Learning device, learning method, inference device, inference method, and program

Abstract

Techniques for building a model using a set of summarized data points are disclosed. One aspect of the present disclosure relates to a learning device having a summary data acquisition unit that acquires summary data of a set of data points, an objective function setting unit that sets a log marginal likelihood calculated from a Gaussian prior distribution based on a sample pseudo-likelihood function and a covariance function based on the summary data, and a hyperparameter adjustment unit that adjusts hyperparameters of the covariance function so as to maximize the log marginal likelihood.

Description

Technical Field

[0001] The present disclosure relates to a learning device, a learning method, an inference device, an inference method, and a program.

Background Art

[0002] Gaussian process regression is known to be useful as a statistical technique for modeling the distribution of data points and making predictions for unknown data points. In Gaussian process regression, it is assumed that the data points follow a Gaussian distribution.

[0003] Under this assumption, a model is constructed to predict the output value of unknown input data and estimate the certainty (variance) of the prediction using sampled data points composed of pairs of input data and output values. Then, the constructed model is used to estimate the output value and the certainty of the prediction from unknown input data. Techniques for determining the parameters of such a model have been proposed.

Prior Art Documents

Patent Documents

[0004]

Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0005] The group of sampled data points can be summarized, for example, from the viewpoint of information protection. For example, position information that can identify an individual can be converted into area information indicating a wider geographical area. In this case, it is necessary to construct a model using the summarized group of data points.

[0006] An object of the present disclosure is to provide a technique for constructing a model using a summarized group of data points.

Means for Solving the Problems

[0007] One aspect of the present disclosure relates to a learning device having a summary data acquisition unit that acquires summary data of a data point group, an objective function setting unit that sets a log marginal likelihood calculated from a sample pseudo-likelihood function based on the summary data and a Gaussian prior distribution based on a covariance function, and a hyperparameter adjustment unit that adjusts hyperparameters of the covariance function so as to maximize the log marginal likelihood.

Advantages of the Invention

[0008] According to the present disclosure, it is possible to provide a technique for constructing a model using a summarized data point group.

Brief Description of the Drawings

[0009]

Figure 1

Figure 2

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Figure 8

Modes for Carrying Out the Invention

[0010] Hereinafter, embodiments of the present disclosure will be described with reference to the drawings.

[0011] In the following examples, a learning device that optimizes hyperparameters of a covariance function using summary data of a data point group, and an inference device that predicts a prediction distribution for a feature quantity of an inference target using a posterior distribution with the optimized hyperparameters are disclosed.

[0012] [Summary of the Present Disclosure] As shown in FIG. 1, a learning device 100 according to an example to be described later determines hyperparameters of a covariance function for predicting a prediction distribution from feature quantities of data points to be predicted using summary data of a data point group. Specifically, when the learning device 100 acquires summary data of a data point group, it adjusts the hyperparameters of the covariance function so as to optimize the log marginal likelihood calculated from a Gaussian prior distribution based on a sample pseudo-likelihood function based on the acquired summary data and the covariance function, and acquires the optimized hyperparameters.

[0013] Here, the summary data may be values obtained by clustering a data point group and acquiring them from each cluster. For example, as shown in FIG. 2, assume that a data point group is composed of five pairs (x1, y1) to (x5, y5) of input x1 to x5 and output y1 to y5 (where x i is a feature quantity or a feature vector, and y i is a parameter obtained by f(x i )) and is clustered into cluster #1{(x1, y1), (x2, y2), (x3, y3)} and cluster #2{(x4, y4), (x5, y5)}. In this case, the summary data is

Number

[0014] The data points (x1, y1) to (x5, y5) may be clustered into two clusters #1 and #2, as shown in FIG. i ,x j A distance such as a similarity may be defined between x and x, and the data points may be clustered based on the distance. i If the location information of individuals is included, data of individuals who live close to each other may be classified into the same cluster. In this case, the representative feature value z i is the x that belongs to the cluster i The summary statistics may be determined as the centroid of

number

[0015] In the following examples, it is assumed that the function f follows a Gaussian process, and the Gaussian process GP is

number

number

[0016] Such summary data

number

[0017] The learning device 100 calculates the log marginal likelihood from the Gaussian prior distribution based on the above-described sample pseudo-likelihood function [Number] and the covariance function k, and optimizes the hyperparameters of the covariance function k so as to maximize the log marginal likelihood. When the optimization is completed, as shown in FIG. 1, the learning device 100 provides the inference device 200 for use in the inference process with the posterior distribution based on the optimized hyperparameters.

[0018] The inference device 200 derives the prediction distribution of the prediction target for the feature amount of the data point to be predicted acquired from a user or the like using the posterior distribution based on the optimized hyperparameters. The derived prediction distribution can be used by the user as the parameter of the prediction distribution for the feature amount.

[0019] Here, the learning device 100 and the inference device 200 may be realized by a calculation device such as a server, a personal computer (PC), a smartphone, a tablet, or the like, or an information processing device, and for example, may have a hardware configuration as shown in FIG. 4. That is, the learning device 100 and the inference device 200 have a drive device 101, a storage device 102, a memory device 103, a processor 104, a user interface (UI) device 105, and a communication device 106 that are interconnected via a bus B.

[0020] Programs or instructions for realizing various functions and processes in the learning device 100 and the inference device 200 may be stored in a removable storage medium such as a CD-ROM (Compact Disk-Read Only Memory) or a flash memory. When the storage medium is set in the drive device 101, the program or instruction is installed from the storage medium via the drive device 101 into the storage device 102 or the memory device 103. However, the program or instruction does not necessarily have to be installed from the storage medium and may be downloaded from any external device via a network or the like.

[0021] The storage device 102 is realized by a hard disk drive or the like and stores files, data, etc. used for the execution of the installed program or instruction together with the installed program or instruction.

[0022] The memory device 103 is realized by a random access memory, a static memory, etc. When the program or instruction is activated, the program or instruction, data, etc. are read from the storage device 102 and stored. The storage device 102, the memory device 103, and the removable storage medium may be collectively referred to as a non-transitory storage medium.

[0023] The processor 104 may be realized by one or more CPUs (Central Processing Units), GPUs (Graphics Processing Units), processing circuitry, etc. that may be composed of one or more processor cores, and executes various functions and processes of the learning device 100 and the inference device 200 according to the program, instruction, data, etc. such as parameters necessary for executing the program or instruction stored in the memory device 103.

[0024] The user interface (UI) device 105 may be composed of input devices such as a keyboard, mouse, camera, microphone, etc., output devices such as a display, speaker, headset, printer, etc., and input / output devices such as a touch panel, and realizes an interface between the user and the learning device 100 and the inference device 200. For example, the user operates the learning device 100 and the inference device 200 by operating a keyboard, mouse, etc. on the GUI (Graphical User Interface) displayed on the display or touch panel.

[0025] The communication device 106 is realized by various communication circuits that execute wired and / or wireless communication processing with communication networks such as external devices, the Internet, a LAN (Local Area Network), and a cellular network.

[0026] However, the above-described hardware configuration is merely an example, and the learning device 100 and the inference device 200 according to the present disclosure may be realized by any other appropriate hardware configuration.

[0027] [Learning device] Next, the learning device 100 according to an embodiment of the present disclosure will be described. FIG. 5 is a block diagram showing the functional configuration of the learning device 100 according to an embodiment of the present disclosure. As shown in FIG. 5, the learning device 100 includes a summary data acquisition unit 110, an objective function setting unit 120, and a hyperparameter adjustment unit 130. Each functional unit of the summary data acquisition unit 110, the objective function setting unit 120, and the hyperparameter adjustment unit 130 may be realized by a computer program stored in the memory device 103 of the learning device 100 being executed by the processor 104.

[0028] The summary data acquisition unit 110 acquires summary data of a data point group. For example, the summary data may be stored in advance in a database, etc., and the summary data acquisition unit 110 may acquire the summary data from the database.

[0029] Here, the data point group is, for example,

Number

Number

[0030] Also, the summary data is generated by clustering a data point group,

Number

Number

Number

[0031] Here, the representative feature amount is the feature amount assigned to each cluster i, and may be determined, for example, as the average, centroid, etc. of the feature amounts of the data point group belonging to the cluster i. Also, the summary statistic amount is the statistic amount calculated from the output of the data point group belonging to each cluster i

Number

[0032] In this embodiment, it is assumed that the function f follows a Gaussian process, that is,

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[0033] Here, the stochastic process

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[0034] For a new input X * (where

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[0035] The objective function setting unit 120 sets the log marginal likelihood calculated from the sample pseudo-likelihood function based on the summary data and the Gaussian prior distribution based on the covariance function. In this embodiment, the hyperparameters of the covariance function are determined using the summary data instead of the data points as the complete data. The conventional likelihood function

Mathematics

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[0036] Taking the prior distribution as p(u) and the likelihood function as

Mathematics

[0037] f * 's posterior distribution is [Number] , provided that [Number] , and K u* is the Gram matrix of (Z, X * ). These marginal likelihoods and posterior distributions are calculated using only the summary data.

[0038] The hyperparameter adjuster 130 adjusts the hyperparameters of the covariance function so as to maximize the log marginal likelihood. That is, the hyperparameter adjuster 130 searches for the hyperparameters of the covariance function k that maximize the log marginal likelihood according to the gradient method.

[0039] To apply the Laplace approximation, [Number] is assumed to hold. Here, [Number] It has unimodality with respect to u and has its mode in the summary statistics. The Laplace approximation has been conventionally used to handle non-Gaussian likelihoods by approximating the posterior distribution with a Gaussian centered at the maximum of the posterior estimate. In this embodiment, the likelihood is replaced with a Gaussian centered at the maximum of the likelihood estimate, and the following holds.

Number

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Number

[0040] Here, when the covariance function k has an error β (∈(0, ∞)) and is multiplied by a matrix of Ο(β),

Number

Number

[0041] Equation (1) shows the asymptotic behavior of the marginal likelihood and the posterior distribution, and this approximation works well when n is sufficiently larger than m. Since the right side of Equation (1) does not depend on the hyperparameters of the covariance function, ε can be optimized using Q. Regarding the posterior distribution, when only summary data is given, since it includes K ff and K *f

Number

[0042] Here, the error β (∈(0, ∞)) of the covariance function k may satisfy the following conditions.

Number

[0043] When summary data is used, the approximation error of the covariance function for the representative feature z i can be evaluated.

Number

[0044] Here, the marginal likelihood and the posterior distribution are approximated using the representative feature z i and the approximation error is evaluated. Initially, the input is

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[0045] The marginal likelihood and posterior distribution of the original model include

Number

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[0046] When β is sufficiently small, the error γ also tends to be small. As the range of the data point group becomes smaller, the integral of p(f|u) over the region within the range of the distance from f = W fu u approaches 1. For simplicity, assume γ≠0, and for δ1∈[0,∞] and δ2∈(0,δ1],

Number

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[0047] Also,

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[0048] The percentile of the chi - square distribution in Equation (5) depends on κ and λ2. The integral in Equation (6) approaches zero as β and γ decrease. The evaluation of the integral acts in the reverse direction with respect to m, and the integral decreases as δ1 increases, and f in R(u,δ1) moves away from fu u. As a result, the error bounds for the marginal likelihood and the posterior distribution can be derived.

Number

Number

[0049] From this, it can be seen that as η decreases, the performance of the approximation improves. Also, η decreases as the summary data becomes more detailed. The length scale of the covariance function significantly affects η.

[0050] According to the learning device 100 described above, the hyperparameters of the covariance function k can be optimized from the summary data according to the sample log-likelihood function, and the posterior distribution using the optimized hyperparameters used in the inference process can be obtained.

[0051] [Learning Process] Next, the learning process according to an embodiment of the present disclosure will be described. The learning process is executed by the learning device 100, and more specifically, may be realized by one or more processors 104 of the learning device 100 executing one or more programs or instructions stored in one or more memory devices 103. FIG. 6 is a flowchart showing the learning process according to an embodiment of the present disclosure.

[0052] In step S101, the learning device 100 acquires summary data of the data point group. For example, the summary data may include representative feature amounts, summary statistics, and the number of data points of each cluster of the data point group. Here, the representative feature amount is a feature amount assigned to each cluster, and may be, for example, the centroid of the data points belonging to each cluster. Also, the summary statistic is a statistic calculated from the outputs of the data points belonging to each cluster, and may be, for example, the statistic of the centroid.

[0053] In step S102, the learning device 100 sets a sample log-likelihood function from the summary data and the covariance function. Here, the sample log-likelihood function

Equation

Equation

[0054] In step S103, the learning device 100 adjusts the hyperparameters of the covariance function so as to maximize the sample pseudo-likelihood function. When the adjustment of the hyperparameters is completed, the learning device 100 provides the inference device 200 with the posterior distribution based on the finally obtained hyperparameters.

[0055] According to the above-described learning process, the hyperparameters of the covariance function k can be optimized from the summary data according to the sample pseudo-likelihood function, and the posterior distribution based on the optimized hyperparameters used in the inference process can be obtained.

[0056] [Inference Device] Next, an inference device 200 according to an embodiment of the present disclosure will be described. FIG. 7 is a block diagram showing the functional configuration of the inference device 200 according to an embodiment of the present disclosure. As shown in FIG. 7, the inference device 200 includes an input point acquisition unit 210 and a hyperparameter calculation unit 220. Each functional unit of the input point acquisition unit 210 and the hyperparameter calculation unit 220 may be realized by a computer program stored in the memory device 103 of the inference device 200 being executed by the processor 104.

[0057] The input point acquisition unit 210 acquires the feature amount of the inference target. Specifically, the input point acquisition unit 210 acquires the feature amount of the inference target extracted from X to which the input feature amount x i , y i belongs in the data point (x i ) used to derive the summary data used in the above-described learning process.

[0058] The parameter calculation unit 220 inputs the feature amount into the covariance function and calculates the predictive distribution for the feature amount. The covariance function here is the one derived by the learning device 100 described above. Specifically, the hyperparameters of the covariance function are adjusted to maximize the log marginal likelihood calculated from the sample pseudo-likelihood function based on the summary data of the data point group and the Gaussian prior distribution based on the covariance function.

[0059] The parameter calculation unit 220 inputs the feature amount acquired from the input point acquisition unit 210 into the covariance function with optimized hyperparameters, and acquires the parameters of the probability distribution for the feature amount to be inferred.

[0060] According to the inference device 200 described above, the probability distribution of the feature amount to be inferred can be predicted by using the posterior distribution with the optimized hyperparameters derived from the summary data.

[0061] [Inference Process] Next, the inference process according to an embodiment of the present disclosure will be described. The inference process is executed by the inference device 200. More specifically, it may be realized by one or more processors 104 of the inference device 200 executing one or more programs or instructions stored in one or more memory devices 103. FIG. 8 is a flowchart showing the inference process according to an embodiment of the present disclosure.

[0062] In step S201, the inference device 200 acquires the feature amount to be inferred. The feature amount may be extracted from X to which the input feature amount x i , y i in the data point (x i that was used to derive the summary data used in the learning process described above belongs.

[0063] In step S202, the inference device 200 inputs the feature amount into the covariance function and calculates the parameters for the feature amount. Specifically, the inference device 200 inputs the feature amount into the covariance function derived by the learning device 100 and calculates the parameters of the probability distribution for the feature amount.

[0064] According to the above-described inference process, the probability distribution of the feature amount to be inferred can be predicted by using the posterior distribution with the optimized hyperparameters derived from the summary data.

[0065] As described above, the embodiments of the present disclosure have been described in detail. However, the present disclosure is not limited to the above-described specific embodiments, and various modifications and changes are possible within the scope of the gist of the present disclosure described in the claims.

Description of Reference Numerals

[0066] 100 Learning device 110 Summary data acquisition unit 120 Objective function setting unit 130 Hyperparameter adjustment unit 200 Inference device 210 Input point acquisition unit 220 Parameter calculation unit

Claims

A learning device including a processor that reads a program from a memory device and executes the program to perform a process of learning hyperparameters of a covariance function so as to maximize a log marginal likelihood, wherein the processor performs a process of acquiring summary data of a data point group from the memory device, performs a process of setting a log marginal likelihood calculated from a sample pseudo-likelihood function based on the summary data and a Gaussian prior distribution based on a covariance function, performs a process of adjusting hyperparameters of the covariance function so as to maximize the log marginal likelihood, and executes, wherein the summary data is a combination of a representative feature amount, a summary statistic amount, and the number of data points of the j-th cluster of the data point group, 【Number 1】 and is represented as, the sample pseudo-likelihood function 【Number 2】 【Number 3】 is defined as, Vuu is 【Number 4】 defined as, u is defined using a function f following a Gaussian process 【Number 5】 as, v is a variance function, a learning device.

2. wherein the representative feature amount is a feature amount assigned to the cluster, and the summary statistic amount is a statistic amount calculated from outputs of data points belonging to the cluster, the learning device according to claim 1.

3. The error β of the covariance function 【Number 6】 satisfies the condition of (where k(·, ·) is a function for obtaining the covariance between data points), where (xi, xj) is a feature amount or a feature vector of a cluster, ω is 【Number 7】 defined as, the learning device according to claim 1.

4. In a computer including a memory device and a processor that reads a program from the memory device and executes the program, a learning method for learning hyperparameters of a covariance function so as to maximize a log marginal likelihood, wherein the processor performs a process of acquiring summary data of a data point group from the memory device, performs a process of setting a log marginal likelihood calculated from a sample pseudo-likelihood function based on the summary data and a Gaussian prior distribution based on a covariance function, performs a process of adjusting hyperparameters of the covariance function so as to maximize the log marginal likelihood, and executes, wherein the summary data is a combination of a representative feature amount, a summary statistic amount, and the number of data points of the j-th cluster of the data point group, 【Number 8】 and is represented as, the sample pseudo-likelihood function 【Number 9】 is 【Number 10】 defined as, Vuu is 【Number 11】 defined as, u is defined using a function f following a Gaussian process 【Number 12】 as, v is a variance function, a learning method. A program stored in a memory device and read and executed by a processor to cause the processor to execute a process of learning hyperparameters of a covariance function so as to maximize a log marginal likelihood, a process of obtaining summary data of a data point group, a process of setting a log marginal likelihood calculated from a sample pseudo-likelihood function based on the summary data and a Gaussian prior distribution based on a covariance function, a process of adjusting hyperparameters of the covariance function so as to maximize the log marginal likelihood, causing the processor to execute, wherein the summary data is a combination of a representative feature amount, summary statistics, and the number of data points of the j-th cluster of the data point group, 【Number 13】 represented as, the sample pseudo-likelihood function 【Number 14】 is 【Number 15】 defined as, Vuu is 【Number 16】 defined as, u is defined using a function f following a Gaussian process 【Number 17】 as, v is a variance function, program. An inference device including a processor that reads a program from a memory device and executes an inference process of calculating a prediction distribution for a feature amount to be inferred by executing the program, wherein the processor executes a process of obtaining a feature amount to be inferred and a process of inputting the feature amount into a covariance function and calculating a prediction distribution for the feature amount, and the covariance function has hyperparameters optimized so as to maximize a log marginal likelihood calculated from a sample pseudo-likelihood function based on summary data of a data point group and a Gaussian prior distribution based on the covariance function, the calculating process calculates the prediction distribution using a posterior distribution with the optimized hyperparameters, the summary data is a combination of a representative feature amount, summary statistics, and the number of data points of the j-th cluster of the data point group, 【Number 18】 represented as, the sample pseudo-likelihood function 【Number 19】 is 【Number 20】 defined as, Vuu is 【Number 21】 defined as, u is defined using a function f following a Gaussian process 【Number 22】 as, v is a variance function, the posterior distribution is 【Number 23】 where f* is 【24 Points】 defined as, and μq and Σq are 【Number 25】 defined as, Ku* is the Gram matrix of (Z, X*), Kuu is the Gram matrix of (Z, Z), K** is the Gram matrix of (X*, X*), X* is 【Number 26】 defined as, Z is 【Number 27】 defined as, inference device. In a computer including a memory device and a processor that reads a program from the memory device and executes the program, an inference method for executing an inference process for calculating a prediction distribution for a feature amount to be inferred, wherein: the processor executes a process of acquiring a feature amount to be inferred, inputs the feature amount into a covariance function, and executes a process of calculating a prediction distribution for the feature amount, and the covariance function has hyperparameters optimized to maximize the log marginal likelihood calculated from a sample pseudo-likelihood function based on summary data of a data point group and a Gaussian prior distribution based on the covariance function, the calculating process calculates the prediction distribution using a posterior distribution with the optimized hyperparameters, the summary data is a combination of a representative feature amount, summary statistics, and the number of data points of the j-th cluster of the data point group, 【Number 28】 and is represented as the sample pseudo-likelihood function 【Number 29】 is 【30 numbers】 defined as Vuu is 【Number 31】 defined as u is defined using a function f following a Gaussian process 【Number 32】 as v is a variance function, and the posterior distribution 【Number 33】 is, where f* is 【Number 34】 defined as, μq and Σq are 【Number 35】 defined as, K u* is the Gram matrix of (Z, X*), K uu is the Gram matrix of (Z, Z), K ** is the Gram matrix of (X*, X*), and X* is 【No. 36】 defined as Z is 【Number 37】 defined as an inference method A program stored in a memory device and read and executed by a processor to cause the processor to execute an inference process for calculating a prediction distribution for a feature amount to be inferred, wherein: the program causes the processor to execute a process of acquiring a feature amount to be inferred, inputs the feature amount into a covariance function, and causes the processor to execute a process of calculating a prediction distribution for the feature amount, and the covariance function has hyperparameters optimized to maximize the log marginal likelihood calculated from a sample pseudo-likelihood function based on summary data of a data point group and a Gaussian prior distribution based on the covariance function, the calculating process calculates the prediction distribution using a posterior distribution with the optimized hyperparameters, the summary data is a combination of a representative feature amount, summary statistics, and the number of data points of the j-th cluster of the data point group, 【Number 38】 and is represented as the sample pseudo-likelihood function 【Number 39】 is 【Number 40】 defined as Vuu is 【Number 41】 is defined as, u is defined using a function f that follows a Gaussian process, 【Number 42】 is defined as, v is a variance function, The posterior distribution is, 【Number 43】 where f* is, 【Number 44】 is defined as, and μq and Σq are, 【Number 45】 is defined as, Ku* is the Gram matrix of (Z, X*), Kuu is the Gram matrix of (Z, Z), and K** is the Gram matrix of (X*, X*), X* is, 【Number 46】 defined by, Z is, 【Number 47】 defined by, program.

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