Analytic system based on multiple task learning with incomplete data

1. A non-transitory computer-readable medium having stored thereon computer-readable instructions that when executed by a computing device cause the computing device to:

for each of a plurality of related tasks,

compute an augmented observation matrix using an observation matrix and a predefined probability value that a value is missing in the observation matrix, wherein the observation matrix includes a plurality of observation vectors, wherein each observation vector includes a plurality of values, wherein each value of the plurality of values is associated with a variable to define a plurality of variables;

compute a plug-in autocovariance matrix using the computed augmented observation matrix and a noise value; and

compute a plug-in covariance vector using a target vector, the computed augmented observation matrix, and the noise value, wherein the target vector includes a target value associated with each of the plurality of observation vectors, the target value is an indicator of a characteristic of the associated observation vector;

compute a weight matrix used to predict the target value for each of the plurality of variables and each of the plurality of related tasks;

(a) compute a gradient descent matrix using the computed plug-in autocovariance matrix, the computed plug-in covariance vector, the computed weight matrix, and a predefined relationship matrix, wherein the predefined relationship matrix defines a relationship between the plurality of related tasks, wherein the gradient descent matrix is computed using {tilde over (W)}=Ŵ t-1 −η({tilde over (∇)}+λŴ t-1 RR T ), where Ŵ t-1 is the computed weight matrix, where η is a predefined step size, λ is a predefined sparsity penalization weight value, R is the predefined relationship matrix, and {tilde over (∇)}=Γ i Ŵ i t-1 −γ i for i=1, . . . , K, where K is a number of the of the plurality of related tasks, Γ i is the plug-in autocovariance matrix, and γ i is the plug-in covariance vector;

(b) compute an updated weight matrix using the computed gradient descent matrix;

repeat (a) and (b) with the computed updated weight matrix as the computed weight matrix until a convergence criterion is satisfied; and

when the convergence criterion is satisfied,

read a new observation vector from a scoring dataset;

compute a predicted target value for each task of the plurality of related tasks using the computed updated weight matrix and the read new observation vector; and

output the computed predicted target value for each task.

2. The non-transitory computer-readable medium of claim 1 , wherein the predefined probability is a predefined probability vector with the predefined probability defined for each task of the plurality of related tasks.

3. The non-transitory computer-readable medium of claim 1 , wherein the noise value is an independent random draw from a distribution function for each task of the plurality of related tasks.

4. The non-transitory computer-readable medium of claim 1 , wherein a number of the plurality of variables is equal for each task of the plurality of related tasks.

5. The non-transitory computer-readable medium of claim 1 , wherein a number of the plurality of observation vectors is different for at least one task of the plurality of related tasks.

6. The non-transitory computer-readable medium of claim 1 , wherein the computed predicted target value is a label.

7. The non-transitory computer-readable medium of claim 1 , wherein the predefined probability value is greater than zero and less than one.

8. The non-transitory computer-readable medium of claim 1 , wherein the augmented observation matrix is computed using

Z

i

=

1

1

-

p

i

⁢

X

i

,

where p i is the predefined probability value for a current task i of the plurality of related tasks, and X i is the observation matrix for the current task i.

9. The non-transitory computer-readable medium of claim 8 , wherein the plug-in autocovariance matrix is computed using

Γ

i

=

1

σ

i

⁢

Z

i

T

⁢

Z

i

-

p

i

σ

i

⁢

diag

⁡

(

Z

i

)

,

where σ i is the noise value for the current task i.

10. The non-transitory computer-readable medium of claim 9 , wherein the plug-in covariance vector is computed using

γ

i

=

1

σ

i

⁢

Z

i

T

⁢

y

i

,

where y i is the target vector for the current task i.

11. The non-transitory computer-readable medium of claim 1 , wherein each variable of the plurality of variables describes a feature used to predict the target value.

12. The non-transitory computer-readable medium of claim 1 , wherein the updated weight matrix is computed using a projected gradient descent method.

13. The non-transitory computer-readable medium of claim 1 , wherein the updated weight matrix is computed using an accelerated proximal gradient descent method.

14. The non-transitory computer-readable medium of claim 1 , wherein the convergence criterion is a number of iterations of repeating (a) and (b).

15. The non-transitory computer-readable medium of claim 1 , wherein the convergence criterion is computed based on a difference value computed between the updated weight matrix computed on a current iteration and a previous iteration of repeating (a) and (b).

16. The non-transitory computer-readable medium of claim 1 , wherein the predefined relationship matrix R describes an undirected graph, where each task is a node, and a pair of tasks are connected if the pair of tasks are related.

17. The non-transitory computer-readable medium of claim 16 , wherein the predefined relationship matrix R has dimension R K,H , where H is a number of edges that connect pairs of tasks, and K is a number of the plurality of related tasks.

18. The non-transitory computer-readable medium of claim 17 , wherein the predefined relationship matrix R is defined using

R

g

,

h

=

{

1

,

if

⁢

⁢

g

<

k

,

and

⁢

⁢

edge

⁢

⁢

h

⁢

⁢

connects

⁢

⁢

nodes

⁢

⁢

g

,

k

,

-

1

,

if

⁢

⁢

g

>

k

,

and

⁢

⁢

edge

⁢

⁢

h

⁢

⁢

connects

⁢

⁢

nodes

⁢

⁢

g

,

k

,

0

,

otherwise

,

where g=1, 2, . . . , K, k=1, 2, . . . , K and h=1, 2, . . . , N, where N is the number of edges.

19. The non-transitory computer-readable medium of claim 16 , wherein the predefined relationship matrix R is computed by computing a statistical distribution for each task, computing a distance between the computed statistical distributions of each pair of tasks, and, when the distance between the pair of tasks is smaller than a predefined threshold distance, an edge is defined to connect the pair of tasks.

20. A computing device comprising:

a processor; and

a non-transitory computer-readable medium operably coupled to the processor, the computer-readable medium having computer-readable instructions stored thereon that, when executed by the processor, cause the computing device to

for each of a plurality of related tasks,

compute an augmented observation matrix using an observation matrix and a predefined probability value that a value is missing in the observation matrix, wherein the observation matrix includes a plurality of observation vectors, wherein each observation vector includes a plurality of values, wherein each value of the plurality of values is associated with a variable to define a plurality of variables;

compute a plug-in autocovariance matrix using the computed augmented observation matrix and a noise value; and

compute a plug-in covariance vector using a target vector, the computed augmented observation matrix, and the noise value, wherein the target vector includes a target value associated with each of the plurality of observation vectors, the target value is an indicator of a characteristic of the associated observation vector;

compute a weight matrix used to predict the target value for each of the plurality of variables and each of the plurality of related tasks;

(a) compute a gradient descent matrix using the computed plug-in autocovariance matrix, the computed plug-in covariance vector, the computed weight matrix, and a predefined relationship matrix, wherein the predefined relationship matrix defines a relationship between the plurality of related tasks, wherein the gradient descent matrix is computed using {tilde over (W)}=Ŵ t-1 −η({tilde over (∇)}+λŴ t-1 RR T ), where Ŵ t-1 is the computed weight matrix, where η is a predefined step size, λ is a predefined sparsity penalization weight value, R is the predefined relationship matrix, and {tilde over (∇)}=Γ i Ŵ i t-1 −γ i for i=1, . . . , K, where K is a number of the of the plurality of related tasks, Γ i is the plug-in autocovariance matrix, and γ i is the plug-in covariance vector;

(b) compute an updated weight matrix using the computed gradient descent matrix;

repeat (a) and (b) with the computed updated weight matrix as the computed weight matrix until a convergence criterion is satisfied; and

when the convergence criterion is satisfied,

read a new observation vector from a scoring dataset;

compute a predicted target value for each task of the plurality of related tasks using the computed updated weight matrix and the read new observation vector; and

output the computed predicted target value for each task.

21. The computing device of claim 20 , wherein the updated weight matrix is computed using a projected gradient descent method.

22. The computing device of claim 20 , wherein the updated weight matrix is computed using an accelerated proximal gradient descent method.

23. The computing device of claim 20 , wherein the augmented observation matrix is computed using

Z

i

=

1

1

-

p

i

⁢

X

i

,

where p i is the predefined probability value for a current task i of the plurality of related tasks, and X i is the observation matrix for the current task i.

24. A method of computing an updated weight matrix used to compute a predicted value, the method comprising:

for each of a plurality of related tasks,

compute, by a computing device, an augmented observation matrix using an observation matrix and a predefined probability value that a value is missing in the observation matrix, wherein the observation matrix includes a plurality of observation vectors, wherein each observation vector includes a plurality of values, wherein each value of the plurality of values is associated with a variable to define a plurality of variables;

compute, by the computing device, a plug-in autocovariance matrix using the computed augmented observation matrix and a noise value; and

compute, by the computing device, a plug-in covariance vector using a target vector, the computed augmented observation matrix, and the noise value, wherein the target vector includes a target value associated with each of the plurality of observation vectors, the target value is an indicator of a characteristic of the associated observation vector;

compute, by the computing device, a weight matrix used to predict the target value for each of the plurality of variables and each of the plurality of related tasks;

(a) compute, by the computing device, a gradient descent matrix using the computed plug-in autocovariance matrix, the computed plug-in covariance vector, the computed weight matrix, and a predefined relationship matrix, wherein the predefined relationship matrix defines a relationship between the plurality of related tasks, wherein the gradient descent matrix is computed using {tilde over (W)}=Ŵ t-1 −η({tilde over (∇)}+λŴ t-1 RR T ), where Ŵ t-1 is the computed weight matrix, where η is a predefined step size, λ is a predefined sparsity penalization weight value, R is the predefined relationship matrix, and Γ i Ŵ i t-1 −γ i for i=1, . . . , K, where K is a number of the of the plurality of related tasks, Γ i is the plug-in autocovariance matrix, and γ i is the plug-in covariance vector;

(b) compute, by the computing device, an updated weight matrix using the computed gradient descent matrix;

repeat, by the computing device, (a) and (b) with the computed updated weight matrix as the computed weight matrix until a convergence criterion is satisfied; and

when the convergence criterion is satisfied,

read, by the computing device, a new observation vector from a scoring dataset;

compute, by the computing device, a predicted target value for each task of the plurality of related tasks using the computed updated weight matrix and the read new observation vector; and

output, by the computing device, the computed predicted target value for each task.

25. The method of claim 24 , wherein the computed predicted target value is a label.

26. The method of claim 24 , wherein the updated weight matrix is computed using a projected gradient descent method.

27. The method of claim 24 , wherein the updated weight matrix is computed using an accelerated proximal gradient descent method.

28. The method of claim 24 , wherein the augmented observation matrix is computed using

Z

i

=

1

1

-

p

i

⁢

X

i

,

where p i is the predefined probability value for a current task i of the plurality of related tasks, and X i is the observation matrix for the current task i.

29. The method of claim 28 , wherein the plug-in autocovariance matrix is computed using

Γ

i

=

1

σ

i

⁢

Z

i

T

⁢

Z

i

-

p

i

σ

i

⁢

diag

⁡

(

Z

i

)

,

where σ i is the noise value for the current task i.

30. The method of claim 29 , wherein the plug-in covariance vector is computed using

γ

i

=

1

σ

i

⁢

Z

i

T

⁢

y

i

,

where y i is the target vector for the current task i.