It is common to reject undesired outputs of Large Language Models (LLMs); however, current methods to do so require an excessive amount of computation, or severely distort the distribution of outputs. We present a method to balance the distortion of the output distribution with computational efficiency, allowing for the generation of long sequences of text with difficult-to-satisfy constraints, with less amplification of low probability outputs compared to existing methods. We show through a series of experiments that the task-specific performance of our method is comparable to methods that do not distort the output distribution, while being much more computationally efficient. Language models sometimes generate undesirable outputs, such as syntactically-incorrect code, hallucinated PII, or profanity. These conditions, which we collectively refer to as errors for the remainder of the paper, can be detected with incremental parsers, regular expression matching, or even simple substring searches. However, once detection occurs, there are several competing methods for mitigating errors in the output. One set of methods, constrained generation (Beurer-Kellner et al., 2024; Geng et al., 2024; Melcer et al., 2024), avoids errors by disabling the generation of any token that immediately leads to such an error. While this method is effective, it can lead to the amplification of low-probability outputs. Another class of methods avoids errors without any amplification of low-probability outputs, at the cost of additional computation. Rejection sampling is the simplest such method; i.e. if the output contains an error, simply generate another sample until the output is acceptable. Adaptive Sampling with Approximate Expected Futures (ASAp) (Park et al., 2024) provides a performance improvement over rejection sampling while maintaining the output distribution by effectively sampling without replacement, but there are still many situations in which it may converge too slowly. In our experiments, we show that our method obtains task-specific performance on par with ASAp, while converging significantly faster when the constraints are difficult to satisfy. We first describe autoregressive language models and their properties.

Large Language Models are powerful tools for program synthesis and advanced auto-completion, but come with no guarantee that their output code is syntactically correct. This paper contributes an incremental parser that allows early rejection of syntactically incorrect code, as well as efficient detection of complete programs for fill-in-the-middle (FItM) tasks. We develop Earley-style parsers that operate over left and right quotients of arbitrary context-free grammars, and we extend our incremental parsing and quotient operations to several context-sensitive features present in the grammars of many common programming languages. The result of these contributions is an efficient, general, and well-grounded method for left and right quotient parsing. To validate our theoretical contributions -- and the practical effectiveness of certain design decisions -- we evaluate our method on the particularly difficult case of FItM completion for Python 3. Our results demonstrate that constrained generation can significantly reduce the incidence of syntax errors in recommended code.