Lasso

In this example, we use the PANOC solver to solve a lasso problem, i.e. least squares with \(\ell_1\)-regularization to promote sparsity.

# %% alpaqa lasso example

from pprint import pprint

import casadi as cs
import numpy as np

import alpaqa as pa

scale = 50
n, m = scale, scale * 2
sparsity = 0.2
rng = np.random.default_rng(0)

# %% Build the problem (CasADi code, independent of alpaqa)

# Quadratic loss plus l1-regularization
# minimize  ½‖Ax - b‖² + λ‖x‖₁

A = rng.uniform(-1, 1, (m, n))
x_exact = rng.uniform(0, 1, n)
x_exact[rng.uniform(0, 1, n) > sparsity] = 0
b = A @ x_exact + rng.normal(0, 0.1, m)
λ = 0.025 * m

# Symbolic solution
x = cs.MX.sym("x", n)
# Objective function is squared norm of Ax - b
f = 0.5 * cs.sumsqr(A @ x - b)

# %% Generate and compile C code for the objective and constraints using alpaqa

# Compile and load the problem
problem = (
    pa.minimize(f, x)
    .with_l1_regularizer(λ)
).compile(sym=cs.MX.sym)

# %% Solve the problem using alpaqa's PANOC solver

direction = pa.LBFGSDirection({"memory": scale})
solver = pa.PANOCSolver({"print_interval": 10}, direction)
# Add evaluation counters to the problem
cnt = pa.problem_with_counters(problem)
# Solve
sol, stats = solver(cnt.problem, {"tolerance": 1e-10})

# %% Print the results

final_f = problem.eval_objective(sol)
print()
pprint(stats)
print()
print("Evaluations:")
print(cnt.evaluations)
print(f"Cost:          {final_f + stats['final_h']}")
print(f"Loss:          {final_f}")
print(f"Regularizer:   {stats['final_h']}")
print(f"FP Residual:   {stats['ε']}")
print(f"Run time:      {stats['elapsed_time']}")
print(stats["status"])

# %% Plot the results

import matplotlib.pyplot as plt

plt.figure(figsize=(8, 5))
plt.plot(x_exact, ".-", label="True solution")
plt.plot(sol, ".-", label="Estimated solution")
plt.legend()
plt.title("PANOC lasso example")
plt.tight_layout()
plt.show()