# Dual Number GPU Integration Plan for v0.9.3
## Overview
This document outlines the GPU acceleration strategy for amari-dual automatic differentiation operations, building on the existing WebGPU/wgpu infrastructure in amari-gpu.
## Current Dual Number Infrastructure Analysis
### Existing Capabilities
- ✅ Single-variable dual numbers (`DualNumber<T>`)
- ✅ Multi-variable dual numbers (`MultiDualNumber<T>`)
- ✅ Complete mathematical function library (sin, cos, exp, etc.)
- ✅ Machine learning activation functions (ReLU, sigmoid, tanh)
- ✅ Chain rule automatic differentiation
- ✅ Comprehensive WASM bindings
### Performance Bottlenecks
- ❌ Sequential gradient computation for large parameter sets
- ❌ CPU-only Jacobian matrix computation
- ❌ No batch processing for automatic differentiation
- ❌ Limited scalability for large neural networks
- ❌ No GPU acceleration for tensor operations
## GPU Acceleration Opportunities
### High Priority (Maximum Impact)
#### 1. Batch Automatic Differentiation
**Current State**: Sequential AD computation on CPU
**GPU Benefits**: Massive parallelization for parameter gradients
**Use Cases**:
- Neural network training with thousands/millions of parameters
- Batch gradient computation for optimization algorithms
- Monte Carlo gradient estimation
**Implementation Strategy**:
```rust
pub async fn batch_forward_reverse_ad(
&self,
functions: &[Function],
inputs: &[Vec<f64>],
variables: &[usize],
) -> Result<Vec<(f64, Vec<f64>)>, GpuError>
```
#### 2. Jacobian Matrix Computation
**Current State**: Sequential computation of partial derivatives
**GPU Benefits**: Parallel computation of all partial derivatives
**Use Cases**:
- Sensitivity analysis for large systems
- Newton-Raphson optimization methods
- Machine learning gradient analysis
**Implementation Strategy**:
```rust
pub async fn compute_jacobian_batch(
&self,
vector_functions: &[VectorFunction],
input_points: &[Vec<f64>],
) -> Result<Vec<Matrix<f64>>, GpuError>
```
#### 3. Neural Network Forward/Backward Pass
**Current State**: CPU-only layer-by-layer computation
**GPU Benefits**: Parallel layer computation with automatic gradients
**Use Cases**:
- Deep learning training acceleration
- Real-time neural network inference with gradients
- Neural architecture search
### Medium Priority (Significant Benefits)
#### 4. Hessian Matrix Computation
**Current State**: Not implemented (would require nested dual numbers)
**GPU Benefits**: Second-order optimization algorithms
**Use Cases**:
- Newton's method optimization
- Uncertainty quantification
- Curvature analysis
#### 5. Tensor Automatic Differentiation
**Current State**: Limited to scalar/vector operations
**GPU Benefits**: Full tensor AD for deep learning
**Use Cases**:
- Convolutional neural networks
- Tensor decomposition with gradients
- Advanced ML model training
#### 6. Stochastic Gradient Computation
**Current State**: Not implemented
**GPU Benefits**: Monte Carlo gradient estimation
**Use Cases**:
- Bayesian neural networks
- Stochastic optimization
- Uncertainty propagation
## Implementation Architecture
### 1. Core GPU Dual Number Infrastructure
```rust
/// GPU-accelerated automatic differentiation operations
pub struct GpuAutoDiff {
device: wgpu::Device,
queue: wgpu::Queue,
// Compute pipelines for different AD operations
forward_ad_pipeline: wgpu::ComputePipeline,
reverse_ad_pipeline: wgpu::ComputePipeline,
jacobian_pipeline: wgpu::ComputePipeline,
hessian_pipeline: wgpu::ComputePipeline,
neural_network_pipeline: wgpu::ComputePipeline,
}
impl GpuAutoDiff {
pub async fn new() -> Result<Self, GpuError>;
// Batch automatic differentiation
pub async fn batch_forward_ad(...) -> Result<Vec<(f64, Vec<f64>)>, GpuError>;
pub async fn batch_reverse_ad(...) -> Result<Vec<(f64, Vec<f64>)>, GpuError>;
// Matrix computations
pub async fn compute_jacobian_batch(...) -> Result<Vec<Matrix<f64>>, GpuError>;
pub async fn compute_hessian_batch(...) -> Result<Vec<Matrix<f64>>, GpuError>;
// Neural network operations
pub async fn neural_network_forward_backward(...) -> Result<NetworkGradients, GpuError>;
pub async fn batch_layer_gradients(...) -> Result<Vec<LayerGradients>, GpuError>;
// Optimization support
pub async fn stochastic_gradient_batch(...) -> Result<Vec<f64>, GpuError>;
pub async fn gradient_descent_step_batch(...) -> Result<Vec<Vec<f64>>, GpuError>;
}
```
### 2. GPU-Optimized Dual Number Representation
```rust
/// GPU-compatible dual number representation
#[repr(C)]
#[derive(Copy, Clone, Pod, Zeroable)]
struct GpuDualNumber {
value: f32,
gradient: f32,
}
/// GPU-compatible multi-dual number (fixed-size for GPU efficiency)
#[repr(C)]
#[derive(Copy, Clone, Pod, Zeroable)]
struct GpuMultiDual<const N: usize> {
value: f32,
gradient: [f32; N],
}
/// Batch operations for efficient GPU computation
#[repr(C)]
#[derive(Copy, Clone, Pod, Zeroable)]
struct GpuDualBatch {
batch_size: u32,
parameter_count: u32,
function_count: u32,
_padding: u32,
}
```
### 3. WGSL Compute Shaders for Automatic Differentiation
#### Forward Mode Automatic Differentiation
```wgsl
// Forward mode AD for batch computation
struct DualNumber {
value: f32,
gradient: f32,
}
@group(0) @binding(0)
var<storage, read> input_duals: array<DualNumber>;
@group(0) @binding(1)
var<storage, read> function_codes: array<u32>; // Encoded function operations
@group(0) @binding(2)
var<storage, read_write> output_duals: array<DualNumber>;
// Dual number arithmetic operations
fn dual_add(a: DualNumber, b: DualNumber) -> DualNumber {
return DualNumber(a.value + b.value, a.gradient + b.gradient);
}
fn dual_mul(a: DualNumber, b: DualNumber) -> DualNumber {
return DualNumber(
a.value * b.value,
a.gradient * b.value + a.value * b.gradient
);
}
fn dual_sin(a: DualNumber) -> DualNumber {
return DualNumber(sin(a.value), a.gradient * cos(a.value));
}
fn dual_exp(a: DualNumber) -> DualNumber {
let exp_val = exp(a.value);
return DualNumber(exp_val, a.gradient * exp_val);
}
@compute @workgroup_size(64)
fn forward_ad_batch(@builtin(global_invocation_id) global_id: vec3<u32>) {
let idx = global_id.x;
if (idx >= arrayLength(&input_duals)) {
return;
}
var current = input_duals[idx];
// Execute function operations encoded in function_codes
for (var i = 0u; i < arrayLength(&function_codes); i++) {
let op_code = function_codes[i];
switch op_code {
case 1u: { // Addition
current = dual_add(current, input_duals[idx + 1u]);
}
case 2u: { // Multiplication
current = dual_mul(current, input_duals[idx + 1u]);
}
case 3u: { // Sine
current = dual_sin(current);
}
case 4u: { // Exponential
current = dual_exp(current);
}
default: {
break;
}
}
}
output_duals[idx] = current;
}
```
#### Jacobian Matrix Computation
```wgsl
@group(0) @binding(0)
var<storage, read> input_batch: array<f32>; // Flattened input vectors
@group(0) @binding(1)
var<storage, read> function_batch: array<u32>; // Function definitions
@group(0) @binding(2)
var<storage, read_write> jacobian_batch: array<f32>; // Flattened Jacobian matrices
@compute @workgroup_size(16, 16)
fn compute_jacobian(@builtin(global_invocation_id) global_id: vec3<u32>) {
let batch_idx = global_id.z;
let output_idx = global_id.y;
let input_idx = global_id.x;
if (batch_idx >= batch_size || output_idx >= num_outputs || input_idx >= num_inputs) {
return;
}
// Create dual number with unit gradient for current input variable
var dual_input = DualNumber(
input_batch[batch_idx * num_inputs + input_idx],
select(0.0, 1.0, input_idx == global_id.x)
);
// Execute function to compute partial derivative
let result = evaluate_function(dual_input, function_batch, output_idx);
// Store partial derivative in Jacobian matrix
let jacobian_offset = batch_idx * num_outputs * num_inputs;
jacobian_batch[jacobian_offset + output_idx * num_inputs + input_idx] = result.gradient;
}
```
#### Neural Network Forward/Backward Pass
```wgsl
struct LayerParams {
weights_offset: u32,
bias_offset: u32,
input_size: u32,
output_size: u32,
activation: u32, // 0=linear, 1=relu, 2=sigmoid, 3=tanh
}
@group(0) @binding(0)
var<storage, read> network_params: array<LayerParams>;
@group(0) @binding(1)
var<storage, read> weights_biases: array<f32>;
@group(0) @binding(2)
var<storage, read> input_batch: array<f32>;
@group(0) @binding(3)
var<storage, read_write> activations: array<f32>;
@group(0) @binding(4)
var<storage, read_write> gradients: array<f32>;
fn activate_dual(x: DualNumber, activation: u32) -> DualNumber {
switch activation {
case 1u: { // ReLU
if (x.value > 0.0) {
return x;
} else {
return DualNumber(0.0, 0.0);
}
}
case 2u: { // Sigmoid
let sig = 1.0 / (1.0 + exp(-x.value));
return DualNumber(sig, x.gradient * sig * (1.0 - sig));
}
case 3u: { // Tanh
let tanh_val = tanh(x.value);
return DualNumber(tanh_val, x.gradient * (1.0 - tanh_val * tanh_val));
}
default: { // Linear
return x;
}
}
}
@compute @workgroup_size(64)
fn neural_network_forward_backward(@builtin(global_invocation_id) global_id: vec3<u32>) {
let batch_idx = global_id.x;
let neuron_idx = global_id.y;
let layer_idx = global_id.z;
if (batch_idx >= batch_size || layer_idx >= num_layers) {
return;
}
let layer = network_params[layer_idx];
if (neuron_idx >= layer.output_size) {
return;
}
// Compute weighted sum with automatic differentiation
var weighted_sum = DualNumber(
weights_biases[layer.bias_offset + neuron_idx],
0.0
);
for (var i = 0u; i < layer.input_size; i++) {
let input_val = if (layer_idx == 0u) {
input_batch[batch_idx * layer.input_size + i]
} else {
activations[(layer_idx - 1u) * batch_size * max_layer_size + batch_idx * max_layer_size + i]
};
let weight = weights_biases[layer.weights_offset + neuron_idx * layer.input_size + i];
let input_dual = DualNumber(input_val, select(0.0, 1.0, i == target_input_idx));
let weight_dual = DualNumber(weight, select(0.0, 1.0, is_weight_target));
weighted_sum = dual_add(weighted_sum, dual_mul(input_dual, weight_dual));
}
// Apply activation function
let activated = activate_dual(weighted_sum, layer.activation);
// Store forward pass result
activations[layer_idx * batch_size * max_layer_size + batch_idx * max_layer_size + neuron_idx] = activated.value;
// Store gradient for backward pass
gradients[layer_idx * batch_size * max_layer_size + batch_idx * max_layer_size + neuron_idx] = activated.gradient;
}
```
### 4. Adaptive Dispatch Strategy
```rust
impl GpuAutoDiff {
/// Determine optimal computation strategy based on problem size
pub fn should_use_gpu_for_ad(operation: ADOperation, problem_size: ADProblemSize) -> bool {
match operation {
ADOperation::ForwardAD => {
problem_size.num_variables >= 100 && problem_size.batch_size >= 32
}
ADOperation::ReverseAD => {
problem_size.num_outputs <= 10 && problem_size.num_variables >= 1000
}
ADOperation::JacobianComputation => {
problem_size.num_variables * problem_size.num_outputs >= 10000
}
ADOperation::NeuralNetworkTraining => {
problem_size.total_parameters >= 1000 && problem_size.batch_size >= 16
}
}
}
}
#[derive(Debug, Clone)]
pub struct ADProblemSize {
pub num_variables: usize,
pub num_outputs: usize,
pub batch_size: usize,
pub total_parameters: usize,
pub function_complexity: usize,
}
```
### 5. Integration with Machine Learning Frameworks
```rust
/// High-level ML integration for GPU-accelerated automatic differentiation
pub struct GpuNeuralNetwork {
gpu_autodiff: GpuAutoDiff,
layers: Vec<LayerDefinition>,
parameters: Vec<f32>,
}
impl GpuNeuralNetwork {
pub async fn forward_with_gradients(
&self,
inputs: &[Vec<f32>],
target_parameters: &[usize],
) -> Result<(Vec<f32>, Vec<f32>), GpuError>;
pub async fn backward_pass(
&self,
loss_gradients: &[f32],
) -> Result<Vec<f32>, GpuError>;
pub async fn gradient_descent_step(
&mut self,
learning_rate: f32,
gradients: &[f32],
) -> Result<(), GpuError>;
}
```
## Performance Targets
### Benchmarking Goals
- **Batch Forward AD**: 10-100x speedup for >1000 variables
- **Jacobian Computation**: 20-200x speedup for large matrices (>100x100)
- **Neural Network Training**: 5-50x speedup for networks >1000 parameters
- **Memory Efficiency**: <3x GPU memory overhead vs minimal CPU implementation
### Scalability Targets
- Support for up to 1M parameters in batch AD
- Jacobian matrices up to 10,000 x 10,000
- Neural networks with >100 layers and >1M parameters
- Batch sizes up to 1024 for training
## Implementation Timeline
### Phase 1: Core Infrastructure (Week 1)
- [ ] Extend amari-gpu with automatic differentiation support
- [ ] Implement `GpuAutoDiff` base structure
- [ ] Create batch forward-mode AD GPU pipeline
- [ ] Add adaptive dispatch for AD operations
### Phase 2: Jacobian Acceleration (Week 2)
- [ ] Implement GPU Jacobian matrix computation
- [ ] Add batch processing for multiple Jacobian computations
- [ ] Optimize memory usage for large matrices
- [ ] Integration with existing amari-dual types
### Phase 3: Neural Network Integration (Week 3)
- [ ] GPU neural network forward/backward pass
- [ ] Batch gradient computation for multiple networks
- [ ] Layer-wise gradient accumulation and optimization
- [ ] Integration with popular ML patterns
### Phase 4: Advanced Operations (Week 4)
- [ ] Hessian matrix computation
- [ ] Second-order optimization algorithms
- [ ] Stochastic gradient computation
- [ ] Comprehensive benchmarking and optimization
## Testing Strategy
### Correctness Verification
- Gradient checking against finite differences
- Cross-validation with existing CPU AD implementations
- Mathematical property verification (chain rule, linearity)
- Precision analysis for f32 vs f64 computations
### Performance Validation
- Scaling benchmarks across different problem sizes
- Memory usage profiling and optimization
- Comparative analysis vs leading AD frameworks
- Real-world machine learning workload testing
### Robustness Testing
- Edge case handling (zero gradients, numerical instabilities)
- Large-scale stress testing (millions of parameters)
- Memory pressure testing
- Cross-platform GPU compatibility
## Risk Mitigation
### Technical Challenges
1. **Numerical Precision**: Implement mixed-precision strategies and validation
2. **GPU Memory Constraints**: Design streaming and chunked computation algorithms
3. **Complex Function Support**: Modular shader design for extensible operations
4. **Debuggability**: Comprehensive logging and gradient verification tools
### Performance Risks
1. **Small Problem Overhead**: Careful threshold tuning for GPU dispatch
2. **Memory Bandwidth**: Optimize data layout and minimize transfers
3. **Divergent Execution**: Design GPU-friendly control flow patterns
## Success Metrics
### Quantitative Goals
- [ ] 10x+ speedup for large-scale automatic differentiation
- [ ] 20x+ speedup for Jacobian matrix computation
- [ ] 5x+ speedup for neural network training
- [ ] <20% GPU memory overhead for typical workloads
### Integration Goals
- [ ] Seamless integration with existing amari-dual API
- [ ] Backward compatibility with all existing AD operations
- [ ] Clear performance characteristics and usage guidelines
- [ ] Comprehensive error handling and fallback mechanisms
## Future Extensions
### Advanced Automatic Differentiation
- Higher-order derivatives (Hessians, tensor derivatives)
- Sparse automatic differentiation for structured problems
- Checkpointing strategies for memory-efficient reverse-mode AD
### Distributed Computation
- Multi-GPU automatic differentiation
- Distributed gradient computation across nodes
- Integration with distributed training frameworks
### Specialized Applications
- GPU-accelerated Bayesian neural networks
- Automatic differentiation for scientific computing
- Real-time optimization with automatic gradients
---
This plan provides a comprehensive roadmap for integrating automatic differentiation operations with GPU acceleration, enabling high-performance machine learning and optimization applications while maintaining the mathematical rigor and correctness of the amari-dual system.