Information processing program, information processing device, and information processing method

Abstract

To improve the efficiency of search processing based on a risk aversion index.SOLUTION: A computer calculates an estimate of a risk aversion index to each of a plurality of candidate points within a search space on the basis of an observation history including one or more observation points and an observation value of a probability variable to each of the one or more observation points, and a kernel function. The computer calculates a value of an acquisition function to each of the plurality of candidate points on the basis of the estimate of the risk aversion index. The computer searches a target point corresponding to a prescribed value of the acquisition function from among the plurality of candidate points on the basis of the value of the acquisition function, and outputs a search result including the target point.SELECTED DRAWING: Figure 4

Description

[Technical Field]

[0001] This invention relates to information processing technology. [Background technology]

[0002] Bayesian optimization is a technique for optimizing an unknown function f, which can only be known through noisy observations, by repeatedly making observations. Bayesian optimization is applied to problems where the function f is optimized while receiving feedback.

[0003] Examples of such problems include material design, drug discovery, recommendation systems, sensor placement for environmental monitoring, hyperparameter tuning for machine learning, and AutoML (Automated Machine Learning). Recommendation systems include systems for web advertising and website optimization.

[0004] In many Bayesian optimizations, a score a(x) is defined for a point x in the search space based on the observation history, and the point with the highest a(x) is selected as the next observation point. a(x) is sometimes called the acquisition function.

[0005] In relation to optimization, methods for optimizing the Conditional Value-At-Risk (CVaR) of a black-box function are known (see, for example, Non-Patent Document 1). Gaussian process optimization in bandit settings is also known (see, for example, Non-Patent Document 2). Spectral bandits for smooth graph functions are also known (see, for example, Non-Patent Document 3). Multi-period trading using convex optimization is also known (see, for example, Non-Patent Document 4). [Prior art documents] [Non-patent literature]

[0006] [Non-Patent Document 1] QP Nguyen et al., “Optimizing Conditional Value-At-Risk of Black-Box Functions”, NeurIPS 2021, 2021. [Non-Patent Document 2] N. Srinivas et al., “Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design”, ICML 2010, 2010. [Non-Patent Document 3] Valko et al., “Spectral Bandits for Smooth Graph Functions”, ICML 2014, 2014. [Non-Patent Document 4] Boyd et al., “Multi-period Trading via Convex Optimization”, Foundations and Trends in Optimization, 2017. [Overview of the project] [Problems that the invention aims to solve]

[0007] In typical Bayesian optimization methods such as GP-UCB (Gaussian Process Upper Confidence Bound) and GP-EI (Gaussian Process Expected Improvement), the search for a point x that maximizes the expected value E[f(x)] of a function f(x) is performed. However, depending on the problem, metrics other than the expected value, such as robustness or risk indicators, may become important.

[0008] For example, in mean-variance optimization, the search for a point x such that the variance V[f(x)] of f(x) is small and the expected value E[f(x)] is large is performed. However, variance treats the cases where the value of the random variable representing f(x) is greater than or less than the expected value the same way. Therefore, when searching for a point x where the value of the random variable is large, the characteristics of the distribution may not be adequately captured by using only the expected value and variance.

[0009] Therefore, CVaR is sometimes used as a metric in Bayesian optimization. CVaR represents the expected value of a random variable conditioned on undesirable events that occur with a probability below a certain level. For example, CVaR is used in finance to evaluate investment strategies. In this case, CVaR represents the average return under undesirable events that occur with a low probability, such as financial crises.

[0010] However, simply searching for a point x that optimizes CVaR does not necessarily guarantee an efficient search.

[0011] Furthermore, this problem arises not only in search processes that optimize CVaR, but also in search processes that optimize various risk avoidance indicators. Risk avoidance indicators refer to indicators used to avoid risk.

[0012] In one aspect, the present invention aims to improve the efficiency of search processing based on risk avoidance indicators. [Means for solving the problem]

[0013] One approach involves having the computer perform the following processes as part of the information processing program.

[0014] The computer calculates an estimate of the risk aversion index for each of several candidate points in the search space, based on an observation history that includes one or more observation points and the observed values ​​of random variables for each of those observation points, and a kernel function. Based on the estimated risk aversion index, the computer calculates the value of the acquisition function for each of the several candidate points.

[0015] The computer searches for a target point corresponding to a predetermined value of the acquisition function from among multiple candidate points based on the value of the acquisition function, and outputs the search results including the target point. [Effects of the Invention]

[0016] From one perspective, it is possible to improve the efficiency of the search process based on risk aversion indicators. [Brief explanation of the drawing]

[0017] [Figure 1] This is a functional configuration diagram of the information processing device according to the embodiment. [Figure 2] This is a flowchart of the search process performed by an information processing device. [Figure 3] This is a functional configuration diagram of the search device. [Figure 4] This figure shows the formula for calculating CVaR(x,t). [Figure 5] This is a flowchart of the first search process. [Figure 6] This figure shows CVaR and UCB in round t. [Figure 7] This figure shows CVaR and UCB in round t+1. [Figure 8] This figure shows CVaR and UCB in round t+2. [Figure 9A] This is the flowchart (part 1) for the second search process. [Figure 9B] This is the flowchart (part 2) for the second search process. [Figure 10] This is a hardware configuration diagram of an information processing device. [Modes for carrying out the invention]

[0018] The embodiments will be described in detail below with reference to the drawings.

[0019] Figure 1 shows an example of the functional configuration of an information processing device (computer) according to an embodiment. The information processing device 101 in Figure 1 includes a calculation unit 111, a search unit 112, and an output unit 113.

[0020] Figure 2 is a flowchart illustrating an example of the search process performed by the information processing device 101 in Figure 1. First, the calculation unit 111 obtains an estimate of the risk aversion index for each of the multiple candidate points in the search space based on the observation history, which includes one or more observation points and the observed values ​​of the random variable for each observation point, and the kernel function (step 201). Next, the calculation unit 111 obtains the value of the acquisition function for each of the multiple candidate points based on the estimate of the risk aversion index (step 202).

[0021] The search unit 112 searches for a target point corresponding to a predetermined value of the acquisition function from among multiple candidate points based on the value of the acquisition function (step 203). The output unit 113 outputs the search results including the target point (step 204).

[0022] According to the information processing device 101 in Figure 1, the efficiency of the search process based on risk avoidance indicators can be improved.

[0023] Figure 3 shows a functional configuration example of a search device corresponding to a specific example of the information processing device 101 in Figure 1. The search device 301 in Figure 3 includes a calculation unit 311, a search unit 312, an output unit 313, and a storage unit 314, and performs either a first search process or a second search process. The first and second search processes are search processes based on Bayesian optimization.

[0024] In the first and second search processes, the search device 301 searches for a target point corresponding to a predetermined value of the acquisition function, which includes a risk avoidance indicator. The predetermined value of the acquisition function may be the maximum or minimum value.

[0025] The calculation unit 311, the search unit 312, and the output unit 313 correspond to the calculation unit 111, the search unit 112, and the output unit 113 in Figure 1, respectively.

[0026] The memory unit 314 stores the search space information 321 and the observation history 322. The search space information 321 indicates the search space W. The observation history 322 includes one or more observation points and the observed values ​​of the random variable for each observation point.

[0027] The calculation unit 311 uses the observation history 322 to calculate an estimated risk aversion index for each of the multiple points in the search space W, and uses the estimated risk aversion index to calculate the value of the acquisition function. The points in the search space W correspond to candidate points.

[0028] The search unit 312 searches for target points based on the calculated acquisition function values, generates a search result 323 including the searched target points, and stores it in the storage unit 314. The output unit 113 outputs the search result 323.

[0029] CVaR is used as an example of a risk aversion indicator. For a random variable X that follows a cumulative distribution function F, CVaR(F) can be expressed, for example, by the following equation.

[0030] CVaR(F)=E X~F [X|X≦F -1 (α)] (1)

[0031] α is a real number between 0 and 1, and represents the risk level. -1 (α) represents the inverse function of F(α). X~F represents that X follows F. Approximately, if we denote the sequence of samples of X generated from F in ascending order as X1, X2, ..., Xn, then the right-hand side of equation (1) corresponds to the mean of X1, X2, ..., X[nα], where [nα] represents the largest integer less than or equal to nα.

[0032] In typical Bayesian optimization models, it is assumed that the function f is modeled by a Gaussian process and a reproducing kernel Hilbert space. In this model, if points x and x' are close together, then f(x) and f(x') are also close together, thus reducing the computational complexity of the search process.

[0033] In contrast, the search device 301 models a probability distribution ρ(x) that depends on x, rather than the function f, using kernel mean embedding (KME). The random variable y representing f(x) follows the probability distribution ρ(x).

[0034] KME is a technique that embeds ρ(x) into a reproducing kernel Hilbert space by vectorizing ρ(x). By vectorizing ρ(x), it is possible to calculate the distance between two probability distributions, the norm of a joint probability distribution, etc. Conditional mean embedding (CME) is a technique that vectorizes conditional probabilities and can be considered a special case of KME.

[0035] The kernel function k(x,x') used in KME represents the similarity between x and x'. The kernel function k can be an RBF (Radial Basis Function) kernel, a Matern kernel, or other types. x and x' can be scalars, vectors, or graphs. For example, if graphs representing the chemical formulas of compounds are used as x and x', then k(x,x') represents the similarity between the two compounds.

[0036] First, let's explain the first search process. The search unit 312 searches for points x in the search space W for rounds t=1,2,...,T. t Select the probability distribution ρ(x t Observed values ​​of the random variable y that follows ) t The following is obtained. t is a control variable representing the time step, and T is an integer greater than or equal to 2 representing the total number of rounds. The observation history H generated in round t is (x1, y1), ..., (x t ,y t) includes. The observation history H corresponds to the observation history 322.

[0037] In round t + 1, the calculation unit 311 calculates an estimated value CVaR(x, t) of CVaR for a point x in the search space W using the observation history H generated in round t and the kernel function k.

[0038] Figure 4 shows an example of the calculation formula for CVaR(x, t). Assuming that all elements of the support of the probability distribution ρ(x) are included in the interval [a, b], the set U is defined by U = {a, b, y1,..., y t}. ν represents any element of U. The right side of the calculation formula in Figure 4 represents the maximum value of the values within {} for each ν included in U.

[0039] α represents the risk level. λ is a positive real number. ψ ν (y) represents max(ν - y, 0), and max(p, q) represents the maximum value of p and q. (ψ ν (y1),..., ψ ν (y t )) represents a t-dimensional row vector whose j-th element is ψ ν (y j ).

[0040] k(x 1:t , x 1:t ) represents a t×t matrix whose (i, j)-th element is k(x i , x j ). 1 t represents a t×t identity matrix. (k(x 1:t , x 1:t ) + λ1 t ) -1 represents the inverse matrix of k(x 1:t , x 1:t ) + λ1 t . k(x 1:t , x) represents a t-dimensional column vector whose i-th element is k(x i , x).

[0041] In round t+1, the calculation unit 311 uses CVaR(x,t) in Figure 4 to calculate the value of the acquisition function UCB for point x, UCB(x,t), using the following formula:

[0042] UCB(x,t) = CVaR(x,t) + (β / α)σ t (x) (2)

[0043] β is a positive real number. t (x) represents the uncertainty of the prediction regarding the random variable y. σ t For (x), for example, the standard deviation in equation (2) of Non-Patent Document 2 can be used.

[0044] When t=1, the search unit 312 randomly selects an observation point x1 from the search space W and observes y1. When t≧2, the search unit 312 selects the point x in the search space W that maximizes UCB(x,t-1) and sets it to the next observation point x t Select as, y t We observe the following. By selecting the point x that maximizes UCB(x,t-1), we can select a point with a large estimate of CVaR, as well as a point with high uncertainty. Selecting a point with high uncertainty facilitates the search.

[0045] The search device 301 can find the optimal solution to various optimization problems by performing a first or second search process. For example, in the problem of optimizing the hyperparameters of a machine learning model, the hyperparameters can be used as x, and the accuracy of the machine learning model on known input data can be used as y.

[0046] Machine learning models such as neural networks, decision trees, logistic regression, and support vector machines are used. In the case of neural networks, hyperparameters may be the number of units, the number of layers, or the learning rate.

[0047] Applications of optimization problems include finance, medicine, healthcare, drug discovery, recruitment, fraud detection, materials design, recommendation systems, and sensor placement. Fraud detection includes detecting fraudulent credit card use and fraudulent insurance claims.

[0048] Finance includes stock trading. For example, in the case of stock trading described in Non-Patent Document 4, a three-dimensional vector with a risk aversion variable, a trade aversion variable, and a holding aversion variable as elements can be used as x, and the daily return obtained from stock trading can be used as y. The larger the risk aversion variable, the more likely traders are to avoid volatile stocks; the larger the trade aversion variable, the less likely traders are to trade; and the larger the holding aversion variable, the less likely traders are to short sell.

[0049] Medical applications include cancer staging. When performing cancer staging using a classifier called XGboost (extreme gradient boosting), a multidimensional vector with multiple hyperparameters as elements can be used as x, and the F-score or similar value from the validation data can be used as y. The hyperparameters may include the learning rate, the maximum depth of the decision tree, the regularization coefficient, and the ratio of the number of randomly selected features when generating the decision tree. XGboost is also used in application fields other than medicine.

[0050] In drug discovery, useful compounds are sometimes found by repeatedly performing simulations or experiments. In this case, the graph representing the chemical formula of the compound can be used as x, and the drug's effect observed through simulation or experimentation can be used as y. The drug's effect is expressed by the bond free energy, dissociation constant, etc.

[0051] Figure 5 is a flowchart showing an example of the first search process. First, the search unit 312 sets an empty set as the observation history H (step 501) and sets t to 1 (step 502).

[0052] Next, the search unit 312 searches the search space W to the observation point x tSelect randomly (step 503), x t Observed value y for t Obtain (step 504). Then, the search unit 312 (x t ,y t Update H by adding ) to H (step 505).

[0053] Next, the search unit 312 compares t with T (step 506). If t is less than T (step 506, NO), the search unit 312 increments t by 1 (step 507).

[0054] Next, the calculation unit 311 calculates CVaR(x,t-1) for each point x in the search space W using the calculation formula in Figure 4 (step 508), and then calculates UCB(x,t-1) using equation (2) (step 509).

[0055] Next, the search unit 312 finds the point x that maximizes UCB(x,t-1) and moves it to the next observation point x t Select as (step 510), and repeat the process from step 504 onwards. The search unit 312 can identify the point x that maximizes UCB(x,t-1) by, for example, random search or brute search.

[0056] If t=T (step 506, YES), the search unit 312 generates the search result 323, and the output unit 113 outputs the search result 323 (step 511). The search result 323 is H={(x1,y1),...,(x T ,y T )} may also be included, (x T ,y T It may contain only ). T This corresponds to the target point.

[0057] In step 506, the search unit 312 may decide whether or not to terminate the search using other termination conditions.

[0058] Observation point x selected in step 510 of round t tThis is an example of the first observation point, and H, updated in step 505, is an example of the updated observation history. Observation point x selected in step 510 of round t+1 t+1 This is an example of a second observation point.

[0059] When optimizing the hyperparameters of a machine learning model, the search unit 312 uses the x of the search result 323. T A machine learning model may be constructed using the hyperparameters indicated. In this case, the search unit 312 can generate a trained machine learning model by inputting training data into the constructed machine learning model and performing machine learning.

[0060] Figure 6 shows examples of CVaR and UCB in round t. The horizontal axis represents the one-dimensional search space W, and the vertical axis represents CVaR and UCB. The dashed lines represent selected observation points x included in the observation history H. Curve 601 represents the true value of CVaR for each point x in the search space W, and curve 602 represents the value of UCB(x,t-1) for each point x in the search space W.

[0061] In this case, the selected observation points x are 0.0, 0.2, 0.4, and 1.0 (4 points). Next observation point x t As such, 0.77 is selected, which corresponds to the maximum value of UCB(x,t-1), and the observed value y corresponds to 0.77. t This is obtained. And, (x t ,y t ) is added to H.

[0062] Figure 7 shows examples of CVaR and UCB in round t+1. Curve 701 represents the true value of CVaR, and curve 702 represents the value of UCB(x,t). The true value of CVaR is not observed, but by using equation (2), the uncertainty in the neighborhood of x=0.77 is reduced by observation, and UCB(x,t) approaches the true value of CVaR.

[0063] In this case, the next observation point x t+1As such, 0.63 is selected, which corresponds to the maximum value of UCB(x,t), and the observed value y corresponds to 0.63. t+1 This is obtained. And, (x t+1 ,y t+1 ) is added to H.

[0064] Figure 8 shows an example of CVaR and UCB in round t+2. Curve 801 represents the true value of CVaR, and curve 802 represents the value of UCB(x,t+1). Observations reduce the uncertainty in the neighborhood of x=0.63, causing UCB(x,t+1) to approach the true value of CVaR.

[0065] According to the first search process, by updating the observation history H and applying the kernel function k to the updated observation history H, the estimated CVaR in the neighborhood of already explored points x is improved, thus improving search efficiency. For example, in the neighborhood of point x where the CVaR is estimated to be small, the search is suppressed, and in the neighborhood of point x where the CVaR is estimated to be large, the search is promoted.

[0066] In conventional Bayesian optimization, which models the function f, the problems for which CVaR can be optimized are limited to specific types of problems. In contrast, Bayesian optimization using KME models the probability distribution, allowing for the optimization of CVaR in more general problems.

[0067] Instead of CVaR, the risk aversion index MV(ρ), which includes the expected value and variance of the random variable y, can be used as a risk aversion index in the first exploration process. MV(ρ) can be expressed, for example, by the following equation.

[0068] MV(ρ) = E[ρ] - γV[ρ] =E y~ρ [y]-γE y~ρ [y 2 ]+γ(E y~ρ [y]) 2 (3)

[0069] γ is a positive real number. E[ρ] represents the expected value of ρ(x), and V[ρ] represents the variance of ρ(x).y~ρ [y] represents the expected value of y, and E y~ρ [y 2 ] is y 2 This represents the expected value. y~ρ indicates that y follows ρ. By performing Bayesian optimization using MV(ρ), we can search for a point x where the expected value E[ρ] is large and the variance V[ρ] is small.

[0070] In this case, the calculation unit 311, in round t+1, uses the observation history H and kernel function k generated in round t to calculate E for point x in the search space W. y~ρ [y r The estimated value m(x,t,r)(r=1,2) of ] is calculated using the following formula.

[0071] m(x,t,r)=(y1 r ,...,y t r ) (k(x 1:t ,x 1:t )+λ1 t ) -1 k(x 1:t ,x) (4)

[0072] (y1 r ,...,y t r ) is when the j-th element is y j r This represents a t-dimensional row vector such that (k(x 1:t ,x 1:t )+λ1 t ) -1 and k(x 1:t The formula for x) is the same as the formula in Figure 4.

[0073] In round t+1, the calculation unit 311 uses m(x,t,r) from equation (4) to calculate the acquisition function UCB for point x. MV Value UCB MV (x,t) is calculated using the following formula.

[0074] UCB MV (x,t) =m(x,t,1)-γm(x,t,2)+γ(m(x,t,1)) 2 +β1σ t (x)+β2(σ t (x)) 2 (5)

[0075] β1 and β2 are positive real numbers. σ t (x) is the same as in Equation (2).

[0076] In the first search process, when using the acquisition function UCB MV the calculation unit 311 calculates m(x,t - 1,r) by Equation (4) in step 508 of FIG. 5, and calculates UCB MV (x,t - 1) by Equation (5) in step 509. Then, the search unit 312 selects, in step 510, the point x that maximizes UCB MV (x,t - 1) as the next observation point x t . Thereby, the index MV(ρ) can be optimized instead of CVaR.

[0077] Next, the second search process will be described. The search space W in the second search process is a finite set. In the second search process, based on the value UCB(x,t) of the acquisition function UCB, points x that do not require search are excluded from the search space W, and the next observation point x t+1 is selected from the remaining points x.

[0078] FIGS. 9A and 9B are flowcharts showing an example of the second search process. First, the search unit 312 sets an empty set as the observation history H (step 901), and sets 1 to t (step 902). Then, the search unit 312 sets 1 to j (step 903), and sets W to the search space W j (step 904).

[0079] Next, the search unit 312 obtains σ j-1 using t(j)=2 t-1 (x,j) (step 905). In step 905, the search unit 312 determines x1,...,x t-1of x t(j) ,...,x t-1 Using only W j Uncertainty σ at each point x within the interior t-1 Define (x) and define σ t-1 (x) σ t-1 Set to (x,j).

[0080] Next, the search unit 312, σ t-1 The point x that maximizes (x,j) is the next observation point x t as W j Select from (step 906), x t Observed value y for t Obtain (step 907). Then, the search unit 312 (x t ,y t Update H by adding ) to H (step 908).

[0081] Next, the search unit 312 increments t by 1 (step 909) and compares t with s (step 910). s represents t(j+1)-1 and the minimum value of T. t(j+1)=2 j If t is less than or equal to s (step 910, YES), the search unit 312 repeats the processing from step 905 onwards.

[0082] If t is greater than s (step 910, NO), the calculation unit 311 calculates W j The estimated CVaR for each point x within the matrix, CVaR(x,j,s), is calculated using the following formula (Step 911).

[0083]

number

[0084] However, U={a,b,y t(j) ,...,y s}. The right-hand side of equation (6) is (x1, y1), ..., (x included in H) t-1 ,y t-1 ) of (x t(j) ,y t(j) ),...,(xs ,y s This represents an estimate of CVaR calculated using only ).

[0085] Next, the calculation unit 311, W j The value of the acquisition function L(x,j,s) for each point x within the matrix is ​​calculated using the following formula (Step 912).

[0086] L(x,j,s) = CVaR(x,j,s)-(β(δ) / α)σ s (x,j) (7) β(δ)=λ 1 / 2 +(log(1 / δ)) 1 / 2 (8)

[0087] λ and δ are positive real numbers. Next, the search unit 312, W j The maximum value Lmax of L(x,j,s) for each point x within is found (step 913). Then, the search unit 312 finds a point x that satisfies the following equation W j By excluding from W j Update the search space W, which includes the remaining points x. j+1 Generate (step 914).

[0088] CVaR(x,j,s)+(β(δ) / α)σ s (x,j) <Lmax (9)

[0089] Next, the search unit 312 compares j with [log2T]+1 (step 915), where [log2T] represents the largest integer less than or equal to log2T. If j is less than [log2T]+1 (step 915, NO), the search unit 312 increments j by 1 (step 916) and returns the W generated in step 914. j+1 Repeat the process from step 905 onwards using [this method].

[0090] If j = [log2T] + 1 (step 915, YES), the search unit 312 generates the search result 323, and the output unit 113 outputs the search result 323 (step 917). The search result 323 is H = {(x1, y1), ..., (x T,y T )} may also be included, (x T ,y T It may contain only ).

[0091] In step 915, the search unit 312 may decide whether or not to terminate the search using other termination conditions.

[0092] The search space W generated in step 914 j+1 This is an example of a first search space, W j+1 The next search space W is generated. j+2 This is an example of a second search space.

[0093] In the second search process, similar to the first search process, applying the kernel function k to the updated observation history H improves the estimated CVaR in the neighborhood of the explored point x, thus improving search efficiency. Furthermore, the search space W j By updating the search target and narrowing it down, it becomes possible to find the optimal solution faster than the first search process.

[0094] The configuration of the information processing device 101 in Figure 1 is merely an example, and some components may be omitted or modified depending on the use or conditions of the information processing device 101. The configuration of the search device 301 in Figure 3 is merely an example, and some components may be omitted or modified depending on the use or conditions of the search device 301.

[0095] The flowcharts in Figures 2, 5, 9A, and 9B are merely examples, and some processes may be omitted or modified depending on the configuration or conditions of the information processing device 101 or the search device 301.

[0096] The calculation formula shown in Figure 4 is merely an example, and the search device 301 may calculate CVaR(x,t) using a different formula. The changes in CVaR and UCB shown in Figures 6 to 8 are merely examples, and CVaR and UCB change depending on the problem to which the search process is applied.

[0097] Equations (1) to (9) are merely examples, and the search device 301 may perform the search process using a different calculation formula.

[0098] Figure 10 shows an example of the hardware configuration of an information processing device used as the information processing device 101 in Figure 1 and the search device 301 in Figure 3. The information processing device in Figure 10 includes a CPU (Central Processing Unit) 1001, memory 1002, input device 1003, output device 1004, auxiliary storage device 1005, media drive device 1006, and network connection device 1007. These components are hardware and are connected to each other by a bus 1008.

[0099] Memory 1002 is, for example, a semiconductor memory such as ROM (Read Only Memory) or RAM (Random Access Memory), and stores the program and data used for processing. Memory 1002 may also operate as the storage unit 314 in Figure 3.

[0100] The CPU 1001 (processor) operates as the calculation unit 111 and search unit 112 in Figure 1, for example, by executing a program using the memory 1002. The CPU 1001 also operates as the calculation unit 311 and search unit 312 in Figure 3, by executing a program using the memory 1002.

[0101] The input device 1003 is, for example, a keyboard, a pointing device, etc., and is used for inputting instructions or information from the user or operator. The output device 1004 is, for example, a display device, a printer, etc., and is used for inquiries or instructions to the user or operator, and for outputting processing results. The processing result may be the search result 323. The output device 1004 may operate as the output unit 113 in Figure 1 or the output unit 313 in Figure 3.

[0102] The auxiliary storage device 1005 is, for example, a magnetic disk drive, an optical disk drive, a magneto-optical disk drive, a tape drive, etc. The auxiliary storage device 1005 may also be a hard disk drive. The information processing device can store programs and data in the auxiliary storage device 1005 and load them into the memory 1002 for use. The auxiliary storage device 1005 may also operate as the storage unit 314 in Figure 3.

[0103] The media drive unit 1006 drives the portable recording medium 1009 and accesses its recorded contents. The portable recording medium 1009 can be a memory device, a flexible disk, an optical disk, a magneto-optical disk, etc. The portable recording medium 1009 may also be a CD-ROM (Compact Disk Read Only Memory), a DVD (Digital Versatile Disk), a USB (Universal Serial Bus) memory, etc. The user or operator can store programs and data on the portable recording medium 1009 and load them into the memory 1002 for use.

[0104] Thus, the computer-readable recording medium that stores the programs and data used in the processing is a physical (non-temporary) recording medium such as memory 1002, auxiliary storage device 1005, or portable recording medium 1009.

[0105] The network connection device 1007 is a communication interface circuit that connects to a communication network such as a WAN (Wide Area Network) or LAN (Local Area Network) and performs data conversion associated with communication. The information processing device can receive programs and data from external devices via the network connection device 1007 and load them into the memory 1002 for use. The network connection device 1007 may also operate as the output unit 113 in Figure 1 or the output unit 313 in Figure 3.

[0106] Note that the information processing device does not need to include all the components shown in Figure 10, and some components may be omitted depending on the intended use or conditions of the information processing device. For example, if the portable recording medium 1009 or a communication network is not used, the media drive device 1006 or the network connection device 1007 may be omitted.

[0107] While embodiments of the disclosure and their advantages have been described in detail, those skilled in the art will be able to make various modifications, additions, and omissions without departing from the scope of the invention as expressly stated in the claims.

[0108] With reference to Figures 1 to 10, the following additional information is disclosed regarding the embodiments described. (Note 1) Based on an observation history including one or more observation points and the observed values ​​of a random variable for each of the one or more observation points, and a kernel function, estimates of risk aversion indicators for each of the multiple candidate points in the search space are obtained. Based on the estimated risk aversion indicator, the value of the acquisition function for each of the multiple candidate points is determined. Based on the value of the acquisition function, a target point corresponding to a predetermined value of the acquisition function is searched from among the plurality of candidate points. Output the search results including the aforementioned target point. An information processing program that causes a computer to perform a task. (Note 2) The information processing program according to Appendix 1, characterized in that the acquisition function includes the risk aversion indicator and the uncertainty of the prediction regarding the random variable. (Note 3) The process of searching for the aforementioned target point is as follows: A process to select a first observation point from among the multiple candidate points based on the value of the acquisition function, A process for obtaining the observed value of the random variable for the first observation point, A process to update the observation history by adding the first observation point and the observed value of the random variable for the first observation point to the observation history, A process to determine the value of the acquisition function based on the updated observation history, A process to select a second observation point from among the multiple candidate points based on the value of the acquisition function obtained based on the updated observation history, A process for searching for the target point using the second observation point, An information processing program according to Appendix 1 or 2, characterized by including the following: (Note 4) The information processing program according to Appendix 3, characterized in that the risk aversion indicator is an indicator that includes a conditional value at risk for the random variable, or an indicator that includes the expected value and variance of the random variable. (Note 5) The process of searching for the aforementioned target point is as follows: A process to generate a first search space by excluding some of the candidate points from the plurality of candidate points based on the value of the acquisition function, A process to obtain the value of the acquisition function based on the first search space, A process to generate a second search space by excluding some of the candidate points from among multiple candidate points in the first search space based on the value of the acquisition function obtained based on the first search space, A process for searching for the target point from among multiple candidate points in the second search space, An information processing program according to Appendix 1 or 2, characterized by including the following: (Note 6) The information processing program according to Appendix 5, characterized in that the risk aversion indicator is an indicator that includes a conditional value at risk with respect to the random variable. (Note 7) A calculation unit that calculates an estimated risk aversion index for each of several candidate points in the search space based on an observation history including one or more observation points and the observed values ​​of a random variable for each of the one or more observation points, and a kernel function, and calculates the value of the acquisition function for each of the several candidate points based on the estimated risk aversion index, A search unit searches for a target point corresponding to a predetermined value of the acquisition function from among the plurality of candidate points based on the value of the acquisition function, An output unit that outputs the search results including the aforementioned target point, An information processing device characterized by comprising: (Note 8) The information processing apparatus according to Appendix 7, characterized in that the acquisition function includes the risk aversion indicator and the uncertainty of the prediction regarding the random variable. (Note 9) The search unit selects a first observation point from among the plurality of candidate points based on the value of the acquisition function, obtains an observed value of the random variable for the first observation point, and updates the observation history by adding the first observation point and the observed value of the random variable for the first observation point to the observation history. The calculation unit determines the value of the acquisition function based on the updated observation history, The information processing device according to Appendix 7 or 8, characterized in that the search unit selects a second observation point from among the plurality of candidate points based on the value of the acquisition function obtained based on the updated observation history, and searches for the target point using the second observation point. (Note 10) The information processing apparatus according to Appendix 9, characterized in that the risk aversion indicator is an indicator that includes a conditional value at risk for the random variable, or an indicator that includes the expected value and variance of the random variable. (Note 11) The search unit generates a first search space by excluding some of the candidate points from the plurality of candidate points based on the value of the acquisition function. The calculation unit determines the value of the acquisition function based on the first search space, The information processing device according to Appendix 7 or 8, characterized in that the search unit generates a second search space by excluding some of the candidate points from a plurality of candidate points in the first search space based on the value of the acquisition function obtained based on the first search space, and searches for the target point from a plurality of candidate points in the second search space. (Note 12) The information processing device according to Appendix 11, characterized in that the risk avoidance indicator is an indicator that includes a conditional value at risk with respect to the random variable. (Note 13) Based on an observation history including one or more observation points and the observed values ​​of a random variable for each of the one or more observation points, and a kernel function, estimates of risk aversion indicators for each of the multiple candidate points in the search space are obtained. Based on the estimated risk aversion indicator, the value of the acquisition function for each of the multiple candidate points is determined. Based on the value of the acquisition function, a target point corresponding to a predetermined value of the acquisition function is searched from among the plurality of candidate points. Output the search results including the aforementioned target point. An information processing method characterized in that the processing is performed by a computer. (Note 14) The information processing method according to Appendix 13, characterized in that the acquisition function includes the risk aversion indicator and the uncertainty of the prediction regarding the random variable. (Note 15) The process of searching for the aforementioned target point is as follows: A process to select a first observation point from among the multiple candidate points based on the value of the acquisition function, A process for obtaining the observed value of the random variable for the first observation point, A process to update the observation history by adding the first observation point and the observed value of the random variable for the first observation point to the observation history, A process to determine the value of the acquisition function based on the updated observation history, A process to select a second observation point from among the multiple candidate points based on the value of the acquisition function obtained based on the updated observation history, A process for searching for the target point using the second observation point, The information processing method described in Appendix 13 or 14, characterized by including the following: (Note 16) The information processing method according to Appendix 15, characterized in that the risk aversion indicator is an indicator that includes a conditional value at risk for the random variable, or an indicator that includes the expected value and variance of the random variable. (Note 17) The process of searching for the aforementioned target point is as follows: A process to generate a first search space by excluding some of the candidate points from the plurality of candidate points based on the value of the acquisition function, A process to obtain the value of the acquisition function based on the first search space, A process to generate a second search space by excluding some of the candidate points from among multiple candidate points in the first search space based on the value of the acquisition function obtained based on the first search space, A process for searching for the target point from among multiple candidate points in the second search space, The information processing method described in Appendix 13 or 14, characterized by including the following: (Note 18) The information processing method according to Appendix 17, characterized in that the risk aversion indicator is an indicator that includes a conditional value at risk with respect to the random variable. [Explanation of Symbols]

[0109] 101 Information Processing Device 111, 311 Calculation section 112, 312 Search Department 113, 313 Output section 301 Search device 314 Storage section 321 Search space information 322 Observation History 323 Search results 601, 602, 701, 702, 801, 802 curve 1001 CPU 1002 memory 1003 Input device 1004 Output device 1005 Auxiliary storage device 1006 Media drive device 1007 Network Connection Device 1008 Bus 1009 Portable recording media

Claims

1. By applying a kernel function using equation (1) to the observation history which includes one or more observation points and the observed values ​​of the random variable for each of the one or more observation points, an estimate of the risk aversion index for each of the multiple candidate points in the search space is obtained. Based on the estimated risk avoidance indicator, the value of the acquisition function for each of the multiple candidate points is determined. Based on the value of the acquisition function, a target point corresponding to a predetermined value of the acquisition function is searched from among the plurality of candidate points. Output the search results including the aforementioned target point. An information processing program that causes a computer to perform a task. [Math 1] However, in equation (1), CVaR(x, t) represents the estimated value, v represents any element of the set U defined by {a, b, y1, ..., yt}, the right-hand side of equation (1) represents the maximum value of the {} for each v included in the set U, α represents the risk level, λ is a positive real number, ψν(y) represents max(v-y, 0), max(p, q) represents the maximum values ​​of p and q, and (ψν(y1), ..., ψν(yt)) is the case where the j-th element is ψν(yj) Here, we have a row vector of t dimensions, where k(x1:t, x1:t) represents a t×t matrix with the element at row i and column j being k(xi, xj), 1t represents the t×t identity matrix, (k(x1:t, x1:t)+λ1t)-1 represents the inverse matrix of k(x1:t, x1:t)+λ1t, and k(x1:t, x) represents a column vector of t dimensions with the i-th element being k(xi, x).

2. The information processing program according to claim 1, characterized in that the acquisition function includes the risk aversion indicator and the uncertainty of the prediction regarding the random variable.

3. The process of searching for the aforementioned target point is as follows: A process to select a first observation point from among the multiple candidate points based on the value of the acquisition function, A process for obtaining the observed value of the random variable for the first observation point, A process to update the observation history by adding the first observation point and the observed value of the random variable for the first observation point to the observation history, A process to determine the value of the acquisition function based on the updated observation history, A process to select a second observation point from among the multiple candidate points based on the value of the acquisition function obtained based on the updated observation history, A process for searching for the target point using the second observation point, An information processing program according to claim 1 or 2, characterized by including the following:

4. The information processing program according to claim 3, characterized in that the risk aversion indicator is an indicator that includes a conditional value at risk for the random variable, or an indicator that includes the expected value and variance of the random variable.

5. The process of searching for the aforementioned target point is as follows: A process to generate a first search space by excluding some of the candidate points from the plurality of candidate points based on the value of the acquisition function, A process to obtain the value of the acquisition function based on the first search space, A process to generate a second search space by excluding some of the candidate points from among multiple candidate points in the first search space based on the value of the acquisition function obtained based on the first search space, A process for searching for the target point from among a plurality of candidate points in the second search space, An information processing program according to claim 1 or 2, characterized by including the following:

6. The information processing program according to claim 5, characterized in that the risk aversion indicator is an indicator that includes a conditional value at risk with respect to the random variable.

7. A calculation unit that applies a kernel function using equation (1) to an observation history including one or more observation points and observed values ​​of random variables for each of the one or more observation points to obtain an estimated value of a risk avoidance index for each of the multiple candidate points in the search space, and calculates the value of an acquisition function for each of the multiple candidate points based on the estimated value of the risk avoidance index, A search unit searches for a target point corresponding to a predetermined value of the acquisition function from among the plurality of candidate points based on the value of the acquisition function, An output unit that outputs the search results including the aforementioned target point, An information processing device characterized by comprising: [Math 1] However, in equation (1), CVaR(x, t) represents the estimated value, v represents any element of the set U defined by {a, b, y1, ..., yt}, the right-hand side of equation (1) represents the maximum value of the {} for each v included in the set U, α represents the risk level, λ is a positive real number, ψν(y) represents max(v-y, 0), max(p, q) represents the maximum values ​​of p and q, and (ψν(y1), ..., ψν(yt)) is the case where the j-th element is ψν(yj) Here, we have a row vector of t dimensions, where k(x1:t, x1:t) represents a t×t matrix with the element at row i and column j being k(xi, xj), 1t represents the t×t identity matrix, (k(x1:t, x1:t)+λ1t)-1 represents the inverse matrix of k(x1:t, x1:t)+λ1t, and k(x1:t, x) represents a column vector of t dimensions with the i-th element being k(xi, x).

8. By applying a kernel function using equation (1) to the observation history which includes one or more observation points and the observed values ​​of the random variable for each of the one or more observation points, an estimate of the risk aversion index for each of the multiple candidate points in the search space is obtained. Based on the estimated risk avoidance indicator, the value of the acquisition function for each of the multiple candidate points is determined. Based on the value of the acquisition function, a target point corresponding to a predetermined value of the acquisition function is searched from among the plurality of candidate points. Output the search results including the aforementioned target point. An information processing method characterized in that the processing is performed by a computer. [Math 1] However, in equation (1), CVaR(x, t) represents the estimated value, v represents any element of the set U defined by {a, b, y1, ..., yt}, the right-hand side of equation (1) represents the maximum value of the {} for each v included in the set U, α represents the risk level, λ is a positive real number, ψν(y) represents max(v-y, 0), max(p, q) represents the maximum values ​​of p and q, and (ψν(y1), ..., ψν(yt)) is the case where the j-th element is ψν(yj) Here, we have a row vector of t dimensions, where k(x1:t, x1:t) represents a t×t matrix with the element at row i and column j being k(xi, xj), 1t represents the t×t identity matrix, (k(x1:t, x1:t)+λ1t)-1 represents the inverse matrix of k(x1:t, x1:t)+λ1t, and k(x1:t, x) represents a column vector of t dimensions with the i-th element being k(xi, x).

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