{"id":"eb3b32c2-9366-4ddf-b4e0-f15aac14cd63","arxiv_id":"2605.17934","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"C-SymmPI reformulates conditional coverage as miscoverage error over a user-specified function class to deliver near-conditional guarantees under group symmetries and distributional invariance.","lead":"The paper introduces C-SymmPI, a framework extending SymmPI to achieve near-conditional coverage for predictive inference on structured data with group symmetries such as networks and clusters. This matters for practitioners needing reliable uncertainty estimates that adapt to data heterogeneity rather than assuming simple exchangeability.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Near-conditional coverage bound depends on how well the user-specified function class approximates the relevant conditional expectation; rates for linear/RKHS classes may not transfer to network/cluster heterogeneity without explicit verification.","rationale":"Reader correctly flags distributional invariance as necessary for the guarantees. However, even under invariance, the conditional claim is only as strong as the approximation quality of F; this is the more direct load-bearing point for 'near-conditional' rather than marginal coverage. The abstract-only review means derivations in the full text (likely §3–4) were not checked, so the concern is raised at the level of the reformulation rather than a specific equation error.","tokens_in":1768,"tokens_out":349,"duration_ms":29586,"concrete_test":"On the hierarchical data experiment, replace the default F with an enriched class that includes explicit cluster-membership indicators; recompute the reported conditional coverage gap. If the gap shrinks by more than 20% while marginal coverage stays intact, the function-class choice is load-bearing for the near-conditional claim.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction reformulates conditional coverage as a miscoverage error over a user-specified class F (inspired by relaxed multi-accuracy). Theoretical guarantees and convergence rates are derived under distributional invariance for linear and RKHS classes, recovering exchangeable results as special cases. For general group symmetries on networks or clusters, if F cannot capture the heterogeneity induced by the group action (e.g., cluster indicators or network features), the 'near' conditional guarantee reduces to a marginal one with large slack. No explicit condition is given on the approximation error of F relative to the group orbit.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces C-SymmPI, a framework for achieving near-conditional coverage in distribution-free predictive inference for data with group symmetries beyond exchangeability. It reformulates conditional coverage as a miscoverage error over a user-specified function class F (inspired by relaxed multi-accuracy), establishes theoretical guarantees and convergence rates under distributional invariance for linear and RKHS classes, develops projection-based and sampling-based algorithms for computational efficiency, and provides empirical results on hierarchical and network data while recovering exchangeable results as special cases.","tokens_in":1933,"tokens_out":538,"duration_ms":37994,"significance":"If the central claims hold with explicit controls on approximation quality, the work would meaningfully extend predictive inference to structured non-exchangeable settings such as networks and clusters, where marginal coverage is often insufficient due to heterogeneity. The derivation of convergence rates for concrete function classes and the recovery of prior exchangeable results as special cases are positive features that strengthen the contribution.","major_comments":[{"comment":"The near-conditional coverage guarantee (abstract and main theoretical section) is stated to hold under distributional invariance, yet the bound depends on how well the user-specified class F approximates the conditional expectation induced by the group action. No explicit condition or bound on the approximation error ||E[· | group orbit] - proj_F|| is provided; without it, the guarantee can reduce to marginal coverage with arbitrarily large slack when F fails to capture cluster- or network-induced heterogeneity.","section":"Theoretical guarantees section (around the reformulation and Theorem on near-conditional coverage)"},{"comment":"Convergence rates are derived for linear and RKHS classes under distributional invariance. It is not shown how these rates extend to general group symmetries on networks or clusters when the function class must represent orbit-specific features (e.g., cluster indicators); the rates may not transfer without additional verification that F is rich enough relative to the group action.","section":"Convergence rates for linear/RKHS classes"}],"minor_comments":[{"comment":"The abstract and introduction use 'near-conditional coverage' without a precise quantitative definition of the slack term; adding an explicit expression for the additive error would improve clarity.","section":"Abstract"},{"comment":"Notation for the group action and the function class F could be introduced earlier with a small illustrative example (e.g., a simple cluster symmetry) to aid readers unfamiliar with the multi-accuracy connection.","section":"Introduction / Preliminaries"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments, which help clarify the scope and limitations of our framework. We address each major comment below and indicate the planned revisions.","responses":[{"response":"We agree that the near-conditional guarantee is expressed in terms of the approximation quality of F to the orbit-conditional expectation, and that this slack can be large for poorly chosen F. This dependence is intentional in the relaxed multi-accuracy style reformulation, allowing users to select F according to known structure (e.g., cluster indicators). To address the concern, we will add an explicit remark in the theoretical guarantees section stating that the bound reduces to marginal coverage when F is the constant class, together with concrete guidance and examples for bounding the approximation error under common group actions such as hierarchical clustering and network symmetries. This is a partial revision that expands existing discussion rather than introducing new theorems.","revision_made":"partial","referee_comment":"[Theoretical guarantees section (around the reformulation and Theorem on near-conditional coverage)] The near-conditional coverage guarantee (abstract and main theoretical section) is stated to hold under distributional invariance, yet the bound depends on how well the user-specified class F approximates the conditional expectation induced by the group action. No explicit condition or bound on the approximation error ||E[· | group orbit] - proj_F|| is provided; without it, the guarantee can reduce to marginal coverage with arbitrarily large slack when F fails to capture cluster- or network-induced heterogeneity."},{"response":"The convergence rates are derived for any linear or RKHS class satisfying the stated boundedness and invariance conditions; they therefore apply whenever the user selects an F that is rich enough to represent the relevant orbit-specific features. In the hierarchical-data experiments we already employ linear classes that include cluster indicators, which are orbit-specific. We will add a short clarifying paragraph in the convergence-rates section that explicitly verifies this richness condition for the network and cluster examples, noting that the rates carry over under the same invariance assumptions once F contains a basis for the orbit features. This revision will be made.","revision_made":"yes","referee_comment":"[Convergence rates for linear/RKHS classes] Convergence rates are derived for linear and RKHS classes under distributional invariance. It is not shown how these rates extend to general group symmetries on networks or clusters when the function class must represent orbit-specific features (e.g., cluster indicators); the rates may not transfer without additional verification that F is rich enough relative to the group action."}],"tokens_in":1478,"tokens_out":541,"duration_ms":50322,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The one thing to know is that this work adds near-conditional coverage to the SymmPI setup by recasting it as miscoverage over a user-specified function class. That lets them target structured data with group symmetries instead of just exchangeable cases. They do a solid job with the theory for linear and RKHS classes, getting convergence rates under distributional invariance, and they recover the exchangeable results as a special case. The two algorithms they propose—one projection-based and one sampling-based—address computational issues for high-dim data or infinite groups. The experiments on hierarchical and network data indicate more stable conditional coverage compared to existing approaches. The soft spot is the reliance on the function class approximating the conditional structure well. Without a good match to the heterogeneity from the group action, the near-conditional guarantee can have large slack and effectively become marginal. The stress test note points out there's no explicit condition on the approximation error relative to the group orbit, and that seems to hold up. Also, while they handle some shift, the core invariance might be a strong assumption for real networks or clusters. This is for people working on predictive inference in non-i.i.d. settings, especially those already familiar with symmetry-based methods. A reader interested in extending conformal prediction to graphs or clustered data would find it useful. It deserves a serious referee. The central construction is a clear step forward even if the practical conditions need more attention in review. Recommendation: Send it for peer review.","headline":"C-SymmPI adds near-conditional coverage to SymmPI by recasting the problem as miscoverage over a user function class, but the practical strength depends on how well that class matches the group-induced heterogeneity.","tokens_in":2402,"tokens_out":380,"would_cite":false,"duration_ms":48441,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"Inspired by the relaxed multi-accuracy perspective, our approach reformulates the conditional coverage as miscoverage error over a user-specified function class... We establish general theoretical guarantees under both distributional invariance and distribution shift settings, and derive convergence rates for linear and RKHS function classes"},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking","paper_passage":"Z d= ρ(G)Z ... V(ρ(g)z) = eρ(g)V(z) ... t_V(z) = Q_{1-α}(ψ(eρ(G)V(z)), G∼U)"}],"headline":"Group-symmetric conformal prediction via relaxed multi-accuracy on orbits; no overlap with J-cost, φ-ladder or distinction-forcing chain","alignment":"orthogonal","rationale":"The paper's core machinery (C-SymmPI threshold via pinball loss on group orbits, reformulation of near-conditional coverage as ε-bounded multi-accuracy over user-specified F, Haar averaging, equivariant V maps) operates entirely within statistical learning theory for structured data. It recovers exchangeable conformal results as the special case G = S_{n+1} but introduces no recognition cost J, golden-ratio identities, 8-tick periodicity, or parameter-free constant derivations. RS theorems on these structures (e.g., reality_from_one_distinction, J-uniqueness via Aczél, Alexander duality for D=3) are therefore neither confirmed nor contradicted.","tokens_in":62118,"confidence":"high","tokens_out":398,"duration_ms":14958,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"C-SymmPI achieves near-conditional coverage for predictive inference under group symmetries beyond exchangeability.","keywords":["conditional predictive inference","group symmetries","near-conditional coverage","structured data","distribution-free methods","networks","clusters","C-SymmPI"],"falsifier":"A dataset constructed so that the distribution changes under the group transformations, where the observed miscoverage rates exceed the rates predicted by the convergence bounds.","tokens_in":2670,"feed_emoji":"","tokens_out":662,"duration_ms":45811,"temperature":0.7,"pith_summary":"The paper introduces C-SymmPI to extend distribution-free predictive inference to structured data with group symmetries such as networks and clusters. It targets near-conditional coverage that accounts for data heterogeneity instead of only marginal averages. The method reformulates conditional coverage as control of miscoverage error over a user-specified function class, inspired by relaxed multi-accuracy. Under the assumption that the data distribution is invariant under the symmetries, it supplies theoretical guarantees and convergence rates for linear and RKHS classes, recovering exchangeable results as special cases. Practical projection and sampling algorithms are given to handle high-dimensional observations and large groups, with demonstrations on hierarchical and network examples showing more stable conditional coverage.","feed_headline":"C-SymmPI achieves near-conditional coverage for symmetric structured data","feed_subtitle":"Reformulating coverage as miscoverage control under distributional invariance extends inference to networks and clusters.","key_machinery":"Reformulation of conditional coverage as miscoverage error over a user-specified function class under distributional invariance.","core_discovery":"C-SymmPI is a framework for distribution-free predictive inference that attains near-conditional coverage for general data structures with group symmetries. It reformulates the conditional coverage goal as minimization of miscoverage error over a user-specified function class. Under distributional invariance, the framework establishes theoretical guarantees and derives convergence rates for linear and reproducing kernel Hilbert space function classes, while recovering prior exchangeable results as special cases. Efficient algorithms are developed for high-dimensional data via projection and for large or infinite groups via sampling, with empirical validation on hierarchical and network data.","pith_inferences":["The approach could support uncertainty quantification for predictors on relational or graph data where symmetries are natural.","Sampling-based computation may extend to continuous symmetry groups such as rotations by drawing finite approximations.","The framework might integrate with black-box models to handle heterogeneity across clusters or sub-populations in practice.","Extensions could address time-series or imaging data with periodic or spatial symmetries if the invariance holds."],"forward_implications":["Near-conditional coverage guarantees become available for network data and cluster-level data.","Convergence rates are obtained for linear and RKHS function classes.","State-of-the-art exchangeable results are recovered as special cases.","Projection-based and sampling-based algorithms enable computation for high-dimensional observations and large groups.","Empirical results indicate more informative and stable conditional coverage with improved accuracy on hierarchical and network data."],"fun_headline_variants":["C-SymmPI attains near-conditional coverage for group symmetric data","C-SymmPI enables near-conditional coverage under group symmetries","C-SymmPI reformulates conditional coverage as miscoverage for symmetries","Distribution-free near-conditional inference for structured symmetric data"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The data distribution is invariant under the group symmetries.","fun_headline_variants_meta":{"raw":{"variants":["C-SymmPI attains near-conditional coverage for group symmetric data","C-SymmPI enables near-conditional coverage under group symmetries","C-SymmPI reformulates conditional coverage as miscoverage for symmetries","Distribution-free near-conditional inference for structured symmetric data"]},"model":"grok-4.3","cost_usd":0.01835,"raw_usage":{"total_tokens":7747,"prompt_tokens":760,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":183503000,"prompt_tokens_details":{"text_tokens":760,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":6918,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":760,"tokens_out":69,"duration_ms":85731,"temperature":1.0,"reasoning_tokens":6918,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-20T01:18:38.791903+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A dataset constructed so that the distribution changes under the group transformations, where the observed miscoverage rates exceed the rates predicted by the convergence bounds.","supporting_citations":[],"review_version":1}