Systems and methods for using deep machine learning to problems corresponding to non-linear generator functions
Systems, apparatuses, methods, and computer program products are disclosed for determining a value corresponding to a composite object. Paths are determined based on initial values of underlying items of the composite object and random numbers. A DNN is trained by determining a final value for each path based on a final set of items corresponding to the path at the final time, iterating the value backward in time using a non-linear generator function from the final value to an initial value, defining a set of initial values comprising the initial value determined for each path and determining a statistical measure based on the set of initial values, and modifying parameters of the DNN based on the statistical measure. Value information comprising the value for the composite object at one or more times is determined based on output of the DNN. The value information is provided such that a user computing device receives it.
1 . A method for determining a value corresponding to a strategy on a composite object comprising a set of items, the method comprising:
determining, by one or more processors, a plurality of paths for the composite object based on an initial set of items and Brownian motion paths, each path corresponding to a set of times comprising an initial time, a final time, and one or more intermediate times between the initial time and the final time, wherein the initial set of items corresponds to the set of items at the initial time;
causing, by the one or more processors, a deep neural network (DNN) of a backward DNN solver to be trained, wherein the DNN comprises a set of parameters, wherein the DNN corresponds to a strategy process for a particular time, wherein a set of inputs to the DNN corresponds to the initial set of items of the composite object at the particular time, wherein the composite object at a time immediately before the particular time is evaluated based on an output of the DNN for the particular time, until a convergence requirement is satisfied by:
for each path, determining a final value for the path based on a final set of items of the composite object corresponding to the path at the final time,
using a Taylor expansion approximation for a non-linear generator function, iterating backward in time from the final value corresponding to the final time to determine intermediate values each corresponding to one of the one or more intermediate times and to determine an initial value corresponding to the composite object at the initial time,
defining a set of initial values comprising the initial value determined for each path and determining one or more statistical measures of spread based on the set of initial values, and
modifying the set of parameters of the DNN based on the one or more statistical measures of spread;
after the convergence requirement is satisfied, determining, by the one or more processors, information corresponding to the composite object at one or more times of the set of times based on an output of the DNN, wherein the information comprises indication of a value corresponding to the composite object, wherein the information corresponding to the composite object is based on a single instance of the DNN; and
causing, by the one or more processors, at least a portion of the information to be provided such that a user computing device receives the at least a portion of the information and provides a representation of the at least a portion of the information via an interactive user interface provided via a display of the user computing device.
2 . The method of claim 1 , wherein the one or more statistical measures of spread comprises a loss function that considers a learned function for the value corresponding to the composite object corresponding to a time of the one or more times.
3 . The method of claim 2 , wherein the loss functions consider a learned function for the value corresponding to the composite object at least one intermediate time of the one or more intermediate times.
4 . The method of claim 1 , wherein the final value is iterated backward in time to determine the intermediate values and the initial value using an analytical solution for the non-linear generator function.
5 . The method of claim 1 , wherein a set of items corresponding to the composite object are associated with at least two different rates.
6 . The method of claim 1 , wherein the set of parameters of the DNN are modified using a stochastic gradient descent training model.
7 . The method of claim 1 , wherein the DNN comprises a plurality of sub-networks, each sub-network corresponding to a date.
8 . The method of claim 1 , wherein the DNN is a feedforward DNN.
9 . The method of claim 1 , wherein the output of the DNN comprises a function for the value at a time of the one or more times as a function of a set of items corresponding to the composite object.
10 . The method of claim 1 , wherein the DNN of the backward DNN solver is a sub-network of the backward DNN solver, wherein the backward DNN solver comprises a plurality of sub-networks, wherein each sub-network of the plurality of sub-networks corresponds to a respective time from the set of times.
11 . An apparatus for determining a value corresponding to a composite object, the apparatus comprising:
processor circuitry configured to:
determine a plurality of paths based on an initial set of items of the composite object and Brownian motion paths, each path corresponding to a set of times comprising an initial time, a final time, and one or more intermediate times between the initial time and the final time;
cause a deep neural network (DNN) of a backward DNN solver to be trained, wherein the DNN comprises a set of parameters, wherein the DNN corresponds to a strategy process for a particular time, wherein a set of inputs to the DNN corresponds to the initial set of items of the composite object at the particular time, wherein the composite object at a time immediately before the particular time is evaluated based on an output of the DNN, until a convergence requirement is satisfied by:
for each path, determining a final value for the path based on a final set of items corresponding to the path at the final time,
using a Taylor expansion approximation for a non-linear generator function, iterating backward in time from the final value corresponding to the final time to determine intermediate values each corresponding to one of the one or more intermediate times and to determine an initial value corresponding to the composite object at the initial time,
defining a set of initial values comprising the initial value determined for each path and determining one or more statistical measures of spread based on the set of initial values, and
modifying the set of parameters of the DNN based on the one or more statistical measures of spread;
after the convergence requirement is satisfied, determine information corresponding to the composite object at one or more times of the set of times based on an output of the DNN, wherein the information comprises indication of a value corresponding to the composite object; and
cause at least a portion of the value information to be provided such that a user computing device receives the at least a portion of the value information and provides a representation of the at least a portion of the value information via an interactive user interface provided via a display of the user computing device.
12 . The apparatus of claim 11 , wherein the one or more statistical measures of spread comprises a loss function that considers a learned function for the value corresponding to the composite object corresponding to a time of the one or more times.
13 . The apparatus of claim 12 , wherein the loss functions consider a learned function for the value for the composite object at least one intermediate time of the one or more intermediate times.
14 . The apparatus of claim 11 , wherein the final value is iterated backward in time to determine the intermediate values and the initial value using an analytical solution for the non-linear generator function.
15 . The apparatus of claim 11 , wherein a set of items corresponding to the composite object are associated with at least two different rates.
16 . The apparatus of claim 11 , wherein the set of parameters of the DNN are modified using a stochastic gradient descent training model.
17 . The apparatus of claim 11 , wherein the DNN comprises a plurality of sub-networks, each sub-network corresponding to a date.
18 . The apparatus of claim 11 , wherein the DNN is a feedforward DNN.