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Mathematics - Data Science - Computer Science

Overview
Calculus
Computing Systems
Data
Graphs
Linear Algebra
Models and Modeling
Organization
Probability
Programming Constructs
- Programming Constructs Overview
- Abstraction
- Array
- Attribute
- Best-first Search
- Binary Search
- Block
- Branch
- Callback
- Class
- Conditional
- Constructor
- Container/Collection
- Dynamic Array
- Encapsulation
- Exception
- Expression
- Function
- Garbage Collection
- Greedy Algorithms
- Hash
- HTTP Request
- Identifier
- Inheritance
- Inner Class
- Instance
- Iterator
- Keyword
- Lambda
- Libraries
- List
- Linked List
- Literal
- Metaclass
- Method
- Mixin
- Object
- Operator
- Overloading
- Overriding
- Package
- Parameter
- Polymorphism
- Primitive
- Programming Process
- Recursion
- Reflection
- Regular Expression
- Reserved Word
- Return
- Sort
- Statement
- Switch
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- Variable
Statistics
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Graph Backpropagation Data Flow

Backpropagation is used to modify the weights and biases applied to data flowing between neural network nodes. The backpropagation function B( ) repeatedly modifies graph weights and biases to reach a minimum value for the error value.

Below is an example flow:

$\\ \mathrm{\textup{ }} v _{1} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, v _{2} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, v _{n} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \bigoplus \overrightarrow{w_{1},b_{1}} \bigoplus \overrightarrow{w_{2},b_{2}} \cdots \bigoplus \overrightarrow{w_{n},b_{n}} \bigoplus \overrightarrow{\varepsilon } B(\varepsilon ,\bar{v},\bar{t},\bar{w},\bar{b}) \\ \mathrm{\textup{ }} \,\,\,\,\,\,\,\,\,\,\,\,\, \Lambda \mathrm{\textup{............ }} \Lambda \mathrm{\textup{................. }} \Lambda \mathrm{\textup{...............}} \vdots$

where:

$\\ \bigoplus \, \, \, \mathrm{\textup{node performing an activation function}} \\ \\ B\left ( \right ) \, \, \mathrm{\textup{backpropagation function}} \\ \\ \rightarrow \: \: \: \: \mathrm{\textup{forward data flow}} \\ \\ \Lambda \,\,\,\,\,\,\,\,\, \mathrm{\textup{update of weights and biases}} \\ \\ \cdots \,\,\,\,\,\, \mathrm{\textup{backward data flow}} \\ \\$

$\\ v_{i} \,\,\,\,\,\,\,\,\,\, \mathrm{\textup{value result of input or activation function}} \\ \\ w_{i} \,\,\,\,\,\,\,\,\, \mathrm{\textup{weight applied to the forward data flow}} \\ \\ b_{i} \,\,\,\,\,\,\,\,\,\,\, \mathrm{\textup{bias applied to the forward data flow}} \\ \\ \bar{v} \,\,\,\,\,\,\,\,\,\,\,\,\, \mathrm{\textup{set of result values}} \\ \\ \bar{t} \,\,\,\,\,\,\,\,\,\,\,\,\,\, \mathrm{\textup{set of training values}} \\ \\ \bar{w} \,\,\,\,\,\,\,\,\,\,\ \mathrm{\textup{set of weights}} \\ \\ \bar{b} \,\,\,\,\,\,\,\,\,\,\,\,\,\, \mathrm{\textup{set of biases}} \\ \\ \varepsilon \,\,\,\,\,\,\,\,\,\,\,\,\,\, \mathrm{\textup{error (loss) between t and }} v_{n}\\ \\$

References

Backpropagation
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