"""Appendix F Settings 1-4 plus the empirical regret-rate fits for Claim 1. Setting 1 (Environment Effects), verbatim: "We consider RCB's robustness in terms of different K = [2, 5, 10], d = [3, 5, 10]. For rest parameters, we set T = 10^5, sigma = 0.05, eps = 0.05, tau_P0 = 0.01, and rho_P0 = 0.95. The prior are set to be beta_{i,0} = 0_d and Sigma_{i,0} = 1/5 Id." (Figure 3 instead shows K = [3, 5, 10] and d = [2, 5, 10]; we run the union K in {2,3,5,10} x d in {2,3,5,10}, the most demanding reading.) Setting 2 (Ad-hoc Design): N fixed to {10, 100, 1000}, everything else as Setting 1. Setting 3 (eps effects): T = 5e4, K = 5, d = 5, eps in {0.01,0.03,0.05}, Sigma_{i,0} = 1/lambda Id, lambda in {3,5,10}. Setting 4 (Prior decay / Assumption 4 mis-specification): T = 5e4, K = 5, d = 5, eps = 0.05, beta_{1,0} = 1_5, beta_{i,0} = 0_5; Sigma_{i,0} in {0.02,0.04,0.1} I; decay modes linear / sqrt / log. """ import argparse, 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, N_eps, L_eps from rcb.env import SyntheticEnv def simulate(T, K, d, sigma=0.05, eps=0.05, tau_P0=0.01, rho_P0=0.95, prior_var=0.2, beta0_first=None, N=None, C_N=None, trust_mode="linear", seed=0, gain_every=50, offset=0.5): rng = np.random.default_rng(seed) beta0 = [np.zeros(d) for _ in range(K)] if beta0_first is not None: beta0[0] = np.full(d, beta0_first) Sigma0 = [prior_var * np.eye(d) for _ in range(K)] env = SyntheticEnv(K, d, sigma, beta0, Sigma0, offset=offset, rng=np.random.default_rng(seed + 99991)) phi0 = env.phi0 if N is None: N = max(1, int(round(N_eps(K, d, sigma, eps, tau_P0, phi0, C=C_N)))) L = L_eps(eps, tau_P0, rho_P0) algo = RCB(K, d, sigma, beta0, Sigma0, N=N, L=L, offset=offset, phi0=phi0, trust_mode=trust_mode, gamma_const=4.0, use_empirical_EF=True, rng=rng) regret = np.zeros(T) gains, gain_t = [], [] cum = 0.0 for t in range(T): x = env.context() mu = env.mean_rewards(x) rec, info = algo.recommend(x) if t % gain_every == 0: gains.append(algo.dbic_gain_expected(x, algo.rec_kernel(x, info))) gain_t.append(t) cum += float(mu.max() - mu[rec]) regret[t] = cum y = env.pull(x, rec) algo.update(x, rec, y, info) Tcold = algo.Tcold if algo.Tcold else T return dict(regret=regret, gains=np.array(gains), gain_t=np.array(gain_t), Tcold=Tcold, N=N, L=L, phi0=phi0, regret_total=float(regret[-1]), regret_exploit=float(regret[-1] - regret[min(Tcold, T - 1)])) def slope(xs, ys): return float(np.polyfit(np.log(xs), np.log(ys), 1)[0]) def main(): ap = argparse.ArgumentParser() ap.add_argument("--out", default="outputs/synthetic.json") ap.add_argument("--seeds", type=int, default=5) ap.add_argument("--N", type=int, default=20, help="cold-start size for the rate fits") ap.add_argument("--only", default="all") ap.add_argument("--C_N3", type=float, default=1e-3) args = ap.parse_args() res, t0 = {}, time.time() S = range(args.seeds) # -------- Is Theorem 1's N(eps) even feasible at the paper's own horizons? -- if args.only in ("all", "feas"): print("== Theorem 1 feasibility: cold-start length K*L*N(eps) vs the paper's T ==") feas = [] for tag, T, K, d, sigma, eps, tau, phi0 in [ ("Setting 1 K=2,d=2", 1e5, 2, 2, 0.05, 0.05, 0.01, 1 / 2), ("Setting 1 K=5,d=5", 1e5, 5, 5, 0.05, 0.05, 0.01, 1 / 5), ("Setting 1 K=10,d=10", 1e5, 10, 10, 0.05, 0.05, 0.01, 1 / 10), ("Setting 3 eps=0.01", 5e4, 5, 5, 0.05, 0.01, 0.01, 1 / 5), ("Setting 3 eps=0.05", 5e4, 5, 5, 0.05, 0.05, 0.01, 1 / 5), ("Warfarin eps=0.025", 5528, 3, 70, 0.054, 0.025, 0.005, 1.81e-4), ("Warfarin eps=0.045", 5528, 3, 70, 0.054, 0.045, 0.005, 1.81e-4)]: N = N_eps(K, d, sigma, eps, tau, phi0) L = L_eps(eps, tau, 0.95) cold = K * L * N feas.append(dict(tag=tag, T=T, N=N, L=L, cold=cold, ratio=cold / T)) print(f" {tag:22s} T={T:<8.0f} N(eps)={N:.3e} L={L:6.1f} " f"K*L*N={cold:.3e} = {cold/T:.3e} x T") res["thm1_feasibility"] = feas # ---------------- Setting 1: full K x d grid at the paper's T = 1e5 ------ if args.only in ("all", "s1"): print("== Setting 1 (T=1e5, K x d grid, eps=0.05) ==") s1 = [] for K in [2, 3, 5, 10]: for d in [2, 3, 5, 10]: rs = [simulate(100_000, K, d, N=args.N, seed=s) for s in S] e = dict(K=K, d=d, N=rs[0]["N"], L=rs[0]["L"], Tcold=float(np.mean([r["Tcold"] for r in rs])), regret=float(np.mean([r["regret_total"] for r in rs])), regret_sd=float(np.std([r["regret_total"] for r in rs])), gain_min=float(np.mean([r["gains"].min() for r in rs])), gain_frac_ok=float(np.mean([(r["gains"] >= -0.05).mean() for r in rs])), regret_curve=np.mean([r["regret"] for r in rs], 0)[::500].tolist()) s1.append(e) print(f" K={K:2d} d={d:2d} Tcold={e['Tcold']:6.0f} R(T)={e['regret']:8.1f}" f" min-gain={e['gain_min']:+.4f} frac(gain>=-eps)={e['gain_frac_ok']:.3f}") res["setting1"] = s1 # rate fits over the grid for name, key, fixed in [("K", "K", "d"), ("d", "d", "K")]: fits = {} for fv in [2, 3, 5, 10]: sub = [e for e in s1 if e[fixed] == fv] fits[fv] = slope([e[key] for e in sub], [e["regret"] for e in sub]) res[f"setting1_slope_{name}"] = fits print(f" regret exponent in {name}: " + ", ".join(f"{fixed}={k}: {v:.3f}" for k, v in fits.items())) # ---------------- regret rate in T (the core of Claim 1) ---------------- if args.only in ("all", "rate"): print("== Regret rate fits: R_exploit(T) vs T, K, d ==") rate = {} Ts = [2 ** k for k in range(13, 18)] for (K, d) in [(3, 5), (5, 5), (10, 5), (5, 10)]: ys = [] for T in Ts: rs = [simulate(T, K, d, N=args.N, seed=s) for s in S] ys.append(float(np.mean([r["regret_exploit"] for r in rs]))) rate[f"K{K}_d{d}"] = dict(T=Ts, regret=ys, slope=slope(Ts, ys)) print(f" K={K} d={d}: exponent in T = {rate[f'K{K}_d{d}']['slope']:.3f} {ys}") # K and d exponents at fixed large T T = 2 ** 17 Ks = [2, 3, 5, 10, 20] yK = [float(np.mean([simulate(T, K, 5, N=args.N, seed=s)["regret_exploit"] for s in S])) for K in Ks] ds = [2, 3, 5, 10, 20] yd = [float(np.mean([simulate(T, 5, d, N=args.N, seed=s)["regret_exploit"] for s in S])) for d in ds] rate["K_sweep"] = dict(K=Ks, regret=yK, slope=slope(Ks, yK)) rate["d_sweep"] = dict(d=ds, regret=yd, slope=slope(ds, yd)) print(f" exponent in K = {rate['K_sweep']['slope']:.3f} (theory 0.5)") print(f" exponent in d = {rate['d_sweep']['slope']:.3f} (theory 0.5)") res["rate"] = rate # ---------------- Setting 2: ad-hoc N ------------------------------------ if args.only in ("all", "s2"): print("== Setting 2 (ad-hoc N in {10,100,1000}) ==") s2 = [] for N in [10, 100, 1000]: for K in [3, 5, 10]: for d in [2, 5, 10]: rs = [simulate(100_000, K, d, N=N, seed=s) for s in S] g = np.concatenate([r["gains"] for r in rs]) e = dict(N=N, K=K, d=d, regret=float(np.mean([r["regret_total"] for r in rs])), gain_min=float(g.min()), frac_violate=float((g < -0.05).mean())) s2.append(e) print(f" N={N:5d} K={K:2d} d={d:2d} R(T)={e['regret']:8.1f}" f" min-gain={e['gain_min']:+.4f} frac(gain<-eps)={e['frac_violate']:.3f}") res["setting2"] = s2 # ---------------- Setting 3: eps effects (Claim 2 tradeoff) -------------- if args.only in ("all", "s3"): print("== Setting 3 (eps x prior variance) ==") s3 = [] for eps in [0.01, 0.03, 0.05]: for lam in [3, 5, 10]: rs = [simulate(50_000, 5, 5, eps=eps, prior_var=1.0 / lam, C_N=args.C_N3, seed=s) for s in S] e = dict(eps=eps, lam=lam, N=rs[0]["N"], L=rs[0]["L"], Tcold=float(np.mean([r["Tcold"] for r in rs])), regret=float(np.mean([r["regret_total"] for r in rs])), gain_min=float(np.mean([r["gains"].min() for r in rs])), frac_ok=float(np.mean([(r["gains"] >= -eps).mean() for r in rs]))) s3.append(e) print(f" eps={eps} 1/lam={1/lam:.3f} N={e['N']:5d} L={e['L']:6.1f}" f" Tcold={e['Tcold']:7.0f} R(T)={e['regret']:8.1f}" f" frac(gain>=-eps)={e['frac_ok']:.3f}") res["setting3"] = s3 # ---------------- Setting 4: Assumption 4 mis-specification -------------- if args.only in ("all", "s4"): print("== Setting 4 (prior decay mis-specification) ==") s4 = [] for pv in [0.02, 0.04, 0.1]: for mode in ["linear", "sqrt", "log", "none"]: rs = [simulate(50_000, 5, 5, eps=0.05, prior_var=pv, beta0_first=1.0, trust_mode=mode, N=args.N, seed=s) for s in S] e = dict(prior_var=pv, mode=mode, regret=float(np.mean([r["regret_total"] for r in rs])), gain_min=float(np.mean([r["gains"].min() for r in rs])), frac_ok=float(np.mean([(r["gains"] >= -0.05).mean() for r in rs]))) s4.append(e) print(f" Sigma0={pv} decay={mode:7s} R(T)={e['regret']:8.1f}" f" min-gain={e['gain_min']:+.4f} frac(gain>=-eps)={e['frac_ok']:.3f}") res["setting4"] = s4 res["wall_clock_s"] = time.time() - t0 os.makedirs("outputs", exist_ok=True) json.dump(res, open(args.out, "w")) print(f"wrote {args.out} [{res['wall_clock_s']:.0f}s]") if __name__ == "__main__": main()