code/clcore.py

"""Core reimplementation of the continual-learning setup of

    "On the Theory of Continual Learning with Gradient Descent for Neural Networks"
    (ICML 2026, OpenReview l35QweVxgn / arXiv:2510.05573v2)

Model (paper Sec. 2.1, matching the authors' notebook
github.com/hosseinta2/continual-learning-with-neural-nets,
`continual_learning_codes-XOR.ipynb`):

    Phi(w, x) = (1/sqrt(m)) * sum_i a_i * phi(x^T w_i),   phi(t) = t^2/2,
    a_i in {+-1} fixed, w_i^{(0)} ~ N(0, I_d).

Data (paper Eq. 3): task k is a d-dimensional XOR cluster with
mu_+^k, mu_-^k orthogonal (across classes *and* across tasks), norms
Theta(1/sqrt(d)), noise sigma = Theta(1/(polylog(d) sqrt(d))).
Following the authors' notebook, task k uses coordinate pair (2k, 2k+1):
    mu_+^k = (e_{2k} + e_{2k+1})/sqrt(d),  mu_-^k = (e_{2k} - e_{2k+1})/sqrt(d).

Training: full-batch gradient descent, T steps per task, step size eta,
sequential over K tasks with no replay and no regularization
(Algorithm 1 of the paper).

Everything is float64 numpy so that the numerical audits are exact to
double precision.
"""

from __future__ import annotations

import os

import numpy as np

# --------------------------------------------------------------------------
# losses.  f(u) acts on the margin u = y * Phi(w, x).
# --------------------------------------------------------------------------


def hinge(u):
    return np.maximum(1.0 - u, 0.0)


def dhinge(u):
    return np.where(u < 1.0, -1.0, 0.0)


def linear(u):
    """The 'linear loss' used in the authors' notebook: f(u) = -u.

    This is the hinge loss restricted to its linear region, which is where
    the paper's analysis operates (see the remark after Thm. 3)."""
    return -u


def dlinear(u):
    return -np.ones_like(u)


def logistic(u):
    # numerically stable log(1 + exp(-u))
    return np.logaddexp(0.0, -u)


def dlogistic(u):
    # d/du log(1+exp(-u)) = -sigmoid(-u)
    return -1.0 / (1.0 + np.exp(np.clip(u, -500, 500)))


LOSSES = {
    "hinge": (hinge, dhinge),
    "linear": (linear, dlinear),
    "logistic": (logistic, dlogistic),
}

# --------------------------------------------------------------------------
# data
# --------------------------------------------------------------------------


def task_means(d: int, k: int):
    """Orthogonal XOR mean pair for task k (0-indexed), norm 1/sqrt(d)."""
    if 2 * k + 1 >= d:
        raise ValueError(f"d={d} too small for {k + 1} orthogonal tasks")
    mp = np.zeros(d)
    mm = np.zeros(d)
    mp[2 * k] = 1.0 / np.sqrt(d)
    mp[2 * k + 1] = 1.0 / np.sqrt(d)
    mm[2 * k] = 1.0 / np.sqrt(d)
    mm[2 * k + 1] = -1.0 / np.sqrt(d)
    return mp, mm


def sample_xor(d: int, k: int, n: int, sigma: float, rng: np.random.Generator):
    """n iid samples from the task-k XOR cluster distribution (Eq. 3).

    Balanced labels; within each class the two antipodal clusters are balanced.
    """
    mp, mm = task_means(d, k)
    n1 = n // 2
    n2 = n - n1
    y = np.concatenate([np.ones(n1), -np.ones(n2)])
    signs1 = np.where(np.arange(n1) < n1 // 2, 1.0, -1.0)
    signs2 = np.where(np.arange(n2) < n2 // 2, 1.0, -1.0)
    centres = np.concatenate(
        [signs1[:, None] * mp[None, :], signs2[:, None] * mm[None, :]], axis=0
    )
    x = centres + sigma * rng.standard_normal((n, d))
    return x, y


# --------------------------------------------------------------------------
# network
# --------------------------------------------------------------------------


# Cap on the number of float64 entries in the (n, m) pre-activation matrix Z.
# Z is the only object in this file whose size grows as n * m, and it is
# consumed row-block by row-block, so bounding it costs nothing numerically
# (the arithmetic is identical, just re-associated) while keeping the peak
# resident set of a worker at ~ZBLOCK * 8 bytes.  4e6 entries = 32 MB.
ZBLOCK = int(os.environ.get("CL_ZBLOCK", 4_000_000))

# Worker-pool size for the sweep drivers.  This box has 12 cores but only
# ~15 GB of RAM (much of it already spoken for), and an over-wide pool sent it
# into swap-death twice, so the default is deliberately conservative.  Raise
# with CL_NPROC on a machine with headroom.
NPROC = int(os.environ.get("CL_NPROC", 3))


def _row_block(m: int, n: int) -> int:
    """Number of samples per chunk so that the (block, m) matrix fits ZBLOCK."""
    return max(1, min(n, ZBLOCK // max(1, m)))


class QuadNet:
    """One-hidden-layer quadratic network with fixed +-1 output layer."""

    def __init__(self, d: int, m: int, rng: np.random.Generator):
        self.d, self.m = d, m
        self.W = rng.standard_normal((m, d))  # w_i^{(0)} ~ N(0, I_d)
        self.a = rng.choice([-1.0, 1.0], size=m)
        self.W0 = self.W.copy()

    def out(self, X):
        """Phi(w, X), computed in row blocks so (n, m) is never materialized."""
        n = X.shape[0]
        bs = _row_block(self.m, n)
        out = np.empty(n)
        scale = 0.5 / np.sqrt(self.m)
        for s in range(0, n, bs):
            Z = X[s:s + bs] @ self.W.T          # (bs, m)
            out[s:s + bs] = scale * ((Z * Z) @ self.a)
        return out

    def dist_from_init(self):
        return float(np.linalg.norm(self.W - self.W0))


def gd_step(net: QuadNet, X, y, eta: float, dloss):
    """One full-batch GD step on (1/n) sum_i f(y_i Phi(w, x_i)).

    Returns the per-sample margins u = y * Phi(w, x).  The gradient is
    accumulated over row blocks, so peak memory is O(ZBLOCK + m*d) rather
    than O(n*m); the result is bit-comparable to the unchunked version up to
    floating-point summation order.
    """
    n = X.shape[0]
    m = net.m
    bs = _row_block(m, n)
    scale = 0.5 / np.sqrt(m)
    u = np.empty(n)
    G = np.zeros((m, net.d))
    for s in range(0, n, bs):
        Xb = X[s:s + bs]
        yb = y[s:s + bs]
        Z = Xb @ net.W.T                        # (bs, m)
        ub = yb * (scale * ((Z * Z) @ net.a))
        u[s:s + bs] = ub
        g = dloss(ub) * yb                      # dL/dPhi per sample, (bs,)
        # grad_{w_i} = (1/n) sum_j g_j (a_i / sqrt(m)) z_{ji} x_j
        Z *= g[:, None]
        G += Z.T @ Xb
    G *= net.a[:, None] / (n * np.sqrt(m))
    net.W -= eta * G
    return u


def eval_task(net: QuadNet, X, y, loss):
    out = net.out(X)
    u = y * out
    return float(np.mean(loss(u))), float(np.mean(u <= 0))


# --------------------------------------------------------------------------
# continual learning driver
# --------------------------------------------------------------------------


def continual_run(
    d=50,
    m=1000,
    K=3,
    n=2500,
    T=200,
    eta=2.0,
    sigma_c=0.1,
    loss_name="linear",
    seed=0,
    n_test=2000,
    track_traj=False,
    n_first=None,
):
    """Run Algorithm 1 and record every quantity the theorems talk about.

    n_first: sample size for task 1 only (paper's Fig. 4 protocol, where the
             first task's n is held fixed while later tasks' n is varied).

    Returns a dict with, for every pair (k, j) with j >= k, the empirical loss
    and misclassification error of task k measured at w_j, plus the test-set
    counterparts, plus ||w_j - w_0|| and the cumulative training losses that
    appear in Theorem 4.
    """
    loss, dloss = LOSSES[loss_name]
    rng = np.random.default_rng(seed)
    sigma = sigma_c / np.sqrt(d)

    ns = [n] * K
    if n_first is not None:
        ns[0] = n_first

    Xs, ys, Xte, yte = [], [], [], []
    for k in range(K):
        xk, yk = sample_xor(d, k, ns[k], sigma, rng)
        Xs.append(xk)
        ys.append(yk)
        xt, yt = sample_xor(d, k, n_test, sigma, rng)
        Xte.append(xt)
        yte.append(yt)

    net = QuadNet(d, m, rng)

    # loss_at[j][k] = empirical loss of task k measured at w_j (after task j+1)
    loss_at = np.full((K, K), np.nan)
    err_at = np.full((K, K), np.nan)
    tloss_at = np.full((K, K), np.nan)
    terr_at = np.full((K, K), np.nan)
    dist = np.zeros(K)
    cum_train_loss = np.zeros(K)      # sum_t Fhat_j(w_j^{(t)}), t = 0..T-1
    traj = [] if track_traj else None

    for j in range(K):
        for t in range(T):
            u = gd_step(net, Xs[j], ys[j], eta, dloss)
            cum_train_loss[j] += float(np.mean(loss(u)))
            if track_traj:
                traj.append([eval_task(net, Xs[k], ys[k], loss)[0] for k in range(K)])
        dist[j] = net.dist_from_init()
        for k in range(K):
            loss_at[j, k], err_at[j, k] = eval_task(net, Xs[k], ys[k], loss)
            tloss_at[j, k], terr_at[j, k] = eval_task(net, Xte[k], yte[k], loss)

    return dict(
        loss_at=loss_at,
        err_at=err_at,
        test_loss_at=tloss_at,
        test_err_at=terr_at,
        dist=dist,
        cum_train_loss=cum_train_loss,
        ns=ns,
        traj=(np.array(traj) if track_traj else None),
        Xs=Xs,
        ys=ys,
        net=net,
        cfg=dict(d=d, m=m, K=K, n=n, T=T, eta=eta, sigma_c=sigma_c,
                 loss=loss_name, seed=seed, n_first=n_first),
    )


def train_forgetting(res, k):
    """F^tr_{k,K} = Fhat_k(w_K) - Fhat_k(w_k)  (k 0-indexed here)."""
    K = res["cfg"]["K"]
    return float(res["loss_at"][K - 1, k] - res["loss_at"][k, k])


def test_forgetting(res, k):
    K = res["cfg"]["K"]
    return float(res["test_loss_at"][K - 1, k] - res["test_loss_at"][k, k])


def gen_gap(res, k):
    """Delayed generalization gap F_k(w_K) - Fhat_k(w_K)."""
    K = res["cfg"]["K"]
    return float(res["test_loss_at"][K - 1, k] - res["loss_at"][K - 1, k])