JP2022124418A - Control method and sampling apparatus - Google Patents

Abstract

To provide a sampling method for efficient sampling from Boltzmann distribution of an Ising model and Boltzmann machine, and an information processing apparatus.SOLUTION: A method substitutes sampling in an interaction model with a complete 3-partite graph structure which includes first auxiliary variable groups formed of N auxiliary variables and second auxiliary variable groups formed of N auxiliary variables, as continuous variables defining N spin groups (N is a natural number) and directions of spins, and has an interactive relation by connecting all the first auxiliary variable groups and all the second auxiliary variable groups to the spins, for sampling for an Ising model. A value of the interactive relation of the interaction model is set with an original interactive relation of the Ising model and a minimum eigenvalue thereof, to a memory of an information processing apparatus. In state transition in the interaction model, all variables in one group of the spin groups, the first auxiliary variable groups, and the second auxiliary variable groups are updated together.SELECTED DRAWING: Figure 7

Description

The present invention relates to control methods and sampling devices. As a non-limiting specific field, we target the Boltzmann distribution of the Ising model including the interaction relationship expressed as a quadratic equation, and based on the Markov chain Monte Carlo method, we collect information so as to sample the next state of the spin. The present invention relates to a control method and the like for operating a processing device.

Patent Literature 1 describes ``Providing a semiconductor device in which constituent elements serving as basic constituent units for obtaining a solution of an interaction model are arranged in an array,'' and ``A value indicating the state of one node of an interaction model. a second memory cell storing an interaction coefficient indicative of interaction from other nodes connected to one node; and for fixing the value of the first memory cell a third memory cell for storing a flag of; a first arithmetic circuit for determining a value indicating the next state of the one node based on the value indicating the state of the other node and the interaction coefficient; and a second arithmetic circuit that determines whether or not to record the value indicating the next state in the first memory cell according to the value of .

Non-Patent Document 1 and Non-Patent Document 2 describe Gaussian integral trick, which is a method of constructing a probability distribution function that is easy to perform sampling by multiplying the Boltzmann distribution of the Ising model by the multivariate normal distribution of the auxiliary variable. is described regarding.

JP 2016-51313 A

J. Martens and I. Sutskever:“Parallelizable Sampling of Markov Random Fields”, Conference on Artificial Intelligence and Statistics (2010). Y. Zhang, Z. Ghahramani, A. J. Storkey and C. A. Sutton:“Continuous Relaxations for Discrete Hamiltonian Monte Carlo”, Advances in Neural Information Processing Systems (2012).

Many physical and social phenomena can be represented by interaction models. This interaction model is defined by a plurality of nodes that make up the model, interactions between the nodes, and biases (weights) for each node if necessary. Various models have been proposed in the fields of physics and social sciences, and all of them can be interpreted as a form of interaction model.

One of the interaction models is the Ising model. The Ising model is a simple model for calculating the collective behavior of electron spins (hereinafter simply referred to as "spins") of atoms that make up a crystal, and is a model of statistical mechanics used to explain magnetic substances. Also, there is a Boltzmann Machine as a model equivalent to the Ising model, which is used as a machine learning model.

Physics simulations using Ising models and learning and inference of Boltzmann machines require sampling from Boltzmann distributions that have them as energy functions. One way to do such sampling is by Markov Chain Monte Carlo. In the Markov chain Monte Carlo method, a desired probability distribution is realized as a stationary distribution by stochastically transitioning between states.

When actual physical phenomena and social phenomena are applied to the Ising model, the interaction between spins (corresponding to nodes) becomes a dense structure, that is, a structure in which each spin is coupled with many other spins. Therefore, in the Markov chain Monte Carlo method, it is not possible to perform probabilistic processing for each spin simultaneously, which makes it difficult to speed up the processing.

The present invention has been made in view of the above problems, and an object of the present invention is to provide a control method and a sampling device that can efficiently perform sampling from the Boltzmann distribution of the Ising model and the Boltzmann machine.

One aspect of the present invention for achieving the above object is
A control method for operating an information processing device so as to sample the next state of a spin based on the Markov chain Monte Carlo method for the Boltzmann distribution of the Ising model including the interaction relationship between spins,
Sampling for the Ising model defines a spin group σ consisting of N (N is a natural number) spins σ _i ( _i = any number from 1 to N) and the direction of each spin σ _i As continuous variables, a first auxiliary variable group x consisting of the N auxiliary variables x _i ( _i = 1 to N) and a second auxiliary variable group consisting of the N auxiliary variables y _i ( _i = 1 to N) y, and all of the first auxiliary variable groups x and all of the second auxiliary variable groups y are connected to the spin σ _i and have an interactive relationship with a complete tripartite graph structure. substitute by sampling at
The value of the interaction relationship of the interaction model is set in the memory of the information processing device with the interaction relationship of the original Ising model and its minimum eigenvalue,
All variables in one of the three variable groups of the spin group σ, the first auxiliary variable group x, and the second auxiliary variable group y are simultaneously updated when the state transition is performed in the interaction model. operating the information processing device to
It is implemented as a control method.

Another aspect of the present invention for achieving the above object is
In order to sample the next states of spins based on the Markov chain Monte Carlo method for the Boltzmann distribution of the Ising model including the interaction relationship between spins,
As statistics for the Ising model, a spin group σ consisting of N (N is a natural number) spins σ _i ( _i = any number from 1 to N) and the direction of each spin σ _i are defined. As continuous variables, a first auxiliary variable group x consisting of the N auxiliary variables x _i ( _i = 1 to N) and a second auxiliary variable group consisting of the N auxiliary variables y _i ( _i = 1 to N) y, and all of the first auxiliary variable groups x and all of the second auxiliary variable groups y are connected to the spin σ _i and have an interactive relationship with a complete tripartite graph structure. a calculator that calculates the transition probability P (σ, σ′) at
Control to simultaneously update all variables in one of the three variable groups of the spin group σ, the first auxiliary variable group x, and the second auxiliary variable group y calculated by the calculating unit a variable control unit;
is implemented as a sampling device comprising:

In addition, the problems disclosed by the present application and their solutions will be clarified by the description of the mode for carrying out the invention and the drawings.

According to the present invention, the Ising model can be sampled by parallel processing, and the Ising model can be efficiently sampled from the Boltzmann distribution. Problems, configurations, and effects other than those described above will be clarified by the following description of the mode for carrying out the invention.

4 is a characteristic graph explaining the concept of the energy landscape in the Ising model; FIG. 10 is a graph diagram showing an interaction relationship between spins of the Ising model as a complete graph; FIG. 10 is a graph diagram showing the interaction relationship between the spin group and the auxiliary variable group in the interaction model for executing sampling as a complete ternary graph. 1 is a block diagram showing a schematic configuration of an information processing apparatus according to an embodiment; FIG. FIG. 5 is a block diagram illustrating the detailed configuration of an arithmetic unit in FIG. 4; 5 is a functional block diagram showing main functions provided in the information processing apparatus of FIG. 4; FIG. FIG. 7 is a flowchart for explaining sampling processing shown in FIG. 6; FIG. 3 is a block diagram showing a detailed configuration example of an arithmetic unit; FIG. 3 is a block diagram showing a circuit configuration example of one unit; FIG.

Hereinafter, embodiments will be described in detail based on the drawings. In the following description, common reference numerals may be assigned to the same or similar configurations, and redundant descriptions may be omitted. Also, when there are a plurality of elements having the same or similar functions, the same reference numerals may be given different suffixes for explanation. Also, when there is no need to distinguish between multiple elements, the subscripts may be omitted.

An information processing apparatus as an embodiment to which the present invention is applied generally replaces sampling of the original Ising model with sampling of an interaction model having an interaction relationship of a complete tripartite graph structure in which spins and auxiliary variables are connected. , stores the energy function of the interaction model, updates the spin and auxiliary variables according to the Markov chain Monte Carlo method based on the energy function, spin and auxiliary variable information, and samples the spin state.

First, the Ising model will be described. The Ising model is a model of statistical mechanics used to explain the behavior of magnetic materials. The Ising model is defined using an interaction coefficient that indicates the state of a spin σ that takes a binary value of +1 or −1 (hereinafter sometimes referred to as “binary spin”) and the interaction between the spins. . The Ising model energy function H(σ) (generally called Hamiltonian) is expressed by the following equation.

In such equation 1, σ is a vector indicating the value of the spin, h is the vector of the external magnetic field coefficient acting on the spin, J is the matrix indicating the interaction coefficient (that is, the force acting between the two spins), and σ and h are The appended T is a sign indicating transposition. Also, the right side of Equation 1 represents the sum of energies resulting from interactions between spins.
Here, when the number of spins to be sampled is N (N is a natural number), it can be defined as follows. That is, in the above formula 1, the value of any i-th (i = any integer from 1 to N) spin is σ _i , and any j-th (j = integer other than i from 1 to N) σ _j , the matrix of interaction coefficients between the i-th spin and the j-th spin is J _i,j , and the external magnetic field coefficient acting on the i-th spin is h _i , respectively. can be done. In the matrix (J) of the interaction coefficients, the diagonal components J _i,i (i=1 to N) are zero.

Note that formula 1 and the various formulas below can include i and j above, even if i and j are not specified hereinafter.

In general, the Ising model is expressed as an undirected graph, without distinguishing between the interaction from the i-th spin to the j-th spin and the interaction from the j-th spin to the i-th spin. , J _i,j =J _j,i .

The Ising model is equivalent to the Boltzmann machine, which is one of the artificial neural networks in the field of machine learning. In Boltzmann machines, binary spin values of 1/0 are often used (instead of +1/−1 as above). Thus, the Ising model and models equivalent to the Ising model are applied in various fields such as physics and machine learning.

In physics simulations using the Ising model and learning and inference of Boltzmann machines, calculation of spin statistics (dispersion value, probability, expected value, etc. of spin values) according to the Boltzmann distribution p(σ) represented by the following equation 2 is required. done.

In Equation 2, _Zσ is a normalization factor called a partition function, which is expressed by Equation 3 below.

The Markov Chain Monte Carlo method (Markov Chain Monte Carlo methods, hereinafter sometimes abbreviated as “MCMC method”) is a method for constructing a Markov chain that converges to a desired stationary distribution. In this embodiment, an operation based on the MCMC method is executed by, for example, a computer (information processing apparatus of the present invention) to perform sampling based on the Boltzmann distribution p(σ) as shown in Equation 2 above.

FIG. 1 is a graph for explaining the concept of the Ising model energy landscape. In FIG. 1, the horizontal axis of the graph indicates the direction or arrangement of spins, and the vertical axis indicates the total energy of the system. In stochastic transitions, the spin repeats stochastic transitions from the current state σ to some state σ' near the state σ. The probability that the spin state (spin value) transitions from the state σ to the state σ′ is hereinafter referred to as transition probability P(σ, σ′).

When applying the MCMC method to the Boltzmann distribution of the Ising model, in order to obtain the transition probability P (in this example, the spin state transition probability P (σ, σ′)), the Metropolis method, Alternatively, a hot bath method (Gibbs sampling) is used. Equation 4 shows a calculation formula for the transition probability P(σ, σ′) in the Metropolis method (sometimes called the Metropolis-Hastings method).

In general, when the Metropolis method is used, as an overview of the operation of the computer (information processing device), the initial state (current state) of each of N spins (spin group) is set, and each Next state candidates of the spin are calculated according to Equation 4, the calculation of the next state candidates is repeated, and the calculation results are output (screen display, printing, etc.) at an appropriate time.

Regardless of which method is used, when the MCMC method is applied to the Boltzmann distribution of the Ising model, the statistics of the next state (σ′) appearing after the current state (σ) of N spins (spin group) A value (probability or expected value) is calculated according to a predefined arithmetic formula.

In the present embodiment described below, it is assumed that the spin value is stochastically determined based on the above-described Equation 4 based on the Metropolis method.

For a plurality of spins that do not interact (that is, do not affect each other) (for example, spins that are not connected, in other words, non-adjacent spins), the state transition based on Equation 4 can be applied at the same time. be. Therefore, for example, as described in Patent Document 1, by preparing a plurality of circuits for realizing stochastic processing based on Equation 4 and updating the values of a plurality of non-adjacent spins in parallel, MCMC method sampling can be performed at high speed.

FIG. 2 is a complete graph (total connection graph) showing the interaction relationship between all spins in the Ising model when the number of spins is 6. In FIG. 6” is attached.

When trying to apply real physical phenomena and social phenomena to the Ising model, the interaction relationship between spins has a dense structure as shown in Fig. 2 (that is, an individual spin (e.g., spin 1) can (In this example, spins 2 to 6) are adjacent (with wire connections)), and have a relationship of mutually influencing each other. Therefore, the probabilistic processing of each spin cannot be performed simultaneously, and it is difficult to speed up the processing based on the MCMC method.

Conversely, for the Ising model with such a close interaction relationship, multiple spins (spins 1 to 6 in the example shown in FIG. 2) are updated simultaneously while satisfying the theoretical background required by the MCMC method. If it can be done, the speed of processing using the MCMC method can be increased by parallel processing.

Based on the above-mentioned viewpoints, the present inventors devised the present embodiment after intensive research, using a complete tripartite graph interaction model that can replace the sampling for the Boltzmann distribution of the original Ising model. to control sampling by the information processing device. The sampling method and the control method of the information processing apparatus according to the present invention will be described below.

As described in Non-Patent Document 1 and Non-Patent Document 2, the Boltzmann distribution is multiplied by the multivariate normal distribution of the auxiliary variable to substitute the probability distribution function for which the MCMC method should be performed (so to speak, "replace"). The method is called the Gaussian integral trick. In this embodiment, by extending the Gaussian integral trick, the multivariate normal distribution of the auxiliary variable group v _i ∈(−∞, ∞) (i=1 to N) and the auxiliary variable group s _i ∈[−1, 1 ] Consider the probability distribution functions represented by the following equations 5 and 6 obtained by multiplying the probability distribution functions consisting of (i=1 to N).

In Equation 6, the matrix W is a positive definite matrix, more specifically, with the smallest eigenvalue −λ of the matrix J (see Equation 1), a positive constant ε, and the identity matrix I, W=J+ is a positive definite matrix defined by (λ+ε)I. Also, the partition function Z (see Equation 3 as appropriate), which is a normalization factor, is a function represented by Equation 7 below.

As shown in Equation 7, the partition function Z can be defined because it takes a finite value.

Then, when the statistic depending only on the current state σ of the spin is calculated based on the above-described Equation 5, it is expressed as Equation 8 below.

Equation 8 shows that the result of sampling for Equation 5 and the result of sampling for Equation 2 match when focusing only on the current state σ of the spin. Therefore, instead of performing the MCMC method on Eq. 2, we may perform the MCMC method on Eq.

Further, if the auxiliary variable x=s+v and the auxiliary variable y=s−v are set for the equation 5, the following equation 9 is obtained.

where |x _i +y _i |≦2 (i=1 to N). Also, constant terms that do not contribute to the probability distribution are ignored.

Equation 9 shows the energy function of the interaction model with respect to the current state σ of the spin and the auxiliary variables x and y. Furthermore, the interaction relationship of this interaction model is represented by the complete tripartite graph shown in FIG. In the graph shown in FIG. 3, the current state of a plurality of spins (six in relation to FIG. 2) is defined as the "spin group σ", and the set of auxiliary variables x for each spin (six in this example) is defined as the auxiliary variable group x, and a set of auxiliary variables y for each spin (also six) is defined as auxiliary variable group y. In other words, σ, x, and y, which should originally be applied to one spin, are substituted (the above-described “replacement”) so that they are applied to a plurality (that is, to a group).

In the above assumptions, there is no interaction inside the spin group (the current state σ) and the two auxiliary variable groups (x, y). Therefore, when executing operations based on the MCMC method for the Boltzmann distribution of the interaction model shown in FIG. Yes, such parallel processing can efficiently perform processing using the MCMC method.

As described above, in the present embodiment, processing and spin sampling using the MCMC method are performed on the Boltzmann distribution having Equation 9 as the energy function instead of Equation 2. FIG.

Next, a method for updating spin σ and auxiliary variables x and y based on the MCMC method will be described. In this embodiment, the Metropolis method is used for the spin σ, and the heat bath method is used for the auxiliary variables x and y.

First, consider the update of any one spin σ _i (state transition to the next state). The energy difference ΔH when the direction of the spin σ _i is reversed can be obtained from the following equation (10).

As described above, the state transition acceptance probability (transition probability) in the Metropolis method is expressed by Equation (4). Therefore, if a uniform random number u (0<u≦1) is used, the condition for accepting a state transition can be expressed by the following equation 11.

Here, consider the case where the spin value σ _i becomes +1 after the state transition. First, if the spin value σ _i before the state transition is +1, then the condition is satisfied when ln(u)>−ΔH from Equation 11, that is, when the following Equation 12 holds.

Similarly, if the spin value σ _i before the state transition is −1, the spin value σ _i after the state transition is +1 when ln(u)≦−ΔH holds, that is, when the following equation 13 holds. becomes.

Therefore, regardless of the value σ _i before the state transition, it can be seen that the spin value σ _i becomes =+1 after the state transition if the following Expression 14 is satisfied.

Consequently, in order to execute a state transition (to the next state) according to the Metropolis method, the next state of the spin value σ _i should be updated according to Equation 15 below. In Equation 15, the sign function sgn(x) is a function that returns +1 if x≧0 and −1 if x<0. In addition, the sign of “←” shown in Equation 15 is an arrow sign indicating the next state (the next state of the spin value σ _i in Equation 15). ←” is also the same.

Next, consider updating the auxiliary variable x. Given σ,y, the conditional cumulative distribution function F(x _i |σ,y) of x _i can be expressed by Equation 16 below.

In Equation 16, α _i is defined by Equation 17 below.

According to the heat bath method based on the inverse function method, x _i should be updated by solving F(x _i |σ, y) = u for x _i as a uniform random number u (0 < u ≤ 1) A value is required. That is, the auxiliary variable x should be updated according to Equation 18 below.

Similarly, when σ and x are given, the auxiliary variable y may also be updated according to Equation 19 below.

However, in Equation 19, α' _i is defined by Equation 20 below.

FIG. 4 is a configuration example of an information processing apparatus for carrying out the sampling method shown in Equations 15, 17, and 19 described above. The information processing device 10 shown in FIG. 4 is a device corresponding to the “sampling device” of the present invention, and includes a processor 11, a main storage section 12, an auxiliary storage section 13, an operation input section 14, an output section 15, and a communication section 16. , one or more computing units 20, and a system bus 5 communicatively connecting these units.

For example, the information processing apparatus 10 may be implemented using virtual information processing resources such as a cloud server (Cloud Server) provided by a cloud system (Cloud System) in whole or in part. good. Further, the information processing apparatus 10 may be realized by, for example, a plurality of communicatively connected information processing apparatuses that operate in cooperation with each other.

The processor 11 plays a role of controlling the entire information processing apparatus 10 (each part described above), and is configured using, for example, a CPU (Central Processing Unit) or an MPU (Micro Processing Unit). In relation to the present invention (sampling device), the processor 11 stores or updates variables (spin group σ, first auxiliary variable group x, and second auxiliary variable group y It plays a role as a "variable control unit" that performs control to simultaneously update all variables in one of the three variable groups of .

The main storage unit 12 and the auxiliary storage unit 13 are devices (hardware storage media) that mainly store programs executed by the processor 11 and an arithmetic unit 20 (to be described later), various data used, and the like.

Among them, the main storage unit 12 includes, for example, ROM (Read Only Memory), SRAM (Static Random Access Memory), NVRAM (Non Volatile RAM), mask ROM (Mask Read Only Memory), PROM (Programmable ROM), etc., RAM (Random Access Memory), DRAM (Dynamic Random Access Memory), and the like.

In addition, the auxiliary storage unit 13 can be any of various storage media having a larger data storage capacity than the main storage unit 12, such as a hard disk drive, a flash memory, an SSD (Solid State Drive), and a CD. (Compact Disc) or an optical storage device such as a DVD (Digital Versatile Disc).

The programs and data stored in the auxiliary storage section 13 are read into the main storage section 12 at any time.

In relation to the present invention, the main storage unit 12 or the auxiliary storage unit 13 may serve as various memories such as "variable memory", "interaction coefficient memory", and "external magnetic field coefficient memory", which will be described later. can. Further, the auxiliary storage unit 13 functions as part of an “output unit” that accumulates sampling results (values in N variable memories 901 to be described later) by the information processing device 10 and outputs them at any time. can also

The operation input unit 14 is a user interface that receives information input from the user, and includes, for example, a keyboard, mouse, card reader, touch panel, and the like.

The output unit 15 is a user interface that provides information to the user. For example, a display device such as an LCD (Liquid Crystal Display) or graphic card that visualizes various information, an audio output device such as a speaker, a printing device such as a printer, and the like. is. The output unit 15 plays a role of outputting the result of sampling by the information processing apparatus 10 at specified timing (for example, at each time interval preset using the user interface).

The communication unit 16 is a communication interface that communicates with other devices, such as a NIC (Network Interface Card), a wireless communication module, a USB (Universal Serial Interface) module, a serial communication module, and the like. In relation to the present invention, the communication section 16 can also play the role of the "output section" described above.

In the information processing apparatus 10 of the present embodiment, the calculation unit 20 is a device (hardware device) that executes the processing related to sampling described above. The calculator 20 corresponds to the "calculator" of the present invention.

The computing unit 20 may take the form of an expansion card attached to the information processing device 10, such as a GPU (Graphics Processing Unit).

The computing unit 20 is configured by hardware such as, for example, a CMOS (Complementary Metal Oxide Semiconductor) circuit, an FPGA (Field Programmable Gate Array), and an ASIC (Application Specific Integrated Circuit). The computing unit 20 includes an external control device or storage device, an interface for connecting to the system bus 5 , etc., and transmits and receives commands and information to and from the processor 11 via the system bus 5 . For example, the calculation unit 20 may be communicably connected to another computer or the like (not shown) via a communication line, and may operate in cooperation with the other computer or the like (not shown). .
Note that the functions realized by the arithmetic unit 20 may be virtually realized by, for example, causing a processor (CPU, GPU, etc.) to execute a program. Then, each time the process is called, it will be executed (activated).

FIG. 5 is a diagram for explaining the principle of operation when the arithmetic unit 20 is configured by hardware resources such as a GPU, and is a block diagram showing a specific example of a circuit that constitutes the arithmetic unit 20. As shown in FIG.
As shown in the figure, the calculation unit 20 includes an interaction coefficient memory 511, an external magnetic field coefficient memory 512, a spin memory 513, a first auxiliary variable memory 514, a second auxiliary variable memory 515, a product-sum calculation device 516, a mathematical function A computing unit 517 is included.

In relation to the present invention, the calculation section 20 corresponds to the "calculation section", the product-sum calculation device 516 to the "product-sum calculation section", and the mathematical function calculation device 517 to the "mathematical function calculation section".

The calculation unit 20 having such a configuration realizes the functions corresponding to the above-described formulas 15, 17, and 19 when functioning as the "calculation unit". The principle of operation of the calculation unit 20 will be described below with reference to FIG. In the following description, N represents the number of spins handled by the calculation unit 20 .

The interaction coefficient memory 511 of the calculation unit 20 stores information representing the interaction matrix W (see Equation 15). The interaction matrix is generally a symmetric matrix, and by utilizing such symmetry, the usage of the interaction coefficient memory 511 can be reduced or minimized.

The external magnetic field coefficient memory 512 stores information representing the vector h (see Equation 15).
The spin memory 513 stores N-dimensional vector information indicating the state of each spin of the spin σ of the complete tripartite graph. The first auxiliary variable memory 514 and the second auxiliary variable memory 515 respectively store N-dimensional vector information indicating the states of the auxiliary variables x and y of the complete tripartite graph (see FIG. 3). .

As shown in FIG. 5, the signal TS and the signal SR are input to the calculation unit 20 . A mathematical function operation device 517 of the operation unit 20 outputs the signals TS and SR, the data (information representing the vector h) read from the external magnetic field coefficient memory 512 described above, and the output of the sum-of-products operation device 516 described later. Data (computed value) are input, and a signal SP is output based on these input data.
Among the above signals, the signal TS is a signal that periodically repeats three values corresponding to the integers 1 to 3, and designates which operation of the equations 15, 17, and 19 is to be executed. In other words, the signal TS is a signal that specifies which of the three values (spin σ, auxiliary variable x, auxiliary variable y) described above is to be updated. In one non-limiting example, signal TS is integer 1 when performing the operation of Equation 15, integer 2 when performing the operation of Equation 17, and integer 3 when performing the operation of Equation 19. to output

The signal SR is a signal representing an N-dimensional vector (that is, a vector of dimensions corresponding to the number of spins handled by the calculation unit 20) whose elements are mutually independent random numbers.

As noted above, the off-diagonal elements of the interaction matrix W are set by the matrix J (see Equation 1 where appropriate), and the diagonal elements by the matrix J's smallest eigenvalue. Calculation of the minimum eigenvalue may be performed by the processor 11 (see FIG. 4) outside the calculation unit 20, or may be calculated by the sum-of-products calculation device 516 in the calculation unit 20 shown in FIG. Well, in either case, it can be done relatively easily (rapidly). For example, the minimum eigenvalue can be efficiently calculated using a commonly known algorithm such as the power method. should be utilized.

As shown in FIG. 5, the sum-of-products operation device 516 according to the present embodiment stores values read from the interaction coefficient memory 511, values read from the spin memory 513, values read from the first auxiliary variable memory 514, , the value read from the second auxiliary variable memory 515, and the signal TS (a signal permitting updating of one of the values) are input. The sum-of-products operation unit 516 then calculates values based on these five inputs and outputs the calculated values to the mathematical function operation unit 517 .

More specifically, Equations 15, 17, and 19 (see also Equation 20, which is related to Equation 19) discussed above each include a multiply-add operation of the matrix W and the vector. Therefore, the sum-of-products operation unit 516 converts the values read from the above-described memories (511, 513, 514, 515) into the equation (equation 15, equation 17, or equation 19) specified by the signal TS. It applies and performs an operation and outputs the calculated value of such operation to the mathematical function calculator 517 .

Furthermore, Equations 15, 17 (see also related Equation 18), and Equation 19 are exponential, logarithmic, and sign functions for scalars such as the outputs of multiply-accumulate operations and uniform random numbers u (0<u≤1). It contains mathematical functions such as In this embodiment, operations on these mathematical functions are performed by mathematical function calculator 517 .

More specifically, the mathematical function operation unit 517 converts the output value of the operation by the sum-of-products operation unit 516 (the output of the sum-of-products operation) and the N-dimensional vector (the vector of dimensions corresponding to the number of spins handled by the operation unit 20) to and the information of the vector h read out from the external magnetic field coefficient memory 512 (see equation 15) are applied to the equation (equation 15, equation 17, or equation 19) specified by the signal TS. is executed, and the calculated value of such calculation is output as a signal SP. According to such a configuration, the output value (signal SP) output from the mathematical function operation unit 517 is the value of spin (σ) or auxiliary variables (x, y) updated according to the MCMC method described above. In this embodiment, the output value (signal SP) output from the mathematical function operation device 517 is output to the spin value reading section 615, which will be described later with reference to FIG. 6 and the like.

In addition, since Formulas 15, 17, and 19 are independent of the variable subscript i (i=1 to N), the calculation by the calculation unit 20 can be performed independently of the subscript i. is. From another point of view, the information processing apparatus 10 is provided with a plurality of calculation units 20 such as those described with reference to FIGS. ) are executed in parallel, the operation (processing) based on the MCMC method can be speeded up.

Next, further details of the information processing apparatus 10, as well as modified examples and the like will be described.

FIG. 6 shows main functions (software configuration) of the information processing apparatus 10 . As shown in the figure, the information processing apparatus 10 includes a storage unit 600, a model coefficient setting unit 611, an eigenvalue calculation unit 612, a variable value initialization unit 613, an interaction calculation execution unit 614, and a spin value reading unit 615. .

These functions are realized by reading and executing the program stored in the main storage unit 12 by the processor 11 described above with reference to FIG. In addition to the functions described above, the information processing apparatus 10 may have other functions such as an operating system, a file system, a device driver, and a DBMS (DataBase Management System).

Among the units (functions) shown in FIG. 6, the storage unit 600 stores the Ising format question data 601 and the arithmetic unit control program 602 in the main storage unit 12 or the auxiliary storage unit 13 described above with reference to FIG.

Here, the Ising format question data 601 is data in which a magnetic material model or a machine learning model is described or input in a predetermined description format (for example, a known programming language). Such Ising question data 601 can be created by a user via a user interface (input device, output device, communication device, etc.) and stored or set in the above-described storage medium of the information processing device 10, for example.

On the other hand, the arithmetic device control program 602 is executed when the interaction arithmetic execution unit 614 controls the arithmetic units 20, or the interaction arithmetic execution unit 614 loads it into each arithmetic unit 20 and causes the arithmetic unit 20 to execute it. It's a program.

The model coefficient setting unit 611 sets the interaction matrix W of the Ising model based on the Ising formal problem data 601 in the interaction coefficient memory 511, and sets the vector h representing the external magnetic field coefficient in the external magnetic field coefficient memory 512. See also Figure 8).

The variable value initialization section 613 performs settings for initializing the values stored in the spin memory 513 , the first auxiliary variable memory 514 , and the second auxiliary variable memory 515 of the calculation section 20 . For example, the spin value (σ) is initialized to +1 or -1 with a 50% probability, and the auxiliary variable values (x, y) are initialized to uniform random numbers from -1 to +1. .

The interaction calculation execution unit 614 performs calculations according to the above-described equations 15, 17, and 19, that is, calculations for updating the state of the interaction model according to the MCMC method (hereinafter referred to as "interaction calculation" ).

Spin value reading unit 615 retrieves the spin value stored in spin memory 513 when interaction calculation executing unit 614 executes Monte Carlo steps (MC steps) a certain number of times (predetermined number of times). Sampling is performed by reading and outputting the read spin value to the output unit 15 and the communication unit 16 (see also FIG. 8 as appropriate).

FIG. 7 is a flowchart for explaining sampling processing performed by the information processing apparatus 10 when sampling the Boltzmann distribution of the Ising model. An example of the sampling process procedure will be described below with reference to the flowcharts (processing steps S711 to S717) shown in FIGS. 1, 4 and 7. FIG. The sampling process shown in FIG. 7 is started, for example, by receiving an instruction or the like from the user via the operation input unit 14 described above with reference to FIG.

In executing the sampling process, first, the model coefficient setting unit 611 sets respective values (model coefficients) in the interaction coefficient memory 511 and the external magnetic field coefficient memory 512 (step S711). The values of the model coefficients in these memories 511 and 512 can also be set or edited by the user via a user interface (for example, realized by the operation input unit 14, the output unit 15, the communication device 16, etc.).

In subsequent step S712, the eigenvalue calculator 612 calculates the minimum eigenvalue of the matrix J stored in the interaction coefficient memory 511, calculates (determines) the diagonal elements of the interaction matrix W, and calculates (determines) the determined pair The corner elements (diagonal components) are stored (set) in the interaction coefficient memory 511 . As mentioned above, this calculation can be performed in the computing unit 20 or in the processor 11 .

Subsequently, the variable value initialization unit 613 initializes the values stored in the spin memory 513, the first auxiliary variable memory 514, and the second auxiliary variable memory 515 (step S713).

Subsequently, the interaction calculation execution unit 614 executes the interaction calculation based on the equations 15, 17, and 19, so that the spin memory 513, the first auxiliary variable memory 514, and the second auxiliary variable memory 515 are The value is updated (step S714).

Subsequently, the interaction calculation execution unit 614 determines whether or not the sampling execution condition is satisfied (for example, whether or not the interaction calculation has been executed a predetermined number of times) (step S715). If the interaction calculation execution unit 614 determines that the sampling execution condition is satisfied (step S715: YES), the process proceeds to step S716. On the other hand, if the interaction calculation execution unit 614 determines that the sampling execution condition is not satisfied (here, the number of executions of the interaction calculation has not yet reached the specified number of times) (step S715: NO), the process returns to step S714. .

Subsequently, the spin value reading unit 615 reads the spin value stored in the spin memory 513 and stores it as the sampling result (step S716). The result of such sampling is stored (data storage) in the auxiliary storage unit 13 shown in FIG.

Subsequently, the spin value reading unit 615 determines whether or not to terminate the process by determining whether or not sampling has been performed a predetermined number of times (step S717). As a result of this determination, if sampling has not been performed the predetermined number of times (step S717: NO), the process returns to step S714 to continue the process. On the other hand, if sampling has been performed the predetermined number of times (step S717: YES), it is determined that the sampling process should be terminated, and the above-described series of processes (steps S711 to S717) ends.

FIG. 8 is a block diagram showing a more detailed configuration example of the computing unit 20, and is a block diagram showing a circuit configuration example when the SRAM technology is applied to the computing unit 20 of this embodiment.

In the example shown in FIG. 8, the arithmetic unit 20 includes an array unit 802, an interaction driver 803, an SRAM interface 804, and a controller 805 in addition to the interaction coefficient memory 511 and the external magnetic field coefficient memory 512 described above with reference to FIG. .

First, the configuration of the array unit 802 among the above blocks will be outlined. The array unit 802 includes a plurality of units 801 (see FIG. 9 as appropriate) having a memory and a processor for calculating and storing values relating to any one spin in a matrix (specifically, N rows (spins)). (number of coefficients)×3 rows (number of coefficients)=3N). Such a configuration of the array unit 802 can be manufactured by applying semiconductor manufacturing technology (for example, SRAM technology).

FIG. 9 is a block diagram showing an example of a circuit configuration in one unit 801 out of a plurality of (3N in this example) units forming the array unit 802 in FIG. As shown in FIG. 9, the unit 801 includes a variable memory 901 for storing one spin (σ _i ) or one of the auxiliary variables x _i and y _i , and the value of the variable memory 901 as It has a configuration for updating (product-sum operation unit 516 and mathematical function operation unit 517). The circuit configuration of the unit 801 as shown in FIG. 9 can be created by applying the circuit configuration of the SRAM. Details of the operation of the sum-of-products operation unit 516, the mathematical function operation unit 517, and the circuit shown in FIG. 9 will be described later.

Returning to the description of the configuration example of FIG. In FIG. 8, an interaction driver 803 is hardware (so to speak, a signal generator) that outputs the signal TS described above in FIG. , the same signal TS is output at once. Further details of interaction driver 803 are provided below.

In this embodiment, the data stored in the interaction coefficient memory 511 shown in FIGS. 5 and 8 are set by the model coefficient setting unit 611 (see also FIG. 8) described above with reference to FIG. That is, the interaction coefficient memory 511 stores the interaction matrix W of the Ising model based on the Ising format problem data 601 (see FIG. 6 as needed).

In the configuration example shown in FIG. 8, this interaction coefficient memory 511 is used in common by all units 801 in order to reduce the circuit scale of the entire arithmetic section 20 . Therefore, the interaction coefficient memory 511 supplies the coefficient W to all the units 801 of the array unit 802, but FIG. 8 omits the illustration of signal lines therefor. In principle, each unit 801 may have an interaction coefficient memory 511 individually.

The interaction driver 803 outputs the signal TS explained in FIG. 5, that is, the signal TS specifying which of the three values (spin σ, auxiliary variable x, auxiliary variable y) is to be updated. Specifically, in the configuration example shown in FIG. 8, the interaction driver 803 designates (selects) one group from the three groups of the spin (σ) group and the auxiliary variables (x, y) as the signal TS. , emits a signal TS authorizing the update of the selected one group. A signal TS sent from the interaction driver 803 is input to each unit 801 of the array unit 802 . By inputting such a signal TS, only one specific group (that is, any group among the group of spins σ, the group of auxiliary variables x, or the group of auxiliary variables y) is simultaneously generated in the array unit 802. is updated to

Overall, the interaction driver 803 corresponds to the "selection signal provider" of the present invention. That is, the interaction driver 803 simultaneously updates the N variables stored in the array unit 802 (that is, the values of the variables stored in the variable memory 901 of each unit 801), the spin group σ and the auxiliary A signal TS is supplied to the array unit 802 to select (designate) one of the variable groups x and y and select a method of updating the values in each of the (N) variable memories 901 .

According to the information processing apparatus 10 (sampling apparatus) of the present embodiment that performs such operations, the Boltzmann distribution of the Ising model can be efficiently sampled by parallel processing.

As shown in FIGS. 8 and 9, the SRAM interface 804 of the arithmetic unit 20 connects each block of the unit 801 of each array unit 802 (that is, the sum-of-products arithmetic unit 516, the mathematical function arithmetic unit 517, and the variables described above). The unit 801 is connected to the memory 901 ), and data is written to and read from the unit 801 . The SRAM interface 804 also outputs the spin value calculated by the array unit 802 to the spin value reading unit 615 as a signal SP. Thus, the spin value read by the SRAM interface 804 is sent to the spin value reading unit 615 after the processing in the array unit 802 is completed. The spin value reading unit 615 outputs the sampling result by appropriately storing (accumulating) the spin value input from the SRAM interface 804 and outputting it to an external device such as a display unit.

The controller 805 of the calculation unit 20 initializes the calculation unit 20 and reports the end of processing according to instructions from the interaction calculation execution unit 614 .

Next, a circuit configuration example of one unit 801 will be described mainly with reference to FIG. As described above, each unit 801 comprises a variable memory 901 that stores one of spin σ _i , auxiliary variable x _i and auxiliary variable y _i . The unit 801 also includes a sum-of-products operation unit 516 and a mathematical function operation unit 517 as a configuration (operation unit or operation processor) for updating the value of the variable memory 901 .

Of these, the sum-of-products operation unit 516 supplies a variable vector (that is, one of (x, y), (σ, y), (σ, y)) supplied from the SRAM interface 804 and an interaction coefficient W is input, the sum-of-products operation is executed based on the input value, and the value of the operation result is supplied to the mathematical function operation device 517 .

On the other hand, the mathematical function operation unit 517 receives the value of the operation result supplied from the sum-of-products operation unit 516, the signal SR representing the N-dimensional vector described above, and the vector h (supplied from the external magnetic field coefficient memory 512) (equation 15 ) is input, a mathematical function operation based on the input is executed, and the value of the operation result is supplied to the variable memory 901 .

More specifically, the sum-of-products operation device 516 of one unit 801 stores values of spin groups and variable groups (the above-described variable vector) is input. These variable vectors are generated (that is, supplied to the sum-of-products operation device 516 of the unit 801) by reading them from the variable memory 901 of the other unit 801 by the SRAM interface 804. FIG. Also, the value of the interaction coefficient W stored in the interaction coefficient memory 511 (see FIG. 8 as appropriate) is input to the sum-of-products operation device 516 of the one unit 801 .

On the other hand, the mathematical function operation unit 517 stores the operation result (output value) by the product-sum operation unit 516, the value of the vector h stored in the external magnetic field coefficient memory 512 (see FIG. 8 as appropriate), and A signal SR (representing an N-dimensional vector in which each element is an independent random number) is input. Further, when the variable memory 901 stores (storage) the values (σ _i ) of the spin group, the variables (σ _i ) stored in the variable memory 901 are also processed (for example, through the SRAM interface 804) for mathematical function operations. Input to device 517 (see dashed arrow in FIG. 9).

With such a configuration, the next state of the variable is output based on the above-described formula 15, formula 17, or formula 19, stored in the variable memory 901, further through the SRAM interface 804 and the spin value reading unit 615, etc. It is possible to appropriately output the sampling result of Boltzmann distribution of the Ising model (for example, how the state of each of N spins transitions).

As described above in detail, according to the information processing apparatus 10 of the present embodiment, the Boltzmann distribution of the Ising model can be efficiently sampled. In addition, the above-described information processing apparatus 10 (including the calculation unit 20) can be realized with a simple configuration, so that it can be manufactured inexpensively and easily.

Although one embodiment has been described in detail above, it goes without saying that the present invention is not limited to the above-described embodiment, and that various changes can be made without departing from the gist of the present invention. For example, the above embodiments have been described in detail in order to explain the present invention in an easy-to-understand manner, and are not necessarily limited to those having all the configurations described. Moreover, it is possible to add, delete, or replace a part of the configuration of the above embodiment with another configuration.

Further, each of the above-described components, functional units, processing units, processing means, etc. may be realized by hardware, for example, by designing them in an integrated circuit. Moreover, each of the above configurations, functions, etc. may be realized by software by a processor interpreting and executing a program for realizing each function. Information such as programs, tables, and files that implement each function can be stored in recording devices such as memories, hard disks, SSDs (Solid State Drives), and recording media such as IC cards, SD cards, and DVDs.

Further, in each of the above drawings, control lines and information lines are those considered to be necessary for explanation, and not all control lines and information lines for implementation are necessarily shown. For example, it may be considered that almost all structures are interconnected in practice.

Moreover, the arrangement form of various functional units, various processing units, and various databases of the information processing apparatus 10 described above is merely an example. The arrangement form of various functional units, various processing units, and various databases can be changed to an optimum arrangement form from the viewpoint of the performance, processing efficiency, communication efficiency, etc. of the hardware and software included in the information processing apparatus 10 .

Also, the configuration of the database (schema, etc.) that stores the various data described above can be flexibly changed from the viewpoint of efficient use of resources, improvement of processing efficiency, improvement of access efficiency, improvement of search efficiency, and the like.

It can be used for a sampling method and an information processing device.

10 information processing device (sampling device)
11 processor (variable controller)
12 main storage device 13 auxiliary storage unit 14 operation input unit 15 output unit 16 communication unit 20 calculation unit (calculation unit)
511 interaction coefficient memory 512 external magnetic field coefficient memory 513 spin memory 514 first auxiliary variable memory 515 second auxiliary variable memory 516 product-sum operation device (product-sum operation unit)
517 Mathematical Function Calculator (Mathematical Function Calculator)
600 storage unit 601 Ising format problem data 602 arithmetic unit control program 611 model coefficient setting unit 612 eigenvalue calculation unit 613 variable value initialization unit 614 interaction calculation execution unit 615 spin value reading unit 801 (3N) units 802 array unit 803 Interaction driver (variable controller)
804 SRAM interface 805 Controller W Interaction coefficient of interaction model h External magnetic field coefficient of Ising model

Claims (7)

A control method for operating an information processing device so as to sample a spin state based on a Markov chain Monte Carlo method for the Boltzmann distribution of an Ising model including an interaction relationship between spins, comprising:
Sampling for the Ising model defines a spin group σ consisting of N (N is a natural number) spins σ _i ( _i = any number from 1 to N) and the direction of each spin σ _i As continuous variables, a first auxiliary variable group x consisting of the N auxiliary variables x _i ( _i = 1 to N) and a second auxiliary variable group consisting of the N auxiliary variables y _i ( _i = 1 to N) y, and all of the first auxiliary variable groups x and all of the second auxiliary variable groups y are connected to the spin σ _i and have an interactive relationship with a complete tripartite graph structure. substitute by sampling at
The value of the interaction relationship of the interaction model is set in the memory of the information processing device with the interaction relationship of the original Ising model and its minimum eigenvalue,
All variables in one of the three variable groups of the spin group σ, the first auxiliary variable group x, and the second auxiliary variable group y are simultaneously updated when the state transition is performed in the interaction model. operating the information processing device to
control method. In order to sample the spin states based on the Markov chain Monte Carlo method for the Boltzmann distribution of the Ising model including the interaction relationship between spins,
As statistics for the Ising model, a spin group σ consisting of N (N is a natural number) spins σ _i ( _i = any number from 1 to N) and the direction of each spin σ _i are defined. As continuous variables, the first auxiliary variable group x consisting of the N auxiliary variables x _i ( _i = 1 to N) and the second auxiliary variable group x consisting of the N auxiliary variables y _i ( _i = 1 to N) and an interaction by a complete tripartite graph structure having an interaction relationship in which all of the first auxiliary variable groups x and all of the second auxiliary variable groups y are connected to the spin σ _i . a calculation unit that calculates the transition probability P (σ, σ′) in the model;
Perform control to simultaneously update all variables in one of the three variable groups of the spin group σ, the first auxiliary variable group x, and the second auxiliary variable group y calculated by the calculating unit a variable control unit;
A sampling device comprising: A sampling device according to claim 2, wherein
N variable memories for storing the value of any one of the three variables of the spin value and the first and second auxiliary variables related to the spin calculated by the calculation unit;
a selection signal supply unit that supplies the same signal (TS) to the N variable memories,
The variable control unit selects one group from the spin group σ and the auxiliary variable groups x and y to update the values in the N variable memories when simultaneously updating all the variables. Perform control to output a signal (TS) for selecting from the selection signal supply unit,
sampling device. A sampling device according to claim 3, wherein
an interaction coefficient memory for storing an interaction coefficient J defining the interaction relationship of the Ising model and an interaction coefficient W of the interaction model defined by its minimum eigenvalue -λ;
an external magnetic field coefficient memory that stores the external magnetic field coefficient h of the Ising model,
The calculation unit uses the values stored in the interaction coefficient memory, the external magnetic field coefficient memory, and the N variable memories to calculate the next states of the N spins.
sampling device. A sampling device according to claim 4, wherein
The interaction coefficient J is a symmetric matrix,
The interaction coefficient W is determined based on the interaction coefficient J and the minimum eigenvalue of the interaction coefficient J,
sampling device. In the information processing device according to claim 5,
The calculation unit includes a sum-of-products operation unit that performs a sum-of-products operation using the values of the spin and the auxiliary variables and the interaction coefficient W as variables.
sampling device. In the information processing device according to claim 6,
The calculation unit includes a mathematical function calculation unit that calculates the next state of the value stored in the corresponding variable memory,
sampling device.

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