MI for DT · Results brief

Prompt-QFM Divergent16

HM multi-dist checkpoint · MCF projection · J ∈ {0, 2, 8} · 10 seeds · Decision-mean ablation · Full heuristic baselines

Checkpoint prompt_qfm_reentrant_param_hm_nolegacy_g1_j4_action_only/best.pt · Jobs pqfm_d16 1630449 (COMPLETED, 1d18h) · pqfm_d16_dm 1632545 (decision-mean ablation, J=0, 16/16 COMPLETED)

1. End-to-End Pipeline

• Metric: cost_rate = total_cost / time (lower better).
• Fixed RTG = −0.02; seeds {0…9}; action projection min_cost_flow.
• Suite: 16 divergent tasks (6 id / 4 near_id / 6 ood) from prompt_eval_divergent16.json.
• Architecture / training details: §2 (dims, embeddings, PoolCount→MCF, HM data).
• Decision-mean ablation: replace DT decision vector d_t by a train-set mean; keep action head + live graph.

2. Model Architecture, Embeddings & Action

2.1 Model dimensions (this checkpoint)

2.2 Input features & embeddings

Prompt-QFM defaults to queue_only=True: before the graph encoder, strip_distribution_priors zeros distribution / cost channels so that λ, μ, h, etc. must be inferred from the trajectory (actions + rewards / RTG), not read from explicit features. Topology needed for legal actions is kept.

What is kept (non-zero structural / state):

• Queue length: dyn_q[...,0] = Q (raw length; upstream stores log1p(Q), then expm1 recovers Q)
• Pool: [c_p, 1] — capacity is an action constraint
• Masks / structure: edge_mask, routing, capacity, q_mask / p_mask, Δt (Time2Vec)
• Trajectory tokens: actions, rewards, RTG

What is explicitly zeroed every forward:

• dyn_q[...,1:] → λ(t), holding cost h → 0
• Entire static_q = [h, λ̄, τ, d^out] → 0
• edge_mu (service rates μ_pi on eligible edges) → 0

So the Fig. A 7-D / μ layout still defines the tensor shapes into the shared MLPs (Linear(7→d), edge MLP on μ, etc.), but those prior channels are constant zeros at train and eval for this checkpoint — not free parameters set to zero. The MLP weights themselves are trained; they just usually see zeros on those inputs.

Spatial pipeline (after stripping): queue MLP on [Q, 0, 0 | 0,0,0,0] · pool MLP on [c_p,1] · local MP with zeroed μ embeddings · 1 global spatial Transformer over [g, h^q, h^p] + type / Δt. Outputs: s_t=g → DT; {h^p}, {h^q} → action head (with μ channel also zero in PoolCount’s W_μμ term).

Temporal token embeddings (into the causal DT):

• s: graph token (prompt stream: linear map with identity init)
• r̂ / r: Linear(1→d)→GELU→Linear(d→d)
• a: ContextualActionEmbedder — each eligible edge (p,q) gets [log(1+A_pq), A_pq/c_p, h^p, h^q] → MLP → edge tokens + CLS → 1-layer Transformer → CLS = action token

2.3 Action output (PoolCount → MCF)

PoolCount head (continuous allocation):

\[ z_{pi}=[d_t,\; h_p,\; h_i,\; W_\mu\mu_{pi}]\in\mathbb{R}^{4d},\quad \ell_{pi}=\mathrm{MLP}_{\rm score}(z_{pi}),\quad \ell_{p,\rm idle}=\mathrm{MLP}_{\rm idle}([d_t,h_p]) \] \[ \pi_p=\mathrm{softmax}([\ell_{p1},\ldots,\ell_{pQ},\ell_{p,\rm idle}]),\qquad \hat a_{pi}=c_p\,\pi_{pi} \]

• Ineligible edges masked; per-pool softmax over queues and idle preserves capacity ∑_i â_pi + â_p,idle = c_p
• Training (this run): action KL only (--rtg-loss-weight 0 --reward-loss-weight 0); supervise continuous â against expert allocations

Rollout projection (this eval): raw edge scores / continuous â → min_cost_flow_pool_action (MCF) to a feasible integer (P×Q) assignment for QGym (work-conserving max-score matching under capacity / queue counts). Alternate option in code: greedy priorities_to_pool_action.

Decision-mean ablation: replace d_t by a train-set mean vector; PoolCount still sees live h^p, h^q, μ — isolates whether the temporal decision content matters.

2.4 Training configuration

3. Eval Setup

4. Full Heuristic Baselines

All four heuristics on the divergent16 suite (mean ± std, 10 seeds). Bold = lowest among heuristics.

Source: j0_vs_baselines.tsv. MaxQ = topology-only M_pi·Q_i. On Exp/hm reentrant tasks cμ often wins; on hh OOD MaxP often wins; MaxQ is strong on n_model-like tasks and weak on deep reentrant / OOD.

Full 56-task representative suite (baseline rollout)

Complete heuristic baseline rollout on the heuristic_baseline_representative.json manifest (20 in-sample param/hm variants + 36 OOD reentrant / hospital / input_switch tasks). Policies: cμ, cμQ, MaxP, MaxQ · 10 seeds · mean ± std (population std over seeds). Bold = lowest cost among the four heuristics per row.

cμ wins most reentrant / param-shift rows; MaxP dominates hh OOD and several criss-cross variants; MaxQ is best on n_model-like tasks but weak on deep reentrant OOD (often 1.3–2× best base). Deterministic hospital/input_switch rows have zero variance (single effective trajectory per seed).

5. Prompt-QFM J-sweep vs All Baselines

Combined view (Model_I_Long style). Bold = lowest among all methods in the row.

vs Best Baseline: Δ and Welch p (two-sided)

Δ = PQFM − best_base (positive = worse than best heuristic). p from Welch two-sample t-test on seed means, using reported mean±std with n=10 each side (unequal variance). Bold red = p<0.05.

Welch t uses only summary statistics (mean, std, n=10); equivalent to testing equality of the two seed-mean distributions under normality. All Δ>0 (PQFM never beats best base on the point estimate). Significant deficits (p<0.05) concentrate on hard reentrant / n_model tasks; several OOD gaps (e.g. J=0 on reentrant_8/10, re-reentrant_8) are not significant. The same Welch protocol is used for all p-values below.

vs MaxQ: Δ and Welch p

Δ = PQFM − MaxQ (negative = better than MaxQ). Bold red = p<0.05.

Prompt J pairwise: Δ and Welch p

Δ = J_a − J_b (negative = longer prompt better). Tests whether changing J shifts mean cost.

Almost no J pairwise difference is significant at α=0.05 (only J8 vs J2 on reentrant_10, p=0.036). Point-estimate prompt gains are real but underpowered / high-variance with 10 seeds.

6. Decision-Mean Ablation (J=0)

Replace d_t by train mean. Status: 16 / 16 COMPLETED (J=0 only). All comparisons: Welch two-sided p, n=10 each side; bold red = p<0.05.

Call = DT helps iff Δ(dm−J0)>0 and p<0.05. Note: on several OOD tasks both J0 and dm beat MaxQ significantly even when dm≈J0 — winning MaxQ does not require a nontrivial decision vector. hh OOD: point estimates favour dm (especially reentrant_6__hh Δ=−22.6, p=0.087), but dm vs J0 is not significant at α=0.05 on any of the three.

Where DT helps: is MaxQ ≈ dm?

7. Summary