Techniques for learning causal graphs

A system for training a causal graph model for machine learning, the system comprising:

• a memory having computer-readable instructions stored thereon;
• and
• a processor that executes the computer-readable instructions to:
• compute an initial sum of squares and cross products (SSCP) matrix from variable values of a plurality of observation vectors, wherein each of the plurality of observation vectors comprises the variable values of a plurality of variables, and wherein a number of the plurality of variables in each of the plurality of observation vectors is n;
• set an initial index value of a first index to be zero;
• set an initial score value of a score to be zero;
• set an initial order list for the plurality of variables;
• and
• learn a topological order of a directed acyclic graph (DAG) by:
• (A) comparing the initial index value of the first index with n;
• (B) responsive to determining that the initial index value of the first index in (A) is less than n, executing (C) or responsive to determining that the initial index value of the first index in (A) is greater than or equal to n, executing (L);
• (C) responsive to determining that the initial index value of the first index in (A) is equal to zero, executing (E) through (K) or responsive to determining that the initial index value in (A) is greater than zero, executing (D) through (K);
• (D) sweeping the initial SSCP matrix based on the initial index value of the first index in (A) to obtain a swept SSCP matrix;
• (E) incrementing the initial index value of the first index in (A) by one to obtain an updated index value of the first index;
• (F) determining an index value of a second index based on the updated index value of the first index in (E), wherein the index value of the second index is determined from the initial SSCP matrix if the initial index value of the first index in (A) is equal to zero or the swept SSCP matrix if the initial index value in (A) is greater than zero;
• (G) computing an updated SSCP matrix based on the updated index value of the first index in (E) and the index value of the second index, wherein the updated SSCP matrix is determined from the initial SSCP matrix if the initial index value of the first index in (A) is equal to zero or the swept SSCP matrix if the initial index value in (A) is greater than zero;
• (H) computing an updated score value as a sum of the initial score value and a value identified from the updated SSCP matrix;
• (I) computing an updated order list from the initial order list based on the updated index value of the first index in (E) and the index value of the second index;
• (J) setting the updated index value of the first index in (E) as the initial index value of the first index in (A), the updated score value as the initial score value, the updated SSCP matrix as the initial SSCP matrix, and the updated order list as the initial order list;
• (K) repeating (A) through (J);
• and
• (L) outputting the updated score value from (H) and the updated order list from (I), wherein the updated order list corresponds to the topological order of the DAG and is stored for implementation in training a causal graph model for machine learning and using the causal graph model for subsequent causal estimation operations, and
• wherein executing by the processor one or more sequences of combinations of (A) through (L) causes an accelerated learning by the processor of the topological order of the DAG with time complexity proportional to n3 seconds using a single central processing unit.

A method for training a causal graph model for machine learning, the method comprising:

• computing, by a processor executing computer-readable instructions stored on a memory, an initial sum of squares and cross products (SSCP) matrix from variable values of a plurality of observation vectors, wherein each of the plurality of observation vectors comprises the variable values of a plurality of variables, and wherein a number of the plurality of variables in each of the plurality of observation vectors is n;
• setting, by the processor, an initial index value of a first index to be zero;
• setting, by the processor, an initial score value of a score to be zero;
• setting, by the processor, an initial order list for the plurality of variables;
• and
• learning, by the processor, a topological order of a directed acyclic graph (DAG) by:
• (A) comparing, by the processor, the initial index value of the first index with n;
• (B) responsive to determining that the initial index value of the first index in (A) is less than n, executing, by the processor, (C) or responsive to determining that the initial index value of the first index in (A) is greater than or equal to n, executing, by the processor, (L);
• (C) responsive to determining that the initial index value of the first index in (A) is equal to zero, executing, by the processor, (E) through (K) or responsive to determining that the initial index value in (A) is greater than zero, executing, by the processor, (D) through (K);
• (D) sweeping, by the processor, the initial SSCP matrix based on the initial index value of the first index in (A) to obtain a swept SSCP matrix;
• (E) incrementing, by the processor, the initial index value of the first index in (A) by one to obtain an updated index value of the first index;
• (F) determining, by the processor, an index value of a second index based on the updated index value of the first index in (E), wherein the index value of the second index is determined from the initial SSCP matrix if the initial index value of the first index in (A) is equal to zero or the swept SSCP matrix if the initial index value in (A) is greater than zero;
• (G) computing, by the processor, an updated SSCP matrix based on the updated index value of the first index in (E) and the index value of the second index, wherein the updated SSCP matrix is determined from the initial SSCP matrix if the initial index value of the first index in (A) is equal to zero or the swept SSCP matrix if the initial index value in (A) is greater than zero;
• (H) computing, by the processor, an updated score value as a sum of the initial score value and a value identified from the updated SSCP matrix;
• (I) computing, by the processor, an updated order list from the initial order list based on the updated index value of the first index in (E) and the index value of the second index;
• (J) setting, by the processor, the updated index value of the first index in (E) as the initial index value of the first index in (A), the updated score value as the initial score value, the updated SSCP matrix as the initial SSCP matrix, and the updated order list as the initial order list;
• (K) repeating, by the processor, (A) through (J);
• and
• (L) outputting, by the processor, the updated score value from (H) and the updated order list from (I), wherein the updated order list corresponds to the topological order of the DAG and is stored for implementation in training a causal graph model for machine learning and using the causal graph model for subsequent causal estimation operations, and
• wherein executing by the processor one or more sequences of combinations of (A) through (L) causes an accelerated learning by the processor of the topological order of the DAG with time complexity proportional to n3 seconds using a single central processing unit.