code/exp4_gengap.py

"""Claims 4 and 5: the delayed generalization gap (Theorems 3 and 4).

Theorem 3 (1-Lipschitz, 1-smooth loss):
    F^gen_{k,K} = E_{D_k}[ F_k(w_K) - Fhat_k(w_K) ]  <~  eta*T*exp(eta*T*(K-k+1)/sqrt(m)) / n

Theorem 4 (additionally self-bounded loss, e.g. logistic):
    F^gen_{k,K}  <~  (eta/n) * E[ exp((eta/sqrt(m)) c_{k,K}) * sum_{t<T} Fhat_k(w_k^{(t)}) ]
    with  c_{k,K} = O( sum_{j>k} sum_{t<T} Fhat_j(w_j^{(t)}) ).

Both are *expectations over the draw of D_k*, so each measurement point
averages over many independent dataset draws; F_k is estimated on a large
fresh test set.  We use the logistic loss (1-Lipschitz, 1/4-smooth,
self-bounded) so that both theorems apply to the same runs and can be compared
head to head.

The headline comparison for Claim 5 is the T-sweep: Theorem 3's right-hand
side grows *linearly* in T, Theorem 4's grows like the cumulative training
loss sum_t Fhat_k(w_k^{(t)}), which for a learnable task is poly-logarithmic
in T.  We fit one global constant per bound and check (i) both remain valid
upper bounds and (ii) Theorem 4's is far tighter and grows sub-linearly.
"""

import json
import os
import sys
import time
from multiprocessing import Pool

import numpy as np

sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
import clcore as C  # noqa: E402

OUT = os.path.join(os.path.dirname(os.path.abspath(__file__)), "results")
os.makedirs(OUT, exist_ok=True)

BASE = dict(d=50, m=1000, K=3, n=400, T=200, eta=100.0, sigma_c=0.1,
            loss_name="logistic", n_test=4000)
# 10 independent draws of D_1..D_K per point.  The generalization gap is a
# difference of two means and is the noisiest quantity in this reproduction, so
# seeds matter here more than anywhere else; 10 keeps the standard error on
# each sweep point at roughly a third of the effect we are fitting, while
# halving the run time relative to the original 20.  n_test 8000 -> 4000
# doubles the Monte-Carlo error on F_k(w) alone, which is small next to the
# seed-to-seed spread.
SEEDS = list(range(10))


def one(job):
    sweep, override, seed = job
    cfg = dict(BASE)
    cfg.update(override)
    cfg["seed"] = seed
    r = C.continual_run(**cfg)
    K, T, eta, n, m = cfg["K"], cfg["T"], cfg["eta"], cfg["n"], cfg["m"]
    k = 0                                     # audit task 1
    rec = dict(sweep=sweep,
               **{kk: vv for kk, vv in cfg.items() if kk != "n_test"})
    rec["gen_gap"] = C.gen_gap(r, k)
    rec["gen_gap_per_k"] = [C.gen_gap(r, kk) for kk in range(K)]
    rec["train_forget"] = C.train_forgetting(r, k)
    rec["test_forget"] = C.test_forgetting(r, k)
    rec["cum_train_loss"] = list(r["cum_train_loss"])   # sum_t Fhat_j(w_j^(t))
    rec["train_loss_end"] = float(r["loss_at"][K - 1, k])
    rec["test_loss_end"] = float(r["test_loss_at"][K - 1, k])
    rec["err_at"] = r["err_at"].tolist()
    rec["test_err_at"] = r["test_err_at"].tolist()
    rec["dist"] = list(r["dist"])
    # Theorem 3 / Theorem 4 right-hand sides (up to the universal constant).
    # We record the exponents separately because they are only benign when
    # eta*T*(K-k+1)/sqrt(m) = O(1) -- see the Claim 4/5 pages.
    ck = float(sum(r["cum_train_loss"][k + 1:]))
    rec["c_kK"] = ck
    rec["exponent_thm3"] = float(eta * T * (K - k) / np.sqrt(m))
    rec["exponent_thm4"] = float(eta * ck / np.sqrt(m))
    rec["rhs_thm3_core"] = float(eta * T / n)                    # eta*T/n
    rec["rhs_thm4_core"] = float((eta / n) * r["cum_train_loss"][k])
    rec["rhs_thm3"] = float(rec["rhs_thm3_core"]
                            * np.exp(min(rec["exponent_thm3"], 700.0)))
    rec["rhs_thm4"] = float(rec["rhs_thm4_core"]
                            * np.exp(min(rec["exponent_thm4"], 700.0)))
    return rec


def build():
    jobs = []
    # (a) sample size -> both bounds predict 1/n
    for n in [50, 100, 200, 400, 800, 1600, 3200]:
        for s in SEEDS:
            jobs.append(("n", dict(n=n), s))
    # (b) training horizon -> Thm 3 linear in T vs Thm 4 polylog in T
    for T in [25, 50, 100, 200, 400, 800, 1600]:
        for s in SEEDS:
            jobs.append(("T", dict(T=T), s))
    # (c) width -> controls the exponential factor in both bounds
    for m in [100, 300, 1000, 3000, 10000]:
        for s in SEEDS:
            jobs.append(("m", dict(m=m), s))
    # (d) number of tasks
    for K in [2, 3, 4, 6]:
        for s in SEEDS:
            jobs.append(("K", dict(K=K), s))
    return jobs


if __name__ == "__main__":
    jobs = build()
    print(len(jobs), "runs", flush=True)
    t0 = time.time()
    with Pool(C.NPROC) as p:
        recs = []
        for i, r in enumerate(p.imap_unordered(one, jobs)):
            recs.append(r)
            if (i + 1) % 50 == 0:
                print(f"  {i+1}/{len(jobs)} {time.time()-t0:.0f}s", flush=True)
    with open(os.path.join(OUT, "exp4_gengap.json"), "w") as f:
        json.dump(recs, f)
    print("wrote exp4_gengap.json", f"{time.time()-t0:.0f}s")