text stringlengths 128 2.05k |
|---|
Proof. With overwhelming probability, the algorithm is initialised within [MATH] 1-bits by Chernoff bounds. Since the distance to any trap point is linear and the probability that SBM flips at least [MATH] bits in a single iteration ( [MATH] ) is exponentially small, so is the probability of mutating into a trap point.... |
[MATH] with [MATH] being an arbitrarily small constant. Since the probability of finding a local optimum from the gradient points or |
Sp points is [MATH] in each step, the probability of not falling into the trap in [MATH] steps is less than [MATH] by the union bound. Overall, the total probability of finding the optimum within [MATH] steps is bigger than [MATH] |
5.3 On Trap Functions In , where Opt-IA was originally introduced, the effectiveness of the algorithm was tested for optimising the following simple trap function: |
[EQUATION] where [MATH] [MATH] [MATH] and the optimal solution is the [MATH] bit string. The reported experimental results were averaged over 100 independent runs each with a termination criterion of reaching [MATH] fitness function evaluations. For all of the results, the population size (i.e., [MATH] ) was 10 and [MA... |
Theorem 16 Opt-IA needs [MATH] expected fitness function evaluations to optimise Simple Trap with [MATH] for [MATH] and [MATH] Proof. |
Given that the number of 1-bits in the best solution is [MATH] , the probability of improving is at least [MATH] if the best solution has more than [MATH] 1-bits and [MATH] otherwise. By following the proof of Theorem , at least one individual will reach [MATH] or |
[MATH] in [MATH] fitness function evaluations in expectation as long as no individual dies due to ageing. The age of an individual reaches [MATH] only if the improvement fails to happen in [MATH] generations which happens with probability at most |
[MATH] since the improvement probability is at least [MATH] . This implies that the expected number of restarts by ageing is exponentially small. |
Given that Opt-IA was tested in also with the parameters suggested by Theorem 16 (i.e., [MATH] [MATH] [MATH] ), we speculate that either FCM was mistakingly not used or the stopping criterion (i.e., the total number of allowed fitness evaluations, i.e., [MATH] was too small. We point out that, for large enough [MATH] a... |
, with or without FCM. In any case, it is not necessary to apply both hypermutations and hypermacromutation together to efficiently optimise Trap functions as reported in |
. On the other hand, the inversely proportional hypermutation operator considered in would fail to optimise this function efficiently because it cannot flip [MATH] bits when on the local optimum. |
Not Allowing Genotype Duplicates None of the algorithms considered in the previous sections use the genotype diversity mechanism. In this section, we do not allow genotype redundancies in the population as proposed in the original algorithm |
This change potentially affects the behaviour of the algorithm. In the following, we will first consider the ageing operator in isolation with genotype diversity (i.e., no genotypic duplicates are allowed in the population). Afterwards we will analyse the complete Opt-IA algorithm with the same diversity mechanism (as ... |
6.1 [MATH] +1) RLS [MATH] with genotype diversity In this subsection, we analyse ( [MATH] +1) RLS [MATH] (Algorithm ) with genotype diversity for optimising Cliff for which ( [MATH] +1) RLS [MATH] without genotype diversity was previously analysed in Section The main difference compared to the analysis there is that ta... |
4: select [MATH] uniformly at random, 5: create [MATH] by flipping one bit of [MATH] 6: Hybrid ageing [MATH] 7: Selection ( [MATH] ), |
8: [MATH] 9: end while Algorithm 9 [MATH] +1) RLS [MATH] with genotype diversity (i.e., [MATH] ). Theorem 17 For constant [MATH] and [MATH] , the ( [MATH] +1) RLS [MATH] with genotype diversity optimises Cliff |
in expected [MATH] fitness function evaluations for any linear [MATH] Proof. By Chernoff bounds , with overwhelming probability the initial individuals are sampled with [MATH] 1-bits for any arbitrarily small [MATH] . Since the population size is constant and there is a constant probability of improving in the first mu... |
If there is a single locally optimal solution in the population, then with probability [MATH] this individual is selected as parent and produces an offspring with fitness [MATH] (i.e., one bit away) with probability |
[MATH] where [MATH] is the number of individuals already with fitness [MATH] In the next generation, with probability [MATH] one of the individuals on the local optimum or one step away from it is selected as parent and produces an offspring on either the local optimum or one bit away with probability at least [MATH] .... |
When the last inferior solution is replaced by an individual on the local optimum, the other individuals have ages in the order of [MATH] Thus, the probability that the rest of the population does not die until the youngest individual reaches age [MATH] , is at least [MATH] the probability that [MATH] individuals above... |
In the first generation when the last individual reaches age [MATH] , with probability [MATH] an offspring is created at the bottom of the cliff (i.e., with a fitness value of [MATH] ) and with probability [MATH] all the parents die together at that step and the offspring survives. The rest of the proof follows the sam... |
Overall, the total expected time to optimise Cliff is dominated by the time to climb the second OneMax slope which takes [MATH] steps in expectation. |
6.2 Opt-IA with genotype diversity In this subsection, we analyse Opt-IA (Algorithm ) with genotype diversity to optimise all the functions for which Opt-IA without genotype diversity was analysed in Section . The following are straightforward corollaries of Theorem and Theorem 16 . Since the ageing mechanism never tri... |
Corollary 2 The upper bounds on the expected runtime of Opt-IA with genotype diversity and ageing parameter [MATH] large enough for OneMax LeadingOnes [MATH] and [MATH] are as follows: |
[MATH] Corollary 3 Opt-IA with genotype diversity needs [MATH] expected fitness function evaluations to optimise Simple Trap with [MATH] [MATH] and [MATH] |
The following theorem shows the same expected runtime for Opt-IA with genotype diversity for HiddenPath as that of the Opt-IA without genotype diversity proven in Theorem 11 . However, we reduce the population size to be constant. The proof follows the main arguments of the proof of Theorem 11 . Here we only discuss th... |
Theorem 18 For [MATH] [MATH] [MATH] and [MATH] , Opt-IA with genotype diversity needs [MATH] expected fitness function evaluations to optimise HiddenPath |
Proof. We follow the analysis of the proof of Theorem 11 for Opt-IA without genotype diversity. Although the analysis did not benefit from genotype duplicates, not allowing them potentially affects the runtime of the events where the population takes over. The potentially affected events are: |
For [MATH] solutions to take over the population, the probabilities are different here since a new [MATH] solution will not be accepted if it is identical to any current [MATH] solutions. Here, after finding the first [MATH] solution, the rest are created and accepted by Opt-IA with probability at least [MATH] . Theref... |
The arguments about the expected time needed for [MATH] solutions to reach the same age after the takeover are the same as in the proof of Theorem 11 without genotype duplicates, the probability of creating another [MATH] is still [MATH] . Hence, the probability of creating two copies in the same generation is still un... |
In the proof of Theorem 11 , the expected time for [MATH] solutions to take over the population of recently initialised individuals is bounded relying on having multiple copies of one [MATH] and one [MATH] solution. This proof strategy cannot be applied with the genotypic diversity mechanism which only allows unique so... |
In order to create a unique [MATH] solution from another [MATH] solution, it is sufficient that the first bit position to be flipped has value [MATH] in the parent bit string and the second position to be flipped has value 0 in all [MATH] solutions currently in the population, including the parent. Such a mutation occu... |
Only the solutions on Sp and [MATH] solutions have better fitness than [MATH] solutions. The rest of the proof of Theorem 11 can still be applied if the only remaining non [MATH] solutions in the population are Sp solutions. So, we will only show that it is unlikely that an [MATH] solution will be sampled before the po... |
[EQUATION] Due to Chebyshev’s Inequality, the probability that such an event happens in [MATH] generations instead of its expectation, which is in the order of [MATH] , is at most [MATH] . Conversely, the probability that such a failure does not occur is [MATH] |
The path solutions have between [MATH] to [MATH] 1-bits, thus the probability that the hypermutation operator yields an [MATH] solution as output given an Sp or [MATH] solution as input is at most [MATH] . The probability that such a mutation occurs in [MATH] generations is at most [MATH] by the union bound. Considerin... |
The rest of the proof of Theorem 11 is not affected by genotype diversity. Conclusion We have presented an analysis of the standard Opt-IA artificial immune system. We first highlighted how both the ageing and hypermutation operators may allow to efficiently escape local optima that are particularly hard for standard e... |
Concerning ageing, we showed for the first time that the operator can be very efficient when coupled with SBM and hypermutations. To the best of our knowledge, the operator allows the best known expected runtime (i.e., [MATH] ) for hard Cliff functions (this expected runtime has recently been matched by a simple hyperh... |
). Afterwards, we presented a class of functions where both the characteristics of ageing and hypermutation are crucial, hence Opt-IA is efficient while standard evolutionary algorithms are inefficient even if coupled with one extra AIS operator (either cloning, ageing, hypermutation or contiguous somatic mutation). Fi... |
Our analysis shows that for easy problems for which local search strategies are efficient, using hypermutations and ageing may be detrimental. We have shown this effect for the simple OneMax and LeadingOnes functions for which we have proven a linear slow-down in the expected runtime compared to local search strategies... |
. However, such Fast AIS still suffer on HyperTrap c/8 On the other hand, for more complicated multimodal functions, with closer characteristics to the optimisation problems that occur in practice, we have shown several advantages of hypermutations and ageing to escape from local optima (i.e., Jump and Cliff ). Further... |
Overall, we believe this work is a significant contribution towards the understanding of which kind of problems it is advantageous to use artificial immune systems on rather than evolutionary algorithms and for which it is detrimental. Future work should focus on providing such advantages and disadvantages for classica... |
Acknowledgement: The research leading to these results has received funding from the EPSRC under grant agreement no EP/M004252/1. References |
# Source: arxiv 1804.01660 # Title: The structure of evolved representations across different substrates for artificial intelligence # Sections: all # Downloaded: 2026-03-03T01:55:21.074551+00:00 |
The structure of evolved representations across different substrates for artificial intelligence Abstract Artificial neural networks (ANNs), while exceptionally useful for classification, are vulnerable to misdirection. Small amounts of noise can significantly affect their ability to correctly complete a task. Instead ... |
Introduction The quest to recreate human-level intelligence within a computational substrate has gained traction in recent years, mostly due to the advent of deep learning methods and convolutional networks (Schmidhuber,, 2015 ; Bengio et al.,, 2015 ; Goodfellow et al.,, 2016 . While the success of these methods across... |
Representations. In Machine Learning, the term “representation” refers to the internal encoding of data in terms of a feature vector. As such, the term “representation” describes a transformation of the image data (for example) into a form that is more suitable for classification. Generally speaking, these representati... |
(Haugeland,, 1991 , quite unlike information present in sensors that always truthfully reflects the environment. Thus, in cognitive science, representations are context-dependent and refer to objects and concepts in the real world, as opposed to the ML term that instead refers to compressed versions of input data. In t... |
Information theory. To quantify the structure of representations, we need to be able to measure them. In previous work (Marstaller et al.,, 2013 we succeeded in giving an information-theoretic foundation to the term representation, as the information that internal brain states have about concepts in the world given the... |
[EQUATION] In Eq. ( ), [MATH] refers to the shared Shannon entropy between world states and brain states, and [MATH] stands for the information shared between world, brain, and sensors, something that Phillips and Singer (1997) have called “coherent information” But while Phillips and Singer assume that evolutionary pr... |
Brains, Dynamical Worlds, and Evolution. While a significant fraction of work in the Deep Learning field deals with the classification of static scenes (excepting Mnih et al., 2015, and similar work on Deep Q-Learning) vertebrate brains must compute sensory outputs in time, in a constantly changing world. Furthermore, ... |
Material and Methods Active Categorical Perception Task. In this classic task (Beer,, 1996 2003 ; van Dartel,, 2005 ; van Dartel et al.,, 2005 ; Marstaller et al.,, 2013 ; Albantakis et al.,, 2014 a mobile agent has to catch or avoid blocks that move towards it. The agent sits on a rail and can only move left or right,... |
The agent perceives the environment and approaching blocks by using four upward facing sensors. The agent is six units wide, and has two sensors on the left and two on the right side of its body leaving a blind spot in the middle. In order to perceive the size and direction of the approaching block properly, the agent ... |
Markov Brains. Markov Brains are networks of logical elements that connect inputs and outputs via internal states (Hintze et al.,, 2017 . Traditionally, the logical elements are deterministic or probabilistic logic gates. Here we only use deterministic logic gates, which we set up to have between 1 and 4 inputs and bet... |
A genome is used to encode the logic and connectivity of each gate. Point mutations (implemented with a per-site mutation rate of [MATH] ), deletions ( [MATH] times genome length), and gene duplications ( [MATH] times genome length) are applied every time an offspring is created by the genetic algorithm to populate the... |
LSTM networks. Long-short-term-memory (Hochreiter and Schmidhuber,, 1997 artificial neural networks (LSTM) implement recurrence in a special way (see Figure ). Instead of “just” looping outputs back to inputs, the LSTM uses two different streams of recurrence ( [MATH] and [MATH] ) as well as more complex modification a... |
Recurrent ANNs. An artificial neural network is typically organized in layers of nodes, where the top layer receives inputs and the bottom layer is interpreted as outputs (Russell et al.,, 2003 . In between can be arbitrarily many and arbitrarily large hidden layers, but all nodes from one layer are always connected to... |
In order to make this ANN recurrent (i.e., an RNN), [MATH] extra nodes are added to the input and output layer. After each update of the ANN the content of the recurrent nodes is copied from the output layer back to the input layer (see Figure ). |
Local and global representations. In order to quantify [MATH] we have to record the states of the sensors, brain (internal) states, as well as the state of the environment over time. While we can simply record the sensor and hidden (brain) states at every update, the world states require more attention. Although the wo... |
To determine how this knowledge (the knowledge about the world stored in the representations) is distributed across nodes, we also partition the brain states into the product random variable [MATH] , one for each hidden or recurrent node in the brain. This allows us to define concept-specific representations in particu... |
Smeared Representations. The representation Matrix shows how much information each node has about each concept. A hand-designed brain that could solve the block catching task would probably take advantage of discrete mappings between nodes and concepts. Specifically, such a brain would probably have one node per concep... |
This allows us to quantify how smeared representations are across nodes (columns), by summing over all nodes [MATH] and for all combinations of concepts [MATH] and [MATH] |
[EQUATION] Similarly, we compute the smearedness of concepts across nodes as the sum over all concepts [MATH] for all combination of nodes [MATH] and [MATH] |
[EQUATION] Robustness. Later, when considering the quality and dispersion of representations, we must also assess how robust these brains and their representations are to external noise. To measure how robust each brain is against noise, each evolved brain is tested over a range of possible noise levels applied to the ... |
Results. We evolved 400 independent populations of 100 agents for each of the three brain types (MB, LSTM, and RNN) for 10,000 generations. After that, the line of descent (Lenski et al.,, 2003 was reconstructed, and we confirm that evolution converged appropriately (see Figure ) over the first 7500 generations, meanin... |
We find that over the course of evolution, LSTMs and RNNs initially adapt faster than MBs but seem to struggle slightly to achieve maximum performance (see Figure ). However, these differences might be explained by the different effects mutations to the genome have in the different systems. |
We find that the total amount of representation for the top performers increases over evolutionary time as expected (see Figure ). We observed before (Marstaller et al.,, 2013 that RNNs change the amount of representation only slightly over time. The RNN is designed to maintain information about the environment, and th... |
Independently of the amount of each representation each brain type has, the brain types might represent the environment differently. We therefore analyzed the amount of representation each node has about each concept individually. This creates a matrix ( [MATH] ) where each row reflects a concept, and each column a hid... |
The difference in representation distribution might be explained by the way these brains work internally. In a MB the logical elements can connect arbitrarily, and the topology of the entire network can evolve, whereas in an LSTM and RNN the topology is predefined, and generally speaking, everything is connected to eve... |
We also find that how representations evolve is different between systems. When comparing, for example, how representations about the concept of the direction the blocks are falling changes over the course of generations, the difference becomes most apparent (see Figure 10 ). In Markov Brains specific hidden states are... |
To quantify how smeared representations are, we compute the overlap between concepts or nodes across the representation matrix [MATH] for all optimally performing agents (Equation and ). We find that representations are smeared across concepts as well as nodes, however in Markov Brains we find the least smeared represe... |
Now that we have shown that indeed representations are more smeared in LSTMs and RNNs than they are in Markov Brains, we will ask how this smeardness relates to performance and robustness. After all, it has been speculated that the deep-learned convolutional networks are easily fooled (Szegedy et al.,, 2014 because the... |
In addition we find that robustness negatively correlates with smearedness across the different brain types (see Figure 13 ). This is another indicator that more smeared representation do indeed make brains more vulnerable to noise, and that systems like Markov Brains, which have more discrete representations, also hav... |
Discussion. A hallmark of intelligence is to reconstruct reality even from a very small set of information. One obvious way to attain this is to use sensory data to trigger models of the world that are stored within a brain, to fill in the missing parts. The information-theoretic concept of representations quantifies p... |
Creating machines that perform reality reconstruction using minimal sensory data is a daunting task, to the point where it has even been suggested that we would be better off creating machines without representations (Brooks,, 1991 Here we showed that Markov Brains, LSTMs, and RNNs can be evolved to perform tasks that ... |
Might the vulnerability to fluctuations in “surface statistics” also be linked to “catastrophic forgetting” (McCloskey and Cohen,, 1989 ; Ratcliff,, 1990 , another plague of conventional systems based on the ANN paradigm? While we have not tested this aspect of information encoding, reasonable arguments can be made tha... |
We should point out that we did not test deep-learned LSTMs or RNNs, but instead used neuroevolution to optimize them, so a direct comparison between the representations that we measured in LSTMS and RNNS and those that might be expected in deep convolutionary networks is still outstanding. There are two reasons for us... |
Acknowledgements. This work was supported in part by Michigan State University through computational resources provided by the Institute for Cyber-Enabled Research. This material is based in part upon work supported by the National Science Foundation under Cooperative Agreement No. DBI-0939454. |
# Source: arxiv 1804.03980 # Title: Emergent Communication through Negotiation # Sections: all # Downloaded: 2026-03-03T02:01:02.083859+00:00 |
Emergent communication through negotiation Abstract Multi-agent reinforcement learning offers a way to study how communication could emerge in communities of agents needing to solve specific problems. In this paper, we study the emergence of communication in the negotiation environment, a semi-cooperative model of agen... |
Introduction How can communication emerge? A necessary prerequisite is a task that requires coordination between multiple agents to solve, and some communication protocol for the agents to exchange messages through (see a review by Wagner et al. ( 2003 on earlier work on emergent communication as well as recent deep re... |
In the context of linguistic communication, previous work on this subject has mainly studied the emergence of communication in cooperative games like referential games, variants of the Lewis signaling game (Lewis, 1969 , where messages are used to disambiguate between different possible referents (Goldman et al., 2007 ... |
In this paper, we study communication in the negotiation game (see Figure ), an established model of non-cooperative games in classical game theory (Nash, 1950b ; Neumann & Morgenstern, 1944 ; Nash, 1950a 1951 ; Schelling, 1960 ; Binmore et al., 1986 ; Peters, 2008 . In this game, agents are asked to establish a mutual... |
Work in classical game theory on negotiation typically uses simple forms of offer / counter-offer bargaining games that do not explicitly address the question of emergent communication (Rubinstein, 1982 Recent work on deep multi-agent reinforcement learning (MARL) has shown great success in teaching agents complex beha... |
We apply techniques from the MARL literature and train agents to negotiate using task success as the only supervision signal. We show that, when communicating via a task-specific communication channel with inherent semantics, selfish agents can learn to negotiate fairly and divide up the item pool to the agents’ mutual... |
The above results are obtained from paired agents interacting exclusively with each other. In more realistic multi-agent scenarios, agents may interact with many other agents within a society. In these cases, cheap talk can have a significant effect on the evolutionary dynamics of the population (Robson, 1990 as well a... |
Game setting 2.1 Negotiation environment The negotiation task is based on the set-up of Lewis et al. ( 2017 , itself based on DeVault et al. ( 2015 . Agents are presented with three types of items: peppers cherries and strawberries . At each round (i) an item pool is sampled uniformly, instantiating a quantity (between... |
Either agent can terminate the negotiation at any timestep [MATH] with a special action, which signifies agreement with the most recent proposal made by the other agent. For example, if agent B terminates the negotiation at [MATH] , agent A gets rewarded by [MATH] while agent B receives reward [MATH] , where [MATH] den... |
We impose an upper limit, [MATH] , on the number of negotiation turns allowed. If the agents get to the limit without agreement, then both agents receive no reward. We experimented with having a fixed upper limit of 10 turns. However, this led to a “first-mover” advantage: if the agents know in advance how many turns t... |
2.2 Communication channels To achieve negotiation, the agents need to communicate. One obvious communication protocol is to directly transmit the proposed division of items [MATH] . We refer to this as the proposal channel. This communication channel is task-specific and pertains to the action space of the agents used ... |
Non-bindingness: Messages sent via this channel do not commit the sender to any course of action, unlike directly transmitting a proposal which binds the sender to the proposal. |
Unverifiability: There is no inherent link between the linguistic utterance and the private proposal made, meaning that the agents could potentially lie. |
We are interested in whether and under what circumstances the agents can make use of this channel to facilitate negotiation and establish a common ground for the symbols. |
In our experiments, we consider 4 separate communication configurations: only the proposal channel, only the linguistic channel, both channels open, and no communication at all. If a channel is closed, we replace any messages in that channel with a fixed dummy symbol to ensure no information is transmitted. If the prop... |
2.3 Agent sociality and reward schemes While purely self-interested agents may learn to divide up items so that only each individual is satisfied, this may not lead to an optimal joint allocation of items, where each item goes to the agent with the highest utility for that item. We introduce a “prosocial” reward [MATH]... |
(Peysakhovich & Lerer, 2017 . For the selfish agent, [MATH] and [MATH] , while for the prosocial agent [MATH] 2.4 Agent architecture and learning |
At each timestep [MATH] , the proposer receives three inputs: The item context [MATH] , a concatenation of the item pool and the proposer’s utilities. |
The utterance [MATH] produced by the other agent in the previous timestep [MATH] . If the linguistic channel is closed, this is simply a dummy message. |
The proposal [MATH] made by the other agent in the previous timestep [MATH] . If the proposal channel is closed, this is again a dummy proposal. |
First, the discrete inputs are turned into dense vectors through an embedding table. We use two embedding tables, one for the item context and the previous proposal, and a separate one for the previous utterance. This is essential as the item context and the previous proposal have predefined numeric semantics, distinct... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.