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GAV: A scientific experimentation environment for PDE agents

Muheng Li, Wenlong Mou

arXiv:2610.30292v1cs.LGcs.AI

TL;DR

PDE agents need reliable numerical feedback to assess accuracy, while fixed datasets alone do not provide new problems as experiments require. GAV learns reference solutions from data and uses manufactured-solution construction to create matching PDE problems with exact references, which the Verifier evaluates. Across numerical and experimental data tasks, GAV reduces mean test error by more than an order of magnitude for three existing agents on numerical Navier–Stokes relative to their original setups.

  • Problem

    PDE agents need reliable numerical feedback to measure accuracy, and fixed datasets alone cannot provide new problems as experiments require.

    Source: §1 · pp. 1–2

  • Method

    GAV learns reference solutions from simulation or measurement data, constructs matching PDE problems using manufactured solutions, and evaluates agent programs against the references.

    Source: §1 · p. 2, §3.2 · p. 4

  • Results

    On numerical Navier–Stokes, GAV reduces the mean test error of CodePDE, AutoNumerics, and PDE-SHARP by more than an order of magnitude relative to their original experimental setups.

    Source: §1 · p. 3

  • Takeaways & Limitations

    GAV’s feedback supports agents as they test ideas, investigate failures, and refine programs, while the analysis establishes evaluation and guided-development guarantees under stated conditions.

    Source: §6 · p. 12

    The guarantees assume the learned problem distribution is close to the target in KL divergence and that loss varies little across generated problems relative to its mean.

    Source: §3.2 · p. 5

Abstract

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Reliable feedback is essential for coding agents to develop programs for partial differential equations (PDEs). Yet evaluating these programs on new PDE problems often requires expensive reference computations, while fixed datasets restrict the range of experiments. We introduce Generator-Agent-Verifier (GAV), a scientific experimentation environment that makes new PDE problems verifiable by construction. The Generator learns a distribution of continuous reference solutions using data from numerical simulations or experimental measurements to represent a target physical regime. Using the method of manufactured solutions, it works backward from the generated reference solutions to construct matching PDE problems without a separate numerical solve. The Agent designs experiments, investigates failures, and iteratively improves its program. The Verifier evaluates candidate programs against the reference solutions and provides numerical feedback on accuracy and computational cost. This feedback can also guide the generation of new problems that probe the current program's weaknesses. We provide a theoretical analysis of evaluation reliability and guided development. Experiments on Navier-Stokes and Vlasov-Poisson-type equations across different data sources and regimes demonstrate the effectiveness of GAV. In particular, on the numerical Navier-Stokes task, GAV reduces the mean test error of three existing PDE agents by more than an order of magnitude relative to their original setups. Code is available at https://github.com/PotatoJnny/PDE-GAV.

1 Introduction

PDE agents need reliable numerical feedback to develop accurate programs, but reference computations can be expensive and fixed datasets cannot provide new problems as experiments require. GAV builds an interactive environment from scientific data, with exact references for generated PDE problems, and provides theoretical guarantees and improved results for existing agents.

Motivation

  • Reliable feedback is needed to assess whether PDE programs both execute successfully and produce accurate results, while high-resolution reference computations can be prohibitively expensive.The environment must also represent data-driven physical regimes and supply new problems and numerical feedback as experiments require.Source: §1 · pp. 1–2

Contribution

  • GAV learns a distribution of reference solutions from simulation or measurement data, constructs matching PDE problems with the method of manufactured solutions, and evaluates programs against exact references.The Agent can request new problems and inputs, receive numerical feedback, and use it to revise programs or guide generation toward weaknesses.Source: §1 · p. 2

Contributions

  • The paper establishes guarantees for evaluation reliability and guided development; under stated assumptions, it bounds fixed-program evaluation error and proves convergence in a convex setting.The introduction also evaluates GAV across Navier–Stokes and Vlasov–Poisson-type equations using numerical and experimental data.Source: §1 · pp. 2–3

Results

  • More than an order of magnitude lower mean test error on numerical Navier–Stokes: GAV improves CodePDE, AutoNumerics, and PDE-SHARP relative to their original experimental setups.Ablations and development case studies also examine how environment components affect program quality and experimental decisions.Source: §1 · p. 3

2 Problem setup

The problem setup specifies the PDE family, available data, program-development task, and Agent setting. The Agent develops an executable program through experiments and numerical feedback, aiming to meet task requirements within its development budget.

PDE specification

  • A PDE specification identifies the equation form, spatial domain, initial and boundary conditions, and, for time-dependent problems, the time interval; a problem is represented by parameters, source term, and data.For Navier–Stokes, these include viscosity and external forcing; Vlasov–Poisson describes a particle distribution coupled to an electric field, with Vlasov–Poisson–Fokker–Planck additionally including collisions.Source: §2 · p. 3

Data

  • The data comprise discrete solution observations from simulations or measurements, potentially with problem inputs, and may include partial observations used to construct continuous references.Complete problem–solution pairs are one possible data format.Source: §2 · p. 3

Program-development task

  • The program-development task specifies inputs, required outputs, and a performance objective, while the Agent chooses its computational method and internal representation.Examples include numerical solution and reduced-model prediction of specified macroscopic quantities, with prediction error and any runtime limit defining the objective.Source: §2 · p. 3

Agent setting

  • The Agent setting specifies the Agent, its development strategy, and its time and computational budget; development outputs an executable program that uses experiments and feedback to meet task requirements.The environment is evaluated through the performance of the final program.Source: §2 · p. 3

3 GAV: A scientific experimentation environment

GAV combines a Generator that constructs PDE problems with reference solutions, a Verifier that returns numerical evaluations, and an Agent that uses feedback to choose experiments and improve its program. The paper analyzes evaluation reliability and idealized guided development, and offers optional generation strategies for probing weaknesses and reducing overfitting.

3.1 Generating PDE problems with reference solutions

  • The Generator learns a model of continuous reference solutions from discrete observations, then uses the method of manufactured solutions to construct matching PDE problems without a separate numerical solve.Continuous representations also let programs query source and boundary data at locations and resolutions needed by their numerical methods.Source: §3.1 · p. 4

3.2 Numerical evaluation and feedback

  • The Verifier compares candidate program outputs with quantities computed from each problem’s exact reference solution and returns numerical feedback under the task’s performance objective.The evaluation analysis uses a fixed test set sampled independently from the learned problem distribution to estimate expected loss under the target distribution.Source: §3.2 · p. 4
  • The fixed-program evaluation guarantee becomes tighter as the learned problem distribution approaches the target and the number of independently evaluated problems increases.The stated assumptions require small KL divergence between distributions and normalized losses that do not vary too much relative to their mean.Source: §3.2 · p. 5

3.3 Agent-directed experimentation

  • Guided generation uses Verifier-error feedback to seek problems with larger predicted program errors, while perturbation modifies generated references subject to PDE constraints to reduce overfitting to the learned distribution.The Agent uses feedback to choose subsequent experiments and revise its program.Source: §3.3 · p. 5
  • In the idealized guided-update setting, the worst expected loss does not increase, and the theorem bounds its gap from the program-family minimum over iterations.The result assumes a compact convex program set, a convex objective, bounded Lipschitz losses, bounded guidance strength, and stabilization parameter τ > β̄G^2.Source: §3.3 · p. 6

4 Experiments

Experiments assess GAV-Full across Navier–Stokes and kinetic tasks, test GAV with existing PDE agents, and examine individual environment components. Results report improved program accuracy across these settings, with ablations showing higher mean error when core features are removed.

4.1 Experimental setup

  • Studies cover 2D incompressible NS and collisionless VP/VPFP, using numerical simulations, PIV measurements, and ARTEMIS observations; agents develop programs over multiple rounds.The shared setup evaluates retained programs on fixed task test sets; Figure 2 shows selected programs and updated memory passing to the next round while both agents’ research notes are retained.Source: §4.1 · pp. 6–7, Figure 2 · p. 7

4.2 Program development across PDE tasks

  • 0.1046% mean test error for GAV-Full on numerical NS, versus 1.6352% for COMSOL; the final program also reaches 0.1256% on independently constructed problems.Table 1 compares mean relative velocity error on the same test set; Figure 3 tracks development on numerical and PIV-derived test sets.Source: Table 1 · p. 8, §4.2 · p. 8, Figure 3 · p. 8
  • All six final kinetic programs improve over their initial programs, but R1-to-R6 error reductions transfer to independent tests in five runs; ARTEMIS Astra’s independent mean rises 5.69-fold.Independent tests use interpolated VPFP references and VP problems from previously unused ARTEMIS profiles.Source: §4.2 · p. 9
  • Across six kinetic runs, final mean error is 25.4%–90.1% lower than first-round error, though Astra’s trajectories include plateaus and regressions.Strong-collision Astra’s error rises in round 5 before falling in round 6; weak-collision Astra reaches its lowest test mean in round 4.Source: Table 2 · p. 9, §4.2 · p. 9, Figure 4 · p. 9
  • For one PIV-derived manufactured problem, the final program reduces full-window vector relative ℓ2 error from 0.0154% to 0.00172%.Figure 5 compares rounds 1 and 5 on the same reference at five times.Source: §4.2 · p. 9, Figure 5 · p. 10

4.3 GAV with existing PDE agents

  • GAV and GAV-Full achieve lower test error than Original for all three PDE agents; GAV reduces mean test error by more than an order of magnitude.Both GAV variants also use less total agent active time than Original.Source: Figure 6 · p. 10, §4.3 · p. 10

4.4 Ablation studies

  • Removing data-based learning, MMS construction, or continuous input queries yields 3.3, 2.2, or 11.8 times GAV’s mean error, respectively.The three ablations are compared with GAV on the same NS test set after three rounds.Source: §4.4 · p. 11, Table 3 · p. 11

5 Related work

GAV builds on PDE-agent and scientific-agent environments by turning scientific data into a continuing source of PDE experiments with reliable numerical feedback, without a reference-solver run for every new problem.

PDE agents

  • Existing PDE agents use numerical reasoning, code generation, and iterative refinement, while GAV enhances their capabilities through environment design.Prior approaches include inference-time search, solver synthesis, and revision using numerical evidence.Source: §5 · p. 11

Environments for scientific agents

  • Scientific-agent environments study experiment choice and model revision, while scientific code-generation work jointly refines test cases and candidate programs.GAV advances this direction by providing reliable feedback without a reference solve for each new problem.Source: §5 · pp. 11–12
  • GAV turns scientific data into a continuing source of PDE experiments with reliable numerical feedback, without running a reference solver for every new problem.This connects scientific-agent experimentation with PDE program improvement.Source: §5 · p. 12

6 Discussion

GAV turns scientific data into new PDE problems with reliable feedback so agents can test ideas, investigate failures, and refine programs. The paper reports theoretical guarantees under stated conditions, substantial gains for existing PDE agents, and plans to extend the approach beyond PDEs.

6 Discussion

  • GAV turns scientific data into new PDE problems with reliable numerical feedback, supporting agents as they test ideas, investigate failures, and refine programs.The paper presents environment design as a way to enhance agents’ ability to solve scientific problems.Source: §6 · p. 12
  • The paper establishes theoretical guarantees for evaluation and guided development under stated conditions, and reports substantial gains for existing PDE agents.These results are presented as evidence for environment design as a way to enhance scientific agents.Source: §6 · p. 12
  • Future work will extend the approach beyond PDEs by coupling data-informed problem generators with task-specific verifiers.The stated aim is to let agents test ideas and improve solutions across scientific domains.Source: §6 · p. 12

AI use statement

The authors report using two agent models in the PDE experiments and AI tools for manuscript and figure preparation.

AI use statement

  • The experiments use gpt-5.6-sol and gpt-6-astra as agent models for PDE tasks.The specified experiments are described in Appendix C.1.Source: AI use statement · p. 12
  • AI tools were also used to polish the manuscript and prepare scientific figures.Source: AI use statement · p. 12

A PDEs and data sources

The studies use Navier–Stokes and kinetic-equation data from numerical simulations and measurements. Separate Generators fit continuous representations, sample reference solutions, and use manufactured solutions to construct matching PDE problems.

A.1 Navier–Stokes equations

  • Navier–Stokes problems specify the domain, viscosity, initial velocity, and boundary conditions; the equations enforce incompressibility and include forcing.The numerical corpus contains 100,000 problems in a unit square with two obstacles at viscosity 0.01, sampled on a 64 × 64 grid at 32 times; manufactured-solution forcing is constructed from sampled velocity and pressure.Source: §A.1 · p. 16
  • The experimental Navier–Stokes source is PIV data for flow past a cylinder at Reynolds number 413, with 8,000 velocity fields sampled at 20 Hz.Problems use a 57 × 39 region with valid measurements, 32 consecutive times over 1.55 s, measured initial and boundary velocities, and forcing constructed from velocity with zero pressure as a convention.Source: §A.1 · pp. 16–17, Figure 7 · p. 16

A.2 Kinetic equations

  • The kinetic studies cover collisionless VP and collisional VPFP; setting the collision rate to zero gives collisionless VP, and the studies use periodic spatial boundaries over t ∈ [0, 1].VPFP studies use d = 2, x ∈ [0, 2π)^2, v ∈ R^2, and temperature 0.8–1.2; the ARTEMIS-informed VP study uses d = 1.Source: §A.2 · p. 17
  • The strong-collision corpus uses simulated VPFP trajectories with sampled collision rates and temperatures, spatially varying Gaussian-mixture velocity distributions, and Gaussian–Hermite source terms.Trajectories are sampled at 33 times on a 16 × 16 spatial grid and 21 × 21 velocity grid, then projected into a continuous representation and paired with manufactured-solution source terms.Source: §A.2 · pp. 17–18
  • The weak-collision kinetic corpus uses coarse numerical solves followed by continuous projection, with 33 times, a 16 × 16 spatial grid, and a 21 × 21 velocity grid.The Generator models macroscopic velocity and stress together with microscopic distribution coefficients.Source: §A.2 · p. 18
  • The ARTEMIS-informed VP Generator uses filtered electron observations to constrain its distribution and supplies continuous periodic reference fields with matching source terms.The observations come from two specified ESA windows; local velocity profiles are represented using 256 bins, and matching sources are constructed with MMS.Source: §A.2 · p. 18

B.1 Reference solutions and problem generation

  • For each data source, a Generator fits a continuous representation, defines a sampling distribution, and applies MMS to sampled reference solutions to obtain problem inputs.Each generated problem has a reference solution without a separate numerical solve, and its continuous representation supplies source and initial/boundary data.Source: §B.1 · p. 18

B.1.1 NS Generator

This section describes generating continuous reference solutions for Navier–Stokes and kinetic problems, constructing matching PDE inputs and verifier targets, and using error-guided sampling and constraint-preserving perturbations alongside a shared multi-turn setup for feedback-driven agent development.

B.1.1 NS Generator

  • NS references use a streamfunction and spectral space-time basis, with boundary masks for numerical data but an extended basis for PIV; regularized fitting yields continuous fields.Source: §B.1.1 · pp. 18–19
  • PCA-compressed, whitened coefficients are modeled with rectified flow; generated velocity and pressure yield matched forcing, initial and boundary data, and verifier values via MMS.Source: §B.1.1 · pp. 19–20

B.1.2 Shared construction for kinetic problems

  • For kinetic problems, the continuous reference distribution determines density, the zero-mean periodic electric field, source, initial data, and verifier quantities; collisionless VP sets νc = 0.Source: §B.1.2 · p. 20

B.1.3 Strong-collision VPFP Generator

  • The strong-collision VPFP Generator uses positive Fourier–Hermite representations with continuous time interpolation and a conditional diffusion model for future coefficients; sampled fields determine the source and verifier moments.Source: §B.1.3 · pp. 20–21

B.1.4 Weak-collision VPFP Generator

  • The weak-collision VPFP Generator combines macro and micro diffusion models in a moving-frame decoder, producing a nonnegative continuous distribution that retains microscopic degrees of freedom.Source: §B.1.4 · p. 21

B.1.5 ARTEMIS-informed VP Generator

  • The ARTEMIS-informed VP Generator learns a continuous nonnegative distribution from conditioned profile data and uses the shared kinetic construction to create new verifiable collisionless VP problems.Source: §B.1.5 · pp. 21–22

B.2 Guided generation

  • Guided generation fits predictors to program-error feedback and uses their gradients to steer sampling toward higher-error problems while keeping the base Generator fixed, then applies MMS to construct each problem.Source: §B.2 · p. 22

B.3 Perturbation

  • Perturbations transform continuous references before MMS construction; NS variants preserve incompressibility, initial behavior, and boundaries, while kinetic variants preserve constraints such as positivity or the initial distribution.Source: §B.3 · pp. 23–24
  • For collisionless VP, a smooth velocity map preserves nonnegativity, bin masses, density, and the Poisson field while recomputing moments and the source.The map fixes bin boundaries and is identity outside the sensor window; it is also applied to the initial distribution.Source: §B.3 · p. 25

C.1 Agent settings and prompts

  • The development loop gives two agents the same current program and memory log: one explores alternatives, while the other refines the current approach using numerical feedback.The selected program, research notes, and evaluation results pass to the next round.Source: §C.1 · p. 25
  • Shared prompts specify the PDE task, public inputs, required outputs, evaluation criterion, and development budget, while assigning exploration and refinement roles.Both roles may change methods, representations, and experiments; the refinement role may replace its current approach.Source: §C.1 · pp. 25–26
  • Numerical NS agents start from a program that holds the queried initial velocity fixed throughout the time interval.The run uses seed 101 and four NVIDIA L40S GPUs, 64 CPU cores, and 160 GiB of host memory.Source: §C.1 · p. 26

C.2 Program selection and final evaluation

The evaluation protocol separates program selection from final testing: selection uses feedback problems, while a fixed, separate test set measures final performance. Shared scoring points and task-specific procedures support comparisons across programs.

C.2 Program selection and final evaluation

  • The retained program is the lowest-error candidate completing evaluation with finite errors; if neither candidate improves, the current program carries forward.Candidates are compared on the same selection problems, evaluation points, and per-problem execution budget.Source: §C.2 · p. 27
  • Final evaluation uses a separate fixed test set after the prescribed last development round, and test results do not determine the reported final round.Programs share test problems and evaluation points within each study.Source: §C.2 · p. 27
  • Navier–Stokes scoring averages errors across unguided, guided, and perturbed problems on generated sets, but equally averages per-problem errors on interpolated tests.Generated NS selection and test sets use 96 and 128 evaluation points per problem, respectively.Source: §C.2 · p. 27
  • Kinetic scoring samples space–time points for VPFP and samples a noninitial native time before spatial locations for VP, with query visibility differing by task.For VP, programs know the time horizon during preparation, but spatial evaluation coordinates remain hidden.Source: §C.2 · p. 27
  • Selection uses fresh unguided, guided, and perturbed problems each round, ranking programs by mean score while providing separate feedback evaluations for the selected program.The selection mix contains 16 problems of each type.Source: §C.2 · p. 27

C.3 Adapting existing PDE agents

GAV lets existing PDE agents request new problems, query inputs, and receive verifier feedback during development. Across the three agents, GAV and GAV-Full achieve lower test error with less active time than Original; the accompanying ablations vary data learning, manufactured-solution construction, and input access.

C.3 Adapting existing PDE agents

  • The three agents retain their established workflows while GAV adds requests for new problems, continuous input queries, and verifier feedback during development.GAV-Full additionally provides guided generation and perturbation; only the Verifier accesses references for newly generated problems.Source: §C.3 · pp. 28–29
  • CodePDE refines candidates, AutoNumerics plans and selects numerical methods, and PDE-SHARP uses judges to nominate and revise solvers.With GAV, these workflows can use further problems and input queries during their respective development stages.Source: §C.3 · p. 29
  • For all three methods, GAV and GAV-Full achieve lower test error with less total agent active time than Original.Table 9 reports active times; Figure 6 reports test error.Source: §C.3 · p. 29, Table 9 · p. 29

C.4 Ablation study details

  • The ablations separately replace the learned Generator with an unfitted Gaussian-process prior, replace MMS references with approximate numerical solves, or restrict development inputs to a fixed 64 × 64 grid.The no-MMS condition uses Nektar++ and incurs additional computation and discretization error; the query restriction applies during development only.Source: §C.4 · p. 30

D.1 Evaluation beyond learned Generators

  • To test transfer beyond the learned Generator distribution, the study evaluates frozen programs on additional references built from numerical solutions or unused observations.The evaluation compares programs on the same problems and scoring points within each task.Source: §D.1 · p. 31

D.1.1 Numerical NS problems

The numerical Navier–Stokes evaluation tests final programs on interpolated problems constructed from unused numerical solutions. GAV-Full completes the full set with low mean relative velocity error, and both GAV variants outperform Original for all three agents on these independent checks.

D.1.1 Numerical NS problems

  • The 96 interpolated problems are constructed by pairing similar unused numerical solutions within each flow scenario and interpolating their fitted streamfunction and pressure fields.MMS supplies the corresponding forcing, and the local interpolation keeps test solutions close to the original data for an independent same-regime check.Source: §D.1.1 · p. 31
  • 0.1256% mean relative velocity error: the final GAV-Full program completes all 96 interpolated test problems, and both GAV variants improve all three agents over Original.Table 10 reports mean relative velocity error (%) on the 96 problems; all programs use the same 128 scoring points per problem.Source: §D.1.1 · p. 31, Table 10 · p. 31
  • In the environment comparison, the condition without continuous input queries has 13.5 times GAV’s error, while GAV has the lowest error among the four conditions.Table 11 reports mean relative velocity error (%) on the 96 interpolated NS test problems.Source: §D.1.1 · p. 31, Table 11 · p. 32

D.1.2 Kinetic problems

Independent manufactured-problem tests show that every final kinetic program improves on its initial program, although one VP run regresses from R1 to R6. The case studies trace how agents use feedback to revise programs, while the visual diagnostics illustrate behavior on selected problems.

D.1.2 Kinetic problems

  • VPFP references interpolate coarse simulated trajectories into normalized continuous distributions, while six VP references use smoothed profiles from twelve unused ARTEMIS observations and manufactured spatial-temporal variation.The VPFP construction fixes mean density to one and evaluates all programs at common sampled space-time points.Source: §D.1.2 · pp. 32–33
  • R6 lowers independent-set mean error versus S0 in every kinetic task, while R1-to-R6 error improves in five of six runs and VP Astra rises 5.69-fold but remains below S0.The independent tests use additional manufactured reference families; their difficulty is not matched to the generated test sets.Source: §D.1.2 · p. 33, Table 12 · p. 33, Figure 8 · p. 34

D.2 Problem difficulty under guided generation

  • At all four guidance updates, the current numerical NS program has higher mean error on newly guided problems than on the current problems.Each comparison holds the program fixed; the authors interpret the observations as showing guided generation can supply harder problems as development proceeds.Source: §D.2 · p. 34, Figure 9 · p. 35

D.3.1 Numerical NS

  • The numerical NS agent uses feedback to select resolutions, replace 61 forcing snapshots with 25 Chebyshev queries, and test a BDF3 change on perturbed problems; the round-4 solver remains final after round 5.The round-5 weighting change helps a small development set but not the selection problems, so the incumbent is retained and its test error is unchanged.Source: §D.3.1 · pp. 35–36

D.3.2 ARTEMIS-informed VP

  • On one shared unguided development problem, the revised sampler reduced error from 0.003964 to 0.001632 and improved all five scored quantities; the common-test score stayed near 0.002854 from R3 through R6.Source: §D.3.2 · p. 37

D.3.3 Weak-collision VPFP

  • On the common test set, retained programs score R4 0.1109, R5 0.1495, and R6 0.1274, so scores do not improve monotonically; rounds 5–6 add midpoint sampling, aliasing corrections, and quadrature changes.Source: §D.3.3 · p. 37

D.3.4 Strong-collision VPFP

  • The round-3 program was retained unchanged in round 4; round 5’s common-test mean was 3.27 × 10−3, and round 6’s selected program scored 6.72 × 10−7 on the common test bank.Source: §D.3.4 · p. 37

D.4 Predicted fields and errors

  • On the fixed PIV-derived example, R4 has lower full-window vector relative ℓ2 error than R5, while VPFP field panels show residual spatial structure across selected rounds and quantities.The visualizations use one unguided test problem per task; full generated-test-bank performance is reported separately.Source: §D.4.1 · p. 41, §D.4 · p. 41, Figure 14 · p. 42

E.1 Proof of Proposition 1

The proof bounds the difference between an empirical mean loss and its target mean by separating distribution shift from sampling variation. It handles zero variance separately to complete the result.

E.1 Proof of Proposition 1

  • The proof normalizes loss under the reference distribution and uses the assumed exponential-moment control to bound the difference between target and reference expected losses.When ε is zero, the means coincide; when ε is positive, a choice of the exponential parameter yields a finite bound.Source: §E.1 · p. 49
  • Independence of the M test problems and exponential-moment bounds give matching upper-tail bounds for the normalized empirical-mean deviation and its negative.Setting x = log(2/δ) and taking a union bound yields a two-sided deviation probability at most δ.Source: §E.1 · pp. 49–50
  • Combining the distribution-shift and sampling bounds by the triangle inequality gives the stated error bound for the empirical mean relative to the target mean.For zero variance, Jensen’s equality forces normalized loss to vanish almost surely under both distributions, so the bound holds exactly.Source: §E.1 · p. 50

E.2 Proof of Theorem 1

The proof establishes that the guided objective matches the robust objective at the current parameter and controls their difference after a parameter change. This quadratic control supports monotone progress and the convergence bound.

E.2.1 Proof of Lemma 1

  • The exponential-tilt lemma bounds the guided objective against the robust objective, with equality at the current parameter and a quadratic penalty for moving away.The bound follows from the common Lipschitz condition on losses and a centered exponential-moment estimate.Source: §E.2 · p. 50, §E.2.1 · p. 52

E.2 Proof of Theorem 1

  • The proof uses the lemma and the minimizing property of each guided update to establish that the robust objective does not increase at each iteration.The argument uses the bounded parameter-domain diameter and chooses τ above the stated constant.Source: §E.2 · p. 51
  • A weighted telescoping sum cancels intermediate optimality gaps and yields the theorem’s convergence bound.The proof chooses α = 2/(t + 2) before summing over iterations.Source: §E.2 · pp. 51–52

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