Evolving Multimodal Robot Behavior via Many Stepping Stones with the Combinatorial Multi-Objective Evolutionary Algorithm

07/09/2018
by   Joost Huizinga, et al.
University of Wyoming
2

An important challenge in reinforcement learning, including evolutionary robotics, is to solve multimodal problems, where agents have to act in qualitatively different ways depending on the circumstances. Because multimodal problems are often too difficult to solve directly, it is helpful to take advantage of staging, where a difficult task is divided into simpler subtasks that can serve as stepping stones for solving the overall problem. Unfortunately, choosing an effective ordering for these subtasks is difficult, and a poor ordering can reduce the speed and performance of the learning process. Here, we provide a thorough introduction and investigation of the Combinatorial Multi-Objective Evolutionary Algorithm (CMOEA), which avoids ordering subtasks by allowing all combinations of subtasks to be explored simultaneously. We compare CMOEA against two algorithms that can similarly optimize on multiple subtasks simultaneously: NSGA-II and Lexicase Selection. The algorithms are tested on a multimodal robotics problem with six subtasks as well as a maze navigation problem with a hundred subtasks. On these problems, CMOEA either outperforms or is competitive with the controls. Separately, we show that adding a linear combination over all objectives can improve the ability of NSGA-II to solve these multimodal problems. Lastly, we show that, in contrast to NSGA-II and Lexicase Selection, CMOEA can effectively leverage secondary objectives to achieve state-of-the-art results on the robotics task. In general, our experiments suggest that CMOEA is a promising, state-of-the-art algorithm for solving multimodal problems.

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I Introduction

A pervasive challenge in reinforcement learning, including evolutionary robotics, is to have agents autonomously learn many qualitatively different behaviors, generally refered to as multimodal behavior [1, 2]. Problems that require such multimodal behavior, which we will refer to as multimodal problems (also known as modal problems [3]), are ubiquitous in real-world applications. A self-driving car will have to respond differently depending on whether it is on a highway, in a city, in a rural area or in a traffic jam. A robot on a search-and-rescue operation will have to behave differently depending on whether it is searching for a victim or bringing a survivor to safety. Even a simple trash-collecting robot will have to behave differently depending on whether it is searching for trash, picking it up, or looking for a place to recharge.

Because multimodal problems require an agent to learn multiple different behaviors, they can be difficult to solve directly. A key insight for solving multimodal problems comes from how natural animals, including humans, learn complex tasks. Rather than learning all aspects of the task at once, we learn simpler, related tasks first. Later, the skills learned in these earlier tasks can be combined and adjusted in order to learn the more complex task at hand. These related tasks thus form the stepping stones towards solving the complete task. Methods that incrementally increase the difficulty of tasks have been successfully applied in animal training [4, 5]

, gradient-descent based machine learning 

[6, 7], and evolutionary algorithms [8, 9, 10, 11, 12, 13, 14]. Unfortunately, defining a proper set of subtasks and an effective ordering is a non-trivial problem, and choosing poor subtasks or presenting them in the wrong order can severely reduce the effectiveness of the learning process [15, 16].

Figure 1: CMOEA can preserve stepping stones that may be lost in other EAs. In this hypothetical example, a four-legged robot has to learn multimodal behavior that involves both running and jumping. Running is initially much easier to learn than jumping, but learning to jump well first is an important stepping stone in order to become excellent at both tasks. Arrows indicate ancestor-descendant relationships that can span many generations. (Left) Example of losing an important stepping stone. Initial generation: Some individuals are better at running while others are better at jumping, but all individuals are evaluated roughly equally by the fitness function. Intermediate generations: Because running is easier to learn than jumping, individuals that are good at running are rated more favorably than individuals that are average at jumping, and those specialized in jumping are not selected for in future generations. Final generation: All individuals have converged to the same local optima, where they are good at running, but only mediocre at jumping. (Right) Example of how CMOEA can preserve important stepping stones. Initial generation: Individuals that specialize in different combinations of tasks are assigned to different bins. Intermediate generations: Individuals that are average at jumping do not compete against individuals that are good at running and they are thus preserved within the population. Final generation: Because jumping turned out to be an important stepping stone, the descendants of individuals that initially specialized in jumping have increased performance on all combinations of tasks.

Population-based Evolutionary Algorithms (EAs) may provide a unique opportunity to combat the problem of choosing and ordering subtasks, because the population as a whole can try many different ways of learning the subtasks and evolutionary selection can preserve the methods that work. For example, imagine training a robot that has to be able to both run and jump, but in order to learn both tasks, it is imperative that it learns how to jump first before it learns how to run. In an evolutionary algorithm, one lineage of robots may start out being better at running, while another lineage may initially be better at jumping. If learning to jump first is an essential stepping stone towards learning to both run and jump, the lineage that started by being good at running will never learn both tasks, but the lineage that started by being good at jumping will, thus solving the problem. However, without proper tuning, most evolutionary algorithms are prone to converge towards the task that is easiest to learn at first, as learning this task will result in the most rapid increase in fitness. For example, if learning to run is much easier than learning to jump, the lineage specialized in running may outcompete the lineage of those specialized in jumping before they have the chance to adapt (Fig. 1 left). As these jumping individuals were an important stepping stone for learning how to both run and jump, this stepping stone is now lost from the population, and because the population is now dominated by runners, it is unlikely that the jumping only behavior will ever be visited again.

It is important to note that we usually have no way of knowing in advance what the important stepping stones are [17, 18]. As such, one of the best ways of preserving stepping stones may be to maintain as many forms of different behavior in the population as possible. Here we introduce111CMOEA was briefly described before in [19], but here we provide a more thorough introduction and a much more detailed experimental investigation. the Combinatorial Multi-Objective Evolutionary Algorithm (CMOEA) [19], a multiobjective evolutionary algorithm specifically designed to preserve the stepping stones of multimodal problems. CMOEA divides the population into separate bins, one for each combination of subtasks, and ensures that there is only competition within bins, rather than between bins. This way, individuals that excel at any combination of subtasks are preserved as potential stepping stones for solving the overall problem (Fig. 1, right). We compare CMOEA against two other multiobjective evolutionary algorithms: the widely applied NSGA-II algorithm [20] and the Lexicase Selection algorithm, which was also specifically designed to solve multimodal problems [3]. We compare the algorithms on both a robotics problem with 6 subtasks and on a multimodal maze-navigation problem with 100 subtasks, and show that CMOEA either outperforms or is competitive with the control treatments. As a separate contribution, we show that adding a linear combination over all objectives as an additional objective to NSGA-II can greatly improve its ability to solve multimodal problems. Lastly, we demonstrate that CMOEA is able to effectively incorporate secondary objectives, also known as auxiliary objectives, that increase the evolvability of individuals by selecting for genotypic and phenotypic modularity. With these secondary objectives, CMOEA achieves state-of-the-art performance on the robotics task, while the controls actually perform worse when these secondary objectives are added. These results indicate that CMOEA is a promising, state-of-the-art algorithm for solving multimodal problems.

Ii Background

Because multimodal problems are ubiquitous in many practical applications, a wide range of strategies have been developed for solving them. Many methods are based on the idea of incremental evolution, where complex problems are solved incrementally, one step at a time [8].

One incremental method is staged evolution, where the evolutionary process is divided into separate stages, each with its own objective function [10, 11]. The process starts in the first, easiest stage, and the population is moved to the next, more difficult stage when the first stage is considered “solved” according to stage-specific success criteria. Staged evolution requires the stages, the order in which the stages are presented, and the success criteria to be defined manually, but the design of the controller being evolved can be fully determined by evolution. Staged evolution can be made more smooth if the environment has parameters that allow for fine-grained adjustments, such as the speed of the prey in a predator-prey simulation [8]. Such fine-grained staging has also been referred to as environmental complexification [14]. Closely related to staged evolution is fitness shaping, where the fitness function is either dynamically or statically shaped to provide a smooth gradient towards a final goal [21, 22]. Fitness shaping provides the benefits of staging without the need to define the stages explicitly, but it does increase the complexity of the fitness function, which needs to be carefully designed by hand.

Another successful incremental-evolution method is behavioral decomposition [14], where a separate controller is optimized for each task and higher level controllers can build upon lower level controllers [12, 13], similar to hierarchical reinforcement learning [23]. Variants of behavioral decomposition techniques frame the process as a cooperative multiagent system, where each agent optimizes the controller for a particular subtasks, such that a group of these agents can combine their controllers and solve the task as a whole [24, 25]. One downside of these behavioral decomposition methods is that it becomes the experimenters responsibility to decide which tasks should get their own controller, which controllers build upon which other controllers, and in what order controllers should be trained, all of which are difficult decisions that can severely impact the effectiveness of the method. In addition, because the controllers are optimized separately, there is no opportunity for the optimization process to reuse information between controllers or find atomic controllers that work well together, meaning computational time may be wasted due to having to reinvent the same partial solutions in several different controllers.

A completely different way of approaching multimodal problems is to focus on behavioral diversity [26]. Behavioral diversity based approaches attempt to promote intermediate stepping stones by rewarding individuals for being different from the rest of the population. As a result, a population will naturally diverge towards many different behaviors, each of which may be a stepping stone towards the actually desired behavior. One canonical example of an algorithm based on behavioral diversity is Novelty Search, where individuals are selected purely based on how different their behaviors are compared to an archive of individuals from previous generations [27]. Novelty Search has been shown to be effective in maze navigation and biped locomotion problems [27], though it is unclear whether these problems required multimodal behavior. By selecting for behavioral diversity and performance, Mouret and Doncieux [28] were able to evolve robots that would exhibit a form of multimodal behavior where a robot would alternate between searching for a ball and depositing it in a predefined goal location. The main drawback of behavioral diversity based approaches is that the space of possible behaviors can be massive, meaning that it may contain many “uninteresting” behaviors that neither resemble a potential solution nor represent a stepping stone towards any other relevant behavior. One method for avoiding the problem of having too many “uninteresting” solutions is by having a fixed number of behavioral niches [29, 18]. By discretizing the behavior space, Cully et al. were able to evolve many different modes of behavior for a hexapod robot, which formed the basis of an intelligent trial-and-error algorithm that enabled the robot to quickly respond to damage [29]

. Similarly, Nguyen et al. were able to evolve a wide range of different looking images by having separate niches for images assigned to different categories by a pre-trained neural network, an algorithm called the

Innovation Engine [18]. CMOEA also builds on the idea of having different niches for different types of solutions, but instead of defining its niches based on different behaviors or different classes, it defines its niches based on different combinations of subtasks.

The strategy of solving multimodal problems considered in this paper revolves around framing it as a multiobjective problem, where each training task is its own objective [14, 22, 3]. We will refer to this strategy as multiobjective incremental evolution and, as with staged evolution, it requires the problem to be decomposed into a set of subtasks, each with its own fitness function. However, in contrast to staged evolution, the subtasks do not have to be explicitly ordered and there is no need to explicitly define success criteria for each stage.

There are many ways to obtain an appropriate set of subtasks. For example, it is possible to use prior knowledge in order to define a separate training task for every mode of behavior that might be relevant for solving the overall problem, such as having separate subtasks for moving and jumping. Alternatively, it is also possible to generate different environments and have each environment be its own training task. Provided that the environments are diverse enough (e.g. some environments include objects that need to be jumped over while other environments feature flat ground that needs to be traversed quickly) they can similarly encourage different modes of behavior. Subtasks could even involve different problem domains, such as image classification for one task and robot locomotion for another task. The main idea is that, as long as a task is unique and somewhat related to the overall problem, it can be added as an objective to promote multimodal behavior.

That said, while it is relatively straightforward to split a multimodal problem into different subtasks, there is no guarantee that classic multiobjective algorithms will perform well on this set. The main reason is that the set of subtasks will often be much larger than the number of objectives generally solved by multiobjective algorithms; rather than a multiobjective problem, which generally refers to problems with three or fewer objectives [30, 31, 32, 33, 34], it becomes a many-objective problem, a term coined for problems that require optimization of many more objectives [31, 32, 33, 34]. For example, the maze navigation problem presented in this paper required at least 100 subtasks in order to promote general maze solving behavior (preliminary experiments with 10 subtasks generalized poorly and even with 100 training mazes generalization is not perfect, SI Sec. S2.2). Many popular multiobjective algorithms have trouble with such a large number of objectives because they are based on the principle of Pareto dominance [30, 31, 32, 33, 34]. According to the definition of Pareto-dominance, an individual dominates an individual only if is not worse on any objective than and is better then on at least one objective [35]. With the help of this Pareto-dominance relation, these algorithms attempt to approximate the true Pareto front, the set of solutions which are non-dominated with respect to all other possible solutions. However, as the number of objectives grows, the number of individuals in a population that are likely to be non-dominated increases exponentially. When nearly all individuals in a population are non-dominated, a Pareto-dominance based algorithm may lose its ability to apply adequate selection pressure. There exist many different methods to increase the maximum number of objectives that these Pareto-based algorithms can handle [36, 37, 31], and these methods have been shown to be effective up to objectives if no assumptions about the problem are made [36], and up to objectives if the majority of those objectives are redundant [31]. However, because of the exponential relationship between the number of objectives and the dimensionality of the Pareto front, it is unlikely that purely Pareto-based methods will be able to scale much further.

There also exist many multiobjective evolutionary algorithms that do not rely on Pareto dominance [35, 38, 3, 39]. Such techniques may be especially relevant for multimodal problems because multimodal problems do not necessarily require an approximation of the true Pareto front. Instead, multimodal problems simply require adequate performance on all objectives, which generally means searching for only a small area or point on the true Pareto front. In theory, searching for a point on a Pareto front can be achieved by simply optimizing a weighted sum of all objectives [35]. The main problem with such a weighted sum approach is that, even when the desired trade-off for the optimal solution is known, the trajectory for finding this optimal solution may not be a straight line, but may instead require the algorithm to find a number of solutions with different trade-offs first [35]. This issue will almost certainly be present in the context of multimodal problems because different modes of behavior can vary greatly in difficulty, meaning that “straight line” optimization (i.e. attempting to learn all modes simultaneously) is likely to fail. As such, multimodal problems may be best tackled by algorithms that do not strictly rely on Pareto dominance, but that still explore many different trade-offs during optimization.

Iii Treatments

To assess the performance of CMOEA relative to other algorithms, we compare CMOEA against two successful multiobjective algorithms, namely NSGA-II [20] and Lexicase Selection [3]. To verify the usefulness of having many CMOEA bins, we also compare CMOEA against a variant that only has a single bin with all of the subtasks. A description of each of these treatments is provided below.

Iii-a Cmoea

The goal of CMOEA is to provide a large number of potential evolutionary stepping stones, thus increasing the probability that some of these stepping stones are on the path to solving the task as a whole. To do so, we define a

bin for every combination of subtasks of our problem. For example, if we have the two subtasks of moving forward and moving backward, there will be one bin for moving forward, one bin for moving backward, and one bin for the combination of moving forward and backward. The algorithm starts by generating and evaluating a predetermined number of random individuals and adding a copy of each generated individual to every bin. Next, survivor selection is performed within each bin such that, afterward, each bin contains a number of individuals equal to some predetermined bin size. For each bin, selection happens only with respect to the subtasks associated with that bin. After this initialization procedure, the algorithm will perform the following steps at each generation: (1) select a number of parents randomly from across all bins, (2) generate one child for each selected parent by copying and mutating that parent (no crossover is performed in the version presented in this paper), (3) add a copy of each child to every bin, and (4) perform survivor selection within each bin (Fig. 2).

For survivor selection to work on a bin with multiple subtasks, we need some way of comparing individuals who may have different performance values on each of these tasks. While there exist many selection procedures specifically designed to work with multiple objectives [40, 41, 42, 35], these multiobjective selection procedures tend to have difficulty with many objectives (see Sec. II

), which is exactly the problem CMOEA was designed to solve. As such, within bins, CMOEA combines performance values on different tasks into a single fitness value by taking their arithmetic mean or by multiplying them. Multiplication can be more effective than taking the arithmetic mean because it requires individuals to obtain at least some non-zero performance on every relevant sub-task within a bin, rather than being able to specialize in a subset of those subtasks while neglecting the others. Note that the same properties can be obtained by taking the geometric mean, but multiplication is computationally more efficient to calculate and both methods result in the same relative ordering. Multiplication does require clipping or shifting values in a way that avoids negatives, as negative values could completely alter the meaning of the combined performance metric. That said, it is generally considered good practice to normalize performance values regardless of whether values are combined by taking the arithmetic mean or through multiplication, as overly large or overly negative values can negatively impact the effectiveness of both aggregation methods.

While CMOEA does not prescribe any particular selection procedure for the survivor selection step within each bin, we implement the multiobjective behavioral diversity method by Mouret and Doncieux [26]. In this method, the multiobjective evolutionary algorithm NSGA-II [20] selects for both performance and behavioral diversity, which allows it to avoid local optima and fitness plateaus [26]. We apply it as a within-bin selection procedure because it ensures that each bin maintains individuals that solve the same subtasks in different ways. In addition, this method outperformed a method based on novelty search with local competition [43], which is another algorithm that optimizes for both performance and diversity (SI S2.3). For any particular bin, the performance objective is the main objective associated with that bin (e.g. move forward, move backward, move forward move backward, etc.). The behavioral diversity of an individual is calculated by first measuring some relevant feature of the behavior of an individual, called the behavior descriptor, and then calculating the mean distance to the behavior descriptors of all other individuals in the same bin. As such, the larger this distance, the more unique the behavior of the individual is with respect to the other individuals in that bin. Behavioral diversity metrics differ per domain, and details can be found in sections IV-B and IV-C. The code for CMOEA, as well as for all experiments and control treatments, is available at: www.evolvingai.org/cmoea.

Figure 2: Overview of CMOEA. At every generation, CMOEA first selects a number of parents ( in this example) randomly from across all bins. It then creates offspring by copying and mutating those parents and one copy of each offspring is added to every bin. Afterward, a local survivor-selection method determines which individuals remain in each bin. In this example, survivor selection is performed by the non-dominated sorting algorithm from NSGA-II, with performance on the tasks associated with the relevant bin as one objective and behavioral diversity within the bin as the other objective.

Iii-B Single Bin CMOEA

To verify that having many CMOEA bins actually provides a practical benefit, we run a control that features only a single CMOEA bin, called the Single Bin treatment. The Single Bin treatment is the same as CMOEA, except that it only has one bin, namely the bin that is associated with all subtasks. To ensure a fair comparison, this bin is resized such that the number of individuals within this bin is equal to the total number of individuals maintained by CMOEA across all bins (the Single Bin treatment with a population size equal to the number of new individuals created at each generation (1000), which is a common default in EAs, performed worse, see SI Fig. S5). In addition, the Single Bin treatment also implements the Pareto-based tournament selection procedure for parent selection from NSGA-II, which is expected to increase the performance of the Single Bin treatment, and thus ensures a comparison against the best possible implementation of this treatment [20].

Iii-C Nsga-Ii

NSGA-II [20] is a Pareto-based multiobjective evolutionary algorithm that, because of its popularity [20, 19, 22, 30, 26, 28, 31, 32, 33, 36, 37, 44, 45, 46]

, functions as a good benchmark to estimate where CMOEA stands with respect to Pareto-based algorithms. Briefly (see 

[20] for details), NSGA-II works by sorting a mixed population of parents and children into ranked fronts where each individual is non-dominated with respect to all individuals in the same and lower ranked fronts. During selection, NSGA-II iteratively adds individuals, starting from the highest ranked front and moving towards the lowest ranked front, until a sufficient number of individuals have been selected to populate the next generation.

Here, in order to apply NSGA-II to multimodal problems, we add every subtask of the problem as a separate objective to NSGA-II. Unfortunately, doing so with plain NSGA-II causes it to perform pathologically poorly on our test problems (Fig. 5 and 9), probably because NSGA-II’s Pareto-based selection pressure is overwhelmed by the large number of objectives in our problems. With such a large number of objectives it becomes almost trivial to become non-dominated, at which point there is no proper selective pressure (see Sec. II for details).

In this paper, we demonstrate that we can alleviate this problem by adding the combined performance on all tasks as an additional objective, which ensures that NSGA-II explicitly selects for individuals that are high performing (or, as a result of the crowding score, diverse) on the objective we actually care about. Because maximizing the combined performance on all tasks is the primary target of our search process, this extra objective will be referred to as the Combined-Target objective, and NSGA-II with a Combined-Target objective will be called Combined-Target NSGA-II.

Iii-D Lexicase Selection

As a recently developed algorithm specifically designed for solving multimodal problems, Lexicase Selection represents the state-of-the-art in terms of multimodal evolutionary algorithms, and it thus presents another good benchmark to compare against [3]. In Lexicase Selection, each individual is selected by first choosing a random order for the objectives, and then selecting the individual that is the best according to the lexicographical ordering that results from the randomly ordered objectives (e.g. if the random order of objectives is {Forward, Backward}, first all individuals that have the maximum performance on the Forward task will be selected and then the performance on the Backward task will serve as a tiebreaker). Because the order of objectives is randomized for every individual being selected, Lexicase selection will select specialists on each of the objectives first, with ties being broken by performance on other objectives.

In the original Lexicase Selection algorithm, evolution happens by selecting a number of parents equal to the population size, and then having the children of these parents replace the old population [3]. To ensure a fair comparison with CMOEA, which maintains a population size that can be much larger than the number of offspring created at each generation, we have similarly decoupled the size of the population and the number of offspring per generation for Lexicase Selection. In our implementation, a predetermined number of parents are selected to produce an equal number of offspring, and then survivors are selected from among the combined population of parents and offspring until the number of remaining individuals equals the intended population size. We apply Lexicase Selection to both parent selection and survivor selection.

Because Lexicase selection is biased towards selecting specialists, it is possible that it will overemphasize them at the cost of selecting generalists. As such, we have also examined the effect of adding a Combined-Target objective to Lexicase Selection, which we call Combined-Target Lexicase Selection, to see whether this improves performance.

Iv Experiments

Iv-a Settings and plots

For all experiments, the number of individuals created at every generation was . Because the population size of CMOEA can not be set directly, as it is partially determined by the number of subtasks, it has to be tuned by setting the bin size. We set the bin size such that each bin would be large enough to allow for some diversity within each bin while keeping the total population size computationally tractable. For a fair comparison, the population size for all other treatments was subsequently set to be equal to the total population size maintained by CMOEA (we also tested the controls with a population size equal to the number of individuals created at every generation, which is a common default in EAs, but those treatments performed worse, see SI Sec. S3.1). All experiments involved the evolution of a network with NEAT mutation operators [47], extended with deletion operators, and the treatments did not implement crossover. Experiment-specific settings are described in the relevant sub-sections.

All line plots show the median over 30 runs with different random seeds. Unless stated otherwise, shaded areas indicate the

bootstrapped confidence interval of the median obtained by resampling 5000 times and lines are smoothed by a median filter with a window size of

. The performance of a run at a particular generation is defined as the highest performance among the individuals at that generation. Symbols in the bar below each plot indicate that the difference between the indicated distributions is statistically significant ( according to the Mann-Whitney-U test). Unless otherwise specified, all statistical comparisons are performed with the Mann-Whitney-U test.

Iv-B Robot domain

Iv-B1 Robot domain experimental setup

We have tested the performance of CMOEA on two different problems. The first is a robotics problem known as the six-tasks problem [19], where a hexapod robot has to learn to perform six different tasks (move forward, move backward, turn left, turn right, jump, and crouch) depending on its inputs (Fig. 3). Neural network controllers (Fig. 4) are evaluated by performing a separate trial for each task, with the information about which task to perform being presented to the inputs. How performance and behavioral diversity are measured on this task follow [19] and are described in SI Sec. S1.1.

(a) Side Top (b) Forward Backward Turn-left Turn-right Jump Crouch
Figure 3: Six-tasks robot and problem. (a)

The hexapod robot has 6 knee joints with one degree of freedom and 6 hip joints with 2 degrees of freedom (up-down, front-back).

(b) The six tasks that need to be learned by the robot.
(a) (b)
Figure 4: The spatial network layout, MSS planes, and associated CPPN for the robotics task. (a)

Spatial layout of the network for the robotics task. Neurons are shown in a cube that extends from -1 to 1 in all directions and neurons are placed such that the extreme neurons lie on the boundaries of this cube. The letter above each of the six input neurons specifies with which task that neuron is associated: forward (F), backward (B), turn-left (L), turn-right (R), jump (J), and crouch (C). Besides these task-indicator neurons, the network has no other inputs. The color of every node and connection indicates to which MSS plane it belongs and it matches the color of the CPPN outputs that determine its parameters.

(b) The CPPN for the robotics task. Colored letters above the CPPN indicate the following outputs: weight output (W), link-expression output (L), bias output (B), and time-constant output (T). There is no bias or time-constant output in the CPPN for the red MSS plane because that plane governs the input neurons of the CTRNN, which do not have bias or time-constant parameters. Inputs to the CPPN are the three coordinates for the source (x1, y1, z1) and target (x2, y2, z2) neurons and a bias input with the constant value of 1.

Because this problem features six subtasks, CMOEA maintains bins (one for each combination of subtasks except the permutation with zero subtasks). For this problem, the size of each bin was set to be , meaning that all controls had a population size of . Following previous work [19]

, the controller was a Continuous-Time Recurrent Neural Network (CTRNN) 

[48] encoded with the HyperNEAT encoding [49], which was extended with a Multi-Spatial Substrate (MSS) [50] and a Link Expression Output [51]. For details about the aforementioned algorithms and extensions, we refer the reader to the cited papers. Parameters for CTRNN and the evolutionary algorithm were the same as in [19] and are listed in the SI for convenience (SI Sec. S1.1).

Figure 5: NSGA-II performs poorly on the robotics task unless a Combined-Target objective is added, while Lexicase selection performs poorly regardless of whether a Combined-Target objective is added. Note that performance values appear to be extremely low because it is the product of six numbers between 0 and 1, but and individual with a fitness greater than 0.001 generally demonstrates some basic competency on all six tasks, while and individual with a fitness smaller than 0.0003 does not (videos are available at: www.evolvingai.org/cmoea).

Iv-B2 Robot domain results

The first thing to note is that Combined-Target NSGA-II significantly and substantially outperformed regular NSGA-II, demonstrating that the Combined-Target objective is an effective method for alleviating the effects that the high-dimensional Pareto-front has on NSGA-II on this particular problem (Fig. 5). This enhancement to the widely used NSGA-II algorithm is an independent contribution of this paper.

Second, Lexicase Selection performs far worse than Combined-Target NSGA-II. The reason that Lexicase Selection performs poorly in this robotics domain is probably because the domain is continuous. Due to its lexicographical ordering, Lexicase Selection will always select individuals that are champions on at least one of the objectives. However, because the probability of an exact tie is low in a continuous domain, it is quite possible that there exists only one champion per objective. Given that Lexicase Selection generates a new random ordering every time it selects an individual, performing Lexicase Selection with six objectives is likely to fill a population with copies of only six different champions, thus greatly reducing the diversity and the number of potential stepping stones in the population. Without the proper stepping stones, Lexicase Selection is then unable to find individuals that perform well on all tasks. While it might be possible to resolve this issue by providing a margin in which two performance values are considered equal, such a margin would have to be tuned correctly, and it is unclear whether a single margin would work for all treatments and throughout all generations. Implementing and testing such a version of Lexicase Selection is an interesting direction for future research.

Adding the Combined-Target objective to Lexicase Selection slightly, though significantly, increases performance, but the resulting performance is still substantially lower than that of Combined-Target NSGA-II. This increased performance is probably because the champion on the combined objective is now explicitly preserved in the population. Unfortunately, this does not resolve the issue of having a reduced population diversity and fewer stepping stones, meaning performance remains low as a result. It is important to note that this is not a flaw in the Lexicase Selection algorithm itself; Lexicase Selection was designed for discrete domains, which do not have this issue. In these experiments, we verify that Lexicase Selection does not generalize to continuous domains, which is another small contribution of this paper Because the Combined-Target versions of both NSGA-II and Lexicase Selection performed better than their regular counterparts, we consider only the Combined-Target versions for the remainder of the robotics task results.

Figure 6: CMOEA performs significantly better than the control treatments in early generations. CMOEA performs significantly better than all control treatments during the first 7000 generations. Afterward, CMOEA still performs significantly better than the Single Bin and Combined-Target Lexicase Selection treatments, but the difference between CMOEA and Combined-Target NSGA-II is no longer significant. Combined-Target NSGA-II significantly outperforms all other controls and the Single Bin treatment significantly outperforms Combined-Target Lexicase Selection.

When we compare the controls with CMOEA, we see that CMOEA performs significantly better than any of the controls for the first 7000 generations (Fig. 6). After those 7000 generations, CMOEA still performs significantly better than Combined-Target Lexicase Selection and the Single Bin treatments, but there is no longer a significant difference between CMOEA and Combined-Target NSGA-II. Combined-Target NSGA-II also performs significantly better than Combined-Target Lexicase Selection and the Single Bin treatments. The fact that the Single Bin treatment performs substantially and significantly worse than both CMOEA and Combined-Target NSGA-II is indicative of the importance of having multiple different objectives. While the Single Bin treatment dedicates all of its resources to the combination of the six objectives, such a strategy did not lead to the best performance on this problem, presumably because it fails to find all the necessary stepping stones required to learn these behaviors. Instead, both CMOEA and Combined-Target NSGA-II dedicate a substantial amount of resources towards optimizing subsets of these objectives and these subsets then form the stepping stones towards better overall performance. Combined-Target Lexicase Selection performs the worst out of all treatments, probably for the reasons discussed earlier.

Given that both CMOEA and Combined-Target NSGA-II were still gaining performance after 15 000 generations and that it was unclear which algorithm would perform better in the long run, we extended the experiments with these treatments up to 75 000 generations (Fig. 7). Other treatments were not extended because it was unlikely that these treatments would catch up with additional generations and extending their runs would have been computationally expensive. While the difference was relatively small, Combined-Target NSGA-II achieved a significantly higher performance than CMOEA after about 65 000 generations. This result suggests that, similar to CMOEA, Combined-Target NSGA-II is capable of maintaining the evolutionary stepping stones required for performing well on this task. In addition, given that Combined-Target NSGA-II maintains only seven different objectives (the six main objectives and the Combined-Target objective), it is likely that Combined-Target NSGA-II is capable of dedicating more resources to individuals that perform well on the Combined-Target objective than CMOEA, thus explaining why Combined-Target NSGA-II eventually outperforms CMOEA. If this is true, the performance of CMOEA can possibly be improved by increasing the relative population size of the bin responsible for the combination of all subtasks. That said, even without such optimization, CMOEA remains competitive with Combined-Target NSGA-II for the majority of the 75 000 generations.

Figure 7: While Combined-Target NSGA-II eventually reached a performance that was significantly higher than that of CMOEA in extended runs, CMOEA performed significantly and substantially better than Combined-Target NSGA-II when secondary objectives were added. Combined-Target NSGA-II performed slightly but significantly better than CMOEA after roughly 65 000 generations. However, CMOEA with secondary objectives for maximizing genotypic and phenotypic modularity performed significantly and substantially better than Combined-Target NSGA-II, obtaining a median performance almost three times higher than that of Combined-Target NSGA-II. Both Combined Target NSGA-II and Combined-Target Lexicase Selection, on the other hand, performed significantly worse when the secondary objectives were added. Note that the experiments for Combined-Target Lexicase Selection are cut off at 15 000 generations because these experiments were not extended. A magnification of Combined-Target Lexicase Selection with and without modularity is provided in the SI to visualize the difference (SI Fig S1).

Previous work has shown that it may be helpful to have secondary objectives, akin to auxiliary tasks [52, 53], which influence the structure of the evolved neural networks, such as by promoting modularity or hierarchy [44, 45, 46]. In particular, the paper that briefly introduced CMOEA [19] demonstrated that selecting for genotypic and phenotypic modularity increases the performance of CMOEA on the six-tasks robotics problem. Modularity may be beneficial on this problem because, once the phenotypic network has developed modules, those modules can be involved in different types of behavior (e.g. there may be a separate module for moving and a separate module for turning), allowing those behaviors to be optimized separately. However, because the network is indirectly encoded [54], a modular phenotype alone may not be sufficient to allow those modules to be separately optimized, as a local change in the genotype may cause a global change in the phenotype. Supporting this hypothesis, previous work demonstrated that performance only increases with simultaneous selection for both genotypic and phenotypic modularity, and not with selection for either genotypic or phenotypic modularity alone [19].

Note that these secondary objectives are different from the subtask objectives in that they are completely unrelated to the problem that needs to be solved, meaning that individuals can gain performance on these objectives without making any progress towards the overall goal. Instead, these objectives promote genotypic and phenotypic structures that increase the evolvability of individuals, thus increasing the potential of these individuals in later generations. As was shown in [19], being able to effectively make use of these secondary objectives can greatly improve the effectiveness of an algorithm. As such, we examined whether Combined-Target NSGA-II and Combined-Target Lexicase Selection could benefit from these same two secondary objectives as well, namely selecting for genotypic and phenotypic modularity.

For CMOEA, these two secondary objectives were added to the NSGA-II selection procedure within every bin, thus ensuring that every individual maintained by CMOEA would be subject to selection for genotypic and phenotypic modularity [19]. However, there does not exist a selection procedure for NSGA-II or Lexicase Selection that is equivalent to CMOEA’s within bin selection procedure. As such, for these controls, the secondary objectives of genotypic and phenotypic modularity were added as additional objectives alongside the primary objectives.

Both Combined-Target NSGA-II and Combined-Target Lexicase Selection performed significantly worse when the modularity objectives were added (Fig. 7). A likely cause for this effect is that, because these secondary objectives are completely separate from the main objectives, the algorithms maintain individuals that are champions at having a modular genotype or a modular phenotype, but not in combination with actually performing well on any of the main objectives. CMOEA avoids this issue by having the secondary objectives be present in every bin, thus forcing all individuals to invest in being modular, regardless of which subtasks they solve. One possible solution could be to add the secondary objectives as part of the Combined-Target objective. Unfortunately, this would involve explicitly defining the trade-off between modularity and performance and it is not unlikely that the optimal trade-off changes over time such that it is impossible to find a single weighting for this problem. As such, CMOEA appears to be more suitable than Combined-Target NSGA-II or Combined-Target Lexicase Selection when it comes to utilizing secondary objectives.

Note that, instead of adding the selection pressure for genotypic and phenotypic modularity to every CMOEA bin, it is possible to allocate additional bins for individuals that combine genotypic and phenotypic modularity with performance (e.g. having one bin for jumping alone and another bin for jumping genotypic and phenotypic modularity). Once again, the main downside of such an approach is that it requires the trade-off between modularity and performance to be explicitly defined. However, because such an approach would increase the number of potential stepping-stones that are preserved, it is possible that doing so would improve the performance of CMOEA even when the modularity and performance objectives are not properly balanced. Examining the effect of adding additional bins that select for a linear combination of modularity and performance remains a topic for future research.

Iv-C Maze domain

Iv-C1 Maze domain experimental setup

The second problem is a maze navigation task, where a wheeled robot is put into a randomly generated maze and has to navigate to a goal location. In contrast to the six-tasks problem discussed before, we do not define the different modalities of the problem explicitly. Instead, we have the modalities arise naturally from the problem instances.

The mazes were generated according to the maze generation algorithm from [55] (originally introduced in [56]), where a grid-based space is repeatedly divided by a wall with a single gap. The mazes for our experiments were generated by dividing a 20 by 20 grid 5 times with the goal placed in the center of a cell randomly selected from the grid. The grid was subsequently converted into a continuous space where each cell in the grid represented an area of 20 by 20 units. The robot had a circular collision body with a radius of 4 units, giving it plenty of space to move within a cell, and always started at the center of the maze facing north. Walls were 2 units wide and the gaps within each wall were 20 units wide. This maze-generation algorithm resulted in mazes with a house-like quality, where the space was divided into separate rooms connected by doorways and the goal positioned somewhere within one of those rooms (Fig. 8a). The robot was simulated with the Fastsim simulator [28, 57].

The robot had two different types of sensors: range-finder sensors, which detect the distance to the closest wall in a certain direction, and goal sensors, which indicate whether the goal lies within a specific quadrant relative to the robot (Fig. 8b). In contrast to previous work [55, 27], our goal sensors did not work through walls. As a result, the problem had two different modes: in the first mode the robot has to traverse different rooms in order to find the room containing the goal, and in the second mode the robot has to move towards a goal that is located in the same room. These two behaviors are different modes because they require the robot to operate in different ways. The most straight-forward method for traversing all rooms is probably a wall-following strategy, as it implicitly implements the classic maze solving strategy of always choosing the left-most or right-most path. However, moving towards the goal requires the robot to leave the wall and exhibit homing behavior instead, as the goal may not be located next to a wall.

(a) (b)
Figure 8: Example maze and robot schematic. (a) Example maze generated by the maze generation algorithm. The green lines represent the rangefinder sensors of the robot. (b) The schematic of the maze exploration robot (adapted from [27]).

In this problem, rather than selecting for each mode of behavior explicitly, we simply generate a large number of mazes that may, by chance, emphasize different modes of behavior. For example, in some random mazes, the robot will start in the same room as the goal, meaning that all it has to do is move straight towards the goal without any wall-following behavior. In other mazes, the robot may be in a different room from the goal, but the goal may be located right next to a wall. In those mazes, wall-following behavior alone can guide the robot to the goal, without any homing behavior being required. Lastly, some mazes will put the robot and the goal in different rooms, and put the goal somewhere in the center of a room, thus requiring both wall-following and homing behavior to be navigated successfully.

This experimental setup is especially relevant because it reflects a practical way of applying CMOEA. While it may be hard to define in advance exactly all the different behavioral modes that are important to solve a particular problem, it is usually much easier to define different instances of the same problem. As with our mazes, different instances of the same problem may emphasize different modalities and, as a result, these different instances may provide effective scaffolding for learning to solve the problem as a whole.

In this experiment, the problem as a whole is not to solve a particular maze, or even any specific set of mazes, but rather to solve these house-like mazes in general. As such, any solution evolved to solve a particular set of mazes has to be tested on a set of unseen mazes to assess its generality. To do this, for every run, we generated a training set of 100 mazes to calculate the fitness of individuals during evolution and we generated a test set of 1000 mazes to assess the generality of individuals. The generality of solutions was evaluated every 100 generations for plotting purposes, but this information was not available to the algorithms itself.

As we performed the experiment with a training set of mazes, each of which represented a different training task, the number of bins required to represent all combinations of subtasks was much larger than what could be realistically processed ( subtasks would require different bins). While it may seem that such exponential scaling would prevent CMOEA from being applied to larger problems, it is important to note that what we really require from CMOEA is that it provides a sufficiently large number of different stepping stones. As long as there is a sufficiently large number of directions in which improvements can be discovered, evolution is unlikely to get stuck in local optima, and can thus continue to make progress. As such, if the number of bins is large, only a subset of bins may be necessary to provide the required stepping stones. To test this theory, we define a maximum number of bins ( with a bin size of in this experiment) to which we assign different sets of subtasks. First, we assign the combination of all subtasks to one of our bins, as this combination represents the problem we are trying to solve. Second, we assign every individual training task to a bin, as those provide the most obvious starting points for our algorithm. Lastly, the remaining bins, which we will call dynamic bins, are assigned random combinations of subtasks. To create a random set of subtasks, we included each training task with a probability of , meaning most bins were associated with about half of the total number of subtasks. We choose this method for its simplicity, even though a different approach could have offered a smoother gradient from bins that govern only a few subtasks to bins that govern many subtasks. Examining the effectiveness of smoothed bin selection methods will remain a topic for future work.

To make sure the algorithm does not get stuck because it was initialized with poor sets of subtasks, we randomly reassign the subtasks associated with one of the dynamic bins every generation. While this does mean that many of the individuals previously assigned to that bin will now be replaced (as the selection criteria may be completely different), some research has suggested that such extinction events may actually help an evolutionary process in various ways [58, 59, 60]. We did not attempt to find the optimal rate at which to reassign the subtasks of the dynamic bins, but we found that the arbitrary choice of one bin per generation performed well in this particular domain.

Performance of an individual on a maze was defined as its distance to the goal divided by the maximum possible distance to the goal for that maze (i.e. the distance from the goal to the furthest corner of the maze). A fitness of one was awarded as soon as the body of the robot would be on top of the goal, at which point the maze was considered solved. Performance on a combination of mazes was calculated as the mean performance over those mazes. We did not calculate the multiplicative performance on this problem because individuals did not seem to over-specialize on easy mazes, probably because no additional fitness could be gained after a maze was solved. Every simulation would last for 2500 time-steps or until the maze was solved. The wheels of the robot had a maximum speed of 3 units per time-step and the speed of each wheel was determined by scaling the output of the relevant output neuron to the range.

The robot controllers were directly-encoded recurrent neural networks with 10 inputs (6 for the rangefinders and 4 for the goal sensors), 2 outputs (one for each wheel), and sigmoid activation functions. Neural network and EA settings followed 

[61] and are listed in SI Sec. S1.2.

Iv-C2 Maze domain results

After 1000 generations, all treatments, except for NSGA-II, have evolved a well-known, general maze-solving solution, which is to pick any wall and follow it in one direction until the goal is reached (see video on www.evolvingai.org/cmoea). Here, the solution is also multimodal, as individuals switch from wall-following behavior to goal-homing behavior when they see the goal.

With respect to performance on the training set, CMOEA and Lexicase Selection performed significantly better than any of the other controls during the first 100 generations and they both quickly converged near the optimal performance of 1 (Fig. 9a). The Combined-Target objective did not have a significant effect on Lexicase Selection, suggesting that it is not a useful extension for Lexicase Selection. The Single Bin treatment converged slightly slower than CMOEA and Lexicase Selection, but it reached a similar near-perfect performance after about 200 generations, indicating that the additional bins were helpful during early generations, but not required for solving this problem. After 500 generations, the Single Bin treatment actually performed significantly better than CMOEA and Lexicase Selection, not in terms of its median performance, but in terms of the number of runs that obtain perfect performance (Fig. 9a inset). This difference is also apparent in terms of the number of mazes solved after 1000 generations (Fig. 10a), as all but three Single Bin runs solved all training mazes perfectly, while the success rate of CMOEA and Lexicase Selection was not as high. Both NSGA-II and Combined Target NSGA-II had much more difficulty finding near-optimal solutions, demonstrating the debilitating effect of 100 objectives on these Pareto-dominance based methods. That said, given the poor performance of NSGA-II on the six-tasks problem, it may be surprising that NSGA-II gained any performance at all on this 100 objective problem. Here, it is important to note that, in terms of mazes solved, NSGA-II still performed much worse than any of the other treatments (Fig. 10a). In addition, and in contrast to the six-tasks domain, solving any maze directly contributes to overall performance, thus providing NSGA-II with a slightly better gradient towards solving the overall problem. As with the six-tasks problem, the Combined-Target objective significantly and substantially increases the performance of NSGA-II in the maze domain, suggesting that the Combined-Target objective may represent a general technique for applying NSGA-II to multimodal problems with many subtasks.

[width=0.49]"figures/100_maze_training_with_inlay_v3".pdf(a)[width=0.49]"figures/100_maze_test".pdf(b)

Figure 9: On the maze domain, CMOEA and Lexicase Selection, irrespective of the Combined-Target objective, significantly outperformed the other treatments during early generations. (a)

CMOEA and Lexicase Selection significantly outperformed the other treatments on the training set during the first 200 generations of evolution. The Single Bin treatment started outperforming all other treatments after 500 generations. The inset shows a zoom in of the indicated area, but instead of showing a confidence interval of the median (which, for many treatments, is too small to be informative in this area), it shows the interquartile range, ordered such that the third quartile is visible for all treatments. The third quartile of the Single Bin treatment converges to 1 around 500 generations, indicating that, in contrast to the other treatments, at least

of its replicates have perfectly solved the problem from this generation forward. (b) Median test-set performance of the individual with the highest training-set performance from each replicate, with ties broken arbitrarily. Because performance on the test set is only evaluated every 100 generations, lines are not smoothed by a median filter. On the test set, CMOEA and Lexicase Selection significantly outperformed the other treatments at generation 100. However, the Single Bin treatment started outperforming all other treatments from generation 300 and onward. While CMOEA seemed to perform better than Lexicase Selection across all generations, the difference was not significant until generation 400. Lastly, the difference between CMOEA and Combined Target NSGA-II is no longer significant after generation 500.

For the most part, observations that held for performance on the training set also held for performance on the test set, both in terms of performance (Fig. 9b) and in terms of mazes solved (Fig. 10b). That is, CMOEA and Lexicase Selection performed significantly better than any of the other treatments in early generations, but the Single Bin treatment outperformed all other treatments in later generations (Fig. 9b) and solved significantly more mazes than all other treatments after the final generation (Fig. 10b). That said, there were two cases for which the results were not equivalent.

First, solutions found by Lexicase Selection did not seem to generalize as well as those found by CMOEA, as CMOEA significantly outperformed Lexicase Selection on the test set after 400 generations, even though there was no significant difference on the training set. One possible explanation is that an individual that exhibits the most commonly useful mode of behavior can start to dominate a population more quickly in Lexicase Selection than in CMOEA and that individuals that learn the most commonly useful mode of behavior first tend to generalize poorly later. For example, imagine a situation where 70 out of the 100 mazes can be solved just by following walls, 10 mazes can be solved by just moving towards the goal, and 20 mazes require a combination of both behaviors. In this case, an individual that learned wall-following behavior first would thus solve 70 mazes, while an individual that learned homing behavior first would solve only 10 mazes. In Lexicase Selection, an individual that solves 70 mazes is more likely to be selected than an individual that solves 10 mazes (assuming equal performance on the remaining 20 mazes) and, given this massive selective advantage, the genome of such an individual would quickly become fixed in the population. In CMOEA, on the other hand, there would always be a small number of bins in which the genome of such an individual would not become fixed, thus leaving room for other strategies, some of which may lead to individuals that generalize better. However, additional experiments will be required to test this hypothesis.

Second, solutions found by Combined-Target NSGA-II performed better than would be expected by their results on the training set, as Combined Target NSGA-II performed significantly worse than CMOEA on the training set, both in terms of performance (Fig. 9a) and in terms of mazes solved (Fig. 10a), but there is no significant difference on the test set (Figs. 9b and 10b). It is unclear why this is the case, but if it is true that a higher diversity in behavioral modes tends to lead to better generality, then the 101-dimensional Pareto-front maintained by Combined-Target NSGA-II may actually be very effective at keeping different behavioral modes in the population, albeit at the expense of fairly slow convergence.

[width=0.49]"figures/boxplot_training_mazes_solved_annotated".pdf(a)[width=0.49]"figures/boxplot_test_mazes_solved".pdf(b)

Figure 10: CMOEA was competitive with Combined Target NSGA-II, Lexicase Selection, and the Single Bin treatment, and substantially outperformed NSGA-II in terms of the number of training and test mazes solved after 1000 generations. (a) The number of training mazes solved by the individual with the highest training-set performance from each replicate, with ties broken arbitrarily. On the training set, CMOEA solved significantly more mazes than NSGA-II and Combined-Target NSGA-II, there was no significant difference between CMOEA and Lexicase Selection, and CMOEA solved significantly fewer mazes than the Single Bin treatment. (b) The number of test mazes solved by the individual with the highest training-set performance, with ties broken arbitrarily. On the test set, CMOEA solved significantly more mazes than Lexicase Selection and NSGA-II, there was no significant difference between CMOEA and Combined-Target NSGA-II, and CMOEA solved significantly fewer mazes than the Single Bin treatment.

These results raise the question of why the Single Bin treatment outperformed all the other treatments on this problem. The answer probably lies in the problem itself. With only two clearly identifiable modes of behavior, this maze navigation problem may not actually require an algorithm specialized in solving multimodal problems. That is, all the necessary stepping-stones may lie along the trajectory followed by an algorithm that attempts to greedily solve all subtasks simultaneously. As such, doing anything other than optimizing all subtasks simultaneously may be a waste of computational resources. By dedicating its entire population towards solving all mazes simultaneously, the Single Bin treatment can be more effective than algorithms that keep dedicating resources to individuals that solve only a subset of mazes.

This observation points us to one of the potential disadvantages of CMOEA and other algorithms that similarly focus on maintaining evolutionary stepping stones; many of the stepping stones probably become obsolete when the population converges close to the global optimum of the search problem. In these situations, CMOEA will spend a lot of computational resources in areas of the search space that are no longer relevant to the problem being solved. However, we argue that this is only a minor disadvantage for most practical problems, as it is unlikely that an evolutionary algorithm will actually get near the true global optimum for a real-world problem. In those problems, diversity and different stepping stones are likely to remain relevant for the entirety of an evolutionary run. However, even if this is not the case, one can switch from CMOEA with many bins to CMOEA with a single bin when there is a belief that additional bins are no longer beneficial. For example, one could switch after a predetermined number of generations or when performance gains slow down. Alternatively, one could estimate the contribution of each bin separately by measuring the number of generations since a child from this bin managed to survive in a different bin, and slowly remove the number of bins over time. Either way, these strategies would allow regular CMOEA to get close to the optimum, while Single Bin CMOEA can perform the final optimizations. Analyzing the effectiveness of such a version of CMOEA is a fruitful topic for future research.

V Conclusion

Many real-world problems are multimodal, from self-driving cars, which need to act differently depending on where they are, to medical robots, which require a wide range of different behaviors to perform different operations. Unfortunately, complex multimodal behavior may be difficult to learn directly and classic evolutionary optimization algorithms tend to rely on manual staging or shaping in order to learn such tasks. Such manual staging or shaping of a task requires extensive domain knowledge because finding the correct stepping stones and the order in which they should be traversed is a difficult problem, making it hard to estimate whether any particular staging or shaping strategy is truly optimal for the problem at hand. In this paper, we have introduced the Combinatorial Multi-Objective Evolutionary Algorithm (CMOEA), an algorithm specifically designed to solve complex multimodal problems automatically, without having to explicitly define the order in which the problem should be learned.

We have shown that CMOEA is effective at solving two different tasks: (1) a multimodal legged robot task and (2) a multimodal maze navigation task. We have also introduced a variant of NSGA-II, called Combined-Target NSGA-II, where we add a Combined-Target objective to the algorithm, which makes it resistant to the problem of having a high-dimensional Pareto-front. On the robotics task, CMOEA outperforms NSGA-II, Lexicase Selection, and a variant of CMOEA with only a single bin, and it is competitive with Combined-Target NSGA-II. On the maze domain, CMOEA outperforms NSGA-II and Combined Target NSGA-II, and it is competitive with Lexicase Selection and Single Bin CMOEA. Lastly, we have shown that, unlike the controls, CMOEA can effective incorporate secondary objectives that increase the evolvability of individuals, and these secondary objectives enable CMOEA to obtain state-of-the-art performance on a multimodal robotics task.

Vi Acknowledgments

We thank Christopher Stanton, Roby Velez, Nick Cheney, and Arash Norouzzadeh for their comments and suggestions. This work was funded by NSF CAREER award 1453549.

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S1 Experimental details

s1.1 Robot experiment

Below is a description of the settings of the multimodal robotics experiment. All settings are from [19].

s1.11 Performance evaluation

Performance for the different robotics tasks is calculated in six separate trials, one for each task. During each trial, the neural network input associated with the task being evaluated is set to 1, and the other inputs are set to 0. At the start of each trial, the robot is moved to its starting position of . Performance values on the forward (), backward (), and crouch () tasks are calculated as:

Where is the center of mass of the robot at time-step , and is the total number of time-steps in an evaluation. is a normalizing constant that was estimated based on the maximum performance reached on this objective in preliminary, single-objective experiments. Performance values on the turning tasks, turn-left () and turn-right (), are calculated as:

Where

is a vector pointing in the forward direction of the robot at time

, is the left angle between and , and is when the robot is upright (the angle between the robot’s up vector and the axis is less than ) and otherwise. In short, turning fitness is defined as the degrees turned while being upright, with a penalty for moving more than one unit away from the start. is a normalizing constant that was estimated based on the maximum performance reached on this objective in preliminary, single-objective experiments. Lastly, jump performance () is defined as:

Where is defined as . During the first half of the evaluation, this equation rewards the robot for jumping towards a target coordinate. During the second half of the evaluation, provided that the robot was able to jump at least half-way towards the target coordinate and that it is upright at the end of the trial, it can obtain additional fitness by returning to the starting position. This second half was added to encourage a proper landing. For bins with multiple subtasks, performance values are multiplied to obtain the fitness of individuals. The number of time steps was for the forward and backward tasks and for the other subtasks.

s1.12 Behavioral diversity

To calculate the behavior descriptor for each individual, we first recorded 6 training-task vectors by setting the input for one of the subtasks to 1, and then binarizing the values of the 18 actuators over 5 time-steps by setting all values

to 1 and other values to , which resulted in 6 binary vectors of 90 elements each. We then created a seventh majority vector by taking the element-wise sum of the 6 training-task vectors, and binarizing the result such that values were set to 1 and others were set to . Lastly, we XORed the majority vector with every training-task vector and concatenated the 6 resulting vectors to create the behavior descriptor. Distances between behavior descriptors were calculated with the hamming distance.

s1.13 Parameters

Parameter Value
Population size
CMOEA bin size
CMOEA number of bins
Add connection prob.
Delete connection prob.
Add node prob.
Delete node prob.
Change activation function prob.
Change weight prob.
Polynomial mutation
Minimum weight CPPN
Maximum weight CPPN
Minimum weight and bias CTRNN
Maximum weight and bias CTRNN
Minimum time-constant CTRNN
Maximum time-constant CTRNN
Activation function CTRNN
Table S1: Parameters of the multimodal robotics task.
Function Definition
Sine
Sigmoid
Gaussian
Linear (clipped)
Table S2: CPPN activation functions for the multimodal robotics task.

The network for the robotics task was represented by the HyperNEAT encoding, meaning that a CPPN genotype detemined the weights of the neural network controller [49]. The CPPN was evolved with the following NEAT mutation operators: add connection, delete connection, add node, delete node, change weight, and change activation function (probabilities are listed in table S1). The change weight and change activation function mutations were per connection and per node, respectively. Weights were mutated with the polynomial mutation operator [62]. The possible activation functions for the CPPN were: sine, sigmoid, Gaussian and linear, where the linear function was scaled and clipped. See table S2 for the definitions of each activation function. Nodes did not have an explicit bias, but a bias input was provided to the CPPN. After mutation, all weights were clipped so they would not fall outside the minimum and maximum values (see table S1). Initial CPPNs were fully connected without hidden neurons and with their weights and activation functions uniformly drawn from their allowable range. The CPPN had separate outputs for the weights, the biases, and the time-constants of the CTRNN, and those outputs were scaled to the fit the minimum and maximum values of the respective CTRNN parameter (see table S1). For the CTRNN, the activation function of the hidden neurons was scaled to to ensure inhibited neurons would not propagate signals.

s1.2 Maze experiment

Below are the settings for the maze experiment. Evolutionary algorithm and neural network settings are from [61].

s1.21 Performance evaluation

As mentioned in the main paper (Sec. IV-C1), performance of an individual on a maze was defined as its distance to the goal divided by the maximum possible distance to the goal for that maze, with a performance of 1 awarded if the robot would hit the goal itself. The equation is:

Here, is the distance between the robot and the goal at the end of the simulation, is the distance between the goal and the furthest corner, and is the radius of the circular robot. A maze would be considered solved as soon as , and the simulation would end immediately when this condition was met.

s1.22 Behavioral diversity

The behavior descriptor for a single maze was defined as the coordinate of the individual at the end of the simulation. The behavior descriptor of an individual over all mazes was a one dimensional vector composed of the final coordinates over all mazes. Distance between behavioral descriptors was defined as the Manhattan distance between those vectors.

s1.23 Parameters

Parameter Value
Population size
CMOEA bin size
CMOEA number of bins
Add connection probability
Delete connection probability
Rewire connection probability
Add node probability
Delete node probability
Change bias probability
Change weight probability
Polynomial mutation
Minimum weight
Maximum weight
Activation function
Table S3: Parameters of the maze navigation task.

In the maze experiment, the controller was a directly encoded recurrent neural network. The controller was evolved with to the following NEAT mutation operators: add connection, remove connection, rewire connection, add node, remove node, change weight, and change bias (probabilities are listed in table S3). The change weight and change bias mutations were per connection and per node, respectively. Weights and biases were mutated with the polynomial mutation operator [62]. After mutation, weights and biases were clipped to lie within their allowable range. To determine whether a rewire connection mutation would be applied, the operator would iterate over all connections, apply the rewire mutation with the indicated probability (Tab. S3), and stop iterating as soon as the mutation was applied once. The ordering of the connections in this process was arbitrary. When applied, it would change either the source () or the target (the other ) of the connection, and randomly draw a new source or target from the available candidates. Multiple connections with the same source and target would not be allowed. Initial networks were created with between 10 and 30 hidden neurons and between 50 and 250 connections and their weights and biases were uniformly drawn from the allowable range.

S2 Preliminary plots and additional analysis

For all plots shown in this SI, lines indicate the median over 30 replicates and shaded areas indicate the bootstrapped confidence interval of the median obtained by resampling 5000 times. Furthermore, the performance of a replicate at a particular generation is defined as the performance of the highest performing individual in the population at that generation. Symbols in the bar below each plot indicate that the difference between the indicated distributions is statistically significant ( on the Mann-Whitney-U test).

s2.1 Combined-Target Lexicase Selection Magnification

Because the performance of Combined-Target Lexicase Selection, both with and without selection for modularity, is very low relative to the other treatments, the difference between Combined-Target Lexicase Selection with selection for modularity and Combined-Target Lexicase Selection without selection for modularity is difficult to see in the original figure (main paper Fig. 7). Here we provide a magnification of that figure, focused on the Combined-Target Lexicase Selection treatments (Fig. S1). The figure shows that, while the extra objectives of maximizing genotypic and phenotypic modularity have little effect during the first 10 000 generations, Combined-Target Lexicase Selection without selection for modularity starts performing significantly better after roughly 13 000 generations. This demonstrates that, even though these secondary objectives technically increase the diversity of the Lexicase population, the fact that these secondary objectives are never combined with the primary objectives means that Lexicase Selection is unable to benefit from these secondary objectives.

Figure S1: Combined-Target Lexicase Selection without selection for genotypic and phenotypic modularity performs significantly better than Combined-Target Lexicase Selection with selection for these secondary objectives. This figure is a magnification of the first 15 000 generations of figure 7 from the main paper. Only the first 15 000 generations are plotted because the Lexicase Selection treatments were never extended to the full 75 000 generations. The Combined-Target Lexicase Selection treatment with modularity only includes 28, rather than 30, replicates because in two replicates the CPPN genomes grew so large that they became computationally intractable to run for the full 15 000 generations.

s2.2 Number of training mazes

In order to evolve general maze solving behavior, it is necessary to have a sufficiently large training set that allows individuals to learn the general behaviors necessary to solve mazes. In initial experiments, we tested CMOEA (without bin-sampling and with a bin size of 10), the Single Bin control, and Combined-Target NSGA-II with a training set of only 10 mazes (Fig. S2). All treatments reach perfect performance on the 10 training mazes before 250 generations, but the highest test set performance is around 0.9, demonstrating that the treatments do not perfectly generalize to other mazes. Interestingly, CMOEA is the first treatment to solve all 10 training mazes, yet it is has the lowest performance on the test set, while Combined-Target NSGA-II is the last treatment to solve all 10 training mazes, but it obtains the highest performance on the test set. This result suggests that, while attempting to maintain a Pareto-front over all objectives slows down progress on the combination of all objectives, such an approaches also preserves more general strategies than the bin-wise approach implemented in CMOEA. We argue that this issue can be resolved by providing CMOEA with a larger number of training mazes, as is presented in the main paper (Sec. IV-C2), as this makes it harder to overfit to the training set.

[width=0.49]"figures/10_maze_training"(a)[width=0.49]"figures/10_maze_test"(b)

Figure S2: A training set of 10 mazes does not lead to general maze-solving behavior on the test set. (a) On the training set of 10 mazes, all treatments quickly converge to the optimal value of 1, suggesting that all treatments can solve all training mazes. (b) On the test set of 1000 mazes, none of the treatments are able to reach a performance of 1, indicating that they are unable to solve all 1000 test mazes and suggesting that the 10 training mazes were insufficient to evolve general maze solving behavior.

Because there exist simple strategies that should generalize well to all mazes (e.g. general wall following behavior combined with homing behavior), we would expect evolution to be able to find such a strategy given a sufficiently large number of training mazes. That said, even when we increased the number of training mazes to 100, individuals that perfectly solved all 100 mazes still did not generalize to all 1000 unseen mazes from the test set, regardless of which algorithm produced those individuals (Fig. S3). Visualizing these individuals on the test mazes that they were unable to solve revealed that most failures happened because of rare sensor values, such as being in the corner of an unusually large room or seeing the goal through the doorway of an adjacent room. These rare sensor values caused inefficient behavior that resulted in the robot not being able to reach the goal in time. A video that includes some of these failed test cases is available at: www.evolvingai.org/cmoea. It is likely that a larger number of training mazes could help these individuals learn how to deal with these corner cases, but doing so is a topic for future work.

Figure S3: Even with a training set of 100 mazes, individuals do not perfectly generalize to 1000 mazes. Plot shows the number of test mazes solved by the individuals from each treatment that were capable of solving all mazes. Below each box is the number of individuals from the relevant treatment that were able to perfectly solve all 100 training mazes.

s2.3 CMOEA bin selection

In early experiments, we examined two survivor selection methods for within CMOEA bins. To ensure that CMOEA bins would not be populated by near-identical copies of the same individual, both selection methods included mechanisms that would be able to preserve within-bin diversity by maintaining individuals that would solve the same combination of tasks in different ways. The selection methods were: (1) NSGA-II’s non-dominated sorting with behavioral diversity as a secondary objective [20], explained in detail in the main paper, and (2) a selection method inspired by Novelty Search With Local Competition [43]. In this second variant, whenever an individual had to be removed in order to reduce the number of individuals in a bin back to the predefined bin size, the algorithm would find the two individuals in the bin that were closest to each other in terms of their behavior (calculated with the same distance metric used when calculating behavioral diversity), and out of those two it would remove the individual with the lowest fitness. As such, this method would promote a diverse set of individuals with fitness values that were high with respect to their behavioral neighborhood.

(a) (b)
Figure S4: (a) CMOEA combined with NSGA-II performed significantly better than CMOEA combined with a Novelty-Search-With-Local-Competition based method in preliminary experiments. (b) The neuron layout for this preliminary experiment was different from the neuron layout of the main experiment. The neurons are depicted in a cube extending from -1 to 1 in all directions. Inputs are positioned as described in the main paper, but the hidden layer consists of 6 rows of 5 neurons, where the rows form the radially distributed spokes of a circle perpendicular to the y-axis with a radius of 1. The first neuron in each row is positioned 0.5 units from the center, and the last neuron is positioned at 1.0 units from the center. Output neurons are positioned similarly, except that all output neurons are positioned at 1 unit from the center. Note that the output neurons for the knee and hip joints have overlapping positions; their positions are differentiated through the Multi-Spatial Substrate technique, which means that they have separate CPPN outputs [50].

In these experiments, NSGA-II’s non-dominated sorting algorithm performed significantly better than the selection method based on Novelty Search With Local Competition (Fig. S4a). One possible reason for this result is that the Novelty-Search-With-Local-Competition based method would lead to a higher diversity at the cost of a lower average performance inside each bin. Given that CMOEA already has its bins as a method of maintaining diversity, within bin performance may be more important than within bin diversity for the purpose of solving multimodal problems.

It is important to note that, for these experiments, the network layout was different from the layout used in the main paper (Fig. S4b). Specifically, input and output neurons were positioned in a radially symmetric pattern corresponding to the physical location of the sensors and actuators of the robot. Other preliminary experiments suggested that the grid based layout presented in the main paper performed better in general, so we did not perform any further experiments with the radial layout. However, as we have no reason to suspect that network layout would interact with the within-bin survivor-selection method, we did not repeat the bin-selection experiments with the grid-based layout.

S3 Control-treatment parameters

To ensure a fair comparison between CMOEA and the control treatments, the control treatments need reasonable parameters and settings. For most parameters, such as the mutation rate and the number of generations, we worked under the assumption that keeping them constant between treatments and not optimizing them for any particular treatment would allow for a fair comparison. However, for some specific parameters and settings, such as the population size and the addition of behavioral diversity, it was initially unclear how the control treatments would interact with these settings. As such, we performed several experiments examining how the control treatments would interact with these particular settings to ensure that we compared against the best possible version of the controls.

s3.1 Population size

In many evolutionary algorithms, including NSGA-II, the population size defines not just the number of individuals maintained by the algorithm at any point in time, but also the number of new individuals produced at every generation. This is not a practical choice for CMOEA, however, because the size of the population is a function of the bin size and the number of objectives to be optimized, which is often too large to be a feasible choice for the number of new individuals to create at every generation. As such, we have similarly decoupled the population size from the number of individuals created at each generation in our control treatments, and allowed our control treatments to have a population size that is larger than the number of offspring created at each generation. In preliminary experiments, we verified that choosing a larger population size did not have unintended negative effects on our NSGA-II based control treatments.

Increasing the population size in NSGA-II has two potential effects. First, it increases the number of Pareto-optimal individuals that are maintained, thus providing a better estimate of the Pareto-front at every generation. Based on this observation, a larger population size could increase the effectiveness of NSGA-II, as a better estimate of the Pareto-front implies a more diverse set of individuals that can serve as the stepping stones towards optimal solutions. However, a larger population size also means that sub-optimal individuals have a higher chance of surviving in the population, thus diluting the pool of parents that supply offspring for the next generation. Including more sub-optimal parents in the population can slow down the evolutionary process, and thus hurt the performance of NSGA-II.

Figure S5: In general, the control treatments perform slightly better with a larger population size. In the legend, CT NSGA-II stands for Combined-Target NSGA-II. The NSGA-II, Combined-Target NSGA-II, and Single Bin treatments all perform significantly better when their population is increased from 1000 (same as the number of offspring created at each generation) to 6300 (same as CMOEA on the six-tasks problem). The only exception is Combined-Target NSGA-II combined with a behavioral diversity objective, where a population size of 1000 seems to perform better than a population size of 6300, though the difference is not significant after 15 000 generations.

Given the large number of objectives presented in our research, we hypothesized that it would require a large population size before non Pareto-optimal individuals would start dominating the population, and thus that increasing the population size should increase NSGA-II’s performance on our problems. This hypothesis was confirmed by our preliminary experiments, which show that most control treatments with a population size of 6300 outperform the same control treatment with a population size of 1000 on the six-tasks robotics problem (Fig. S5). The one exception is when Combined-Target NSGA-II is combined with behavioral diversity, as Combined-Target NSGA-II Behav. Div. with a population size of 1000 outperforms Combined-Target NSGA-II Behav. Div. with a population size of 6300, though the difference is not significant after 15 000 generations. The reason for this effect is unclear but, because behavioral diversity tends to reduce the effectiveness of Combined-Target NSGA-II (Sec. S3.2), we decided not to include behavioral diversity in our NSGA-II controls, meaning that this effect was not important for the results presented in this paper. In light of these results, the population size for all control treatments presented in the main paper was set to be equal to the population size of CMOEA.

s3.2 Behavioral diversity

Figure S6: Behavioral diversity either has no significant effect or actively hurts the performance of the many-objective NSGA-II control treatments. In the legend, CT NSGA-II stands for Combined-Target NSGA-II. For Combined-Target NSGA-II with a population size of 6300, adding behavioral diversity significantly reduces the performance on the six-tasks robotics problem. This effect is also present when genotypic and phenotypic modularity (the CT NSGA-II Pop. 6300 Mod. treatments) are added as additional objectives. For Combined-Target NSGA-II and regular NSGA-II with a population size of 1000, the addition of behavioral diversity as an objective has no significant effect.

Previous work has demonstrated that adding behavioral diversity as an additional objective to NSGA-II can greatly increase its performance on problems with one or two objectives [63]. However, it was unclear whether these benefits would also be present on problems with six or more objectives. While a behavioral diversity objective could aid the evolutionary process on a many-objective problem by increasing the diversity of the population, and thus increasing the number of potential stepping stones, it is also possible that adding yet another dimension to the already high-dimensional space of a many-objective problem would only hurt the performance of the algorithm. To examine whether behavioral diversity would increase the performance of NSGA-II on a many-objective problem, we ran preliminary experiments with behavioral diversity added to different variants of NSGA-II on the six-tasks robotics problem.

The results show that adding behavioral diversity significantly hurts the performance of Combined-Target NSGA-II with a population size of 6300, both with and without modularity objectives (Fig. S6). Furthermore, behavioral diversity has no observable effect on regular NSGA-II or Combined-Target NSGA-II with a population size of 1000. These results suggest that behavioral diversity does not increase the performance of NSGA-II when applied to many-objective optimization problems. As such, the NSGA-II based controls presented in the main paper are implemented without behavioral diversity.