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ea3a71e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 | """Sharper tests for FedDPO / DecDPO convergence claims: fitted rate exponents
and parameter-free ratio checks, at larger scale than the first pass.
Log-linear DPO: reward r(x) = theta^T phi(x); on a preference pair (w, l) the
loss is -log sigmoid(beta (r_w - r_l)). Clients hold heterogeneous preference
data generated by client-specific ground-truth rewards.
"""
import json
import numpy as np
RESULTS = {}
D, BETA = 32, 1.0
def make_clients(N, kappa, seed, n_per=200):
rng = np.random.default_rng(seed)
base = rng.normal(size=D); base /= np.linalg.norm(base)
cl = []
for i in range(N):
t = base + kappa * rng.normal(size=D) / np.sqrt(D)
t /= np.linalg.norm(t)
W = rng.normal(size=(n_per, D)); L = rng.normal(size=(n_per, D))
flip = (W - L) @ t < 0
W2 = np.where(flip[:, None], L, W); L2 = np.where(flip[:, None], W, L)
cl.append((W2, L2, t))
return cl, base
def grad(th, W, L):
z = BETA * ((W - L) @ th)
s = 1.0 / (1.0 + np.exp(z))
return -BETA * ((W - L) * s[:, None]).mean(axis=0)
def loss(th, cl):
tot = 0.0
for W, L, _ in cl:
z = BETA * ((W - L) @ th)
tot += float(np.mean(np.log1p(np.exp(-z))))
return tot / len(cl)
def fed_dpo(cl, R=200, E=5, S=None, lr=0.5, q_max=0, seed=0):
N = len(cl); S = S or N
rng = np.random.default_rng(seed)
th = np.zeros(D); buf = {}
hist = []
for r in range(R):
sel = rng.choice(N, size=S, replace=False)
deltas = []
for i in sel:
local = th.copy()
for _ in range(E):
local -= lr * grad(local, cl[i][0], cl[i][1])
d = local - th
delay = int(rng.integers(0, q_max + 1)) if q_max else 0
buf.setdefault(r + delay, []).append(d)
for d in buf.pop(r, []):
deltas.append(d)
if deltas:
th = th + np.mean(deltas, axis=0)
hist.append(loss(th, cl))
return np.array(hist)
def metropolis(adj):
n = adj.shape[0]; deg = adj.sum(1)
Wm = np.zeros((n, n))
for i in range(n):
for j in range(n):
if i != j and adj[i, j]:
Wm[i, j] = 1.0 / (1 + max(deg[i], deg[j]))
Wm[i, i] = 1 - Wm[i].sum()
ev = np.sort(np.abs(np.linalg.eigvals(Wm)))[::-1]
return Wm, float(ev[1])
def dec_dpo(cl, Wm, R=200, E=5, lr=0.5):
N = len(cl)
TH = np.zeros((N, D)); hist = []
for r in range(R):
for i in range(N):
for _ in range(E):
TH[i] -= lr * grad(TH[i], cl[i][0], cl[i][1])
TH = Wm @ TH
hist.append(float(np.mean(np.linalg.norm(TH - TH.mean(0), axis=1))))
return np.array(hist)
def claim2_participation():
rows = []
N = 20
cl, _ = make_clients(N, 0.8, seed=1)
for S in (2, 5, 10, 20):
finals = [fed_dpo(cl, R=150, S=S, seed=100 + s)[-20:].mean() for s in range(5)]
var = [np.var(fed_dpo(cl, R=150, S=S, seed=200 + s)[-20:]) for s in range(5)]
rows.append({"S": S, "N": N, "final_loss": float(np.mean(finals)),
"tail_variance": float(np.mean(var)),
"one_over_S": 1.0 / S})
print(" S=%-3d final loss=%.6f tail var=%.3e (1/S=%.3f)" %
(S, rows[-1]["final_loss"], rows[-1]["tail_variance"], 1.0 / S), flush=True)
ls = np.log([r["one_over_S"] for r in rows]); lv = np.log([max(r["tail_variance"], 1e-16) for r in rows])
RESULTS["claim2_participation"] = {
"rows": rows, "loglog_slope_var_vs_1_over_S": round(float(np.polyfit(ls, lv, 1)[0]), 4),
"var_ratio_S2_over_SN": round(rows[0]["tail_variance"] / max(rows[-1]["tail_variance"], 1e-16), 2)}
print(" variance slope vs 1/S = %.3f ; S=2 vs S=N ratio = %.1fx" %
(RESULTS["claim2_participation"]["loglog_slope_var_vs_1_over_S"],
RESULTS["claim2_participation"]["var_ratio_S2_over_SN"]), flush=True)
def claim3_staleness():
rows = []
cl, _ = make_clients(10, 0.8, seed=3)
for q in (0, 1, 2, 5, 10):
f = [fed_dpo(cl, R=150, q_max=q, seed=300 + s)[-20:].mean() for s in range(5)]
rows.append({"q_max": q, "final_loss": float(np.mean(f)),
"sd": float(np.std(f))})
print(" q_max=%-3d final loss=%.6f +- %.6f" % (q, rows[-1]["final_loss"], rows[-1]["sd"]), flush=True)
base = rows[0]["final_loss"]
RESULTS["claim3_staleness"] = {
"rows": rows, "monotone_in_q": all(rows[i+1]["final_loss"] >= rows[i]["final_loss"] - 1e-9
for i in range(len(rows) - 1)),
"penalty_at_qmax10": round(rows[-1]["final_loss"] - base, 6)}
def claim5_topology():
N = 8
cl, _ = make_clients(N, 0.8, seed=5)
tops = {}
ring = np.zeros((N, N), int)
for i in range(N):
ring[i, (i + 1) % N] = ring[(i + 1) % N, i] = 1
tops["ring"] = ring
star = np.zeros((N, N), int); star[0, 1:] = star[1:, 0] = 1
tops["star"] = star
full = np.ones((N, N), int) - np.eye(N, dtype=int)
tops["complete"] = full
path = np.zeros((N, N), int)
for i in range(N - 1):
path[i, i + 1] = path[i + 1, i] = 1
tops["path"] = path
rows = []
for name, adj in tops.items():
Wm, rho = metropolis(adj)
h = dec_dpo(cl, Wm, R=120)
rows.append({"topology": name, "rho": round(rho, 4),
"one_over_1_minus_rho2": round(1.0 / (1 - rho ** 2), 3),
"final_consensus_error": float(h[-1]),
"mean_tail_consensus": float(h[-20:].mean())})
print(" %-9s rho=%.4f 1/(1-rho^2)=%8.2f consensus err=%.4e" %
(name, rho, rows[-1]["one_over_1_minus_rho2"], rows[-1]["mean_tail_consensus"]), flush=True)
x = np.log([r["one_over_1_minus_rho2"] for r in rows])
y = np.log([max(r["mean_tail_consensus"], 1e-16) for r in rows])
sl, ic = np.polyfit(x, y, 1)
r2 = 1 - np.var(y - (sl * x + ic)) / np.var(y)
RESULTS["claim5_topology"] = {"rows": rows,
"loglog_slope_consensus_vs_1_over_1_minus_rho2": round(float(sl), 4),
"r2": round(float(r2), 4)}
print(" consensus error vs 1/(1-rho^2): slope %.3f, R2 %.3f" % (sl, r2), flush=True)
if __name__ == "__main__":
claim2_participation(); claim3_staleness(); claim5_topology()
json.dump(RESULTS, open("dpo_results.json", "w"), indent=1)
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