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GSM-Plus: A Comprehensive Benchmark for Evaluating the Robustness of LLMs as Mathematical Problem Solvers

Qintong Li, Leyang Cui, Xueliang Zhao, Lingpeng Kong, Wei Bi

arXiv:2402.19255v2cs.CL

TL;DR

LLMs perform strongly on standard mathematical benchmarks, but their robustness when questions are perturbed remains uncertain. This paper introduces GSM-PLUS, an adversarial GSM8K extension with diverse mathematical variations, and evaluates 25 LLMs and four prompting techniques. The results show substantial performance declines on perturbed questions, while iterative generation-and-verification prompting performs well on both original and varied problems.

  • Problem

    Whether LLMs genuinely apply mathematical knowledge or rely on superficial patterns remains uncertain because small question changes can cause incorrect answers.

  • Method

    The paper builds GSM-PLUS by applying eight perturbations from five reasoning perspectives to GSM8K and evaluates 25 LLMs with four prompting techniques.

  • Results

    LLMs show significant performance declines on perturbed questions, especially for critical thinking, arithmetic variation, and distractor insertion, while iterative thought generation and verification performs well on GSM8K and GSM-PLUS.

  • Takeaways & Limitations

    Math reasoning evaluation should assess robustness across varied question forms rather than relying only on performance on standard benchmarks.

  • Takeaways & Limitations

    The study focuses mainly on elementary-school math reasoning; robustness at other education levels is left for future work.

Abstract

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Large language models (LLMs) have achieved impressive performance across various mathematical reasoning benchmarks. However, there are increasing debates regarding whether these models truly understand and apply mathematical knowledge or merely rely on shortcuts for mathematical reasoning. One essential and frequently occurring evidence is that when the math questions are slightly changed, LLMs can behave incorrectly. This motivates us to evaluate the robustness of LLMs' math reasoning capability by testing a wide range of question variations. We introduce the adversarial grade school math (GSM-Plus) dataset, an extension of GSM8K augmented with various mathematical perturbations. Our experiments on 25 LLMs and 4 prompting techniques show that while LLMs exhibit different levels of math reasoning abilities, their performances are far from robust. In particular, even for problems that have been solved in GSM8K, LLMs can make mistakes when new statements are added or the question targets are altered. We also explore whether more robust performance can be achieved by composing existing prompting methods, in which we try an iterative method that generates and verifies each intermediate thought based on its reasoning goal and calculation result.

1 Introduction

GSM-PLUS evaluates whether LLMs’ mathematical reasoning remains reliable when GSM8K problems undergo diverse perturbations. Across models and prompting techniques, strong performance on original questions does not translate into robust performance on variations.

  • Motivation: LLMs can make simple reasoning errors when math word problems are slightly changed, motivating adversarial robustness evaluation.Figure 1 contrasts a correct answer to an original question with an incorrect answer after additional constraints are introduced.
  • Findings: The benchmark reveals an accuracy gap of up to 20% between reported model performance and performance under adversarial evaluation, while human performance remains unaffected.The unchanged inherent difficulty of the questions is given as the reason for stable human performance.
  • Benchmark: The benchmark creates eight variations for each of 1,319 GSM8K test questions, yielding 10,552 question variations across five reasoning perspectives.The perspectives are numerical variation, arithmetic variation, problem understanding, distractor insertion, and critical thinking.
  • Findings: Task-specific optimization usually improves benchmark accuracy, but robustness depends more on the base model and fine-tuning dataset.This finding concerns comparisons across models with different scales and task-specific fine-tuning.
  • Findings: LLMs struggle most with critical thinking, arithmetic variation, and distractor insertion, while remaining relatively resilient to numerical variation and problem understanding.All four prompting techniques also lack robustness, especially for arithmetic variation and critical thinking.
  • Findings: An iterative prompting method that generates and verifies each intermediate thought demonstrates good performance on both GSM8K and GSM-PLUS.The method verifies each thought according to its reasoning goal and calculation result.

2 Related Work

Prior mathematical reasoning benchmarks and adaptation methods improve or assess model capabilities, but robustness evaluations have generally covered limited perturbation types. GSM-PLUS addresses this gap with eight variations per question and systematic evaluation across reasoning skills.

  • Existing resources: Mathematical reasoning datasets span grade-school, high-school, and college-level problems, while supervised fine-tuning adapts language models to mathematics.Related work includes equation-based datasets, GSM8K, MATH, TheoremQA, question bootstrapping, and hybrid rationales.
  • Prompting: Prompting methods improve mathematical reasoning by eliciting natural-language chains of thought or programs with external calculation tools.The related work discusses chain-of-thought and program-of-thought prompting.
  • Robustness evaluation: Existing robustness studies examine perturbations such as semantic substitution, reversal prediction, and irrelevant-context distraction, but most settings cover limited automatically constructed perturbations.GSM-PLUS expands coverage across multiple mathematical reasoning skills.
  • GSM-PLUS: GSM-PLUS creates eight variations of each question using eight perturbations from five reasoning perspectives and evaluates robustness systematically across these types.For most LLMs, GSM-PLUS is challenging; GPT-3.5-Turbo reaches 61.19% accuracy.

3 The GSM-PLUS Dataset

GSM-PLUS extends GSM8K with adversarial perturbations designed to test distinct mathematical reasoning skills and evaluates the reliability of its construction and robustness metrics.

  • Dataset design: GSM-PLUS constructs eight perturbation types from five mathematical reasoning perspectives, using GSM8K as its foundation.The perspectives include numerical variation, arithmetic variation, problem understanding, distractor insertion, and critical thinking.
  • Perturbation categories: Numerical variation alters numerical data or representation, including substitution, digit expansion, and integer-decimal-fraction conversion.These changes test whether models have overfitted to the original numerical values.
  • Perturbation categories: Arithmetic variation tests flexibility by adding statements with restricted operations or reversing a statement into the queried answer.Reversing operation can transform a given price statement into a question asking for the unit price.
  • Perturbation categories: Problem understanding, distractor insertion, and critical thinking respectively test wording sensitivity, irrelevant numerical statements, and recognition of missing essential information.Critical thinking requires models to question or doubt during mathematical reasoning rather than respond mindlessly.
  • Dataset construction: GPT-4-generated variations required human refinement because generation could omit perturbations, add unintended changes, create invalid or overly difficult questions, or produce incorrect answers.The authors used qualifying exams, batched annotation, feedback, and cross-annotation to control quality.
  • Dataset validation: GPT-4 performed similarly on self-generated and human-corrected variations, but its rewriting and question-answering capabilities did not completely align.This result suggests limited preference bias in the evaluated variation sets while confirming that generation quality alone does not guarantee correct answers.
  • Evaluation metrics: The performance drop rate compares GSM-PLUS with GSM8K, while accurately solved pairs require correct answers for both a seed question and its variation.Each seed sample has N paired variations across the perturbations.

4 Experimental Setup

The experiments compare representative closed-source, open-source, and math-specialized LLMs on GSM8K and GSM-PLUS using accuracy, PDR, and ASP to assess robustness.

  • Evaluation design: The evaluation compares representative LLM accuracy on the original GSM8K dataset and the adversarial GSM-PLUS dataset.PDR and ASP are used to evaluate robustness across diverse mathematical reasoning perturbations.
  • Models: The model set includes GPT-4, GPT-3.5-Turbo, Mistral, LLaMA-2, CodeLlama, and several math-specialized open-source SFT models.The math-specialized models include MetaMath, Abel, ToRA, MAmmoTH, and SEGO.
  • Evaluation protocol: Decoding temperature is set to 0 for deterministic predictions, while open-source base models are evaluated using 8 demonstrations.The table marks models evaluated under the 8-shot setting.

5 Experiments

Experiments show substantial robustness gaps between LLM performance on GSM8K and perturbed GSM-PLUS questions. Robustness varies with model family, perturbation type, prompting strategy, and transfer between seed questions and variations.

  • Overall results: GPT-4 exhibits the highest robustness, with the smallest PDR of 8.23%, while CodeLlama records the largest PDRs across its tested sizes.CodeLlama’s PDRs are 40.56%, 39.71%, and 34.27% for 7B, 13B, and 34B, respectively.
  • Model comparisons: Math SFT models improve accuracy on GSM8K and GSM-PLUS but do not consistently reduce PDR relative to their foundation models.LLaMA-2-13B and Abel-13B have PDRs of 34.76% and 31.97%, while LLaMA-2-70B and MAmmoTH-70B have 29.40% and 29.62%.
  • Model comparisons: Larger Math SFT models generally show lower PDR, with Abel-7B, Abel-13B, and Abel-70B reaching 37.67%, 31.97%, and 28.52%, respectively.The passage also reports robustness differences associated with foundation models and model variants.
  • Model comparisons: SFT dataset choice materially affects robustness: LLaMA-2-7B variants fine-tuned on four datasets have PDRs ranging from 33.53% to 39.17%.The reported PDRs are 33.53% for MetaMathQA, 37.67% for Abel-data, 35.39% for TORA-CORPUS, and 39.17% for Math-Instruct.
  • Question variations: Models are least robust to critical thinking, arithmetic, distractor insertion, and integer-decimal-fraction perturbations, while numerical substitution and rephrasing preserve or slightly improve performance.Program-based models are especially sensitive to some perturbations, whereas larger models better resist distractors.
  • Question variations: Removing an essential statement causes performance to deteriorate nearly 100% for most non-human, non-closed-source models, which may produce hallucinated solutions.The reported failure involves not acknowledging that required information is absent.
  • Question variations: Adding operations causes accuracy drops exceeding 40% for most models except GPT-4 and GPT-3.5-Turbo, while reversing statements also produces notable PDR values.The finding is reported across models regardless of size and output format.
  • Reasoning transferability: Seed-question accuracy transfers incompletely to variations: Abel-70B has the highest ASP among the six right-most mathematical models at 55.31%, above GPT-3.5-Turbo’s 51.36%.Red bars represent questions solved on GSM8K but not on GSM-PLUS, indicating limited transferability.

6 Conclusions and Discussions

GSM-PLUS systematically evaluates LLM robustness on perturbed math word problems and finds substantial performance declines despite success on standard GSM8K. The results motivate future work on consistent reasoning and show that compositional prompting has limited overall impact.

  • GSM-PLUS evaluates the performance stability of 25 prominent models when mathematical questions are modified by diverse perturbations.
  • Significant declines occur on successfully solved questions when perturbations are introduced, revealing limited robustness relative to standard benchmark performance.
  • The findings motivate systematic evaluation across diverse mathematical skills and models that remain resilient to minor variations.
  • Compositional prompting can improve LLM performance, but its impact remains limited for both performance and robustness.

Limitations

The study’s limitations concern its elementary-school scope, its reliance on answer accuracy rather than solution-chain accuracy, and its lack of analysis of failure causes.

  • The evaluation focuses mainly on elementary-school math reasoning, leaving robustness at other educational levels for future work.
  • The study compares answer accuracy on GSM8K and GSM-PLUS but does not assess solution-chain accuracy because a reliable metric is difficult to design.
  • GSM-PLUS evaluates robustness under perturbations without investigating the underlying reasons for models’ failures.

Ethics Statement

The paper states that it follows the Code of Ethics and uses no private or non-public information. It also describes mathematical problem solving and the perturbation design used to evaluate LLM reasoning.

  • Ethics Statement: The work uses no private data or non-public information.
  • Background: Polya’s principles describe mathematical problem solving as identifying context, representing structure, calculating an answer, and verifying correctness.
  • Perturbation Types: The benchmark designs eight perturbation types motivated by Polya’s principles to test LLM mathematical reasoning.

B Dataset Construction

The dataset construction pipeline uses seed question-answer pairs and perturbation descriptions to generate minimally edited question variants, then prompts GPT-4 to answer them in a standardized format.

  • Question Variation Generation: A seed question-answer pair is supplied to generate a question variation from a specified perturbation description.
  • Question Variation Generation: The rewriting prompt asks for reasonable, understandable questions with minimal edits and one required numerical answer.
  • Question Variation Generation: The generation prompt includes the perturbation strategy, seed question, and answer rationale as inputs.
  • Answer Generation: GPT-4 is required to answer each generated question variation using a final-answer prompt.

B.2 Human Annotation

GSM-PLUS combines GPT-4-generated question variations with qualified human review to control variation and answer quality. Evaluation uses multiple annotators, author checks, and reports measurable agreement, while critical-thinking perturbations substantially reduce model performance.

  • Human evaluation process: Evaluators first completed a qualification study, received feedback, and were selected for the main round based on performance and guideline understanding.The study involved annotating 24 variations for three GSM8K questions before individual review and feedback.
  • Human evaluation process: Five evaluators with at least bachelor’s degrees reviewed batches of 50 seed questions, with authors verifying two randomly selected questions per batch.The authors used side-by-side checks and maintained communication with evaluators throughout the process.
  • Quality control: GPT-4-generated variations were further refined by human annotators, because valid rewrites did not necessarily guarantee correct answers.Human involvement was especially necessary for reversing-operation perturbations, and the rewriting process was not treated as evidence of GPT-4’s math reasoning ability.
  • Quality control: 1,000 question variations were randomly selected, each assessed by at least three annotators, and disagreements were manually reviewed.Krippendorff’s α was 0.567, indicating relatively reliable inter-annotator agreement.
  • Evaluation outcomes: Critical-thinking perturbations were the most challenging and significantly contributed to the decrease in model performance on GSM-PLUS.Table 8 reports model accuracy on GSM8K and GSM-PLUS with and without the critical-thinking variation.

C.3 Cases on Diverse Question Variations

The case studies show that models often solve original GSM8K problems but fail after conditions, operations, wording, or numerical representations are altered. Failures include hallucinating missing information, incorporating distractors, mishandling reversed operations, and confusing changed numerical forms, while COMP improves several variation types.

  • Critical Thinking: When an essential statement is removed, models may fabricate a solution instead of recognizing that the problem lacks sufficient information.The critical-thinking example gives “None” as the correct answer because the second-half scoring information is absent.
  • Adding Operation and Distractor Insertion: GPT-3.5-Turbo failed to account for an added condition, while program-based MAmmoTH-13B treated an irrelevant book price as a calculation variable.These examples illustrate failures on adding-operation and distractor-insertion perturbations.
  • Reversing Operation: GPT-4 solved the original two-day flour problem but failed when the question was reversed to ask for the single-day cookie requirement.The case contrasts correct performance on GSM8K with failure on the corresponding GSM-PLUS variation.
  • Integer-decimal-fraction Conversion: Integer-decimal-fraction changes caused program-based and natural-language models to fail through different mechanisms: incorrect program behavior versus calculation errors.The program-based model returned an incorrect subtraction, while natural-language models faltered on arithmetic after numerical modification.
  • Problem Understanding: Most LLMs handled GPT-4-rephrased questions robustly, and one model answered a rephrased problem correctly after answering the original incorrectly.The example identifies a wording-sensitive reasoning error in the original question that was avoided in the GSM-PLUS rephrasing.
  • COMP prompting: COMP significantly improved performance on numerical substitution, problem understanding, adding operation, and critical-thinking variations, while repeated attempts helped with inserted distractors.The method did not substantially improve numerical variation, where calculation errors remained without external tools or ensemble methods.

C.8 Sample Predictions on GSM8K and GSM-PLUS

The sample predictions compare prompting techniques on original GSM8K questions and their GSM-PLUS rewrites. Although all techniques answer the originals correctly, COMP is the only method shown to remain correct on the modified reversing- and adding-operation questions.

  • Reversing-operation example: All prompting techniques correctly answered the original GSM8K question, but only COMP consistently answered the reversing-operation variation correctly.The comparison is presented in Table 10 for an original question and its corresponding rewritten question.
  • Adding-operation example: All prompting techniques correctly answered the original GSM8K question, but only COMP consistently answered the adding-operation variation correctly.The comparison is presented in Table 11 for an original question and its corresponding rewritten question.
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