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

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.

Revisiting Convergence of AdaGrad with Relaxed Assumptions

In this study, we revisit the convergence of AdaGrad with momentum (covering AdaGrad as a special case) on non-convex smooth optimization problems. We consider a general noise model where the noise magnitude is controlled by the function value gap together with the gradient magnitude. This model encompasses a broad range of noises including bounded noise, sub-Gaussian noise, affine variance noise and the expected smoothness, and it has been shown to be more realistic in many practical applications. Our analysis yields a probabilistic convergence rate which, under the general noise, could reach at (\tilde{\mathcal{O}}(1/\sqrt{T})). This rate does not rely on prior knowledge of problem-parameters and could accelerate to (\tilde{\mathcal{O}}(1/T)) where (T) denotes the total number iterations, when the noise parameters related to the function value gap and noise level are sufficiently small. The convergence rate thus matches the lower rate for stochastic first-order methods over non-convex smooth landscape up to logarithm terms [Arjevani et al., 2023]. We further derive a convergence bound for AdaGrad with mometum, considering the generalized smoothness where the local smoothness is controlled by a first-order function of the gradient norm.

On Convergence of Adam for Stochastic Optimization under Relaxed Assumptions

The Adaptive Momentum Estimation (Adam) algorithm is highly effective in training various deep learning tasks. Despite this, there's limited theoretical understanding for Adam, especially when focusing on its vanilla form in non-convex smooth scenarios with potential unbounded gradients and affine variance noise. In this paper, we study vanilla Adam under these challenging conditions. We introduce a comprehensive noise model which governs affine variance noise, bounded noise and sub-Gaussian noise. We show that Adam can find a stationary point with a $\mathcal{O}(\text{poly}(\log T)/\sqrt{T})$ rate in high probability under this general noise model where $T$ denotes total number iterations, matching the lower rate of stochastic first-order algorithms up to logarithm factors. More importantly, we reveal that Adam is free of tuning step-sizes with any problem-parameters, yielding a better adaptation property than the Stochastic Gradient Descent under the same conditions. We also provide a probabilistic convergence result for Adam under a generalized smooth condition which allows unbounded smoothness parameters and has been illustrated empirically to more accurately capture the smooth property of many practical objective functions.

High Probability Convergence of Adam Under Unbounded Gradients and Affine Variance Noise

In this paper, we study the convergence of the Adaptive Moment Estimation (Adam) algorithm under unconstrained non-convex smooth stochastic optimizations. Despite the widespread usage in machine learning areas, its theoretical properties remain limited. Prior researches primarily investigated Adam's convergence from an expectation view, often necessitating strong assumptions like uniformly stochastic bounded gradients or problem-dependent knowledge in prior. As a result, the applicability of these findings in practical real-world scenarios has been constrained. To overcome these limitations, we provide a deep analysis and show that Adam could converge to the stationary point in high probability with a rate of $\mathcal{O}\left({\rm poly}(\log T)/\sqrt{T}\right)$ under coordinate-wise "affine" variance noise, not requiring any bounded gradient assumption and any problem-dependent knowledge in prior to tune hyper-parameters. Additionally, it is revealed that Adam confines its gradients' magnitudes within an order of $\mathcal{O}\left({\rm poly}(\log T)\right)$. Finally, we also investigate a simplified version of Adam without one of the corrective terms and obtain a convergence rate that is adaptive to the noise level.