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, this event happens with probability at most [MATH] The expected number of restarts until this event happens is at least [MATH] Since every restart clearly leads to at least one function evaluation, the expected optimisation time of local search algorithms on Hurdle problems is [MATH] |
4.2 (1+1) Evolutionary Algorithm In this section, we prove a tight bound [MATH] on the expected optimisation time of the (1+1) EA on the Hurdle problems with an arbitrary hurdle width [MATH] This result implies that the (1+1) EA is not efficient on Hurdle . Our rigorous analysis complements the non-rigorous arguments g... |
Note in particular that starting from an initial search point with at least [MATH] zeros, the first generation will end in a local optimum after at most [MATH] iterations of the local search with [MATH] Since this has a small |
additive contribution to the overall runtime, we can assume that a local optimum is the current search point. Theorem 2 The expected optimisation time of the (1+1) EA on the |
Hurdle problem with hurdle width [MATH] is [MATH] Proof. Assume that a local optimum [MATH] be the current search point with [MATH] zeros. Lemma yields that the nearest search points with a better fitness than [MATH] are all local optima with [MATH] zeros. For such a mutation we just need to flip at least [MATH] zeros ... |
[MATH] since [MATH] and [MATH] , Appendix C] The expected number of generations until the global optimum found is now bounded from above by |
[EQUATION] Now we need to calculate the sum [MATH] Let us consider the function [MATH] Since [MATH] [MATH] is a monotonically decreasing function in [MATH] . So we can approximate the upper bound of this sum using a method called approximation by integrals , i.e. |
[MATH] , Appendix A] Applying this method yields [EQUATION] Substituting this into ( ), we get [MATH] To derive the lower bound, we again focus on [MATH] for the same line of arguments as in Theorem |
Theorem 3 The expected optimisation time of the (1+1) EA on Hurdle problems with hurdle width [MATH] is [MATH] Proof. We first show that a search point with [MATH] zeros is reached with probability [MATH] . The initial number of zeros follows a binomial distribution with parameters |
[MATH] and [MATH] . As [MATH] , the probability that the initial search point will have at least [MATH] zeros is at least [MATH] (by symmetry of the binomial distribution). It may be possible to “jump over” search points with |
[MATH] zeros, i.e. to make a transition from [MATH] zeros to [MATH] zeros, so long as the Hurdle value with [MATH] zeros is less or equal to that of [MATH] zeros. However, Lemma 9 in |
states that the conditional probability of standard bit mutation reaching a search point with [MATH] zeros, given that a search point with at most |
[MATH] zeros is reached, is at least [MATH] Once the (1+1) EA has reached such a search point with [MATH] zeros, the expected remaining optimisation time is |
[MATH] as the optimum is the only search point with a higher fitness (see Lemma ) and the probability of jumping to the optimum from any such search point is |
[MATH] . By the law of total expectation, the expected optimisation time is at least [MATH] We remark that the (1+1) EA can be slightly sped up by increasing the mutation rate. As the above analysis has shown, the expected optimisation time is dominated by the time to locate the global optimum from a search point with ... |
for the (1+1) EA. However, even choosing an optimal mutation rate does not help much as, even if a mutation does flip exactly [MATH] bits, the mutation needs to select the right bits to flip, and the chance of choosing exactly the [MATH] bits that differ from the optimum is [MATH] leading to an expected time of at leas... |
[MATH] from such a local optimum (and [MATH] from random initialisation). This still results in a superpolynomial expected time if [MATH] , that is, [MATH] grows with [MATH] |
Memetic Algorithms Are Efficient We now show that, in contrast to local search on its own, and evolutionary algorithms, the (1+1) MA can find the global optimum efficiently, for both BILS and FILS . The main result of this section is as follows. Note in particular we consider BILS and FILS with local search depth [MATH... |
Theorem 4 The expected number of function evaluations of the (1+1) MA on Hurdle with any hurdle width [MATH] is [MATH] for BILS and [MATH] for FILS , both using [MATH] |
In order to prove Theorem , we first prove upper bounds of [MATH] and [MATH] , respectively, and then we prove lower bounds of [MATH] and [MATH] , respectively, showing that the upper bounds are asymptotically tight. |
To prove the upper bounds, we first provide an upper bound on the expected number of generations needed. This does not include the function evaluations made during local search, which will be bounded separately. |
Theorem 5 The expected number of generations of the (1+1) MA using BILS or FILS with [MATH] on Hurdle with any hurdle width [MATH] is |
[MATH] Proof. Assume a local optimum [MATH] is the current search point with [MATH] zeros in the bitstring. The fitness landscape around [MATH] is illustrated in Fig. . Lemma yields that the nearest search points with a better fitness than [MATH] are the local optima with exactly [MATH] zeros. Let us consider the situa... |
[MATH] to [MATH] ,…, [MATH] to [MATH] (say event B). Clearly, event A happens with probability [EQUATION] Given event A, event B occurs with unity probability, i. e. [MATH] as the local search always locates the next local optimum. Hence, the probability of reaching [MATH] from [MATH] |
is bounded from below by [MATH] , and the expected number of generations is at most [MATH] The expected number of generations until the global optimum is found is |
[EQUATION] We have, using [MATH] [EQUATION] Substituting into ( yields [MATH] In order to bound the number of function evaluations made during local search, we distinguish between local searches that result in a strict improvement over the previous current search point, and those that don’t. This is an example of the a... |
, Chapter 17.2] , where function evaluations are charged to one of two accounts and the total costs are bounded separately for each account. Adding the two bounds will yield an upper bound on the total number of function evaluations made during local search. |
We first bound the number of function evaluations spent in any improving local search. Lemma 3 Call a local search improving if it terminates with a search point that has a strictly better fitness than the current search point of the (1+1) MA; otherwise, the local search is called non-improving The number of function e... |
Proof. As Hurdle is a function of unitation, while no local optimum is found, local search will either decrease the number of ones in each iteration, or increase the number of ones in every iteration. In both cases a local optimum will be found after at most [MATH] iterations. Once a local optimum has been reached, at ... |
As BILS makes [MATH] function evaluations in every iteration, it makes at most [MATH] function evaluations in total. For FILS , after one iteration of the outer for loop (see Algorithm a local optimum will be found, and then [MATH] further evaluations are needed before it stops. |
The expected number of function evaluations for non-improving local searches can be bounded as follows. Lemma 4 In the setting of Theorem , the expected number of function evaluations spent by BILS and FILS during any non-improving local search is [MATH] |
Proof. The lower bound [MATH] is trivial as both local searches make at least [MATH] function evaluations before stopping. Let [MATH] denote the number of zeros in the current search point of the (1+1) MA and [MATH] denote the number of zeros in the search point after mutation, from which local search is called. |
If [MATH] , that is, [MATH] is a local optimum, [MATH] will lead to an improving local search (see Fig. ), and [MATH] may either be improving or go back to a search point with [MATH] ones in one iteration. If [MATH] then local search will make at most [MATH] iterations. |
If [MATH] , that is, [MATH] is not a local optimum, [MATH] leads to an improving local search, whereas [MATH] will stop after at most [MATH] iterations. |
In all these cases, local search is either improving, or it makes at most [MATH] iterations. Note that a necessary condition of mutating a search point with [MATH] ones into one with [MATH] ones is that at least [MATH] bits flip. The probability for this event is at most [MATH] . The expected number of iterations in a ... |
[EQUATION] The number of function evaluations made during a local search that stops after [MATH] iterations is at most [MATH] . Hence the expected number of function evaluations is at most [MATH] |
Putting the previous results together, we are now prepared to prove the upper bounds claimed in Theorem Proof of the upper bounds from Theorem |
By Theorem the expected number of generations is bounded by [MATH] . The expected number of function evaluations spent in any non-improving local search is [MATH] according to Lemma . Together, the number of function evaluations in all non-improving local searches is at most [MATH] |
By Lemma , the number of function evaluations in any improving local search is at most [MATH] for BILS and at most [MATH] for FILS. Since every improving local search ends in a local optimum with a better fitness than the current search point of the (1+1) MA, there can only be [MATH] improving local searches as this is... |
To prove the lower bounds from Theorem , we first show a very general lower bound of [MATH] for the (1+1) MA with BILS . It holds for all functions with a unique global optimum and may be of independent interest. |
Theorem 6 The (1+1) MA using BILS makes at least [MATH] function evaluations, with probability [MATH] and in expectation, on any function with a unique global optimum. |
Proof. It suffices to show the high-probability statement as the expectation is at least [MATH] We show that with probability [MATH] one of the following events occurs. |
[MATH] The (1+1) MA spends at least [MATH] generations before finding the optimum. [MATH] BILS makes a total of at least [MATH] iterations before finding the optimum. |
Each event implies a lower bound of [MATH] as each iteration of BILS makes [MATH] function evaluations, and each generation leads to at least one iteration of BILS |
In order for none of these events to occur, the (1+1) MA must find the optimum within [MATH] generations, using fewer than [MATH] iterations of BILS in total. For this to happen, one of the following rare events must occur: |
[MATH] the (1+1) MA is initialised with a search point that has a Hamming distance less than [MATH] to the unique optimum or [MATH] |
the initial search point has a Hamming distance of at least [MATH] to the optimum, and the algorithm decreases this distance to 0 during the first [MATH] generations, using fewer than [MATH] iterations of BILS |
The reason is that, if none of the events [MATH] and [MATH] occur, then this implies [MATH] By contraposition, [MATH] and [MATH] by the union bound. |
Event [MATH] has probability [MATH] by Chernoff bounds. For [MATH] , note that each iteration of local search can decrease the Hamming distance to the optimum by at most 1. Hence all iterations of BILS can only decrease the Hamming distance to the optimum by [MATH] in total, and so the remaining distance of [MATH] need... |
Together, we have by the union bound, [EQUATION] This completes the proof. Proof of the lower bounds from Theorem A bound of [MATH] for the (1+1) MA with BILS follows from Theorem |
We prove lower bounds [MATH] for both local searches by considering the remaining time when the (1+1) MA has reached a local optimum with [MATH] zeros. Theorem reveals that the (1+1) EA reaches such a local optimum with probability [MATH] , and it is obvious that the same statement also holds for the (1+1) MA. Then a l... |
From such a local optimum, the (1+1) MA with BILS has to flip at least two zeros in one mutation. Otherwise, the offspring will have at least [MATH] zeros, and BILS will run back into a local optimum with [MATH] zeros (or a worse local optimum). The probability for such a mutation is at most [MATH] , and the expected n... |
The same statement holds for the (1+1) MA with FILS : here it is necessary to either flip at least two zeros as above, or to create a search point with [MATH] zeros and to have FILS find a search point with [MATH] zeros as the first improvement. In the latter case, FILS will find the global optimum. The probability of ... |
In every generation, both BILS and FILS make at least [MATH] evaluations. Hence we obtain [MATH] as a lower bound on the number of function evaluations. |
Conclusions and Future Work We have provided a rigorous runtime analysis, comparing the simple (1+1) EA with the (1+1) MA using two local search algorithms, FILS and BILS , on the class of Hurdle problems. Our main results are tight bounds of [MATH] on the expected number of function evaluations of the (1+1) MA using B... |
[MATH] for the (1+1) MA using FILS . On the other hand, the (1+1) EA and local search algorithms on their own take time [MATH] and |
[MATH] , respectively. For [MATH] the latter times are superpolynomial, whereas the expected number of function evaluations for the (1+1) MA is always polynomial, regardless of the hurdle width [MATH] |
The Hurdle problem hence represents an illustrative problem where a hybrid algorithm drastically outperforms both of the individual search algorithms it contains, when these are run on their own. |
A surprising conclusion is also that the Hurdle problem class becomes easier for the (1+1) MA as the hurdle width [MATH] grows. The reason is that while the (1+1) EA has to jump to the global optimum by mutation, for the (1+1) MA it suffices to jump into the basin of attraction of the global optimum. Increasing the hur... |
More specifically, our analysis has shown that the (1+1) MA can efficiently reach a better local optimum by flipping two 0-bits during mutation, as then the resulting mutant is located in the basin of attraction of a better local optimum. The expected optimisation time is dominated by the time spent in the last local o... |
Amongst problems with a “big valley” structure, Hurdle has a favourable landscape for the (1+1) MA as local optima have a very small Hamming distance to search points in the basin of attraction of better local optima. This makes it easy to transition from one local optimum to another by mutation and local search. A pro... |
Another avenue for future work could be to rigorously analyse the expected running time of genetic algorithms with crossover on the Hurdle problem class, to investigate how their performance compares against that of the (1+1) MA. Experimental results in suggest that crossover provides a substantial advantage over the (... |
# Source: arxiv 1804.08676 # Title: Gesture based Human-Swarm Interactions for Formation Control using interpreters # Sections: all # Downloaded: 2026-03-03T02:01:50.961956+00:00 |
Gesture based Human-Swarm Interactions for Formation Control using interpreters Abstract We propose a novel Human-Swarm Interaction (HSI) framework which enables the user to control a swarm shape and formation. The user commands the swarm utilizing just arm gestures and motions which are recorded by an off-the-shelf we... |
keywords: Human-Swarm Interaction, Distributed Control, Dynamic Average Consensus, Formation Control, Human Interpretable Dynamics, Gesture Decoding, Hidden Markov Models, Kalman Filter, GUI Design |
Introduction Motivation. Due to recent advances in technology, the field of swarm robotics has become pervasive in the research community while slowly permeating to the industry. Although the coordination of multiple robots such as foraging, coverage, and flocking ( Olfati-Saber et al. ( 2006 ); Jadbabaie et al. ( 2003... |
Related Work. According to recent surveys on HSI ( Kolling et al. ( 2016 ) and human multi-agent systems ( Franchi ( 2017 ), humans either take a supervisory( Savla and Frazzoli ( 2012 ), direct( Setter et al. ( 2015 ), shared ( Franchi et al. ( 2012 ) or environmental( Wang and Schwager ( 2016 ) control role in an HSI... |
Gromov et al. ( 2016 . These works rely on proximal multi-modal interaction schemes which require complex hardware setup to interpret the human gestures, which is not practical for large scale swarms. We rely on a single wearable device without any other external electronics, which makes the implementation more practic... |
Statement of Contributions. We propose a novel HSI framework where we consider both a human agent and a dynamic swarm, with an interpreter acting as an bridge between the two. By means of it, the user can communicate their intentions intuitively and naturally, without having an in depth understanding of the swarm dynam... |
Paper Organization. Section presents preliminary concepts required to build our framework in Section , which includes the problem statement. We then describe our approach taken to solve each aspect of the problem statement in Section . Next, we state and discuss our results using our proposed approach in Section . We f... |
Preliminary Concepts This section introduces the basic notation and concepts used to construct our HSI framework. 2.1 Basic Notations |
We let denote the space of real numbers, and [MATH] the space of positive integers. Also, and M×n denote the [MATH] -dimensional real vector space and the space of [MATH] real matrices, respectively. We use |
[MATH] to denote the set of [MATH] dimensional polygonal shapes. In what follows, [MATH] are column vector of ones, [MATH] is the identity matrix, and [MATH] denotes a matrix of zeros. In what follows, [MATH] |
denotes the Euclidean norm. Given a matrix [MATH] its eigenvalues are denoted by [MATH] enumerated by their increasing real parts. The [MATH] row of a matrix [MATH] is denoted by [MATH] |
2.2 Graph Theory Notions Here, we introduce some basic Graph Theory notations which will be used in the sequel. Readers can refer |
Bullo et al. ( 2009 ); Godsil and Royle ( 2001 for more details on Graph Theory and its application to robotics. Consider a swarm of [MATH] agents in . Let [MATH] denote the position and velocity respectively of the |
[MATH] agent at time [MATH] . We denote by [MATH] the position of the whole swarm defined by [MATH] We model the communication among agents by means of an undirected |
[MATH] -disk communication graph [MATH] where [MATH] denotes the set of agents (vertices of the graph), and [MATH] , denotes the set of edges. In particular, [MATH] if and only if [MATH] . The entries of the associated adjacency matrix [MATH] become: |
[EQUATION] The neighbor set [MATH] for the [MATH] agent is given by [MATH] . Associated with [MATH] , we consider a weight-balanced weighting [MATH] , where [MATH] is the metropolis weight matrix corresponding to the communication graph [MATH] ; see Xiao and Boyd ( 2004 With [MATH] |
being the out degree of the [MATH] agent, [MATH] is given by: [EQUATION] Since we consider an undirected graph the matrix [MATH] is symmetric and doubly stochastic. From equation ( ) the graph |
[MATH] is balanced as [MATH] . We denote by [MATH] the diagonal degree matrix of [MATH] with [MATH] , the degree of node [MATH] , being the [MATH] diagonal entry of [MATH] . The Laplacian matrix [MATH] of the graph [MATH] is given by [MATH] , and the normalized laplacian matrix is given by [MATH] . Similarly the weight... |
[MATH] of the Laplacian matrix [MATH] . We can also express connectivity in terms of [MATH] and [MATH] . We can say that the respective graph is connected if [MATH] , and connectivity increases with increase in |
[MATH] Proposed Framework and Problem Formulation Here, we first introduce the various timescales involved in the interactions, and propose a new HSI framework, while providing a description of its components. Later, we identify the various problems to be solved to implement this framework. |
Timescales Involved. We assume that the interactions between the human, interpreter and the swarm, and the dynamic update of the swarm, may occur at time scales that go from coarser to finer resolution. In this way, human and interpreter may interact at discrete times that are a multiple of [MATH] , the interpreter and... |
[MATH] , while the swarm dynamic update times occur at multiples of [MATH] . In what follows, we identify [MATH] (resp. [MATH] , and [MATH] ) and we distinguish these integers as belonging to [MATH] (resp. [MATH] , and [MATH] .) We use the time variable [MATH] for the wearable device as it operates at a fast rate, simi... |
Proposed Framework. The user specifies their intentions which are translated by the interpreter and in turn communicated to the swarm. The human uses a wearable device called the MYO armband which observes the human intended swarm command. By means of it, the user specifies a desired formation shape [MATH] centroid [MA... |
[MATH] , where [MATH] The MYO armband receives the human intention [MATH] as Electromyography (EMG) signals [MATH] and Inertial Measurement Unit IMU signals [MATH] , where [MATH] |
The interpreter first uses a decoder (Section 4.1 ) to translate human intentions [MATH] into [MATH] . Then it translates [MATH] in [MATH] to desired relative agent positions [MATH] which best depicts the swarm shape. The swarm also has an operation mode |
[MATH] corresponding to [MATH] different communication ranges for each agent of the swarm. We have the notion of swarm operating cost involving [MATH] as a trade-off between network connectivity and network maintenance costs. We also introduce the notion of Human Interpretable Dynamics (HID), which represents easily un... |
Now, Given a desired formation [MATH] and the current state [MATH] the interpreter then determines the set of switching intermediate goals [MATH] with [MATH] [MATH] and |
[MATH] being the time horizon for switching. These intermediate goals [MATH] follow the HID and are optimal with respect to the swarm operating costs. These intermediate goals represent way points and intermediate shapes which will be communicated to the swarm. These parameters constitute the high-level commands that t... |
Problem 1 (Human Intention Decoder). Given the observations [MATH] and [MATH] from the Myo armband, design a decoder to get the desired human intention [MATH] |
Problem 2 (Behavior Specifier). Given the desired human intention [MATH] design an algorithm to produce the goal behavior [MATH] which can be understood by the swarm. |
Problem 3 (Planning Algorithm). Given the goal behavior [MATH] , generate the set of optimal intermediate behavior subgoals [MATH] with [MATH] , and [MATH] denoting the time horizon, and [MATH] |
which follow human-interpretable dynamics and minimize swarm operating costs. Problem 4 (Distributed Swarm Controller). Given the command [MATH] , for some [MATH] , design a distributed algorithm to drive the swarm to the intermediate shape [MATH] with scaling |
[MATH] , rotation [MATH] and centroid [MATH] using operation mode [MATH] from some initial position [MATH] Problem 5 (User Interface Design and Feedback). |
Develop a Graphical user interface (GUI) for the human to communicate their intention [MATH] to the interpreter and receive feedback about the decoded intention [MATH] and the state of the swarm. |
We propose solutions to the above problems in Section Technical Approach The following subsections describe the proposed solutions to the problems of Section |
4.1 Problem 1:Intention Decoding The user conveys their intention [MATH] through gestures and arm movement which are recorded by the Myo armband as EMG signals. There are [MATH] |
spatial EMG sensors on the Myo armband which generate EMG signals [MATH] at every time [MATH] . The 9 DoF IMU provides 3D acceleration, 3D angular velocity, and 3D angular orientation values. We only consider the planar angular velocity and orientation signals and hence the relevant IMU signals |
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