Ratio prediction using machine learning models employing strict convex loss functions and proximal point optimization

1 . A method comprising:

receiving, by a processor, a training data set, the training data set comprising data representing consumer interactions and actions taken by consumers after the consumer interactions;

executing, by the processor, a training run using a predictive model, the predictive model including a plurality of trainable parameters;

computing, by the processor, a loss of the training run using a loss function, the loss function comprising a strict convex function wherein the derivative of the loss function equals a ratio of summed outcomes to summed interactions at optimum;

optimizing, by the processor, the plurality of trainable parameters based on an output of the loss function using a proximal point method that iteratively updates parameters by minimizing a combination of the loss function and a regularization term penalizing parameter changes, wherein the proximal point method comprises updating the plurality of trainable parameters by iteratively solving a regularized subproblem at each iteration, the regularized subproblem including a proximity term that limits deviation of updated parameters from parameters of a previous iteration and a penalty term that limits magnitude of the updated parameters;

storing, by the processor, the trainable parameters as an inference model wherein the predicted ratio corresponds to an inverse of the derivative of the strict convex function; and

predicting, by the processor, a future ratio using the inference model.

2 . The method of claim 1 , wherein the loss function is selected from a family of functions within a function space defined by a difference between outcomes of actions taken after consumer interactions multiplied by a strict convex function of the predicted outcome and the predicted outcome of the consumer interactions.

3 . The method of claim 2 , wherein the outcomes of actions are computed by computing a total number of actions within a predetermined time period and where the consumer interactions are computed by summing a total number of consumer interactions in the predetermined time period.

4 . The method of claim 1 , wherein optimizing the plurality of trainable parameters based on an output of the loss function comprises minimizing a regularized version of the loss function after the training run.

5 . The method of claim 4 , wherein minimizing a regularized version of the loss function after the training run comprises utilizing a proximal point method.

6 . The method of claim 5 , wherein the plurality of trainable parameters are updated to next trainable parameters by finding a minimum value of a combination of the loss function, a first regularization term that represents a deviation between the plurality of trainable parameters and previous trainable parameters, and a second regularization term that evaluates a magnitude of the plurality of trainable parameters.

7 . The method of claim 1 , wherein the method further comprises updating a bid for a future user interaction using the future ratio.

8 . A non-transitory computer-readable storage medium for tangibly storing computer program instructions capable of being executed by a computer processor, the computer program instructions defining steps of:

receiving a training data set, the training data set comprising data representing consumer interactions and actions taken by consumers after the consumer interactions;

executing a training run using a predictive model, the predictive model including a plurality of trainable parameters;

computing a loss of the training run using a loss function, the loss function comprising a strict convex function wherein the derivative of the loss function equals a ratio of summed outcomes to summed interactions at optimum;

optimizing the plurality of trainable parameters based on an output of the loss function using a proximal point method that iteratively updates parameters by minimizing a combination of the loss function and a regularization term penalizing parameter changes, wherein the proximal point method comprises updating the plurality of trainable parameters by iteratively solving a regularized subproblem at each iteration, the regularized subproblem including a proximity term that limits deviation of updated parameters from parameters of a previous iteration and a penalty term that limits magnitude of the updated parameters;

storing the trainable parameters as an inference model wherein the predicted ratio corresponds to an inverse of the derivative of the strict convex function; and

predicting a future ratio using the inference model.

9 . The non-transitory computer-readable storage medium of claim 8 , wherein the loss function is selected from a family of functions within a function space defined by a difference between outcomes of actions taken after consumer interactions multiplied by a strict convex function of the predicted outcome and the predicted outcome of the consumer interactions.

10 . The non-transitory computer-readable storage medium of claim 9 , wherein the outcomes of actions are computed by computing a total number of actions within a predetermined time period and where the consumer interactions are computed by summing a total number of consumer interactions in the predetermined time period.

11 . The non-transitory computer-readable storage medium of claim 8 , wherein optimizing the plurality of trainable parameters based on an output of the loss function comprises minimizing a regularized version of the loss function after the training run.

12 . The non-transitory computer-readable storage medium of claim 11 , wherein minimizing a regularized version of the loss function after the training run comprises utilizing a proximal point method.

13 . The non-transitory computer-readable storage medium of claim 12 , wherein the plurality of trainable parameters are updated to next trainable parameters by finding a minimum value of a combination of the loss function, a first regularization term that represents a deviation between the plurality of trainable parameters and previous trainable parameters, and a second regularization term that evaluates a magnitude of the plurality of trainable parameters.

14 . The non-transitory computer-readable storage medium of claim 8 , wherein the steps further comprise updating a bid for a future user interaction using the future ratio.

15 . A device comprising:

a processor; and

a storage medium for tangibly storing thereon logic for execution by the processor, the logic comprising instructions for:

receiving a training data set, the training data set comprising data representing consumer interactions and actions taken by consumers after the consumer interactions,

executing a training run using a predictive model, the predictive model including a plurality of trainable parameters,

computing a loss of the training run using a loss function, the loss function comprising a strict convex function wherein the derivative of the loss function equals a ratio of summed outcomes to summed interactions at optimum,

optimizing the plurality of trainable parameters based on an output of the loss function using a proximal point method that iteratively updates parameters by minimizing a combination of the loss function and a regularization term penalizing parameter changes, wherein the proximal point method comprises updating the plurality of trainable parameters by iteratively solving a regularized subproblem at each iteration, the regularized subproblem including a proximity term that limits deviation of updated parameters from parameters of a previous iteration and a penalty term that limits magnitude of the updated parameters,

storing the trainable parameters as an inference model wherein the predicted ratio corresponds to an inverse of the derivative of the strict convex function, and

predicting a future ratio using the inference model.

16 . The device of claim 15 , wherein the loss function is selected from a family of functions within a function space defined by a difference between outcomes of actions taken after consumer interactions multiplied by a strict convex function of the predicted outcome and the predicted outcome of the consumer interactions.

17 . The device of claim 16 , wherein the outcomes of actions are computed by computing a total number of actions within a predetermined time period and where the consumer interactions are computed by summing a total number of consumer interactions in the predetermined time period.

18 . The device of claim 15 , wherein optimizing the plurality of trainable parameters based on an output of the loss function comprises minimizing a regularized version of the loss function after the training run.

19 . The device of claim 18 , wherein minimizing a regularized version of the loss function after the training run comprises utilizing a proximal point method.

20 . The device of claim 19 , wherein the plurality of trainable parameters are updated to next trainable parameters by finding a minimum value of a combination of the loss function, a first regularization term that represents a deviation between the plurality of trainable parameters and previous trainable parameters, and a second regularization term that evaluates a magnitude of the plurality of trainable parameters.