Machine learning device, machine learning program, and machine learning method
Patent Information
- Application Number
- PCT/JP2023/028810
- Authority / Receiving Office
- WO · WO
- Patent Type
- Applications
- Current Assignee / Owner
- Filing Date
- 2023-08-07
- Publication Date
- 2025-02-13
Smart Images
Figure JP2023028810_13022025_PF_FP_ABST
Abstract
Description
Machine learning device, machine learning program, and machine learning method
[0001] The present disclosure relates to a machine learning device, a machine learning program, and a machine learning method, and in particular to a machine learning device, a machine learning program, and a machine learning method that use a steepest descent method to derive a solution.
[0002] Machine learning can handle processes including event prediction and classification. This can be achieved by artificially constructing a probability distribution P that matches the distribution (empirical distribution) Q of a random variable X obtained from real-world experience. A Boltzmann machine is a machine learning algorithm that constructs a distribution similar to the empirical distribution from observed data using the Boltzmann distribution (also known as the canonical or Gibbs distribution) used in statistical mechanics. The canonical distribution is a theoretical distribution that describes a physical system in thermal equilibrium at a constant temperature T. At the high-temperature limit T = ∞, all states of the system are equally likely to be realized, and the probability of a low-energy state increases exponentially as the temperature decreases.
[0003] The Ising model is used as a specific physical system in thermal equilibrium, which is represented by a set of spins σ, which are random variables with two degrees of freedom. σ can take any realizable value, but is usually {-1, 1} or {0, 1}. The energy of the Ising model takes the following form:
[0004]
[0005]
[0006] In the above formulas (6) and (7), the average <σ i σ jThe subscript P indicates the theoretical model average based on the Boltzmann-distributed Ising model. The subscript Q indicates the data average based on the empirical distribution obtained from experience (training data). Therefore, when the moments based on the empirical distribution match the moments based on the model distribution, the KL divergence reaches a minimum. This condition is called moment matching. To calculate the model average of moments, sampling averages obtained through numerical calculations such as the Monte Carlo method are generally used. Thus, the Boltzmann machine is a machine learning algorithm developed based on the Boltzmann distribution, which describes physical systems in thermal equilibrium. The Boltzmann distribution is a type of exponential distribution, and high-energy states become exponentially less likely to occur and can be ignored. In other words, the Boltzmann machine, as well as its extensions, such as deep Boltzmann machines and deep neural networks, can be considered machine learning algorithms that originate from physical systems in thermal equilibrium.
[0007] Arnaud Mignan and Marco Broccardo, "Neural Network Applications in Earthquake Prediction (1994-2019): Meta‐Analytic and Statistical Insights on Their Limitations", Seismological Research Letters 91, 2330, (2020). Lu Yang, He Jian, Qing Song, and Jun Guo, "A Survey on Long-Tailed Visual Recognition", International Journal of Computer Vision, 130, 1837 (2022). Constantino Tsallis, "Introduction to Nonextensive Statistical Mechanics" (Springer Science+Business Media, LLC, 2009). Hiroki Suyari, "Basic Mathematics for Complex Systems - Mathematics of Power Laws and Tsallis Entropy -", Makino Shoten (Tokyo, 2010). I. Fukuda and H. Nakamura, "Tsallis Dynamics Using the Nose-Hoover Approach", Physical Review E 65, 026105 (2002).AR Lima, JS Sa Martins, and TJP Penna, Physica A 268, 533 (1999).T. Inagaki, Y. Haribara, et. al., "A coherent ising machine for 2000-node optimization problems", Science 354, 603--606 (2016).
[0008] Machine learning has shown remarkable progress in recent years. However, machine learning based on Boltzmann machines has some areas where it struggles. A typical example is problems involving power-law distributions. The Boltzmann distribution, or exponential distribution, is often hidden behind the theoretical background of known machine learning. Exponential distributions capture typical events and ignore outliers. This characteristic is expected to work well for problems where typical examples exist. On the other hand, problems involving power-law distributions cannot ignore outliers, making prediction and classification difficult and potentially increasing the cost of collecting the data required for training. For example, predicting the date and time of earthquakes is a challenging problem that cannot be solved even with current technology (Non-Patent Document 1). This problem is thought to be essentially related to the emergence of a power-law distribution between earthquake frequency and energy. Furthermore, while training data used in image recognition is often normalized to ensure the same number of events regardless of type, collecting data on events with low power-law frequency presents a challenge (Non-Patent Document 2).
[0009] The present disclosure has been made to achieve the above object, and relates to a machine learning device, a machine learning program, and a machine learning method that can learn events that follow a power distribution.
[0010] In order to achieve the above object, a machine learning device of one embodiment of the present disclosure includes a sampling unit that inputs model parameters and outputs spin states of an Ising model that conforms to a Tsallis distribution, and a learning unit that inputs the spin states and outputs the Ising model parameters, and the sampling unit and the learning unit repeatedly input and output the spin states and the Ising model parameters, thereby changing the model parameters until the number of repetitions exceeds a specified number or the model parameters converge.
[0011] A machine learning program according to one embodiment of the present disclosure is a program executed in the above-described machine learning device, which causes a computer to realize the following functions: simulating the amplification of input light; simulating the measurement of at least a portion of the light amplified by the resonator; simulating the coupling of light based on the measurement results; sampling the spin states of an Ising model that follows the Tsallis distribution; and learning the model parameters. The learning function calculates the changes in the model parameters from the sampled spin states using the steepest descent method, and determines the convergence of the model parameters from the results of the calculation.
[0012] A machine learning method according to one embodiment of the present disclosure includes the steps of amplifying input light, measuring at least a portion of the amplified light, combining light based on the measurement results of the step of measuring at least a portion of the light, sampling spin states of an Ising model that follows a Tsallis distribution, and learning model parameters, wherein the learning step calculates changes in the model parameters from the sampled spin states using a steepest descent method, and determines the convergence of the model parameters from the results of this calculation.
[0013] According to the above aspects, it is possible to provide a machine learning device, a machine learning program, and a machine learning method that can learn events that follow a power distribution.
[0014] It is a schematic diagram for explaining a machine learning device of an embodiment of the present disclosure. It is a schematic diagram for explaining a sampling unit shown in Figure 1. It is a flowchart for explaining a machine learning method of an embodiment of the present disclosure.
[0015] An embodiment of the present disclosure will be described below with reference to the drawings. The drawings used in this embodiment are intended to explain the configuration, function, action, effect, and technical concept of the present disclosure, and are not intended to limit the specific configuration or shape.
[0016] [Summary] Before describing the present disclosure in detail, an overview of the present disclosure will be described.
[0017] In non-equilibrium states, even if the occurrence of low-energy states is dominant in frequency, there are phenomena in which high-energy states cannot be ignored. Examples of such phenomena include earthquakes and stock market crashes. In these complex and critical systems, a distribution that can be described by the following q-exponential function, which appears in Tallies statistics, is often observed. In the specification of this disclosure, such a power-law distribution is referred to as the "Tallies distribution." Such a Tallies distribution is described, for example, in Non-Patent Document 3.
[0018]
[0019] [Embodiment] (Machine Learning Device) FIG. 1 is a schematic diagram illustrating a machine learning device 100 used in this embodiment. The machine learning device 100 shown in FIG. 1 includes a sampling unit 11 and a learning unit 13. The sampling unit 11 samples spin states of an Ising model that follows a Tsallis distribution. The learning unit 13 learns Ising model parameters. The sampling unit 11 outputs the sampled spin states as a set of spin states. The learning unit 13 inputs the set of spin states output by the sampling unit 11, calculates the average of the moments of the spin states, and obtains changes in the model parameters through a calculation that determines their time evolution. Then, the convergence of the model parameters is determined from the results of the calculation.
[0020] 2 is a schematic diagram illustrating a coherent Ising machine constituting the sampling unit 11 shown in FIG. 2. The coherent Ising machine shown in FIG. 2 functions as a computing device that generates the spin state of an Ising model with a Tsallis distribution, and shows an example of a physical system that simulates the electron spin degrees of freedom of a magnetic material. The coherent Ising machine simulates the binary degrees of freedom of the Ising spin using an optical parametric oscillation phenomenon (degenerate optical parametric oscillator (DOPO)), and also implements the spin-spin coupling (interaction between spins) of the Ising model using an optical measurement and feedback mechanism.
[0021] In FIG. 2 , the coherent Ising machine includes a PSA (Phase Sensitive Amplifier) 2 provided in a ring-shaped optical fiber that functions as ring resonator 1, an optical coupling unit 5 that extracts a portion of the optical pulse from ring resonator 1, a measurement unit 6 that measures the extracted optical pulse, a calculator 7 that calculates a coupled signal using the measurement result of measurement unit 6, and an external pulse input unit 8 that inputs an external optical pulse corresponding to the coupled signal to ring resonator 1 via optical coupler 9.
[0022] That is, by injecting pump light 3, the PSA 2 generates an optical pulse train 4 in the ring resonator 1, consisting of optical pulses whose number corresponds to the number of sites (spins) in the Ising model. When each optical pulse in the optical pulse train generated in the ring resonator 1 completes one circuit and reaches the PSA 2, the pump light 3 is input again to the PSA 2, amplifying each optical pulse in the optical pulse train 4. The optical pulses in the optical pulse train 4 generated by the initial injection of pump light are weak pulses with an indeterminate phase, and are amplified by the PSA 2 each time they circulate within the ring resonator 1. As a result, the phase state of each optical pulse in the optical pulse train 4 gradually becomes determinate. In this case, the PSA 2 amplifies each optical pulse with a phase of 0 or π relative to the phase of the pump light source, so that the optical pulses are determinately in one of these phase states. At this time, if the intensity of the pump light 3 is made sufficiently greater than the oscillation threshold, known as the oscillation point, the number of photons in each optical pulse increases significantly, resulting in the oscillation of a coherent optical pulse train 4 with a phase of 0 or π. In this way, the coherent Ising machine is implemented by making the spins 1 and -1 in the Ising model correspond to the phases 0 and π of the optical pulse.
[0023] As described above, the optical pulses circulating within the ring resonator 1 are subjected to measurement and feedback to impart an interaction (coupling constant) between pulses (i.e., between spins). That is, the optical coupling unit 5 extracts a part of each optical pulse each time the optical pulse circulates. Then, the measurement unit 6 measures the signed amplitude x of each optical pulse in the phase 0 or π direction. iThe sign is positive in the phase 0 direction and negative in the phase π direction. The calculator 7 is given information on the coupling constant between the optical pulses in advance, and calculates the coupling signal to be fed back using the measurement results from the measurement unit 6. The coupling signal Δx fed back to the i-th optical pulse is i is expressed by the following equation (10).
[0024] In equation (10), ξ is the dimensionless feedback intensity. Next, the external light pulse input unit 8 outputs the calculated combined signal Δx i and inputs the external optical pulse into ring resonator 1 via optical coupler 9. The measurement and feedback control described above allows a correlation to be imparted to the phases of the optical pulses that make up the optical pulse train. While imparting this correlation, the optical pulse train is circularly amplified within ring resonator 1, and a stable state, i.e., time evolution is advanced until a steady state in which there is essentially no change in the state of the optical pulses, is reached. The phases 0 and π of the optical pulses that make up the optical pulse train are measured. The final measurement results are sent to a computer (not shown) and used for the desired purpose.
[0025] In the above-described coherent Ising machine, known coherent Ising machines measure the phase in a stable state by setting the amplitude of the pump light 3 to be greater than the oscillation point, i.e., the oscillation threshold (amplitude) (Non-Patent Document 7). In contrast, in this embodiment, the amplitude of the pump light 3 is kept below the oscillation point and measurement is performed while maintaining this value. This increases the influence of quantum fluctuations and makes it possible to sample the spin state of the Ising model that follows a power distribution that can be approximated by a Tsallis distribution.
[0026] Furthermore, the sampling unit 10 of this embodiment may be an electronic calculator that simulates the above-described coherent Ising machine. The electronic calculator may be a so-called computer in which hardware such as a CPU (Central Processing Unit), memory, and user interface cooperates with software such as a program that runs on the hardware to function.
[0027] Furthermore, the sampling unit 11 of the machine learning device 100 of this embodiment is not limited to the configuration shown in Fig. 2. The sampling unit 11 may be, for example, an electronic computer that outputs the spin state of an Ising model that follows the Tsallis distribution by the Monte Carlo method.
[0028] In this embodiment, the learning unit 13 is configured as an electronic computer. The learning unit 13 as an electronic computer implements a machine learning method, which will be described later, by means of a program. Furthermore, in this embodiment, the learning unit 13 is not limited to being arranged as shown in FIG. 1, but may be configured to be incorporated into the calculation unit 7 shown in FIG. 2. In this case, the spin state output by the sampling unit 11 is used to change the coupling constant between optical pulses provided to the calculation unit 7, i.e., the model parameters. A specific method of change will be described in the machine learning method below.
[0029] (Machine Learning Method) Next, a description will be given of a machine learning method performed using the above-described machine learning device 100. The machine learning method of this embodiment is a computational process performed in the learning unit 13 shown in Fig. 1. In this specification, prior to a specific description of the computational process, the concept of the machine learning method of this embodiment will be described.
[0030] In this embodiment, it is assumed that the random variable vector x follows the Tsallis distribution shown in equation (11).
[0031]
[0032]
[0033]
[0034]
[0035]
[0036]
[0037] In the known Boltzmann machine, the KL divergence is used as the potential function, but there are many other divergences that quantify the difference between probability distributions. Here, we use α divergence, D (α) (Q||P)=4 / (1-α2 )(1-Σ x Q (1-α) / 2 (x)P (1+α) / 2 (x)) is used. α is an arbitrary parameter, and α divergence corresponds to various other divergences depending on the value of α. Here, α = 1-2q is used. Therefore, in this embodiment, the following equation (25) holds.
[0038]
[0039] In equation (27), the first term on the right-hand side is the model mean of the vector-valued function due to the q-escort distribution of the Tsallis distribution, which arises from the differentiation of the generalized Mathieu potential. The second term on the right-hand side represents the data mean of the vector-valued function calculated using the observed data.
[0040] Therefore, the Tsallis distribution, which was first introduced for theoretical construction, does not need to be used for specific optimization. In summary, the following equation, obtained by the steepest descent method, is used to optimize the model parameters.
[0041] As described above, the machine learning method of this embodiment makes it possible to improve, i.e., learn, the theoretical distribution using statistical data observed in the past. Furthermore, this embodiment uses the Tsallis distribution instead of the Boltzmann distribution, thereby enabling prediction and classification that also accommodates events that include power-law distributions. This corresponds to generalizing a Boltzmann machine that has the well-known Boltzmann distribution as its theoretical background and creating a "Tsallis machine" that has the Tsallis distribution as its theoretical background.
[0042] The machine learning method described above is formulated for an arbitrary energy function formula 12. Next, a specific example using the Ising energy formula 1 as the energy function formula 12 will be described.
[0043] Here, equation (1) is presented again.
[0044] When there are N spin variables, equation (1) can be expressed as θ1=J11 ,…,θ N =J N1 , θ N+1 =J 12 ,…, θ N*N =J NN ,θ N*N+1 =B1,…, θ N*(N+1) =B N Therefore, the vector in the following equation is adopted as the model parameter θ.
[0045] Similarly, equation (1) can be expressed as N(N+1) arbitrary functions of equation (12), f1(σ)=σ1σ1,...,f N (σ)=σ N σ1, f N+1 (σ)=σ N σ2, …,f N*N (σ)= σ N σ N , f N*N+1 (σ) = σ1,…, f N*(N+1) (σ)=σ N is adopted and g(θ) = 0. Therefore, the specific time evolution equation of the Ising model parameters, which corresponds to the time evolution equation (29) of the model parameters according to the steepest descent method, is as follows:
[0046] The flowchart in Fig. 3 illustrates an embodiment of machine learning in which the energy of an Ising model expressed using a set of spins σ, which are random variables with two degrees of freedom, is used as the arbitrary energy in equation (12). In this embodiment, the flowchart in Fig. 3 is executed by the sampling unit 11 and the learning unit 13 shown in Fig. 1.
[0047]
[0048] The learning unit 13 calculates a candidate θ′ for the next model parameter θ according to the time evolution equations (32) and (33) of the model parameters (step S105).
[0049] The learning unit 13 determines that convergence has occurred if the vector norm |θ-θ'| is equal to or less than a certain threshold value or if the model parameters have already been changed a predetermined number of times (step S106: YES), and outputs the optimized model parameters θ (step S107).
[0050] If the learning unit 13 determines that convergence has not occurred (step S106: NO), it substitutes the value of θ′ into the model parameter θ and updates it (step S108).
[0051] In the present embodiment described above, the learning unit 13 calculates the model mean of moments using sampling of spin states of an Ising model that follows a Tsallis distribution, obtains a data mean of the moments, determines the convergence of the Ising model parameters from these results, and changes the Ising model parameters if necessary. Therefore, this embodiment can provide a machine learning device, machine learning program, and machine learning method that can learn, using observed data, even about phenomena that include a power-law distribution, and predict and classify them with high accuracy.
[0052] REFERENCE SIGNS LIST 1 Ring resonator 2 PSA 3 Pump light 4 Pulsed light 5, 9 Optical coupler 6 Measurement unit 7 Calculator 8 External pulse input unit 11 Sampling unit 13 Learning unit
Claims
1. A machine learning device comprising: a sampling unit that inputs model parameters and outputs a spin state of an Ising model that follows a Tsallis distribution; and a learning unit that inputs the spin state and outputs the Ising model parameters, wherein the sampling unit and the learning unit repeatedly input and output the spin state and the Ising model parameters, thereby changing the model parameters until the number of repetitions exceeds a specified number or the model parameters converge.
2. The machine learning device of claim 1, wherein the sampling unit includes: a resonator that amplifies input light; a measurement unit that measures at least a portion of the light amplified by the resonator; and a calculator that calculates the coupling of the light based on the measurement results of the measurement unit.
3. The machine learning device according to claim 1, wherein the learning unit performs calculations using spin states of an Ising model that follows a Tsallis distribution and determines the convergence of the model parameters from the results of the calculations.
4. The machine learning device according to claim 1, wherein the learning unit uses alpha divergence as a potential function indicating the time change of the model parameters.
5. The learning unit optimizes the model parameters using an equation obtained by a steepest descent method, the equation being: θ model parameters 6. The machine learning device according to claim 2, wherein the learning unit is incorporated in the computing unit of the sampling unit.
7. A machine learning device, which is an electronic computer simulating the machine learning device according to any one of claims 1 to 6.
8. A program executed in the machine learning device of claim 7, which causes a computer to realize a function of simulating the amplification of input light, a function of simulating the measurement of at least a portion of the light amplified by the resonator, a function of simulating the coupling of light based on the measurement results, a function of sampling the spin state of an Ising model that follows the Tsallis distribution, and a function of learning model parameters, wherein the learning function calculates the change in the model parameters from the sampled spin states using the steepest descent method, and determines the convergence of the model parameters from the results of the calculation.
9. A machine learning method comprising: a step of amplifying input light; a step of measuring at least a portion of the amplified light; a step of combining light based on a measurement result of the step of measuring at least a portion of the light; a step of sampling spin states of an Ising model that follows a Tsallis distribution; and a step of learning model parameters, wherein the learning step calculates changes in the model parameters from the sampled spin states using a steepest descent method, and determines the convergence of the model parameters from the results of the calculation.