Machine learning efficient numerical sampling method based on under-damped ranque-hugoniot system

Abstract

This invention discloses an efficient numerical sampling method for machine learning based on underdamped ensemble systems, belonging to the field of machine learning sampling technology. Addressing the problems of existing sampling methods failing to maintain the intrinsic geometry and ergodicity of the system when the potential function exhibits super-quadratic growth, and also suffering from low convergence order, this invention first establishes an underdamped ensemble system with a super-quadratic growth potential function; then, it constructs a tamed potential function and its corresponding tamed gradient function; finally, it constructs a fully explicit tamed sampling model based on the tamed gradient function for iterative sampling. The method of this invention can simultaneously and strictly maintain the conformal symplectic geometry and numerical ergodicity of the original system, effectively avoiding dynamic distortion in long-term sampling, and achieving long-term second-order uniform convergence accuracy. This significantly improves the efficiency and accuracy of sampling high-dimensional complex distributions, providing strong technical support for machine learning applications such as Bayesian statistical inference and Bayesian neural networks.

Description

Technical Field

[0001] This invention belongs to the field of machine learning sampling technology. It proposes an efficient numerical sampling method for machine learning based on an underdamped Langevin system for complex high-dimensional problems. Background Technology

[0002] Achieving efficient and high-precision sampling from complex and high-dimensional probability distributions is a key technical problem in machine learning fields such as Bayesian statistical inference and Bayesian neural networks. To achieve efficient sampling in machine learning, the Markov chain Monte Carlo (MCMC) method based on stochastic dynamical evolution is a widely used and important sampling technique. However, traditional MCMC methods, such as the unadjusted Langevin algorithm (ULA) based on overdamped Langevin systems, are prone to inefficient exploration through random walks when dealing with high-dimensional complex problems, thus limiting sampling efficiency.

[0003] To overcome this challenge, recent research has further proposed MCMC methods based on underdamped Nissan systems to improve high-dimensional sampling efficiency. Unlike overdamped Nissan systems, underdamped Nissan systems, which incorporate momentum variables, are dissipative stochastic Hamiltonian systems. This structural characteristic of underdamped Nissan systems significantly improves the convergence speed of sampling methods, resulting in a substantial increase in the efficiency of MCMC methods based on underdamped Nissan systems in handling complex high-dimensional machine learning applications.

[0004] For drift coefficient For cases satisfying the global Lipschitz condition, existing numerical sampling methods have achieved relatively complete results. However, in applications such as complex high-dimensional machine learning, These nonlinear systems often exhibit super-quadratic growth, where the gradient no longer satisfies the global Lipshitz condition. If standard explicit sampling methods are directly applied to these systems, the superlinear growth of the system's drift term can lead to numerical solution explosion. To suppress this explosion, the traditional approach is to introduce a Metropolis-Hastings rejection step (such as the MALA algorithm), but this results in extremely low sample acceptance rates in high-dimensional data spaces, severely impacting sampling efficiency. Therefore, constructing efficient explicit sampling methods independent of path rejection steps under non-global Lipshitz conditions is currently an important research direction in sampling techniques.

[0005] Currently, for the potential function Some progress has been made in constructing and analyzing the convergence of sampling algorithms for the super-quadratic growth case. For example, under strong convexity and local Lipshitz conditions, an explicit compliant sampling algorithm has been constructed, and the convergence order analysis results between the numerical solution process distribution and the target distribution have been given. There is also a sampling method using split scalar auxiliary variables, and corresponding long-term weak convergence results have been established. However, currently, for potential functions... In the case of super-quadratic growth, the convergence order of existing sampling algorithms remains limited, and higher-order convergence sampling algorithms need to be developed to improve simulation accuracy and efficiency. Furthermore, based on the ergodicity and conformal symplectic geometry of underdamped Langevin systems, how to construct higher-order sampling methods that can simultaneously preserve their intrinsic structural properties to improve long-term numerical simulation results remains an important technical problem to be solved. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides an efficient numerical sampling method for machine learning based on an underdamped Langevin system, solving the following technical problems in the prior art: (1) Lack of inheritance of ergodicity and conformal symplectic geometry: Existing machine learning sampling methods cannot maintain the inherent conformal symplectic geometry of underdamped Langevin systems, which leads to the phenomenon of phase flow deviating from the original dynamics in long-term sampling simulations; in addition, the lack of explicit numerical schemes that take into account both geometric structure and numerical ergodicity can easily lead to distortion of sampling methods during long-term sampling. (2) The accuracy of long-term simulation needs to be improved: The convergence order of existing machine learning sampling methods is limited, making it difficult to achieve high-order convergence. Therefore, in high-dimensional and complex applications such as machine learning, the computational efficiency of existing methods is low, which is not conducive to the realization of efficient and high-precision numerical sampling.

[0007] This invention discloses an efficient numerical sampling method for machine learning based on an underdamped Langevin system, comprising the following steps: Step 1: For the target distribution, establish an underdamped Langevin system with a super quadratic growth potential function; Step 2: For the super quadratic growth potential function, construct the taming potential function and its corresponding taming gradient function; Step 3: Based on the tamed gradient function, construct a tamed sampling model for iteratively obtaining the sampled numerical solution; The expression for the tamed sampling model is:

[0008] in, express The sampled numerical solution of the intermediate step in the n-step iterative process of the position component direction; express The sampled numerical solution of the position component direction after n iterations; express The sampled numerical solution for the momentum component direction iteratively in n+1 steps; express The sampled numerical solution of the momentum component direction iteratively over n steps; express The sampled numerical solution of the position component direction iteration n+1 steps; Indicates the coefficient of friction; Represents the natural base; These are standard Gaussian random variables; This represents the discrete time step.

[0009] Optionally, the expression for the taming potential function in step two is:

[0010] in, Represents the docility potential function; Represents the original potential function; Indicates the taming radius; express ; Represents the polishing function; This indicates the positional independent variable.

[0011] Optionally, the polishing function is a smoothing function, whose function value is a constant one when the absolute value of the independent variable is less than or equal to one, and zero when the absolute value of the independent variable is greater than or equal to two.

[0012] Optionally, the expression for the underdamped Langevin system in step one is:

[0013] in, It is a moment momentum components Location ; It is Brownian motion. exist The value at time; express d The real-valued space of dimension.

[0014] Compared with the prior art, the present invention has at least the following beneficial effects: (1) It can simultaneously inherit the conformal symplectic geometric structure and ergodicity of the original system, overcoming the difficulty of existing explicit sampling methods in avoiding numerical explosion and destroying the intrinsic structural properties of the system, thus enabling the sampling method to have better numerical performance over long periods of time.

[0015] (2) This invention provides a construction scheme for a high-order explicit sampling method for the case of super-quadratic growth potential function. This enables the target distribution to be simulated with higher accuracy under the same computational step size, thereby effectively improving the sampling accuracy. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the embodiments will be briefly introduced below. The features and advantages of the present invention can be more clearly understood by referring to the accompanying drawings. The accompanying drawings are schematic and should not be construed as limiting the present invention in any way. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0017] Figure 1 This is a conformal symplectic geometry diagram of the sampling method of the present invention; Figure 2 This is a numerical ergodic evolution diagram of the sampling method of the present invention; Figure 3 This is a long-term consistent convergence order diagram of the sampling method of the present invention; Figure 4 This is an empirical distribution diagram of the sampling method of the present invention; Figure 5 This is a reference distribution diagram for the present invention; Figure 6 This is a graph showing the order of the error of the sampling method of the present invention with respect to the dimension. Detailed Implementation

[0018] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other.

[0019] A specific embodiment of the present invention, such as Figure 1-6 As shown, a machine learning numerical sampling method based on an underdamped Langevin system is disclosed, which specifically includes the following steps: Step 1: Establish an underdamped RANZUAN system.

[0020] Considering sampling problems in machine learning fields such as Bayesian models, the target distribution for sampling is... for:

[0021] in, This represents the potential function or loss function. Indicates the positional independent variable; express d 3D real-valued space, The normalization constant is .

[0022] Therefore, in order to determine the target distribution Sampling is performed to determine the smooth potential function for the polynomial growth of each derivative. And establish an underdamped Langevin system, the expression of which is:

[0023] in, It is a moment momentum components Location ; It is the coefficient of friction. It is Brownian motion. exist The value at time; express d A real-valued space of dimension, when When, abbreviated as .

[0024] Furthermore, the potential function or loss function and its gradient The polynomial growth orders are respectively ,in, For real numbers, At this point, from the underdamped Langevin system (1), we can obtain: the invariant distribution of the system. exist The marginal density along the component direction is the target distribution of the sample. Therefore, the target distribution of the sample will be... The sampling problem is transformed into a computational problem of system (1) that is efficient and preserves the intrinsic structural properties.

[0025] Step 2: For the underdamped Langevin system, construct the controlled gradient function under the controlled potential function. Furthermore, the expression for the taming potential function is:

[0026] in, Indicates the discrete time step; Indicates the taming radius; express ; This represents the polishing function.

[0027] Furthermore, taming radius Coefficients of the quadratic function term It is a constant. For the polishing function, such that hour , hour , Representation function The independent variable.

[0028] Furthermore, based on the taming potential function, the expression for the taming gradient function is constructed as follows: .

[0029] Specifically, the expansion of the taming gradient function is:

[0030] in, Represents the polishing function The first derivative of .

[0031] Step 3: Construct a docile sampling model based on the docile gradient function.

[0032] Furthermore, a docile sampling model is constructed based on the docile gradient function, and the docile sampling model is used to iteratively obtain the sampled numerical solution. The expression for a single-step iteration is:

[0033] in, express The sampled numerical solution of the intermediate step in the n-step iterative process of the position component direction; express The sampled numerical solution of the position component direction after n iterations; express The sampled numerical solution for the momentum component direction iteratively in n+1 steps; express The sampled numerical solution of the momentum component direction iteratively over n steps; express The sampled numerical solution of the position component direction iteration n+1 steps; Indicates the coefficient of friction; Represents the natural base; These are mutually independent standard Gaussian random variables.

[0034] Furthermore, for a given spatial dimension d, we set... These are the initial momentum and position components.

[0035] The sampling model of this invention has the following important properties: Property 1: It can inherit the conformal symplectic geometric structure of the original system. That is: .

[0036] Property 2: The sampling model preserves numerical ergodicity. (Based on Lyapunov functions) ,in, Represents the Lyapunov function; Indicates the independent variable of momentum. Indicates the positional independent variable; Since the coefficients are constant, it can be proved that the uniform moments of the numerical solution are bounded, that is, for any given order... They all and ,in, Represents the mathematical expectation; This represents a positive constant; it also implies the Lyapunov property of numerical solutions. For space... Any two distinct points in Due to the existence of a solution using the explicit method, the Brownian motion increments corresponding to the first and second steps of the numerical method can be uniquely determined. and , such that for a given initial value It can achieve , Representation space The point in the middle; Representation space The point in the middle; express Initial value; express Initial value; express exist The value of ; express exist The value of is determined. Therefore, for any given starting and ending positions, the numerical solution can always be reached with a positive probability. Furthermore, the constant matrix representing the noise intensity... Ensures that for any point and Borel Collection For any positive integer n, from Depart and enter Borel Settlement after 2n steps. transition probability It has a continuous density. Therefore, the numerical solution process corresponding to the sampling model can be obtained with a unique invariant distribution. The sampling model maintains numerical ergodicity, that is, for the test function... ,have ,in, Represents positive integers that tend towards infinity; Represents the mathematical expectation; This represents the test function; Indicates that the value remains unchanged. .

[0037] Property 3: Long-term uniform convergence.

[0038] Based on the aforementioned taming strategy, it can be proven that the sampling model has consistent convergence results over a long period of time, with numerical values ​​remaining unchanged. With exact solution invariant distribution The convergence order is 2. That is, for any given test function... ,have: ,in, , Indicates distribution , Next, regarding test functions The points.

[0039] Step 4: Sampling and verification based on the taming sampling model from Step 3.

[0040] Specifically, for a given spatial dimension d, set Set the termination step number K and the number of sample paths M for the initial momentum and position components.

[0041] use Let represent the sampled numerical solution obtained after k iterations for the m-th sample path. It is understandable that a family of sampled numerical solutions is obtained by iterating K times from the initial value. The approximate distribution that follows the target distribution is obtained through a tamed sampling model. The final numerical sampling result.

[0042] Furthermore, based on the numerical ergodicity and long-term consistent convergence properties of the docile sampling model obtained in step three, a test function is selected. Dirac function A smooth approximation is obtained, and the numerical density is calculated using the Monte Carlo method. The sampling numerical results obtained from the plotted numerical density follow the sampling target distribution. The results demonstrate that the docile sampling model of the present invention has good simulation capability for the sampling target. Among them, M This indicates the number of sample orbitals in the Monte Carlo method; K Indicates the total number of steps; Indicates at any given point The Dirac function takes a value of infinity at one position and a value of 0 at other positions; Indicates the first m Under the sample orbits, the numerical solution of the sampling model in the position component is... The value of step.

[0043] To illustrate the effectiveness of the method proposed in this invention, the following detailed description of the above technical solution is provided through a specific embodiment. The specific implementation steps are as follows: Step 1: Establish an underdamped RANZUAN system.

[0044] For machine learning problems such as Bayesian models, consider a typical sampling objective function determined by model features such as the posterior distribution of model parameters. , The corresponding potential function is a pure fourth-order potential well function with super-quadratic growth. For this function, the coefficient of friction is selected. Establish an underdamped Nielsen system:

[0045] Step 2: Construct the taming function.

[0046] Specific definition of polishing function ,when definition ,when Time definition ,when When, define .here,

[0047] Define the taming potential function:

[0048] At this point, the constant , , Correspondingly, the taming gradient function is defined. .

[0049] Step 3: Construct a docile sampling algorithm.

[0050] Based on the taming strategy proposed in step two, a tamed underdamped Langevin system is obtained, and a sampling algorithm is constructed. The following sections numerically verify the conformal symplectic structure, simulate numerical ergodicity, and calculate the long-term uniform convergence order of the proposed sampling algorithm.

[0051] Property 1: Conformal symplectic geometric structure.

[0052] Set dimensions Step length Let the initial value lie on the unit circle, satisfying For different times Plot the curves corresponding to the sampling algorithms respectively. Figure 1 The conformal symplectic geometry of the sampling algorithm proposed in this invention is shown, namely the geometric property of the phase space area decaying over time.

[0053] Property 2: Numerical ergodicity.

[0054] Set dimensions Step length Given 5 initial values, respectively The finish time is Select a test function. and adopt Monte Carlo simulations were performed on the sample paths. For each time step, the calculations were performed. Evolutionary behavior over time. Here, It is the current time step, when hour, and This indicates that the numerical solution corresponding to the proposed sampling algorithm is in the th order under different sample orbits. The value of step. Figure 2 This demonstrates the numerical ergodicity of the sampling algorithm proposed in this invention, that is, its evolutionary property that tends to stabilize over a long period of time.

[0055] Property 3: Long-term uniform convergence.

[0056] Set dimensions initial value , .use Monte Carlo simulation was performed on the sample paths, and a termination time was set. The step size is selected as ,in and use test functions Calculation error To test the convergence order of the proposed sampling algorithm. Here Indicates the numerical solution in The value of the position, Using the same numerical sampling algorithm at a very small step size The reference solution generated below is The value of the position. For example... Figure 3 As shown, the long-term uniform convergence order of the explicit sampling algorithm proposed in this invention is 2.

[0057] Step 4: Calculate the numerical sampling results.

[0058] Based on the docility potential function ,formula exact solution invariant distribution The corresponding marginal density in the Q direction is:

[0059] Set dimensions initial value , The step size is selected as .use 100 sample points, and take the termination time. The taming sampling model obtained from step three is as follows. , Indicates the first One sample point, here This represents the total time step. The resulting family of sampled numerical solutions... That is, it approximately follows the target distribution. The numerical sampling results.

[0060] Set dimensions initial value , The step size is selected as .use Monte Carlo simulations were performed on independent sample paths, and the termination time was recorded. Numerical density was calculated using the Monte Carlo method. ,here This represents the total time step. For example... Figure 4 As shown, after sufficient evolution, the numerical density generated by the sampling algorithm proposed in this invention is similar to... Figure 5 The distribution of the sampling targets shown is highly consistent, demonstrating the excellent simulation capability of the sampling algorithm proposed in this invention for the sampling targets.

[0061] Furthermore, select the dimension. initial value for A zero-dimensional vector with a step size of 1. End time and adopt Monte Carlo simulations are performed on independent sample paths. A test function is selected. and , Indicates the first The components at each dimension position, for each dimension Calculate the error respectively To test the order of the proposed sampling algorithm with respect to dimension. Here Indicates the numerical solution in The value of the position, Using the same numerical sampling algorithm at a very small step size The reference solution generated below is The value of the position. For example... Figure 6 As shown, the sampling algorithm proposed in this invention exhibits linear error growth with increasing dimensionality, demonstrating its ability to handle complex high-dimensional problems.

[0062] This invention is based on the MCMC sampling method for underdamped Langevin systems, which enables sampling of target distributions with superquadratic growing potential functions to simultaneously achieve: low-overhead fully explicit computation, strict preservation of the system's conformal symplectic geometry and numerical ergodicity, and second-order uniform convergence accuracy. This method overcomes the sampling distortion problem caused by the destruction of the system's intrinsic dynamic properties and geometry in existing sampling methods, improving sampling efficiency and accuracy, and providing new technical support for complex high-dimensional machine learning applications.

[0063] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A machine learning-efficient numerical sampling method based on an underdamped Langevin system, characterized in that, Includes the following steps: Step 1: For the target distribution, establish an underdamped Langevin system with a super quadratic growth potential function; Step 2: For the super quadratic growth potential function, construct the taming potential function and its corresponding taming gradient function; Step 3: Based on the tamed gradient function, construct a tamed sampling model for iteratively obtaining the sampled numerical solution; The expression for the tamed sampling model is: in, express The sampled numerical solution of the intermediate step in the n-step iterative process of the position component direction; express The sampled numerical solution of the position component direction after n iterations; express The sampled numerical solution for the momentum component direction iteratively in n+1 steps; express The sampled numerical solution of the momentum component direction iteratively over n steps; express The sampled numerical solution of the position component direction iteration n+1 steps; Indicates the coefficient of friction; Represents the natural base; These are standard Gaussian random variables; This represents the discrete time step.

2. The efficient numerical sampling method for machine learning based on an underdamped Langevin system according to claim 1, characterized in that, The expression for the taming potential function in step two is: in, Represents the docility potential function; Represents the original potential function; Indicates the taming radius; express ; Represents the polishing function; This indicates the positional independent variable.

3. The efficient numerical sampling method for machine learning based on an underdamped Langevin system according to claim 2, characterized in that, The polishing function is a smoothing function, whose function value is a constant one when the absolute value of the independent variable is less than or equal to one, and zero when the absolute value of the independent variable is greater than or equal to two.

4. The efficient numerical sampling method for machine learning based on an underdamped Langevin system according to claim 1, characterized in that, The expression for establishing the underdamped Langevin system in step one is as follows: in, It is a moment momentum components Location ; It is Brownian motion. exist The value at time; express d The real-valued space of dimension.

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