Unitary Matrix Angle Parametrization for Parallel JVP Backpropagation
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Solution Overview
Problem
Existing methods for training neural networks using orthogonal matrices face challenges such as limited generalizability, high computational complexity, and lack of parallelization, particularly when dealing with large models.
Innovation Solution
The use of special orthogonal matrices (SO(n)) and Givens rotations to parametrize unitary matrices, allowing for parallel computation of unitary matrices and Jacobian-vector products (JVP) through parallel processing units, reducing computational complexity to O(n) and O(n log n) respectively.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If orthogonal matrices are used to encode neuron weights, then the vanishing gradient problem and exploding gradient problem are avoided, but the computational complexity increases and parallelization is limited
Solution Approach 1:
The orthogonal matrix is segmented into multiple Givens rotation blocks, each operating on specific coordinate pairs. This segmentation allows independent computation of each rotation block, enabling parallelization while maintaining the gradient stability benefits of orthogonal matrices.
Solution Approach 2:
The patent introduces a dynamic parametrization scheme where orthogonal matrices are represented through learnable Givens rotation angles. This dynamic approach replaces static weight matrices with parameterized rotations that can be efficiently optimized while preserving orthogonality constraints throughout training.
2Reliability
If orthogonal matrices are used to encode neuron weights, then gradient normalization is achieved, but generalizability and parallelization capability are reduced
Solution Approach 1:
The computational process is divided into independent Givens rotation operations on coordinate pairs, which can be executed in parallel. Each rotation block operates independently on specific matrix elements, enabling efficient parallelization across multiple processing units while maintaining gradient normalization properties.
Solution Approach 2:
The patent transforms the weight representation from arbitrary real values to parameterized Givens rotation angles. This parameter change enables both gradient normalization through orthogonal constraints and improved parallelization, as the rotation parameters can be independently optimized without violating orthogonality.
3Reliability
If traditional backpropagation is used with orthogonal matrices, then training is possible, but computational time is excessive for large models
Solution Approach 1:
The backpropagation computation is segmented into efficient Givens rotation operations that exploit the structured form of orthogonal matrices. By computing gradients through parameterized rotations rather than full matrix operations, the computational time is significantly reduced while maintaining training feasibility for large models.
Solution Approach 2:
The patent replaces traditional matrix multiplication and inversion operations with Givens rotation-based computations. This substitution leverages the geometric properties of rotations to achieve the same training objectives with reduced computational complexity and faster execution time.
Data Source
AI summary
A computing system, including a processor configured to train a machine learning model in a plurality of backpropagation iterations. Each backpropagation iteration may include generating a coordinate pair sequence. Each coordinate pair may be unique within the coordinate pair sequence and may include non-matching coordinates. The backpropagation iteration may further include receiving parametrizing angles respectively associated with the coordinate pairs. The backpropagation iteration may further include computing a unitary matrix parametrized by the parametrizing angles, computing a loss gradient matrix, and computing a Jacobian-vector product (JVP). Computing the JVP may include computing a rotated unitary matrix and a rotated loss gradient matrix for each coordinate pair. The JVP may be computed from the rotated unitary matrix and the rotated loss gradient matrix. The backpropagation iteration may further include updating the parametrizing angles based at least in part on the JVP.