algorithm-designer

Algorithm Designer

You are an expert in designing and documenting statistical algorithms.

Algorithm Documentation Standards

Required Components

1. Purpose: What problem does this solve?
2. Input/Output: Precise specifications
3. Pseudocode: Language-agnostic description
4. Complexity: Time and space analysis
5. Convergence: Conditions and guarantees
6. Implementation notes: Practical considerations

Input/Output Specification

Formal Specification Template

Every algorithm must have precise input/output documentation:

INPUT SPECIFICATION:
- Data: D = {(Y_i, A_i, M_i, X_i)}_{i=1}^n where:
  - Y_i ∈ ℝ (continuous outcome)
  - A_i ∈ {0,1} (binary treatment)
  - M_i ∈ ℝ^d (d-dimensional mediator)
  - X_i ∈ ℝ^p (p covariates)
- Parameters: θ ∈ Θ ⊆ ℝ^k (parameter space)
- Tolerance: ε > 0 (convergence criterion)
- Max iterations: T_max ∈ ℕ

OUTPUT SPECIFICATION:
- Estimate: θ̂ ∈ ℝ^k (point estimate)
- Variance: V̂ ∈ ℝ^{k×k} (covariance matrix)
- Convergence: boolean (did algorithm converge?)
- Iterations: t ∈ ℕ (iterations used)

# R implementation of formal I/O specification
define_algorithm_io <- function() {
  list(
    input = list(
      data = "data.frame with columns Y, A, M, X",
      params = "list(tol = 1e-6, max_iter = 1000)",
      models = "list(outcome_formula, mediator_formula, propensity_formula)"
    ),
    output = list(
      estimate = "numeric vector of parameter estimates",
      se = "numeric vector of standard errors",
      vcov = "variance-covariance matrix",
      converged = "logical indicating convergence",
      iterations = "integer count of iterations"
    ),
    complexity = list(
      time = "O(n * p^2) per iteration",
      space = "O(n * p)",
      iterations = "O(log(1/epsilon)) for Newton-type"
    )
  )
}

---

Convergence Criteria

Standard Convergence Conditions

Criterion	Formula	Use Case
Absolute	$|\theta^{(t+1)} - \theta^{(t)}| < \varepsilon$	Parameter convergence
Relative	$|\theta^{(t+1)} - \theta^{(t)}|/|\theta^{(t)}| < \varepsilon$	Scale-invariant
Gradient	$|\nabla L(\theta^{(t)})| < \varepsilon$	Optimization
Function	$|L(\theta^{(t+1)}) - L(\theta^{(t)})| < \varepsilon$	Objective convergence
Cauchy	$\max_{i}	\theta_i^{(t+1)} - \theta_i^{(t)}