Gradient Descent Algorithm Near-Zero Learning Rate
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Solution Overview
Problem
Gradient descent algorithms are unable to support zero or near-zero learning rate values, leading to divide-by-zero errors and inefficient convergence to optimal solutions.
Innovation Solution
The method involves using a product value, based on the accumulated linear gradient value and learning rate, to update parameters, avoiding division by zero errors by incorporating a scaling value and predetermined fixed values in the parameter function.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Stability of the object's composition
If learning rate is set to zero or near-zero to avoid initial large adjustments, then parameter stability is improved, but the algorithm produces divide-by-zero errors and cannot execute
Solution Approach 1:
The patent extracts the learning rate from the denominator of the parameter update function. Instead of having the learning rate appear as a divisor (which causes divide-by-zero errors), the algorithm uses a product value formulation where the learning rate is multiplied with gradient values in the numerator. This extraction eliminates the harmful division operation while preserving the stabilizing effect of small learning rates on parameter updates.
Solution Approach 2:
The patent changes the mathematical formulation from a division-based update rule to a multiplication-based product value formulation. By transforming the parameter update function to use products rather than quotients, the algorithm can accept zero or near-zero learning rate values without producing errors, while still maintaining controlled parameter adjustments through the multiplicative scaling effect.
2Productivity
If learning rate starts at non-zero value to enable algorithm execution, then algorithm can run, but initial large adjustments occur before optimal path is discovered
Solution Approach 1:
By removing the learning rate from the denominator position, the algorithm can start with zero or near-zero learning rates without causing execution errors. The learning rate still performs its stabilizing function through multiplication in the numerator, but now the formulation allows the algorithm to execute immediately with small learning rate values rather than requiring non-zero initialization.
3Speed
If learning rate is increased to improve convergence speed, then convergence to optimal solution is faster, but parameter adjustments become too large and unstable
Solution Approach 1:
The patent transforms the parameter update mechanism to use a product value formulation where the learning rate multiplies gradient values rather than dividing them. This allows the learning rate to continue serving as a scaling factor that controls adjustment magnitude, enabling faster convergence when increased, while the multiplicative nature of the formulation maintains stability by directly scaling the gradient-based update steps.
Data Source
AI summary
A system and method for iteratively updating a parameter according to a gradient descent algorithm. In a given nth iteration of the method, one or more processors may determine a gradient value of a gradient vector of the parameter in a first dimension, determine a product value based at least in part on a sum of (i) the product value determined in an n−1th iteration and (ii) a product of the determined gradient value and a learning rate of the gradient descent algorithm, determine an updated parameter value according to a function including the product value, and update the parameter to equal the updated parameter value.