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Optimizing the Score, Losing Sight of the Task: Reward Hacking Across Weights, Selection, and Prompts

Vansh Wahi

arXiv:2609.25848v1cs.AI

TL;DR

A higher evaluation score may reflect task improvement or exploitation of evaluator mistakes across weights, selection, and persistent prompts. The paper compares these substrates using reachable behavior, optimization budgets, proxy error, and defenses. Its analysis and illustration show why distance alone cannot establish a universal vulnerability ranking.

  • Problem

    Evaluation scores can reward behavior exploiting evaluator weaknesses, making measured progress an unreliable distinction between solving the task and satisfying the evaluator.

    Source: Introduction · p. 1

  • Method

    The paper develops a comparative framework for weights, selection, and persistent prompts using budgeted policy classes, behavioral reach, proxy error, and cross-substrate defenses.

    Source: Introduction · p. 2

  • Results

    The analysis gives a distance-dependent bound and a separate capacity ordering, while a counterexample and finite-output illustration show that distance alone does not rank vulnerability universally.

    Source: Introduction · p. 2

  • Takeaways & Limitations

    Trustworthy improvement requires controlling accessible failure modes and preserving evidence of task quality independent of the score being optimized.

    Source: Introduction · pp. 1–2

    The abstraction is not exhaustive: some agent changes require an enlarged policy or explicit environment changes, which must be reported as part of the intervention.

    Source: What the three-substrate abstraction… · p. 3

Abstract

from arXiv · show

A higher evaluation score does not always mean a better language model system. When optimization exploits an evaluator's mistakes, measured progress can conceal unchanged or deteriorating task performance. This failure can arise through parameter updates, selection among generated outputs, or revisions to persistent prompts. We develop a comparative framework for reward hacking across these three optimization substrates: weights, selection, and text. Building on the Proxy Compression Hypothesis and research on inference-time and in-context reward hacking, we examine how reachable behavior, optimization budgets, and persistent adaptation shape exposure to proxy error. We formalize a distance-dependent upper bound on evaluator disagreement and a capacity ordering for nested policy classes, then show why distance alone cannot establish a universal ranking of vulnerability. An exact finite-output illustration demonstrates how the location of a scoring defect changes the behavior favored by each method. We also map representative defenses across substrates, identifying which mechanisms transfer directly and which offer only functional analogies. Persistent prompts receive particular attention: their contents are inspectable, but the behavior induced by a small textual change may be difficult to anticipate. The formal analysis, numerical illustration, and published evidence together provide a basis for comparing optimization methods and identifying the conditions under which their defenses transfer. The resulting framework connects optimization choices to verification requirements: reliable improvement depends on controlling accessible failure modes and preserving evidence of task quality independent of the score being optimized.

Introduction

A higher evaluation score can reflect improved task performance or better exploitation of evaluator weaknesses, across weight updates, output selection, and persistent prompt revisions. The paper compares these substrates through reachable behavior, proxy error, and defenses, while distinguishing formal capacity from practical discovery.

Introduction

  • Parameter updates, best-of-n selection, and persistent prompt optimization can all reward behavior that exploits evaluator weaknesses rather than solving the task.A prompt mutation raised judge-alignment pass rate from 23.1% to 80.0% while defect-identification precision remained unchanged.Source: Introduction · p. 1
  • Weights, selection, and persistent text are distinct ways to change which outputs a system produces or accepts in response to imperfect reward.Weights modify parameters, selection changes returned-candidate distributions, and prompts revise conditioning instructions, examples, or memory.Source: Introduction · pp. 1–2
  • Reward hacking draws on shared mechanisms, but the substrates differ in reachable behaviors, controls, and the inspectability of persistent text artifacts.Text can be read directly even when the behavior induced by it remains uncertain.Source: Introduction · p. 2
  • Vulnerability depends on how reachable policies relate to evaluator errors and on resources available for searching them, so reach alone cannot rank substrates universally.A distant policy may move toward evaluator errors or toward regions where the evaluator remains accurate; a small change may concentrate probability on a nearby exploit.Source: Introduction · p. 2
  • The paper formalizes budgeted policy classes, a conditional proxy-gap bound, and a separate nested-class ordering, then uses a finite-output illustration to compare substrate responses to scoring defects.It also maps defenses across substrates, distinguishing direct transfer from functional analogy.Source: Introduction · p. 2
  • The framework combines published observations, mathematical analysis, and a reproducible numerical illustration, but reports no new language-model experiments.Its practical aim is to connect optimization choices with the evidence needed to trust improvements.Source: Introduction · p. 2

Background and Scope

The section frames reward hacking through proxy compression and compares weights, selection, and persistent text as ways to construct evaluated behavior. It distinguishes reusable prompt artifacts from one-off refinement and notes that the abstraction does not cover every change to an agent system.

The Proxy Compression Hypothesis

  • The Proxy Compression Hypothesis organizes reward hacking around objective compression, optimization amplification, and evaluator–policy co-adaptation.Compression maps a rich objective to a restricted signal, while optimization can concentrate behavior on proxy errors and repeated policy–evaluator changes can produce co-adaptation.Source: The Proxy Compression Hypothesis · p. 3
  • Proxy compression alone does not establish hacking; risk requires missing or misvalued information together with an optimizer able to exploit it.Proxy error includes omitted requirements and incorrect assessments of requirements the evaluator attempts to measure.Source: The Proxy Compression Hypothesis · p. 3

What the three-substrate abstraction covers

  • The three-substrate abstraction covers parameter updates, candidate selection, and persistent prompt revisions, but is not an exhaustive inventory of agent-system changes.Tool access, retrieval, code, routing, and permissions may require an enlarged policy or an explicit environment change; agentic comparisons should hold environment and actions fixed.Source: What the three-substrate abstraction… · p. 3

Persistent prompt optimization and in-context refinement

  • Persistent prompt optimization differs from one-conversation refinement because its resulting instructions or memory are reused on later inputs.A shared prompt exploit can affect a class of future answers, while the artifact can also be versioned, inspected, compared, and rolled back.Source: Persistent prompt optimization and… · p. 4

Three Substrates, One Formal Setup

The framework compares weights, selection, and persistent prompts as budgeted policy classes, separating what each can represent from what an optimizer can discover. It distinguishes evaluator disagreement from task-performance decline and identifies where behavioral reach and evaluation assumptions limit comparisons.

Policies, rewards, and budgets

  • Independent task evaluation may use imperfect tests or expert review, so studies should state what their measure assesses rather than treat it as an oracle.Source: Policies, rewards, and budgets · p. 4
  • Proxy performance is non-decreasing as the feasible distance budget expands, a nested-set property that does not imply practical search improves at every iteration.Source: Policies, rewards, and budgets · p. 4

What counts as hacking in the comparison

Weight-space

  • Weight updates search an architecture-dependent policy class and can make behavior practically accessible and reusable, but do not generally establish support expansion or unbounded KL.Source: Weight-space · p. 5

Selection-space

  • Best-of-n returns the highest-scoring of n sampled candidates; finite N bounds behavioral divergence, while locating a rare exploit depends on its base probability and draw count.Source: Selection-space · p. 5
  • Selection can change distribution shape and concentration, and soft-selection variants should be modeled explicitly rather than assumed equivalent to best-of-n.Source: Selection-space · p. 5

Prompt-space

  • Prompt-length limits alone do not provide a useful behavioral-KL bound, and artifact constraints such as edit limits are not interchangeable with behavioral constraints.Persistent text is inspectable, but a readable instruction can interact differently across inputs and still introduce regressions.Source: Prompt-space · p. 6

Shared Mechanisms and Different Dynamics

The paper traces proxy exploitation across weights, selection, and prompts while distinguishing the source of evaluator error from the way optimization exposes it. It also separates fixed-evaluator adaptation from co-adaptation, where the evaluator or oversight process changes in response.

Proxy error is available to all three substrates

Amplification and the shape of the curve

  • Proxy imperfection does not determine a single overoptimization curve: constant additive miscalibration preserves rankings, while prompt exploitation can raise rationale alignment without improving defect-identification precision.Wahi (2026) reports rationale alignment rising from 23.1% to 80.0% through lexical mimicry; a separate corrupted-label incident led to removal of correct compliance rules.Source: Amplification and the shape of the curve · p. 7

Fixed-evaluator adaptation versus co-adaptation

  • Best-of-n can exploit a fixed scorer without persistent learning, while tuned selection, revised judge prompts, or shared memory can carry adaptation across calls.Policy adaptation to a fixed evaluator differs from co-adaptation, which also revises the evaluator or oversight process.Source: Fixed-evaluator adaptation versus… · pp. 7–8

Reach, Geometry, and Hacking Capacity

The framework relates reward-hacking capacity to reachable behavior and evaluator errors, but neither distance nor substrate labels alone establish a universal vulnerability ranking.

A distance-dependent upper bound

  • Under a common non-decreasing error envelope, maximum behavioral distance bounds the available proxy gap, but does not imply that farther policies have larger errors.The envelope is practically useful only when informative at relevant distances; estimating or validating it for a real evaluator is a separate empirical problem.Source: A distance-dependent upper bound · p. 8

A concrete bound when proxy error is bounded

  • When proxy error has bounded range, behavioral divergence bounds additional evaluator disagreement across substrates only if that divergence is actually controlled.An instruction-edit count alone is insufficient, and the bound does not guarantee task improvement and can be loose.Source: A concrete bound when proxy error is… · pp. 8–9

When a capacity ordering is justified

  • Policy-class inclusion orders worst-case gap capacity, but does not order discovered exploits under matched compute or follow from substrate labels alone.A weight class containing all policies induced by a prompt family has at least that family's gap capacity; whether a real architecture and budget provide this inclusion is not established by the substrate name.Source: When a capacity ordering is justified · p. 9

The interpretation for language models

  • For language models, selection, prompts, and parameter updates differ in reachable behavior and search geometry, so practical comparisons must identify accessible exploits and evidence that distinguishes them.Distance helps when informative about evaluator error, but is not a substitute for identifying the error.Source: The interpretation for language models · pp. 9–10

A finite-output numerical illustration

  • Under proxy A, flexible optimization and selection first improve true reward then deteriorate, while bounded conditioning ends at true reward 0.603 and proxy reward 0.792; under proxy B, bounded conditioning reaches true reward approximately 0.042 and proxy reward 0.870.The illustration computes rewards exactly over five outputs without LLM calls or Monte Carlo estimates; its full-simplex class has maximum signed gap 0.9 for either proxy, while a finite regularization sweep need not reach it.Source: A finite-output numerical illustration · pp. 10–11

Description length as an operational constraint

  • Prompt edit limits can constrain inserted evaluation-specific information, but edit volume and per-iteration caps do not bound cumulative information or prevent compact scoring shortcuts.The paper recommends recording artifact length, added examples, cumulative change, and evaluation-content reuse, and measuring behavioral drift on an independent probe set when possible.Source: Description length as an operational… · p. 11

A Defense Correspondence Across Substrates

Defenses can improve evaluation, constrain optimization, detect concerns, or control acceptance, but these functions are complementary. Similar mechanisms may transfer across weights, selection, and prompts without carrying equivalent guarantees.

A Defense Correspondence Across Substrates

  • Defenses improve the proxy, restrict policies, detect disagreement, or gate deployment; none alone establishes that an accepted policy generalizes.A deployment gate does not establish that an accepted policy generalizes.Source: A Defense Correspondence Across… · p. 11

Improve or diversify the evaluation signal

  • Rare prompt-space errors can dominate an optimized policy despite good calibration on ordinary examples, so evaluation components need validation inside the search loop.Uncertainty estimates should not be assumed to identify outputs an adaptive optimizer preferentially discovers or to remain reliable under search-induced distribution shift.Source: Improve or diversify the evaluation… · p. 12

Constrain optimization pressure and admissible behavior

  • Optimization constraints range from behavioral-drift limits and candidate caps to prompt-edit budgets and hard checks, but their guarantees depend on what they control.A KL penalty directly constrains behavioral divergence only when KL is accurately evaluated; an edit budget controls text, whose behavioral effect still needs testing.Source: Constrain optimization pressure and… · pp. 12–13

Worked correspondence: ensembles and judge panels

  • Evaluator panels can make single-judge exploitation less attractive, but shared preferences and repeated optimization can preserve blind spots.Assess panels on exploit transfer and task quality, not only internal agreement; reviewing a patch's text does not replace evaluating its resulting behavior.Source: Worked correspondence: ensembles and… · p. 13

Worked correspondence: holdouts and information boundaries

  • Holdouts help across substrates, but information can reach optimizers through training, benchmark tuning, or prompt artifacts, and input independence differs from evaluator independence.A final untouched set can help when monitoring data has entered development; protocols should state which kind of independence they provide.Source: Worked correspondence: holdouts and… · p. 13

Worked correspondence: canaries and integrity monitoring

  • Canaries can be integrated into acceptance and audit procedures across substrates, but a pass is an alarm rather than proof of cheating.Evaluate detection coverage and false alarms separately from ordinary task accuracy; selective exploits may avoid a canary.Source: Worked correspondence: canaries and… · p. 14

Detection, authority, and acceptance

  • Detection and authority are separate: gates can reject proposals or trigger review, but each check supports conclusions only about requirements it tests.A passed parser check says little about semantic correctness, and mechanical checks do not capture every failure.Source: Detection, authority, and acceptance · p. 14

Choosing and Combining Substrates

Weights, selection, and prompts suit different kinds of change, but the framework does not rank their risks universally. Comparisons and hybrid pipelines depend on reachable behavior, search, feedback, and how exploits move between stages.

Choose according to the required change

  • Selection suits tasks where the base model produces useful candidates and scoring distinguishes them; prompts suit reusable instructions, while weights suit behavior hard to elicit or needed across many inputs.Selection adds inference cost; prompts are inspectable and reversible but can induce unintended behavior, while weight changes can be harder to inspect and attribute.Source: Choose according to the required change · pp. 14–15

Interpreting comparisons of prompting and reinforcement learning

  • Reported prompting and reinforcement-learning comparisons vary by task and do not establish a universal risk ranking across substrates.Search efficiency, feedback, model family, optimizer strength, and compute accounting can all contribute; the cited 25× gap is specific to Liu et al.'s social-media engagement example.Source: Interpreting comparisons of prompting… · p. 15

Hybrid systems and transferred exploits

  • In prompt-plus-selection pipelines, a prompt can make an exploit more common before selection amplifies it, and later distillation can persist it in weights.The best-of-n KL bound is relative to the prompt-conditioned generator, not automatically the original baseline; validate the combined policy and track transitions.Source: Hybrid systems and transferred exploits · pp. 15–16

Evidence and Relationship to Prior Work

Prior work supplies evidence of reward exploitation across settings, but differing tasks and measures support shared mechanisms and informative comparisons rather than a universal law. This paper brings those strands into a three-way comparison with formal analysis, an inspectable example, and a defense map.

What the existing evidence supports

  • On a 47-case contract-analysis suite, Wahi (2026) reports a 100% exploited pass rate versus a 68.1% clean baseline; these observations were not a controlled substrate comparison.The code-quality calibration benchmark combines 15 human-labeled directories with 39 labeled by a human-calibrated model, so its agreement figures are not direct human agreement on every item.Source: What the existing evidence supports · p. 16
  • Wahi (2026) reports that rationale-before-score ordering improved agreement in one calibration campaign, while several rubric refinements did not.This is a measured intervention in one campaign, not evidence that rubric changes are generally ineffective or that output ordering universally resolves judge bias.Source: What the existing evidence supports · p. 16

Relation to the main research strands

  • The paper connects reward-gaming, inference-time alignment, prompt optimization, and evaluator-robustness research through a comparison centered on persistent text optimization.Its contribution combines a common formal setup, reach and capacity analysis, an inspectable numerical example, and a defense map that distinguishes established components from proposed transfers.Source: Relation to the main research strands · pp. 16–17

Limitations

The framework's conclusions are bounded by assumptions about policy classes, environments, and evidence of task quality. Its numerical example and defense map illustrate comparisons but do not establish universal rankings or cross-task efficacy.

Limitations

  • The framework does not establish a universal safety ranking because real architectures, prompt interfaces, and optimizers need not have a simple class-inclusion relation.Even when inclusion holds, optimization efficiency remains separate.Source: Limitations · p. 17
  • The five-output illustration omits compositional generation, long trajectories, stochastic environments, and learned evaluators, and is not evidence about deployed-LLM hacking frequency or severity.Its flexible distribution path is an analytic surrogate, and its bounded conditioning family is designed for illustration.Source: Limitations · p. 17
  • True reward is difficult to observe, and independent tests or expert labels can be incomplete or wrong.Empirical studies must describe limitations of their operational reference.Source: Limitations · p. 17
  • Exact induced-policy KL may be unavailable for hosted models, while sequence-level estimation can be expensive or noisy.Artifact-level constraints remain useful but should not be described as measured behavioral bounds unless evaluated as such.Source: Limitations · p. 17
  • The defense map is a synthesis rather than a meta-analysis and does not establish cross-task effect sizes or equal efficacy across substrates.It compares functions and access requirements for selected methods, distinguishing established components from proposed transfers.Source: Limitations · p. 17
  • The fixed-environment abstraction does not fully analyze changes such as tool additions, retrieval changes, altered permissions, or evaluator tampering.Some interventions change the effective problem rather than merely moving a policy within it.Source: Limitations · p. 17

Conclusion

Reward hacking spans weights, selection, and persistent text, but each substrate changes which behaviors are reachable, persistent, inspectable, or constrained. The framework separates distance-based bounds and capacity ordering from the geometry and search that shape actual vulnerability.

  • Reward hacking can arise through weight updates, output selection, or persistent text revisions, each changing reachable behaviors, persistence, inspectability, or constraints.Source: Conclusion · p. 18
  • A distance-dependent error envelope bounds evaluator disagreement, while class inclusion orders capacity; neither alone determines which substrate is safest.Accessible-behavior geometry and search effectiveness also shape what a system finds.Source: Conclusion · p. 18
  • Practitioners should improve evaluators, constrain accepted behaviors, preserve independent task-quality evidence, and specify how to handle disagreement with the optimizing score.Source: Conclusion · p. 18
  • Judge improvement by its effect on the task, using evidence independent of the score that produced it.A shared framework supports reuse across settings while keeping guarantees distinct.Source: Conclusion · p. 18

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