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Then, each of the input sequences is encoded using an LSTM (Hochreiter & Schmidhuber, 1997 , one for each input. This results in 3 fixed-size vectors: [MATH] [MATH] and [MATH] . These vectors are concatenated and fed through a feedforward layer, followed by a ReLU non-linearity (Nair & Hinton, 2010 , to give [MATH] , t... |
[MATH] is the policy for the termination action. If this action is taken by an agent, both agents receive reward according to the last proposal made by the other agent. This is a binary decision, and we parametrise [MATH] as a single feedforward layer, with the hidden state as input, followed by a sigmoid function, to ... |
[MATH] is the policy for the linguistic utterances. This is parametrised by an LSTM, which takes the hidden state of the agent as the initial hidden state. For the first timestep, a dummy symbol is fed in as input; subsequently, the model prediction from the previous timestep is fed in as input at the next timestep, in... |
[MATH] is the policy for the proposals the agent generates. This is parametrised by 3 separate feedforward neural networks, one for each item type, which each take as input [MATH] and output a distribution over [MATH] indicating the proposal for that item. |
The overall policy for the agent, [MATH] , is a combination of the separate policies. The action that the agent takes at turn [MATH] can be summarised by a triple [MATH] , where [MATH] is a binary variable indicating whether the agent took the termination action, [MATH] is a sequence of symbols produced from [MATH] , a... |
During training, each agent [MATH] tries to independently find the policy [MATH] that maximises the following objective function: |
[EQUATION] where [MATH] is the reward agent [MATH] receives from the trajectory [MATH] and [MATH] is an entropy regularisation term to encourage the agent to explore during training. Note that each agent has its own objective function, and the objectives of both agents are coupled by the trajectory sampled from both ag... |
(Williams, 1992 update rule with an exponentially smoothed mean baseline. Full hyperparameter details are in the appendix. Experiment 1: Can self-interested agents learn to negotiate? |
3.1 Experiment description In our first experiment, we test whether purely self-interested agents can learn to negotiate and divide up items fairly, and investigate the effect of the various communication channels on negotiation success. We train self-interested agents to negotiate for 500k episodes. Each episode corre... |
3.2 Results Self-interested agents can learn to negotiate fairly Figure 2(a) shows that self-interested agents learn to divide up the items fairly when they exchange proposals directly. We see that the proportion of total utility each agent receives is roughly equal, and above 50%, suggesting that the agents have learn... |
Table presents an illustrative example of two self-interested agents negotiating using the proposal channel. While the initial proposals made by both agents are over-optimistic, over the course of the negotiation, they compromise to less items. This allows them to achieve higher joint reward, as seen in Figure 2(b) , a... |
Self-interested agents do not appear to ground cheap talk When using the linguistic channel, agents do not negotiate optimally. Instead, the agents randomly alternate taking all the items, which is borne out by the oscillations in Figure 2(a) . Indeed, examination of the messages exchanged by self-interested agents sho... |
We conjecture that this finding is partially due to the fact that the agents operate in a non-iterated environment (i.e., they have no explicit memory of previous interactions). In the classic prisoner’s dilemma game, rational agents playing a single-shot version of the game learn to always defect, whereas rational age... |
Experiment 2 - Can prosocial agents learn to coordinate? 4.1 Experiment description In the previous experiment, we showed that self-interested agents can divide up items fairly such that each agent gets equal reward. However, this does not correspond to an optimal joint allocation where items go to the agent with the h... |
4.2 Results Cheap talk helps agents coordinate For prosocial agents with aligned interests, theoretical results suggest that communication using cheap talk is a Nash equilibrium. This is indeed what we observe, as the linguistic channel results in much better task success than any other communication scheme. The inform... |
Table illustrates an example of negotiation between the two prosocial agents. Interestingly, we observe that agent A over the course of the negotiation game adjusts its proposal (which is however hidden and not revealed) based on the utterances of agent B. |
Prosocial agents can still use the proposal channel to co-ordinate With random termination, agents communicating with the proposal channel do even worse than the no-communication baseline. However, when given the full 10 turns to exchange information (see Figure in the appendix), prosocial agents manage to outperform t... |
Analysis of linguistic communication 5.1 Symbol usage The symbol unigram and bigram distributions of messages exchanged by prosocial agents (see Figures 3(a) and 3(b) respectively) show that agent A, the agent who initiates the negotiation, is not transmitting any information using the linguistic channel. On the other ... |
In contrast, the selfish agents do not show evidence of grounded symbol usage. The unigram and bigram statistics show that most messages consist solely of strings of a single fixed symbol, regardless of the item context, and hence no information is being exchanged. |
5.2 Content of the messages To interpret the information that is transmitted through the linguistic channel, we use the message transcript and the item pool to create probe classifiers that predict the hidden utilities of each agent and the (hidden) accepted proposal. We use an LSTM to encode the sequence of symbols be... |
We also include two baselines. The first, reported as the “random baseline” in Table , attempts to predict the accepted proposal and agent utilities from a message transcript and item pool of all 0’s (indicating chance performance at the task). The other predicts the accepted proposal from the item pool and a message t... |
We can see from Table that not only is there variety in the symbols used, but the messages also have semantic content: they contain information about the hidden utilities of agent B, and about the proposal that was made. This shows that our agents have learned to give meaning to the symbols, and can use them to transmi... |
Experiment 3: A society of agents In more realistic scenarios, we learn and practice negotiation in environments with diverse agent populations and levels of prosociality. In this setting, maximising one’s reward requires identifying which agents are most useful in achieving one’s aims. In practice, this involves ident... |
6.1 Experimental protocol We trained a fixed agent against a community of 10 agents, with varying proportions of selfish and prosocial agents. For each training episode, we randomly sample one agent from the community, play a batch of negotiation games, and then update both agents. We do this for both fixed agent A (th... |
At test time, we care primarily about whether self-interested agents can exploit prosocial agents and whether prosocial agents can cooperate with other prosocial agents. Hence, we only test the fixed agent against the prosocial agents in the community. We generate 10 batches of 128 games each batch, and play each game ... |
If opposing agents are identifiable (i.e. each agent has a name-tag), the fixed agent receives this information as a one-hot vector, which is then used to look up an embedding table with trainable parameters with one embedding per opposing agent. This opponent embedding is then concatenated with the encoded inputs, bef... |
6.2 Analysis Identifying agents Our results show that providing agent ID information does help the fixed agent achieve its aims. This effect is particularly pronounced for a selfish fixed agent; here, providing ID information uniformly improves performance. For cooperative agents, the results are mixed; for a fixed age... |
We also computed the 2D PCA projections of the learnt opponent embeddings for a fixed agent A under a variety of channels and prosocial levels (see Figure ). Even in cases where the agent ID does not aid negotiation, we find that the embeddings cluster according to the reward scheme of the other agent. This shows that ... |
Community linguistic phenomena In one of our experimental settings, a community of prosocial agents developed a language and were able to use this to achieve better negotiation success. This is the outcome in the starred cell in Table . For a visualisation of the bigram usage statistics in each interaction pair in this... |
Moreover, we find that when prosocial agents make use of the linguistic channel, the communication protocol differs within the community. We calculate the Spearman correlation [MATH] between bigram ranks of different pairs of agents, all of them with the same fixed agent A, and show the results in Table . Even though a... |
Discussion We showed that by communicating through a verifiable and binding communication channel, self-interested agents can learn to negotiate fairly by reinforcement learning, using only task success as the reward signal. Moreover, cheap talk facilitated negotiation in prosocial but not in self-interested agents, co... |
A related paper from Lewis et al. ( 2017 takes a top-down approach to learning to negotiate by leveraging dialogue data. We demonstrated a bottom up alternative towards learning communicative behaviours directly from interaction with peers. This opens up the exciting possibility of learning domain-specific reasoning ca... |
Acknowledgements We would like to thank Mike Johanson for his insightful comments on an earlier version of this paper, as well as Karl Moritz Hermann and the rest of the DeepMind language team for many fruitful discussions over the course of the project. |
Appendix A Additional figures and tables Appendix B Hyperparameter details Embedding sizes, and all neural network hidden states, had dimension 100. We used the ADAM optimizer (Kingma & Ba, 2014 , with default parameter settings, to optimize the parameters of each agent. Each agent had a separate optimizer. We used a s... |
# Source: arxiv 1804.06173 # Title: Memetic Algorithms Beat Evolutionary Algorithms on the Class of Hurdle Problems # Sections: all # Downloaded: 2026-03-03T01:57:32.086427+00:00 |
Memetic Algorithms Beat Evolutionary Algorithms on the Class of Hurdle Problems Abstract Memetic algorithms are popular hybrid search heuristics that integrate local search into the search process of an evolutionary algorithm in order to combine the advantages of rapid exploitation and global optimisation. However, the... |
We provide a rigorous runtime analysis of a simple memetic algorithm, the (1+1) MA, on the Hurdle problem class, a landscape class of tuneable difficulty that shows a “big valley structure”, a characteristic feature of many hard problems from combinatorial optimisation. The only parameter of this class is the hurdle wi... |
Index terms— Evolutionary algorithms, hybridisation, iterated local search, local search, memetic algorithms, running time analysis, theory |
Introduction 1.1 Motivation Memetic Algorithms ( MAs ), also known as evolutionary/genetic local search or global-local search hybrids, are hybrid stochastic search methods that incorporate one or more intensifying local search algorithms into an evolutionary framework. The motivation behind this hybridisation is to cr... |
In , three advantages for memetic algorithms are pointed out: 1. Local search can quickly find solutions of high quality due to its rapid exploitation. |
2. Selection is only performed after local search has had a chance to improve on new offspring; this is beneficial for low-fitness offspring located in the basin of attraction of a high-fitness local optimum, as in a conventional evolutionary algorithm such low-fitness offspring would be removed by selection. This effe... |
3. Local search can include problem-specific knowledge; this is often possible since local search strategies are typically easy to design, even when it is hard to design a global problem-specific strategy |
A challenge when dealing with memetic algorithms and hybrid algorithms, in general, is that the search dynamics can be very hard to understand, in particular due to the interplay of different operators. It is not well understood when and why memetic algorithms perform well, when they do not, and how to design memetic a... |
There are many different variants of memetic algorithms, from algorithms that only rarely apply local search, with a fixed local search frequency to iterated local search algorithms where local search is applied in every generation |
In the latter scenario, local search turns all search points into local optima, and evolution acts on the sub-space of local optima. The hope is that mutation can lead a memetic algorithm to leave its current local optimum, and to reach the basin of attraction of a better one. |
We demonstrate that this strategy works very effectively on a class of problems introduced by Prügel-Bennett as example problems where genetic algorithms using crossover perform better than hill climbers. The Hurdle problem class (formally defined in Section ) is a function of unitation with an underlying gradient lead... |
with non-rigorous arguments based on simplifying assumptions, that led to approximations of the expected time for finding the global optimum. |
Here we provide a rigorous analysis for the expected optimisation time of the (1+1) EA: we give a tight bound of [MATH] for the expected optimisation time , confirming that the performance degrades very rapidly with increasing hurdle width. For hurdle widths growing with [MATH] [MATH] , this expected time is superpolyn... |
In contrast, we show that memetic algorithms perform very effectively on this problem class due to their combination of evolutionary operators and local search. We study a simple iterated local search algorithm called (1+1) MA with two different local searches, First-Improvement Local Search FILS ) and Best-Improvement... |
, and show that the (1+1) MA with BILS takes expected time [MATH] and the (1+1) MA with FILS takes expected time [MATH] to find the optimum. These times are polynomial for all choices of the hurdle width. |
Note that the term [MATH] decreases with the hurdle width, hence the surprising conclusion is that larger hurdle widths make the problem much harder for evolutionary algorithms, while making the problem easier for memetic algorithms. |
The Hurdle problem, albeit having been defined for a very different purpose , turns out to be an ideal example for showcasing the power of memetic algorithms and iterated local search. This finding is particularly significant in the light of “big valley” structures, an important characteristic of many hard problems fro... |
, where “many local optima may exist, but they are easy to escape and the gradient, when viewed at a coarse level, leads to the global optimum” |
The Hurdle problem is a perfect and very illustrative example of a big valley landscape. By explaining how the (1+1) MA easily solves the Hurdle problem class, we hope to gain insight into how memetic algorithms perform on big valley structures, which may help to explain why state-of-the-art memetic algorithms perform ... |
1.2 Related Work There are other examples of functions where memetic algorithms were theoretically proven to perform well (see Sudholt |
for a more extensive survey). In examples of constructed functions were given where the (1+1) EA, the (1+1) MA, and Randomised Local Search ( RLS ) can mutually outperform each other. The paper |
investigates the impact of the local search depth , which is often used to limit the number of iterations local search is run for. The author gives a class of example functions where only specific choices for the local search depth are effective, and other parameter settings, including plain evolutionary algorithms wit... |
Sudholt showed for instances of classical problems from combinatorial optimisation that memetic algorithms with a different kind of local search, variable-depth local search can efficiently cross huge fitness valleys that are nearly impossible to cross with evolutionary algorithms. Witt |
further analysed the performance of a memetic algorithm, iterated local search, for the Vertex Cover problem. Sudholt and Zarges |
investigated the use of memetic algorithms for the graph colouring problem. Finally, Wei and Dineen analysed memetic algorithms for solving the Clique problem, investigating the choice of the fitness function |
as well as the choice of the local search operator Gießen presented another example function class based on a discretised version of the well-known Rastrigin function. He designed a memetic algorithm using a new local search method called opportunistic local search , where the search direction switches between minimisa... |
Another line of research is work on hyperheuristics that combine different operators. Alanazi and Lehre demonstrated the usefulness of hyperheuristics for artificial functions, and Lissovoi, Oliveto, and Warwicker |
presented novel, provably efficient hyperheuristic algorithms. The difference to memetic algorithms is that while hyperheuristics typically apply one operator, while learning which operator performs best, memetic algorithms apply different operators, variation and local search, in sequence. The interplay of variation a... |
1.3 Outline The paper is structured as follows. Section introduces the (1+1) EA, (1+1) MA as well as the two local searches. The class of Hurdle problems are then formally defined in Section , which also includes detailed description about their properties. Section points out the inefficiency of the (1+1) EA and two lo... |
Preliminaries 2.1 (1+1) Evolutionary Algorithm In order to focus on the main differences between evolutionary algorithms and memetic algorithms, and to facilitate a rigorous theoretical analysis, we consider simple bare-bones algorithms from these two paradigms. |
The (1+1) EA is the simplest evolutionary algorithm, operating with a population of size one and using only mutation. The mutation operator flips each bit independently with mutation probability [MATH] , with the default choice being [MATH] where [MATH] is the length of the bitstring. The fitness function is defined as... |
returns a new bitstring resulting from flipping bits in [MATH] independently with probability [MATH] [MATH] repeat [MATH] if [MATH] then |
[MATH] until some stopping condition is fulfilled. Algorithm 1 (1+1) EA Practical implementations of Evolutionary Algorithms in particular and other search metaheuristics in general require to specify some stopping condition. The simplest is to stop when a fixed number of generations has been exceeded. The theoretical ... |
expected optimisation time of the algorithm, defined as the mean of the number of fitness (or function) evaluations performed by the algorithm until a global optimum is found. |
2.2 (1+1) Memetic Algorithm Algorithm outlines the typical procedure of the (1+1) MA, the simplest memetic algorithm. The algorithm consists of a population of one individual and produces an offspring in each generation by independently flipping each bit in the current search point with mutation probability [MATH] The ... |
[MATH] repeat [MATH] [MATH] if [MATH] then [MATH] until some stopping condition is fulfilled. Algorithm 2 (1+1) MA 2.3 Local Searches |
We consider the following two local searches in the context of the (1+1) MA. Both local searches are common practice and have also been analysed in |
input : a bitstring [MATH] for [MATH] iterations do create a random permutation Per of set [MATH] [MATH] for [MATH] to [MATH] do |
[MATH] if [MATH] then [MATH] [MATH] if BetterFound = false then return [MATH] return [MATH] Algorithm 3 FILS 2.3.1 First Improvement Local Search |
FILS ), shown in Algorithm , adapted from Wei and Dinneen , takes advantage of the first improvement it finds while searching the neighbourhood. The algorithm runs for [MATH] iterations. Bits are flipped according to a random permutation |
Per of length [MATH] (to avoid any search bias due to the choice of bit positions), and newly generated individuals are then scored by the fitness function. Here [MATH] returns a new bitstring resulting from flipping the [MATH] -th bit in [MATH] The current search point is replaced by the first neighbour found with a b... |
iterations of the outer for loop have been performed or after visiting all [MATH] neighbours of the current search point without any improvement. |
2.3.2 Best Improvement Local Search BILS ), shown in Algorithm , adapted from Wei and Dinneen , searches the whole neighbourhood and then picks a search point giving the best improvement. |
The algorithm runs for [MATH] iterations, and in each iteration a neighbour with the largest improvement in the fitness among all [MATH] neighbours of the current search point is picked to be the next search point. In order to keep track of the progress so far, it stores the best neighbour(s) and best fitness into CurB... |
CurBestFit , respectively. This means that whenever a neighbour with better fitness compared to CurBestFit has been found, the algorithm performs update on the two variables. At the end of an iteration, if there are more than one neighbours with the same fitness value that is better than [MATH] , then the next search p... |
input : a bitstring [MATH] for [MATH] iterations do [MATH] [MATH] for [MATH] to [MATH] do [MATH] if [MATH] then [MATH] [MATH] else if [MATH] then |
[MATH] if [MATH] then return [MATH] else [MATH] return [MATH] Algorithm 4 BILS Class of Hurdle Problems The Hurdle function class was introduced back in 2004 by Prügel-Bennett |
as an example class where genetic algorithms with crossover outperform hill climbers. Here we give a formal definition and discuss basic properties of the function that will be used in the subsequent analyses. |
The objective is to find a bitstring that maximises the fitness function [MATH] The value of the fitness function at a given bitstring |
[MATH] is [EQUATION] In this function, [MATH] is the number of zeros in the bitstring [MATH] [MATH] is called the hurdle width and is the only parameter of the Hurdle problems. Note that [MATH] may be a function of [MATH] . Finally, [MATH] is the remainder of [MATH] divided by [MATH] , while |
[MATH] is the ceiling function. Lemma 1 The global optimum for the Hurdle problem is [MATH] Proof. For every [MATH] [MATH] since both [MATH] and [MATH] cannot be negative. The equality happens if and only if both |
[MATH] and [MATH] equal zero, or equivalently [MATH] Note in particular that [MATH] can also be viewed as the Hamming distance [MATH] between the current solution [MATH] and the global optimum [MATH] The fitness landscapes close to the global optimum are shown in Fig. |
It can be clearly seen that the global optimum [MATH] coincides with the origin where both [MATH] and [MATH] In the following lemma, the term nearest refers to the scale of [MATH] , i. e. the most similar number of zeros. Note that this relates to Hamming distances as follows: any search point with [MATH] zeros has Ham... |
Lemma 2 Given a Hurdle problem with hurdle width [MATH] and a local optimum [MATH] as the current search point, the nearest search points with fitness larger than [MATH] are all search points with [MATH] zeros: [MATH] |
Proof. The current local optimum [MATH] contains [MATH] zeros where [MATH] (see Fig. ). Let us consider a search point [MATH] which is the nearest search point with [MATH] and [MATH] |
where [MATH] . Here, we exclude the case [MATH] as the next local optimum corresponds to [MATH] , and its fitness value is already known to be better than [MATH] |
Now we need to calculate the fitness values for two search points, [MATH] and [MATH] Note that [MATH] , and [MATH] , then [EQUATION] |
On the other hand, [MATH] , and [MATH] as we can rewrite [MATH] , then [EQUATION] Now we consider two different cases as follows. If [MATH] , then [MATH] and [MATH] otherwise, [MATH] , and [MATH] , then |
[MATH] For all [MATH] , we only have [MATH] if and only if [MATH] , and then [MATH] where [MATH] This result implies that [MATH] must be a local optimum. This proof also shows that the difference in the fitness values of two consecutive local optima is exactly one. |
Why Is Hybridisation Necessary? 4.1 Local Searches In this section, we show that local search algorithms in general are unable to optimise the Hurdle problems, unless the initial search point is chosen from a specific regions in the fitness landscape. |
Let [MATH] denote the number of zeros in the initial search point. It is obvious that if [MATH] , then the local search algorithm cannot locate the global optimum as it gets stuck at a local one forever. Otherwise, the global optimum can be found with some probability. However, if the local search is allowed to run onl... |
Theorem 1 The expected optimisation time of local search algorithms BILS and FILS with [MATH] restarting after [MATH] iterations of the local search, on Hurdle problems with hurdle width [MATH] for some constant [MATH] |
is [MATH] We focus on [MATH] for some constant [MATH] as, otherwise, the majority of search points would lie in the basin of attraction of the global optimum, resembling the function OneMax |
Proof of Theorem The local search algorithm flips one bit and only accepts new search points with strictly better fitness value compared to the current one in each iteration; therefore, the initial search point |
decides whether the global optimum can be reached. It is clear that this search point needs to have at most [MATH] zeros in order for the algorithm to be able to optimise the problem (see Fig. ). By Chernoff bounds |
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