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

1. 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

1. 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)

1. Integrate with GPU and parallel execution

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

1. Write tests

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

1. 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