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[EQUATION] One special case of the reverse accumulation principle is the backpropagation of errors in multilayer perceptron, a commonly encountered problem in artificial intelligence and neural networks in particular. |
Complexity of AD modes Unlike in the forward mode, an obvious drawback in the use of the reverse mode is the need to store the evaluated computational graph to be used during the reverse sweep for accumulating the gradient. A naive implementation of the reverse mode AD may lead to a hike in the storage that is proporti... |
favours more computations. It requires partial evaluations and partial storage at the same time. In other words, checkpointing involves re-evaluating the values of the graph vertices rather than storing the entire graph structure. |
When dealing with functions of type ( 4.27 ), theoretical analysis by Bischof et al. revealed that if [MATH] operations are needed to evaluate the value of function [MATH] , then, the total computational requirement associated with the reverse accumulation of its gradient is not more than [MATH] operations . But, the b... |
Besides, if the multivariate problem under consideration is a vector function of length [MATH] such that: [EQUATION] then, the relative cost of using forward or reverse accumulation becomes similar. In fact, in such cases, it is often hard to determine the appropriate balance between the forward and reverse computation... |
Ultimately, a comparison of AD techniques with the method of divided differences in reveals that the divided differences method can only be as fast as the AD methods when the problem size is small. This is however not the case as problem size grows. Furthermore, there is no guarantee that the derivatives obtained via t... |
4.7.3 Implementation Techniques of Automatic Differentiation As mentioned earlier, the key to successful implementation of AD tool is the simultaneous evaluation of the function value and its derivative at every vertex of the computational graph. There are two approaches for implementing AD tool: source transformation ... |
to yield a very efficient AD implementation as it allows compile time optimization. The operator overloading approach which is the method adopted in this work requires a new user defined data type (AD object) that combines the function value; it’s gradient and Hessian in a single object. The function is decomposed into... |
Presently, a number of available AD implementations include the ADOL-C, ADIFOR, Tapenade, ADIC mostly in C, and the ADiMaT and MAD which are Matlab based |
implementations built on source code transformation and overloaded operators respectively. 4.7.4 The Proposed Matlab Implementation of the AD Algorithm |
Matlab is a computational tool that has both the object oriented programming capability and operator overloading facility. Therefore, it is a suitable platform for the proposed AD implementation. The method adopted here is based on the vector-mode approach of forward accumulation where for a multivariate function of th... |
[EQUATION] the scalar augmented derivative term [MATH] is redefined to a matrix [MATH] such that: [EQUATION] where for the input variables, the gradient [MATH] with respect to the input variables is what is initialized as [MATH] array of identity matrices [MATH] . Because the problems to be addressed here are of the fo... |
[EQUATION] The exact gradient value is obtained by updating this initial gradient [MATH] as the evaluation propagates from the input variables to the output variable. |
Similarly, the expression for the second derivative (i.e., the Hessian) is as follows: [EQUATION] where the Hessian of the input variables is what is initialized as [MATH] which is an [MATH] dimensional array of zeros. This is also updated as the evaluation propagates from the input variables, through the intermediate ... |
Therefore, defining the AD objects in this way guarantees that evaluation of both the function value and its derivatives is completed in a single forward sweep. The three fundamental steps involved in the development of the AD algorithm via overloading method are: |
i. Defining a new data type (Class) the instances (objects) of which possess separate fields (properties) for the values and the derivatives and can execute the user-defined overloaded functions (methods). |
ii. Defining a constructor function that can create an instance of the above class and automatically initialize its properties. iii. |
Creating overloaded functions for all the arithmetic operators and all functions (trigonometric, logarithmic, exponential etc.) upon which the AD object get its properties evaluated. |
Thus, for the proposed design, each instance of the AD class will have the following three property fields: (i) A function value field: [MATH] |
(ii) A function derivative field: [MATH] , and (iii) A function Hessian field: [MATH] To realize a vector-form implementation for the forward accumulation, when initializing any given function of dimension [MATH] at an initial point [MATH] , the constructor class for the AD objects will always initialize the object’s p... |
i) [MATH] is initialized to the initial point [MATH] ii) [MATH] is initialized to an identity matrix of size [MATH] , i.e., [MATH] |
iii) [MATH] is initialized to an array of zeros i.e., [MATH] Thus, the AD object [MATH] will then be a data type that holds the function value, the derivative and the Hessian right at a spot. This model conforms to the classic extended differentiation arithmetic model proposed by Rall |
. Hence, [EQUATION] where according to the extended differentiation arithmetic, any independent variable [MATH] is defined as a triplet having a component for its own value [MATH] , one for its derivative [MATH] and another one for its second derivative or Hessian [MATH] , such that: |
[EQUATION] In the same way, any constant [MATH] is a triplet with value [MATH] , derivative, [MATH] and second derivative [MATH] , such that: |
[EQUATION] Hence, the corresponding values for the intermediate and dependent variables are obtained by operating on the above independent variables (initialized as AD objects) based on the chain rule ( 4.20 ). |
An algorithmic template for the AD class structure is shown in . It is made up of a method that defines the three property fields for the AD objects (lines [MATH] ) and a constructor function (lines [MATH] ) that construct and initialize the AD objects based on the dimensionality [MATH] of the problem (derived at line ... |
Algorithm 2 A constructor class template for an automatic differentiation object 1: classdef ADClass % Defining the AD object’s properties - Comment |
2: properties 3: [MATH] 4: [MATH] 5: [MATH] 6: end % The Class constructor function - Comment 7: methods 8: [MATH] 9: [MATH] % Initialization of the object’s properties - Comment |
10: [MATH] 11: [MATH] 12: [MATH] 13: end 14: end 15: end In the following section, some overloaded arithmetic operators and functions will be presented. Using a simple example (section 4.7.6 ), a demonstration of how this automatic differentiation method can be used to evaluate the value, first and second derivatives o... |
4.7.5 Some Overloaded Operators and Functions for AD Objects Having constructed the AD objects with their property fields initialized, it is possible to execute all arithmetic operations on them so long as the built-in operators and functions (i.e., the standard real arithmetic operators and other mathematical function... |
Arithmetic operators–Both the binary and unary versions Logarithmic and exponential operators Trigonometric functions Norm (ABS), etc. |
The following collection presents the process of overloading some basic operators and functions. Remember that for all the arithmetic operators, basic chain rule of differentiation ( 4.20 ) will be applied, but for any arbitrary differentiable functions such as trigonometric, logarithmic, exponential etc., a more gener... |
Binary Addition/Subtraction: Suppose the function of interest is a [MATH] dimensional function [MATH] , such that: [EQUATION] Then, the simple graph with root vertices corresponding to each of the two independent (input) variables and a top vertex corresponding to the dependent (output) variable represents the evaluati... |
Let the vertices be: [MATH] and [MATH] , then, based on equation ( 4.33 ), we have: [EQUATION] and [EQUATION] However, since the proposed implementation seek to vectorize the definition for each of these vertices , we initialize their fields based on the dimensionality [MATH] of the function under consideration, such t... |
[EQUATION] and [EQUATION] Similarly, [EQUATION] and [EQUATION] Therefore, the input vertices [MATH] and [MATH] can now be redefined as: |
[EQUATION] [EQUATION] Then, the output vertex [MATH] is: [EQUATION] Notice how a mere addition/subtraction of the two AD objects (i.e., vertices: [MATH] and [MATH] ) leads to evaluation of the value, derivative and Hessian of the function [MATH] 4.34 ) as components of the output AD object [MATH] . Such single sweep ex... |
Multiplication Operator: Let the function of interest be defined as: [EQUATION] Then, the AD objects can also be seen as the vertices of the graph and their components can be defined as in equations ( 4.39 ) and ( 4.40 ). The output vertex [MATH] is therefore: |
[EQUATION] Now, multiplying the two AD objects based on the chain rule ( 4.20 ) gives: [EQUATION] The Sine Function Now consider the following trigonometric function (sine): |
[EQUATION] The root vertex [MATH] and the top vertex [MATH] as shown in the graph can be defined as: [EQUATION] and therefore, the AD variables will be defined as: |
[EQUATION] All other operators and functions are overloaded based on the principles above. In the following, an example is provided to further illustrate the application of the AD method described above by comparing it with the classical symbolic method of differentiation. |
4.7.6 Example: Comparing Symbolic and Forward mode AD methods In order to compare the computational approach of symbolic differentiation method with forward mode AD technique, consider the [MATH] dimensional function ( 4.43 ), in the following, the value and derivatives of this function will be evaluated at [MATH] both... |
[EQUATION] Symbolic Differentiation: This entails direct substitution of the solution point [MATH] after evaluating the formula for the value and the derivatives using chain rule. |
Value: For the problem under consideration ( 4.43 ), [EQUATION] Gradient : The gradient is evaluated in two stages: first, the formula is derived by applying chain rule ( 4.20 ) on the partial derivatives of problem ( 4.43 ). Second, the solution point is substituted into the obtained formula to get the gradient. |
[EQUATION] [EQUATION] Hessian : In the same way, the Hessian is obtain as follows: [EQUATION] [EQUATION] Notice how the above symbolic approach requires the computer to explicitly evaluate and store the formula before substituting the values and solving via the basic real arithmetic. In the following, the AD forward mo... |
Forward mode AD: Here we will redefine the problem variables (both dependent and independent) in terms of AD objects. Then, the differentiation arithmetic described above will be used to concurrently evaluate the function value and derivatives algorithmically. It will be interesting to realize how this can be achieved ... |
[EQUATION] then, these vertices can be initialized as AD objects based on the solution point [MATH] , such that: [EQUATION] Hence, problem ( 4.43 ) is now |
[EQUATION] Therefore, the value and derivatives of problem ( 4.43 ) can now be obtained by evaluating equation ( 4.44 ) via the differentiation arithmetic (i.e. the AD approach). |
[EQUATION] Therefore, [EQUATION] which is similar to the solution obtained via the traditional symbolic method above. Now, from the solution [MATH] which is an AD object (i.e., a structure), one can extract the function value, derivative and Hessian respectively as follows: |
Function Value = [MATH] Function Derivative = [MATH] , and Function Hessian = [MATH] The elegance of this approach is in its suitability for algorithmic computation in computer. Notice how the final solution yields the exact results for the function value, gradient and the Hessian. Using such exact Hessians, the propos... |
4.8 Contribution The earlier analysis of the characteristics of various gradient based local search algorithms in this chapter has helped us picked the sequential quadratic programming (SQP) algorithm which is a Newton based method. The choice was made based upon the following two findings: |
First, since in the proposed hybrid setup (details in the next chapter), the SQP algorithm will be invoked after sufficient convergence of the global algorithm, it is quite certain that the local algorithm will be initialized with a solution that is always in the vicinity of the global optimum point. Thus, whenever ini... |
Second, to further minimize the number of iterations required by this local search method, an automatic differentiation algorithm based on the proposed vectorized forward accumulation method is used to cheaply evaluate the derivatives. This improves the quality of the evaluated search directions and alleviates the need... |
This ultimately leads to the realization of the proposed local search algorithm that will ensure rapid convergence to the optimum solution by taking long but few steps. |
4.9 Remarks This chapter begins with an investigation of various types of gradient based local search algorithms. Then, the advantages and disadvantages of the steepest descent, Newton, quasi-Newton and conjugate gradient methods were analyzed. Thereafter, various methods for evaluating search directions and step sizes... |
Chapter 5 The Proposed Hybrid Evolutionary Algorithm Besides a brief treatment of the motivation behind the need for the development of hybrid optimization methods, this chapter will begin by introducing the current trends in hybridizing evolutionary algorithms. Taxonomies of various categories of hybrid algorithms wil... |
5.1 Why the need for hybrid algorithms In the last few decades, it has become well understood that population based search methods like the evolutionary algorithms (EAs) are effective in exploring search spaces even when faced with problems having high-dimensionality, nonconvexity, multimodality, isolated optima, nonun... |
. On the other hand, local search methods are generally more effective when it comes to exploiting specific regions of the search space, i.e., they can easily converge to better solutions in the vicinity of any given solution. Therefore, the notion of hybridizing various categories of algorithms with the aim of establi... |
Prior to the development of any hybrid system, it is imperative to address the following issues in order to ascertain whether a hybrid system is needed, and if so, which kind of hybridization approach is suitable for the problem under consideration; the issues are: |
i. Understand the type of problem at hand and based on the optimization goal, one can decide whether to use only approximate, exact or a hybrid of the two algorithms. Typically, when dealing with simple convex problems of lower dimensions, local algorithms like the gradient based methods can suffice. Also, when the qua... |
ii. Determine what algorithms to combine and which type of combination of these algorithms might work well for the class of problem at hand and why. |
iii. Ascertain what role enhancing the capabilities of the individual algorithms can play to the success of the proposed hybrid system. |
iv. Determine how to fine tune the hybridized system to optimality for the category of problems under consideration. Unfortunately, not all of the above questions have direct or simple answers, in fact, the previous chapters in this work have so far concentrated on selection and tuning of the individual algorithms to b... |
5.2 Taxonomy of Hybrid Evolutionary Algorithms As noted by , it is not possible to exhaustively enumerate the various types of hybrid algorithms in the literature. This is true because the notion of hybridization in itself lacks a precise definition or a specific framework that clearly defines what should constitute th... |
, lack of precise boundary in the area of hybrid algorithms is what made the research field very rich and versatile. In other words, raising rigid boundaries between related fields of research often impedes creative thinking and exploration of new research directions. |
Yet, a noteworthy effort made by Raidl and colleagues categorizes the various aspects of hybrid algorithms. They proposed a classification that attempt to unify the general framework of hybridizing algorithms. A concise schematic for the classification is shown in figure 5.1 . The four major features depicted in this f... |
Although as earlier highlighted, in recent years, a lot of research works involving hybrid algorithms are reported in the literature. Another view of classifying hybrid approaches entails the following: |
i. Hybrid metaheuristics: Though this can be seen as a superset of the rest of the hybridization classes, it particularly refers to the combination of two or more approximate algorithms. Thus, it involves combining various nature inspired algorithms like EAs and/or non-nature inspired algorithms like tabu search, itera... |
a. The development of problem specific representation b. The development of additional genetic operators ; and c. The incorporation of domain specific knowledge or features of classical algorithms |
ii. Memetic Algorithms: May be seen as a subset of the above class, but they uniquely consist of combination of approximate algorithms like EAs with exact methods like gradient based local search algorithms. The main focus of this class is to facilitate rapid convergence to the optimum solution (i.e., exploitation of t... |
argue that although this class yields very successful hybrid algorithms, not much work exist in this direction. iii. Algorithms portfolio: This notion is based on the intuition that executing many short runs of one or more algorithms (in a parallel or interleaved manner) over a prescribed solution period can provide im... |
formulated the computational cost of a portfolio as a random variable having a probability distribution and evaluated its mean (i.e., the expected evaluation cost) and standard deviation (i.e. the dispersion or the risk involved). They argue that a skilful schedule of multiple copies of a single algorithm can outperfor... |
highlighted that a portfolio essentially comprises of both scheduling and machine learning aspect. They divided the scheduling aspect into either a restart or task-switching schedules. The former is a schedule for a single randomized heuristic, executed in a restart model. While the latter consist of a set of one or mo... |
In essence, each one of the above three classes have its merits and demerits. The first class favours exploration of problem space at the expense of exploiting high quality regions. Although the second class tried to alleviate this problem, a naive design of a memetic algorithm that is overly focused on exploitation ma... |
In table 5.1 we present some recent hybrid algorithms reported in the literature. The table summarizes the methods from different perspectives ranging from the type, class and numbers of the combined algorithms, the adopted control/switching mode, the feature that is optimized, the validation method employed and the in... |
5.3 The Proposed Task-Switching Hybrid Evolutionary Algorithm As a first attempt, we propose combining a global evolutionary algorithm (EC) presented in chapter with the Newton based local search algorithm (SQP) presented in chapter in a collaborative manner. In essence, the two algorithms will run sequentially retaini... |
In addition, realizing that there is yet a slight tendency for the final solution returned by the local search algorithm to be a sub-optimal one , a validation loop will be utilized to kick-start an additional round of a global search by the EC algorithm. While in the validation loop, the initial population will also b... |
Algorithm 1 The proposed hybrid EC/SQP algorithm 1: begin 2: [MATH] 3: initialize population [MATH] : size [MATH] 4: get Global optimum solution: [MATH] invoke EC [MATH] % see Algorithm LABEL: |
5: get Local optimum solution: [MATH] invoke SQP [MATH] % see Algorithm LABEL: 6: re-initialize population [MATH] : size [MATH] 7: validate [MATH] [MATH] invoke EC [MATH] [MATH] is the optimum solution |
8: end Algorithm demonstrates the working of the proposed hybrid switching procedure. The proposal entails designing a memetic algorithm that relies on a task switching procedure to transfer control between the global and local algorithms. It begins with an initial randomly created population [MATH] of size [MATH] from... |
5.4 Experiments An initial evaluation of the proposed hybrid algorithm is done by applying it on some benchmark numerical optimization problems. Details of the parameter settings of the hybrid algorithm used for the experiments are presented in table 5.2 |
5.4.1 Characteristics of Global Optimization Test Problems Before evaluating any global optimization algorithm, it is important to seek a suite of test problems that satisfy some of the following qualities: |
1. Nonlinearity: The function should not be linear 2. Scalability: The dimensionality of the test function must be extendible to medium or large sizes |
3. Non-separability: A function is separable if it has no nonlinear interaction between its variables 4. Multimodality and Nonconvexity: The function must possess many sub-optimal peaks/valleys |
5. Non-symmetricity: The global optimum should not be equidistant from any oppositely located local optima pair 6. High Dispersion: Dispersion indicates how far from being convex is the global topology of the function. It was argued that |
, a highly dispersed function can be more difficultly to global optimization algorithms than just a multimodal function having convex-like global structure. |
Detailed analysis and evaluation of the above features and more for test problems can be found in 5.4.2 Selected Benchmark Test Problems |
To evaluate the performance of the proposed hybrid algorithm, a series of tests have been carried out on the Ackley, Rastrigin and Schwefel benchmark functions for global optimization. |
i. Ackley function: The generalized form of Ackley function is defined as: [EQUATION] where [MATH] and global optimum [MATH] Ackley function is a multi dimensional nonlinear global optimization problem. Although the function is multimodal having many peaks/valleys forming several sub-optimal solutions, it is has low di... |
ii. Rastrigin function: The generalized form of Rastrigin function is defined as: [EQUATION] where: [MATH] and global optimum [MATH] |
Rastrigin function is a scalable, nonlinear and highly multimodal function with many valleys increasing in depth when approaching the global optimum point. Although it is separable and has a global topology (i.e. pseudo-convexity) it is actually flatter and of higher dispersion compared to Ackley function and therefore... |
iii. Schwefel function: The generalized form of the Schwefel function is: [EQUATION] where: [MATH] and global optimum [MATH] Also nonlinear, separable and multimodal, Schwefel function is highly dispersed and lacks any global topology. Moreover, it is non-symmetric and scalable to higher dimensions. |
If we sort the complexity of the above problems based on their degree of dispersion, the Ackley function is the least and the Schwefel function is the most difficult. |
5.4.3 Results and Discussions Two different sets of experiments were carried out. The first experiment investigates how the proposed hybrid algorithm behaves under increasing problem size. While the second experiment is aimed at evaluating the performance of the hybrid algorithm (EC/SQP) by comparing it with a standard... |
Experiment 1: Scalability Test With the aim of testing the robustness of the proposed method under increasing problem size, we seek to optimize the Ackley test problem using the EC/SQP algorithm with the problem size set to [MATH] [MATH] and [MATH] dimensions. The results of this experiment are shown in figure 5.2 . Th... |
Experiment 2: Performance Comparison Test This is a fitness comparison test where the proposed EC/SQP algorithm is compared to a standard evolutionary algorithm EC and the well known evolutionary strategy method called CMA-ES algorithm. The results of this experiment are shown in figure 5.3 . Similar to experiment 1, w... |
Discussions This study provided additional evidence on the fact that a skilful design of a hybrid system that combines the strengths of population-based and local search methods could yield an efficient and robust optimization system. The proposed hybrid EC/SQP algorithm combines a population-based global algorithm (EC... |
which may simply be put as: two hands are better than one . Recall that prior to the development of evolutionary methods and their hybrids, the notion of developing widely applicable optimization methods barely exists. Researchers were often focused on optimization methods that are mainly tailored to a specific problem... |
A careful look at the results of experiment 1 (see figure 5.2 ) reveals that for virtually all the varying sizes of the three different test problems, up to [MATH] of the maximum attainable fitness level was reached within the first [MATH] functions evaluations. This is indicative of the speed at which the proposed EC/... |
A possible explanation of the discrepancy shown in plot 5.2 (a) by the Ackley test function might be related to the fact that it is a pseudo-convex function with several local optima induced via its cosine component (see equation ( 5.1 )). These local optimum valleys become shallow and smooth as its dimensionality incr... |
Turning now to the findings from experiment [MATH] , the simulation results for the comparison tests are depicted in the six plots shown in figure 5.3 . Notice that for the three test problems under consideration (see left column), in overall, the proposed hybrid EC/SQP algorithm shows quite a significant performance i... |
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