US 6,516,309 B1Grant
Method and apparatus for evolving a neural network
Issue Date:2003-02-04
•20 Claims
•7 Drawing Sheets
Abstract
A method of evolving a neural network that includes a plurality of processing elements interconnected by a plurality of weighted connections includes the step of obtaining a definition for the neural network by evolving a plurality of weights for the plurality of weighted connections, and evolving a plurality of activation function parameters associated with the plurality of processing elements. Another step of the method includes determining whether the definition for the neural network may be simplified based upon at least one activation function parameter of the plurality of activation function parameters. Yet another step of the method includes updating the definition for the neural network in response to determining that the definition for the neural network may be simplified. The method utilizes particle swarm optimization techniques to evolve the plurality of weights and the plurality of activation parameters. Moreover, the method simplifies activation functions of processing elements in response to corresponding activation parameters meeting certain criteria, and removes processing elements from the definition of the neural network in response to corresponding activation parameters satisfying certain criteria. Various apparatus are also disclosed for implementing network evolution and simplification.
Metadata
Assignee
- Advanced Research & Technology Institute
Inventors
- Russell C. Eberhart
- Yuhui Shi
Application Information
Application Number:US 09/353,743
Filing Date:1999-07-14
Priority Date:1998-07-17
Art Unit:7
Classifications
IPC:
G06F 1518
Field of Search:
706 27706 14706 19706 25709206707 2382157382232382158713600
Patent Drawings (7 sheets)
Description
[0002] This application claims the benefit of U.S. Provisional Application Ser. No. 60/093,241 filed Jul. 17, 1998.
Field of the Invention
[0003] The present invention relates generally to artificial neural networks, and more particularly to a method and apparatus for evolving an artificial neural network.
Definitions
[0004] An artificial neural network (ANN) is used herein to refer to an analysis paradigm that is roughly modeled after the massively parallel structure of a biological neural network such as the human brain. An ANN is typically implemented with many relatively simple processing elements (PEs) that are interconnected by many weighted connections in order to obtain a computational structure that simulates the highly interconnected, parallel computational structure of biological neural networks. Hereinafter, the terms network and neural network are used interchangeably with the term artificial neural network.
[0005] The terms neural network topology and topology are used herein to refer to the number of PE layers of a neural network, number of PEs per PE layer of the neural network, and the interconnections between PEs of the neural network.
[0006] The term neural network architecture is used herein to refer to the neural network topology, activation functions implemented by the PEs of the neural network, and learning algorithms specified for a neural network.
[0007] The term evolutionary computation is used herein to refer to machine learning optimization and classification paradigms that are roughly based on evolutionary mechanisms such as biological genetics and natural selection. The evolutionary computational field includes genetic algorithms, evolutionary programming, genetic programming, and evolution strategies.
[0008] A swarm is used herein to refer to a population of interacting elements that collaboratively search through a problem space in order to optimize some global objective. Interactions between relatively local (topologically) swarm elements are often emphasized. Moreover, a swarm tends to have a general stochastic (or chaotic) characteristic that causes swarm elements to move toward a center of mass in the population located on critical dimensions, thus resulting in convergence on an optimum for the global objective of the swarm.
[0009] A particle swarm, as used herein, is similar to a genetic algorithm (GA) in that the system is initialized with a population of randomized positions in hyperspace that represent potential solutions to an optimization problem. However, each particle of a particle swarm, unlike a GA, is also assigned a randomized velocity. The particles (i.e. potential solutions) are then “flown” through hyperspace based upon their respective velocities in search of an optimum solution to a global objective.
Background of the Invention
[0010] Artificial neural networks and evolutionary computational techniques are effective in solving certain classes of problems. For example, artificial neural networks are good at mapping input patterns to output patterns in such applications as diagnostic systems. Moreover, evolutionary computational techniques are good at optimizing an objective in such applications as scheduling systems. In light of the fact that artificial neural networks and evolutionary computational techniques excel at different classes of problems, engineers and scientists have combined artificial neural networks and evolutionary computational techniques in order to develop hybrid computational tools that are even more effective than either methodology by itself.
[0011] For example, in Russ Eberhart, et al., Computational Intelligence PC Tools (1996) a particle swarm technique is described which evolves weights for weighted connections of a neural network. Use of the particle swarm technique has proven to be highly successful and efficient at accurately evolving neural network weights. However, the particle swarm technique described in Computational Intelligence PC Tools does not evolve the activation functions used by PEs of the neural network structure nor does the described particle swarm technique evolve aspects of the neural network topology such as the number of PE's used to implement the neural network. Accordingly, while the described particle swarm technique may successfully train a neural network, a simpler neural network (i.e. fewer PEs and/or less complex activation functions) may be obtainable if the particle swarm technique were extended to evolve additional aspects of the neural network.
[0012] A need, therefore, exists for a method and apparatus which evolve neural network weights and other neural network parameters to obtain a simpler neural network than achievable by evolving only neural network weights.
Summary of the Invention
[0013] In accordance with one embodiment of the present invention, there is provided a method of evolving a neural network that includes a plurality of processing elements interconnected by a plurality of weighted connections. One step of the method includes obtaining a definition for the neural network by evolving a plurality of weights for the plurality of weighted connections, and evolving a plurality of activation function parameters associated with the plurality of processing elements. Another step of the method includes determining whether the neural network definition may be simplified based upon at least one activation function parameter of the plurality of activation function parameters. The method also includes the step of updating the definition for the neural network in response to determining that the neural network definition may be simplified.
[0014] Pursuant to another embodiment of the present invention, there is provided another method of evolving a neural network that includes a plurality of processing elements interconnected by a plurality of weighted connections. The method includes the step of initializing a swarm of particles in which each particle includes (i) a velocity vector that represents motion of the particle through a hyperspace, and (ii) a position in the hyperspace that represents a plurality of weights for the plurality of weighted connections and a plurality of activation function parameters associated with the plurality of processing elements. The method also includes the step of determining for each particle of the swarm, a fitness value for a respective definition of the neural network that includes the respective plurality of weights defined by the particle and the plurality of activation function parameters defined by the particle. Another step of the method includes determining based upon the fitness values whether termination criteria have been satisfied. Moreover the method includes the steps of updating for the each particle of the swarm, a personal best value and a personal best position based upon the respective fitness value for the each particle, updating for the each particle of the swarm, a local best value and a local best position based upon fitness values associated with a respective group of the particles, and updating for the each particle of the swarm, the position and the velocity vector for the particle based upon the personal best position for the particle, the local best position for the particle, and the velocity vector for the particle. Finally, the method includes the step of repeating the above determining and updating steps until the termination criteria have been satisfied.
[0015] Pursuant to yet another embodiment of the present invention, there is provided a computer readable medium for evolving a neural network that includes a plurality of processing elements interconnected by a plurality of weighted connections. The computer readable medium includes code which when executed by a network evolution system causes the network evolution system to obtain a definition for the neural network by evolving a plurality of weights for the plurality of weighted connections, and evolving a plurality of activation function parameters associated with the plurality of processing elements. Moreover, the code of the computer readable medium when executed by the network evolution system further causes the network evolution system to determine whether the neural network definition may be simplified based upon at least one activation function parameter of the plurality of activation function parameters. The code of the computer readable medium when executed by the network evolution system also causes the network evolution system to update the definition for the neural network in response to determining that the neural network definition may be simplified.
[0016] Pursuant to a further embodiment of the present invention, there is provided a computer readable medium for evolving a neural network that includes a plurality of processing elements interconnected by a plurality of weighted connections. The computer readable medium includes code which when executed by a network evolution system causes the network evolution system to initialize a swarm of particles in which each particle includes (i) a velocity vector that represents motion of the particle through a hyperspace, and (ii) a position in the hyperspace that represents a plurality of weights for the plurality of weighted connections and a plurality of activation function parameters associated with the plurality of processing elements. The code of the computer readable medium when executed by the network evolution system further causes the network evolution system to determine for each particle of the swarm, a fitness value for a respective definition of the neural network that includes the respective plurality of weights defined by the particle and the plurality of activation function parameters defined by the particle, and determine based upon the fitness values whether termination criteria have been satisfied.
[0017] Furthermore, the code of the computer readable medium when executed by the, network evolution system causes the network evolution system to (i) update for the each particle of the swarm, a personal best value and a personal best position based upon the respective fitness value for the each particle, (ii) update for the each particle of the swarm, a local best value and a local best position based upon fitness values associated with a respective group of the particles, and (iii) update for the each particle of the swarm, the position and the velocity vector for the particle based upon the personal best position for the particle, the local best position for the particle, and the velocity vector for the particle. Moreover, the code of the computer readable medium when executed by the network evolution system causes the network evolution system to repeat the above determining and updating actions until the termination criteria have been satisfied.
[0018] Pursuant to yet a further embodiment of the present invention, there is provided a network evolution system for evolving a neural network that includes a plurality of processing elements interconnected by a plurality of weighted connections, the network evolution system. The network evolution system includes a network evolver and a network simplifier. The network evolver is operable to obtain a definition for the neural network by evolving a plurality of weights for the plurality of weighted connections, and evolving a plurality of activation function parameters associated with the plurality of processing elements. The network simplifier is operable to (i) determine whether the definition may be simplified based upon at least one activation function parameter of the plurality of activation function parameters, and (ii) update the definition for the neural network in response to determining that the neural network definition may be simplified.
[0019] It is an object of the present invention to provide a new and useful method and apparatus for evolving neural networks.
[0020] It is also an object of the present invention to provide an improved method and apparatus for evolving neural networks.
[0021] It is another object of the present invention to provide a method and apparatus for evolving both connection weights and processing element activation functions of neural networks.
[0022] It is yet another object of the present invention to provide a method and apparatus for simplifying a neural network topology.
[0023] It is a further object of the present invention to provide a method and apparatus for simplifying processing element activation functions of a neural network architecture.
[0024] It is a further object of the present invention to provide a method and apparatus for evolving a neural network that may directly process non-normalized (i.e. raw) data input signals.
[0025] The above and other objects, features, and advantages of the present invention will become apparent from the following description and the attached drawings.
Brief Description of the Drawings
[0026] FIG. 1 shows a block diagram of an exemplary neural network;
[0027] FIG. 2 shows a block diagram of an exemplary network evolution system suitable for evolving parameters of the neural network shown in FIG. 1;
[0028] FIGS. 3A-3B show a flowchart of a network evolution method implemented by the exemplary network evolution system of FIG. 2; and
[0029] FIG. 4 shows a flowchart of a network simplification method implemented by the exemplary network evolution system of FIG. 2;
[0030] FIG. 5 illustrates the manner by which the network simplification method of FIG. 4 removes a processing element of the exemplary neural network of FIG. 1; and
[0031] FIG. 6 shows a block diagram of a general purpose computer system which is suitable for implementing the exemplary neural network of FIG. 1 and the network evolution system of FIG. 2.
Detailed Description of Exemplary Embodiments
[0032] While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that there is no intent to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.
[0033] Exemplary Neural Network
[0034] Referring now to FIG. 1, there is illustrated an exemplary neural network 100. As depicted, the neural network 100 includes an input layer Fxof processing elements PEx0, PEx1, . . . PExn, a hidden layer Fyof processing elements PEy0, PEy1, . . . PEyq, and an output layer Fzof processing elements PEz1, PEz2, . . . PEzp. In general, each processing element PExhof the input layer Fxis coupled to each processing element PEyiof the hidden layer Fyvia a matrix of weighted connections W. Moreover, each processing element PEyiof the hidden layer Fyis coupled to each processing element PEzjof the output layer Fzvia a matrix of weighted connections U.
[0035] The exemplary neural network 100 of FIG. 1 is commonly referred to as a fully connected feed-forward neural network since a weighted connection exists between each processing element of adjacent layers and no signal path of the neural network 100 passes through a processing element more than once. While the exemplary neural network 100 is a fully connected feed-forward neural network, it should be appreciated by those skilled in the art that features of the present invention may be applied to sparsely connected neural network topologies, randomly connected neural network topologies, single layer network topologies, and/or feed-back neural network topologies.
[0036] Referring to FIG. 1 in more detail, the input layer Fxof the neural network 100 includes a biasing processing element PEx0and input processing elements PEx1, PEx2, . . . PExn. The biasing processing element PEx0essentially provides an internal bias for each of the hidden processing elements PEy1, PEy2, . . . PEyq. To this end, the biasing processing element PEx0of the input layer Fxis operable to generate a constant output signal x0(e.g. a signal having a value of 1) which is propagated to the hidden processing elements PEy1, PEy2, . . . PEyqvia weighted connections W10, W20, . . . Wq0of the weighted connections matrix W.
[0037] The input processing elements PEx1, PEx2, . . . PExnessentially distribute input signals a1k, a2k, . . . ankof an input pattern Akto the hidden processing elements PEy1, PEy2, . . . PEyq. To this end, each input processing element PExhis operable to (i) receive a single input signal ahkof the input pattern Ak, and (ii) generate a corresponding output signal xhwhich is propagated to the hidden processing elements PEy1, PEy2, . . . PEyqvia the weighted connections matrix W. More specifically, each input processing element PExhof the exemplary neural network 100 is operable to generate an output signal xhthat is equal to its respective input signal ahk. Moreover, the weighted connections W1h, W2h, . . . Wqhassociated with each input processing element PExhare operable to propagate the generated output signal xhto each of the hidden layer processing elements PEy1, PEy2, . . . PEyq.
[0038] The hidden layer Fyof the neural network 100 includes a biasing processing element PEy0and the hidden processing elements PEy1, PEy2, . . . PEyq. The biasing processing element PEy0essentially provides an internal bias for each of the output processing elements PEz1, PEz2, . . . PEzp. To this end, the biasing processing element PEy0of the hidden layer Fyis operable to generate a constant output signal y0(e.g. a signal having a value of 1) which is propagated to the output processing elements PEz1, PEz2, . . . PEzpvia weighted connections u01, u02, . . . u0pof the weighted connections matrix U.
[0039] The hidden processing elements PEy1, PEy2, . . . PEyqessentially generate output signals y1, y2, . . . yqthat are a function of the received output signals x0, x1, . . . xnand weighted connections W. More specifically, each hidden processing element PEy1generates an output signal yithat is a function of the received output vector X (i.e. output signals x0, x1, . . . xn) and associated weighted connections vector Wi(i.e weighted connections wi0, wi1, . . . win. Accordingly, each output signal yiof the hidden processing elements PEy1, PEy2, . . . PEyqmay be represented mathematically as follows:
[0040] yi=F(X, Wi) (1)
[0041] where F represents the processing element function which is also commonly referred to as the activation function of the processing element.
[0042] In implementing the activation function F, each hidden processing element PEyiof the exemplary neural network 100 performs a combinatory function c( ) of its corresponding inputs X and Wiand passes the resulting combinatory value c through a threshold function f( ). More specifically, each hidden processing element PEyiof the exemplary neural network 100 performs a linear combination (i.e. dot product) of its corresponding inputs X and Wiand passes the resulting dot product value c through a sigmoid threshold function. The following hidden layer signal equation (2) represents the output signal yias a threshold function f( ) of the combinatory function c( ) where the combinatory function c( ) is implemented as the linear combination of inputs X and Wi:
[0043] The following equation (3) represents the sigmoid threshold function used by each hidden processing element PEyi.
[0044] where α represents a slope factor of the sigmoid function that in essence scales the inputs X and Wiof the hidden processing element PEyi.
[0045] The output layer Fzof the neural network 100 includes the output processing elements PEz1, PEz2, . . . PEzp. The output processing elements PEz1, PEz2, . . . PEzpessentially generate output signals z1, z2, . . . Zpthat are a function of the received hidden layer output signals y0, y1, . . . yqand the weighted connections matrix U. More specifically, each output processing element PEzjgenerates an output signal zjthat is a function of a received output vector Y (i.e. output signals y0, y1, . . . yq) and associated weighted connection vector Uj(i.e. weighted connections uj0, uj1, . . . ujp). Accordingly, each output signal zjof the output processing elements PEz1, PEz2, . . . PEzpmay be represented mathematically as follows:
[0046] zj=F(Y, Uj) (4)
[0047] where F represents the activation function of the processing element function.
[0048] In implementing the activation function F, each output processing element PEzjof the exemplary neural network 100 performs a combinatory function c( ) of its corresponding inputs Y and Ujand passes the resulting combinatory value c through a threshold function f( ). More specifically, each output processing element PEzjof the exemplary neural network 100 performs a linear combination (i.e. dot product) of its corresponding inputs Y and Ujand passes the resulting dot product value c through a sigmoid threshold function. The following output layer signal equation (5) represents the output signal zjas a threshold function f( ) of the combinatory function c( ) where the combinatory function c( ) is implemented as the linear combination of inputs Y and Uj:
[0049] where f( ) represents the above sigmoid threshold function which is presented again with a slope factor of β instead of α so that the slope factors β1, β2: . . . βpof the output processing elements PEz1, PEz2, . . . PEzpare easily distinguishable from the slope factors 1, α2: . . . αqof the hidden processing elements PEy1, PEy2, . . . PEyq:
[0050] During operation, the neural network 100 essentially receives an input pattern Akand generates a respective output pattern Zkbased upon the processing element activation functions F and weighted connections matrices W and U. More specifically, the input layer Fxreceives an input pattern Akof input signals a1k, a2k, . . . ank, and generates a corresponding input layer signals x1, x2, . . . xnthat are propagated to the hidden layer Fyvia the weighted connections matrix W. Moreover, the input layer Fxgenerates a biasing signal x0that is also propagated to the hidden layer Fyvia the weighted connections matrix W. The hidden layer Fythen generates hidden layer signals y1, y2, . . . yqthat are based upon the received biasing signal x0, the input layer signals x1, x2, . . . xn, the weighted connections matrix W, and the activation functions F of the hidden processing elements PEy1, PEy2, . . . PEyq. Moreover, hidden layer Fygenerates a biasing signal y0which is propagated along with the hidden layer signals y1, y2, . . . yqto the output layer Fzvia the weighted connections matrix U. The output layer Fzthen generates output signals z1, z2, . . . zpbased upon the received biasing signal y0, the hidden layer signals y1, y2, . . . yn, weighted connections matrix U, and the activation functions F of the output processing elements PEz1, PEz2, . . . PEzp.
[0051] While the input processing elements PEx1, PEx2, . . . PExnof the exemplary neural network 100 essentially implement an identity activation function F( ) that generates an output signal that is equal to a received input signal, the input processing elements PEx1, PEx2, . . . PExnmay also be implemented in a manner similar to the processing elements of the hidden layer Fyand the output layer Fz. Moreover, while the processing elements of the hidden layer Fyand the output layer Fzutilize a linear combination function and a sigmoid threshold function, it should be appreciated by those skilled in the art that the activation functions F( ) of the processing elements may be implemented in several other known manners. More specifically, the activation function F( ) may use a different combinatory function c( ) and pass the combinatory result c through a different threshold function f( ). For example, TABLE 1 discussed below lists several alternatives for the threshold function f( ). Moreover, it should be appreciated that while the exemplary processing elements combine the received signals and pass the result through a threshold function, the activation function F( ) may be implemented in such a manner as to generate an output signal directly from the received signals without first performing a combinatory function of the received inputs.
[0052] Exemplary Network Evolution System
[0053] The operation of the neural network 100 is directly related to the connection weights W and U, the activation functions F of the processing elements, and the number of processing elements that implement the neural network 100. Therefore, in order for the neural network 100 to generate appropriate output patterns Zkfrom received input patterns Ak, the exemplary neural network 100 must have properly defined weighted connections matrices W and U, activation functions F, and processing element layers Fx, Fy, and Fz. To this end, a network evolution system 200 is shown in FIG. 2 that defines and/or evolves these parameters of the exemplary neural network 100 in order to obtain a trained neural network suitable for processing input data patterns Ak.
[0054] The network evolution system 200 includes a network evolver 202, a network simplifier 204, and a network verifier 206. In general, the network evolver 202 is operable to (i) define an initial neural network architecture for the neural network 100, and (ii) continually adjust the architecture of the neural network 100 until the network evolver 202 obtains a definition for the neural network 100 that meets predefined criteria. More specifically, the network evolver 202 in an exemplary embodiment is operable to (i) apply input patterns Akof a training pattern set TRAINSETto the input layer Fxof the neural network 100, and (ii) adjust parameters of the neural network 100 based upon output patterns Zkgenerated by the neural network 100 in response to the input patterns Ak. As will be explained in more detail in reference to FIG. 3A-3B, the network evolver 202 in an exemplary embodiment includes a particle swarm optimizer which is operable to adjust parameters of the neural network 100 in such a manner so as to achieve a trained neural network 100 that generates appropriate output patterns Zkin response to processing input patterns Ak.
[0055] The network simplifier 204 of the network evolution system 200 is generally operable to simplify a definition for the neural network 100. As will be explained in more detail with reference to the network simplification method 400 of FIG. 4, the network simplifier 204 in an exemplary embodiment is operable to (i) receive a definition for the neural network 100 from the network evolver 202, (ii) redefine certain processing elements of the neural network 100 such that the processing element implements a simpler activation function, and (iii) remove unnecessary processing elements from the definition of the neural network 100.
[0056] Finally, the network verifier 206 of the network evolution system 200 is generally operable to verify the accuracy of the obtained simplified network definition for the neural network 100. More specifically, the network verifier 206 in an exemplary embodiment is operable to (i) apply input patterns Akof a test pattern set TESTSETto the input layer Fxof the simplified definition of the neural network 100, and (ii) generate a fitness value that is indicative of how well the trained neural network 100 as defined by obtained simplified definition is able to produce output patterns Zkin response to processing input patterns Ak. In an exemplary embodiment, the network verifier 206 generates the fitness value for the trained neural network by calculating an average sum-squared error between the generated output patterns Zkand expected patterns Bkof the test pattern set TESTSET. (See, below equation (8) for details on performing an average sum-squared error calculation.)
[0057] From the produced fitness value for the simplified network definition, the network evolution system 200 is operable to determine whether the neural network 100 has been successfully trained. In particular, the network evolution system 200 in an exemplary embodiment determines that the neural network 100 has been successfully trained if the fitness value (e.g. average sum-squared error) for the simplified definition has a predetermined relationship to (e.g. less than) a fitness threshold value FITTHR(e.g. 0.01).
[0058] Exemplary Network Evolution Method
[0059] Referring now to FIGS. 3A-3B, a flowchart of a network evolution method 300 that illustrates in detail the operation of the network evolver 202 of the exemplary network evolution system 200. As illustrated in FIGS. 3A-3B, the network evolver 202 begins in step 302 by initializing an iteration counter ITER and determining various parameters to be used in evolving the neural network 100. More specifically, the network evolver 202 in an exemplary embodiment initializes the iteration counter ITER by setting the iteration counter to a value of 1.
[0060] Moreover, the network evolver 202 in an exemplary embodiment obtains various parameter values from a configuration file that define (i) a number of particles P# (e.g. 20) for a particle swarm S, (ii) a maximum particle velocity value VMAX(e.g. 10.0), (iii) a maximum position value POSMAX(e.g. 10.0), (iv) a dynamic initial lower limit value LOWER0(e.g. −0.5), (v) a dynamic initial upper limit value UPPER0(e.g. 0.5), (vi) a starting inertia weight wi0(e.g. 0.9), (vii) a slope upper limit SLOPEULIM(e.g. 90), an error cutoff ECUT(e.g. 0.025), (vii) a maximum number of iterations ITERMAX(e.g. 1000), a training pattern set TRAINSET(e.g. a file that includes training pairs of input patterns Akand corresponding expected output patterns Bk), a number of pattern pairs PAT# in the training pattern set TRAINSET(e.g. 500), a testing pattern set TESTSET(e.g. a file that includes testing pairs of input patterns Akand corresponding expected output patterns Bk), and a number of pattern pairs TPAT# in the testing pattern set TESTSET(e.g. 300).
[0061] After obtaining the above parameters, the network evolver 202 in step 304 generates an initial topology for the neural network 100. More specifically the network evolver 202 in an exemplary embodiment defines an appropriate three layer, fully connected, feed-forward network topology for the neural network 100. (See, FIG. 1.) To this end, the network evolver 202 determines the number p of output signals bjkthat each expected output pattern Bkincludes, and the number n of input signals ahkthat each input pattern Akincludes. From this information, the network evolver 202 defines an initial topology for the neural network 100 that includes (i) an input layer Fxhaving a biasing processing element PEx0and n input processing elements PExh, (ii) an output layer Fzhaving p output processing elements PEzj, and (iii) a hidden layer Fyhaving a biasing processing element PEy0and q hidden processing elements PEyi.
[0062] A suitable number q of hidden processing elements can vary widely according to the application and bears a relationship to the number of statistically significant factors that exist in the input data. If there are too few hidden processing elements PEyi, the network evolver 202 will probably fail to train the neural network 100. If there are just barely enough, the network evolver 202 may successfully train the neural network 100, but the resulting neural network 100 may fail to generate appropriate output patterns Zkfor input patterns Akthat were not part of the training pattern set TRAINSET. Moreover, the resulting neural network 100 will probably not handle noisy data well. Conversely, if there are too many hidden processing elements PEyi, the resulting neural network 100 probably will not generalize very well. In other words, the resulting neural network 100 probably will not generate appropriate output patterns Zkfor input patterns Akthat were not part of the training pattern set TRAINSET.
[0063] Accordingly, the neural network may need to be trained several different times with different values of q until a suitable number q of hidden processing elements PEyiis found. A suitable number q of hidden processing elements PEyi, however, may often be obtained by taking the square root of the number n of input processing elements PExhsquared plus the number p of output processing elements squared PEzjplus a few additional processing elements. This relationship for q is represented mathematically by the following equation (7):
[0064] q=ceil({square root over (no (7)
[0065] where n represents the number of input processing elements PExh, p represents the number of output processing elements PEzj, ceil( ) represents a ceiling function which rounds a non-integer number up to the next integer number, and o represents a small integer with respect to ceil({square root over (n2+p2)}).
[0066] In an exemplary embodiment of the present invention, the values n, q, and p are supplied by a user via a configuration file that contains appropriate values for n, q, and p as well as the above discussed parameters. However, it should be appreciated that a user may alternatively supply the values n, q, and p as well as the above discussed parameters via an input device such as a mouse or a keyboard. Accordingly, the network evolver 202 in step 304 may determine appropriate values for n, q and p based upon (i) the input patterns Akand output pattern Bkof the training pattern set TRAINSET, and/or (ii) user supplied parameters received via an input device and/or configuration file.
[0067] After defining an initial network topology for the neural network 100, the network evolver 202 in step 306 initializes a swarm S of particles P0, P1, . . . Pp# that represent P# possible definitions for the neural network 100. More specifically, the network evolver 202 defines for each particle Pxof the swarm S, (i) a position POSxin D-dimensional hyperspace, and a velocity vector Vxthrough the D-dimensional hyperspace. More specifically, the network evolver 202 defines the D-dimensional hyperspace for the neural network 100 such that each dimension of the D-dimensional hyperspace represents a weighted connection whiof the weighted connections matrix W, a weighted connection uijof the weighted connections matrix U, a slope factor αiof a slope vector A, or a slope factor βjof a slope vector B.
[0068] For example, if the neural network 100 is initially defined to include an (i) input layer Fxhaving a biasing processing element PEx0and two input processing elements PEx1, and PEx2, a hidden layer Fyhaving a biasing processing element PEy0and four hidden processing elements PEy1, PEy2, . . . and PEy4, and an output layer Fzhaving a single output processing element PEz1, then the neural network 100 would have a weighted connections matrix W consisting of 12 weighted connections w10, w20, . . . w40, w11, w21, . . . w41, w12, w22, . . . w42, a weighted connections matrix U consisting of 5 weighted connections u10, u11, . . . u14, a slope vector A consisting of 4 slope factors α1, α2, . . . α4, and a slope vector B consisting of 1 slope factor β1. Therefore, the network evolver 202 in this example, would define a 22-dimensional hyperspace in which each position in the 22-dimensional hyperspace represents a possible solution for the 12 weighted connections of the weighted connections matrix W, the 5 weighted connections of the weighted connection matrix U, the 4 slope factors of the slope vector A, and the 1 slope factor of the slope vector B.
[0069] In an exemplary embodiment, the network evolver 202 in step 308 randomly assigns each particle Pxof the particle swarm S an initial position POSxand velocity vector Vx. More specifically, the network evolver 202 randomly assigns the initial positions POS1, POS2, . . . POSP# of the particles P1, P2, . . . PP# such that the weight connections whiand uijrepresented by the positions POS1, POS2, . . . POSP# are between the initial lower limit LOWER0and initial upper limit UPPER0. Moreover, the network evolver 202 assigns the initial positions POS1, POS2, . . . POSP# of the particles P1, P2, . . . PP# such that the slope factors αiand βjare initially equal to 1. Furthermore, the network evolver randomly assigns the D-dimensional velocity vectors V1, V2, . . . VP# of the particles P1, P2, . . . PP# such that each dimensional velocity component vx1, vx2, vxDof a velocity vector Vxis between the initial lower limit LOWER0and the initial upper limit UPPER0.
[0070] After initializing the positions POS1, POS2, . . . POSP# of the particles P1, P2, PP#, the network evolver 202 in step 310 determines a personal best value PBESTx, a personal best position PBESTXx, a local best value LBESTx, and a local best position LBESTXxfor each particle the particles P1, P2, . . . PP#. More specifically, each personal best value PBESTxrepresents the corresponding best definition obtained by the particle Pxfor the neural network 100, and each personal best position PBESTXxrepresents the position in hyperspace where the particle Pxobtained its corresponding particle best value PBESTx. Similarly, each local best value LBESTxrepresents the corresponding best definition obtained by a particle group PGxthat includes particles Px−L, . . . Px−1, Px, Px+1, . . . Px+L, and each local best position LBESTXxrepresents the position in hyperspace where the particle group PGxobtained its corresponding local best value LBESTx.
[0071] In an exemplary embodiment, the particle groups PG1, PG2, . . . PGP# are defined in a circular array fashion based upon a local neighbor parameter L. More specifically, if the local neighbor parameter L is equal to 2 and the number of particles is equal to 20, then the first particle group PG1would include particles P19, P20, P1, P2, and P3and the second particle group PG2would include the particles P20, P1, P2, P3, and P4. It should be appreciated by those skilled in the art that if the local neighbor parameter L is equal to or greater than one-half the number of particles P#, then a single local best value LBEST and corresponding local best position LBESTX may be used since all of the particle groups PG1, PG2, . . . PGP# would include every particle P1, P2, . . . PP# of the particle swarm S. This special case is referred to as a global version of the particle swarm optimizer implemented by the network evolver 202. Moreover, the single local best value LBEST and corresponding local best position LBESTX in this special case are referred to as the global best value GBEST and the global best position GBESTX, respectively.
[0072] It has been found that the larger the local neighbor parameter L becomes the quicker (i.e. less iterations) on average the particle swarm optimizer of the network evolver 202 converges to an optimum. However, as the local neighbor parameter L becomes larger, the particle swarm optimizer of the network evolver 202 is more likely to converge on a local optimum instead of a global optimum. In other words, as the local neighbor parameter L becomes larger, the more likely the particle swarm optimizer of the network evolver 202 will fail to obtain a definition for the neural network 100 that achieves a desired level of performance. Accordingly, in an exemplary embodiment, the network evolver 202 utilizes a local neighbor parameter L of 2 which has been found to cause the particle swarm optimizer to converge on a global optimum as opposed to a local optimum at highly successful rate.
[0073] In order to determine a particle best value PBESTxand a particle best position PBESTXxfor each particle Pxof the particle swarm S, the network evolver 202 computes a fitness value FVxfor each definition of the neural network 100 as defined by the particle positions POS1, POS2, . . . POSP#. More specifically, the network evolver 202 in an exemplary embodiment calculates a fitness value FVxfor each particle Pxbased upon a fitness function FIT( ) of (i) the output patterns Z1, Z2, . . . ZPAT# generated in response to propagating the input patterns A1, A2, . . . APAT# of the training pattern set TRAINSETthrough the neural network 100 as defined by the selected particle Px, and (ii) the corresponding expected output patterns B1, B2, . . . BPAT# of the training pattern set TRAINSET. In an exemplary embodiment of the present invention, the network evolver 202 calculates the fitness value FVxof a particle Pxbased upon the following fitness function FIT( ) which computes the average sum-squared error between the output patterns Z1, Z2, . . . ZPAT# and the expected output patterns B1, B2, . . . BPAT#:
[0074] where q represents the number of output processing elements of the neural network output layer Fz, zkjrepresents the output signal of the output processing element PEzjin response to the input pattern Akbeing applied to the neural network input layer Fx, bkjrepresents the corresponding expected output signal of the output processing element PEzj, and PAT# represents the number of patterns of the test pattern set TRAINSET.
[0075] To this end of generating fitness values FV1, FV2, . . . FVP# for the particles FV1, FV2, . . . FVP#, the network evolver 202 in step 312 initializes a particle index N to a value of 1 in order to cause the particle index N to identify the first particle P1of the particle swarm S. The network evolver 202 then in step 314 selects the particle PNidentified by the particle index N.
[0076] After selecting the particle PNidentified by the particle index N, the network evolver 202 in step 316 calculates a fitness value FVNfor the particle definition of the neural network 100 via the above fitness function FIT( ). In particular, the network evolver 202 in an exemplary embodiment generates output patterns Z1, Z2, . . . ZPAT# in response to applying the input patterns A1, A2, . . . APAT# of the training pattern set TRAINSETto a definition of the neural network 100 as defined by the position POSNof the selected particle PN. More specifically, the network evolver 202 generates the output patterns Z1, Z2, . . . ZPAT# based upon a definition of the neural network 100 in which the neural network 100 has a weighted connections matrix W, a weighted connections matrix U, a slope vector A, and a slope vector B as defined by the position POSNof the selected particle PN. Moreover, the network evolver 202, in the exemplary embodiment, generates a fitness value FVNfor the selected particle PNbased upon above equation (8) which causes the network evolver 202 to calculate the average sum-squared error between the generated output patterns Z1, Z2, . . . ZPAT# and the expected output patterns B1, B2, . . . BPAT# of the test pattern set TRAINSET.
[0077] After obtaining the fitness value FVNfor the selected particle PN, the network evolver 202 in step 318 determines whether the personal best value PBESTNand personal best position PBESTXNfor the selected particle PNneed to be updated. To this end, the network evolver 202 determines whether the obtained fitness value FVNfor the selected particle PNis better than the current personal best value PBESTNfor the selected particle PN. If the obtained fitness value FVNfor the selected particle PNis better than the current personal best value PBESTNfor the selected particle PN, then the network evolver 202 proceeds to step 320 in order to update the personal best value PBESTNand personal best position PBESTXNfor the selected particle PN. Otherwise, if the obtained fitness value FVNfor the selected particle PNis not better than the current personal best value PBESTNfor the selected particle PN, then the network evolver 202 proceeds to step 322 in order to determine whether a fitness value FVxhas been obtained for each particle Px.
[0078] In an exemplary embodiment, the network evolver 202 attempts to minimize the fitness value FVx(i.e. average sum-squared error) for the neural network 100. Accordingly, in the exemplary embodiment, the network evolver 202 determines that an obtained fitness value FVNis better than a personal best value PBESTNif the fitness value FVNis less than the personal best value PBESTN. However, it should be appreciated that the fitness function FIT( ) may be defined in such a way that the network evolver 202 needs to maximize the fitness values FVxin order to properly train the neural network 100. Accordingly, in such a maximization environment, the network evolver 202 would determine that the a fitness value FVNis better than a personal best value PBESTNif the fitness value FVNis greater than the personal best value PBESTN. Moreover, it should be appreciated that the fitness functions of equation (8) is merely exemplary and that other fitness functions may be used.
[0079] After determining that the fitness value FVNis better than the personal best value PBESTNfor the selected particle PN, the network evolver 202 in step 320 updates the personal best value PBESTNand the personal best position PBESTXNfor the selected particle PN. More specifically, the network evolver 202 sets the personal best value PBESTNequal to the calculated fitness value FVNfor the selected particle PN. Moreover, the network evolver 202 sets the personal best position PBESTNequal to the position POSNof the selected particle PN.
[0080] It should be noted that in an exemplary embodiment, the personal best values PBESTxare initially set to poor values so that network evolver 202 during the first iteration through the network evolution method 300 (i.e. iteration counter ITER equal to 1) will update the personal best values PBESTNwith the calculated fitness value FVNfor the selected particle PN.
[0081] In step 322, the network evolver 202 determines whether a fitness value FVxhas been obtained for each particle P1, P2, . . . PP# of the particle swarm S. If the network evolver 202 determines that a fitness value FVxhas been obtained for each particle P1, P2, . . . PP# of the particle swarm S, then the network evolver 202 proceeds to step 326 of the network evolution method 300. However, if the network evolver 202 determines that a fitness value FVxhas not been obtained for each particle P1, P2, . . . PP# of the particle swarm S, then the network evolver 202 proceeds to step 324 of the network evolution method 300. In an exemplary embodiment of the present invention, the network evolver 202 determines that a fitness value FVxhas been obtained for each particle P1, P2, . . . PP# of the particle swarm S if the particle index N is greater than the number of particles P# included in the particle swarm S.
[0082] After determining that a fitness value FVxhas not been obtained for each particle P1, P2, . . . PP#, the network evolver 202 in step 324 updates the particle index N and returns to step 314 in order to select the particle PNidentified by the updated particle index N and obtain a fitness value FVNfor the newly selected particle PN. In an exemplary embodiment, the network evolver 202 updates the particle index N by incrementing the particle index N by a value of 1.
[0083] After determining that a fitness value FVxhas been obtained for each particle P1, P2, . . . PP#, the network evolver 202 updates the local best value LBESTxand local best position LBESTXxfor each particle P1, P2, . . . PP#. To this end, the network evolver 202 in step 326 initializes a group index M to a value of 1 in order to obtain a group index M that identifies a first particle group PG1.
[0084] Then, the network evolver 202 in step 328 determines whether any particle PM−L, . . . PM−1, PM, PM+1, . . . PM+Lof the identified group PGMhas a personal best value PBESTxthat is better than the current local best value LBESTMfor the identified particle group PGM. If any of the personal best values PBESTM−L, . . . PBESTM−1, PBESTM, PBESTM+1, . . . PBESTM+Lof the identified group PGMis better than the current local best value LBESTMfor the identified particle group PGM, then the network evolver proceeds to step 330 in order to update the local best value LBESTMfor the identified particle group PGM. Otherwise, if none of the personal best values PBESTM−L, . . . PBESTM−1, PBESTM, PBESTM+1, . . . PBESTM+Lof the identified group PGMare better than the current local best value LBESTMfor the identified particle group PGM, then the computer system proceeds to step 332 in order to determine whether all particle groups PG1, PG2, . . . PGP# have been processed.
[0085] After determining that at least one personal best value PBESTM−L, . . . PBESTM−1, PBESTM, PBESTM+1, . . . PBESTM+Lof the identified group PGMis better than the current local best value LBESTM, the network evolver 202 in step 330 updates the local best value LBESTMand the local best position LBESTXMfor the identified particle group PGM. More specifically, the network evolver 202 sets the local best value LBESTMequal to the best, personal best value PBESTBof the identified group PGM. Moreover, the network evolver 202 sets the local best position LBESTMequal to the personal best position PBESTXBassociated with the best, personal best value PBESTBof the identified group PGM.
[0086] It should be noted that in an exemplary embodiment, the local best values LBESTxare initially set to poor values so that the network evolver 202 during the first iteration through the network evolution method 300 (i.e. iteration counter ITER equal to 1) will update the local best values LBESTxwith one of the personal best values PBESTM−L, . . . PBESTM−1, PBESTM, PBESTM+1, . . . PBESTM+Lof its respective particle group PGx. In another exemplary embodiment, the network evolver 202 during the first iteration through the network evolution method 300 sets each LBESTxof a particle Pxequal to its respective personal best value PBESTxin step 320. Under either exemplary embodiment, each local best value LBESTxshould be equal to the best, personal best value PBESTX−L, . . . PBESTX−1, PBESTx, PBESTx+1, . . . PBESTx+Lof its respective particle group PGxafter the first iteration through the network evolution method 300.
[0087] In step 332, the network evolver 202 determines whether all of the particle groups PG1, PG2, . . . PGP# have been processed. If all of the particle groups PG1, PG2, . . . PGP# have been processed, then the network evolver 202 proceeds to step 336 in order to determine whether termination criteria have been satisfied. However, if all of the particle groups PG1, PG2, . . . PGP# have not been processed, then the network evolver 202 proceeds to step 334. In an exemplary embodiment, the network evolver 202 determines that all of the particle groups PG1, PG2, . . . PGP# have been processed if the particle group index M is greater than the number of particle P# of the swarm S.
[0088] In step 334, the network evolver 202 updates the particle group index M and returns to step 328 in order to process the next particle group PGM. In an exemplary embodiment, the network evolver 202 updates the particle group index M by incrementing the particle group index M by a value of 1.
[0089] After processing all of the particle groups PG1, PG2, . . . PGP#, the network evolver 202 in step 336 determines whether defined termination criteria have been satisfied. If the network evolver 202 determines that the defined termination criteria have been satisfied, then the network evolver 202 proceeds to step 344 in order define the weighted connections matrix W, the weighted connections matrix U, the slope vector A, and the slope vector B. However, if the network evolver 202 determines that the defined termination criteria have not been satisfied, then the network evolver 202 proceeds to step 338 in order to update the velocity vectors V1, V2, . . . VP# associated with the particles P1, P2, . . . PP#.
[0090] In an exemplary embodiment of the present invention, the termination criteria are defined by a maximum number of iterations ITERMAXand an error cutoff ECUT. More specifically, the network evolver 202 in an exemplary embodiment determines to terminate the network evolution method 300 in response to (i) the iteration counter ITER having a predetermined relationship to the maximum number of iterations ITERMAX, or (ii) the best of the local best values LBEST1, LBEST2, . . . LBESTP# having a predetermined relationship to the desired error cutoff ECUT. For example, in an exemplary embodiment which attempts to minimize the fitness values FVx, the network evolver 202 may be implemented to terminate the network evolution method 300 if either (i) the iteration counter ITER is equal to the maximum number of iterations ITERMAX, or (ii) the best of the local best values LBEST1, LBEST2, . . . LBESTP# is less than the desired error cutoff ECUT.
[0091] In step 338, the network evolver 202 updates the velocity vector Vxfor each particle Pxof the swarm S. More specifically, the network evolver 202 updates the velocity vector Vxof a particle Pxbased upon (i) an inertia weight wi, (ii) the personal best position PBESTXxof the particle Px, and (iii) the local best position LBESTXxfor the particle group PGxto which the particle Pxbelongs. In an exemplary embodiment, the network evolver 202 updates each velocity component vx1, vx2, . . . vxDof the velocity vector Vxfor a particle Pxbased upon the following velocity equation (9):
[0092] vxd′=wi*vxd+c1*rand( )*(pbestxxd−posxd)+c2*Rand( )*(lbestxxd−posxd) (9)
[0093] In the above velocity equation (9), vxd′ represents an updated velocity component of the velocity vector Vxin the dthdimension, vxdrepresents the current velocity component of the velocity vector Vxin the dthdimension, wi represents the inertia weight parameter, and c1and c2represent acceleration constants. Moreover, rand( ) and Rand( ) represent random functions that each generate a random number in the range between 0 and 1. Furthermore, posxdrepresents the position of the particle Pxin the dthdimension, pbestxxdrepresents the position of the personal best value PBESTxin the dthdimension, and lbestxxdrepresents the position of the local best value LBESTxin the dthdimension.
[0094] The inertia weight parameter wi and the acceleration constants c1and c2may be used to control the tension in the system. Lower values (less than 1) for c1and c2tend to restrict particles Pxto roam farther from their respective personal best values PBESTxand local best values LBESTx, whereas higher values (greater than 1) for the acceleration constants c1and c2tend to allow particles Pxto explore regions nearer their respective personal best values PBESTxand local best values LBESTx. Similarly, high values for the inertia weight parameter wi tend to allow particles Pxto roam farther from their respective personal best values PBESTxand local best values LBESTx, whereas low values for the inertia weight parameter wi tend to restrict particles Pxto regions nearer their respective personal best values PBESTxand local best values LBESTx.
[0095] In an exemplary embodiment of the present invention, the network evolver 202 sets the acceleration constants equal to a value of 2. Moreover, the network evolver 202 decreases the inertia weight wi linearly over the defined maximum iterations ITERMAXfrom the initial inertia weight wi0to a final inertia weight such as 0.4. The advantage of adjusting the inertia weight wi from a high value of 0.9 to a low value of 0.4 is that the particles P1, P2, . . . PP# initially perform a more global exploration and move gradually toward a more local exploration.
[0096] Moreover, in an exemplary embodiment, the network evolver 202 clips the updated velocity components vx1′, vx2′, . . . vxD′ such that no updated velocity component vxd′ is greater than the maximum velocity parameter VMAX. For example, if the network evolver 202 obtains an updated velocity component vxd′ equal to 11.3 and the maximum velocity parameter VMAXis set to 10.0, then the network evolver 202 would set the updated velocity component vxd′ equal to 10.0 instead of 11.3.
[0097] After updating the velocity vector Vxfor each particle Pxof the particle swarm S, the network evolver 202 in step 340 updates each position POSxof each particle Px. In particular, the network evolver 202 updates each position component posx1, posx2, . . . posxDof each position POSxbased upon the updated velocity vector Vxfor corresponding particle Px. To this end, the network evolver 202 in an exemplary embodiment updates each position component posx1, posx2, . . . posxDof a particle position POSxbased upon the following position equation (10).
[0098] posxd′=posxd+vxd′ (10)
[0099] where posxdrepresents the current position component of the particle position POSxin the dthdimension, posxd′ represents the updated position component of the particle position POSxin the dthdimension, and vxd′ represents the updated velocity component for the particle Pxin the dthdimension. Moreover, in an exemplary embodiment, the network evolver 202 clips the updated position components posx1′, posx2′, . . . posxD′ such that no updated position component posxd′ is greater than the maximum position parameter POSMAX. For example, if the network evolver 202 obtains an updated position component posxd′ equal to −11.3 and the maximum position parameter POSMAXis set to 10.0, then the network evolver 202 would set the updated position component posxd′ equal to −10.0 instead of −11.3. However, it should be appreciated that the network evolver 202 may also be implemented such that the updated positions components posx1′, posx2′, . . . posxD′ are not limited to a specified range.
[0100] In step 342, the network evolver 202 updates the iteration counter ITER and returns to step 310 in order to process the updated particles P1, P2, . . . PP# of the particle swarm S. In particular, the network evolver 202 in an exemplary embodiment increments the iteration counter ITER by a value of 1 before returning to step 310.
[0101] After the termination criteria are satisfied, the network evolver 202 in step 344 obtains a neural network definition for the neural network 100 from the particle swarm S. More specifically, the network evolver 202 obtains the position in the D-dimensional hyperspace that achieved that best fitness value and uses this position to define the weighted connections matrix W, the weighted connections matrix U, the slope matrix A, and the slope matrix B of the neural network 100. To this end, the network evolver 202 in an exemplary embodiment (i) obtains the best, personal best position PBESTXBassociated with the best of the personal best values PBEST1, PBEST2, . . . PBESTP#, (ii) sets each weighted connection whiof the weighted connections matrix W, each weighted connection uijof the weighted connections matrix U, each slope factor α of the slope factor A, and each slope factor β of the slope vector B equal to a respective position component pbestxB1, pbestxB2, . . . pbestBDof the best, personal best position PBESTXB.
[0102] Similarly, in another exemplary embodiment, the network evolver 202 (i) obtains the best, local best position LBESTXBassociated with the best of the local best values LBEST1, LBEST2, . . . LBESTP#, (ii) sets each weighted connection whiof the weighted connections matrix W, each weighted connection uijof the weighted connection matrix U, each slope factor α of the slope factor A, and each slope factor β of the slope vector B equal to a respective position component lbestxB1, lbestxB2, . . . lbestBDof the best, local best position PBESTXB. It should be appreciated that either of the two exemplary embodiments should achieve the same results for the weighted connections matrix W, the weighted connections matrix U, the slope vector A, and the slope vector B.
[0103] Exemplary Network Simplification Method
[0104] Referring now to FIG. 4, there is illustrated a flowchart of an network simplification method 400. In general, the network simplification method 400 when executed by the network simplifier 204 causes the network simplifier 204 to simplify a definition for the neural network 100 in order to obtain a less complex definition of the neural network 100. More specifically, the network simplifier 204 in implementing the network simplification method 400 generally (i) removes unnecessary processing elements from the neural network 100, and/or (ii) replaces complex activation functions of certain processing elements with less computationally complex activation functions.
[0105] To this end, the network simplifier 204 in step 402 determines based upon the slope vector A and the slope vector B whether any of the activation functions of the hidden processing elements PEy1, PEy2, . . . PEyqor the output processing elements PEz1, PEz2, . . . PEzpmay be simplified. In an exemplary embodiment, each hidden processing element PEyiis initially implemented with the sigmoid threshold function of equation (3) and each output processing element PEzjis initially implemented with the sigmoid threshold function of equation (6). As can be seen from equations (3) and (6), if the slope factors α and β are positive and have a sufficiently large magnitude, the sigmoid activation function essentially becomes the following step threshold function (11):
[0106] Similarly, if the slope factors α and β are negative and have a sufficiently large magnitude, the sigmoid threshold function essentially becomes the following step threshold function (12):
[0107] As a result of the above properties of the sigmoid threshold function, the network simplifier 204 in an exemplary embodiment determines that activation functions of the neural network 100 may be simplified if any of the slope factors α and β of the slope vectors A and B has a magnitude greater than a slope upper limit SLOPEULIM. It has been found that a slope upper limit SLOPEULIMas small as 10 often results in a simplified neural network 100 with minimal effect on the accuracy of the output patterns Zkgenerated by the neural network 100.
[0108] In step 404, the network simplifier 204 redefines the activation functions for those processing elements PE which may be implemented with a simpler activation function. In particular, for each hidden processing element PEyiwith a positive slope factor αihaving a magnitude greater than the slope upper limit SLOPEULIM, the network simplifier 204 in an exemplary embodiment replaces the sigmoid threshold function of the hidden processing element PEyiwith the step threshold function of equation (11). Moreover, for each hidden processing element PEyiwith a negative slope factor αihaving a magnitude greater than the slope upper limit SLOPEULIM, the network simplifier 204 replaces the sigmoid threshold function of the hidden processing element PEyiwith the step threshold function of equation (12). Similarly, for each output processing element PEzjwith a positive slope factor βjhaving a magnitude greater than the slope upper limit SLOPEULIM, the network simplifier 204 in an exemplary embodiment replaces the sigmoid threshold function of the output processing element PEzjwith the step threshold function of equation (11). Moreover, for each output processing element PEzjwith a negative slope factor βjhaving a magnitude greater than the slope upper limit SLOPEULIM, the network simplifier 204 replaces the sigmoid threshold function of the output processing element PEzjwith the step threshold function of equation (12).
[0109] The network simplifier 204 then in step 406 determines whether any of the hidden processing elements PEy1, PEy2, . . . PEyqmay be removed from the neural network 100. If the network simplifier 204 determines that at least one of the hidden processing elements PEy1, PEy2, . . . PEyqmay be removed from the neural network 100, then the network simplifier 204 proceeds to step 408. However, if the network simplifier 204 determines that none of the hidden processing elements PEy1, PEy2, . . . PEyqmay be removed from the neural network 100, then the network simplifier proceeds to step 410.
[0110] As stated above, each hidden processing elements PEyi, of the neural network 100 in an exemplary embodiment is initially implemented with the sigmoid threshold function of equation (3). As can be seen from the sigmoid threshold function of equation (3), the output of the sigmoid threshold function is roughly equal to a constant value of 0.5 when the magnitude of the slope factor α is less than a slope lower limit SLOPELLIM. Accordingly, the network simplifier 204 in an exemplary embodiment determines that a hidden processing element PEyimay be removed from the neural network 100 if the slope factor αiassociated with the hidden processing element PEyiis less than the slope lower limit SLOPELLIM. Appropriate values for the slope lower limit SLOPELLIMare dependent upon the application; however, the network simplifier 204 in an exemplary embodiment uses a lower slope limit SLOPELLIMof 0.1 which may be sufficient for many different types of applications.
[0111] In step 408, the network simplifier 204 removes those hidden processing elements PEyiidentified in step 406 as being appropriate for removal. To this end, for each hidden processing element PEyiwith a slope factor αihaving a magnitude less than the slope lower limit SLOPELLIM, the network simplifier 204 (i) removes the identified hidden processing element PEyi, (ii) removes the weighted connections vector Wifrom the weighted connections matrix W associated with the removed hidden processing element PEyi, (iii) updates the biasing weight connections u10, u20, . . . up0in order to replicate the function of the removed hidden processing element PEyi, and (iv) removes the weighted components u1i, u2i, . . . upifrom the weighted connections matrix U associated with the removed hidden processing element PEyi.
[0112] More specifically, since the removed hidden processing element PEyiessentially generated a constant output signal yiof 0.5, the removed hidden processing PEyiessentially affected each of the output processing elements PEz1, PEz2, . . . PEzpby an amount of 0.5 times the respective weighted connections u1i, u2i, . . . upi. Accordingly, the network simplifier 204 in an exemplary embodiment replicates the effect of the removed hidden processing element PEyiby increasing the biasing weighted connections u10, u20, . . . up0from the biasing processing element PEy0by 0.5 times the removed weight connections u1i, u2i, . . . upi. The following equation (13) represents this update of the biasing weight connections u10, u20, . . . up0:
[0113] uk0′=uk0+0.5*uki (13)
[0114] where uk0′ represents the updated kthweighted connection from the biasing processing element PEy0, uk0represents the current kthweighted connection from the biasing processing element PEy0, and ukirepresents the kthweighted connection associated with the removed hidden processing element PEyi. For example, FIG. 5 illustrates a simplified neural network 100′ in which the hidden processing element PEy1has been removed from the neural network 100 of FIG. 1, and the biasing weighted connections u10, u20, . . . up0have been updated to replicate the effect of the removed hidden processing element PEy1.
[0115] In step 410, the network simplifier 204 transfers the simplified definition of the neural network 100 to the network verifier 206. In particular, the network simplifier 204 transfers the obtained weighted connections matrix W, the weighted connections matrix U, the slope vector A, and the slope vector B to the network verifier 206. The network verifier 206 may then test the obtained simplified definition for the neural network 100 by applying test input patterns Akof the test pattern set TESTSET, and calculating a fitness value for the simplified definition based upon generated output patterns Zkand expected output patterns Bkof the test pattern set TESTSET.
[0116] Exemplary Implementations Of Neural Networks And Network Evolution Systems
[0117] It should be appreciated by those skilled in the art that the neural network 100 and/or the network evolution system 200 may be implemented with various hardware components such a digital signal processors, digital logic components, and analog components. Moreover, it should be appreciated that the neural network 100 and/or the network evolution system 200 may be implemented with properly programmed general purpose computer systems, multiprocessor computer systems, and distributed clusters of computer systems.
[0118] For example, FIG. 6 illustrates a general processing system 600 which is suitable for implementing the neural network 100 and network evolution system 200 of the present invention. In particular, the general processing system 600 includes a processor 602, memory 604, mass storage device 606, video display 608, and input device 610. Moreover, the general processing system 600 includes a mass storage controller 612 for controlling the mass storage device 606, a video controller 614 for controlling the video display 608, an I/O controller 616 for controlling the input device 610, and a system bus 618. The system bus 618 operably couples the processor 602 to the memory 604, the mass storage controller 612, the video controller 614, and the I/O controller 616.
[0119] The memory 604 includes random access memory (RAM) such as SRAM (static RAM), DRAM (dynamic RAM), and SDRAM (synchronous DRAM) which store software routines obtained from computer readable media such as a floppy disk, CD-ROM disc, DVD disc, and hard disks. The memory 604 may also include nonvolatile computer readable media such as PROM (programmable read only memory), EPROM (electrically PROM), EEPROM (electrically erasable PROM), and flash memory that store software routines.
[0120] The processor 602 is operable to execute the software and/or firmware routines stored in the memory 604, and communicate with the mass storage device 606, the video display 608, and the input device 610 via the mass storage controller 612, the video controller 614, and the I/O controller 616 respectively. Most importantly, the processor 602 is operable to execute software and/or firmware routines of the memory 604 which cause the processor 602 to implement the network evolution system 200 of FIG. 2.
[0121] It should be appreciated by those skilled in the art that since the slope factors α and β may become arbitrarily large and the input signal a1k, a2k, and ankmay be arbitrarily large precautions must be taken with a computer system implementation of the neural network 100 and/or the network evolution system 200 to ensure against overflow and underflow errors. For example, in an exemplary computer system embodiment of the present invention, the computer system 600 in calculating the sigmoid activation function of equations (3) and (6) first test to see if the product of the slope factor α or β and the resulting combinatory value c is greater than a threshold number such as 90. If the product is greater than 90, then the computer system 600 generates a value of 0 for the result of the sigmoid activation function instead of performing the rest of the sigmoid activation calculation. This threshold test insures that the computer system 600 will not encounter an overflow or underflow error due to computational limits inherent to digital computations.
[0122] Alternative Threshold Functions And Simplifications
[0123] It should be appreciated by those skilled in the art that the above exemplary network evolution system 200 may be used to train neural networks having processing elements that utilize different activation functions. For example, the above exemplary network evolution system 200 may be used to evolve and simplify neural networks that utilize the threshold functions of the TABLE 1. More specifically, the network simplifier 206 may replace a processing element threshold function with the simpler threshold function if the slope factor a of the processing element meets the criteria of the TABLE 1. Moreover, the network simplifier 206 may remove hidden layer processing elements of the neural network 100 if the simplified threshold function generates a constant output.
| TABLE 1 | ||
|---|---|---|
| Function Name | Function | Simplification |
| Hyperbolic tangent | For small α, f(x) = 0 | |
| for large positive α, | ||
| for large positive α, | ||
| Hyperbolic secant | For small α, f(x) = 1 for large α, f(x) = 0 | |
| Gaussian Function, or Radial Basis Function | For small α, f(x) = 0 for large α, f(x) = 1 | |
| Augmented Ratio of Squares | For small α, f(x) = 0 for large α, | |
[0124] While the invention has been illustrated and described in detail in the drawings and foregoing description, such illustration and description is to be considered as exemplary and not restrictive in character, it being understood that only the exemplary embodiments have been shown and described and that all changes and modifications that come within the spirit of the invention are desired to be protected.
Claims
What is claimed is:
1. A method of evolving a neural network comprising a plurality of processing elements interconnected by a plurality of weighted connections, comprising the steps of:
a) obtaining a definition for said neural network by evolving a plurality of weights for said plurality of weighted connections, and evolving a plurality of activation function parameters associated with said plurality of processing elements,
b) determining, based upon a first activation function parameter of said plurality of activation function parameters, whether a first processing element of said plurality of processing elements may be removed from said neural network to simplify the definition for said neural network, and
c) updating said definition for said neural network by removing said first processing element from said definition of said neural network in response to determining that said first processing element may be removed.
2. A method of evolving a neural network comprising a plurality of processing elements interconnected by a plurality of weighted connections, comprising the steps of:
a) obtaining a definition for said neural network by evolving a plurality of weights for said plurality of weighted connections, and evolving a plurality of activation function parameters associated with said plurality of processing elements,
b) determining whether said definition for said neural network may be simplified by determining whether a first slope factor of said plurality of activation function parameters has a predetermined relationship to a slope threshold, and
c) updating said definition for said neural network by removing a first processing element implementing a sigmoid activation function from said definition of said neural network in response to determining that said first slope factor has said predetermined relationship to said slope threshold.
3. The method of claim 1, wherein:
step b) further comprises the step of determining that a first processing element of said plurality of processing elements substantially generates a substantially constant output signal regardless of received input signals based upon a first slope factor of a said plurality of activation functions parameters that is associated with a first activation function implemented by said first processing element; and
step c) further comprises the steps of:
c1) removing said first processing element from said definition of said neural network in response to determining that said first processing element generates said substantially constant output signal, and
c2) updating biasing weighted connections of said plurality of weighted connections associated with a biasing processing element of said plurality of processing elements based upon first weighted connections of said plurality of weighted connections associated with said first processing element in order to substantially reproduce an effect said first processing element had on said plurality of processing elements prior to said first processing element being removed from said definition of said neural network.
4. A method of evolving a neural network comprising a plurality of processing elements interconnected by a plurality of weighted connections, comprising the steps of:
a) obtaining a definition for said neural network by evolving a plurality of weights for said plurality of weighted connections, and evolving a plurality of activation function parameters associated with said plurality of processing elements, said obtaining the definition for said neural network further comprising
a1) initializing a swarm of particles in which each particle has a position in a hyperspace that represents a separate definition for said neural network, and a velocity vector that represents motion of said particle through said hyperspace,
a2) determining for each particle of said swarm, a fitness value for said respective definition of said neural network,
a3) determining based upon said fitness values whether termination criteria have been satisfied,
a4) updating for said each particle of said swarm, a personal best value and a personal best position based upon said respective fitness value for said each particle,
a5) updating for said each particle of said swarm, a local best value and a local best position based upon fitness values associated with a respective group of said particles,
a6) updating for said each particle of said swarm, said position and said velocity vector for said particle based upon said personal best position for said particle, said local best position for said particle, and said velocity vector for said particle, and
a7) repeating steps a2), a3), a4), a5), and a6) until said termination criteria have been satisfied;
b) determining whether said definition for said neural network may be simplified based upon at least one activation function parameter of said plurality of activation function parameters; and
c) updating said definition for said neural network in response to determining that said definition for said neural network may be simplified.
5. The method of claim 4, wherein step a6) comprises the step of:
updating said each particle of said swarm such that said velocity vector has more effect on early updates of said each particle than said velocity vector has on later updates of said each particle.
6. A computer readable medium for evolving a neural network comprising a plurality of processing elements interconnected by a plurality of weighted connections, said computer readable medium comprising code which when executed by a network evolution system causes said network evolution system to:
obtain a definition for said neural network by evolving a plurality of weights for said plurality of weighted connections, and evolving a plurality of activation function parameters associated with said plurality of processing elements;
determine whether said definition for said neural network may be simplified based upon at least one activation function parameter of said plurality of activation function parameters; and
update said definition for said neural network in response to determining that said definition for said neural network may be simplified.
7. The computer readable medium of claim 6, wherein said code when executed by said network evolution system further causes said network evolution system to:
determine whether said definition for said neural network may be simplified by determining, based upon a first activation function parameter of said plurality of activation parameters, whether a first activation function of a first processing element of said plurality of processing elements may be implemented with a less complex activation function, and
update said definition for said neural network by replacing said first activation function with said less complex activation function in response to determining that said first activation function may be implemented with said less complex activation function.
8. The computer readable medium of claim 6, wherein said code when executed by said network evolution system further causes said network evolution system to:
determine whether said definition for said neural network may be simplified by determining whether a first slope factor of said plurality of activation function parameters has a predetermined relationship to a slope threshold, and
update said definition for said neural network by replacing a sigmoid activation function of a first processing element with a step activation function in response to determining that said first slope factor has said predetermined relationship to said slope threshold.
9. The computer readable medium of claim 6, wherein said code when executed by said network evolution system further causes said network evolution system to:
determine whether said definition for said neural network may be simplified by determining, based upon a first activation function parameter of said plurality of activation parameters, whether a first processing element of said plurality of processing elements may be removed from said neural network, and
update said definition for said neural network by removing said first processing element from said definition of said neural network in response to determining that said first processing element may be removed.
10. The computer readable medium of claim 6, wherein said code when executed by said network evolution system further causes said network evolution system to:
determine whether said definition for said neural network may be simplified by determining whether a first slope factor of said plurality of activation function parameters has a predetermined relationship to a slope threshold, and
update said definition for said neural network by removing a first processing element implementing a sigmoid activation function from said definition of said neural network in response to determining that said first slope factor has said predetermined relationship to said slope threshold.
11. The computer readable medium of claim 6, wherein said code when executed by said network evolution system further causes said network evolution system to:
determine whether said definition for said neural network may be simplified by determining that a first processing element of said plurality of processing elements substantially generates a substantially constant output signal regardless of received input signals based upon a first slope factor of a said plurality of activation functions parameters that is associated with a first activation function implemented by said first processing element; and
update said definition for said neural network by (i) removing said first processing element from said definition of said neural network in response to determining that said first processing element generates said substantially constant output signal, and (ii) updating biasing weighted connections of said plurality of weighted connections associated with a biasing processing element of said plurality of processing elements based upon first weighted connections of said plurality of weighted connections associated with said first processing element in order to substantially reproduce an effect said first processing element had on said plurality of processing elements prior to said first processing element being removed from said definition of said neural network.
12. The computer readable medium of claim 6, wherein said code when executed by said network evolution system further causes said network evolution system to obtain a definition for said neural network by:
a) initializing a swarm of particles in which each particle has a position in a hyperspace that represents a separate definition for said neural network, and a velocity vector that represents motion of said particle through said hyperspace;
b) determining for each particle of said swarm, a fitness value for said respective definition of said neural network;
c) determining based upon said fitness values whether termination criteria have been satisfied;
d) updating for said each particle of said swarm, a personal best value and a personal best position based upon said respective fitness value for said each particle;
e) updating for said each particle of said swarm, a local best value and a local best position based upon fitness values associated with a respective group of said particles;
f) updating for said each particle of said swarm, said position and said velocity vector for said particle based upon said personal best position for said particle, said local best position for said particle, and said velocity vector for said particle; and
g) repeating b), c), d), e), and f) until said termination criteria have been satisfied.
13. The computer readable medium of claim 12, wherein said code when executed by said network evolution system further causes said network evolution system to update said position and said velocity vector for said each particle of said swarm by:
updating said each particle of said swarm such that said velocity vector has more effect on early updates of said each particle than said velocity vector has on later updates of said each particle.
14. A network evolution system for evolving a neural network comprising a plurality of processing elements interconnected by a plurality of weighted connections, said network evolution system comprising:
a network evolver operable to obtain a definition for said neural network by evolving a plurality of weights for said plurality of weighted connections, and evolving a plurality of activation function parameters associated with said plurality of processing elements; and
a network simplifier operable to (i) determine whether said definition for said neural network may be simplified based upon at least one activation function parameter of said plurality of activation function parameters, and (ii) update said definition for said neural network in response to determining that said definition for said neural network may be simplified.
15. The network evolution system of claim 14, wherein said network simplifier is further operable to:
determine whether said definition for said neural network may be simplified by determining, based upon a first activation function parameter of said plurality of activation parameters, whether a first activation function of a first processing element of said plurality of processing elements may be implemented with a less complex activation function, and
update said definition of said neural network by replacing said first activation function with said less complex activation function in response to determining that said first activation function may be implemented with said less complex activation function.
16. The network evolution system of claim 14, wherein said network simplifier is further operable to:
determine whether said definition for said neural network may be simplified by determining whether a first slope factor of said plurality of activation function parameters has a predetermined relationship to a slope threshold, and
update said definition of said neural network by replacing a sigmoid activation function of a first processing element with a step activation function in response to determining that said first slope factor has said predetermined relationship to said slope threshold.
17. The network evolution system of claim 14, wherein said network simplifier is further operable to:
determine whether said definition for said neural network may be simplified by determining, based upon a first activation function parameter of said plurality of activation parameters, whether a first processing element of said plurality of processing elements may be removed from said neural network, and
update said definition of said neural network by removing said first processing element from said definition of said neural network in response to determining that said first processing element may be removed.
18. The network evolution system of claim 14, wherein said network simplifier is further operable to:
determine whether said definition for said neural network may be simplified by determining whether a first slope factor of said plurality of activation function parameters has a predetermined relationship to a slope threshold, and
update said definition of said neural network by removing a first processing element implementing a sigmoid activation function from said definition of said neural network in response to determining that said first slope factor has said predetermined relationship to said slope threshold.
19. The network evolution system of claim 14, wherein said network simplifier is further operable to:
determine whether said definition for said neural network may be simplified by determining that a first processing element of said plurality of processing elements substantially generates a substantially constant output signal regardless of received input signals based upon a first slope factor of a said plurality of activation functions parameters that is associated with a first activation function implemented by said first processing element, and
update said definition of said neural network by (i) removing said first processing element from said definition of said neural network in response to determining that said first processing element generates said substantially constant output signal, and (ii) updating biasing weighted connections of said plurality of weighted connections associated with a biasing processing element of said plurality of processing elements based upon first weighted connections of said plurality of weighted connections associated with said first processing element in order to substantially reproduce an effect said first processing element had on said plurality of processing elements prior to said first processing element being removed from said definition of said neural network.
20. The network evolution system of claim 14, wherein said network evolver is further operable to obtain said definition of said neural network by:
a) initializing a swarm of particles in which each particle has a position in a hyperspace that represents a separate definition for said neural network, and a velocity vector that represents motion of said particle through said hyperspace;
b) determining for each particle of said swarm, a fitness value for said respective definition of said neural network;
c) determining based upon said fitness values whether termination criteria have been satisfied;
d) updating for said each particle of said swarm, a personal best value and a personal best position based upon said respective fitness value for said each particle;
e) updating for said each particle of said swarm, a local best value and a local best position based upon fitness values associated with a respective group of said particles;
f) updating for said each particle of said swarm, said position and said velocity vector for said particle based upon said personal best position for said particle, said local best position for said particle, and said velocity vector for said particle; and
g) repeating b), c), d), e), and f) until said termination criteria have been satisfied.
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