code/exp8_nonvacuous.py

"""Claims 4 & 5: find the corner where the bounds are actually non-vacuous.

Theorems 3 and 4 both carry an exponential prefactor,

    Thm 3:  F^gen <=  C * (eta*T/n)             * exp( eta*T*(K-k+1)/sqrt(m) )
    Thm 4:  F^gen <=  C * (eta/n) * sum_t Fhat  * exp( eta*c_kK      /sqrt(m) )

On the main exp4 grid (eta = 100) those exponents run from 71 to 15179, so both
right-hand sides exceed the measured gap by 30 to 300+ orders of magnitude.
"The bound holds" is then true but empty: it would hold for any conceivable
measurement.  The question worth answering is whether there is *any* reachable
setting in which the bounds say something, and whether they hold there.

The exponent is O(1) when eta*T*K / sqrt(m) is O(1).  That is reachable with a
small step size and a wide network: eta = 1, T <= 100, m up to 1e4 gives
eta*T*K/sqrt(m) in [0.2, 6.7], i.e. an exp factor of at most ~800 rather
than 10^103.

Grid: d=50, K=3, logistic loss, eta=1, T in {10,25,50,100},
m in {2000,10000}, n in {200,800,3200}, 6 seeds  =  144 runs.
Cheap: K*T <= 300 GD steps, and both the width and sample size stay modest.
"""

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)

D, K, ETA, SIGMA_C = 50, 3, 1.0, 0.1
TS = [10, 25, 50, 100]
MS = [2000, 10000]
NS = [200, 800, 3200]
SEEDS = list(range(6))
NPROC = int(os.environ.get("CL_NPROC", 3))


def one(job):
    T, m, n, seed = job
    r = C.continual_run(d=D, m=m, K=K, n=n, T=T, eta=ETA, sigma_c=SIGMA_C,
                        loss_name="logistic", seed=seed, n_test=4000)
    k = 0                                        # audit task 1
    gap = C.gen_gap(r, k)
    ck = float(sum(r["cum_train_loss"][k + 1:]))       # c_{k,K} of Theorem 4
    ex3 = float(ETA * T * (K - k) / np.sqrt(m))
    ex4 = float(ETA * ck / np.sqrt(m))
    core3 = float(ETA * T / n)
    core4 = float((ETA / n) * r["cum_train_loss"][k])
    return dict(
        T=T, m=m, n=n, seed=seed, d=D, K=K, eta=ETA,
        gen_gap=gap,
        train_forget=C.train_forgetting(r, k),
        test_forget=C.test_forgetting(r, k),
        c_kK=ck,
        exponent_thm3=ex3, exponent_thm4=ex4,
        rhs_thm3_core=core3, rhs_thm4_core=core4,
        rhs_thm3=float(core3 * np.exp(min(ex3, 700.0))),
        rhs_thm4=float(core4 * np.exp(min(ex4, 700.0))),
        train_loss_end=float(r["loss_at"][K - 1, k]),
        test_loss_end=float(r["test_loss_at"][K - 1, k]))


if __name__ == "__main__":
    jobs = [(T, m, n, s) for T in TS for m in MS for n in NS for s in SEEDS]
    print(len(jobs), "runs", flush=True)
    t0 = time.time()
    recs = []
    with Pool(NPROC) as p:
        for i, r in enumerate(p.imap_unordered(one, jobs)):
            recs.append(r)
            if (i + 1) % 12 == 0:
                print(f"  {i+1}/{len(jobs)}  {time.time()-t0:.0f}s", flush=True)
    with open(os.path.join(OUT, "exp8_nonvacuous.json"), "w") as f:
        json.dump(recs, f)
    print("wrote exp8_nonvacuous.json", f"{time.time()-t0:.0f}s", flush=True)