This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2021-036691, filed on Mar. 8, 2021, the entire contents of which are incorporated herein by reference.
The embodiments discussed herein are related to an optimization apparatus, a non-transitory computer-readable storage medium, and an optimization method.
Information processing is performed in various fields in currently society. Such information processing is performed by arithmetic apparatuses such as computers, which perform operations and reorganization on various data and obtain meaningful results to perform prediction, determination, control, and the like. Optimization processing is one approach of this information processing and has become an important field.
One approach of the optimization processing is to solve a discrete optimization problem. In a large-scale multivariable discrete optimization problem, the number of combinations increases explosively, and the calculation time sometimes does not fall within a realistic range in the technique of exhaustively performing calculations to work out all combinations.
Examples of a method for solving such a large-scale multivariable discrete optimization problem include simulated annealing (SA) using an Ising-type energy function. In this SA, calculation is performed by replacing a problem to be calculated with an Ising model, which is a model representing behavior of spins of a magnetic material.
In the optimization calculation using the Ising model, bits are stochastically inverted to calculate an energy change in a case where one bit is inverted, and whether or not to accept the bit inversion according to the energy change is adopted, thereby searching for an optimum solution that minimizes the energy.
Japanese Laid-open Patent Publication No. 2019-145010, Japanese Laid-open Patent Publication No. 2019-46038, International Publication Pamphlet No. WO 2015/190593, and U.S. Patent Application Publication No. 2019/0087388 are disclosed as related art.
According to an aspect of the embodiments, an optimization apparatus includes a memory configured to store one or more coupling coefficients that represent interaction of a plurality of variables corresponding to a plurality of bits included in an energy function; and a processor configured to select, based on a difference of a value of the energy function associated with inversion of a value of each of the plurality of bits, adoption or rejection of bit inversion to perform optimization, wherein the processor is further configured to: specify a coupling coefficient corresponding to an auxiliary variable from the one or more coupling coefficients stored in the memory, the auxiliary variable being a product of variables corresponding to respective bits from which a variable corresponding to a specific bit in the energy function is excluded, and execute calculation of a term of a third-order or higher of a difference associated with inversion of the specific bit using the auxiliary variable and the coupling coefficient . . . .
The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention.
The related art described above has a problem that, in a case of searching for a minimum value of an energy function E(x) including a higher-order term of a third-order or higher, the number of parameter elements to be transferred from a memory to a computational resource becomes enormous.
For example, an example of describing the energy function E(x) related to all orders, where k represents an order, is as expressed by the following formula (1), where a k-th order coupling coefficient represents a k-th order array W1, 2, . . . , k. The coupling coefficient represents the strength of mutual coupling of each of a plurality of spins in the Ising model in which each of a plurality of variables is regarded as each of the plurality of spins of a magnetic material.
Here, j1 . . . jk represent the element position in each order, and x represents a binary (0, 1) or a spin (1, −1). W represents an array different for each order, which is generally a multidimensional array.
The formula (1) is rewritten as the following formula (2). ZK represents a coupling coefficient of the k-th order term, which is K-dimensional array.
Here, a formula (3), which is a calculating formula of an energy change ΔEk (x, i) of the K-th order term associated with inversion of the i-th element of x, and a formula (4), which is a total calculating formula, are as follows. Note that E0 disappears as it is a change.
For example, in the third-order term, x′j1x′j2x′j3 obtained by inverting xj1xj2xj3 by 1 bit is calculated. Accordingly, when the inverted part is set as m and a bit difference is set as Δxm, the third-order term is the sum of calculated Zm,j2,j3Δxmxj2xj3, Zj1,m,j3xj1Δxmxj3, and Zj1,j2,mxj1xj2Δxm.
That is, for example, the amount of memory access to perform calculation needs the hyperplane of Z by the number of orders. A case of a fourth-order or higher applies in a similar manner. The values of K×N{circumflex over ( )}(K−1) elements are transferred from the memory to the computational resource.
The arithmetic processing unit 201 includes a reading unit 203, operation units 204 to 206, and an addition unit 207. The arithmetic processing unit 201 determines a position (i) for bit inversion, inputs a vector (x) including a difference value Δxi of xi before and after the inversion, reads a predetermined part from the K-th order coupling coefficient ZK stored in the memory 202, and outputs the energy difference ΔEk of the term.
For a certain dimension of the multidimensional array Z, the reading unit 203 selects a hyperplane in which the position (i) of x for the bit inversion is fixed for each dimension, and reads it from the memory 202 to prepare K pieces. This reading from the memory 202 is the location where the transfer load to the arithmetic processing unit 201, which is the computational resource, is generated.
The operation units 204 to 206 and the addition unit 207 perform a product-sum operation for each hyperplane output from the reading unit 203. For example, in a case where K=3, the operation in the operation unit 204 is as follows.
The addition unit 207 adds up the calculation results of the operation units 204 to 206 to output ΔEk.
As illustrated in
After the operations by the arithmetic processing units 201a to 201c and the multiplication unit 208, the addition unit 209 adds up the respective results to create ΔE (x, i). An adoption/rejection determination unit 210 determines adoption/rejection of the bit changed according to the publicly known metropolis standard or the like based on ΔE (x, i). The adoption/rejection determination unit 210 sets Enext=E+ΔE (x, i) for the current energy E in the case of adoption, and sets Enext=E in the case of rejection.
For example, in a case where the order K=3 and the number of bits is N, the number of 3N2 elements is transferred from the memory to the computational resource. Accordingly, in a case of searching for the minimum value of the energy function E(x) including a higher-order term, the transfer amount increases, and the transfer time may be long. In a case where the order is fourth or higher and N is larger, the transfer amount further increases, and the transfer time increases.
In one aspect, it aims to provide an optimization apparatus, an optimization program, and an optimization method capable of suppressing a data transfer amount at a time of an optimization operation.
Hereinafter, an optimization apparatus, an optimization program, and an optimization method according to embodiments will be described with reference to the drawings. Configurations with the same functions in the embodiments are denoted by the same reference signs, and redundant descriptions will be omitted. Note that the optimization apparatus, the optimization program, and the optimization method to be described in the embodiments below are merely examples, and do not limit the embodiments. Furthermore, the embodiments below may be appropriately combined unless otherwise contradicted.
In an embodiment, an information processing apparatus that performs various kinds of arithmetic processing is exemplified as an example of the optimization apparatus. In this information processing apparatus, in the simulated annealing using an Ising-type energy function, adoption/rejection of bit inversion is selected based on a difference in the energy function accompanying the bit inversion to perform optimization as one arithmetic processing. Specifically, for example, the information processing apparatus stochastically inverts one or more bits and calculates an energy change in a case where the bits are inverted, and adopts whether or not to accept the bit inversion according to the energy change, thereby searching for an optimum solution that minimizes the energy. Note that, the embodiment is not limited to the case of using the Ising model to solve the optimization problem, and is also applicable to a case of using a model including a higher-order term of a third-order or higher to solve the optimization problem.
The information processing apparatus according to the embodiment uses the Monte Carlo method in which one or more bits are randomly inverted to search for the optimum solution to obtain a state x, which is the minimum energy of an energy function (E). Here, a change amount(difference) of the energy function (E) from the energy before inversion due to the inversion of an i-th bit, which is a candidate for the inversion, is set to ΔEi. The information processing apparatus according to the embodiment calculates ΔEl, and then determines (selects) adoption/rejection of the inversion in the i-th bit using the metropolis standard or the like on the basis of the ΔEi.
Here, the calculation of ΔEl in the information processing apparatus according to the embodiment will be described. First, with respect to the formula (2) mentioned above, the terms of the third-order or higher are enclosed in Z, and it is rewritten as the following formula (5).
Here, considering an energy difference ΔE caused by the bit inversion, E0 in the formula (5) is a constant, and thus it is unnecessary. Furthermore, while a variable x is a discrete value and is impossible to be subject to differentiation, a primary difference is set as Δx focusing on a difference (primary difference) due to the bit change in the coupled x. With the number of variables of coupled x reduced by one using the Δx, ΔEi becomes as expressed in the following formula (6).
Next, in the formula (6), an auxiliary bit (also referred to as auxiliary variable) yi,m that can be calculated from x is introduced. The auxiliary bit yi,m is the product obtained by excluding a variable xi of the i-th bit to be inverted from the product of all variables x (xj1, . . . , xjk). Since y can be calculated from x, the introduction of y does not increase the bit itself. With the auxiliary bit yi,m introduced, the formula (6) can be rewritten as the following formula (7).
Here, Zl,m is introduced. Zi,m is a coupling coefficient corresponding to yi,m. Accordingly, ΔEi can be expressed as in the following formula (8). In the operation of the formula (8), it is possible to reduce the number of elements of Z referenced to by gi by one order.
In a case of adopting bit inversion, there is a possibility that values of a plurality of auxiliary variables yi,m change due to inversion of a variable xj for energy update. With the change set as Δyi,m, the energy update is expressed by the following formula (9).
In the information processing apparatus according to the embodiment, the terms of the third-order or higher in the calculation of ΔEi are calculated as in the formulae (8) and (9). Specifically, for example, the information processing apparatus calculates ΔEi using the auxiliary variable yi,m, which is the product of the variables corresponding to respective bits excluding the variable corresponding to the bit to be a candidate for the inversion and the coupling coefficient Zi,m corresponding to yi,m. At this time, the information processing apparatus reads the coupling coefficient Zi,m corresponding to the auxiliary variable from the memory that stores the coupling coefficients corresponding to all the bits related to the energy function. Therefore, the information processing apparatus is enabled to reduce the number of elements to be read from the memory by one order, whereby it becomes possible to suppress the data transfer amount at the time of operation.
The UI 1 is a user interface (UI) that performs data input from a user, output of processing results, and the like. Parameters given from the outside, such as the coupling coefficient Z corresponding to all the bits related to the energy function and a temperature, are input by the user via the UI 1.
The minimum solution search unit 2 is a processing unit that executes arithmetic processing related to optimization and searches for an optimum solution that minimizes the energy, and includes a memory 10 and an arithmetic processing unit 20. The general-purpose CPU 3 is a general-purpose CPU that handles an application for processing of the UI 1 and the arithmetic processing unit 20, and the like.
The memory 10 is a hard disk drive (HDD), a random access memory (RAM), or the like, which stores parameters such as a coupling coefficient W input via the UI 1.
The arithmetic processing unit 20 is a processing unit that calculates ΔEi according to the formulae (8) and (9) and determines, based on the ΔEi, adoption/rejection of the inversion in the i-th bit using the metropolis standard or the like to search for an optimum solution. A dedicated hardware apparatus such as an application specific integrated circuit (ASIC), a field-programmable gate array (FPGA), a graphics processing unit (GPU), a general-purpose CPU, or the like may be applied to the arithmetic processing unit 20.
For example, the arithmetic processing unit 20 reads and executes a program stored in the memory 10, thereby executing arithmetic processing related to optimization. Note that, the program to be executed by the arithmetic processing unit 20 is not necessarily stored in the memory 10. For example, a program stored in a storage medium that can be read by the information processing apparatus 100 may be read, and the arithmetic processing unit 20 may execute it. The storage medium that can be read by the information processing apparatus 100 corresponds to, for example, a portable recording medium such as a CD-ROM, a DVD disk, or a universal serial bus (USB) memory, a semiconductor memory such as a flash memory, a hard disk drive, or the like. Furthermore, the program may be prestored in an apparatus connected to a public line, the Internet, a LAN, or the like, and the information processing apparatus 100 may read the program from them to execute it. Furthermore, the various processing functions implemented by the arithmetic processing unit 20 may be executed by a plurality of computers in cooperation through cloud computing.
The operation unit 21 reads, from the coupling coefficient Z corresponding to all the bits stored in the memory 10, Zi,m excluding the dimension corresponding to the bit to be a candidate for the inversion, and performs operation according to the formula (8).
The operation unit 22 calculates Δy,m from the variable x and the candidate bit i to be changed.
The operation unit 23 and the addition unit 24 perform calculation related to hi according to the formula (8). Note that, whether or not to update hi depends on the adoption/rejection determination by the adoption/rejection determination unit 28, and hi is updated in a case where the adoption/rejection determination unit 28 determines adoption.
The addition unit 25 performs calculation related to gi according to the formula (8) on the basis of the operation result of the operation unit 21. Note that whether or not to update gi depends on the adoption/rejection determination by the adoption/rejection determination unit 28, and gi is updated in a case where the adoption/rejection determination unit 28 determines adoption.
The addition unit 26 adds the operation results of the addition units 24 and 25. The multiplication unit 27 multiplies the operation result of the addition unit 26 by −Δxi. For example, in the operation of the addition unit 26 and the multiplication unit 27, ΔE; is calculated according to the formula (8).
The adoption/rejection determination unit 28 determines adoption/rejection of the bit changed according to the metropolis standard based on ΔE (x, i), which is the operation result up to the multiplication unit 27.
As illustrated in
Next, the arithmetic processing unit 20 prepares an initial value hi(0) of hi according to the following formula (11) (S2). Here, hi represents a vector of N bits (N: number of bits).
Next, the arithmetic processing unit 20 prepares an initial value gi(0) of gi according to the following formula (12) (S3). Here, gi represents a vector of N bits.
Next, the arithmetic processing unit 20 randomly selects 1 bit (i) to be inverted at x, and sets the difference between the selected bits before and after the inversion as Δxi (S4).
Next, the arithmetic processing unit 20 calculates ΔEi from Δxi, hi, and gi according to the following formula (13) (S5).
[Math. 14]
ΔEi=−(hi+gi)Δxi (13)
Next, the adoption/rejection determination unit 28 of the arithmetic processing unit 20 determines, using the calculated ΔEi, adoption/rejection of bit inversion based on the metropolis standard or the like (S6). For example, in the case of the metropolis standard, the adoption/rejection determination unit 28 generates a uniform random number rand in the section 0≤rand≤1 as an example. Next, the adoption/rejection determination unit 28 determines that the bit inversion is to be adopted if rand>exp (ΔE×β). Here, β represents an inverse temperature.
In the case of rejection (rejection in S6), the arithmetic processing unit 20 proceeds to S10 in the process. In the case of adoption (adoption in S6), the arithmetic processing unit 20 updates the energy E according to the following formula (14) (S7). The updated energy E (x, t+1) is a value obtained by adding E before the update and ΔEi of the formula (13), which is the operation result of the multiplication unit 27.
[Math. 15]
E(x,t+1)←E(x,t)−(hi(t)+gi(t))Δxj (14)
Next, the arithmetic processing unit 20 updates hi according to the following formula (15) based on the operation of the operation unit 23 and the addition unit 24 (S8).
[Math. 16]
h
i(t+1)←hi(t)+Vi,jΔxj (15)
Next, the arithmetic processing unit 20 updates gi according to the formula (16) based on the operation of the operation unit 21, the operation unit 22, and the addition unit 25 (S9). Here, Δy,m is Δy, m=ΔxjΔxk.
Next, the arithmetic processing unit 20 determines whether or not termination conditions, such as the arithmetic processing of S4 to S9 is repeated a predetermined number of times or predetermined energy is obtained, are satisfied (S10). If the termination conditions are satisfied (Yes in S10), the arithmetic processing unit 20 terminates the process assuming that the optimization has been achieved. If the termination conditions are not satisfied (No in S10), the arithmetic processing unit 20 returns the process to S4. In this manner, the arithmetic processing unit 20 repeats the bit inversion until the predetermined number of times or the predetermined energy is obtained.
As described above, the information processing apparatus 100 performs calculation for the terms of the third-order or higher of the difference accompanying the inversion of the specific bit (i) in the energy function using the auxiliary variable (yi,m), which is the product of the variables corresponding to respective bits excluding the variable corresponding to the bit to be a candidate for the inversion and the coupling coefficient (Zi,m) corresponding to the auxiliary variable. Then, at the time of this calculation, the information processing apparatus 100 reads the coupling coefficient corresponding to the auxiliary variable from the memory 10 that stores the coupling coefficient (Z) representing the interaction of the plurality of variables corresponding to all of the bits related to the energy function. As a result, the information processing apparatus 100 is enabled to reduce the number of elements to be read from the memory 10 by one order, whereby it becomes possible to suppress the data transfer amount at the time of operation.
Furthermore, in a case of adopting inversion of a specific bit, the information processing apparatus 100 calculates a change in the auxiliary variable due to the inversion of the variable corresponding to the specific bit (S8 and S9). As a result, in the case of adopting inversion of a specific bit, the information processing apparatus 100 is capable of updating the value of the auxiliary variable in accordance with the bit inversion.
As illustrated in
In the conventional optimum solution search, the data transfer amount from the memory to the computational resource associated with 1-bit inversion is kN(k-1), where the number of spins (number of bits) is N and the order is k, for each calculation of a difference in the energy. In the optimum solution search according to the present embodiment, N(k-1) or less may be set for one calculation of the difference in the energy.
All examples and conditional language provided herein are intended for the pedagogical purposes of aiding the reader in understanding the invention and the concepts contributed by the inventor to further the art, and are not to be construed as limitations to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although one or more embodiments of the present invention have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.
Number | Date | Country | Kind |
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2021-036691 | Mar 2021 | JP | national |