Information processing device and information processing method

Abstract

This technology provides a way to solve high-dimensional optimization problems more effectively than before. [Solution] An information processing device for finding the optimal solution of an objective function represented by n-dimensional variables, comprising a processor and a memory storing a program, wherein the program is provided to the processor by a d-dimensional (d

Description

Technical Field

[0001] The present invention relates to an information processing apparatus and an information processing method, and more particularly to an information processing apparatus and an information processing method for effectively solving an optimization problem.

Background Art

[0002] An optimization problem is a problem of finding a solution that maximizes or minimizes an objective function among given conditions, and is applied to various scenarios. As an example, the present inventor has proposed a method for optimizing cardboard sizes in an EC logistics company in Non-Patent Document 1.

[0003] In Non-Patent Document 1, the optimization problem is solved using Bayesian optimization. It is known that Bayesian optimization becomes difficult to optimize when the dimension increases, for reasons such as falling into a local solution or the maximization of the acquisition function not working well. Therefore, it is necessary to perform optimization after reducing the dimension of the objective function by some method.

[0004] In Non-Patent Document 1, dimensionality reduction is performed using a method called REMBO (Non-Patent Document 2). REMBO is a method that embeds a low-dimensional space using a random matrix in a high-dimensional search space and performs Bayesian optimization. By such dimensionality reduction, a high-dimensional optimization problem can be appropriately solved.

Prior Art Documents

Non-Patent Documents

[0005]

Non-Patent Document 1

Non-Patent Document 2

[0006] In REMBO, the entire low-dimensional space is used as the search space, but there is room for improvement in the selection of the search space. If the search space can be appropriately limited, it will be possible to more appropriately handle high-dimensional optimization problems.

[0007] The present disclosure aims to provide a technique that can more appropriately solve high-dimensional optimization problems than before. [Means for Solving the Problems]

[0008] One aspect of the present disclosure is an information processing apparatus for obtaining a solution for optimizing an n-dimensional objective function, including a processor and a memory storing a program, the program causing the processor to generate a random matrix for embedding a d-dimensional (d < n) low-dimensional space into the n-dimensional space, obtain an optimal solution of the objective function using a Bayesian optimization method with a subspace of the low-dimensional space as a search space, and execute the steps, where the search space is a hollow sphere or a spherical region in the low-dimensional space, and the information processing apparatus is characterized by this.

[0009] Another aspect of the present disclosure is an information processing method executed by an information processing apparatus for obtaining a solution that optimizes an n-dimensional objective function, the method including: generating a random matrix for embedding a d-dimensional (d < n) low-dimensional space into the n-dimensional space; using a Bayesian optimization method to obtain an optimal solution of the objective function with a subspace of the low-dimensional space as a search space; wherein the search space is a hollow sphere or a spherical region in the low-dimensional space. The information processing method is characterized in that.

Effect of the Invention

[0010] According to the present invention, it is possible to more appropriately solve a high-dimensional optimization problem than in the prior art.

Brief Description of the Drawings

[0011] [Figure 1] FIG. 1 is a diagram for explaining a search range in a low-dimensional space in the optimization method according to the present disclosure. [Figure 2] FIG. 2 is a diagram showing that the distribution when a random variable following a uniform distribution in a high-dimensional space is projected into a low-dimensional space by the inverse matrix of a random matrix can be approximated by a standard normal distribution. [Figure 3] FIG. 3 is a diagram showing that a random variable following a standard normal distribution in a high-dimensional space is distributed near a hypersphere with a radius of √n. [Figure 4] FIG. 4 is a flowchart showing the flow of the optimization process according to the present disclosure. [Figure 5] FIG. 5 is a diagram showing the configuration of the optimization apparatus according to the present disclosure. [Figure 6] FIG. 6 is a diagram showing the configuration according to an application example of the optimization apparatus according to the present disclosure. [Figure 7] FIG. 7 is a flowchart showing the flow of a process for evaluating an objective function in an application example of the optimization apparatus according to the present disclosure. [Figure 8]Figure 8 illustrates the effect of optimization in an example of the application of the optimization device described in this disclosure. [Figure 9] Figure 9 is a diagram illustrating Bayesian optimization. [Figure 10] Figure 10 is a diagram illustrating REMBO. [Modes for carrying out the invention]

[0012] Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the drawings. However, the present invention is not limited to the embodiments described. Furthermore, not all components described in the embodiments are necessarily essential to the present invention. Also, while the problem of finding a solution that minimizes the objective function is used as an example below, the problem of finding a solution that maximizes the objective function can be treated similarly.

[0013] (Bayesian optimization and REMBO) First, we will explain Bayesian optimization and REMBO. Figures 9(A) and 9(B) illustrate Bayesian optimization. In the figures, graph 901 represents the true objective function (unknown) to be optimized. If one or more observations are obtained, a Gaussian process model can be applied to obtain a prediction of the objective function and the range in which the function is likely to exist (called the prediction interval or confidence interval). In Figure 9(A), the function prediction 905 and the prediction interval 906 are obtained from the values ​​of observation points 902 to 904.

[0014] From the function prediction 905 and prediction interval 906, the acquisition function 908, which represents the probability that a solution with optimal conditions exists, can be calculated. The probability of obtaining optimal conditions is highest at point 909, where the acquisition function 908 is maximized, so point 909 is selected as the next observation point. When the observation value 910 at point 909 is obtained, the function prediction 905' and prediction interval 906' are updated, as shown in Figure 9(B).

[0015] Bayesian optimization is a method that searches for the optimal solution by repeatedly applying such Gaussian process models and updating the acquisition function.

[0016] REMBO (Random EMbedding Bayesian Optimization; Non-Patent Document 2) is a method that performs Bayesian optimization by embedding a d-dimensional low-dimensional space into an n-dimensional original search space using a random matrix. Here, d is a positive integer such that d < n.

[0017] REMBO is an effective method when there exists an effective dimension for the objective function f. The effective dimension refers to the dimension d e when the value of the function f depends only on a d-dimensional linear subspace. e is meant. By setting d such that d e < d, it is shown in Non-Patent Document 2 that the minimum value of the objective function f can be obtained with probability 1.

[0018] Figure 10(A) is a diagram for explaining the effective dimension. The objective function 1001 changes in the x-axis direction but does not change in the y-axis direction. In this case, the effective dimension is 1, and by performing optimization with any straight line 1002 passing through the origin excluding the straight line x = 0 as the search range, the global minimum value can be obtained.

[0019] Figure 10(B) (quoted from Non-Patent Document 2) is a diagram for explaining the embedding of the low-dimensional space. Region 1011 represents the high-dimensional (2-dimensional) original search space, and region 1012 represents the low-dimensional (1-dimensional) space to be embedded. That is, this example is a case where n = 2 and d = 1. In REMBO, a Gaussian process is constructed on the low-dimensional space 1012, and optimization is performed within this space. At this time, for any point y in the low-dimensional space, the objective function is evaluated using the random matrix A. Here, the random matrix is a matrix whose each component independently follows a normal distribution N(0,1). When the embedded Ay is located outside the high-dimensional space 1011, a projection onto the high-dimensional space 1011 is performed.

[0020] In REMBO, a hypercube within the d-dimensional low-dimensional space is used as the search range.

[0021] (Proposed Method) In this method, the search space is limited to a hollow sphere region (or a sphere region in some cases) instead of a hypercube in a low-dimensional space. Below, the theoretical basis for the functionality of such a search space selection will be explained.

[0022] First, consider optimizing the objective function f n on the hypercube K = [-√3, √3] in a high-dimensional space. Here, assume that the function f has a minimum point inside K. If necessary, scaling can be done without loss of generality for the discussion. The reason for choosing [-√3, √3] here is that the uniform distribution on it becomes a random variable with mean 0 and variance 1.

[0023] Now, consider what subset of R † the region K is mapped to by the Moore - Penrose inverse matrix √n A of the (n, d)-random matrix 1 / √n A d .

[0024] When each element of X=(X1, X2, ..., X n )T follows a random variable with a uniform distribution on [-1, 1] n , for a sufficiently large n, approximately,

Equation

[0025] Figures 2(A) to (E) are diagrams explaining that this holds.

[0026] Figures 2(A) and (B) are samples from the uniform distribution on the 100-dimensional cube [-√3, √3] 100 by √nA †The graphs show the mappings to one and two dimensions, respectively. Note that in the one-dimensional mapping, matrix A is a (100, 1) matrix, and in the two-dimensional mapping, matrix A is a (100, 2) matrix. As the density function of the standard normal distribution is superimposed on each figure, it can be seen that the mappings follow a standard normal distribution.

[0027] Also, Figures 2(C) to (E) show [-√3, √3] n The uniform distribution above is √nA † These are projections onto a two-dimensional plane, representing the cases where n=3, 10, and 50, respectively. As shown in Figure 2(C), when n=3, it is a projection of a normal cube onto a plane, resulting in a hexagon, but as n increases, It can be seen that the distribution approaches a circle.

[0028] Next, the inventors found that when dimension n is sufficiently large, random variables following a standard normal distribution in n-dimensional space are distributed near a hypersphere of radius √n. More precisely, Z=(Z1, ..., Z n ) T When N is ~N(0, I / 3), for sufficiently large n>0, approximately,

number

[0029] Figure 3 shows the results of an experimental observation of this proposition, illustrating the histogram of the Euclidean norm of a probability vector following a normal distribution with mean 0 and variance 1 / 3. Here, graphs 301-305 show the cases where the number of dimensions n is 1, 5, 15, 30, and 50, respectively. Furthermore, for each dimension... The mean μ = √(n / 3) is shown by the dashed line. As can be seen from the figure, this proposition holds true for n = 15 or greater.

[0030] From Proposition 1 and Proposition 2 above, [-1, 1] nThe above uniform distribution is the Moore-Penrose inverse of the random matrix 1 / √n A of (n,d)-random matrix √n A. † This shows that it maps to a normal distribution of N(√(n / 3), 2 / 3).

[0031] The interesting point here is that the variance of the distribution is independent of the dimension. From this proposition, R d The above N(0,I d It can be seen that random variables following ( / 3) are concentrated around a hypersphere of radius √(n / 3). Therefore, by focusing our search on that area, we can obtain the optimal solution for the objective function f. This can be understood. Furthermore, the reason the variance of the distribution is dimensional independent is because a standardized random matrix 1 / √n A is used. This is the effect of what is being done.

[0032] Figure 1 is a diagram illustrating this method. In this method, the low-dimensional space R d (102) The standardized randomization matrix 1 / √n A gives a high-dimensional space R n Embed in (101). Here, A high-dimensional space R with sufficiently large dimension n n (101) Random variables following a standard normal distribution √n A † This results in a low-dimensional space R d Projecting onto (102), as described above, the distribution is radius Hypersphere of √n √n S n-1 It is distributed in the neighborhood of (403). Therefore, in optimization, It becomes clear that we should focus our search on the periphery of the hypersphere.

[0033] Search space B H d For example, from a sphere with radius √(π / 3)+4 / 3 to a sphere with radius max(√(π / 3)-4 / 3, 0) The sphere can be a d-dimensional hollow sphere or a d-dimensional sphere with the sphere removed. Note that while the search space can be a 2σ range here, other ranges such as a σ range or a 3σ range may also be used.

number

[0034] Search space B H d This can be expressed using polar coordinates as follows:

number

[0035] Therefore, [R low , R up ]×[0, π] d-2 Optimization should be performed in the range ×[0, 2π]. Note, R low = max(√(π / 3)-4 / 3, 0), R up =√(π / 3)+4 / 3. This search space is d-dimensional. Since it is a supercuboid, it is easy to implement.

[0036] [R low , R up ]×[0, π] d-2 Optimization is performed on ×[0, 2π], but the kernel function is B H d The calculation needs to be based on the above distance. Therefore, the polar coordinate transformation is φ: [R low , R up ]×[0, π] d-2 ×[0, 2π]→B H d Therefore, for a kernel function k, we can use k(φ(x), φ(y)) as the kernel function to be used in this method.

[0037] (process) The optimization process using this method will be explained below. Figure 4 is a flowchart showing the flow of the optimization process using this method. The optimization process is performed by an optimization device 500 having the hardware configuration shown in Figure 5.

[0038] The optimization device 500 has the configuration of a typical computer (information processing device), and specifically includes an arithmetic processor 501 such as a CPU, a communication device 502 capable of wireless or wired communication, an input device 503 such as a keyboard or mouse, an output device 504 such as a display, and a storage device 505 including main memory and auxiliary storage. By the processor 501 executing a computer program stored in the storage device 505, the computer 500 functions as an optimization device and performs the optimization process shown in Figure 4. Note that some or all of the processes may be executed by dedicated hardware devices. Furthermore, the optimization device may be realized by multiple computers that can communicate with each other.

[0039] The optimization process using this method will be explained below with reference to Figure 4. Here, the dimension of the objective function to be optimized is n.

[0040] In step S401, the optimization device 500 embeds a low-dimensional space into an n-dimensional high-dimensional space. Determine the dimension d of the space. Here, as mentioned above, dimension d must be greater than or equal to the effective dimension of the objective function. Dimension d can be, for example, a value input or set by the user. It is possible.

[0041] In step S402, the optimization device 500 generates an (n,d) random matrix 1 / √n × A for embedding a d-dimensional low-dimensional space into an n-dimensional high-dimensional space. Here, matrix A is a matrix in which each component independently follows a standard normal distribution N(0, 1) with mean 0 and variance 1.

[0042] In step S403, the optimization device 500 implements a kernel function. The kernel function is not particularly limited, and can be an RBF (Radial Bass Function) kernel, a Matern kernel, a linear kernel, etc. As an example, a Matern 5 / 2 kernel with ν = 5 / 2 in the following equation can be used. Nell's function is used. Note that Γ is the gamma function and Kν is the modified Bessel function of the second kind.

number

[0043] However, since this method performs optimization using polar coordinate variables, a function represented by K(φ(x), φ(y)) is used as the kernel function in Bayesian optimization. Here, x and y are points in a low-dimensional interval expressed in polar coordinates, and φ represents the coordinate transformation from polar coordinates to Cartesian coordinates.

[0044] In step S404, the optimization device 500 determines the subspace Y = B of the d-dimensional low-dimensional space. H d We start Bayesian optimization using the search space Y = B. H d This is a lower-order This is a hollow sphere or a region of a sphere in the original space. In this embodiment, a point in the lower-dimensional space is given polar coordinates (r, θ1, …, θ d-1 ) is used to represent it, and [R low , R up ]×[0, π] d-2 The implementation uses the region ×[0, 2π] as the search space. By using polar coordinate variables, each variable can be selected independently within the above range. The program for solving Bayesian optimization problems is Optuna, Bayesian. While libraries such as Optimization (bayes_opt) and Scikit-optimize (skopt) are available, they are generally designed to perform searches on a hypercube. Although libraries that enable searches in other spaces also exist, all of them support searches on a hypercube at least. In the implementation of this invention, by using polar coordinate variables to make each variable an independent variable on the hypercube, there are no restrictions on the libraries that can be used, and implementation with existing libraries becomes easier.

[0045] The process up to step S403 corresponds to the preprocessing for Bayesian optimization, and the process from step S404 onward corresponds to the process of finding the optimal solution of the objective function using Bayesian optimization.

[0046] In step S405, the optimization device 500 determines one or more candidate points xi on Y Evaluate the value of the objective function f. Specifically, a point xi on Y, represented by polar coordinate variables, is transformed into a Cartesian coordinate variable by polar coordinate transformation φ, and then 1 / √n A x is calculated using a random matrix. i Projecting onto the x, f(1 / √n A x i Evaluate the value of ).

[0047] After this, the loop process L1 consisting of steps S406 to S408 is repeated a predetermined number of times. In step S406, the optimization device 500 selects candidate points x i From the value of the objective function f, The gain function is calculated. The gain function is not particularly limited and can be the expected improvement (EI). ), stochastic improvement (PI), lower confidence bound (LCB), etc. can be used. In step S407, the optimization device 500 will use the following options Complementary point x i+1 Select the point where the acquisition function takes its maximum value. Specifically, select the next candidate point x. i+1 Select as such. Methods for searching for the point where the acquisition function takes its maximum value include, for example, gradient descent, grid search, and random search. In step S408, the optimization device 500 selects the next candidate point x i+1 Evaluate the value of the objective function f. Then proceed to step S406. Return and repeat the above process. Loop L1 may be repeated a predetermined number of times, or it may be repeated until the optimization converges.

[0048] In step S409, the optimization device 500 outputs the point x that gives the minimum (or maximum) value obtained so far as the solution to the optimization problem. More precisely, in a low-dimensional space The solution is the point obtained by projecting the internal point x onto the original higher-dimensional space by 1 / √n A x.

[0049] In this way, a solution that gives the minimum (or maximum) value of the objective function f is obtained. Here, the search range for the low-dimensional space is Y=B H d Because it is limited to this, it may get stuck in a local minimum or the acquisition function may be limited This method avoids situations where the Taika transformation fails, enabling more appropriate optimization. Furthermore, since the above method represents points in low-dimensional space using polar coordinate variables, the search range is [R low , R up ]×[0, π] d-2 It becomes a hypercube of size [0, 2π], and each variable can be changed independently within a predetermined range. Another advantage is that it can be simplified, making implementation easier.

[0050] (Examples of application) The following describes an example of applying this optimization method to a specific problem. Here, we deal with optimizing cardboard box sizes for an e-commerce logistics company. In an e-commerce logistics company, when a customer places an order for goods, the company selects a cardboard box from several types of boxes prepared in advance that can accommodate the ordered goods, packs the goods in that box, and delivers them. The size of the cardboard box (for example, the sum of the three sides) affects the delivery cost, so the delivery cost changes depending on the size of the cardboard boxes prepared in advance. Therefore, it is desirable to design a cardboard box size that minimizes the delivery cost. In this application example, we will optimize this cardboard box size. That is, we will find the cardboard box size that minimizes the delivery cost, with delivery cost as the objective function.

[0051] Figure 6 shows the functional configuration of the optimization device 600 in this application example. As shown in the figure, the optimization device 600 includes a shipping cost calculation unit 610, an optimization unit 615, an order data storage unit 620, a box size data storage unit 630, and a shipping cost data storage unit 640.

[0052] The order data storage unit 620 stores multiple order data, and each order data contains one or more product information. The product information includes at least the size of the product. The order data stored in the order data storage unit 620 can be, for example, actual data for one week from a single warehouse (shipping center).

[0053] The box size data storage unit 630 stores data on the sizes of pre-prepared cardboard boxes. Here, the cardboard box is assumed to be a rectangular prism, and its size is specified by its length, width, and height. The box size data is the data to be optimized by the optimization unit 615. For example, if there are 10 types of pre-prepared cardboard boxes, in this case the optimization problem becomes a 30-dimensional problem.

[0054] The shipping cost data storage unit 640 stores data showing the relationship between the size of the cardboard box and the shipping cost. Although actual shipping costs also vary depending on the shipping distance, here we want to consider the effect of the cardboard box size, so the shipping cost will be assumed to be determined solely by the size of the cardboard box.

[0055] The shipping cost calculation unit 610 calculates the shipping cost when shipping an order included in the order data storage unit 620, given that there are multiple cardboard boxes of sizes defined in the box size data storage unit 630. The shipping cost calculation unit 610 can be considered a functional unit that calculates the value of the objective function at a specific observation point (i.e., a combination of cardboard box sizes). The shipping cost calculation unit 610 includes an order data acquisition unit 611, a box selection unit 612, and a calculation unit 613 as sub-functional units.

[0056] Figure 7 is a flowchart showing the flow of the shipping cost calculation process by the shipping cost calculation unit 610. In loop processing L2, the shipping cost calculation unit 610 calculates the shipping cost for each of the order data.

[0057] Specifically, in step S701, the order data acquisition unit 611 acquires one order data from the order data storage unit 620, and the box selection unit 612 selects the smallest box capable of storing the set of products included in that order. The packing of products into packing boxes has been extensively studied as a 3D bin-packing problem. Since the 3D bin-packing problem is NP-hard, many methods for deriving approximate solutions using heuristic techniques have been proposed. While there are no particular limitations on the solution method for the 3D bin-packing problem that can be used in this example, for example, the method described in Non-Patent Document 3 can be employed.

[0058] In step S702, the calculation unit 613 calculates the shipping cost for the box selected in step S701 by referring to the shipping cost data storage unit 640.

[0059] The loop process L2, consisting of steps S701 and S702 described above, is repeated until all order data in the order data storage unit 620 has been processed. Once the shipping costs for all order data have been determined, the process proceeds to step S703, and the shipping cost calculation unit 610 outputs the sum of the shipping costs for all orders as the final shipping cost.

[0060] The optimization unit 615 optimizes the cardboard box size according to the flowchart in Figure 4. Here, the evaluation of the value of the objective function f in steps S405 and S408 is performed using the shipping cost calculation unit 610. Therefore, the optimization unit 615 can find the cardboard box size that minimizes the overall shipping cost (more precisely, the size of each of the three sides of the pre-prepared cardboard box). As a constraint, in order to suppress extreme changes in box size, the allowable range of change from the initially prepared cardboard box size may be limited to a predetermined range (e.g., ±5%). Given the allowable range of change for each variable, each variable is defined as a variable with a range of [-1, 1]. Variable transformations are possible. For example, a variable L with an initial value L0 is given by L=(1+0.05t)L0, which gives a range [-1, 1]. It can be represented by the variable t. By representing all variables with values ​​in the range [-1, 1], the process described above can be easily implemented.

[0061] Figure 8 shows the optimization process when the optimization device 600 performs the optimization. Here, actual data for one week from a specific warehouse (shipping center) was used as order data, and the sizes of the cardboard boxes available at that warehouse were used as box size data. Furthermore, the Matern5 / 2 kernel was used as the kernel function, and EI (Expected Improvement) was used as the acquisition function. Furthermore, the dimension of the lower-dimensional space embedded in the higher-dimensional space was set to d=5. Also, iteracy The number of iterations (the number of repetitions of loop processing L1 in Figure 4) was set to 20.

[0062] In the first iteration, shipping costs are evaluated without changing the pre-prepared cardboard box size, and the box size is changed in subsequent iterations. Figure 8 shows the shipping costs for each box size in each iteration, relative to the initial shipping cost. As can be seen from the figure, changing the box size reduces shipping costs by approximately 0.55%. This shows that a 0.55% reduction is a significant improvement that can yield substantial cost savings, especially when the annual delivery costs are very large.

[0063] Optimizing cardboard box sizes based on order data is just one example of this method's application. More generally, it can be applied to problems that determine the size of boxes or containers that optimize an objective function, given a given set of items. Furthermore, the applicable problems are not limited to optimizing box or container sizes; it can be applied to any problem where the variable takes continuous values, for example, to determine the inventory quantity of each product in each warehouse. Of course, it can also be applied to any problem outside the logistics field. [Explanation of Symbols]

[0064] 500: Optimization device (information processing device) 501: Processor 505: Memory device 600: Optimization unit 610: Shipping cost calculation unit 611: Order data acquisition unit 612: Box selection unit 613: Calculation unit 614: Optimization unit 620: Order data acquisition unit 630: Box size data storage unit 640: Shipping cost data storage unit

Claims

1. An information processing device for finding an optimal solution for an n-dimensional objective function, It includes a processor and memory that stores the program, The program is provided to the processor: A step of generating a random matrix for embedding a d-dimensional (d < n) low-dimensional space into the n-dimensional space, The steps include: using a subspace of the low-dimensional space as the search space, and finding the optimal solution to the objective function using a Bayesian optimization method; This will cause it to execute, The search space is a region of a hollow sphere or sphere in the lower-dimensional space. An information processing device characterized by the following:

2. The search space is a d-dimensional hollow sphere or d-dimensional sphere obtained by removing a sphere with radius max(√(π / 3)-4 / 3, 0) from a sphere with radius √(π / 3)+4 / 3. The information processing apparatus according to feature 1.

3. In the step of finding the optimal solution, the point in the low-dimensional space is given polar coordinates (r, θ) 1 , …, θ d-1 ) is used to represent it, and [R low , R up ]×[0, π] d-2 The region ×[0, 2π] is defined as the search space. (However, 0≦R) low <R up ), The information processing apparatus according to feature 1.

4. R low = max(√(π / 3)-4 / 3, 0), R up = √(π / 3)+4 / 3 The information processing apparatus according to claim 3.

5. In the step of finding the optimal solution, when points in the low-dimensional space are represented in polar coordinates as x and y, and the coordinate transformation from polar coordinates to Cartesian coordinates is represented as φ, a function represented as K(φ(x), φ(y)) is used as the kernel function in the Bayesian optimization. The information processing apparatus according to claim 3.

6. The aforementioned random matrix is ​​1 / √n × A, where matrix A is an (n, d) matrix whose components each independently follow a standard normal distribution with mean 0 and variance 1. The information processing apparatus according to feature 1.

7. The step of finding the optimal solution is: A first step is to evaluate the value of the objective function at a point in the n-dimensional space corresponding to one or more points in the search space, A second step involves calculating the acquisition function based on the results of the above evaluation, A third step involves determining the next evaluation point from within the search space based on the value of the acquisition function, A step of repeating the first to third steps a predetermined number of times, including, The information processing apparatus according to feature 1.

8. The aforementioned information processing device performs optimization to determine multiple box sizes to be prepared in advance so as to minimize the shipping cost corresponding to a given order data. The step of evaluating the value of the objective function is, Steps to retrieve multiple order data, For each of the aforementioned multiple order data, the step of selecting a box from a predetermined set of boxes in which the products included in the order data should be stored, For each of the aforementioned multiple order data, the step of determining the shipping cost for the selected box, The steps include outputting the total shipping costs for the multiple order data, The information processing apparatus according to claim 1, which includes the following:

9. An information processing method performed by an information processing device to find a solution that optimizes an n-dimensional objective function, A step of generating a random matrix for embedding a d-dimensional (d < n) low-dimensional space into the n-dimensional space, The steps include: using a subspace of the low-dimensional space as the search space, and finding the optimal solution to the objective function using a Bayesian optimization method; Includes, The search space is a region of a hollow sphere or sphere in the lower-dimensional space. An information processing method characterized by the following:

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