] Leo Gugerty and Gary Olson. Debugging by skilled and novice program-
mers. In Proceedings of the SIGCHI conference on human factors in com-
puting systems , pages 171{174, 1986.
37
[21] Paul Luo Li, Andrew J Ko, and Jiamin Zhu. What makes a great software
engineer? In 2015 IEEE/ACM 37th IEEE International Conference on
Software Engineering , volume 1, pages 700{710. IEEE, 2015.
[22] Joel Lehman, Jay Chen, Je Clune, and Kenneth O Stanley. Safe mu-
tations for deep and recurrent neural networks through output gradients.
InProceedings of the Genetic and Evolutionary Computation Conference ,
pages 117{124. ACM, 2018.
[23] John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp
Moritz. Trust region policy optimization. In International Conference on
Machine Learning , pages 1889{1897, 2015.
[24] Edgar Galv an-L opez, James McDermott, Michael O'Neill, and Anthony
Brabazon. Towards an understanding of locality in genetic programming.
InProceedings of the 12th annual conference on Genetic and evolutionary
computation , pages 901{908, 2010.
[25] Rafal Salustowicz and J urgen Schmidhuber. Probabilistic incremental pro-
gram evolution. Evolutionary computation , 5(2):123{141, 1997.
[26] Lee Spector and Alan Robinson. Genetic programming and autoconstruc-
tive evolution with the push programming language. Genetic Programming
and Evolvable Machines , 3(1):7{40, 2002.
[27] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert:
Pre-training of deep bidirectional transformers for language understanding.
arXiv preprint arXiv:1810.04805 , 2018.
[28] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones,
Aidan N Gomez,  Lukasz Kaiser, and Illia Polosukhin. Attention is all you
need. Advances in neural information processing systems , 30, 2017.
[29] Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma,
Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles
Sutton, Sebastian Gehrmann, et al. Palm: Scaling language modeling with
pathways. arXiv preprint arXiv:2204.02311 , 2022.
[30] Dan Hendrycks, Steven Basart, Saurav Kadavath, Mantas Mazeika, Akul
Arora, Ethan Guo, Collin Burns, Samir Puranik, Horace He, Dawn Song,
et al. Measuring coding challenge competence with apps. arXiv preprint
arXiv:2105.09938 , 2021.
[31] Alex Ray and Sam McCandlish. Independent contribution: Training di
models, 2020.
[32] Tim Taylor. Exploring the concept of open-ended evolution. In Arti-
cial Life 13 (Proceedings of the Thirteenth International Conference on the
Simulation and Synthesis of Living Systems) , pages 540{541, Cambridge,
MA, 2012. MIT Press.
38
[33] Tim Taylor, Mark Bedau, Alastair Channon, David Ackley, Wolfgang
Banzhaf, Guillaume Beslon, Emily Dolson, Tom Froese, Simon Hickin-
botham, Takashi Ikegami, et al. Open-ended evolution: Perspectives from
the oee workshop in york. Articial life , 22(3):408{423, 2016.
[34] Joel Lehman and Kenneth O. Stanley. Abandoning objectives: Evolution
through the search for novelty alone. Evolutionary Computation , 19(2):
189{223, 2011. URL http://eplex.cs.ucf.edu/papers/lehman ecj11.pdf.
[35] J.-B. Mouret and St ephane Doncieux. Encouraging behavioral diversity in
evolutionary robotics: An empirical study. Evolutionary computation , 20
(1):91{133, 2012.
[36] Deepak Pathak, Pulkit Agrawal, Alexei A Efros, and Trevor Darrell.
Curiosity-driven exploration by self-supervised prediction. In International
conference on machine learning , pages 2778{2787. PMLR, 2017.
[37] Christopher Stanton and Je Clune. Curiosity search: producing gener-
alists by encouraging individuals to continually explore and acquire skills
throughout their lifetime. PloS one , 11(9):e0162235, 2016.
[38] Djordje Grbic, Rasmus Berg Palm, Elias Najarro, Claire Glanois, and Se-
bastian Risi. Evocraft: A new challenge for open-endedness. In Interna-
tional Conference on the Applications of Evolutionary Computation (Part
of EvoStar) , pages 325{340. Springer, 2021.
[39] Sam Earle, Julian Togelius, and LB Soros. Video games as a testbed for
open-ended phenomena. In 2021 IEEE Conference on Games (CoG) , pages
1{9. IEEE, 2021.
[40] Rui Wang, Joel Lehman, Aditya Rawal, Jiale Zhi, Yulun Li, Jerey Clune,
and Kenneth O. Stanley. Enhanced poet: Open-ended reinforcement learn-
ing through unbounded invention of learning challenges and their solu-
tions. In International Conference on Machine Learning , pages 9940{9951.
PMLR, 2020.
[41] L. B. Soros and Kenneth O. Stanley. Identifying minimal conditions for
open-ended evolution through the articial life world of chromaria. In
Proceedings of the Fourteenth International Conference on the Synthesis
and Simulation of Living Systems , pages 793{800, Cambridge, MA, 2014.
MIT Press.
[42] Michael Dennis, Natasha Jaques, Eugene Vinitsky, Alexandre Bayen, Stu-
art Russell, Andrew Critch, and Sergey Levine. Emergent complexity and
zero-shot transfer via unsupervised environment design. Advances in Neu-
ral Information Processing Systems , 33:13049{13061, 2020.
[43] Jonathan C Brant and Kenneth O Stanley. Minimal criterion coevolution:
a new approach to open-ended search. In Proceedings of the Genetic and
Evolutionary Computation Conference , pages 67{74. ACM, 2017.
39
[44] Detailed description of unied format. https://www.gnu.org/software/
diutils/manual/html node/Detailed-Unied.html.
[45] Open AI blogpost: New GPT-3 capabilities: Edit and insert. https://
openai.com/blog/gpt-3-edit-insert/, 2022.
[46] Long Ouyang, Je Wu, Xu Jiang, Diogo Almeida, Carroll L Wainwright,
Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex
Ray, et al. Training language models to follow instructions with human
feedback. arXiv preprint arXiv:2203.02155 , 2022.
[47] Ben Wang and Aran Komatsuzaki. GPT-J-6B: A 6 Billion Param-
eter Autoregressive Language Model. https://github.com/kingo
olz/
mesh-transformer-jax, May 2021.
[48] Melanie Mitchell. An introduction to genetic algorithms . MIT Press, 1999.
[49] Kenneth A. De Jong. Evolutionary Computation: A unied approach . MIT
Press, Cambridge, MA, 2006.
[50] Justin K Pugh, Lisa B. Soros, and Kenneth O. Stanley. Quality di-
versity: A new frontier for evolutionary computation. 3(40), 2016.
ISSN 2296-9144. URL http://www.frontiersin.org/evolutionary-robotics/
10.3389/frobt.2016.00040/abstract.
[51] Jean-Baptiste Mouret and Je Clune. Illuminating search spaces by map-
ping elites. ArXiv e-prints , abs/1504.04909, 2015. URL http://arxiv.org/
abs/1504.04909.
[52] Rui Wang, Joel Lehman, Je Clune, and Kenneth O. Stanley. Poet:
Open-ended coevolution of environments and their optimized solutions.
InProceedings of the Genetic and Evolutionary Computation Conference ,
GECCO '19, page 142{151, New York, NY, USA, 2019. Association for
Computing Machinery. ISBN 9781450361118. doi: 10.1145/3321707.
3321799. URL https://doi.org/10.1145/3321707.3321799.
[53] Jonathan C. Brant and Kenneth O. Stanley. Diversity preservation in
minimal criterion coevolution through resource limitation. In Proceedings
of the 2020 Genetic and Evolutionary Computation Conference , GECCO
'20, page 58{66, New York, NY, USA, 2020. Association for Computing
Machinery. ISBN 9781450371285. doi: 10.1145/3377930.3389809. URL
https://doi.org/10.1145/3377930.3389809.
[54] Bowen Baker, Ingmar Kanitscheider, Todor Markov, Yi Wu, Glenn Powell,
Bob McGrew, and Igor Mordatch. Emergent tool use from multi-agent
autocurricula. arXiv preprint arXiv:1909.07528 , 2019.
[55] Antoine Cully, Je Clune, Danesh Tarapore, and Jean-Baptiste Mouret.
Robots that can adapt like animals. Nature , 521(7553):503{507, 2015.
40
[56] Silja Meyer-Nieberg and Hans-Georg Beyer. Self-adaptation in evolutionary
algorithms. In Parameter setting in evolutionary algorithms , pages 47{75.
Springer, 2007.
[57] Oliver Kramer. Evolutionary self-adaptation: a survey of operators and
strategy parameters. Evolutionary Intelligence , 3(2):51{65, 2010.
[58] Nikolaus Hansen. The cma evolution strategy: a comparing review. To-
wards a new evolutionary computation , pages 75{102, 2006.
[59] Daan Wierstra, Tom Schaul, Jan Peters, and Juergen Schmidhu-
ber. Natural evolution strategies. In Evolutionary Computation, 2008.
CEC 2008.(IEEE World Congress on Computational Intelligence). IEEE
Congress on , pages 3381{3387. IEEE, 2008.
[60] Kenneth O. Stanley and Risto Miikkulainen. A taxonomy for articial
embryogeny. Articial Life , 9(2):93{130, 2003. URL http://nn.cs.utexas.
edu/keyword?stanley:alife03.
[61] Kenneth O. Stanley. Compositional pattern producing networks: A novel
abstraction of development. Genetic Programming and Evolvable Machines
Special Issue on Developmental Systems , 8(2):131{162, 2007.
[62] Josh C. Bongard and Rolf Pfeifer. Repeated structure and dissociation
of genotypic and phenotypic complexity in articial ontogeny. In Lee
Spector, Erik D. Goodman, Annie Wu, W. B. Langdon, Hans-Michael
Voigt, Mitsuo Gen, Sandip Sen, Marco Dorigo, Shahram Pezeshk, Max H.
Garzon, and Edmund Burke, editors, Genetic and Evolutionary Com-
putation Conference , pages 829{836, 2001. ISBN 1-55860-774-9. URL
http://www-illigal.ge.uiuc.edu:8080/gecco-2001/.
[63] Petet J. Bentley and S. Kumar. Three ways to grow designs: A comparison
of embryogenies for an evolutionary design problem. In Proceedings of the
Genetic and Evolutionary Computation Conference (GECCO-1999) , pages
35{43, 1999.
[64] Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An
Introduction . 1998. ISBN 0-262-19398-1.
[65] Nitish Shirish Keskar, Bryan McCann, Lav R Varshney, Caiming Xiong,
and Richard Socher. CTRL: A conditional transformer language model for
controllable generation. arXiv preprint arXiv:1909.05858 , 2019.
[66] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual
learning for image recognition. arXiv preprint arXiv:1512.03385 , 2015.
[67] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and
Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint
arXiv:1707.06347 , 2017.
41
[68] John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter
Abbeel. High-dimensional continuous control using generalized advantage
estimation. arXiv preprint arXiv:1506.02438 , 2015.
[69] Paul Christiano, Jan Leike, Tom B Brown, Miljan Martic, Shane Legg,
and Dario Amodei. Deep reinforcement learning from human preferences.
arXiv preprint arXiv:1706.03741 , 2017.
[70] Nisan Stiennon, Long Ouyang, Je Wu, Daniel M Ziegler, Ryan Lowe,
Chelsea Voss, Alec Radford, Dario Amodei, and Paul Christiano. Learn-
ing to summarize from human feedback. arXiv preprint arXiv:2009.01325 ,
2020.
[71] Lili Chen, Kevin Lu, Aravind Rajeswaran, Kimin Lee, Aditya Grover,
Michael Laskin, Pieter Abbeel, Aravind Srinivas, and Igor Mordatch. De-
cision transformer: Reinforcement learning via sequence modeling. arXiv
preprint arXiv:2106.01345 , 2021.
[72] M. Sipper. Tiny genetic programming in python. https://github.com/
moshesipper/tiny gp, 2019.
A Comparing Mutation Operators
This section gives more details on experiments using dierent mutation opera-
tors to x bugs in a single step of perturbation. A simple form of GP mutation
is implemented with Tiny GP [72], a tree-based GP implementation. In par-
ticular, the mutation operator is restricted to mutating nodes, which oers it
the highest chance of success given the nature of the bugs introduced (which
do not introduce new structure, but instead each bug in eect swaps an incor-
rect node for a correct one). For Tiny GP, The mutation rate was tuned by
hand for each problem. While many more sophisticated mutation operators in
GP exist [25, 26], the motivation for these experiments is mainly to highlight
the potential for LLM-based directed mutation operators to make sophisticated
movements along the manifold of code.
The experimental setup for each task (described next) is that each mutation
operator is given many independent trials, where the operator perturbs a single
buggy parent (with potentially several bugs), and the resulting child is tested
for correctness. First, a Python 3 version of each function was written, then (for
perturbation by GP) it was translated by hand into a Tiny GP tree. The commit
message for the di models for these tasks is \Fixed bugs." Note that GP
cannot make use of the plain-text doc-string that describes what the function is
intended to do, which highlights another advantage of LLMs for perturbation, in
that they can use (and create) language-based comments to guide the evolution
of code. For prompt engineering, the following prompt format was used (with
\fproblemg" replaced with the buggy implementation code):
42
1# A buggy implementation
2{problem}
3
4# Fixed Bugs
5def
Two benchmark tasks were explored: 4-Parity, where the objective is to
calculate the parity of a 4-bit sequence, and Quadratic, where the objective is
to calculate the result of a quadratic function, given the values of the coecients
a,b,c, and the independent variable x. The motivation for 4-Parity is that bit
parity is a common GP benchmark task [3] and provides a simple test-bed
for whether LLMs can make multiple coordinated (and eective) changes to
code. Quadratic provides another simple test-bed, and unlike 4-Parity, the bugs
introduced are more ambiguous (i.e. the function description does not imply that
the function must be of any canonical form), making it more similar to the use
case of this paper, wherein undirected yet semantically meaningful changes are
desired. Note that the nature of the introduced bugs for each task are described
in the next section along with the source code for the correct implementation
(into which bugs are introduced).
A.1 Python Source for Benchmark Tasks
A.1.1 4-Parity
1#!/usr/bin/python3
2def parity(b1,b2,b3,b4):
3 """ Return binary parity of a sequence of input bits.
4 Return 0 for even parity, 1 for odd parity """
5 bit_sum = sum([b1,b2,b3,b4])
6 return bit_sum % 2
Bugs were incrementally introduced in the following way: For the rst four
mutations, each variable with a \b\ prex was renamed with a \c\ prex (e.g.
rst \b1" is changed to \c1", then \b2" is changed to \c2", etc.). For the
fth mutation, the \modulus two" is replaced with modulus three. For GP,
additional \c"-prexed terminals were introduced.
A.1.2 Quadratic
1#!/usr/bin/python3
2def quadratic(a,b,c,x):
3 """ Return quadratic: a,b,c are coefficients
43
Figure 20: Comparing di mutation to GP mutation at xing bugs in
the Quadratic task. The plot shows how the ability of a single mutation to
produce correct solutions changes as bugs are incrementally added to a working
implementation that solves the Quadratic task. Note that success percentage
is shown in log scale , i.e. success for GP mutation decreases signicantly when
the second bug is added, while di mutation is unaected. The conclusion is
that this task adds more evidence that LLM-based mutation can make multiple
sensible coupled changes to code.
4 and x is the independent variable."""
5 return a*pow(x,2)+b*x+c
A maximum of two bugs was introduced to the Quadratic task by individu-
ally replacing the + operators with operators, from left to right.
A.2 Comparing GP and Di Mutation
Performance plots that compare GP to di mutation for the 4-Parity task are in
Figure 1 in the main text. Figure 20 in this appendix shows the same comparison
for the Quadratic task, where performance is similar to 4-Parity, i.e. the di
mutation's performance is both greater than GP mutation for both cases of
bugs, and degrades in a dierent pattern (here, the performance of di mutation
is unaected by the introduction of the second bug).
A.3 Comparing API-based Mutations
Performance plots that compare di mutation to mutations possible through
the OpenAI API for the 4-Parity task can be seen in Figure 2 in the main
text. Figure 21 here shows the same comparison for the Quadratic task, which
highlights similar results: There are multiple options for mutation operators
available through the OpenAI API that perform as well or better than the di
model applied in this paper's experiments.
44
Figure 21: Comparing alternate LLM-based mutations at xing bugs
in the Quadratic task. The performance of dierent mutation operators in
xing bugs is shown as bugs are incrementally added to an correct implementa-
tion for the Quadratic task. Both edit mode and prompt-engineering approaches
outperform the 300M di model applied in this paper's experiments. The con-
clusion is that this additional task adds evidence that there exist multiple viable
options to build upon the work in this paper.
B Seed Source Code
This section contains the source code of the seed programs for ELM. Videos
for the Sodaracers these programs represent are available at: https://y2u.be/
jeP8Nsulu48.
B.1 CPPN Seeds
There are two CPPN-like seeds. CPPN-Fixed does not allow the core func-
tionality of the CPPN encoding (encapsulated in the query cppn function) to
change, whereas CPPN-Mutable includes the source code for that function,
thereby enabling the CPPN encoding itself also to evolve.
B.1.1 CPPN-Fixed
1def make_walker():
2 wc = walker_creator()
3
4 def connect(x1,y1,x2,y2):
45
5 if ((x1-x2)**2+(y1-y2)**2)>4.5:
6 return False
7 return True
8
9 def amp(x1,y1,x2,y2):
10 return max(abs(x1-x2),abs(y1-y2))
11
12 def phase(x1,y1,x2,y2):
13 return np.sign(x1)
14
15 joints = query_cppn(wc,8,3,1.5,connect,amp,phase)
16
17 return wc.get_walker()
B.1.2 CPPN-Mutable
1def query_cppn(wc, xgrid,ygrid,scale,connect_func,amp_func,
2 phase_func):
3 """ Create a grid of points and functionally connect them. """
4 joints = {}
5 for x in range(xgrid):
6 for y in range(ygrid):
7 joints[(x,y)] = wc.add_joint(x*scale,y*scale)
8
9 for x1 in range(xgrid):
10 for y1 in range(ygrid):
11 for x2 in range(x1,xgrid):
12 for y2 in range(y1,ygrid):
13 if x1==y1 and x2==y2:
14 continue
15 if connect_func(x1,y1,x2,y2):
16 amp = amp_func(x1,y1,x2,y2)
17 phase = phase_func(x1,y1,x2,y2)
18 wc.add_muscle(joints[(x1,y1)],
19 joints[(x2,y2)],False,amp,phase)
20
21 return joints
22
23def make_walker():
24 wc = walker_creator()
25
26 def connect(x1,y1,x2,y2):
27 if ((x1-x2)**2+(y1-y2)**2)>4.5:
28 return False
46
29 return True
30
31 def amp(x1,y1,x2,y2):
32 return max(abs(x1-x2),abs(y1-y2))
33
34 def phase(x1,y1,x2,y2):
35 return x1 if x1%2==1 else -x1
36
37 joints = query_cppn(wc,8,3,1.5,connect,amp,phase)
B.2 Square Seed
1def make_square(wc, x0, y0, x1, y1):
2 """ Make a square with top left x0,y0 and top right x1,y1 """
3
4 j0 = wc.add_joint(x0, y0)
5 j1 = wc.add_joint(x0, y1)
6 j2 = wc.add_joint(x1, y1)
7 j3 = wc.add_joint(x1, y0)
8
9 return j0, j1, j2, j3
10
11
12def make_walker():
13
14 wc = walker_creator()
15
16 # the main body is a square
17 sides = make_square(wc, 0, 0, 10, 10)
18 center = wc.add_joint(5, 5)
19
20 # connect the square with distance muscles
21 for k in range(len(sides)-1):
22 wc.add_muscle(sides[k], sides[k+1])
23 wc.add_muscle(sides[3], sides[0])
24
25 # one prong of the square is a distance muscle
26 wc.add_muscle(sides[3], center)
27
28 # the other prongs from the center of the square are active
29 wc.add_muscle(sides[0], center, False, 5.0, 0.0)
30 wc.add_muscle(sides[1], center, False, 10.0, 0.0)
31 wc.add_muscle(sides[2], center, False, 2.0, 0.0)
32
47
33 return wc.get_walker()
B.3 Radial Seed
1def make_circle(wc, cx,cy,radius,num_points):
2 """ Approximate a circle with center (cx,cy) square with
3 num_points points """
4 joints = []
5
6 tot_ang = 3.14*2.0
7
8 for idx in range(num_points):
9 ang = tot_ang/(num_points-1)*idx
10 x = math.cos(ang) * radius + cx
11 y = math.sin(ang) * radius + cy
12 joints.append(wc.add_joint(x,y))
13
14 return joints
15
16
17def make_walker():
18
19 wc = walker_creator()
20
21 num_points = 8
22 rad = 5.0
23 cx,cy = (5,5)
24 # the main body is a square
25 points = make_circle(wc, cx,cy,rad,num_points)
26 center = wc.add_joint(cx,cy)
27
28 for k in range(num_points):
29 wc.add_muscle(points[k], points[(k+1)%num_points])
30 wc.add_muscle(points[k], center,False,float(k)/num_points,
31 float(k)/num_points)
32
33 return wc.get_walker()
C Model Architectures
Architectural details for the language models applied in this paper are shown in
Table 1. Models are based on the GPT-3 architecture, and further description
of architecture and hyperparameters can be found in Brown et al. [2].
48
nparams nlayers dmodel nheads dhead
0.1M 2 64 4 16
85M 12 768 12 64
350M 24 1,024 16 64
760M 24 1,536 16 96
Table 1: Model architectures. The table shows hyperparameters that de-
scribe the architectures of the models used in this paper, including parameter
count ( nparams ), number of layers ( nlayers), number of units in each bottleneck
layer ( dmodel ), number of attention heads ( nheads), and dimension of each atten-
tion head ( dhead).
D Seed Robustness
A subtle issue came to light when bringing together the full pipeline, which is
that there are complex interactions between the kind of seed that kicks o ELM
in Stage 1 and the performance of RL models trained in Stage 3. In particular,
some seeds (like the Radial seed) attain high QD scores in Stage 1, but fail to
provide good jumping-o points to adapt to novel terrains in Stage 3. When
examining the products of the Radial seed, many of them exhibited chaotic
dynamics that appeared overly sensitive to initial conditions. Similarly chaotic
results were observed with the CPPN-Mutable seed trained with the pretrained
di model. The conclusion is that QD score does not entirely capture what
enables generalization and adaptation to novel terrains. Understanding this
issue may be important for further research.
Possible ideas for biasing seeds towards producing generalizable inventions
include disallowing precise setting of joint position and oscillatory parameters,
introducing stochasticity to prevent overtting to initial conditions, and incre-
mentally adjusting the seed. Preliminary results in disallowing precise setting
of parameters provided mixed results.
One promising result came from incremental seed design. With the CPPN-
Mutable seed (where the logic describing the CPPN encoding was able to be
evolved), the pretrained di model behaves similarly to the Radial seed (it cre-
ates inventions with high quantitative performance but which exploit chaotic
dynamics). However, when the di model is ne-tuned on the products of the
CPPN-Fixed seed (where the core CPPN logic is conserved), further CPPN-
Mutable runs retain qualitative characteristics of the CPPN-Fixed seed while
outperforming it quantitatively. That is, the CPPN-Fixed seed provided \train-
ing wheels" for learning how to modulate the encoding itself in the CPPN-
Mutable seed. In this way, an incremental approach to seed design (potentially
involving interactive evolution) may be a promising approach to qualitatively
shaping the outputs of ELM; alternatively, the notion of QD score could be
expanded or changed to better align with robust downstream performance.
49
E Final Map Approach to Stage 2
There are a variety of ways to distill the raw data generated by Stage 1 into
a dataset upon which a model can be trained. This section details a natural
alternative approach to the percentage threshold method used in the paper,
called the nal map approach. The method is to concatenate from all runs the
solutions from their nal MAP-Elites maps, i.e. the best quality solutions for
each niche at the end of a run.
This approach strikes a dierent trade-o between quantity and quality of
data samples than the percentage threshold method. The percentage threshold
approach normalizes performance across runs for each niche, and then includes
all reasonably-high quality solutions. The nal map approach, on the other
hand, is agnostic to the performance of a given run or seed (it does not normal-
ize across runs), and for each run takes only the highest-quality data for each
discovered niche.
The nal map dataset naturally consists of fewer examples (only 13K exam-
ples). Models trained on the nal map generally perform worse than percentage
threshold models on QD score. Lower QD results from the fact that the per-
formance across the nal map varies signicantly across seeds (e.g. the Square
seed performs very strongly in certain niches, but fails to nd solutions in others,
while the CPPN-like seed discovers solutions in nearly all niches, but generally
with weaker performance). As a result, the average sample from the nal map
dataset performs worse than those from the percentage threshold dataset (re-
sulting in lower QD score in the dataset, and also in trained models).
Additionally, preliminary Stage 3 experiments proved unstable when using
models trained on the nal map dataset. In eect, the nal map dataset appears
to be too small to serve as a reliable jumping-o point for further RL.
F Source Code for Completion Targets
This section includes the source code for the three inventions that serve as out-
of-distribution completion tests for trained models in Stage 2. Videos for these
inventions is shown at: https://y2u.be/8C2K5fk28HI.
ELM often adds structure to the seed, as in the nested loop of the Wheel,
or the multiple added loops in the Galloper, and also reuses function calls (e.g.
calling make sensor several times in the Galloper; note that make sensor is a
renamed (and modied) version of the make square function included in the
Square seed.
Nonsensical comments are often inserted (as in \acrylic of current (m)" in
the Runner's source), although parsimony pressure in the MAP-Elites algorithm
tends to eventually strip them out (e.g. there are no comments in the Wheel
invention). In some situations the seed's original comments are preserved, as in
the comment \connect the square with distance muscles" in the source code of
the Galloper.
50
F.1 Wheel
1import math
2def make_circle(wc, cx,cy,radius,num_points):
3 joints = []
4 tot_ang = 3.14*2.0
5 for idx in range(num_points):
6 ang = tot_ang/(num_points+1) * idx
7 x = math.cos(ang) * radius + 0.5
8 y = math.sin(ang) * radius + cy
9 joints.append(wc.add_joint(x,y))
10 return joints
11def make_walker():
12 wc = walker_creator()
13 num_points = 8
14 rad = 3.0
15 cx,cy = (11,5)
16 points = make_circle(wc, 0.6, -0.5,rad/2,num_points)
17 center = wc.add_joint(cx+1,cy+1)
18 for j in range(num_points):
19 for i in range(num_points-5):
20 wc.add_muscle(points[j], points[(i+j)%num_points],
21 0.0, 1.0, (j+1)/num_points)
22 wc.add_muscle(points[j], center,False,3,(j+1)/num_points)
23 return wc.get_walker()
F.2 Galloper
1def make_sensor(wc, x0, y0, x1, y1, d):
2 return wc.add_joint(x0, y0), wc.add_joint(x1, y1),
3 wc.add_joint(x1, y0), wc.add_joint(x0, y1),
4 wc.add_joint(d, 0.5), wc.add_joint(x1, 0.5)
5
6def make_walker(dx=0.0, dy=0.0, ddr=0, ddc=1.6, sid=8.0,
7 s_influence=0.2, s_side_width=0.0,
8 first_center=5.0, last_center=15.0):
9 wc = walker_creator()
10 ends = [make_sensor(wc, 5 + dx, -1 + dy, ddr, ddc, 4.5),
11 make_sensor(wc, 0, -0.1, sid, 9.5, 0.03),
12 make_sensor(wc, 5.5, -0.001, 5.0, 4.86 +0.8, 0.07),
13 make_sensor(wc, 5.5, -3.0, 6.0, 4.86 + 0.8, 0.07),
14 make_sensor(wc, 0, dx, ddr, ddc, 1.0)]
15
51
16 sides = ends[0] + ends[1] + ends[2] + ends[-1] + ends[-2]
17 + ends[-3]
18
19 center = wc.add_joint(dx, dy)
20 # connect the square with distance muscles
21 for k in range(len(sides)-6):
22 wc.add_muscle(sides[k], sides[k+1], True, 30, 0.5)
23 wc.add_muscle(sides[2], sides[4], False, 4.0, 0.8)
24 for k in range(len(sides)-2):
25 wc.add_muscle(sides[k], sides[k + 2], True, 18.0,
26 60.0 / 5.5)
27
28 for k in reversed(range(len(sides)-6)):
29 wc.add_muscle(sides[k], sides[k + 5], False, 4.0,
30 20.0 / 9.0)
31 wc.add_muscle(center, sides[7], False, 2.0, 90.0 / 9.0)
32 return wc.get_walker()
F.3 Runner
1import math
2import numpy as np
3
4def make_walker(p_scale=1): # acrylic of current (m)
5 wc = walker_creator()
6
7 def connect(x1,y1,x2,y2):
8 if -2*x1+x2*2>2:
9 return True
10 return x1<= abs(y1-y2)
11
12 def amp(x,y,x2,y2):
13 return abs(x-x2) + abs(y-y2)
14
15 def phase(x1,y1,x2,y2):
16 return -x1/2 - math.cos(math.pi/9)
17
18 joints = query_cppn(wc,5,7+p_scale,2,connect,amp,phase)
19 return wc.get_walker()
52
G Source Code for Selected Stage 1 Sodaracers
G.1 Blob (from CPPN Seed)
A video of the Sodaracer represented by the code below can be seen at: https:
//y2u.be/JDUAI8yrNcY.
1import math
2
3def walker():
4 wc = walker_creator()
5
6 def connect(x1,y1,x2,y2):
7 return (x1-x2)**2+5*y1**2-4*x2**2+y2**2 > 2.5
8 def amp(x1,y1,x2,y2):
9 return (x1-x2)**2+x2**2 + 1 - y2**2 < 2
10
11 def phase(x1,y1,x2,y2):
12 return math.sin(x1)*math.cos(y1)**2 + 1
13
14 joints = query_cppn(wc,5,6,2.1,connect,amp,phase)
15 return wc.get_walker()
G.2 Hopper (from Square Seed)
A video of the Sodaracer represented by the code below can be seen at: https:
//y2u.be/noSPGFX5m3M.
1def make_square(wc, x0, y0, x1, y1, length):
2 j0 = wc.add_joint(x0, y0)
3 j1 = wc.add_joint(x0, y1)
4 j2 = wc.add_joint(x1, y1)
5 j3 = wc.add_joint(x1, y0)
6
7 return j0, j1, j2, j3
8
9
10def make_walk(n=6):
11
12 wc = walker_creator()
13
14 # the main body is a square
15 sides_2_theta = make_square(wc, 0.0, 0.0, 5.6, 9.4, 2.4)
16 sides_1_theta = make_square(wc, 0.5, 0.8, 6.5, 13.1, 1.3)
53
17 sides_2_theta += make_square(wc, -0.8, -0.6, 6.7, 13.0, 2.3)
18 sides_2_theta += make_square(wc, -0.9, -0.6, 8.4, 12.5, 0.7)
19 sides_2_theta += make_square(wc, 0.0, -0.5, 0.2, 12.4, 1.7)
20 sides = sides_1_theta + sides_2_theta + sides_1_theta
21 center = wc.add_joint(2, 2)
22
23 # connect the square with distance muscles
24 for k in range(len(sides)-2):
25 wc.add_muscle(sides[k], sides[k+1])
26 wc.add_muscle(sides[k+2], sides[k], False, 30.0, 30.0)
27
28 # similarities of the Squares with":
29 for k in range(len(sides)-2):
30 wc.add_muscle(sides[k], sides[k], True)
31
32 for n in range(k, len(sides)):
33 wc.add_muscle(sides[k], sides[n], False)
34 wc.add_muscle(sides[3], center)
35 # the other prongs from the center of the square are active
36 wc.add_muscle(sides[2], center, False, 25.0, 25.0-0.7)
37 wc.add_muscle(sides[3], center, False, 20.0, 30.0+0.4)
38
39 return wc.get_walker()
G.3 Centipede (from Radial Seed)
A video of the Sodaracer represented by the code below can be seen at: https:
//y2u.be/zhMsPzo22do.
1import math
2
3
4def make_circle(wc, cx,cy,radius,num_points,eccentricity=1.4):
5 joints = []
6
7
8 tot_ang = math.pi*2.0*eccentricity
9
10 for idx in range(1,num_points):
11 x = math.cos(3.14*(idx+num_points)*tot_ang/(num_points))
12 * radius + cx
13 y = math.sin(3.14*(idx+num_points)*tot_ang/(num_points))
14 * radius + cy
15 joints.append(wc.add_joint(x,y))
54
16
17 return joints
18
19def make_walker(num_points=300,rad=3.25,f=3,max_rad=3):
20 wc = walker_creator()
21
22 cx,cy = (0,0)
23 body_size = rad*1.625
24
25 points = make_circle(wc, 0,0,body_size,num_points)
26 center = wc.add_joint(cx,cy)
27
28 for k in range(1,num_points-1):
29 wc.add_muscle(points[((k%10) - 1) % 10], points[k], False,
30 int(f*k/float(10)), k/10.)
31 wc.add_muscle(points[(k%10)], points[k], True, 1, k/10.)
32
33 return wc.get_walker()
H Probing Stage 2 Models
One hope for the models trained in Stage 2 is that they will learn not only
to memorize the training data (Python examples of Sodaracers), but also to
internalize the underlying structure of the domain (e.g. how in general to mix
together springs and masses to create functional Sodarace inventions). This
section discusses some preliminary observations of informal experiments that
change the training procedure in Stage 2 to explore what the model is capable
of learning. In particular, Sodarace examples are augmented with additional
comments (either as a prex or postx) that contain both the Sodaracer's tness
and its behavior characterization (its width, height, and mass).
The idea is that after training, the model can be asked to predict e.g. the
tness of an unseen invention (if trained with postx comments), or to generate a
walker with desired properties (if trained with prex comments). For example,
a prex-trained model can be conditionally sampled based on a prex that
species the desired height, width, and mass of a Sodaracer, to see how reliably
samples can match those properties when evaluated in the domain.
Preliminary experiments with both prex and postx 300M parameter mod-
els highlighted that the model was able to make such associations within the
training distribution, e.g. when a postx model was queried with heights, widths,
and masses taken from test set examples (held out from the same distribution),
it was able to consistently generate a Sodaracer with those properties. It was
slightly less reliable when conditioned on tness, re
ecting that this is a much
more complicated association (e.g. unlike width and height, tness depends on
the physical dynamics of the generated walker).
55
However, when taken out of distribution the model was less robust. For
example, a prex model struggled to targetedly generate Sodaracers within a
band of width and height that was deliberately held out from the training set.
Interestingly, while it was not reliable in generating Sodaracers of particular
held-out widths and heights, samples from the model did in eect cover the
holdout area, suggesting that the variation accessible within the model is enough
for interpolation or slight extrapolation, which is an important property for
enabling continual open-ended elaboration.
More starkly, a postx model had very limited ability to predict the tness
of Sodaracers taken from the Radial seed, which was not seen in training (there
was a Spearman correlation of only 0 :08). One hypothesis that is left to future
work to explore, is that larger models, trained with much more generated data,
may have more robust performance when taken out-of-distribution. If true, this
would support that scaling can benet open-ended learning, just as it does in
unsupervised and supervised learning.
A more speculative line of thought emerging from these experiments relates
to how Stage 2 structures the knowledge about the domain, which may signi-
cantly impact the dynamics of how RL in Stage 3 unfolds. That is, by training
the model to associate Sodaracers with their properties (through a prex or
postx), it may be more likely that Stage 3 can smoothly interpolate in the
space of those properties, which otherwise the model would have no explicit
knowledge about. However, when a prex-trained model was tested in the in-
terpolation setup of Appendix I, it did not perform any better than those trained
without prexes. While such prex-training did not have the desired impact,
it remains an open question how to include within Stage 2 information that
intuitively seems highly-relevant to RL (like tness) in a way that maximally
benets such RL.
Overall, the conclusion is that (at least with 300M parameter models and
the current amount of training data), Stage 2 models demonstrate modest ca-
pabilities to learn structure within Sodarace, but are not yet robust when taken
out-of-distribution. The implication for open-endedness is unclear (whether or
not this poses a problem for future research): For example, it may be that
stronger generalization capabilities may more naturally emerge when the exist-
ing pipeline (which is mainly a proof of concept) is extended such that Stage 3
is embedded within an open-ended process. Indeed, at least in human processes
of innovation, general insight does seemingly often emerge from continual open-
ended accumulation of initially-disparate examples of phenomena that are only
later unied.
I Interpolation Experiments
This section discusses experiments probing how well the conditional inventor
(the product of Stage 3) is able to understand the domain of the invention, by
exploring whether the model can appropriately adjust its output in response to
structured changes in the environment. That is, adjusting inventions in response
56
to smooth variations in the environment requires a deeper understanding of the
structure of the domain, and could potentially enable inventors to generalize
beyond the environments observed during training.
To examine this capability, an environment distribution with smoothly vary-
ing features is created, in particular, by varying the heights of tunnel terrains.
The motivation for this distribution is the observation that while larger Sodarac-
ers are unable to navigate low tunnels, they tend to locomote more quickly on

at terrains. Thus, the model is incentivized to adapt the height of the produced
Sodaracer to the height of the tunnel in the terrain, using \taller" Sodaracers
that locomote quickly for the taller tunnels, and shorter, slower-moving Sodarac-
ers for the lower tunnels. The ability to achieve such a solution would imply
that the model has learned about the underlying structure of the domain, in
that it is able to tweak the height of the produced inventions, and has captured
this relationship between the height and speed of the Sodaracer. To enable the
model to potentially learn a smooth mapping from the height of the tunnel to
the produced Sodaracer, the ResNet TEN architecture is employed.
In the experiments, however, the model repeatedly converged on solutions
outputting the same Sodaracer regardless of the height of the tunnel, i.e. an
unconditional solution. Examples of such solutions are shown at https://y2u.
be/gt1Z0lnjAuE.
These results point towards a subtle characteristic of the invention pipeline
introduced in this work. The models are not exhibiting a deep understanding
of the domain, nding a local, unconditional optimum that works \reasonably"
well on almost all terrains in the distribution. Particularly concerning is that
the produced Sodaracer is not able to navigate all terrains in the distribution,
highlighting the suboptimality of the learned solution. This property confounds
probing the interpolation capabilities of the inventors, and it remains unclear if
the invention pipeline can produce complex solutions that are able to smoothly
vary the produced inventions in response to smooth variations in the environ-
ment. Conversely, the experiments presented in the main body of this document
imply that the model is able to produce conditional solutions when no uncon-
ditional solutions are sucient.
We speculate that unconditional local optima are simpler and easier to learn
using RL methods, such that the models \gravitate" towards them when such so-
lutions exist. However, in future work, the invention pipeline could be deployed
in more complex, open-ended processes where unconditional solutions should
be rendered insucient. In such settings, it is conceivable that the pipeline will
output conditional inventors that have a deeper understanding of the domain
structure, as such solutions will allow the inventors to achieve signicantly higher
rewards in the domain, negating the concern regarding unconditional solutions.
Another avenue for future research would attempt to make the learning task
posed in Stage 3 easier by exploring maximum likelihood learning methods when
bootstrapping the conditional inventor (Stages 1 and 2). Here, the assumption is
that the exploration task in Stage 3, coupled with the necessity of incorporating
the new modality, is quite challenging for RL procedures. A simple approach
to this could be to sample from the unconditional LLM multiple times, and use
57
the best-performing samples for each terrain in the distribution as a supervised
(terrain-Sodaracer pairs) dataset to ne-tune both the LLM and the TENs.
Stage 3 could include terrain-distributions incorporating terrains unseen during
Stage 2, encouraging the inventor to further generalize and explore the space of
inventions. Looking even further down the line, it is conceivable to replace the
MAP-Elites-based ELM procedure of Stage 1 with a POET-style algorithm [40],
which would produce a supervised dataset of this form during Stage 1, relieving
pipeline designers of the need to hand-specify terrain distributions on which to
train the conditional inventor in Stage 2.
58