Differential Evolution and Bayesian Optimisation for Hyper-Parameter Selection in Mixed-Signal Neuromorphic Circuits Applied to UAV Obstacle Avoidance
Llewyn Salt, David Howard, Giacomo Indiveri, Yulia Sandamirskaya
JJOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1
Differential Evolution and Bayesian Optimisationfor Hyper-Parameter Selection in Mixed-SignalNeuromorphic Circuits Applied to UAV ObstacleAvoidance
Llewyn Salt, David Howard,
Member, IEEE,
Giacomo Indiveri,
Senior Member, IEEE,
Yulia Sandamirskaya,
Member, IEEE,
Abstract —The Lobula Giant Movement Detector (LGMD) is aan identified neuron of the locust that detects looming objects andtriggers its escape responses. Understanding the neural principlesand networks that lead to these fast and robust responses can leadto the design of efficient facilitate obstacle avoidance strategiesin robotic applications. Here we present a neuromorphic spikingneural network model of the LGMD driven by the output ofa neuromorphic Dynamic Vision Sensor (DVS), which has beenoptimised to produce robust and reliable responses in the face ofthe constraints and variability of its mixed signal analogue-digitalcircuits. As this LGMD model has many parameters, we use theDifferential Evolution (DE) algorithm to optimise its parameterspace. We also investigate the use of Self-Adaptive DifferentialEvolution (SADE) which has been shown to ameliorate thedifficulties of finding appropriate input parameters for DE. Weexplore the use of two biological mechanisms: synaptic plasticityand membrane adaptivity in the LGMD. We apply DE and SADEto find parameters best suited for an obstacle avoidance system onan unmanned aerial vehicle (UAV), and show how it outperformsstate-of-the-art Bayesian optimisation used for comparison.
Index Terms —Differential Evolution, Bayesian Optimisation,Self-adaptation, STDP, Neuromorphic Engineering
I. I
NTRODUCTION S Tate-of-the-art robotic systems are less power efficientand robust than their natural counterparts. Indeed, a beeis capable of robust flight, obstacle avoidance, and cognitivecapabilities with a brain that only consumes 10 µW of power.On the other hand, vehicles in the DARPA Desert and Urbanchallenges consume around 1 kW of power [1]. Using natureas inspiration, neuromorphic engineers have attempted tobridge the power-consumption gap through hardware solu-tions [1]. Neuromorphic processors allow for the hardwareimplementation of spiking neural networks (SNNs) [2], [3].These mixed-signal analog/digital chips are low power andprovide an attractive alternative to current digital hardwareused in mobile applications such as robotics.Another successful neuromorphic solution is the DynamicVision Sensor (DVS) [4], [5]. The DVS is analogous to acamera, except instead of integrating light in a pixel array L. Salt is with the School of Information Technology and ElectricalEngineering, University of Queensland, Queensland, Australia.D. Howard is with the Autonomous Systems Lab, CSIRO, Queensland,Australia.G. Indiveri and Y. Sandamirskaya are with the Institute of Neuroinformatics,University of Zurich and ETH Zurich, Zurich, Switzerland. for a period of time and then converting it to an image, itdetects local changes in luminance at each pixel and transmitsthese as events pixel by pixel, as they are produced, and withmicrosecond latency [6]. This leads to a reduction in power,bandwidth, and overhead in post processing.Typically, high-speed agile manoeuvres, such as juggling,pole acrobatics, or flying through thrown hoops use exter-nal motion sensors and high powered CPUs to control theUAVs [7]–[9]. A system with sensors and image processingin-situ on the UAV is an essential step for autonomous UAVsystems in GPS restricted environments. Due to its hightemporal precision, the DVS also does not suffer from blurringas a standard-frame based camera when conducting high-speedmanoeuvres on an unmanned aerial vehicle (UAV) [10]. Thismakes it ideal as an on-board sensor for high-speed agilemanoeuvres.A model that has shown promise for collision avoidance inrobotics is the locust lobula giant motion detector (LGMD).The locust uses the LGMD to escape from predators bydetecting whether a stimulus is looming (increasing in size inthe field of view) or not [11]. It should be robust to translation,which is why it is an ideal candidate for obstacle avoidance.Previous implementations of this model used frame basedcameras and simplified neural models for embedded roboticapplications [11]–[13].Salt et al. [14] modified the LGMD model to use Adap-tive Exponential Integrate and Fire (AEIF) neuron equationswhich have been shown to be biologically plausible [15]and readily implementable in hardware neuromorphic proces-sors [2]. The LGMD Neural Network (LGMDNN) was alsomodified to make it compatible with the Reconfigurable On-Line Learning Spiking (ROLLS) neuromorphic processor [16].Coupling the LGMDNN with the EIF neural equations yields11 user-defined parameters after making simplifying assump-tions based on the constraints of the neuromorphic processor.Identifying promising parameter sets for robust functionaloperation of this model is the focus of this work.Optimising this parameter space is challenging as it con-tains up to 18 hyper-parameters that have complex inter-dependencies. Due to the computational resources and timerequirements involved in evaluation (approximately 1 to 4minutes per LGMDNN), an exhaustive search is infeasible.We are therefore motivated to investigate the use of efficient a r X i v : . [ c s . N E ] N ov JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 stochastic optimisation algorithms.Differential Evolution (DE) [17] is particularly suited to ourapplication. DE is a simple and efficient stochastic vector-based real-parameter optimisation algorithm with performance(at least) comparable to other leading optimisation algo-rithms [18], [19]. DE has only two user-defined rates [17],[20], [21], however their optimal values are problem specificand can drastically affect algorithmic performance [22]. Thishas prompted research into Self-Adaptation (SA), which al-lows the rates to vary autonomously in a context-sensitivemanner throughout an optimisation run. Self-Adaptive DE(SADE) has been shown to perform at least as well asDE on benchmarking problems [22], [23]. Importantly, SAhas been shown to reduce the number of evaluations re-quired per optimisation in resource-constrained scenarios withprotracted evaluation times [24], compared to non-adaptivesolutions [25]. Here, we compare DE and SADE to BayesianOptimisation (BO), which is also well suited to this task.Spiking networks are particularly amenable to a form ofunsupervised learning called Spike-Time Dependent Plasticity(STDP) [26], which allows synaptic weights to change au-tonomously in response to environmental inputs. STDP hasbeen shown to provide faster responses compared to non-plastic networks in dynamic environments [27], which mo-tivates our investigations into its use in our LGMD networks.Our hypothesis is that these adaptivity mechanisms arebeneficial to the optimisation process. To test this hypothesis,we evaluate the performance of our algorithms (DE, SADE,and BO, with and without STDP) when optimising loomingresponses in LGMD networks which are stimulated by (i)simple and (ii) complex DVS recordings on the UAV.The original contributions of this work are (i) developmentof an objective function that accurately describes the desiredLGMD behaviour, (ii) statistical comparisons of three leadingalgorithms in optimising LGMD response, and (iii) the firstuse of STDP and adaptation in spiking neuromorphic LGMDnetworks. II. M
ODEL
This section will describe the background for the model set-up and the specific equations that were used in the experiment.
A. LGMD
We implement the model as described by Salt et al. [14].The LGMD model consists of a photoreceptor (P), a summinglayer (S), an intermediate photoreceptor (IP), an intermediatesumming layer (IS), and an LGMD neuron layer. The interme-diate layers can be seen as analagous to sum-pooling layers indeep convolutional neural networks [28]–[30]. These layers areconnected by excitatory (E), inhibitory (I), and feed-forward(F) connections, which are modelled as AEIF neurons. Fig. 1shows the topology of the network [14].The feed-forward neurons (F) are intended to inhibit trans-lational motion. The inhibitory connections (I) from the pho-toreceptor to the summing layer inhibit non-looming stimuli.The weights of the inhibitory connections are assigned based SP LGMDISIPFig. 1: The neuromorphic LGMD model. Solid lines: excita-tory connections; Inhibitory connections; dashed lines: slowerinhibitions; dotted lines: faster inhibitions.on their distance from the central excitatory neuron. Thisconnection configuration spans the P layer like a kernel.The intermediate layers were added to make the model com-patible with the CXQuad neuromorphic processor describedin [3]. However, Salt et al. [14] found that the addition of theintermediate (sum-pooling) layer before the LGMD neuronincreased the performance of the network on all but slowcircular stimuli.
1) Adaptive Exponential Integrate and Fire Spiking Net-works:
We use Adaptive Exponential Integrate and Fire(AEIF) networks; the respective neuron equations follow (1)and (2): dVdt = − g L ( V − E L ) + g L ∆ T exp( V − V T ∆ T ) + IC , (1) I = I e − I iA − I iB − I ad , (2)where C is the membrane capacitance, g L is the leak con-ductance, E L is the leak reversal potential, V T is the spikethreshold, ∆ T is the slope factor, V is the membrane potential, I e is an excitatory current, I ad is the adaptation current, and I iA and I iB describe fast/slow inhibitory current [15]. Whena spike is detected ( V > V T ) the voltage resets ( V = V r ), andthe post-synaptic neuron receives a current injection from thepre-neuron firing given by: I al + = q al , (3) I ad + = b, (4) HELL et al. : BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 3 where the subscript l corresponds to the post-synaptic layer, q al is the current, b is the spike-triggered adaptation, and thesubscript a refers to either excitation or inhibition. To simplifythe model for embedded implementation, inhibitory currentswere bound as a ratio of the excitatory current: q il ( A | B ) = inh ( A | B ) l · q el , (5)where the ( A | B ) notation indicates either A or B type inhi-bition. The decay of the excitatory or inhibitory currents isdescribed by: dI a dt = − I a τ a , (6)where I a is the current and τ a is the time constant for thedecay. The subscript a refers to either inhibition or excitation.Finally, the decay of the adaptation current is described by: dI ad dt = a ( V − E L ) − I ad τ ad , (7)where a is the sub-threshold adaptation and τ ad is the timeconstant for the decay.Initially, the adaptation current is set to 0, which serves as acomparative baseline when investigating the use of adaptation. B. Spike Time Dependent Plasticity
Spike Time Dependent Plasticity (STDP) is a realisationof Hebbian learning based on the temporal correlations be-tween pre- and post-synaptic spikes. This synaptic plasticityis thought to be fundamental to adaptation, learning, andinformation storage in the brain [31], [32].Considering an arbitrary neuron, receipt of a pre-synapticspike closely before a post-synaptic spike increases efficacyof the synapse, with the reverse being true if a post-synapticspike is received in close proximity to a pre-synaptic spike.Long term potentiating (LTP, synaptic weight increase) of thesynapse occurs in the former case, long term depression (LTD,synaptic weight decrease) occurs in the latter case. Fig. 2shows the effect of the difference of the post- and pre- synapticspikes on the synaptic weight. STDP modifies the synapticFig. 2: The impact of STDP on the synaptic weights. If thepre-synaptic spike arrives before the post synaptic spike, thenthe strength of the weights is increased. If the post synapticspike arrives first than the strength of the synapse is weakened. current injection given in (3) by multiplying it by a weight w .If a pre-synaptic spike occurs then: I al + = wq al , (8) A pre + = ∆ pre, (9) w + = A post . (10)If a post-synaptic spike occurs then: A post + = ∆ post , (11) w + = A pre . (12)(13) A pre | post are the amount by which the weight w is strength-ened or weakened, and ∆ pre | post is a user-defined value forincreasing A pre | post each time a spike occurs. At each spikeevent: dA pre dt = − A pre τ pre , (14) dA post dt = − A post τ post . (15)Each time a spike occurs, A pre | post decays according to thefunction given above.III. O PTIMISATION T ECHNIQUES
In this Section, we describe the three optimisation tech-niques that we compare: DE, SADE, and BO, and how theyare applied to optimising the LGMDNN parameter space. Eachindividual is a parametrisation of the LGMDNN, given by: x = [ τ e , τ iA , τ iB , q eP , q eS , q eIP , q eIS , q eL , inhA S , inhB S , inhA L , [[ a , b , τ w adapt ]], (( τ pre , τ post , ∆ pre , ∆ post ))] A. Differential Evolution
DE is an efficient and high performing optimiser for real-valued parameters [17], [20]. As it is based on evolutionarycomputing, it performs well on multi-modal, discontinuousoptimisation landscapes. DE performs a parallel direct searchover a population of size
N P , where each population member x is a D -dimensional vector. for each generation G : x i,G = 1 , , . . . , N P. (16)We use the canonical DE/rand/1/bin to describe the al-gorithmic process. The initial population is generated fromrandom samples drawn from a uniform probability distributionof the parameter space, bounded to the range of the respectivevariable. These bounds are shown in Subsubsection IV-B2.The fitness of each vector in the population is then calculatedby the objective function, as described in Section IV-A.In each generation, each parent generates one offspring byway of a ‘donor’ vector, created following Eq. (17): v i,G +1 = x r ,G + F · ( x r ,G − x r ,G ) , (17)where r (cid:54) = r (cid:54) = r (cid:54) = i ∈ [1 , N P ] index randomunique population members, and differential weight F ∈ [0 , determines the magnitude of the mutation. The final offspringis generated by probabilistically merging elements of the JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 parent with elements of the donor vector. The new vector u i,G +1 = ( u i,G +1 , . . . , u Di,G +1 ) is found by: u ji,G +1 = (cid:40) v ji,G +1 , if rand ( j ) ≤ CR or j = R,x ji,G , otherwise , (18)where j ∈ (1 , . . . , D ) , CR ∈ [0 , is the crossover rate, rand ( j ) ∈ [0 , is a uniform random number generator, and R ∈ (1 , . . . , D ) is a randomly chosen index to ensure thatat least one parameter changes. The value of index i is thencalculated as: x i,G +1 = (cid:40) u i,G +1 , if f ( u ji,G +1 ) > f ( x i,G ) ,x i,G , otherwise . (19)Once all offspring are generated, they are evaluated on theobjective function, and selected into the next generation if theyscore better than their parent. Otherwise, the parent remainsin the population. B. Self-Adaptive DE
Storn and Price [17] showed that DE/rand/1/bin outper-formed several other stochastic minimisation techniques inbenchmarking tests whilst requiring the setting of only twoparameters, CR and F . Many different mutation schemeswere subsequently suggested for DE, named following theconvention DE/x/y/z , where x denotes the vector to bemutated (in this case a random vector), y denotes the numberof vectors used, and z denotes the crossover method (bincorresponds to binomial).Brest et al. [22] present the first widely-used self-adaptiverate-varying DE, which is expanded by Qin et al., to allowthe mutation scheme to be selected (from four predeter-mined schemes) alongside the rates [23], based on previously-successful settings. Different rates/schemes are shown to workbetter on different problems, or in different stages of a singleoptimisation run. The strategy for a given candidate is selectedbased on a probability distribution determined by the successrate of a given strategy over a learning period LP . A strategy isconsidered successful when it improves the candidate’s value.In the interest of brevity, we refer the interested reader to [23]for a full algorithmic description.Rates are adapted as follows. Before G > LP , CR iscalculated by randomly selecting a number from a normaldistribution, N (0 . , . , with a mean of 0.5 and a standarddeviation of 0.3. Afterwards it is calculated by a randomnumber from N ( CR mk , . where CR mk is the median valueof the successful CR values for each strategy K . F is simplyselected from a normal distribution N (0 . , . , which willcause it fall on the interval [ − . , . with a probability of0.997 [23]. C. Bayesian Optimisation
Bayesian optimisation (BO), e.g. [33], is a probabilisticoptimisation process that typically requires relatively fewevaluations [34]–[36], although the evaluations themselves are computationally expensive. When parallelised, BO is shown tolocate hyper-parameters within set error bounds significantlyfaster than other state-of-the-art methods on four challengingML problems [37], in one case displaying 3% improvedperformance over state-of-the-art expert results. As such, BOcan be considered an extremely challenging optimiser as acomparator for DE and SADE, and as SNNs have many hyper-parameters, they are ideal candidates for optimisation.BO assumes the network hyper-parameters are sampledfrom a Gaussian process (GP), and updates a prior distributionof the parameterisation based on observations. For LGMDNN,observations are the measure of generalization performanceunder different settings of the hyper-parameters we wish tooptimise. BO exploits the prior model to decide the next setof hyper-parameters to sample.BO comprises three parts: (i) a prior distribution, (ii) anacquisition function, and (iii) a covariance function.
1) Prior:
We use a Gaussian Process (GP) prior, as itis particularly suited to optimisation tasks [34]. A GP isa distribution over functions specified by its mean, m , andcovariance, k , which are updated as hyper-parameter sets areevaluated. The GP returns m and k in place of the standardfunction f : f ( x ) ∼ GP ( m ( x ) , k ( x, x (cid:48) )) . (20)
2) Covariance Function:
The covariance function deter-mines the distribution of samples drawn from the GP [33],[37]. Following [37], we select the 5/2 ARD Mat´ern kernel(21), where θ is the covariance amplitude. k m ( x i , x j ) = aexp ( − (cid:113) r ( x i , x j )) , (21)where: a = θ (1 + (cid:113) r ( x i , x j ) + 53 r ( x i , x j )) , (22)where: r ( x i , x j ) = x i − x j θ . (23)
3) Acquisition Function:
An acquisition function is a func-tion that selects which point in the optimisation space toevaluate next. We evaluate the three acquisition functions,which select the hyper-parameters for the next experiment:Probability of Improvement (PI), Expected Improvement (EI)[34], and Upper Confidence Bound (UCB) [38] — see [33] forfull implementation details. Briefly, the PI can be calculated,given our current maximum observation of the GP, x + , by: P I ( x ) = P ( f ( x ) ≥ f ( x + ) + ζ )= Φ( µ ( x ) − f ( x + ) − ζσ ( x ) ) . (24)Here, ζ ≥ is a user-defined trade-off parameter that balancesexploration and exploitation [39]. EI maximises improvement with respect to f ( x + ) : I ( x ) = max { , f ( x ) − f ( x + ) } . (25) HELL et al. : BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 5
The new sample is found by maximising the expectation of I ( x ) : x = arg max x E ( I ( x ) |{ X , F } ) . (26)Following [40], EI is evaluated by: EI ( x ) = (cid:40) a + σ ( x ) φ ( Z ) , if σ ( x ) > , , otherwise ; (27) a = ( µ ( x ) − f ( x + ) − ζ )Φ( Z ); (28) Z = (cid:40) µ − f ( x + ) − ζσ ( x ) , if σ ( x ) > , , otherwise , where φ and Φ correspond to the probability and cumulativedistribution functions of the normal distribution, respectively. UCB maximises the upper confidence bound:
U CB ( x ) = µ ( x ) + κσ ( x ) , (29)where κ ≥ balances exploration and exploitation [37], andis calculated per evaluation as: κ = √ ντ t , (30)where ν is the user tunable variable and: τ t = 2 log( t d +2 π δ ) . (31) δ ∈ { , } , d is the number of dimensions in the function and t is the iteration number.IV. T EST P ROBLEM
This section will outline the rationale of the objective func-tion, the experimental set-up, and assumptions. It is importantto note that the motivation behind the model simplificationsand objective function is for the work to be directly transfer-able to the neuromorphic processors described in [16] oncethey are readily available.
A. Objective Function
The function to optimise was formulated as a weightedmulti-objective function [41]. We direct the interested readerto [42] for a detailed formulation of the objective function, F Acc ( λ ) , which is calculated by: F Acc ( λ ) = × F ( λ ) , if F ( λ ) > and Acc = 1 ,Acc × F ( λ ) , if F ( λ ) > , , if Acc = 1 and F ( λ ) < ,F ( λ ) , otherwise . (32)Here, Acc is the accuracy of the LGMDNN output and F ( λ ) is the fitness function. The LGMD network is said to havedetected a looming stimulus if the output neuron’s spike rateexceeds a threshold SL . This can be formalised by: Looming = (cid:40) True , if SR > SL,
False , Otherwise , (33) where SR can be calculated by: SR = t +∆ T (cid:88) i = t S i , (34)where ∆ T is the time over which the rate is calculated and S i is whether or not there is a spike at time i ; a spike is definedto occur if at time i the membrane potential exceeds V T .The looming outputs are categorised into true positives(
T P ), false positives (
F P ), true negatives (
T N ), and falsenegatives (
F N ). Output accuracy is then:
Acc = T P + T NT P + T N + F P + F N . (35) F ( λ ) can be calculated by: F ( λ ) = Score ( λ ) + SSEOS ( λ )2 , (36)where Score is a scoring function based on the timing ofspiking outputs and
SSEOS is the sum squared error of theoutput signal.The score is calculated by difference of the penalties’ andreward functions’ sums over the simulation:
Score ( λ ) = N (cid:88) i =1 R i − N (cid:88) i =1 P i . (37)The reward can at a given time can be calculated by: R ( t ) = (cid:40) k exp( t ∆ t ) + 1 , if looming and spike , , otherwise . (38)The punishment can be calculated by: P ( t ) = ( l − c ) t ∆ t + c, if not looming andspike and t < ∆ t ;( l − c ) − ( t − ∆ t ) ∆ t + c, if not loomingand spike and t > ∆ t ;0 , otherwise . (39)In these equations t and ∆ t remain consistent with the otherobjective functions and k , l , and c are all adjustable constantsto change the level of punishment or reward.To calculate SSEOS ( λ ) , the signal was first processed sothat every spike had the same value. This was done so that theideal voltage and the actual voltage would match in loomingregions, as the voltage can vary for a given spike. Ultimately,the only criterion is that the voltage has crossed the spikingthreshold. In the non-looming region the ideal signal was takento be the resting potential, which was negative for the AEIFmodel equation. The signal error was calculated at every timestep as: SSEOS ( λ ) = − N (cid:88) i =1 ( V iactual − V iideal ) . (40) V actual could be obtained directly from the state monitorobject of the LGMD output neuron in the SNN simulator(Brian2). N in this case is the length of the simulation and JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 i indicated each recorded data point at each time step of thesimulation. V ideal was given by: V ideal = (cid:40) V spk , if looming ,V r , otherwise , (41)where V spk is the normalised value given to each spike and V r is the resting potential.Overall, this gives an objective function that takes intoaccount the expected spiking behaviour, whilst penalisingthe system for deviating from plausible voltage values andrewarding it for accurately categorising looming and non-looming stimuli. B. Experimental Set-up
The model was set-up using Brian2 spiking neural networksimulator [43].
1) Data Collection:
Data was collected using a DVS in-situon a quadrotor UAV (QUAV). Two types of data were col-lected: simple and real world. The simple data was synthesisedusing PyGame to generate black shapes on a white backgroundthat increased in area in the field of view of the DVS. Thisincluded: a fast and slow circle, a fast and slow square, anda circle that loomed then translated while increasing in speed(composite). The laptop playing the stimuli was placed in frontof the hovering QUAV and the stimuli were recorded. Thiswas done to maintain any noise that might be generated bythe propellers of the QUAV.To challenge the model, real stimuli were also recorded: awhite ball on a black slope was rolled towards the DVS from3 different directions; a cup was suspended in the air and theQUAV flew towards and away from the QUAV; and a handwas moved towards and away from the DVS on the hoveringQUAV. These are increasing in complexity in terms of theshapes that are presented.Two looming and two non-looming events (˜25s) from thecomposite stimulus were used to optimise the model and thenthe optimised model was evaluated on the other stimuli. Thestimuli were chosen to show that the model generated is bothshape and speed invariant.
2) Hyper-parameter Constraints:
The hyper-parameterswere all continuous and could range from zero to infinity.There were many regions of the parameter space that werenot computable even when using a cluster with 368GB ofRAM. To mitigate some of the computational difficultiesthe temporal resolution of the simulation was set to 100 µs .Bayesian optimisation using the expected improvement utilityfunction (BO-EI) was used over 20 eight hour runs to findfeasible regions of the optimisation space. C , g L , E L , V T , and ∆ T were set as constants as theyappeared to have little to no co-dependencies and model per-formance was not impacted by setting these values and appro-priately optimising the other parameters [42]: C = 124 . pF , g L = 60 . nS , E L = − . mV , V T = − . mV , and ∆ T = 6 . mV .Table I shows the constraints found for the rest of the hyper-parameters. TABLE I: The constraints of the optimisation space. Parameter Min Max τ e τ iA τ iB q eP q eS q eIP
84 230 q eIS
119 270 q eL
29 472 inhA S inhB S inhA L a b
40 141 τ w adapt τ pre τ post ∆ pre ∆ post
3) Comparing Optimisers:
SADE, DE, BO-EI, BO-PI, andBO-UCB were evaluated thirty times on the same inputstimulus, so that they could be statistically compared usinga Mann-Whitney U test. The input stimulus included a blackcircle on a white background performing a short translation tothe right, followed by a half loom, a full recession, and thena full loom (The first two non-loom to loom transitions ofthe composite stimulus). The stimulus was selected because itconsisted of a 50:50 looming to not looming ratio. The valuesof the user defined parameters were selected as: • BO-EI and BO-PI: ζ = 0 . ; • BO-UCB: κ = 2 . ; • DE:
N P = dim , F = 0 . , CR = 0 . ; • SADE: LP = 3 , N P = dim , where dim is the numberof hyper parameters.The tests were run using the non-adaptive and non-plasticmodel with the bounds from Table I. They were defined ashaving converged if they had not improved for × N P eval-uations. This meant three generations for the DE algorithmsand the same number of BO evaluations. The population sizewas two more than what is recommended by [21] for the DEalgorithm. This size was chosen as it is relatively small andtime was an issue. The short convergence meant that the SADEalgorithm needed to have a short LP. The processor time wasnot included as a metric for this as the tests were run onthree different computers so the results would not have beencomparable.
4) Comparing Models:
Once the best optimiser was found(a comparison of optimisers can be found in Subsection V-A),the best performing optimiser, SADE, was used to optimisethe following models:
LGMD:
Neuromorphic LGMD; A: LGMD with adaptation; P: LGMD with plasticity;
HELL et al. : BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 7
AP:
LGMD with adaptation and plasticity.The SADE variables were set to: LP = 3 and N P =10 dim . The optimisation process was run 10 times and the bestoptimiser from these ten runs was selected. The model wasthen tested on each input case for ten looming to non-loomingor non-looming to looming transitions. The performance ofeach model is reported in Subsection V-C.Plasticity was found to degrade the performance sometimesso we experimented clamping it from 0% to 100% of theoriginal synaptic strength. This allowed it to range from zeroto double the original values when at 100% to no variation at0%. V. R
ESULTS AND D ISCUSSION
The results are split into two subsections. First, we will com-pare the optimisers and then we will compare the addition ofadaptation, plasticity, and adaptation and plasticity combinedto the baseline model.The models are evaluated on their accuracy (Acc), sensitiv-ity (Sen), Precision (Pre), and Specificity (Spe). Acc is definedin Subsection IV-A. The other metrics can be found in [44].
A. Optimiser Comparison and Statistical Analysis
Table II shows that the SADE algorithm achieved the bestfitness, accuracy, precision, and specificity. The BO-PI algo-rithm converged on its solution in the least number of objectivefunction evaluations and the DE algorithm achieved the bestsensitivity but the worst fitness, precision, and specificity.TABLE II: Optimisation algorithm metrics.
Meth Fit Eva Acc Sen Pre SpeBPI -197.1 DE -675.4 238.8 0.62 BEI -454.0 181.3 0.63 0.57 0.80 0.69
SADE -84.9 -533.3 180.0 0.62 0.61 0.78 0.64Table III shows the statistical significance of the results fromTable II. The method in the comparison column is comparedto each method in the subsequent column. A + indicatesstatistically significant values and a . indicates no statisticalsignificance. Statistical significance was defined as p ≤ . .The Mann-Whitney U test was used to determine statisticalsignificance because it does not require normally distributedsamples.SADE’s better fitness is statistically significant comparedto all optimisers other than BO-PI. However, to achieve thisfitness it also performed the most evaluations when comparedto the others. This difference is significant compared to allthe optimisers except for DE, which has almost the samenumber of evaluations. SADE also has significantly worsesensitivity than all but the BO-PI algorithm. Both SADE andBO-PI scored the best fitness values whilst exhibiting thesignificantly lowest sensitivity values when compared to theother algorithms.BO-PI was significantly better than DE and BO-UCB forfitness. It also had significantly less evaluations than DE and TABLE III: Comparison of the statistical significance of theresults. Meth Fit Eva Acc Sen Pre SpeBUCB
BPI + . . + . .DE . + . . . .BEI . . . . . .SADE + + . + + + DE BPI + + . + . .BEI . + . . . .SADE + . + + + +BUCB . + . . . .
BEI
BPI + . . + . .DE . + . . . .SADE + + . + . .BUCB . . . . . .
SADE
BPI . + . . + +DE + . + + + +BEI + + . + . .BUCB + + . + + +
BPI
DE + + . + . .BEI . . . + . .SADE . + . . + +BUCB + . . + . .SADE. Its precision and specificity is significantly less thanthe SADE algorithm.BO-EI has significantly worse fitness and sensitivity whencompared to BO-PI and SADE.DE took significantly more evaluations to converge whencompared to all algorithms but SADE. It also had significantlyworse fitness, accuracy, precision and specificity than SADEbut significantly higher sensitivity. It had significantly worsefitness but significantly better sensitivity than BO-PI.BO-UCB had significantly worse fitness but better sensi-tivity than SADE and BO-PI. It also had significantly worseprecision and specificity than SADE.A possible reason that DE underperformed is that the F values provided in [21] are not appropriate for this problem.The population size may have also been too small. Beforethe SADE algorithm was implemented, doubling the recom-mended population size made DE find better results than whenit had a smaller population. When the population size is toosmall, whole regions of the parameter space can be missedresulting in poor performance.SADE removes the need to find control parameters and hasbeen shown to perform as well or better than DE even when thecontrol parameters are well selected [23]. The generalisabilitythat comes with finding the right control parameters on-the-flyis also appealing.The addition of the various mutation functions to SADE alsoseems to help it find better results. This is probably due to thedesirable properties of each mutation function cancelling outthe undesirable properties of other mutation functions.A surprising result was that of the BO algorithms BO-PIseemed to perform the best. This is contrary to what the JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 authors in [33] found. They suggested that it tended to havethe worst performance of the three.
B. SADE Averages
The SADE algorithm performed the best out of all ofthe algorithms. Fig. 3a shows the average
F itness mod ofthe population over 19 generations. The average
F itness mod converged by five iterations. The max
F itness mod starts offat 0. This indicates that a 100% accuracy candidate was foundin the initialisation period. The max
F itness mod then rises to400 which is not visible as the range of the average score is-50000 to -1500.The F average results in Fig. 3b are quite interesting. Theystart off at 1 as they are selected from U ([0 , and thendrop down to 0.5 as they are selected from U ([0 , afterthe first generation. Once the learning period has finishedall of the F values have converged to less than 0.1. Thisindicates that the F values that are having the most successare small and therefore taking advantage of exploration ratherthan exploitation. It was unexpected that the algorithm wouldfind a min/max within so few generations. This could bewhy the authors select F from N (0 . , . forcing F torange from -0.4 to 1.4. With F this small the algorithmwould effectively be performing gradient descent. However,this could be because the function on the restricted spacedoesn’t have many local maxima. Indeed, these results docome from the best performing LGMD model found.Fig. 3c shows how the CR for each function changes overtime. For the first nine generations, the CR values are selectedfrom U ([0 , and so the mean stays at 0.5. However, as withthe F mean values once the learning period is over, all ofthe CR values go down to less than 0.1. This means that lessthan 10% of the mutations will generally take place. Froma set of 11 hyper-parameters this means that probabilisticallyone value will change in addition to the random index that ischosen. CR is generally associated with convergence.The probability of each function being chosen is shownin Fig. 3d. The probabilities are fixed at 0.25 for the first 9generations and then they vary based on their success. It is in-teresting to see that in spite of the F and CR values suggestingthat the algorithm is converging on a solution, the DE/Rand-to-Best/2/Bin algorithm is the least successful. The DE/Curr-to-Rand/1 algorithm performs relatively well until about 16generations where it tapers off. The DE/Rand/2/bin algorithmdips initially but then increases as DE/Curr-to-Rand/1 starts todrop off. The DE/Rand/1/bin remains relatively high during theentire algorithm only to be overtaken by The DE/Rand/2/binin the last generation. C. Comparison of Models
Table IV shows the selected final parameters of each model.These values were all found by the SADE algorithm, due tothe superior quality of its results. The (1) tag in the parametercolumn indicates that the variable is unit-less.In both models with plasticity, the clamping value c was setto 0.05, or 5%. (a)(b)(c)(d) Fig. 3: Averages F acc ( λ ) , F , CR , and p for the SADE pop-ulation over 19 generations. The dotted vertical line indicatesthat the learning period has ended. HELL et al. : BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 9
TABLE IV: Parameters used by each model.
Parameter LGMD A P AP τ e ( ms ) τ iA ( ms ) τ iB ( ms ) q eP ( pA ) q eS ( pA ) q eIP ( pA ) q eIS ( pA ) q eL ( pA ) inhA S ( ) inhB S ( ) inhA L ( ) a ( ) - 0.79 - 0.79 b ( ) - 14.51 - 14.51 τ w adapt ( ms ) - 30.00 - 30.00 τ pre ( ms ) - - 1.56 1.56 τ post ( ms ) - - 10.03 10.03 ∆ pre ( ) - - 0.031 0.031 ∆ post ( ) - - 0.027 0.027 c ( ) - - 0.05 0.05As expected, all of the models have a τ iA < τ iB whichmeans that the B inhibitions will persist for longer andhave slower dynamics relative to the A inhibitions. Whatis unexpected is that the B inhibitions also have strongercurrent injection than the A inhibitions. On top of this,both of the inhibitory current injections are actually strongerthan the excitatory connections. Whereas the model in [12]with discrete dynamics had relatively low inhibitory currentinjections, with inhA S = 0 . and inhB S = 0 . of theexcitation strength. Clearly, there is a difference between theneuron models that are used, but this is an interesting outcomenonetheless.Table V shows the accuracy, sensitivity, precision, andspecificity for each LGMD model for a given simple stimulus.The stimuli can be described as follows: composite: A standard test bench stimulus that consists of ablack circle on a white background that translates andlooms at increasing speeds. Fig. 4a shows the compositeinput. circleFast/Slow:
A purely looming black circle on whitebackground at high or low speeds. Collected on hoveringQUAV. Fig. 4b shows the circleFast/Slow stimulus. squareFast/Slow:
A purely looming black square on a whitebackground at high/low speeds. Fig. 4c shows the square-Fast/Slow stimulus.The results in Table V show that the models performedwell (
Accuracy ≥ . ) on most of the stimuli. LGMD and A perform poorly on the circleSlow test, missing two out offive of the looming stimuli. P misses one looming stimulus,and AP detects all stimuli accurately. The plasticity increasesthe weights of important connections and the adaptation filtersout over excited neurons.These results show that the models are capable of detect- (a) Filtered Composite Input (P Layer Raster Plot).(b) Filtered circleSlow Input (P Layer Raster Plot).(c) Filtered squareFast Input (P Layer Raster Plot). Fig. 4: The input layer for the simple stimuli.The whiteand coloured backgrounds indicate non-looming and loomingrespectively.ing looming stimuli of varying speeds and of differentiatingbetween translation and looming stimuli for the most part. AP scored 100% in every test besides the composite stimuluswhere it misclassified the first short translation as a loom. Thiscan probably be attributed to the network not starting in itsresting/equilibrium state.After performing the simulated experiments of computergenerated shapes, real objects moving towards and away fromthe camera were recorded. These stimuli can be described as: ballRoll[1-3]: Three different runs of a white ball rollingtowards the camera on a black platform at different anglesand speeds. This is a purely looming stimulus. Fig. 5ashows one of the three ball rolls. cupQUAV:
A QUAV flying towards a cup suspended in A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
TABLE V: Quality metrics of the performance of different LGMD models for different simulated looming stimuli.
Stimulus Model Accuracy Sensitivity Precision Specificitycomposite LGMD A P AP circleSlow LGMD A P AP circleFast LGMD A P AP squareSlow LGMD A P AP squareFast LGMD A P AP Hand:
A Hand moving towards and away from the hoveringQUAV. Fig. 5c shows the looming hand stimulus.Fig. 5a, Fig. 5b, and Fig. 5c show that the real stimuli tendto have more noise and do not adhere to a strong patternwhen compared to Fig. 4a, Fig. 4b, and Fig. 4c. Table VI showsthat the models do not perform as well on real world stimuli.ballRoll[1-3] is the simplest real stimulus, and as such P and AP achieved full accuracy. LGMD and A missed one roll.Surprisingly good results come from the cupQUAV stimu-lus: 70% accuracy for all models except for AP , which had80%. It is worth noting that AP performed consistently wellwhen compared with the other models.The possibility of detecting the hand by stochasticallydropping pixel-events, was investigated. Dropping 50% ofthe DVS events and re-optimising the network gave 100%accuracy for the hand and cupQuad stimulus. However, indoing this, the network was no longer robust to the speedchanges in the composite benchmark test. Indeed, even usingall of the pixels, the network could be optimised to work on thereal world stimuli. The inhibition values went up and the gainvalues went down, meaning the network struggled to spikeon stimuli that weren’t noisy or event heavy. Some sort ofadditional pre-filtering could be useful in getting the loomingnetwork to be fully robust in all situations.
1) The Effect of Changing c on Plasticity: Fig. 6a, Fig. 6b,and Fig. 6c show how changing the bounds of the plasticityclamping changes the LGMD ( P model) accuracy for thecomposite, cicleSlow, and hand stimuli respectively.Interestingly, for the two simulated stimuli increasing theclamping to beyond 25% caused the accuracy to drop to 50%.The sensitivity dropped to 0% indicating that it was no longer detecting looms and that the synaptic weights were no longercausing the LGMD neuron to fire.Increasing the clamping to 45% increases the accuracy forboth the P and AP models on the hand stimulus. This showsthat plasticity is a double edged sword that can both improveand degrade the performance of the model. Knowledge aboutthe nature of your input can help to determine what level ofplasticity you require. In all cases, a small contribution ofplasticity improved the performance. This could be due to thefact that the amount of noise in the simulated stimuli was farless than the noise in the real stimuli.
2) Weight Visualisation:
Fig. 7, Fig. 8, and Fig. 9 showsnapshots of the weights at the end of each looming or non-looming sequence. We used the P model with c = 0 . on thecomposite stimulus. This was done because it achieved 100%accuracy and 25% clamping has greater weight variation than10%.Fig. 7 is interesting as it most obviously correlates to theinput. We can see in the first non-looming snapshot that theP-IP layer is strongly inhibiting a circle translating from rightto left. In the looming section, the circle is moving outwardsand the central weights have the highest density of low values.This shows that the centre of the circle is not associated withthe output. In the second non-looming snapshot, the densityof high values is in the centre of the circle showing that it hashigher inhibitions.Fig. 8 and Fig. 9 show the IP and IS connections to theLGMD layer. The IP-LGMD snapshots tend to have higherweights during looming than non-looming stimuli. Interest-ingly, in both figures, the highest value is one, meaning thatthe weights have only become weaker than they initially were,at least for these selected times. HELL et al. : BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 11
TABLE VI: Quality metrics of the performance of different LGMD models for different real looming stimuli
Stimulus Model Accuracy Sensitivity Precision SpecificityballRoll[1-3] LGMD A P AP cupQUAV LGMD A P AP hand LGMD A P AP (a) Filtered ballRoll2 Input (P Layer Raster Plot).(b) Filtered cupQUAV Input (P Layer Raster Plot).(c) Filtered Hand Input (P Layer Raster Plot). Fig. 5: Complex real stimuli. The white and coloured back-grounds indicate non-looming and looming respectively. VI. C
ONCLUSIONS
We implemented a neuromorphic model of the locustLGMD network using recordings from a UAV equipped witha DVS sensor as inputs. The neuromorphic LGMDNN wascapable of differentiating between looming and non-loomingstimuli. It was capable of detecting the black and white simplestimuli correctly regardless of speed and shape. Real-worldstimuli performed relatively well using the parameters foundby the optimiser for synthesised stimuli. However, when re-optimised, the real-world stimuli performed comparably tothe synthesised stimuli. This was mainly because real-worldstimuli tend to contain a higher number of luminance changesand therefore the magnitude parameters needed to be reduced.We showed that BO, DE, and SADE are capable of findingparameter values that give the desired performance in theLGMDNN model. It can be seen that SADE statistically sig-nificantly outperformed DE on all metrics besides sensitivityand the number of evaluations, although the only metrics thatformed part of the objective function were fitness and accuracy.Once a suitable objective function was found that accuratelydescribed the desired output of the LGMDNN, BO, DE andSADE outperformed hand-crafted attempts. The algorithmswere able to achieve 100% accuracy on black and white simplestimuli of varying shapes and speeds. SADE performed wellin this task and we have shown that it is suitable for theoptimisation of a multi-layered LGMD spiking neural network.This could save time when developing biologically plausibleSNNs in related applications.In the future, we would like to apply the optimisation al-gorithms directly to tuning the neuromorphic processors usingthe neuromorphic model, with the end goal being a closedloop control system on a UAV. Showing that optimisation iseffective for selecting parameters on neuromorphic hardwarewill increase their usability.A
CKNOWLEDGMENT
We are grateful to Prof. Claire Rind, who provided valuablecomments and feedback on the definition of the neuromorphicmodel, and acknowledge the CapoCaccia Cognitive Neuro-morphic Engineering workshop, where these discussions andmodel developments took place. A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 (a) Effect of changing the c clamping value on the learning weight w for the composite stimulus(b) Effect of changing the c clamping value on the learning weight w for the circleSlow stimulus(c) Effect of changing the c clamping value on the learning weight w for the hand stimulus Fig. 6: The effect of changing the clamping value on variousstimuliWe would also like to thank INILABs for use of the DVSsensor and the Institute of Neuroinfoamrtics (INI), Universityof Zurich and ETH Zurich for its neuromorphic processordevelopments. Part of this work was funded by the EUERC Grant “neuroP” (257219) and EU H2020-MSCA-IF-2015 grant “ECogNet” (707373).R
EFERENCES[1] S. C. Liu and T. Delbruck, “Neuromorphic sensory systems,”
CurrentOpinion in Neurobiology , vol. 20, no. 3, pp. 288–295, 2010.[2] E. Chicca, F. Stefanini, C. Bartolozzi, and G. Indiveri, “Neuromorphicelectronic circuits for building autonomous cognitive systems,”
Proceed-ings of the IEEE , vol. 102, no. 9, pp. 1367–1388, 2014.
Fig. 7: The weights of the synapses from the P to the IP layerat the end of a looming or non-looming sequence.Fig. 8: The weights of the synapses from the IP to the LGMDlayer at the end of a looming or non-looming sequence.Fig. 9: The weights of the synapses from the IS to the LGMDlayer at the end of a looming or non-looming sequence. [3] G. Indiveri, F. Corradi, and N. Qiao, “Neuromorphic architectures forspiking deep neural networks,” in . IEEE, 2015, pp. 4–2.[4] P. Lichtsteiner, C. Posch, and T. Delbruck, “A 128x128 120 dB 15 µ slatency asynchronous temporal contrast vision sensor,” IEEE Journal ofSolid-State Circuits , vol. 43, no. 2, pp. 566–576, Feb 2008.[5] T. Serrano-Gotarredona and B. Linares-Barranco, “A 128 ×
128 1.5%contrast sensitivity 0.9% FPN 3 µ s latency 4 mW asynchronous frame-free dynamic vision sensor using transimpedance preamplifiers,” IEEEJournal of Solid-State Circuits , vol. 48, no. 3, pp. 827–838, 2013.[6] T. Delbruck, “Frame-free dynamic digital vision,” in
Proceedings of Intl.Symp. on Secure-Life Electronics, Advanced Electronics for Quality Lifeand Society , 2008, pp. 21–26.[7] M. M¨uller, S. Lupashin, and R. D’Andrea, “Quadrocopter ball juggling,”in
HELL et al. : BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 13
Systems . IEEE, 2011, pp. 5113–5120.[8] D. Brescianini, M. Hehn, and R. D’Andrea, “Quadrocopter pole acrobat-ics,” in . IEEE, 2013, pp. 3472–3479.[9] D. Mellinger and V. Kumar, “Minimum snap trajectory generation andcontrol for quadrotors,” in
Robotics and Automation (ICRA), 2011 IEEEInternational Conference on . IEEE, 2011, pp. 2520–2525.[10] E. Mueggler, B. Huber, and D. Scaramuzza, “Event-based, 6-dof posetracking for high-speed maneuvers,” in . IEEE, 2014, pp. 2761–2768.[11] R. D. Santer, R. Stafford, and F. C. Rind, “Retinally-generated saccadicsuppression of a locust looming-detector neuron: investigations using arobot locust,”
Journal of The Royal Society Interface , vol. 1, no. 1, pp.61–77, 2004.[12] S. Yue, R. D. Santer, Y. Yamawaki, and F. C. Rind, “Reactive directioncontrol for a mobile robot: a locust-like control of escape directionemerges when a bilateral pair of model locust visual neurons areintegrated,”
Autonomous Robots , vol. 28, no. 2, pp. 151–167, 2010.[13] R. Stafford, R. D. Santer, and F. C. Rind, “A bio-inspired visual collisiondetection mechanism for cars: combining insect inspired neurons tocreate a robust system,”
BioSystems , vol. 87, no. 2, pp. 164–171, 2007.[14] L. Salt, G. Indiveri, and S. Yulia, “Obstacle avoidance with LGMD neu-ron : towards a neuromorphic UAV implementation .” in
InternationalSymposium on Circuits and Systems , 2017.[15] R. Brette and W. Gerstner, “Adaptive exponential integrate-and-firemodel as an effective description of neuronal activity,”
Journal ofneurophysiology , vol. 94, no. 5, pp. 3637–3642, 2005.[16] N. Qiao, H. Mostafa, F. Corradi, M. Osswald, F. Stefanini, D. Sum-islawska, and G. Indiveri, “A reconfigurable on-line learning spikingneuromorphic processor comprising 256 neurons and 128K synapses,”
Frontiers in Neuroscience , vol. 9, pp. 1–17, 2015.[17] R. Storn and K. Price, “Differential evolution–a simple and efficientheuristic for global optimization over continuous spaces,”
Journal ofglobal optimization , vol. 11, no. 4, pp. 341–359, 1997.[18] R. Dong, “Differential evolution versus particle swarm optimization forpid controller design,” in
Natural Computation, 2009. ICNC ’09. FifthInternational Conference on , vol. 3, Aug 2009, pp. 236–240.[19] N. Karaboga and B. Cetinkaya, “Performance comparison of geneticand differential evolution algorithms for digital fir filter design,” in
Advances in Information Systems , ser. Lecture Notes in ComputerScience, T. Yakhno, Ed. Springer Berlin Heidelberg, 2005, vol. 3261,pp. 482–488.[20] S. Das and P. N. Suganthan, “Differential evolution: A survey ofthe state-of-the-art,”
IEEE transactions on evolutionary computation ,vol. 15, no. 1, pp. 4–31, 2011.[21] M. E. H. Pedersen, “Good parameters for differential evolution,”
MagnusErik Hvass Pedersen , 2010.[22] J. Brest, S. Greiner, B. Boskovic, M. Mernik, and V. Zumer, “Self-adapting control parameters in differential evolution: a comparativestudy on numerical benchmark problems,”
IEEE transactions on evo-lutionary computation , vol. 10, no. 6, pp. 646–657, 2006.[23] A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolutionalgorithm with strategy adaptation for global numerical optimization,”
IEEE transactions on Evolutionary Computation , vol. 13, no. 2, pp.398–417, 2009.[24] G. D. Howard, “On self-adaptive mutation restarts for evolutionaryrobotics with real rotorcraft,” in
Proceedings of the 17th Annual Con-ference on Genetic and Evolutionary Computation . ACM, 2017, p. Inpress.[25] D. Howard and T. Merz, “A platform for the direct hardware evolution ofquadcopter controllers,” in
Intelligent Robots and Systems (IROS), 2015IEEE/RSJ International Conference on . IEEE, 2015, pp. 4614–4619.[26] G.-Q. Bi and M.-M. Poo, “Synaptic modifications in cultured hip-pocampal neurons: Dependence on spike timing, synaptic strength, andpostsynaptic cell type.”
J. Neurosc , vol. 77, no. 1, pp. 551–555, 1998.[27] G. Howard, E. Gale, L. Bull, B. de Lacy Costello, and A. Adamatzky,“Evolution of plastic learning in spiking networks via memristiveconnections,”
IEEE Transactions on Evolutionary Computation , vol. 16,no. 5, pp. 711–729, 2012.[28] L. Liu, C. Shen, and A. van den Hengel, “The treasure beneathconvolutional layers: Cross-convolutional-layer pooling for image clas-sification,” in
Proceedings of the IEEE Conference on Computer Visionand Pattern Recognition , 2015, pp. 4749–4757.[29] A. Babenko and V. Lempitsky, “Aggregating local deep features forimage retrieval,” in
Proceedings of the IEEE international conferenceon computer vision , 2015, pp. 1269–1277. [30] B. Fernando, E. Gavves, J. Oramas, A. Ghodrati, and T. Tuytelaars,“Rank pooling for action recognition,”
IEEE transactions on patternanalysis and machine intelligence , vol. 39, no. 4, pp. 773–787, 2017.[31] S. Song, K. D. Miller, and L. F. Abbott, “Competitive hebbian learn-ing through spike-timing-dependent synaptic plasticity,”
Nature neuro-science , vol. 3, no. 9, pp. 919–926, 2000.[32] J. Sj¨ostr¨om and W. Gerstner, “Spike-timing dependent plasticity,”
Spike-timing dependent plasticity , p. 35, 2010.[33] E. Brochu, V. M. Cora, and N. De Freitas, “A tutorial on bayesianoptimization of expensive cost functions, with application to activeuser modeling and hierarchical reinforcement learning,” arXiv preprintarXiv:1012.2599 , 2010.[34] J. Mockus, “Application of bayesian approach to numerical methodsof global and stochastic optimization,”
Journal of Global Optimization ,vol. 4, no. 4, pp. 347–365, 1994.[35] D. R. Jones, “A taxonomy of global optimization methods based onresponse surfaces,”
Journal of global optimization , vol. 21, no. 4, pp.345–383, 2001.[36] M. J. Sasena, “Flexibility and efficiency enhancements for constrainedglobal design optimization with kriging approximations,” Ph.D. disser-tation, General Motors, 2002.[37] J. Snoek, H. Larochelle, and R. P. Adams, “Practical bayesian optimiza-tion of machine learning algorithms,” in
Advances in neural informationprocessing systems , 2012, pp. 2951–2959.[38] N. Srinivas, A. Krause, S. M. Kakade, and M. Seeger, “Gaussian processoptimization in the bandit setting: No regret and experimental design,” arXiv preprint arXiv:0912.3995 , 2009.[39] D. J. Lizotte,
Practical bayesian optimization . University of Alberta,2008.[40] D. R. Jones, M. Schonlau, and W. J. Welch, “Efficient global optimiza-tion of expensive black-box functions,”
Journal of Global optimization ,vol. 13, no. 4, pp. 455–492, 1998.[41] K. Deb,
Multi-Objective Optimization Using Evolutionary Algorithms .Wiley, 2001.[42] L. Salt, “Optimising a Neuromorphic Locust Looming Detector for UAVObstacle Avoidance,” Master’s thesis, The University of Queensland,2016.[43] D. F. Goodman, M. Stimberg, P. Yger, and R. Brette, “Brian 2:neural simulations on a variety of computational hardware,”
BMCNeuroscience , vol. 15, no. 1, p. 1, 2014.[44] E. Alpaydin,