CN113836482A - Code distributed computing system - Google Patents

Code distributed computing system Download PDF

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CN113836482A
CN113836482A CN202110873853.6A CN202110873853A CN113836482A CN 113836482 A CN113836482 A CN 113836482A CN 202110873853 A CN202110873853 A CN 202110873853A CN 113836482 A CN113836482 A CN 113836482A
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matrix
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distributed computing
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CN113836482B (en
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童艳荔
代明军
王兰
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Shenzhen University
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization

Abstract

The invention provides a code distributed computing system, wherein a quasi-Newton polynomial code is applied to the code distributed computing system, in a computer cluster, one computer serves as a main node, other computers serve as working nodes, a matrix AB is calculated in the computer cluster in a distributed mode for multiplication, the main node divides the matrix A and the matrix B into K blocks, then a coding strategy is executed, the matrix A and the matrix B are coded respectively, and the AB can be decoded after the main node receives the calculation results of the K working nodes with the highest calculation speed. The invention has the beneficial effects that: the invention has lower condition number, effectively improves the problem of unstable Van der Monte matrix decoding value in matrix multiplication, improves the decoding accuracy, and has the functions of fault tolerance and backward node improvement.

Description

Code distributed computing system
Technical Field
The invention relates to the technical field of distributed systems, in particular to a coding distributed computing system.
Background
Distributed computing is the heart of modern machine learning and data analysis, where millions of very high-dimensional data points are being processed. The problem of dequeued nodes, which are working nodes that return computation results much slower than average because of their poor computing power or communication link, naturally exists in a distributed system. Much research in the field of "coding computing" has focused on integrating redundancy into distributed computing, addressing these challenges based on strategies of coding heuristics. The core idea of the distributed system is to obtain (n, k) Combination Property (CP) by coding, i.e. k original independent tasks are coded into n (n ≧ k), and among the n task results, any k can recover the original k calculation results.
The theoretical research directions of CDC mainly include three categories, including multiplication of coding matrix and vector, multiplication of coding matrix and matrix, and application of CDC in machine learning. Matrix multiplication (especially large matrix multiplication) is ubiquitous in the fields of data analysis (mining), machine learning, image processing, and the like. In the context of application to matrix multiplication, in polynomial-based methods, data is partitioned and each worker node stores a linear coded combination of these data partitions. The linear combination coefficients are selected according to the result of a suitable polynomial calculation, the node then calculates the coded packets of data, the master node aggregates the outputs of these calculations and restores the overall calculation by a decoding process of polynomial interpolation, such as Reed-Solomon codes, MatDot codes, etc.
Despite great success, one of the major challenges limiting the application of polynomial-based methods to practice is their numerical stability problem. The decoding process inevitably involves interpolation of polynomials, the main reason for instability of interpolation being that interpolation effectively solves for linear systems whose transforms are characterized by Vandermonde (Vandermonde) matrices. It is known that the condition number of the Vandermonde matrix with real-valued nodes grows exponentially in matrix size, and the large condition number means that small perturbations of the Vandermonde matrix due to numerical precision problems can cause singular matrices, thereby affecting the decoding result to be unreliable or even fail to decode.
Disclosure of Invention
The invention aims to solve the problem of unstable numerical value in the polynomial interpolation decoding process when the matrix is multiplied by the matrix. The numerical instability problem of polynomial interpolation occurs mainly because the interpolation basis used is a simple monomial. In matrix multiplication, the invention replaces the monomial basis of Van der Monte with Newton polynomial according to the advantages of the Newton polynomial basis, constructs a new Newton polynomial-like code by jointly designing codes, and obtains a code matrix instead of a Van der Monte matrix by using the Newton polynomial.
The invention provides a coding distributed computing system, wherein a Newton-like polynomial code is applied to the coding distributed computing system, in a computer cluster, one computer serves as a Master node (Master), other computers serve as working nodes, a matrix AB multiplication is calculated in a distributed mode in the computer cluster, the Master node (Master) divides a matrix A and a matrix B into K blocks, then a coding strategy is executed, and the matrix A and the matrix B are coded respectively to obtain the coding distributed computing system
Figure BDA0003189691360000021
And
Figure BDA0003189691360000022
master node handle
Figure BDA0003189691360000023
And
Figure BDA0003189691360000024
sending the data to N working nodes, wherein N is more than or equal to K, and the working nodes execute
Figure BDA0003189691360000025
Then at each working node pair
Figure BDA0003189691360000026
Interpolation is carried out on different real numbers, and the slave nodes can receive the calculation results of the K working nodes with the highest calculation speed
Figure BDA0003189691360000027
And (3) decoding the AB.
As a further improvement of the present invention, the Master node (Master) blocks the matrix a and the matrix B as follows:
Figure BDA0003189691360000028
the result to be calculated now becomes A0B0、A0B1、A1B0、A1B1。
As a further improvement of the invention, A is calculated by a plurality of working nodes0B0、A0B1、A1B0、A1B14 results of (1).
As a further improvement of the invention, the matrices A and B are jointly encoded using Newton polynomials and then multiplied by the code packet
Figure BDA0003189691360000029
Figure BDA00031896913600000210
Figure BDA00031896913600000211
As a further improvement of the invention, the master node sends the coding result to each working node, and the working nodes respectively calculate
Figure BDA00031896913600000212
Then, at a plurality of working nodes, respectively pair
Figure BDA00031896913600000213
Interpolating 5 different real numbers x0,x1,x2,x3,x4，
Figure BDA0003189691360000031
As a further improvement of the invention, the matrix is written in the form of a matrix
Figure BDA0003189691360000032
As a further improvement of the present invention,
Figure BDA0003189691360000033
after calculation at each working node, the left matrix is a 5 x 4 dimensional Newton-like polynomial interpolation matrix, each row represents one working node, 4 rows of results, namely 4 working nodes, are arbitrarily returned to form a 4 x 4 dimensional matrix, and the matrix is multiplied to the right matrix after inversion to decode
Figure BDA0003189691360000034
A laggard node or a faulty computer can be accommodated at this point.
As a further improvement of the invention, the number of the main nodes is one, and the number of the working nodes is five.
The invention has the beneficial effects that: the invention has lower condition number, effectively improves the problem of unstable Van der Monte matrix decoding value in matrix multiplication, improves the decoding accuracy, and has the functions of fault tolerance and backward node improvement.
Drawings
FIG. 1 is a diagram of a CDC execution matrix and matrix multiplication system model;
FIG. 2 is a graph of the condition number growth of Newton-like codes and polynomial codes under the same conditions when N is 2K;
FIG. 3 is a graph of Newton-like code and polynomial code condition number growth under the same conditions when N is 10K;
FIG. 4 is a graph of the increase of the relative error of Newton-like code and polynomial code under the same condition when N is 2K;
FIG. 5 is a graph of the increase of the relative error of Newton-like code and polynomial code under the same condition when N is 10K;
fig. 6 is a vandermonde matrix image.
Detailed Description
Aiming at the problems in the background art, the invention researches a coding scheme with numerical stability under a CDC framework and provides a novel Newton-like polynomial code, and in the multiplication of two-dimensional matrixes, two Newton-like polynomial codes are subjected toThe large matrix is divided into (A) horizontal and (B) vertical, the two matrixes are respectively coded and multiplied by a coding formula based on Newton polynomial joint design, and a brand new coefficient matrix is obtained after interpolation (for example, when the order is 8, the data of the Van der Monte matrix is x)8The data of the Newton-like polynomial matrix is (x-x)0)2(x-x1)2(x-x2)(x-x3)(x-x4)(x-x5) The new coefficient matrix constructed by the present invention still satisfies the (n, k) combination property. The quasi-Newton polynomial matrix is interpolated into a step matrix, which is obviously more sparse than a Van der Monte matrix in structure, and the quasi-Newton polynomial constructed by the invention has a lower condition number, can effectively improve the problem of unstable decoding values of the Van der Monte matrix in matrix multiplication, improves the decoding accuracy, and has the functions of fault tolerance and improving laggard nodes.
The following is a detailed description:
1. the mathematical model is as follows:
consider a computing system having a master node and N worker nodes as shown in fig. 1. The invention contemplates calculating C ═ AB, where the matrix
Figure BDA0003189691360000041
Matrix array
Figure BDA0003189691360000042
Figure BDA0003189691360000043
Representing the real number domain. The matrix A is divided laterally into m sub-matrices, including
Figure BDA0003189691360000044
To
Figure BDA0003189691360000045
The matrix B is divided longitudinally into q sub-matrices, including
Figure BDA0003189691360000046
To
Figure BDA0003189691360000047
Here we assume that H and L are divisible by m and q, respectively (otherwise we zero-fill a and B so that the number of rows of a and the number of columns of B are multiples of m and q, respectively). Under such blocking, the calculation result we want becomes
Figure BDA0003189691360000048
Calculating C is equivalent to calculating C simultaneously1To Cmq. From the multiplication with the matrix for the encoded distributed matrix, we set K to mq.
As explained before, directly mix C1To CmqIs allocated to K mq working nodes (working node i calculates Ci) The computation may suffer from a problem of a dequeued node. To remove the problem of the dequeued node, a CDC technique incorporating two dimensions, where the two dimensions respectively correspond to the pair A, is requirediCoded dimension sum pair BiThe dimension of the code. This process can be divided into three stages in total.
1) And (3) an encoding stage: by the pair A1To AmExecuting a certain coding strategy, obtaining N & gtK & ltmq coding packets, and marking as
Figure BDA0003189691360000051
To
Figure BDA0003189691360000052
In a similar manner, for B1To BqBy implementing a certain coding strategy, we can obtain a coded packet
Figure BDA0003189691360000053
To
Figure BDA0003189691360000054
2) A calculation stage: the master node will encode the pair
Figure BDA0003189691360000055
Sent to the working node
Figure BDA0003189691360000056
The working node performs multiplication
Figure BDA0003189691360000057
And returns the calculation result to the master node.
3) And a decoding stage: need to design the pair A togetheriAnd BiTo ensure (N, K) CP, i.e.
Figure BDA0003189691360000058
To
Figure BDA0003189691360000059
Any K calculation results in (1) are recorded as
Figure BDA00031896913600000510
C can be reconstructed.
2. Polynomial code:
examples are: when m is equal to 2, the compound is,
Figure BDA00031896913600000511
encoding a matrix
Figure BDA00031896913600000512
Figure BDA00031896913600000513
Figure BDA00031896913600000514
Are respectively paired
Figure BDA00031896913600000515
Interpolating 4 different real numbers x0,x1,x2,x3
Written in matrix form as
Figure BDA00031896913600000516
Inverting the left matrix can decode.
3. Newton interpolation polynomial:
the Newton interpolation method is an improvement of the Lagrange interpolation method, has 'attack-bearing property', overcomes the defect that the whole calculation work restarts when one node is added compared with the Lagrange interpolation polynomial, and can save the times of multiplication and division operation.
Suppose that in the interval [ a, b]In which n +1 real number interpolation points x exist0…xnNewton interpolation selected basis:
Figure BDA00031896913600000517
known to have properties from the substrate
Figure BDA00031896913600000518
By pij(x) Polynomial form found for linear combination of substrates
Figure BDA00031896913600000519
Wherein a isk(k ═ 0, 1, 2.., N) is the undetermined coefficient, which we denote as Nn(x) This form of interpolation polynomial is called a Newton interpolation polynomial.
Interpolating by substituting n +1 different real values
π0(x0)a0+π1(x0)a1+...+πn(x0)an＝Nn(x0)
π0(x1)a0+π1(x1)a1+...+πn(x1)an＝Nn(x1)
…
π0(xn)a0+π1(xn)a1+...+πn(xn)an＝Nn(xn)
Written in matrix form
Figure BDA0003189691360000061
The objective is to solve the undetermined coefficient a0～anFrom the above equation, the coefficient solving matrix for newton interpolation is a lower triangular matrix.
4. Newton-like polynomial code encoding scheme
We use the above mathematical framework to calculate C ═ AB, where H ═ L, and then m ═ q.
Interval [ a, b]In which N +1 real number points x exist0…xnThe matrixes A and B are respectively transversely and longitudinally cut into m blocks,
Figure BDA0003189691360000062
the two matrices are encoded as follows
Figure BDA0003189691360000063
Figure BDA0003189691360000064
Wherein
Figure BDA0003189691360000065
Figure BDA0003189691360000066
Figure BDA0003189691360000067
The master node will encode the pair
Figure BDA0003189691360000071
Sending to the working nodes, calculating after interpolation of each working node
Figure BDA0003189691360000072
Expansion type
Figure BDA0003189691360000073
For this formula, it is divided into m parts, denoted
Figure BDA0003189691360000074
i∈{1,2,…，m},
Figure BDA0003189691360000075
Wherein M is used1To represent
Figure BDA0003189691360000076
The first m terms of the interpolation coefficient term, c1To represent
Figure BDA0003189691360000077
The first m items of coefficients to be solved are processed the rest items in the same way,
Figure BDA0003189691360000078
can be written as
Figure BDA0003189691360000079
By x0…xnTo pair
Figure BDA00031896913600000710
Interpolation is carried out to obtain
Figure BDA00031896913600000711
Figure BDA00031896913600000712
…
Figure BDA00031896913600000713
Interpolating n +1 points x0…xnDivided into m +1 sets xj，j∈{1,2,…,m,m'}，x1～xmEach comprising m points, where x1Data point of (1) is x0…xm-1，xmComprises the elements of
Figure BDA00031896913600000714
The remaining n-m2+1 points are all placed at xm'In the specification, are
Figure BDA00031896913600000715
By MijRepresents MiAt xjFrom the result of interpolation, e.g. M11Represents M1At x1The interpolated matrix is processed and then processed,
Figure BDA0003189691360000081
in the same way, M21Represents M2At x1The interpolated matrix is obviously known as a 0 matrix and has a property Mij＝0,
Figure BDA0003189691360000082
By x2Point pair M in1The interpolated matrix is denoted as M12，
Figure BDA0003189691360000083
After all the interpolated results are processed in the above manner, they can be written in the form of a lower matrix
Figure BDA0003189691360000084
Similar to newton polynomials, the coding matrix can still be treated as a trapezoid, which we call newton-like interpolation polynomials.
For example, when m is 2
Figure BDA0003189691360000085
Encoding a matrix
Figure BDA0003189691360000086
Figure BDA0003189691360000087
Figure BDA0003189691360000088
Are respectively paired
Figure BDA0003189691360000089
Interpolating 4 different real numbers x0,x1,x2,x3
Working node computation
Figure BDA00031896913600000810
Figure BDA00031896913600000811
Figure BDA0003189691360000091
Figure BDA0003189691360000092
Written in matrix form as
Figure BDA0003189691360000093
Figure BDA0003189691360000094
After being calculated at each working node, the left matrix is subjected to inverse multiplication to the right matrix, and then decoding can be carried out.
The Newton interpolation matrix is a lower triangular matrix, and the Newton interpolation matrix is more sparse than the Van der Monte matrix from the structure, but the matrix constructed by the invention is a variant of the Newton interpolation matrix after interpolation and has the property of a Newton polynomial, and the Newton-like polynomial has lower condition number, can improve the decoding accuracy and effectively improve the problem of unstable decoding value of the Van der Monte matrix in matrix multiplication.
In summary, assuming we have a distributed system with one master node and 5 working nodes, we want to compute the multiplication of a large matrix AB, and first the master node blocks the two matrices, for example, assuming that they are both two, as follows
Figure BDA0003189691360000095
The result to be calculated now becomes A0B0，A0B1，A1B0，A1B1
We use 5 nodes to compute these 4 results
Jointly encoding the matrices A and B using Newton polynomials and multiplying by the encoded packets
Figure BDA0003189691360000096
Figure BDA0003189691360000097
Figure BDA0003189691360000098
The main node sends the coding result to each working node, and the working nodes calculate respectively
Figure BDA0003189691360000099
Then, at 5 working nodes, respectively for
Figure BDA00031896913600000910
Interpolating 5 different real numbers x0,x1,x2,x3,x4，
Figure BDA0003189691360000101
Figure BDA0003189691360000102
Figure BDA0003189691360000103
Figure BDA0003189691360000104
Figure BDA0003189691360000105
The extracted data is written into matrix form
Figure BDA0003189691360000106
Figure BDA0003189691360000107
After the calculation on each working node, it can be seen that the left matrix is a 5 x 4 dimensional newton-like polynomial interpolation matrix, each row represents one working node, the result of 4 rows, i.e. 4 working nodes, is arbitrarily returned to form a 4 x 4 dimensional matrix, and the matrix is inverted and multiplied to the right matrix to be decoded
Figure BDA0003189691360000108
A laggard node or a faulty computer can be accommodated at this point.
The beneficial effects of the present invention can be demonstrated by the following experimental data.
Interpolation point selection [0,3] random number
m Number of divided blocks 4 5 6 7 8 9
kRecovery threshold 16 25 36 49 64 81
Results iterate 30 times to average, the expression of two matrix condition numbers
When n is 2k, it is as shown in fig. 2. When the working node is far larger than the recovery threshold, n is 10k, as shown in fig. 3. The relative error of the two matrix results, when n is 2k, is shown in fig. 4; n-10 k as shown in fig. 5.
The application field of the invention is a coding distributed system, which is divided into a Coding Distributed Storage (CDS) system and a Coding Distributed Computing (CDC) system. Newton-like polynomial codes are applied in Coded Distributed Computing (CDC) systems. In practice, the encoding distributed computing system is widely applied in the fields of machine learning, edge calculation, image processing and the like. In the era of blowout-type growth of data, there are often millions of data points with very high dimensionality that need to be processed simultaneously, for example, machine-learned data sets are often so large, implementing and expanding distributed data processing across a large number of nodes presents lag, communication bottlenecks, and security issues. The invention constructs a new coding matrix multiplication technology with stable numerical value, has the functions of fault tolerance and backward node improvement, and experimental results show that the structure of the invention has obviously lower condition number and numerical value relative error and can improve the accuracy of decoding results.
The technical difficulty of the invention is that: one of the major challenges in the practice of encoding distributed computing in polynomial-based approaches is their numerical stability problem. Because the decoding process inevitably involves the interpolation of polynomials, the existing polynomial codes eventually evolve into the inversion of the vandermonde matrix after interpolation, the vandermonde matrix is very dense and has neither sparseness nor symmetry, the condition number of the vandermonde matrix grows exponentially along with the size of the matrix, and the large condition number influences the decoding result and even leads to decoding failure. An encoding scheme which has stable numerical value, ensures reliability and has a fault-tolerant function and can improve laggard nodes is urgently needed.
Vandermonde matrix:
Figure BDA0003189691360000111
as shown in fig. 6, in the interval [0, 1 ]]x19，x20The middle curves are almost coincident, as is the case with two columns of orders 19 and 20, and too large a linear correlation results in an irreversible matrix.
The invention designs a new coding mode, the matrix obtained after interpolation is not a Van der Waals matrix but a brand new matrix with the property of a Newton interpolation matrix, the matrix is a multi-zero sparse ladder type matrix, the density is not as dense as the Van der Waals matrix, and the data is not x19，x20Such a monomial (e.g. at an order of 8, the Van der Monte moment)Data of the array is x8The data of the Newton-like polynomial matrix is (x-x)0)2(x-x1)2(x-x2)(x-x3)(x-x4)(x-x5))，x0...x5The random real number points in a certain interval subtract different real numbers, effectively open the difference of high-order data and reduce the linear correlation. The invention has lower condition number, effectively improves the problem of unstable Van der Monte matrix decoding value in matrix multiplication, improves the decoding accuracy, and has the functions of fault tolerance and backward node improvement.
The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.

Claims (8)

1. A coding distributed computing system is characterized in that Newton-like polynomial codes are applied to the coding distributed computing system, in a computer cluster, one computer serves as a main node, other computers serve as working nodes, matrixes AB are calculated in the computer cluster in a distributed mode for multiplication, the main node divides the matrixes A and B into K blocks, then coding strategies are executed, and the matrixes A and B are coded respectively to obtain
Figure FDA0003189691350000011
And
Figure FDA0003189691350000012
master node handle
Figure FDA0003189691350000013
And
Figure FDA0003189691350000014
sending to N working nodes, N is more than or equal to K, workingNode execution
Figure FDA0003189691350000015
Then at each working node pair
Figure FDA0003189691350000016
Interpolation is carried out on different real numbers, and the slave nodes can receive the calculation results of the K working nodes with the highest calculation speed
Figure FDA0003189691350000017
And (3) decoding the AB.
2. The coded distributed computing system according to claim 1, wherein the Master node (Master) blocks matrix a and matrix B as follows:
Figure FDA0003189691350000018
the result to be calculated now becomes A0B0、A0B1、A1B0、A1B1。
3. The encoded distributed computing system of claim 2, wherein a is computed using a plurality of worker nodes0B0、A0B1、A1B0、A1B14 results of (1).
4. The encoded distributed computing system of claim 3, wherein matrices A and B are jointly encoded using Newton polynomials and then multiplied by encoded packets
Figure FDA0003189691350000019
Figure FDA00031896913500000110
Figure FDA00031896913500000111
5. The coded distributed computing system of claim 4, wherein the master node sends the coded results to each of the worker nodes, the worker nodes computing individually
Figure FDA00031896913500000112
Then, at a plurality of working nodes, respectively pair
Figure FDA00031896913500000113
Interpolating 5 different real numbers x0,x1,x2,x3,x4，
Figure FDA00031896913500000114
6. The encoded distributed computing system of claim 5, wherein the extracted and then written into matrix form is
Figure FDA0003189691350000021
7. The encoded distributed computing system of claim 6,
Figure FDA0003189691350000022
after calculation at each working node, the left matrix is a 5 x 4 dimensional Newton-like polynomial interpolation matrix, each row represents a working node, 4 rows, namely 4 working nodes, are arbitrarily returned to form a 4 x 4 dimensional matrix,after inversion, multiplication to the right matrix can be decoded
Figure FDA0003189691350000023
A laggard node or a faulty computer can be accommodated at this point.
8. The encoded distributed computing system of any of claims 1 to 7, wherein there is one master node and five worker nodes.
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