{"id":"e2ff57f3-a5cf-48aa-9d57-64c43d08b697","arxiv_id":"2507.18961","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The paper introduces HGT, a batched bandit algorithm for network formation that learns latent types and complementarities via variational inference and updates pairings by maximum-weight matching.","lead":"An adaptive policy for network formation repeatedly pairs agents, learns which hidden agent types work best together, and re-optimizes pairings each batch. Simulations indicate it approaches the performance of a fully informed oracle within a few batches, with theory supporting the learning step.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (5.2) treats each batch's LVA as an independent categorical likelihood for Z_i, but Section 4 provides no calibration result for q_{t,i}; if q_{t,i} is miscalibrated or data-dependent, the Thompson-in-Z step and the matching policy can be misled.","rationale":"The reader's weakest_assumption identifies the same Eq. (5.2) concern, and I agree with it. The paper's simulation evidence is real and suggests the algorithm works in the tested regimes, which is why I do not call for rejection. But the theoretical foundation for the Z-signal is absent: Section 4's LAN and variational-equivalence results are for the global parameter and for whole-experiment likelihood ratios, not for the calibration of per-agent LVAs as likelihoods. The proof of Theorem 2 establishes that J_{pi,theta;q-hat}(Y) equals log f_{pi,theta}(Y,Z) plus o_P(1), but this asymptotic likelihood-ratio equivalence does not make each q_i(a) a valid categorical signal for a sequential Bayes update; whole-experiment equivalence and per-agent signal calibration are different statements. Since the algorithm's exploration in Z relies directly on Eq. (5.2), the conditional verdict remains appropriate pending the calibration check. I therefore keep the reader's CONDITIONAL verdict and classify the change as UNCHANGED.","tokens_in":26332,"tokens_out":11506,"duration_ms":123047,"concrete_test":"Run a calibration check for Eq. (5.2): in the K=2 setting with n=16, fix true Z over T=6 batches and 1,000 replications; after each batch, given the fitted theta-hat_t and pi-hat_t used to produce q_{t,i}, enumerate (2^15 terms per agent) the exact marginal likelihood L_{t,i}(a)=P(Y_t | Z_i=a, theta-hat_t, pi-hat_t) and compute the exact sequential posterior omega^exact_{t,i} proportional to omega^exact_{t-1,i} L_{t,i}(a). Compare with Eq. (5.2)'s omega_{t,i}; if average total-variation distance exceeds 0.05 or exact-likelihood regret at batch 6 is more than 10% lower than HGT's, the categorical-signal interpretation fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that batched LVA estimates can be treated as categorical signals enabling principled Bayesian updating is used in Eq. (5.2): omega_{t,i}(a) is proportional to omega_{t-1,i}(a) q_{t,i}(a). For this to be a valid likelihood update, q_{t,i} must be, or approximate, the conditional likelihood of the batch data given Z_i=a, calibrated across agents and independent of the information already in omega_{t-1,i}. Section 4 does not establish this. Theorem 2 proves asymptotic equivalence of the variational estimator to the complete-graph MLE for (omega, theta) and, at best, consistency of q_i toward the true indicator; consistency does not imply calibration. Moreover, q_{t,i} is computed from the same batch data and from the same theta-hat_t that is separately used in Eq. (5.1), so the two signals are correlated. The manuscript itself says q is used as categorical approximations to the posterior distribution of Z, not as a likelihood. If q is miscalibrated, Eq. (5.2) compounds the error across batches and the maximum-weight matching in (5.3) may pair agents on the basis of incorrect type draws. This is the load-bearing weak point of the algorithm's theoretical foundation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes an online, batched network formation algorithm for weighted stochastic block models in which agents' latent types are unobserved and only pairwise edge outcomes are observed. The authors establish a batched local asymptotic normality result for the complete-graph model (Proposition 1), an asymptotic equivalence between the marginal likelihood and the complete-data likelihood under sparse network asymptotics (Theorem 1), and an asymptotic equivalence between the variational ELBO and the complete-data likelihood (Theorem 2). They use these results to justify treating each batch's variational estimates of θ and the local variational approximations q_{t,i} as Gaussian and categorical 'signals,' respectively, in a Bayesian updating scheme (Section 5.1), and feed the resulting posteriors into a maximum-weight matching problem to form the next batch's network (HGT algorithm, Section 5.3). Monte Carlo simulations in Section 6 show declining false labeling rates and regret moving toward the oracle benchmark, including with worker turnover.","tokens_in":26706,"tokens_out":8149,"duration_ms":84689,"significance":"If the theoretical link between the variational estimates and the Bayesian signals is made rigorous, the paper makes a genuinely useful contribution: it brings Le Cam's asymptotic theory for batched adaptive experiments into a network setting with latent heterogeneity, and it shows how feasible variational inference can be embedded in an online allocation policy. The paper is also commendable for providing proofs in Appendix A, for explicitly modeling sparse network asymptotics, and for calibrating the simulations to real-world studies (Rohrer et al. 2021; Xu et al. 2024). The main weakness is that the asymptotic results in Section 4 justify consistency and asymptotic normality of point estimates, but they do not establish the calibration and independence properties that the Bayesian update in Eq. (5.2) requires. This gap is directly relevant to the paper's central claim that variational estimates can be treated as normal/categorical signals, and it needs to be addressed before the theoretical foundation claim is supported.","major_comments":[{"comment":"This update treats each batch's local variational approximation q_{t,i}(a) as an independent categorical likelihood for Z_i. Section 4 shows consistency of q_i toward the true indicator and asymptotic equivalence of the variational estimator to the complete-graph MLE, but consistency does not imply calibration: no result in the paper establishes that q_{t,i}(a) equals or approximates the conditional probability of batch t's data given Z_i=a, nor that q_{t,i} is independent of the information summarized in ω_{t-1,i}. In fact, q_{t,i} is computed from the same batch data and from the same θ-hat_t used in Eq. (5.1), so the θ and Z signals from a batch are correlated. Because Eq. (5.2) compounds the approximation error across batches, and because the Thompson-in-Z draws enter the matching problem (5.3), this is the load-bearing step for the paper's claimed 'theoretical foundation for treating variational estimates as normal signals.' The authors should either prove a calibration result for q under Assumptions 1-5, or recast the update as a heuristic and provide a regret or risk analysis that does not rely on exact likelihood interpretation.","section":"Section 5.1, Eq. (5.2)"},{"comment":"The proof of Theorem 1 depends on high-level Assumptions 4 and 5 (the concentration inequalities (A.4)-(A.5), Lipschitz continuity, and the negative directional derivative condition) as well as on the unimodality and identifiability restrictions stated immediately before the theorem. The text asserts at the end of Section 4.2 that these are 'readily verified' for the models used, but the paper does not actually verify them for the binary-outcome WSBM used in Section 6. Unimodality of f_{π,θ} is a strong global condition that is not obviously implied by the local conditions in Assumption 2. Since Theorem 2 inherits these assumptions and Theorem 2 is the theoretical foundation for the Gaussian-signal update (5.1), the manuscript should provide the verification (or a precise citation with the conditions checked) for the simulation DGP and for the classes of p_θ advertised in Section 2.","section":"Section 4.2, Theorems 1-2; Appendix A.2"},{"comment":"The agent-level matching problem uses realized draws \\tilde Z_i from the posterior as if they were the true types, both in the objective (5.3) and in the clipping constraint (5.6). Since \\tilde Z_i are random, the resulting policy need not satisfy the intended type-pair-level constraints (2.5)-(2.6) for the actual latent types, and the expected network output is evaluated under a single posterior draw rather than under the posterior distribution. This weakens the claim that the HGT policy solves the constrained optimization problem (2.7). The paper should justify the plug-in approach, for example by a consistency argument for the matching solution as the posterior concentrates, or by reformulating the constraints in terms of the posterior probabilities of type assignments.","section":"Section 5.2, Eqs. (5.3)-(5.6)"}],"minor_comments":[{"comment":"The objective in Eq. (2.3) appears to be missing the summation over type pairs; the expression 'max_{ψ∈Ψ} ψ(a,b)Υ(a,b)' should be 'max Σ_{a,b} ψ(a,b)Υ(a,b)' in display (2.3) and in (2.7).","section":"Section 2.2, Eq. (2.3)"},{"comment":"There is a duplicated word 'that' in the sentence comparing production models; it reads 'production models that that impose structure.'","section":"Section 1, Introduction"},{"comment":"The legends in Figures 2 and 3 label the algorithm 'HTG'; elsewhere in the paper it is 'HGT'. Please correct.","section":"Figures 2 and 3"},{"comment":"There is a typo 'in these dateset' that should be 'in these datasets.'","section":"Section 4.2, after Assumption 3"},{"comment":"The standard error bse_{t,ab} is introduced in Eq. (5.1) without a formula or implementation details; the simulations should state how it is obtained from the variational estimate (e.g., a plug-in of the asymptotic variance Q^{-1}_{θ,t}).","section":"Section 5.1, Eq. (5.1)"},{"comment":"The statement of Theorem 2(i) asserts the o_P(1) term is uniform over π and θ, but the proof in Appendix A.3 only demonstrates pointwise equality; the proof or statement should be adjusted to track uniformity.","section":"Theorem 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is serious and the simulation section is well designed. My main concern for the editor is the mismatch between the strong claims in the abstract and Section 5.1 and what the theorems in Section 4 actually establish; a revision that either proves the calibration result or tempers the claims would resolve the issue. I see no citation or novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper deserves a serious referee, not because it is airtight but because it is doing something genuinely new: it connects Hirano-Porter batched asymptotics to weighted stochastic block models and turns them into a practical adaptive pairing algorithm. The HGT heuristic—greedy in theta, Thompson-sampled in Z—is sensible, and the simulation results are encouraging, including the nonstationary turnover case.\n\nThe theoretical contribution is real. Proposition 1 gives batched LAN for the complete-graph WSBM cleanly, and Theorem 2's asymptotic equivalence of variational estimates to complete-data MLE under sparse asymptotics is a useful extension of Bickel et al. That said, the proof of Theorem 2 rests on Assumptions 4–5 which are stated but not checked; for binary outcomes they are plausible from Bickel et al., but the paper should verify or explicitly cite the verification. The reader's concern about Equation (5.2) is on target: q_t,i is shown to be consistent for the latent label, but consistency is not calibration. Treating it as an independent categorical likelihood for Z_i in the Bayes update is a heuristic. The same q is also computed from the same batch and the same theta-hat as the theta signal, so the two updates are not independent. This is a genuine gap, but not a fatal one—the simulations suggest the algorithm still works, and the gap could be closed by a calibration argument or by treating (5.2) as an approximation with a clear caveat.\n\nOther soft spots are more minor: the standard errors bse_{t,ab} in (5.1) are never defined, the clipping constraint (5.6) may be infeasible with small m, and the initial policy psi_1 is unspecified. The Monte Carlo lacks any baseline other than the oracle; a greedy-one-shot or a non-adaptive uniform policy would help calibrate how much of the gain is from adaptation. The caption typos (\"HTG\", stray \"33\") are cosmetic.\n\nOverall: the central idea is sound, the asymptotic scaffolding is mostly credible, and the weakness is in the bridge from asymptotics to the Bayesian updating of Z. This is addressable. I would send it to referee with a request to tighten that step and add a baseline.","headline":"A genuinely useful bridge between WSBM network econometrics and batched bandits, with a promising algorithm and a real gap between the asymptotic theory and the Bayesian type-updating step.","tokens_in":27164,"tokens_out":2043,"would_cite":true,"duration_ms":23138,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"By proving that batched variational estimates behave like Gaussian signals, this paper turns adaptive network formation into a Bayesian bandit whose policy improves toward oracle performance within a few batches.","keywords":["adaptive network formation","algorithm design","weighted stochastic block model","batched multi-armed bandits","batched local asymptotic normality","variational approximation","Bayesian updating","maximum weight matching"],"falsifier":"Run a simulation with true latent types known and compare the batch-by-batch posterior (5.2) with the oracle posterior; if the posterior's probability on the true type decreases or its credible-set coverage of the true type falls well below nominal as batches grow, the categorical-signal premise is refuted. Alternatively, artificially sharpen the LVA by exponentiating $q_{t,i}$ and observe whether label accuracy and network value degrade.","tokens_in":26135,"feed_emoji":"📈","tokens_out":9176,"duration_ms":89812,"temperature":0.7,"pith_summary":"An organization that must repeatedly pair agents—work teams, classroom seatmates, platform recommendations—cannot observe the agents' latent types directly, only the outcomes of each pairing. This paper proposes to learn the pair-production parameters and the agent types online, batch by batch, and uses them to improve the next pairing policy. The theoretical core is a batched local asymptotic normality result for the weighted stochastic block model, together with an asymptotic equivalence between the intractable likelihood and its mean-field variational approximation. Together these results justify treating each batch's variational estimates as Gaussian signals for the production parameters and each agent's local variational approximation as a categorical signal for its latent type. That justification converts the learning problem into a Bayesian bandit, and the paper's simulations show the resulting Hybrid Greedy-Thompson policy approaches an oracle's network value within a few batches.","feed_headline":"Network policy hits near-oracle in six batch updates","feed_subtitle":"A theory of Gaussian batch signals plus Bayesian matching reduces simulated regret by over 90 percent in six batches.","key_machinery":"The central object is the equivalence of three likelihoods under batched sparse asymptotics: the complete-graph likelihood with observed agent types, the marginal likelihood with types integrated out, and the mean-field variational ELBO all have the same local Gaussian limit, with per-type-pair variance inversely proportional to the policy's sampling frequency times the Fisher information (Proposition 1 and Theorems 1–2). This identity is what licenses the paper's central move: treat each batch's variational estimator as a Gaussian signal for $\\theta$, treat each agent's local variational approximation as a categorical signal for $Z$, and combine them across batches by Gaussian and categorical Bayesian updating (equations 5.1 and 5.2). The resulting posteriors are plugged into a constrained maximum-weight matching problem whose solution is the next batch's pairing policy, and the Hybrid Greedy-Thompson algorithm is the specific greedy-in-$\\theta$, Thompson-in-$Z$ instance of that machinery.","core_discovery":"The paper's claim is that, under batched sparse network asymptotics, a weighted stochastic block model with unobserved latent types is asymptotically equivalent to a Gaussian shift experiment at the level of likelihood ratios, and the computationally feasible mean-field variational ELBO inherits the same limit as the full marginal likelihood. Consequently, batched variational estimates of the complementarity parameters can be treated as Gaussian signals, and the per-agent local variational approximations can be treated as categorical signals for the latent types. The paper then aggregates these signals by Bayesian updating and converts the resulting posteriors into a network policy by solving a constrained maximum-weight matching problem; in the proposed Hybrid Greedy-Thompson algorithm the production parameters are used at their posterior means while agent types are sampled from their categorical posteriors. Monte Carlo experiments with two and three latent types show the policy's expected total output rising toward the oracle benchmark within about six batches, with false labeling rates declining steadily, and with the algorithm remaining effective—though slower to converge—when agents enter and exit the pool.","pith_inferences":["A direct calibration test of equation (5.2)—checking whether the posterior's coverage of true latent types matches nominal levels in simulations where $Z$ is known—would show whether the LVA can be used at face value as a categorical likelihood or needs tempering.","Nothing in the Gaussian-signal argument fixes the greedy-in-$\\theta$ choice; posterior sampling or upper-confidence-bound rules on $\\theta$ would inherit the same asymptotic justification, making HGT one point on a family of viable policies.","Because the proof relies on local asymptotic normality plus sparse-degree asymptotics rather than on details of the WSBM, the same batched design should transfer to other pairwise latent-type models, such as mixed-membership or dynamically evolving types, if the analogous equivalence theorems hold.","The small finite-sample biases visible in the posterior histograms suggest a practical extension: measuring how bias decays with per-batch sample size would tell practitioners when batched updating dominates a single offline estimate."],"forward_implications":["If the Gaussian/categorical signal interpretation holds, the algorithm's expected total network output approaches the oracle benchmark within a few batches, as the simulations demonstrate for $K=2$ and $K=3$ latent types.","The same asymptotic results justify offline estimation: treating the full sample as a single batch gives variational estimates that are asymptotically normal and equivalent to complete-graph maximum likelihood.","Practical constraints—per-agent workload limits and clipping rates that keep every type pair observable—are built into the policy as integer constraints in the matching problem, so exploration is never starved.","In nonstationary pools with agent entry and exit, the algorithm keeps improving but needs additional batches to relearn new agents' types; its regret converges to a higher floor at higher turnover.","Posteriors for both $\\theta$ and $Z$ sharpen over batches, with false labeling rates roughly halved or better in the simulated settings."],"supporting_citations":[{"why":"defines the weighted stochastic block model of team production with unobserved discrete types and complementarity parameters that this paper builds on.","marker":"Bonhomme (2021)"},{"why":"supplies likelihood-modularity and consistency techniques for stochastic block models that the equivalence proofs extend.","marker":"Bickel and Chen (2009)"},{"why":"establishes asymptotic normality of maximum likelihood and variational estimates for stochastic blockmodels, the technical template for Theorems 1–2.","marker":"Bickel et al. (2013)"},{"why":"provides the multi-stage asymptotic representation theorem for batched adaptive experiments used to justify the Gaussian bandit approximation.","marker":"Hirano and Porter (2025)"},{"why":"supplies the local asymptotic normality definitions and asymptotic-normal theory invoked in Assumption 2 and Proposition 1.","marker":"Van der Vaart (2000)"},{"why":"contributes the posterior-sampling principle that the algorithm applies to agent latent types.","marker":"Thompson (1933)"},{"why":"frames nonparametric identification and estimation of stochastic block models from many small networks, supporting the batched sparse regime.","marker":"Jochmans (2024)"}],"fun_headline_variants":["Batched bandit builds near-oracle networks in six rounds","Six batches to near-optimal network formation","Adaptive policy nears oracle network output in six updates","Batched Bayesian matching achieves near-optimal in six batches"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithm's type-learning step assumes each batch's local variational approximation acts as an independent, well-calibrated categorical signal for that agent's latent type; the paper proves global asymptotic normality of parameter estimates but does not prove this per-agent signal interpretation, and miscalibrated or data-correlated signals would mislead the matching policy.","fun_headline_variants_meta":{"raw":{"variants":["Batched bandit builds near-oracle networks in six rounds","Six batches to near-optimal network formation","Adaptive policy nears oracle network output in six updates","Batched Bayesian matching achieves near-optimal in six batches"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000448,"raw_usage":{"total_tokens":2299,"prompt_tokens":1022,"completion_tokens":1277,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":1212}},"tokens_in":638,"tokens_out":1277,"duration_ms":9709,"temperature":1.0,"reasoning_tokens":1212,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:05:04.806839+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a simulation with true latent types known and compare the batch-by-batch posterior (5.2) with the oracle posterior; if the posterior's probability on the true type decreases or its credible-set coverage of the true type falls well below nominal as batches grow, the categorical-signal premise is refuted. Alternatively, artificially sharpen the LVA by exponentiating $q_{t,i}$ and observe whether label accuracy and network value degrade.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies likelihood-modularity and consistency techniques for stochastic block models that the equivalence proofs extend."},{"cited_title":"(2013), Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels, The Annals of Statistics, 41, 1922--1943","cited_arxiv_id":null,"evidence_quote":"establishes asymptotic normality of maximum likelihood and variational estimates for stochastic blockmodels, the technical template for Theorems 1–2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the local asymptotic normality definitions and asymptotic-normal theory invoked in Assumption 2 and Proposition 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"contributes the posterior-sampling principle that the algorithm applies to agent latent types."},{"cited_title":"(2024), Nonparametric Identification And Estimation of Stochastic Block Models From Many Small Networks, TSE Working Paper","cited_arxiv_id":null,"evidence_quote":"frames nonparametric identification and estimation of stochastic block models from many small networks, supporting the batched sparse regime."}],"review_version":1}