  Maximum likelihood estimation (MLE) is a statistical method used to estimate
the parameters of a probability distribution that best explain the observed
data. In the context of text generation, MLE is often used to train generative
language models, which can then be used to generate new text. However, we argue
that MLE is not always necessary and optimal, especially for closed-ended text
generation tasks like machine translation. In these tasks, the goal of model is
to generate the most appropriate response, which does not necessarily require
it to estimate the entire data distribution with MLE. To this end, we propose a
novel class of training objectives based on convex functions, which enables
text generation models to focus on highly probable outputs without having to
estimate the entire data distribution. We investigate the theoretical
properties of the optimal predicted distribution when applying convex functions
to the loss, demonstrating that convex functions can sharpen the optimal
distribution, thereby enabling the model to better capture outputs with high
probabilities. Experiments on various text generation tasks and models show the
effectiveness of our approach. It enables autoregressive models to bridge the
gap between greedy and beam search, and facilitates the learning of
non-autoregressive models with a maximum improvement of 9+ BLEU points.
Moreover, our approach also exhibits significant impact on large language
models (LLMs), substantially enhancing their generative capability on various
tasks. Source code is available at
\url{https://github.com/ictnlp/Convex-Learning}.
