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Direct Inference on Admissible Sets (Diona)

Lorenzo Sabug Jr., Eric Kerrigan

arXiv:2610.31526v1math.OCcs.LGcs.SYeess.SY

TL;DR

Reconstructing functions from costly samples calls for accurate estimates and uncertainty bounds. Diona uses local sample constraints to estimate values and derivatives without offline training. Across six functions, it improved noiseless accuracy over GPs, especially on denser data, while its bounds covered true values at more than 95% of query points in four functions.

  • Problem

    Function reconstruction from samples matters when acquiring samples is expensive or challenging, and applications need fast, accurate, and reliable estimates.

    Source: §1 · p. 2

  • Method

    Diona uses local Taylor-based constraints on latent values and derivatives to estimate a query state and its admissible bounds through per-query optimization.

    Source: §1 · p. 3, §2.2 · p. 7, §7 · p. 22

  • Results

    In six functions, Diona was more accurate than GPs on noiseless data, especially with denser data sets, and its bounds covered true values at more than 95% of query points in four functions.

    Source: §1 · pp. 3–4

  • Takeaways & Limitations

    Diona’s latency advantage applies when data sets update often and relatively few predictions are needed between updates.

    Source: §1 · p. 4

    Diona needs the entire data set available for fetching, with storage demands growing and solve time scaling polynomially in dimension; huge static data sets favor global training.

    Source: §6.4 · p. 22

Abstract

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We propose a new regression method named Direct Inference on Admissible Sets (Diona) that directly uses samples to estimate the state and its derivatives at a query point, along with their admissible bounds, without long global training. By casting the values and derivatives at the query and neighbouring samples as latent variables in a constrained optimisation problem, the burden of long training loops becomes one of on-demand local solves. We establish Diona's theoretical properties including finite-sample coverage and, in the noiseless case, convergence to the true state and derivatives under dense sampling. In a nonlinear regression benchmark on six functions of three to eight dimensions, Diona was more accurate than a Gaussian process (GP) on noiseless data especially with denser data sets. Furthermore, as with GP, it produced uncertainty bounds that contained the true value at more than 95% of query points in four functions. Diona's time to a first prediction after a data set update was three to four orders of magnitude shorter than that of the GP, and two to three orders shorter than that of a neural network, which was more accurate but offers no uncertainty quantification. This low latency is paid for by on-demand predictions, so this latency advantage applies when relatively few predictions are needed between data updates.

1. Introduction

Diona addresses costly sample-driven function reconstruction with local inference that estimates query values, derivatives, and bounds without offline training. Its analysis and benchmark results characterize when this approach is useful, including the trade-off of on-demand prediction.

1. Introduction

  • The motivation is low-latency reconstruction where sampling is costly and updated data must inform decisions; in exploration, uncertainty is a mission decision metric.Source: §1 · pp. 2–3
  • Diona jointly estimates admissible query values and derivatives from function-value samples, returning point estimates and uncertainty bounds through per-query solves rather than offline training.The method treats values and derivatives at the query and neighboring samples as latent variables in a constrained optimization problem.Source: §1 · p. 3
  • The analysis derives information radii for query values and derivatives and proves noiseless consistency as sampling becomes dense.The method uses a Lipschitz-continuous Hessian with latent gradients and Hessians to account for local curvature.Source: §1 · p. 3
  • In six functions, Diona beat GP in noiseless accuracy, especially on densest data sets, and bounds covered true values at more than 95% of query points in four functions.The benchmark spans functions of three to eight dimensions.Source: §1 · pp. 3–4
  • Relative to preliminary DCBR [13], this version simplifies the formulation, treats uncertainty, investigates theory, and expands benchmark analysis.Source: §1 · p. 4

Notation

  • Notation uses query point x′ and state triples that collect value, gradient, and Hessian information at query and sample locations.Source: Notation · pp. 4–5

2. Theoretical Framework

The theoretical framework considers an unknown scalar function over a compact convex region and seeks estimates of its state and derivatives at an interior query point. It uses regularity assumptions and sampled function values, which may have bounded noise.

2. Theoretical Framework

  • The framework considers an a priori unknown black-box scalar function on a closed, compact, convex D-dimensional region.Source: §2.1 · p. 5
  • It assumes the function has a Lipschitz-continuous Hessian with an a priori known constant LH.Source: §2.1 · p. 5
  • Samples provide function values with a known finite noise bound, including the noiseless case; derivative observations are unavailable.Source: §2.1 · p. 6
  • For an interior query point, the problem is to estimate its state triple from the sample data and prior information.Source: §2.1 · p. 6

bounding set T ′⟨n⟩satisfying the finite-sample coverage on the true triple,

The paper combines Taylor expansions and set membership to bound the query-point state and derivatives from nearby samples. In the noiseless case, the estimate and admissible set converge to the true triple as sampling becomes dense.

bounding set T ′⟨n⟩satisfying the finite-sample coverage on the true triple,

2.2. Taylor-based uncertainty quantification and estimation

  • Taylor relations and set membership define linear inequalities over unknown states and derivatives at the query and neighboring samples.The fixed sample displacements make the inequalities linear in the unknowns despite their quadratic Taylor relations.Source: bounding set T ′⟨n⟩satisfying the… · p. 6, §2.2 · pp. 7–8
  • The admissible set yields upper and lower bounds for the query value, whose midpoint is the estimate and whose difference is the uncertainty measure.Source: §2.2 · pp. 8–9

2.3. Neighbour samples

  • Neighbor samples are considered at a trial radius, and the section introduces a µ0-poised neighbor-set condition.Source: §2.3 · pp. 9–10

Dh is µ0-poised if for a given µ0 > 0, σ(Φ) ≥µ0

The section assumes access to a µ0-poised neighbor set for an interior query point.

Dh is µ0-poised if for a given µ0 > 0, σ(Φ) ≥µ0

3. Theoretical properties

Under Assumptions 1–2, the true query triple lies in the admissible set, and the set’s diameter bounds the spread among admissible states and derivatives. Under the stated noiseless and sampling conditions, that set converges to the true triple.

3. Theoretical properties

  • Under Assumptions 1–2, the true query triple is contained in the admissible set for finite samples.Source: §3 · p. 10
  • The worst-case diameter result bounds pairwise differences between admissible query states and derivatives under Assumptions 1–2.Source: §3 · p. 11

µ0-poised neighbour set N,

  • In the noiseless case, halving the neighborhood radius tightens the state-value bound eightfold, while noise leaves that bound with a floor as the radius shrinks.With noise, the passage says the derivative bounds grow again when considering increasingly small neighborhoods.Source: µ0-poised neighbour set N · p. 11
  • The admissible set shrinks to the true triple under Assumptions 1–3, zero noise, and a poised-neighbor radius that shrinks with sampling.Source: µ0-poised neighbour set N · p. 12

4. Implementation notes

The implementation estimates unknown noise and Hessian-Lipschitz constants from data and uses a greedy procedure to select poised neighbors. Estimating the constants invalidates the formal theoretical guarantees.

4. Implementation notes

4.1. Parameter estimation

  • The method estimates the Hessian-Lipschitz constant by solving a linear program over active points.Source: §4.1 · pp. 12–13
  • Estimating the Hessian-Lipschitz and noise bounds from data invalidates Diona’s formal theoretical guarantees.The paper reports competitive performance despite using estimated constants.Source: §4.1 · p. 13

4.2. Practical neighbour selection

  • A greedy neighbor-selection procedure builds a full-rank feature matrix and adds samples iteratively to achieve µ0-poisedness.The exact minimum-radius selection is combinatorial and can be computationally prohibitive.Source: §4.2 · p. 13

5. Direct Inference on Admissible Sets (Diona)

Diona selects neighbouring samples and uses local constraints to formulate an optimisation problem for estimating a query’s state and uncertainty. This replaces global training with an on-demand solve for each query.

5. Direct Inference on Admissible Sets (Diona)

  • Diona assembles local constraints over the query and neighbouring samples into an optimisation problem that returns a state estimate and uncertainty.Each query is handled by a local solve rather than a long global training process.Source: §5 · pp. 13–14
  • The practical neighbour-selection procedure builds a full-rank set of m points, expanding the candidate pool when needed.Here, m = 1 + D + D(D+1)/2, and column-pivoted QR is used to construct the proposed set.Source: §5 · p. 14
  • If the selected neighbours remain rank-deficient after all samples are exhausted, the procedure returns NeighbourFailure.The algorithm also returns NeighbourFailure when the candidate pool is exhausted without a suitable set.Source: §5 · p. 14

6. Comparative study

The comparative study evaluates Diona across benchmark functions and compares its accuracy, uncertainty informativeness, and computational burden with other regressors. Its results expose trade-offs between on-demand prediction and globally trained models.

6.1. Benchmark setup

  • The benchmark covers functions of three to eight dimensions, with 500D, 1000D, and 2000D Sobol-layout samples and noiseless or 20 dB noisy data.The study also evaluates relative L2 error at 5000D Sobol-distributed query points.Source: §6.1 · p. 15

6.2. Results: regression accuracy

  • On noiseless benchmarks, Diona, NSM, RBF, and IDW errors decreased with denser samples, while GP errors stayed mostly within 7 × 10^-3 to 8 × 10^-3.The study reports that Diona was more accurate than GP especially on the densest noiseless data; Figure 2 shows relative L2 error across six functions.Source: §6.2 · p. 16, Figure 2 · p. 19

6.2.1. Results: uncertainty informativeness

  • Diona produced tighter uncertainty widths than NSM and widths either better than or within one order of magnitude of GP, but its noisy-data coverage dropped especially for Park and Friedman.Figure 3 compares median relative uncertainty width with truth coverage; lower-left markers indicate tighter intervals that include the truth.Source: §6.2.1 · p. 18, Figure 3 · p. 20

6.3. Results: empirical computational burden

  • Diona had zero training time but the longest per-query time, while GP and NN training took minutes; denser data increased both times for GP, IDW, NSM, and RBF.Diona’s per-query solve explains its higher inference time; Figure 4 compares median per-query and training times.Source: §6.3 · pp. 19–20, Figure 4 · p. 21

6.4. Synthesis

  • Diona can be preferable when data arrive in real time and only a few predictions are needed before the next update.The study frames this as a trade-off between Diona and other methods in accuracy, uncertainty informativeness, and computational demands.Source: §6.4 · p. 21
  • Diona requires the entire data set to be available for fetching, creating storage demands that can become unreasonable; its per-query solve time also scales polynomially with dimension.For huge static data sets, the authors contrast this with globally trained surrogates that compress data and are quick to evaluate.Source: §6.4 · p. 22

7. Conclusions and outlook

The conclusion presents Diona as a local, Taylor-constraint-based approach to function estimation and uncertainty quantification. It reports benchmark advantages in noiseless settings while emphasizing the online-versus-offline computational trade-off.

7. Conclusions and outlook

  • Diona uses local information and Taylor-relation constraints to estimate an unknown function and quantify uncertainty through linear programs.This formulation removes the need for a long offline training stage by paying computational costs per query.Source: §7 · p. 22
  • In six benchmark functions of three to eight dimensions, Diona was more accurate than GP on noiseless data, especially at the highest sample densities.Like GP, its bounds contained true values at more than 95% of query points in four functions; time to a first prediction was orders of magnitude shorter, but each query cost more than evaluating a trained model.Source: §7 · p. 22
  • Future work includes hybrid offline-online approaches to reduce per-query computation and storage needs while retaining responsiveness to new samples.The authors also identify incorporating physics information to support extrapolation beyond the data-set domain as an avenue for future work.Source: §7 · pp. 22–23

Proof of Lemma 1

The proofs establish that the true query triple belongs to the admissible projection, derive a bound using a poised neighbourhood, and use dense sampling to complete the convergence argument.

Proof of Lemma 1

  • The true query triple belongs to the admissible projection because the true collection vector satisfies the assumptions and projects onto the query coordinates.The proof places the true collection vector in A by construction.Source: Proof of Lemma 1 · p. 25

Proof of Lemma 2

  • The Lemma 2 proof normalizes neighbour displacements so the matrix Φ depends on their normalized geometry, then uses poisedness to establish full column rank and derive (13).The resulting bound is stated to be independent of neighbourhood radius through Φ; the excerpt does not give the bound's expression.Source: Proof of Lemma 2 · pp. 25–26

Proof of Lemma 3

  • The Lemma 3 proof uses dense samples to obtain a sufficiently poised reference configuration and recursively bound the neighbourhood radius.It assumes, without loss of generality, that one sample is added at each iteration.Source: Proof of Lemma 3 · pp. 26–27

Proof of Theorem 1

  • Theorem 1 combines Lemma 1 with Lemma 2 at ε = 0, then applies Lemma 3 to the true triple to obtain (14).The proof first establishes that the true triple is in the admissible set.Source: Proof of Theorem 1 · p. 27
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