automatic-differentiation-advanced

name	automatic-differentiation-advanced
description	Extend Sounio's automatic differentiation capabilities to higher‑order derivatives, Hessians, differentiation through control flow, and GPU‑accelerated gradient computation, enabling state‑of‑the‑art scientific machine learning.

Advanced Automatic Differentiation

When to use this skill

Use this skill when you need to:

• Add support for higher‑order derivatives (second‑order, Hessians, Jacobians)
• Enable differentiation through loops, conditionals, and recursion
• Optimize gradient computation for GPU or distributed execution
• Integrate with existing neural‑network libraries (QNN, ONN, SNN)
• Implement novel AD techniques (checkpointing, source‑to‑source transformation)

When NOT to use this skill

• For simple forward‑mode AD that already works (use the existing autodiff module)
• For changes unrelated to differentiation (e.g., adding new math functions)
• For performance tuning of non‑AD code

Inputs required

• Specification of the AD extension (which order, which differentiation mode)
• Mathematical definitions of the new derivative operators
• Expected performance characteristics and memory usage
• Integration points with the existing AD infrastructure

Workflow

1. Understand the current AD implementation

   • Read stdlib/autodiff/ and self‑hosted/check/autodiff.sio (if present)
   • Review the AD test suite under tests/autodiff/
   • Examine the intermediate representation used for gradient computation

2. Design the extension

   • Decide whether to extend forward‑mode, reverse‑mode, or both
   • Determine how higher‑order derivatives will be represented (nested tangents, Hessian matrices)
   • Plan for control‑flow differentiation (e.g., taping loops, unrolling)
   • Consider memory/performance trade‑offs

3. Implement the core differentiation logic

   • Add new AD operators to the appropriate standard library modules
   • Extend the compiler's AD lowering passes if necessary
   • Ensure correctness for edge cases (discontinuities, non‑differentiable points)

4. Integrate with GPU and parallel execution

   • If targeting GPU, modify the GPU backend to support AD primitives
   • Add parallel gradient accumulation for distributed settings

5. Write tests

   • Add unit tests for each new differentiation operator
   • Include numerical correctness tests against finite‑difference approximations
   • Test differentiation through control‑flow constructs

6. Update documentation

   • Extend the AD user guide (docs/qnn/AUTO_DIFFERENTIATION_GUIDE.md)
   • Provide examples of higher‑order derivative usage
   • Note any limitations or performance considerations

Examples

Hessian computation via nested forward‑mode

// Function to differentiate
fn rosenbrock(x: f64, y: f64) -> f64 {
    (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x)
}

// Compute Hessian using nested forward‑mode AD
let (value, grad, hess) = diff2_forward(rosenbrock)(0.5, 1.5)
// grad is (df/dx, df/dy)
// hess is [[d²f/dx², d²f/dxdy], [d²f/dydx, d²f/dy²]]

Differentiation through a while loop

// Loop that accumulates a value
fn accumulate(n: i64) -> f64 {
    var i = 0
    var sum = 0.0
    while i < n {
        sum = sum + f(i)
        i = i + 1
    }
    return sum
}

// Gradient with respect to n (requires tape‑based reverse‑mode)
let dsum_dn = grad_reverse(accumulate)(10)

GPU‑accelerated Jacobian

// Vector‑valued function
fn vector_func(x: [f64; 3]) -> [f64; 2] {
    [x[0] * x[1], sin(x[2])]
}

// Jacobian computed on GPU
kernel fn jacobian_kernel(x: [f64; 3]) -> [[f64; 3]; 2] with GPU {
    // GPU‑friendly Jacobian computation
}

Troubleshooting / edge cases

• Expression blow‑up: higher‑order derivatives can produce huge intermediate expressions; consider simplification passes.
• Control‑flow non‑differentiability: loops with variable iteration counts may require special handling (e.g., unrolling up to a bound).
• Numerical stability: higher‑order finite‑difference validation may be noisy; use symbolic differentiation for verification where possible.
• GPU memory: reverse‑mode on GPU may need careful memory management for the tape.

Related files

• stdlib/autodiff/ – existing automatic differentiation modules
• self‑hosted/check/autodiff.sio – AD‑related type‑checking (if present)
• tests/autodiff/ – AD test suite
• docs/qnn/AUTO_DIFFERENTIATION_GUIDE.md – AD user documentation

Next steps

After implementing advanced AD, consider:

• Adding automatic batching (vmap) for vectorized gradient computation
• Integrating with optimisation libraries (gradient descent, Newton‑based solvers)
• Creating a differentiable programming tutorial that shows end‑to‑end scientific ML