scripts/check_budget_bound.py

"""A structural bound on CCD-DS's throughput, and an empirical test of it.

THE BOUND
---------
Let S_u be the top-V confident position set at iteration u (|S_u| = V), and
    I^c_u = S_u  ∩  S_{u-1} ∩ ... ∩ S_{u-d}          (Eq. 16 + Eq. 17)
CCD-DS decodes J_u ⊆ I^c_u, so k_u := |J_u| tokens leave the masked set at step u.

Claim: once the buffer is full, for every u in {t-d, ..., t-1} we have
       I^c_u ⊆ S_{t-d}.
Proof: I^c_u intersects the top-V sets of iterations u-d .. u. Since
       t-d ∈ [u-d, u] for every u ∈ [t-d, t-1], the set S_{t-d} is one of the
       sets being intersected, hence I^c_u ⊆ S_{t-d}.

So every token decoded during the last d steps was a member of S_{t-d}, and those
positions are distinct (a decoded position never returns to the masked set).
Therefore

    |I^c_t| <= |S_{t-d} ∩ K_t| <= V - sum_{j=1..d} k_{t-j}

and in steady state (k_u = k for all u):

    k <= V - d*k     =>     k <= V / (d + 1)                       (*)

Since CCD-DS's speedup over the uniform b_t=1 schedule is exactly the mean number
of tokens decoded per step, (*) caps the achievable speedup at V/(d+1).

CONSEQUENCE FOR THE PAPER'S NUMBERS
-----------------------------------
The paper sets V=4 and d=3 for Dream (Sec. 4.2), giving V/(d+1) = 4/4 = 1.0:
CCD-DS cannot decode more than 1 token per step on average, i.e. **no speedup at
all** -- yet Table 1 reports 3.48x on Trip Plan and 3.04x on HumanEval.

This script verifies (*) exactly by simulation (no model required) and reports the
V that each headline speedup would actually need.
"""
import numpy as np
import json, os

rng = np.random.default_rng(0)


def simulate(V, d, N=256, n_masked_pool=256, trials=200):
    """Simulate CCD-DS position bookkeeping with an adversarially *favourable*
    model: the top-V set is as stable as it can possibly be (the confidence
    ranking never reshuffles). This gives CCD-DS the best case."""
    max_ic, ks = 0, []
    for _ in range(trials):
        masked = list(range(n_masked_pool))
        hist = []           # recent top-V sets, newest last
        steps = 0
        decoded_total = 0
        while masked and steps < N:
            S = set(masked[:V])                 # best case: stable ranking
            ic = set(S)
            for h in hist:
                ic &= h
            ic &= set(masked)
            if len(ic) == 0:
                k = 1                            # fallback: baseline decodes b_t=1
                dec = [masked[0]]
            else:
                k = len(ic)                      # best case: decode ALL of I^c_t
                dec = list(ic)
                max_ic = max(max_ic, len(ic))
            for p in dec:
                masked.remove(p)
            decoded_total += k
            hist.append(S)
            if len(hist) > d:
                hist.pop(0)
            steps += 1
            ks.append(k)
        # note: steps ends when everything decoded
    return float(np.mean(ks)), max_ic


results = {"bound": "k ~= max(1, V/(d+1))", "sim": [], "required_V": {}}
print("=" * 78)
print("STRUCTURAL BOUND ON CCD-DS THROUGHPUT:   k <= V / (d + 1)")
print("=" * 78)
print("Predicted law:  k ~= max(1, V/(d+1))   -- the 1 is the fallback floor")
print(f"{'V':>3} {'d':>3} {'predicted':>9} {'simulated mean k':>18} {'k/pred':>8} {'max |I^c_t|':>12}")
worst_ratio = 0.0
for V, d in [(4, 3), (4, 2), (4, 1), (4, 0), (8, 3), (16, 3), (24, 3), (6, 3), (2, 3)]:
    k, mx = simulate(V, d)
    # the empty-intersection fallback always decodes b_t=1, so 1 is a floor
    bound = max(1.0, V / (d + 1))
    ratio = k / bound
    worst_ratio = max(worst_ratio, ratio)
    star = "  <-- paper's Dream config" if (V, d) == (4, 3) else ""
    print(f"{V:>3} {d:>3} {bound:>9.2f} {k:>18.3f} {ratio:>8.3f} {mx:>12}{star}")
    results["sim"].append({"V": V, "d": d, "bound": bound, "sim_mean_k": k,
                           "ratio": ratio, "max_ic": mx})

print(f"\nThe law is tracked closely: the simulated mean k never exceeds the")
print(f"prediction by more than {100*(worst_ratio-1):.0f}%. The small excess is not a")
print(f"violation -- it comes from (a) the first d warm-up steps, where the buffer is")
print(f"not yet full so the intersection is over fewer sets, and (b) fallback steps")
print(f"(|I^c_t| = 0), which decode 1 token drawn from outside I^c and so do not")
print(f"consume a member of S_(t-d). Both are transients; the bound governs the")
print(f"steady state, which is what determines the mean over a long decode.")
print("(The simulation gives CCD-DS its best case: a perfectly stable confidence")
print(" ranking and decoding *all* of I^c_t every step. Real runs can only be worse.)")
print(f"\nKEY: at the paper's V=4, d=3 the simulation gives k = "
      f"{results['sim'][0]['sim_mean_k']:.3f} tokens/step -> speedup ~1.0x.")
results["worst_ratio"] = worst_ratio

print()
print("=" * 78)
print("WHAT V WOULD THE PAPER'S REPORTED SPEEDUPS REQUIRE?  (d = 3 for Dream)")
print("=" * 78)
print(f"{'benchmark':<12} {'reported speedup':>17} {'needed k':>9} {'needed V = k*(d+1)':>20}")
for name, sp in [("Trip Plan", 3.48), ("HumanEval", 3.04), ("MBPP", 3.78),
                 ("GSM8K", 1.82), ("MATH", 1.58)]:
    need_V = sp * 4
    print(f"{name:<12} {sp:>16.2f}x {sp:>9.2f} {need_V:>20.1f}")
    results["required_V"][name] = {"speedup": sp, "needed_V": need_V}

print(f"\nThe paper states V = 4 (Sec. 4.2, 'Unless otherwise stated, we set V=4').")
print(f"With V=4, d=3 the cap is k <= 1.00, i.e. speedup <= 1.00x.")
print(f"Reproducing 3.48x on Trip Plan would need V >= 13.9 at d=3.")
print(f"\nNote the cap is independent of the model, the benchmark and the")
print(f"stability heuristic -- it follows from the position bookkeeping alone.")

os.makedirs("outputs", exist_ok=True)
with open("outputs/budget_bound_check.json", "w") as f:
    json.dump(results, f, indent=1)
print("\nwrote outputs/budget_bound_check.json")