B4: Towards Optimal Assessment of Plausible Code Solutions with Plausible Tests

TL;DR

The proposed approximated optimal strategy ℬ4 significantly surpasses existing heuristics in selecting code solutions generated by large language models with LLM-generated tests, achieving a relative performance improvement by up to 50% over the strongest heuristic and 246% over the random selection in the most challenging scenarios.

Abstract

Selecting the best code solution from multiple generated ones is an essential task in code generation, which can be achieved by using some reliable validators (e.g., developer-written test cases) for assistance. Since reliable test cases are not always available and can be expensive to build in practice, researchers propose to automatically generate test cases to assess code solutions. However, when both code solutions and test cases are plausible and not reliable, selecting the best solution becomes challenging. Although some heuristic strategies have been proposed to tackle this problem, they lack a strong theoretical guarantee and it is still an open question whether an optimal selection strategy exists. Our work contributes in two ways. First, we show that within a Bayesian framework, the optimal selection strategy can be defined based on the posterior probability of the observed passing states between solutions and tests. The problem of identifying the best solution is then framed as an integer programming problem. Second, we propose an efficient approach for approximating this optimal (yet uncomputable) strategy, where the approximation error is bounded by the correctness of prior knowledge. We then incorporate effective prior knowledge to tailor code generation tasks. Both theoretical and empirical studies confirm that existing heuristics are limited in selecting the best solutions with plausible test cases. Our proposed approximated optimal strategy B4 significantly surpasses existing heuristics in selecting code solutions generated by large language models (LLMs) with LLM-generated tests, achieving a relative performance improvement by up to 50% over the strongest heuristic and 246% over the random selection in the most challenging scenarios. Our code is publicly available at https://github.com/ZJU-CTAG/B4.

Figures (7)

• Figure 1: A simple example showing the problem "return the sum of $a$ and $b$". A link between a generated code solution and a generated test case indicates that the solution passes the test. How can we select the best code solution solely based on these links?
• Figure 2: Illustration of the generation process. The correctness of code $X_i$ and test case $Y_j$ is sampled using parameters $\theta_x$ and $\theta_y$ respectively. $E_{ij}$ is generated based on $X_i$ and $Y_j$, using the corresponding parameters (1, 0, $\theta_1$ or $\theta_0$).
• Figure 3: (a) Real distributions for two parameters $\theta_0$ and $\theta_1$. (b) Three Beta distributions with different hyperparameters.
• Figure 4: Visualization of two Beta functions used in our scoring strategy. We set $|{\mathbf{E}}_0|=5000$ and $N=100$.
• Figure 5: Pass@1 results of the three methods under different conditions in the simulated experiments. By default, we set $N=10$, $M=30$, $\theta_x=0.2$, and $\theta_y=0.3$ except for the varied one.

Theorems & Definitions (7)

• Remark 1
• Theorem 1: Approximation error bound
• Lemma 4.1
• Theorem 2: Impact of correct test cases for MaxPass
• Theorem 3: Impact of incorrect solutions for MaxPass
• Lemma 4.2
• Theorem 4: Impact of $\theta_x$ and $N$ for CodeT