Making Decisions with Cost Functions

by River Kanies

How do we make decisions?

To make decisions we:

• List options
• Weigh variables
• Score options (with weighted variables)

Example:

How to choose a sandwich

• List sandwiches to choose from
• Weigh variables such as taste + nutritional value
• Score sandwiches by summing weighted variables (you choose the sandwich with the best score)

Decision Matrix

Cost Function

C is the "cost" or score of the choice

Wa is weight for variable A

Wb is weight for variable B

What can we use cost functions for?

• Every day decisions (what link to click next)
• Design (best alternative)
• Machine Learning (linear regression + error minimization)
• Optimization of simulations

So, choosing from a small number of discrete options is simple...

But what if we have a continuous range of options?

(in multiple dimensions)

Typical Continuous Cost Function

Here the "optimal" choice for x is approximately 200

where C(x) is minimum

How do we calculate optimal value?

• Inspection (if approximate is good enough)
• Analytically: the derivative is 0 at min and max points
• Computational methods:
  • Newton's method
  • gradient descent

Optimization

Cost functions give us a method to choose the seemingly optimal choice in complex scenarios.

But assumptions must always be made. Models are never perfect.

Cost function vs. Model

• The model is the set of assumptions (variables and calculations) used to model the system
• The cost function is the method of evaluating the effectiveness of a particular set of input data (design decisions), using the model
• Optimize your model by minimizing cost function
• Cost function is commonly straight forward while model can be very complex

Simulation

• Develop cost function
• Track relevant data
• Model system
• Simulate stochastically
• Minimize cost function

It's a bit more complicated than before...

Specifically: Discrete Event Simulation (DES)

Simple medical office

• Wphys is the total physician wait time
• Wpat is the total patient wait time
• The 5 means physician time is 5 times as valuable as patient time

Model System

Simulate

• assume data distributions
• model:
  • stochastic
  • data --> wait times
• evaluate cost

Cost

FACTOR

(this represents scheduling time offset. Optimize this value to determine best scheduling times)

Simple Cost Function Plot

Linear regression and error minimization

(often, error = cost)

Multidimensional Cost Function

There are 2 independent variables in a simple line

above: theta values

left: b and a

Optimize Multidimensional Cost Functions with Gradient Descent

Uses partial derivatives

What if?!

Global vs. Local optimum

Guess and check

Good heuristics

Simulated annealing 

So essentially...

Any decision can be made with cost function optimization

Even super complex ones...

Classification, Neural Nets

• Forward Propagation:

Input --> Weighted Nodes --> Output

Each output is a "class"

• Cost Calculation

Hypothesis - Example

• Back Propagation

Adjust weights

(similar to gradient descent)

2 layer cost function

X

Y

Z

Practical Neural Nets

Ex:                       

Question --> Redirect link

Natural Language Classifier:

• Instantiate classifier with training set
• make api calls to use