scripts/claim3_structure.py

"""Claim 3 [ALGORITHMIC]: mechanism structure + complexity.

Two parts:
 (A) Structural verification of the two-stage mechanism against the paper's own
     pseudocode and the invariants its proofs rely on.
 (B) Empirical complexity: the paper's "Complexity Analysis" states the cold start
     lasts O(K L N) rounds with per-round cost "only the sorting of prior means",
     and the exploitation stage is dominated by the offline oracle (O(d^3) inversion).
"""
import json, os, sys, time
import numpy as np

os.chdir(os.path.abspath(os.path.join(os.path.dirname(__file__), "..")))
sys.path.insert(0, os.path.join(os.path.dirname(__file__), ".."))
from rcb.core import RCB, ipgs, L_eps, m0_eps
from rcb.env import SyntheticEnv

RES, FAIL = {"checks": {}}, []


def check(name, cond, detail=""):
    RES["checks"][name] = {"pass": bool(cond), "detail": detail}
    print(f"[{'PASS' if cond else 'FAIL'}] {name}  {detail}")
    if not cond:
        FAIL.append(name)


# ============================================================ (A) structure
def structural():
    rng = np.random.default_rng(0)

    # -- IPGS is a valid kernel and p(b_t) >= 1/K (asserted in proof Eq. C.14/C.15)
    worst_pb, ok_simplex = 1.0, True
    for _ in range(200000):
        K = int(rng.integers(2, 15))
        gamma = float(10 ** rng.uniform(-2, 5))
        p, b = ipgs(rng.normal(size=K), gamma)
        ok_simplex &= bool(np.all(p >= -1e-12) and abs(p.sum() - 1) < 1e-10)
        worst_pb = min(worst_pb, p[b] * K)
    check("IPGS is a probability kernel for all gamma>0, K>=2", ok_simplex,
          "2e5 random draws: p >= 0 and sums to 1")
    check("IPGS satisfies p_t(b_t) >= 1/K (used in Part I of Eq. C.15)",
          worst_pb >= 1.0 - 1e-9, f"min over 2e5 draws of K*p(b_t) = {worst_pb:.6f} >= 1")

    # -- Cold start: organic pulls must NOT increment N_i or S_i (Section 3.1)
    K, d = 4, 3
    beta0 = [np.zeros(d) for _ in range(K)]
    beta0[0] = np.full(d, 0.3)
    S0 = [0.2 * np.eye(d) for _ in range(K)]
    env = SyntheticEnv(K, d, 0.05, beta0, S0, rng=np.random.default_rng(3))
    a = RCB(K, d, 0.05, beta0, S0, N=15, L=8.0, offset=0.5, phi0=1 / d,
            use_empirical_EF=True, rng=np.random.default_rng(4))
    phases, organic_counted = {"MPASC": 0, "RASC-promote": 0, "RASC-organic": 0, "IPGS": 0}, 0
    for t in range(20000):
        x = env.context()
        before = a.Ni.copy()
        rec, info = a.recommend(x)
        phases[info["phase"]] = phases.get(info["phase"], 0) + 1
        a.update(x, rec, env.pull(x, rec), info)
        if info["phase"] == "RASC-organic" and not np.array_equal(before, a.Ni):
            organic_counted += 1
    check("Organic recommendations do not increment N_i or S_i", organic_counted == 0,
          f"{phases['RASC-organic']} organic rounds, {organic_counted} wrongly counted")
    check("Both stages of the mechanism are exercised",
          phases["MPASC"] > 0 and phases["RASC-promote"] > 0 and phases["IPGS"] > 0,
          f"MPASC={phases['MPASC']} RASC-promote={phases['RASC-promote']} "
          f"RASC-organic={phases['RASC-organic']} IPGS={phases['IPGS']}")
    check("Every arm reaches the saturation threshold N before Stage 2",
          bool(np.all(a.Ni >= 15)), f"N_i at transition = {a.Ni.tolist()}, N = 15")
    RES["phases"] = phases

    # -- RASC explores with frequency exactly 1/L (Algorithm 1, q_t ~ Ber(1/L))
    prom = org = 0
    for sd in range(40):
        env2 = SyntheticEnv(K, d, 0.05, beta0, S0, rng=np.random.default_rng(1000 + sd))
        a2 = RCB(K, d, 0.05, beta0, S0, N=60, L=8.0, offset=0.5, phi0=1 / d,
                 use_empirical_EF=True, rng=np.random.default_rng(2000 + sd))
        while a2.stage == 1:
            x = env2.context(); r, i = a2.recommend(x)
            a2.update(x, r, env2.pull(x, r), i)
            prom += i["phase"] == "RASC-promote"; org += i["phase"] == "RASC-organic"
    n_r = prom + org
    frac = prom / n_r
    se = np.sqrt(0.125 * 0.875 / n_r)
    check("RASC explores with frequency 1/L", abs(frac - 0.125) < 3 * se,
          f"measured {frac:.4f} over {n_r} RASC rounds vs 1/L = 0.1250 (3 s.e. = {3*se:.4f})")

    # -- epoch schedule m0 = ceil(2 + log2 N)
    ok = all(m0_eps(N) == int(np.ceil(2 + np.log2(N))) for N in [1, 5, 10, 100, 1000, 175781])
    check("Exploitation starts at epoch m0 = ceil(2 + log2 N)", ok, "checked N in {1..1.8e5}")

    # -- Cold-start DBIC inequality Eq. (C.7): with L >= 1 + (Delta0-eps)/(tau*rho+eps),
    #    (1-1/L) E[G|G>0]Pr(G>0) + (1/L)(mu0_i - mu0_j) >= -eps
    worst = np.inf
    for _ in range(200000):
        eps = float(rng.uniform(0, 0.3)); tau = float(rng.uniform(1e-3, 0.2))
        rho = float(rng.uniform(0.5, 1.0)); D0 = float(rng.uniform(0, 1.0))
        L = L_eps(eps, tau, rho, delta_max=D0)
        lhs = (1 - 1 / L) * (tau * rho) + (1 / L) * (-D0)
        worst = min(worst, lhs + eps)
    check("Cold-start DBIC bound Eq. (C.7) holds at the prescribed L", worst >= -1e-9,
          f"min over 2e5 random (eps,tau,rho,Delta0) of LHS+eps = {worst:.3e} >= 0")


# ============================================================ (B) complexity
def complexity():
    def cold_len(K, L, N, seed=0, d=3):
        beta0 = [np.zeros(d) for _ in range(K)]; beta0[0] = np.full(d, 0.3)
        S0 = [0.2 * np.eye(d) for _ in range(K)]
        env = SyntheticEnv(K, d, 0.05, beta0, S0, rng=np.random.default_rng(seed + 5))
        a = RCB(K, d, 0.05, beta0, S0, N=N, L=L, offset=0.5, phi0=1 / d,
                use_empirical_EF=True, rng=np.random.default_rng(seed))
        t = 0
        while a.stage == 1 and t < 5_000_000:
            x = env.context(); rec, info = a.recommend(x)
            a.update(x, rec, env.pull(x, rec), info); t += 1
        return t

    def sl(xs, ys):
        return float(np.polyfit(np.log(xs), np.log(ys), 1)[0])

    # fit away from the small-K boundary (at K=2 only one arm needs RASC at all)
    Ks = [8, 16, 32, 64]
    yK = [np.mean([cold_len(K, 10.0, 20, s) for s in range(3)]) for K in Ks]
    Ls = [4.0, 8.0, 16.0, 32.0]
    yL = [np.mean([cold_len(5, L, 20, s) for s in range(3)]) for L in Ls]
    Ns = [10, 20, 40, 80]
    yN = [np.mean([cold_len(5, 10.0, N, s) for s in range(3)]) for N in Ns]
    sK, sL, sN = sl(Ks, yK), sl(Ls, yL), sl(Ns, yN)
    check("Cold start is linear in K (paper: O(K L N))", abs(sK - 1.0) < 0.15,
          f"fitted exponent in K = {sK:.3f}")
    check("Cold start is linear in L", abs(sL - 1.0) < 0.15, f"fitted exponent in L = {sL:.3f}")
    check("Cold start is linear in N", abs(sN - 1.0) < 0.15, f"fitted exponent in N = {sN:.3f}")
    RES["cold_start_complexity"] = dict(K=dict(x=Ks, y=list(map(float, yK)), slope=sK),
                                        L=dict(x=Ls, y=list(map(float, yL)), slope=sL),
                                        N=dict(x=Ns, y=list(map(float, yN)), slope=sN))

    # per-round wall-clock of Algorithm 2 vs d, and epoch-refit cost vs d
    def timing(d, T=8000, K=5):
        beta0 = [np.zeros(d) for _ in range(K)]; S0 = [0.2 * np.eye(d) for _ in range(K)]
        env = SyntheticEnv(K, d, 0.05, beta0, S0, rng=np.random.default_rng(1))
        a = RCB(K, d, 0.05, beta0, S0, N=5, L=4.0, offset=0.5, phi0=1 / d,
                use_empirical_EF=True, rng=np.random.default_rng(2))
        for _ in range(2000):
            x = env.context(); r, i = a.recommend(x); a.update(x, r, env.pull(x, r), i)
        t0 = time.perf_counter()
        for _ in range(T):
            x = env.context(); r, i = a.recommend(x); a.update(x, r, env.pull(x, r), i)
        return (time.perf_counter() - t0) / T * 1e6      # microseconds per round

    ds = [2, 4, 8, 16, 32, 64]
    yt = [timing(d) for d in ds]
    st = sl(ds, yt)
    check("Algorithm 2 per-round cost grows sub-cubically in d (per-round work is O(Kd))",
          st < 1.6, f"fitted exponent in d = {st:.3f}; per-round us = "
                    f"{[round(v,1) for v in yt]}")
    RES["per_round_us"] = dict(d=ds, us=yt, slope=st)

    # offline-oracle refit: separate the O(d^3) linear solve from the O(n d^2) Gram
    def timeit(f, reps):
        f(); t0 = time.perf_counter()
        for _ in range(reps):
            f()
        return (time.perf_counter() - t0) / reps * 1e6

    def solve_us(d, reps=50):
        rng = np.random.default_rng(0)
        A = rng.normal(size=(d, d)); A = A @ A.T + d * np.eye(d); b = rng.normal(size=d)
        return timeit(lambda: np.linalg.solve(A, b), reps)

    def gram_us(d, n=4000, reps=10):
        rng = np.random.default_rng(0)
        X = rng.normal(size=(n, d))
        return timeit(lambda: X.T @ X, reps)

    # fit in the asymptotic regime; below d~128 the measurement is dominated by
    # constant LAPACK/Python call overhead rather than by arithmetic
    ds2 = [128, 256, 512, 1024]
    ysolve = [solve_us(d) for d in ds2]
    ygram = [gram_us(d) for d in ds2]
    s_solve, s_gram = sl(ds2, ysolve), sl(ds2, ygram)
    check("Oracle linear solve scales close to the paper's stated O(d^3)",
          2.2 < s_solve < 3.4, f"fitted exponent in d = {s_solve:.3f} (theory 3, wall-clock fit); "
                               f"us = {[round(v,1) for v in ysolve]}")
    check("Oracle Gram formation scales close to O(n d^2), the true dominant cost",
          1.6 < s_gram < 2.6, f"fitted exponent in d = {s_gram:.3f} (theory 2, wall-clock fit); "
                              f"us = {[round(v,1) for v in ygram]}")
    RES["refit_us"] = dict(d=ds2, solve_us=ysolve, gram_us=ygram,
                           slope_solve=s_solve, slope_gram=s_gram)


if __name__ == "__main__":
    structural(); complexity()
    RES["n_fail"] = len(FAIL); RES["failed"] = FAIL
    os.makedirs("outputs", exist_ok=True)
    json.dump(RES, open("outputs/claim3_structure.json", "w"), indent=1)
    print(f"\n{len(RES['checks'])} checks, {len(FAIL)} failed")
    sys.exit(1 if FAIL else 0)