The amortized optimization framework can learn a mapping from market condition parameters to optimal portfolio allocations, replacing expensive convex optimization at inference time.

A neural network trained via amortized optimization can learn a parametric mapping f_θ: Ω → Δ^n (where Ω is the space of market condition parameters and Δ^n is the n-asset simplex) such that for any market condition ω ∈ Ω, f_θ(ω) approximates the optimal Markowitz/CVaR portfolio allocation w*(ω) within ε = 1% tracking error (measured by out-of-sample Sharpe ratio degradation), while achieving ≥100× inference speedup over a cold-start convex solver (e.g., CVXPY with OSQP backend) on identical hardware.

Minimum Viable Test (MVT): Train an amortized optimization network on synthetic market condition parameters drawn from a calibrated multivariate distribution, using exact CVXPY solutions as supervision targets. Evaluate on held-out synthetic data and two real historical periods (2008 financial crisis, 2020 COVID crash) using n=50, 100, 200 assets. Compare Sharpe ratio, turnover, constraint satisfaction, and wall-clock inference time against CVXPY+OSQP baseline.

Full Validation: Extend to n=500 assets, 5 market regimes, rolling walk-forward backtest from 2000–2024, ablation over network architectures (MLP, attention-based, hypernetwork), and comparison against warm-started solvers and differentiable optimization layers (e.g., OptNet, CVXPY layers).

# AMORTIZED PORTFOLIO OPTIMIZATION - IMPLEMENTATION SKETCH
# Architecture: KKT-Penalized Amortized Network (KPAN)

import torch
import torch.nn as nn
import cvxpy as cp
import numpy as np
from torch.utils.data import DataLoader, TensorDataset

# ============================================================
# 1. DATA GENERATION
# ============================================================
def generate_market_params(n_assets, n_samples, seed=42):
    """Generate synthetic market condition parameters."""
    np.random.seed(seed)
    params, solutions, solve_times = [], [], []
    
    for _ in range(n_samples):
        # Sample expected returns from N(0.08, 0.15^2) annualized
        mu = np.random.normal(0.08, 0.15, n_assets)
        
        # Sample covariance via Wishart distribution
        A = np.random.randn(n_assets, n_assets + 10) * 0.1
        Sigma = A @ A.T / (n_assets + 10) + np.eye(n_assets) * 0.01
        
        # Risk aversion parameter
        lam = np.random.uniform(0.5, 5.0)
        
        # Solve with CVXPY
        w = cp.Variable(n_assets)
        obj = cp.Minimize(lam * cp.quad_form(w, Sigma) - mu @ w)
        constraints = [cp.sum(w) == 1, w >= 0]  # long-only
        prob = cp.Problem(obj, constraints)
        
        import time
        t0 = time.time()
        prob.solve(solver=cp.OSQP, warm_start=True)
        solve_times.append(time.time() - t0)
        
        if prob.status == 'optimal':
            # Encode omega as flattened feature vector
            omega = np.concatenate([mu, Sigma.flatten(), [lam]])
            params.append(omega)
            solutions.append(w.value)
    
    return np.array(params), np.array(solutions), np.array(solve_times)

# ============================================================
# 2. NETWORK ARCHITECTURE
# ============================================================
class AmortizedPortfolioNet(nn.Module):
    def __init__(self, n_assets, hidden_dim=512, n_layers=4):
        super().__init__()
        # Input: mu (n) + Sigma (n^2) + lambda (1)
        input_dim = n_assets + n_assets**2 + 1
        
        layers = [nn.Linear(input_dim, hidden_dim), nn.LayerNorm(hidden_dim), nn.GELU()]
        for _ in range(n_layers - 2):
            layers += [nn.Linear(hidden_dim, hidden_dim), nn.LayerNorm(hidden_dim), nn.GELU(), nn.Dropout(0.1)]
        layers += [nn.Linear(hidden_dim, n_assets)]
        
        self.network = nn.Sequential(*layers)
        self.n_assets = n_assets
    
    def forward(self, omega):
        logits = self.network(omega)
        # Softmax ensures sum-to-1 and non-negativity (long-only)
        weights = torch.softmax(logits, dim=-1)
        return weights

# ============================================================
# 3. LOSS FUNCTION WITH KKT PENALTY
# ============================================================
class KKTPenalizedLoss(nn.Module):
    def __init__(self, alpha_kkt=0.1, alpha_obj=1.0):
        super().__init__()
        self.alpha_kkt = alpha_kkt  # KKT violation penalty weight
        self.alpha_obj = alpha_obj  # Objective gap penalty weight
    
    def forward(self, w_pred, w_star, mu, Sigma, lam):
        # Supervised loss: L2 distance to optimal solution
        supervised_loss = torch.mean(torch.sum((w_pred - w_star)**2, dim=-1))
        
        # KKT violation: sum-to-one constraint (already enforced by softmax)
        sum_violation = torch.mean((torch.sum(w_pred, dim=-1) - 1.0)**2)
        
        # Objective gap: portfolio variance difference
        # w^T Sigma w for predicted vs optimal
        obj_pred = torch.einsum('bi,bij,bj->b', w_pred, Sigma, w_pred)
        obj_star = torch.einsum('bi,bij,bj->b', w_star, Sigma, w_star)
        obj_gap = torch.mean(torch.relu(obj_pred - obj_star))  # penalize only if worse
        
        total_loss = supervised_loss + self.alpha_kkt * sum_violation + self.alpha_obj * obj_gap
        
        return total_loss, {
            'supervised': supervised_loss.item(),
            'kkt_violation': sum_violation.item(),
            'obj_gap': obj_gap.item()
        }

# ============================================================
# 4. TRAINING LOOP
# ============================================================
def train_amortized_model(model, train_loader, val_loader, n_epochs=100, lr=1e-3):
    optimizer = torch.optim.Adam(model.parameters(), lr=lr)
    scheduler = torch.optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=n_epochs)
    loss_fn = KKTPenalizedLoss()
    
    best_val_loss = float('inf')
    patience_counter = 0
    
    for epoch in range(n_epochs):
        model.train()
        for omega, w_star, mu, Sigma, lam in train_loader:
            optimizer.zero_grad()
            w_pred = model(omega)
            loss, metrics = loss_fn(w_pred, w_star, mu, Sigma, lam)
            loss.backward()
            torch.nn.utils.clip_grad_norm_(model.parameters(), max_norm=1.0)
            optimizer.step()
        
        # Validation
        val_loss = evaluate(model, val_loader, loss_fn)
        scheduler.step()
        
        # Early stopping
        if val_loss < best_val_loss * 0.99:
            best_val_loss = val_loss
            patience_counter = 0
            torch.save(model.state_dict(), 'best_model.pt')
        else:
            patience_counter += 1
        
        # ABORT CHECKPOINT: no improvement for 10 epochs
        if patience_counter >= 10:
            print(f"Early stopping at epoch {epoch}")
            break
    
    return model

# ============================================================
# 5. EVALUATION & BENCHMARKING
# ============================================================
def benchmark_inference_speed(model, test_params, n_trials=1000):
    import time
    model.eval()
    
    # Amortized model speed
    t0 = time.time()
    with torch.no_grad():
        for i in range(n_trials):
            _ = model(test_params[i:i+1])
    amortized_time = (time.time() - t0) / n_trials
    
    return amortized_time  # seconds per portfolio

def compute_sharpe_degradation(w_amortized, w_optimal, returns_matrix):
    """Compute relative Sharpe ratio degradation."""
    sharpe_amortized = compute_sharpe(w_amortized, returns_matrix)
    sharpe_optimal = compute_sharpe(w_optimal, returns_matrix)
    degradation = (sharpe_optimal - sharpe_amortized) / abs(sharpe_optimal)
    return degradation  # positive = amortized is worse

def compute_sharpe(weights, returns):
    portfolio_returns = returns @ weights
    return np.mean(portfolio_returns) / np.std(portfolio_returns) * np.sqrt(252)

# ============================================================
# 6. MAIN EXPERIMENT RUNNER
# ============================================================
if __name__ == "__main__":
    N_ASSETS = 50  # MVT; scale to 100, 200, 500 in full validation
    N_TRAIN = 400000
    N_VAL = 50000
    N_TEST = 50000
    
    # Generate data
    print("Generating training data...")
    params, solutions, solve_times = generate_market_params(N_ASSETS, N_TRAIN + N_VAL + N_TEST)
    print(f"Mean CVXPY solve time: {np.mean(solve_times)*1000:.1f}ms")
    
    # Train model
    model = AmortizedPortfolioNet(n_assets=N_ASSETS)
    model = train_amortized_model(model, train_loader, val_loader)
    
    # Benchmark
    amortized_ms = benchmark_inference_speed(model, test_params) * 1000
    speedup = np.mean(solve_times) / (amortized_ms / 1000)
    print(f"Amortized inference: {amortized_ms:.2f}ms | Speedup: {speedup:.0f}x")
    
    # Quality evaluation
    degradation = compute_sharpe_degradation(w_amortized, w_optimal, test_returns)
    print(f"Sharpe degradation: {degradation*100:.2f}%")
    
    # SUCCESS/FAILURE determination
    assert speedup >= 100, f"FAIL: Speedup {speedup:.0f}x < 100x threshold"
    assert degradation < 0.05, f"FAIL: Sharpe degradation {degradation:.2%} > 5% threshold"
    print("ALL SUCCESS CRITERIA MET")