Gram-based CD/BCD/FISTA solvers for (group)Lasso when n_samples >> n_features

gram_test.py

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+# data available at  https://www.dropbox.com/sh/32b3mr3xghi496g/AACNRS_NOsUXU-hrSLixNg0ja?dl=0
+
+
+import time
+import numpy as np
+from celer import GroupLasso
+from skglm.solvers.gram import gram_fista_group_lasso, gram_group_lasso
+
+X = np.load("design_matrix.npy")
+y = np.load("target.npy")
+groups = np.load("groups.npy")
+weights = np.load("weights.npy")
+grps = [list(np.where(groups == i)[0]) for i in range(1, 33)]
+
+
+alpha_ratio = 1e-2
+n_alphas = 10
+tol = 1e-8
+
+# Case 1: slower runtime for small alphas
+# alpha_max = 0.003471727067743962
+alpha_max = np.max(np.linalg.norm((X.T @ y).reshape(-1, 5), axis=1)) / len(y)
+alpha = alpha_max / 100
+clf = GroupLasso(fit_intercept=False, tol=tol,
+                 groups=grps, weights=weights, alpha=alpha, verbose=1)
+
+t0 = time.time()
+clf.fit(X, y)
+t1 = time.time()
+
+print(f"Celer: {t1 - t0:.3f} s")
+
+t0 = time.time()
+res = gram_group_lasso(X, y, alpha, groups=grps, tol=tol, weights=weights, max_iter=10_000,
+                       check_freq=50)
+t1 = time.time()
+
+print(f"skglm gram: {t1 - t0:.3f} s")
+
+
+# FISTA Gram for very small alphas
+alpha = alpha_max / 1e-4
+clf = GroupLasso(fit_intercept=False, tol=tol, groups=grps, weights=weights, alpha=alpha, 
+                 verbose=1)
+
+t0 = time.time()
+clf.fit(X, y)
+t1 = time.time()
+
+print(f"Celer: {t1 - t0:.3f} s")
+
+t0 = time.time()
+res = gram_fista_group_lasso(X, y, alpha, groups=grps, tol=tol, weights=weights, max_iter=10_000,
+                             check_freq=50)
+t1 = time.time()
+print(f"skglm fista gram: {t1 - t0:.3f} s")

skglm/gram_solver.py

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+from time import time
+import numpy as np
+from numpy.linalg import norm
+from celer import Lasso, GroupLasso
+from benchopt.datasets.simulated import make_correlated_data
+from skglm.solvers.gram import gram_fista_group_lasso, gram_fista_lasso, gram_lasso, gram_group_lasso
+
+
+n_samples, n_features = 100, 300
+X, y, w_star = make_correlated_data(
+    n_samples=n_samples, n_features=n_features, random_state=0)
+alpha_max = norm(X.T @ y, ord=np.inf)
+
+# Hyperparameters
+max_iter = 10_000
+tol = 1e-8
+reg = 0.1
+group_size = 3
+
+alpha = alpha_max * reg / n_samples
+
+weights = np.random.normal(2, 0.4, n_features)
+weights_grp = np.random.normal(2, 0.4, n_features // group_size)
+
+# Lasso
+print("#" * 15)
+print("Lasso")
+print("#" * 15)
+start = time()
+w = gram_lasso(X, y, alpha, max_iter, tol, weights=weights)
+gram_lasso_time = time() - start
+clf_sk = Lasso(alpha, weights=weights, tol=tol, fit_intercept=False)
+start = time()
+clf_sk.fit(X, y)
+celer_lasso_time = time() - start
+start = time()
+w_fista = gram_fista_lasso(X, y, alpha, max_iter, tol, weights=weights)
+gram_fista_lasso_time = time() - start
+np.testing.assert_allclose(w, clf_sk.coef_, rtol=1e-4)
+np.testing.assert_allclose(w, w_fista, rtol=1e-4)
+
+print("\n")
+print("Celer: %.2f" % celer_lasso_time)
+print("CD Gram: %.2f" % gram_lasso_time)
+print("FISTA Gram: %.2f" % gram_fista_lasso_time)
+print("\n")
+
+# Group Lasso
+print("#" * 15)
+print("Group Lasso")
+print("#" * 15)
+start = time()
+w = gram_group_lasso(X, y, alpha, group_size, max_iter, tol, weights=weights_grp)
+gram_group_lasso_time = time() - start
+start = time()
+w_fista = gram_fista_group_lasso(X, y, alpha, group_size, max_iter, tol,
+                                 weights=weights_grp)
+gram_fista_group_lasso_time = time() - start
+
+np.testing.assert_allclose(w, w_fista, rtol=1e-4)
+
+clf_celer = GroupLasso(group_size, alpha, tol=tol, weights=weights_grp,
+                       fit_intercept=False)
+start = time()
+clf_celer.fit(X, y)
+celer_group_lasso_time = time() - start
+np.testing.assert_allclose(w, clf_celer.coef_, rtol=1e-4)
+
+print("\n")
+print("Celer: %.2f" % celer_group_lasso_time)
+print("BCD Gram: %.2f" % gram_group_lasso_time)
+print("FISTA Gram: %.2f" % gram_fista_group_lasso_time)
+print("\n")

skglm/solvers/gram.py

@@ -0,0 +1,254 @@
+import numpy as np
+from numba import njit
+from numpy.linalg import norm
+from celer.homotopy import _grp_converter
+
+from skglm.utils import BST, ST, ST_vec
+
+
+@njit
+def primal(alpha, r, w, weights):
+    n_features = len(weights)
+    p_obj = (r @ r) / (2 * len(r))
+    pen = 0.
+    for j in range(n_features):
+        if weights[j] == np.inf:
+            continue
+        pen += np.abs(w[j] * weights[j])
+    return p_obj + alpha * pen
+
+
+@njit
+def primal_grp(alpha, norm_r2, r, w, grp_ptr, grp_indices, weights):
+    p_obj = norm_r2 / (2 * len(r))
+    for g in range(len(grp_ptr) - 1):
+        if weights[g] == np.inf:
+            continue
+        w_g = w[grp_indices[grp_ptr[g]:grp_ptr[g + 1]]]
+        p_obj += alpha * norm(w_g * weights[g], ord=2)
+    return p_obj
+
+
+@njit
+def dual(alpha, norm_y2, theta, y):
+    d_obj = - np.sum((y / (alpha * len(y)) - theta) ** 2)
+    d_obj *= 0.5 * alpha ** 2 * len(y)
+    d_obj += norm_y2 / (2 * len(y))
+    return d_obj
+
+
+@njit
+def dnorm_l1(theta, X, weights):
+    n_features = X.shape[1]
+    scal = 0.
+    for j in range(n_features):
+        Xj_theta = X[:, j] @ theta
+        scal = max(scal, Xj_theta / weights[j])
+    return scal
+
+
+@njit
+def dnorm_l21(theta, grp_ptr, grp_indices, X, weights):
+    scal = 0.
+    n_groups = len(grp_ptr) - 1
+    for g in range(n_groups):
+        if weights[g] == np.inf:
+            continue
+        tmp = 0.
+        for k in range(grp_ptr[g], grp_ptr[g + 1]):
+            j = grp_indices[k]
+            Xj_theta = X[:, j] @ theta
+            tmp += Xj_theta ** 2
+        scal = max(scal, np.sqrt(tmp) / weights[g])
+    return scal
+
+
+@njit
+def create_dual_point(r, alpha, X, y, weights):
+    theta = r / (alpha * len(y))
+    scal = dnorm_l1(theta, X, weights)
+    if scal > 1.:
+        theta /= scal
+    return theta
+
+
+@njit
+def create_dual_point_grp(r, alpha, y, X, grp_ptr, grp_indices, weights):
+    theta = r / (alpha * len(y))
+    scal = dnorm_l21(theta, grp_ptr, grp_indices, X, weights)
+    if scal > 1.:
+        theta /= scal
+    return theta
+
+
+@njit
+def dual_gap(alpha, norm_y2, y, X, w, weights):
+    r = y - X @ w
+    p_obj = primal(alpha, r, w, weights)
+    theta = create_dual_point(r, alpha, X, y, weights)
+    d_obj = dual(alpha, norm_y2, theta, y)
+    return p_obj, d_obj, p_obj - d_obj
+
+
+@njit
+def dual_gap_grp(y, X, w, alpha, norm_y2, grp_ptr, grp_indices, weights):
+    r = y - X @ w
+    norm_r2 = r @ r
+    p_obj = primal_grp(alpha, norm_r2, r, w, grp_ptr, grp_indices, weights)
+    theta = create_dual_point_grp(r, alpha, y, X, grp_ptr, grp_indices, weights)
+    d_obj = dual(alpha, norm_y2, theta, y)
+    return p_obj, d_obj, p_obj - d_obj
+
+
+@njit
+def compute_lipschitz(X, y):
+    n_features = X.shape[1]
+    lipschitz = np.zeros(n_features, dtype=X.dtype)
+    for j in range(n_features):
+        lipschitz[j] = (X[:, j] ** 2).sum() / len(y)
+    return lipschitz
+
+
+@njit
+def prox_l21(w, u, weights, grp_ptr, grp_indices):
+    n_groups = len(grp_ptr) - 1
+    out = w.copy()
+    for g in range(n_groups):
+        idx = grp_indices[grp_ptr[g]:grp_ptr[g + 1]]
+        grp_nrm = norm(w[idx], ord=2)
+        scaling = np.maximum(1 - u / grp_nrm * weights[g], 0)
+        out[idx] *= scaling
+    return out
+
+
+def gram_lasso(X, y, alpha, max_iter, tol, w_init=None, weights=None, check_freq=100):
+    n_features = X.shape[1]
+    norm_y2 = y @ y
+    grads = X.T @ y / len(y)
+    G = X.T @ X
+    lipschitz = compute_lipschitz(X, y)
+    w = w_init.copy() if w_init is not None else np.zeros(n_features)
+    weights = weights if weights is not None else np.ones(n_features)
+    for n_iter in range(max_iter):
+        cd_epoch(X, G, grads, w, alpha, lipschitz, weights)
+        if n_iter % check_freq == 0:
+            p_obj, d_obj, d_gap = dual_gap(alpha, norm_y2, y, X, w, weights)
+            print(f"iter {n_iter} :: p_obj {p_obj:.5f} :: d_obj {d_obj:.5f}" +
+                  f" :: gap {d_gap:.5f}")
+            if d_gap < tol:
+                print("Convergence reached!")
+                break
+    return w
+
+
+def gram_fista_lasso(X, y, alpha, max_iter, tol, w_init=None, weights=None,
+                     check_freq=100):
+    n_samples, n_features = X.shape
+    norm_y2 = y @ y
+    t_new = 1
+    w = w_init.copy() if w_init is not None else np.zeros(n_features)
+    z = w_init.copy() if w_init is not None else np.zeros(n_features)
+    weights = weights if weights is not None else np.ones(n_features)
+    G = X.T @ X
+    Xty = X.T @ y
+    L = np.linalg.norm(X, ord=2) ** 2 / n_samples
+    for n_iter in range(max_iter):
+        t_old = t_new
+        t_new = (1 + np.sqrt(1 + 4 * t_old ** 2)) / 2
+        w_old = w.copy()
+        z -= (G @ z - Xty) / L / n_samples
+        w = ST_vec(z, alpha / L * weights)
+        z = w + (t_old - 1.) / t_new * (w - w_old)
+        if n_iter % check_freq == 0:
+            p_obj, d_obj, d_gap = dual_gap(alpha, norm_y2, y, X, w, weights)
+            print(f"iter {n_iter} :: p_obj {p_obj:.5f} :: d_obj {d_obj:.5f} " +
+                  f":: gap {d_gap:.5f}")
+            if d_gap < tol:
+                print("Convergence reached!")
+                break
+    return w
+
+
+def gram_group_lasso(X, y, alpha, groups, max_iter, tol, w_init=None, weights=None,
+                     check_freq=100):
+    n_features = X.shape[1]
+    grp_ptr, grp_indices = _grp_converter(groups, X.shape[1])
+    n_groups = len(grp_ptr) - 1
+    norm_y2 = y @ y
+    grads = X.T @ y / len(y)
+    G = X.T @ X
+    lipschitz = np.zeros(n_groups, dtype=X.dtype)
+    for g in range(n_groups):
+        X_g = X[:, grp_indices[grp_ptr[g]:grp_ptr[g + 1]]]
+        lipschitz[g] = norm(X_g, ord=2) ** 2 / len(y)
+    w = w_init.copy() if w_init is not None else np.zeros(n_features)
+    weights = weights if weights is not None else np.ones(n_groups)
+    for n_iter in range(max_iter):
+        bcd_epoch(X, G, grads, w, alpha, lipschitz, grp_indices, grp_ptr, weights)
+        if n_iter % check_freq == 0:
+            p_obj, d_obj, d_gap = dual_gap_grp(y, X, w, alpha, norm_y2, grp_ptr,
+                                               grp_indices, weights)
+            print(f"iter {n_iter} :: p_obj {p_obj:.5f} :: d_obj {d_obj:.5f} " +
+                  f":: gap {d_gap:.5f}")
+            if d_gap < tol:
+                print("Convergence reached!")
+                break
+    return w
+
+
+def gram_fista_group_lasso(X, y, alpha, groups, max_iter, tol, w_init=None,
+                           weights=None, check_freq=100):
+    n_features = X.shape[1]
+    norm_y2 = y @ y
+    grp_ptr, grp_indices = _grp_converter(groups, X.shape[1])
+    n_groups = len(grp_ptr) - 1
+    t_new = 1
+    w = w_init.copy() if w_init is not None else np.zeros(n_features)
+    z = w_init.copy() if w_init is not None else np.zeros(n_features)
+    weights = weights if weights is not None else np.ones(n_groups)
+    G = X.T @ X
+    Xty = X.T @ y
+    L = np.linalg.norm(X, ord=2) ** 2 / len(y)
+    for n_iter in range(max_iter):
+        t_old = t_new
+        t_new = (1 + np.sqrt(1 + 4 * t_old ** 2)) / 2
+        w_old = w.copy()
+        z -= (G @ z - Xty) / L / len(y)
+        w = prox_l21(z, alpha / L, weights, grp_ptr, grp_indices)
+        z = w + (t_old - 1.) / t_new * (w - w_old)
+        if n_iter % check_freq == 0:
+            p_obj, d_obj, d_gap = dual_gap_grp(y, X, w, alpha, norm_y2, grp_ptr,
+                                               grp_indices, weights)
+            print(f"iter {n_iter} :: p_obj {p_obj:.5f} :: d_obj {d_obj:.5f} " +
+                  f":: gap {d_gap:.5f}")
+            if d_gap < tol:
+                print("Convergence reached!")
+                break
+    return w
+
+
+@njit
+def cd_epoch(X, G, grads, w, alpha, lipschitz, weights):
+    n_features = X.shape[1]
+    for j in range(n_features):
+        if lipschitz[j] == 0. or weights[j] == np.inf:
+            continue
+        old_w_j = w[j]
+        w[j] = ST(w[j] + grads[j] / lipschitz[j], alpha / lipschitz[j] * weights[j])
+        if old_w_j != w[j]:
+            grads += G[j, :] * (old_w_j - w[j]) / len(X)
+
+
+@njit
+def bcd_epoch(X, G, grads, w, alpha, lipschitz, grp_indices, grp_ptr, weights):
+    n_groups = len(grp_ptr) - 1
+    for g in range(n_groups):
+        if lipschitz[g] == 0. and weights[g] == np.inf:
+            continue
+        idx = grp_indices[grp_ptr[g]:grp_ptr[g + 1]]
+        old_w_g = w[idx].copy()
+        w[idx] = BST(w[idx] + grads[idx] / lipschitz[g], alpha / lipschitz[g]
+                     * weights[g])
+        diff = old_w_g - w[idx]
+        if np.any(diff != 0.):
+            grads += diff @ G[idx, :] / len(X)