Settling: Equilibrium Inference for Non-Convex Validity Sets

Lyes Saad Saoud

Source-linked AI summary

Settling: Equilibrium Inference for Non-Convex Validity Sets

Lyes Saad Saoud

arXiv:2609.09682v1Sep 9, 2026cs.LG

TL;DR

The paper studies how squared-loss point inference can produce invalid conditional means when valid alternatives occupy non-convex sets. It introduces Settling, an equilibrium-based operator that refines mean-seeking proposals through consistency dynamics, and reports strong controlled geometric performance while leaving learned high-dimensional validation open.

Problem
Squared-loss point inference can average valid alternatives into an invalid output when admissible outputs form a non-convex or disconnected set.
Method
Settling separates proposal generation, consistency evaluation, and equilibrium selection, refining an initialized hypothesis toward a locally stable, constraint-consistent equilibrium.
Results
In the 100-context diagnostic, the mean-seeking baseline succeeds in 0/100 contexts, stochastic denoising in 100/100, and Settling in 99/100 with lower trajectory roughness.
Takeaways & Limitations
The study provides a transparent operator-level proof of concept and identifies local gradient alignment, stable basin geometry, and sufficient refinement as conditions for learned realization.
Takeaways & Limitations
Learned high-dimensional validation remains open because the study does not establish that a learned critic will satisfy the required alignment and basin conditions.

Abstract

from arXiv · show

Many learning systems return a single point estimate even when admissible outputs form disconnected or non-convex sets. Under squared loss, an ambiguous conditional distribution can therefore have a Bayes-optimal conditional mean that is invalid. We formalize this failure as conditional mean collapse and introduce Settling, an equilibrium-based inference operator that separates proposal generation, consistency evaluation, and test-time equilibrium selection. The operator treats a mean-seeking proposal as an initialization and refines it toward a locally stable configuration; conditional on initialization, refinement is deterministic. We establish exact-gradient descent, local convergence, and an inexact-gradient robustness condition relevant to learned consistency critics. In a reproducible 100-context geometric diagnostic, the mean-seeking baseline succeeds in 0/100 contexts, stochastic denoising in 100/100, and Settling in 99/100 while producing substantially lower trajectory roughness. A 1,200-run sensitivity study yields 97-100% success across obstacle-jitter ranges up to 0.20 and 94-100% across one-time initialization perturbations from 0.05 to 0.50. Cross-domain panels remain mechanism illustrations; learned high-dimensional validation remains an open empirical test.

1. Introduction

The paper identifies conditional mean collapse: under ambiguity, squared-loss point inference can average valid alternatives into an invalid output in a non-convex validity set. It introduces Settling, which treats the aggregate as an initialization and refines it through consistency-driven equilibrium selection.

1. Introduction: Under squared loss, the Bayes-optimal conditional mean can be invalid when valid outputs form a disconnected or non-convex set.Two valid trajectory alternatives can have an arithmetic mean that intersects an obstacle region.
1. Introduction: Settling separates proposal generation, consistency evaluation, and equilibrium selection to refine an initially invalid aggregate toward a locally stable configuration.The operator is presented as complementary to mean-seeking inference rather than as a new optimizer, density model, or universal critique of attention.
1. Introduction: The reproducible study combines a 100-context geometric mechanism evaluation with a 1,200-run robustness sweep and explicit conditions for learned transfer.The analytic setting isolates proposal geometry, validity, and consistency dynamics, while learned critics require local gradient alignment, recoverable basins, and sufficient refinement budget.
1. Introduction: The paper formalizes this failure as an inference-operator property rather than a consequence of model capacity, training data, architectural depth, or optimization procedure.Theorem 1 states that increasing capacity without changing the mean-seeking objective cannot remove the failure for the stated context.
1. Introduction: The paper limits its claim to a single convex aggregate used as a terminal point estimate and does not generalize the failure to complete Transformer architectures.Mixture heads, search, discrete decoding, nonlinear transformations, and explicit constraint modules may preserve or recover multimodality.

3. Settling as an Inference Paradigm

Settling reframes inference as dynamical resolution: a proposal is refined through a context-conditioned consistency landscape until a stable equilibrium is selected. Its three-part decomposition supports deterministic mode selection in non-convex validity spaces.

Settling treats inference as evolution toward a stable, constraint-consistent equilibrium rather than a single-step output statistic.
Consistency Landscape: Low-energy regions represent internally consistent configurations, while the consistency functional is not a globally minimized task objective or likelihood.
Equilibrium Selection: Settling replaces aggregation with deterministic equilibrium selection: initialization and local landscape geometry determine which valid interpretation is reached.
Modular Structure: The operator separates hypothesis generation, constraint evaluation, and equilibrium selection, with each component necessary for deterministic inference.
Hypothesis Generation: The proposal provides an initial hypothesis but may be mean-seeking; settling keeps it revisable through structured relaxation over the consistency landscape.

4. A Practical Instantiation of Settling Inference

Settling-Based Inference operationalizes the paradigm as a modular test-time operator combining proposal generation, constraint evaluation, and equilibrium selection. Its dynamics correct mean-seeking proposals without relying on stochastic refinement or complex validity projection.

SBI realizes settling as a modular operator composed of hypothesis generation, constraint evaluation, and equilibrium selection.
Operational Stages: The first stage constructs a context-consistent latent configuration, while the second induces competition among interpretations for deterministic equilibrium selection.
Consistency Evaluation: The constraint energy defines non-convex validity and rejects invalid configurations, while proposal anchoring preserves context relevance.
Validity Enforcement: Projection enforces only simple domain bounds; complex validity, including obstacles and task logic, is enforced through the constraint energy.
Settling Dynamics: Conditional on initialization, SBI evolves deterministically toward a stable fixed point, using a one-time perturbation only to break exact symmetry.
Necessity of the Decomposition: Removing any one component is insufficient: proposal-only inference collapses toward means, constraint-only search is inefficient, and dynamics without anchoring lose context.

5. Theoretical Properties of Settling

The theory establishes local progress, robustness, convergence, and deterministic selection for settling dynamics under explicit regularity conditions. These guarantees are local and conditional rather than global or probabilistic.

Settling has four analyzed properties: exact-gradient improvement, bounded-error descent robustness, local convergence, and deterministic selection conditional on initialization.
Monotonic Improvement: For an L-smooth consistency functional, step sizes η < 2/L make exact-gradient settling decrease consistency monotonically at every iteration.
Monotonic Improvement: The consistency value provides an ordered test-time progress measure tied to the validity-oriented functional used for refinement.
Inexact Gradients: A learned critic need only maintain sufficiently small relative gradient error for descent to persist locally, but this is not a learnability theorem.
Local Convergence: Under local strong convexity, gradient Lipschitzness, and a forward-invariant basin, refinement converges linearly to the corresponding stable valid equilibrium.
Scope: The guarantees do not identify the basin from arbitrary initialization or correct a learned critic whose stable point is invalid.

6. Relation to Existing Paradigms

Settling differs from energy-based, diffusion, planning, and equilibrium-model paradigms primarily in inferential purpose: it refines one initialized hypothesis toward a stable, constraint-consistent equilibrium rather than modeling distributions or solving user-defined objectives.

6. Relation to Existing Paradigms: Unlike conventional EBMs, Settling uses energy-like consistency functions for test-time selection rather than density modeling or generative inference.The distinction concerns inferential role, not merely the presence of an energy function.
6. Relation to Existing Paradigms: Unlike diffusion, Settling is deterministic after initialization and does not aim to represent distributional diversity through stochastic sampling.Any noise is confined to initialization, where it can break symmetric saddle points.
6. Relation to Existing Paradigms: Unlike classical planning and trajectory optimization, Settling does not minimize a user-defined task objective; its dynamics resolve inconsistency between proposals and validity constraints.The dynamics are a mechanism for inference rather than the problem-solving objective itself.
6. Relation to Existing Paradigms: The claimed novelty is the conditional-mean-collapse formulation and operator-level inference interpretation, not fixed-point iteration or gradient-based relaxation themselves.The comparison therefore makes no claim of task-level superiority over neighboring paradigms.
6. Relation to Existing Paradigms: Settling uses a consistency landscape to refine one initialized hypothesis toward a stable equilibrium, making constraint satisfaction explicit.Its output is a fixed point of a context-conditioned dynamical system.

7. Analytic Validation and a Learnable Extension

The analytic validation isolates Settling’s inference mechanism with known proposal and validity geometry, while the learnable extension specifies compatible proposal and consistency components without claiming learned high-dimensional transfer.

7. Analytic Validation and a Learnable Extension: A learnable realization separates an amortized proposal initialization from a consistency critic that ranks valid states below invalid ones for each context.The proposal may be invalid initially because its role is to provide a fast, context-aligned state for later revision.
7. Analytic Validation and a Learnable Extension: Mixture-valued, set-valued, autoregressive, and mode-seeking proposals fall outside the mean-collapse statement and should be included as comparison baselines.The failure claim applies specifically to mean-seeking proposals under the analyzed ambiguous distributions.
7. Analytic Validation and a Learnable Extension: The paper does not evaluate whether proposal and critic training should be independent, joint, or alternating.At inference, the required ingredients are an initialization and a consistency landscape whose local minima sufficiently match valid interpretations.
7. Analytic Validation and a Learnable Extension: Transfer to learned systems is conditional on recoverable initialization, sufficiently aligned critic gradients, and a refinement budget that reaches a stable state at acceptable cost.These are hypotheses for future evaluation, not empirical evidence established by this paper.
7. Analytic Validation and a Learnable Extension: The evaluated diagnostic uses an analytic conditional-mean proposal and obstacle-defined consistency landscape, keeping invalidity and refinement trajectories inspectable and reproducible.This design isolates equilibrium selection from representation learning, model capacity, and training instability.

8. Controlled Evidence and Diagnostics

The controlled geometric study isolates operator-level behavior under non-convex validity, comparing mean-seeking, stochastic, direct-descent, and Settling procedures across randomized contexts. Settling achieves high validity and smoothness, with robustness diagnostics and ablations clarifying its dependence on initialization and refinement dynamics.

8.1. Evidence Organization and Claim Scope: The geometric evidence is quantitative and reproducible, whereas semantic panels and numerical diagnostics are mechanism illustrations rather than formal evidence or certificates.Figure 13 is a binary single-run basin-selection diagnostic, and Figure 12 is not a formal certificate of the smooth-gradient theorem.
8.2.1. Operator baselines and inference budgets: Settling treats the analytical mean as an initialization, applies one-time symmetry breaking, and deterministically refines a spline toward a consistency-based equilibrium.Noise is applied only at initialization; refinement uses 120 settling steps in the released implementation.
8.2.2. Metrics and quantitative results: 99/100 Settling success on randomized obstacle contexts contrasts with 0/100 for the mean-seeking aggregate and 100/100 for stochastic denoising.Success requires every waypoint to satisfy the declared clearance rule; the single Settling failure has minimum clearance 0.01674 versus the 0.02 threshold.
8.2.2. Metrics and quantitative results: 0.00092 versus 0.02631 second-difference roughness shows Settling produces substantially smoother trajectories than stochastic denoising.The comparison uses mean squared second-difference smoothness, which is a geometric-regularity diagnostic rather than a task reward.
8.2.3. Robustness to context perturbation and tie-breaking amplitude: 100/100, 99/100, 97/100, and 98/100 success across obstacle-jitter ranges from 0.00 to 0.20 indicate broad but non-uniform robustness.The median minimum clearance remains approximately 0.0495, while a lower tail crosses the success threshold at harder settings.
8.2.3. Robustness to context perturbation and tie-breaking amplitude: 100/100 success for σ ≤ 0.20 falls to 99/100 at σ = 0.30 and 94/100 at σ = 0.40 and 0.50.The sweep identifies initialization amplitude as an operating parameter rather than tuning it away.

9. Discussion, Limitations, and Claim Boundaries

The discussion narrows the paper’s claims to a reproducible operator-level proof of concept and distinguishes supported geometric evidence from illustrative or idealized analyses. It identifies inference cost, initialization, learned critics, and consistency landscapes as important boundaries for transfer.

9. Discussion, Limitations, and Claim Boundaries: Learned high-dimensional deployment remains untested because representation error, spurious equilibria, basin coverage, gradient alignment, and inference cost remain empirical questions.The inexact-gradient result narrows transfer conditions but does not establish that learned critics satisfy them.
9. Discussion, Limitations, and Claim Boundaries: The study does not establish superiority over state-of-the-art planners, diffusion policies, learned EBMs, mixture predictors, or set-valued predictors.The stochastic and direct-descent comparisons are stylized operator baselines, while cross-domain illustrations are not quantitative benchmark evidence.
9. Discussion, Limitations, and Claim Boundaries: Determinism is conditional on initialization, and larger tie-breaking perturbations reduce reliability from 100/100 for σ ≤ 0.20 to 94/100 at σ = 0.40 and 0.50.The released implementation uses a one-time random perturbation before deterministic refinement.
9. Discussion, Limitations, and Claim Boundaries: The monotonicity theorem applies to unprojected L-smooth gradient updates under a stated step-size bound, not automatically to clipped or reduced practical implementations.Clipping, endpoint constraints, numerical gradients, and parameterization changes separate the idealized theorem from implementation details.
9. Discussion, Limitations, and Claim Boundaries: Selected-equilibrium quality depends on the consistency landscape, whose misspecification can create spurious minima or semantically incorrect basins.The framework shifts responsibility from averaging toward explicit validity representation.
9. Discussion, Limitations, and Claim Boundaries: Settling introduces domain-dependent inference-time computation, so future applications should report stopping conditions, iteration counts, and wall-clock costs.Adaptive stopping and learned basin shaping are identified as future directions.

10. Conclusion

The conclusion frames Settling as an inference-level decomposition that makes an invalid aggregate revisable and selects a stable equilibrium. The geometric study supports a transparent proof of concept, while realistic learned transfer remains the next empirical test.

10. Conclusion: Settling separates proposal generation, consistency evaluation, and test-time refinement, treating an invalid aggregate as a revisable internal hypothesis.The contribution is the operator-level decomposition rather than gradient descent or fixed-point computation alone.
10. Conclusion: 99/100 Settling success, versus 0/100 for the analytical mean and 100/100 for stochastic denoising, accompanies much lower trajectory roughness in the reproducible geometric diagnostic.The 1,200-run study reports 97–100% success across obstacle-jitter ranges and 94–100% across initialization perturbations.
10. Conclusion: Learned proposals and consistency critics must preserve useful basin geometry and gradient alignment while remaining competitive in task quality and inference cost.The conclusion identifies this as the next empirical test rather than an established result.

Data and Code Availability

The paper identifies a public GitHub repository as the planned archive for the quantitative study, code-generated figures, diagnostics, and supplementary animation. It reports no external funding and limited generative-AI use for language refinement.

Data and Code Availability: The quantitative geometric study, 1,200-run robustness sweep, code-generated figures, diagnostics, and supplementary animation.gif are planned for public GitHub archival.The repository URL is referenced but not included in the supplied passage.
Data and Code Availability: Generative AI was used only to refine English and readability, with the author reviewing and approving the final manuscript.
Data and Code Availability: The research received no external funding.

Appendix A. Proof of Conditional Mean Collapse

The appendix proves that squared-loss point estimation can select an invalid conditional mean, establishing a structural mismatch that added model capacity cannot remove. It also gives descent conditions for exact and inexact gradient updates.

The Bayes-optimal squared-loss point predictor can be invalid when the conditional mean lies outside the validity set, and greater capacity cannot change this solution.
For an L-smooth energy, unprojected gradient descent strictly decreases energy for 0 < η < 2/L whenever the gradient is nonzero.The result does not directly cover projected or clipped implementations without additional assumptions.
Under a relative gradient-error bound, strict energy decrease holds when 0 < η < 2(1 −δ)/(L(1 + δ)^2).

Appendix D. Local Convergence

The appendix specifies Settling as iterative energy-based refinement from a proposal and proves local linear convergence under forward invariance, strong convexity, and smoothness. The geometric landscape visualization is illustrative rather than quantitative evidence.

Local convergence: Within a forward-invariant neighborhood where the energy is strongly convex and smooth, Settling converges linearly to the unique valid stationary point.
Settling inference via equilibrium relaxation: Settling initializes from a proposal, applies energy-gradient updates for a fixed budget, optionally projects onto constraints, and returns the final iterate.
The implementation uses a low-dimensional spline with hard-pinned start and goal, while tie-breaking makes refinement deterministic only conditional on the initialized state.
Figure 14 shows a reduced two-dimensional consistency landscape for intuition, whereas the evaluated inference state is a multi-control-point trajectory.The figure is not used as quantitative evidence.