Add automatic differentiation support to sample_points() function (Proposal Type - Dev)

[ChinmayRout9040895625]

Summary

This PR exposes the automatic differentiation (autodiff) functionality of volesti to the R interface, allowing users to pass only function definitions without explicitly providing gradient implementations.

1. R Function Enhancements (R/sample_points.R)

• Added use_autodiff parameter (default: FALSE for backward compatibility)
• Added grad_function parameter for function/gradient specification
• Added parameter validation
• Routing logic to appropriate C++ backend

2. Rcpp Bindings (NEW FILES)

• src/sample_points_standard.cpp: Refactored existing implementation

  • Handles standard (manual gradient) sampling path

• src/sample_points_autodiff.cpp: NEW automatic differentiation path

  • Wraps user functions for autodiff types (autodiff::Real)
  • Integrates AutoDiffFunctor from volesti C++ library
  • Automatically computes gradients via autodiff library

3. Documentation (man/sample_points.Rd)

• Updated function signature documentation
• Added parameter descriptions for use_autodiff and grad_function
• Added comprehensive examples showing both approaches

4. Examples (examples/autodiff_example.R) NEW

• Demonstrates autodiff sampling with log-concave distribution
• Shows how to use new use_autodiff=TRUE parameter
• Compares autodiff vs manual gradient approaches

5. Tests (tests/testthat/test_autodiff.R) NEW

• Unit tests for autodiff functionality
• Validates autodiff produces valid samples
• Compares autodiff vs manual gradient (should be identical with same seed)
• Tests with different polytope types
• Parameter validation tests

User-Facing Changes

Before (still works):

# Users had to manually compute gradient
neg_log_prob <- function(x) sum(x^2) / 2
neg_grad <- function(x) -x

samples <- sample_points(P, grad_function = neg_grad, N=1000)

**After**

# Users can now just define the function
neg_log_prob <- function(x) sum(x^2) / 2

# Gradient computed automatically!
samples <- sample_points(P, use_autodiff=TRUE, 
                        grad_function=neg_log_prob, N=1000)