Title: Selective Generation for Controllable Language Models
Venue: NeurIPS 2024
Accepted: yes
Keywords: trustworthy AI

selective classification

conformal prediction

hallucination

LLM
Abstract: Trustworthiness of generative language models (GLMs) is crucial in their deployment to critical decision making systems. Hence, certified risk control methods such as selective prediction and conformal prediction have been applied to mitigating the hallucination problem in various supervised downstream tasks. However, the lack of appropriate correctness metric hinders applying such principled methods to language generation tasks. In this paper, we circumvent this problem by leveraging the concept of textual entailment to evaluate the correctness of the generated sequence, and propose two selective generation algorithms which control the false discovery rate with respect to the textual entailment relation (FDR-E) with a theoretical guarantee: $\texttt{SGen}^{\texttt{Sup}}$ and $\texttt{SGen}^{\texttt{Semi}}$. $\texttt{SGen}^{\texttt{Sup}}$, a direct modification of the selective prediction, is a supervised learning algorithm which exploits entailment-labeled data, annotated by humans. Since human annotation is costly, we further propose a semi-supervised version, $\texttt{SGen}^{\texttt{Semi}}$, which fully utilizes the unlabeled data by pseudo-labeling, leveraging an entailment set function learned via conformal prediction. Furthermore, $\texttt{SGen}^{\texttt{Semi}}$ enables to use more general class of selection functions, neuro-selection functions, and provides users with an optimal selection function class given multiple candidates. Finally, we demonstrate the efficacy of the $\texttt{SGen}$ family in achieving a desired FDR-E level with comparable selection efficiency to those from baselines on both open and closed source GLMs. Code and datasets are provided at https://github.com/ml-postech/selective-generation.

Selective Generation for Controllable Language Models

Abstract

Trustworthiness of generative language models (GLMs) is crucial in their deployment to critical decision making systems. Hence, certified risk control methods such as selective prediction and conformal prediction have been applied to mitigating the hallucination problem in various supervised downstream tasks. However, the lack of appropriate correctness metric hinders applying such principled methods to language generation tasks. In this paper, we circumvent this problem by leveraging the concept of textual entailment to evaluate the correctness of the generated sequence, and propose two selective generation algorithms which control the false discovery rate with respect to the textual entailment relation (FDR-E) with a theoretical guarantee: SGen Sup and SGen Semi . SGen Sup , a direct modification of the selective prediction, is a supervised learning algorithm which exploits entailment-labeled data, annotated by humans. Since human annotation is costly, we further propose a semisupervised version, SGen Semi , which fully utilizes the unlabeled data by pseudolabeling, leveraging an entailment set function learned via conformal prediction. Furthermore, SGen Semi enables to use more general class of selection functions, neuro-selection functions , and provides users with an optimal selection function class given multiple candidates. Finally, we demonstrate the efficacy of the SGen family in achieving a desired FDR-E level with comparable selection efficiency to those from baselines on both open and closed source GLMs. Code and datasets are provided at https://github.com/ml-postech/selective-generation .

1 Introduction

Generative language models (GLMs) [1, 2, 3, 4] have garnered significant attention for their ability to generate human-level language [5] primarily due to underlying transformer architectures [6]. However, GLMs raise concerns about generating hallucinated facts [7], which is an undesirable property when they are used as knowledge retrieval sources. This issue can be mitigated by finetuning with human feedback [7, 8], but it remains expensive in terms of training and labeling costs. Certified risk control methods such as selective prediction [9] and conformal prediction [10] are promising cost-efficient alternatives, which have been applied to the hallucination mitigation in various supervised downstream tasks [9, 10, 11, 12, 13, 14].

Q:WhatistheobjectiveofTourdeFrance?

Atrue:The objective of Tour deFranceisbicyclerace.

Atrue:ThebookofGenesismentionsSodomandGomorrah.

LLM

Dcal

LLM

A:Theobjectiveof TourdeFranceistodetermine thebest

↑

overallcyclistthroughaseriesofstagesandchallenges.

A:TheBible mentionsSodom and Gomorrahin the book of Genesis.

SelectionFunction

(Entailment)

(Exact Match)

(Entailment)

YES

↑

↑

NO

Is Atrue=A?

YES

Does A imply Atrue?

↑

↑

↑

A

IDK

A

(Il don't know.)

PDa {P(Ais wrong| Do not abstain)≤ 0} ≥ 1 - 8

Twodifferentselectionfunctiontraining()schemes,using existing

Inferenceusinganentailment-awareselectivegenerator,

(left)and proposed (right) methods with a frozen LLM(),respectively.

whichreferstoanLLM&aselectionfunctionpair.

The main bottleneck in applying such certified methods to language generation tasks is that provided risk control guarantees require correctness labels during the learning process. Specifically, in classification, high-quality correctness labels can be directly acquired by comparing true and predicted labels using exact match (EM). However, this is not the case for language generation tasks, since multiple valid answers can exist for the same question. As correctness metrics such as EM and F1-score do not account for the multiple valid answers, directly applying them to language generation tasks results in a significant gap between the true and measured correctness, which we call the metric misalignment . Thus, a correctness evaluation metric that accounts for multiple answers is required.

In this paper, we resolve the metric misalignment problem by leveraging textual entailment to evaluate the correctness of generated answers and define the false discovery rate with respect to the textual entailment relation (FDR-E). Given two ordered sequences, a premise and a hypothesis, we say that the premise entails the hypothesis if the hypothesis is true given the premise. Based on this notion of entailment, we propose two selective generation algorithms, SGen Sup and SGen Semi , which are generalized versions of selective classification [9] to control the FDR-E by abstaining from returning an answer when a GLM is uncertain of its answer.

In particular, SGen Sup , a direct modification of [9], is a supervised selective generator learning algorithm which requires entailment labels. This necessitates human annotations on textual entailment, where a generated answer is the premise and a true answer is the hypothesis. As labeling is expensive and SGen Sup solely relies on entailment-labeled data, we propose a semi-supervised method, SGen Semi , which enables the exploitation of entailment-unlabeled data in learning a selective generator by pseudo-labeling textual entailment using an entailment set function learned via conformal prediction [10]. Based on an entailment classifier originally developed for the natural language inference problem [15, 16], the estimated entailment set function approximates a true entailment set function, which returns all entailed answers if a true answer is given as a hypothesis.

Additionally, SGen Semi introduces the general class of selection functions for selective generation, called neuro-selection functions . In selective prediction, learning a selective predictor is equivalent to learning a selection function, which is an indicator function to decide whether to abstain from returning a prediction. The standard selective prediction algorithm [9] considers the class of singlethreshold indicator functions using a pre-specified confidence-rate function. For the same risk level, the better the confidence-rate function quantifies the model's uncertainty, the less likely the selective predictor is to abstain from making a prediction. We refer to this as selection efficiency henceforth. As appropriate confidence calibration for language generation remains challenging, optimizing a single-threshold indicator function with a poorly calibrated confidence-rate function leads to low selection efficiency. Instead, we generalize the selection function by using a multiple-threshold indicator function with trainable features. Furthermore, SGen Semi provides a user with an optimal class of selection functions among possible candidates in terms of the FDR-E.

Finally, we empirically demonstrate the efficacy of SGen Semi over open and closed source GLMs, where we consider SGen Sup as one of our baselines as it is a direct modification of [9]. To validate

our method and its theoretical guarantee, we create a new dataset on textual entailment using the Natural Questions (NQ) dataset [17] for each GLM. Given a question and answer pair, the textual entailment is labeled by letting a generated answer as a premise and the true answer in declarative form as a hypothesis. As communities lack human-annotated entailment-labeled data for language generation, we believe that our dataset contributes to the hallucination evaluation of GLMs. For both open and closed source GLMs, SGen Semi is effective in achieving a desired FDR-E level with better selection efficiency compared to baselines.

1.1 Related Work

We introduce two main research directions to mitigate hallucination in GLMs.

Heuristics for hallucination mitigation. The hallucination in language generation usually refers to the situation where a GLM generates wrong answers with high confidence, which hinders the reliable deployment of GLMs. As fine-tuning methods are expensive, heuristics for hallucination mitigation without tuning have been proposed [18, 19]. Notably, [19] proposes a performant hallucination detection method, which quantifies the self-consistency among multiple generated answers for the same question using textual entailment models to detect the hallucination. However, these methods do not provide certified control over the occurrence of hallucinated contents.

Certified methods for hallucination mitigation. Conformal prediction outputs a prediction set that is guaranteed to contain a true label with high probability, where a provided coverage guarantee is model-agnostic under a mild assumption on a data [10]. Although this property enables the safe deployment of complex models and has made conformal prediction popular [10, 12, 13, 20, 21, 22], the constructed prediction sets in language generation are often less-informative due to an unbounded label space, which frequently renders the coverage guarantee ineffective [23, 24]. To restrict the prediction set to a moderate size, [23] constructs the prediction set over answers by sampling them sequentially, while still satisfying the coverage guarantee. Still, post-selection of answers from the prediction set is necessary for final decision making, which may result in the selection bias [25, 26]. [27, 28] decompose generated answers into alignment-labeled sub-claims and return a set of sub-claims that contains no contradiction with high probability via conformal prediction. Even though the post-selection is unnecessary, it requires expensive alignment labels for every sub-claim.

Unlike conformal prediction, selective prediction directly manages target risk at a desired level by introducing an abstaining option on unsure predictions. [9] proposes a selective prediction method mainly for classification, which learns a threshold-based selection function that controls the false discovery rate (FDR) to a desired level. [24] generalizes the selective prediction to language generation. However, their theoretical guarantee is not focused on the target risk to control, but on a consistency property of a surrogate loss function with respect to a true loss function in optimization process. [29], concurrently published with our paper, proposes a certified selective generation method for context-given language generation which controls the FDR. Unlike [9] which takes the number of selected samples as constraint in learning the selection function, [29] set the power as constraint. However, as [24] does, they require an additional calibration set for training an entailment scoring function. Importantly, while existing selective generation methods are supervised learning methods, we propose a semi-supervised learning algorithm that can fully leverage entailment-unlabeled data.

2 Background

While we consider general language generation tasks, we confine our scope to the open-ended question-answering task and define the notation accordingly for the sake of clarity and for maintaining consistency in descriptions on the experiment. Specifically, let W denote a token space constructed using a tokenizer, such as Byte Pair Encoding [30], and let W ∗ denote a token sequence space, defined as W ∗ := ∪ ∞ i =0 W i . Let ( x , y ) ∈ X × Y be a question and answer sequence pair, where X := W ∗ and Y := W ∗ refer to the token sequence spaces of questions and answers, respectively. We assume the answer sequence is in a declarative form. Finally, x i : j refers to the sub-sequence of x from the i -th to the j -th token.

2.1 Language Generation

Given a question as input, a GLM generates an answer through the sequential process called decoding, which we call language generation. Here, we consider the greedy decoding, a deterministic generation process described as follows. Let p M : X × W → R ≥ 0 denote a GLM which returns a next-token

distribution given the input sequence x , where ∑ w ∈W p M ( w | x ) = 1 for all x ∈ X . A language generator G : X → Y using greedy decoding sequentially generates tokens from the GLM as follows: ˆ y i := arg max w ∈W p M ( w | ( x , ˆ y 1: i -1 )) for i ≥ 2 and ˆ y 1 := arg max w ∈W p M ( w | x ) . The generator G returns a generated answer ˆ y := G ( x ) and terminates the decoding process when the end-of-sequence ( EOS ) token is returned. Here, the conditional probability of the answer ˆ y is defined as f M ( x , ˆ y ) := p M (ˆ y 1 | x ) ∏ | ˆ y | i =2 p M (ˆ y i | ( x , ˆ y 1: i -1 )) , commonly used as its uncertainty measure.

2.2 Selective Prediction

Selective prediction refuses to make a prediction by returning 'I don't know' ( IDK ) if the prediction is uncertain. In classification, the selective classifier ˆ S consists of a pair of a classifier ˆ y and a selection function ˆ s , and is defined as follows: ˆ S ( x ) := { G ( x ) if ˆ s ( x ) = 1 IDK otherwise , where ˆ y ( x ) :=

̸

̸

arg max y ∈Y f ( x , y ) . Here, f ( x , y ) refers to an estimated likelihood of the given input x for being a class y , determined by an underlying classification model f . Although the selection function can be of arbitrary form, the common choice is a single threshold indicator function using the maximum likelihood as the confidence-rate function, i.e., ˆ s ( x ) := ✶ ( f ( x , ˆ y ) ≥ τ ) . Here, the confidence-rate function is defined to quantify the uncertainty of the model's prediction. Under the independent and identically distributed (i.i.d.) assumption, [9] proposed the certified threshold learning algorithm which controls the false discovery rate (FDR) with respect to the EM metric with the PAC guarantee, where the FDR is defined as R EM ( ˆ S ) := /a80 { ˆ y ( x ) = y | ˆ S ( x ) = IDK } . Since EM considers the answer ˆ y ( x ) to be correct when it is exactly the same as the reference answer y , it is an inappropriate correctness metric for language generation problems that can have multiple valid sequences for the same input. This results in learning a too conservative and vacuous selection function for language generation, which is empirically verified by our experiments. Thus, we leverage the textual entailment to evaluate the correctness of the generated sequence to alleviate the metric misalignment problem.

2.3 Textual Entailment

Natural language inference (NLI), also denoted as recognizing textual entailment, predicts whether one sequence implies another. The former refers to a premise ( p ), and the latter refers to a hypothesis ( h ). Since the release of two large-scale benchmarks of ordered sequence pairs labeled with textual entailment [15, 16], a number of transformer-based entailment classifiers have been proposed and shown impressive results. Each pair is classified into one of three categories: entailment if h is true given p ; contradiction if h is false given p ; and neutral otherwise. In this paper, we define the entailment scoring function as f E ( G ( x ) , y ) := 1 -p E ( contradict | p = G ( x ) , h = y ) to estimate and pseudo-label the correctness of G ( x ) , where p E ( contradict | p = G ( x ) , h = y ) is the likelihood that G ( x ) contradicts y . While pseudo-labeling enables the full exploitation of unlabeled data to learn a selection function, controlling the mislabeling error remains as a challenge.

2.4 Conformal Prediction

Conformal prediction [10] outputs a prediction set to quantify the uncertainty of a given model with a model-agnostic correctness guarantee under minimal assumptions on data generating process. Specifically, under the i.i.d. assumption, PAC conformal prediction [11] incorporates the interpretation of tolerance regions [31] and training-conditional inductive conformal prediction [20] through the lens of PAC learning theory [32]. In this paper, we adopt the PAC prediction set learning algorithm to control the rate of mislabeling error in pseudo-labeled samples used to learn a selection function for selective generation. See Section A.1 for detailed discussion on conformal prediction.

Scalar-parameterized Conformal Set. In this paper, we consider a conformal set C : X → 2 Y parameterized by a scalar [11, 33] as C ( x ) := { y ∈ Y | f ( x , y ) ≥ τ } , where τ ∈ H is a scalar parameter to learn, H is a hypothesis space ( e.g., H a finely discretized non-negative real numbers), and f : X × Y → R ≥ 0 is called a scoring function . The scoring function corresponds to a target model whose uncertainty is to be quantified, where the softmax output is a common choice in classification. Specifically, f ( x , y ) measures the likelihood of y as a response given x as input.

PAC Guarantee. The PAC prediction set learning algorithm outputs a conformal set ˆ C which upper bounds a miscoverage rate R MC ( ˆ C ) := /a80 { y / ∈ ˆ C ( x ) } to a desired level ε ∈ (0 , 1) , where the miscoverage rate can be generalized to risk R 01 ( ˆ C ) := ❊ { ℓ 01 ( ˆ C, x , y ) } , on any indicator losses that are monotonic with respect to τ . The algorithm is probably approximately correct (PAC) in

the sense that it provides a calibration data-conditional guarantee at every risk and confidence level. Specifically, it controls the risk to a desired level irrespective of which calibration data is used to learn ˆ C with a desired confidence δ ∈ (0 , 1) as follows: /a80 {R 01 ( ˆ C ) ≤ ε } ≥ 1 -δ, where the probability is taken over the calibration set Z ∼ D n to learn the conformal set. In this paper, we leverage the PAC conformal set for a pseudo-labeling function such that the guarantee on the labeling quality provides the overall PAC guarantee in semi-supervised selective generator learning algorithm.

Algorithm. The PAC conformal set learning algorithm A Binom : ( X × Y ) ∗ →H [11, 20, 34] returns the conformal set parameter ˆ τ , where H is a finely-discretized R ≥ 0 . Specifically, the algorithm returns ˆ τ = max τ ∈H τ subject to U Binom ( k τ ; n, δ ) ≤ ε , where k τ := ∑ n i =1 ℓ 01 ( ˆ C, x i , y i ) . Letting F ( k ; n, θ ) be a cumulative distribution function of a binomial distribution with n trials and success probability θ , U Binom ( k ; n, δ ) := inf { θ ∈ [0 , 1] | F ( k ; n, θ ) ≤ δ } ∪ { 1 } is an upper binomial tail bound that satisfies /a80 {R 01 ( ˆ C ) ≤ U Binom ( k τ ; n, δ ) } ≥ 1 -δ, where δ is the desired confidence. Note that we similarly denote a lower binomial tail bound by L Binom. If optimization in the algorithm A Binom is infeasible, the algorithm returns ˆ τ = 0 , a vacuous conformal set. Thus, the algorithm is PAC, and see Section A.1 for proof.

2.5 Calibration

In classification, calibration aims to adjust the classifier's maximum likelihood response, or confidence, to be correct. We say the classifier response f : X × Y → R ≥ 0 is perfectly calibrated with respect to a distribution D over X ×Y and a classifier ˆ y if /a80 { y = ˆ y ( x ) | f ( x , ˆ y ( x )) = t } = t for all t ∈ [0 , 1] [35, 36]. Calibration aims to find the classifier response such that it is perfectly calibrated asymptotically. In this paper, we make an interesting connection between calibration and selective generation. In particular, given the definition of the perfect calibration for a language scoring function f M , we formally provide a sufficient condition for a selective generator to control the FDR with respect to the textual entailment relation at any desired risk level.

3 Problem: Selective Generation

Let x ∈ X be a question and y ∈ Y be an answer, assuming that each question has a desired answer. Here, we assume ( x , y ) i.i.d. ∼D ′ , where D ′ is a data generating process of question-answering pairs. Then, given a generator G : X → Y , we consider a selective generator ˆ S : X → Y ∪ { IDK } which refuses to return G ( x ) if a selection function ˆ s ( x , G ( x )) ∈ { 0 , 1 } deems uncertain as follows:

Our main goal is to learn a selective generator ˆ S to control a generalized false discovery rate (FDR) with respect to a relation R as

̸

Here, the probability is taken over examples ( x , y , e, v ) , where e := ✶ (( G ( x ) , y ) ∈ R ) is an additional label to be annotated due to unknown R and v ∈ { 0 , 1 } is a visibility flag of e for semisupervised learning. For the data generation of ( x , y , e, v ) , we assume that a label e is observed with an unknown success probability of p v , independent of the generative process of ( x , y , e ) , i.e., ( x , y , e, v ) ∼ D := D ′ ·V , where D ′ is a distribution over X×Y×{ 0 , 1 } and V := Bernoulli ( p v ) . Note that the definition of e , D ′ varies by generator G even with the same data generating distribution of ( x , y ) . In this paper, we design a learning algorithm A that returns a selective generator ˆ S to control the generalized FDR with respect to R within a desired level ε ∈ (0 , 1) with probability at least 1 -δ ∈ (0 , 1) , i.e., /a80 {R R ( A ( Z )) ≤ ε } ≥ 1 -δ. Here, the probability is taken over a calibration set Z ∼ D n . This guarantee is called a probably approximately correct (PAC) guarantee [32]. Among selective generators that satisfies the PAC guarantee, we choose one that minimizes the ratio of IDK -answers with the highest selection efficiency . The main challenge is to find a sample and selection efficient PAC algorithm for any ε and δ along with designing a relation R for structured labels, as in question-answering. Frequently, we may not obtain a PAC algorithm for any ε , so in this paper, we use a relaxed notion of controllable instead of correct if the algorithm provides minimum achievable risk beoyond a given ε .

4 Semi-Supervised Learning for Controllable Selective-Generation

In this paper, we leverage the textual entailment as the evaluation metric in language generation to consider multiple valid answers in a principled way, and propose two selective generator learning algorithms which control FDR with respect to the textual entailment: SGen Sup and SGen Semi .

4.1 False Discovery Rate via Textual Entailment (FDR-E)

A textual entailment relation R E is an ordered subset of Y × Y where ( y ′ , y ) ∈ R E if y ′ entails y . In question-answering as an example, the generated answer G ( x ) is correct if the reference answer y is a logical consequence of G ( x ) . In other words, G ( x ) is valid if G ( x ) ∈ E true ( y ) , where the true entailment set function E true : Y → 2 Y is defined as follows: E true ( y ) := { y ′ ∈ Y | ( y ′ , y ) ∈ R E } . Then, an FDR with respect to the entailment relation R E (FDR-E) that we aim to control is as follows:

̸

R R E ( ˆ S ) := /a80 { G ( x ) / ∈ E true ( y ) | ˆ S ( x ) = IDK } , where the probability is taken over labeled examples, i.e., ( x , y , e ) ∼ D . Here, the label e is specifically called an entailment label, i.e., e := G ( x ) ∈ E true ( y ) . Then, for any G , D , V , and ˆ S , the FDR-E can be decomposed as follows:

̸

4.2 FDR-E Bound for Supervised Learning

where /a80 D ˆ S {·} := /a80 {· | ˆ S ( x ) = IDK ) . Note that as ( x , y , e ) and v are independent, (A), (C), and (E) in (2) are of the same quantity, which is the target risk that we aim to find an upper bound.

We first propose the supervised learning algorithm SGen Sup (Algorithm 8), a direct modification of [9] to language generation tasks. In particular, SGen Sup is a supervised method in the sense that it solely exploits labeled examples Z E := { ( x , y , e ) | ( x , y , e, v ) ∈ Z ∧ v = 1 } to learn a selective generator that controls the upper bound (C) in (2). Note that for supervised learning, we assume that (B) in (2) is always 1, so we only consider the the upper bound (C) via the binomial tail bound as [9].

4.3 FDR-E Bound for Semi-Supervised Learning

As SGen Sup requires human annotations for entailment labels and makes no use of abundant unlabeled examples Z U := { ( x , y ) | ( x , y , e, v ) ∈ Z ∧ v = 0 } , we further propose a novel semi-supervised learning algorithm SGen Semi (Algorithm 5), which fully exploits both Z E and Z U while controlling the FDR-E in (2). In particular, we (1) estimate a true entailment set E true via conformal prediction with labeled examples Z E and then (2) use the estimated entailment set ˆ E to annotate pseudo-labels on Z U . Finally, we (3) use both labeled and pseudo-labeled examples to learn a selective generator. Interestingly, this heuristic-looking algorithm could be a rigorous algorithm that controls the FDR-E of a selective generator, which will be described in the following sections.

4.3.1 FDR-E Decomposition

SGen Semi leverages unlabeled examples by estimating an entailment set as a pseudo-labeling function. However, the estimation error introduces wrong pseudo-labels. Here, we consider a rigorous way to derive the FDR-E upper bound by controlling the estimation error of the pseudolabeling function. In particular, two different types of estimation errors of an estimated entailment set ˆ E are illustrated in Figure 2, i.e., a false negative entailment rate (FNER) and a false entailment rate (FER). This results in the following decomposition.

Lemma 1. (E) in (2) is decomposed as follows:

Ω

Ω

E

true

TD

Ω

ˆ

E

TD

FNER

FER

NER

Here, the first two terms are related to the entailment label estimation error and the last term is the approximate FDR-E using pseudo-labels. As three terms are inter-related, we choose to control the FER term to control (E) in (2) via conformal prediction in the following section.

4.3.2 Pseudo-labeling via Conformalized Entailment Set Learning

SGen Semi leverages the PAC conformal prediction for the entailment label estimation to control the mislabeling error. Specifically, we estimate the true entailment set function E true via an estimated entailment set ˆ E using Z E , where we use the entailment scoring function f E as a scoring function, i.e., ˆ E ( y ) := { y ′ ∈ Y | f E ( y ′ , y ) ≥ τ E } . Here, the corresponding loss ℓ ( ˆ E, x , y , e ) := ✶ ( e = 0 ∧ G ( x ) ∈ ˆ E ( y )) is a monotonically non-increasing function with respect to τ E , so we can use the PAC conformal set learning algorithm. Given a desired risk ε E and confidence δ E level, the corresponding algorithm A FER ( i.e., Algorithm 1) returns the estimated entailment set function ˆ E which controls the false entailment rate (FER) of pseudo-labeled examples R FER ( ˆ E ) := /a80 D ˆ S { e = 0 ∧ G ( x ) ∈ ˆ E ( y ) } with the following PAC guarantee, where the probability is taken over training examples from D ˆ S .

4.3.3 FDR-E Bound

We then bound the FDR-E for semi-supervised learning, i.e., (E) in (2), via the PAC guarantee by the conformal set learning on Z E and the binomial tail bound on Z E and Z U . In particular, the FER is upper-bounded by ε E , the FNER is lower-bounded by the binomial tail bound using Z E , and NER is upper-bounded by the binomial tail bound using Z U . These bounds hold with high probability, and are therefore combined via a union bound, as in the following lemma. See Appendix G for a proof.

̸

̸

Lemma 2. Let ˆ Z E := { ( x , y , e ) ∈ Z E | ˆ S ( x ) = IDK } and ˆ Z U := { ( x , y ) ∈ Z U | ˆ S ( x ) = IDK } . For any G , D , V , and ˆ S , if ˆ E := A FER ( ˆ Z E ) satisfies ˆ Z E {R FER ( ˆ E ) ≤ ε E } ≥ 1 -δ ′ E / 2 , we have with probability at least 1 -δ ′ E -δ ′ S , where the probability is taken over Z . Here, ˆ k := ∑ ( x , y ,e ) ∈ ˆ Z E ( e = 1 ∧ G ( x ) / ∈ ˆ E ( y )) and ˆ l := ∑ ( x , y ) ∈ ˆ Z U ( G ( x ) / ∈ ˆ E ( y )) .

✶ ✶ Notably, each of three bounds holds over a conditional distribution D ˆ S , but Lemma 2 relaxes this to an unconditional distribution D for our final FDR-E guarantee.

Optimizing the FDR-E Bound (5). Lemma 2 introduces a hyper-parameter ε E , which controls a trade-off between the FER and other terms. To find a best trade-off, we optimize ε E to minimize the upper bound (5) among Q candidates of ε E via A U SSL-Opt, described in Algorithm 3. This optimization algorithm can find a tighter FDR-E bound, as in the following lemma. See Appendix H for a proof.

Lemma 3. Let U SSL be as in (5) and Q be the Q candidates of ε E . Then, we have with probability at least 1 -δ ′ E /Q -δ ′ S /Q , where the probability is taken over Z .

Note that for semi-supervised learning, the upper bound of (B), (C), (D), and (E) in (2) should be provided. The upper bound of (E) is provided in (5), which we denote by U SSL. The upper bound of (B), (C), and (D) are denoted by w SL , U SL, and w SSL, respectively, each of which is computed by the binomial tail bound. See Algorithm 4 and the proof of Theorem 1 for details.

4.4 Neuro-selection Functions

The FDR-E bounds for both supervised and semi-supervised learning are crucial for controlling the final FDR-E of a selective generator given a selection function ˆ s . But, the choice of the selection function is critical for a good selection efficiency and here we discuss a better selection function than the standard one, i.e., ˆ s ( x ) := ✶ ( f M ( x , G ( x )) ≥ τ S ) for τ S ∈ R ≥ 0 . In particular, certified selective classification [9] considers the single-threshold indicator function using the maximum likelihood as the confidence rate function. For the language generation, the conditional probability of the answer ˆ y , i.e., f M 1 ( x , ˆ y ) , would be a natural and commonly-used candidate. However, as it is known to be poorly calibrated [37], an alternative would be a self-consistency

score, i.e., f M 2 ( x , G ( x )) := 1 K ∑ K k =1 f E (˜ y k , G ( x )) , where ˜ y k are generated answers with the same question x but different random seeds. It is empirically shown that the self-consistency score properly quantifies uncertainty when a language model is uncertain of an answer [19]. The importance of score calibration with respect to the true entailment relation is demonstrated in Lemma 4, which provides the sufficient condition for the selective generation algorithm using the single-threshold indicator function (Algorithm 5) to control the FDR-E at any level. See Appendix J for a proof.

Lemma 4. If we have access to E true and f M is perfectly calibrated with respect to E true, the FDR-E is monotonically non-increasing in τ S .

However, as [37] points out, calibrating the language scoring function remains an uneasy task, os it is still an active research area. Therefore, we propose a general class of selection functions, neuroselection functions , which is the multiple-threshold indicator function using possibly learnable feature map Φ : x ↦→ R v as follows: ˆ s ( x ; Φ , W , b ) := ∧ u i =1 ( W Φ( x )) i + b i ≥ 0 , where W ∈ R u × v and b ∈ R u × 1 are linear proejction and bias terms, respectively. In this paper, we only consider two specific sub-classes of neuro-selection functions, where the former reduces to learning the single-threshold selection function using a scoring function (Algorithm 5) and the latter reduces to learning the bi-threshold selection function using two scoring functions (Algorithm 6). Only the bias term b is the learnable parameter for both algorithms, where the others set as hyperparameters. Specifically, W = I 1 , Φ 1 ( x ) = [ f M ( x , G ( x ))] , and b = -τ S for Algorithm 5, while W = I 2 , Φ 2 ( x ) = [ f M 1 ( x , G ( x )) f M 2 ( x , G ( x ))] T , and b = -[ τ S, 1 , τ S, 2 ] T for Algorithm 6 if two promising scoring functions exist. Here, developing a selection function learning algorithm where W and Φ( · ) are also fully learning parameters is left as future work. In the following section, we introduce our algorithm that chooses the optimal combination of scoring functions via neuro-selection functions.

4.5 Semi-Supervised Selective Generator Learning Algorithm with Neuro-Selection

SGen Semi is a semi-supervised learning algorithm for certified selective generation, which fully exploits unlabeled data in learning a selection function via certified pseudo-labeling and uses a neuro-selection function for choosing an optimal combination of scoring functions. In particular, SGen Semi solves the following optimization problem over selective generators H such that ˆ S closely satisfies the equality in the constraint, as described in Algorithm 7:

Here, ˆ S ∈ H has a selection function ˆ s ( x ; Φ 2 ( x ) , diag ( w ) , b ) , where w ∈ { [1 , 0] T , [0 , 1] T , [1 , 1] T } and b ∈ R 2 ≤ 0 . Note that SGen Semi returns an additional term ˆ U , which is the FDR-E bound given the selective generator ˆ S ( i.e., Algorithm 4) and informs the infeasibility of the optimization. The proposed Algorithm 7 satisfies the following controllability guarantee. See Appendix I for a proof.

Theorem 1. A SGen Semi satisfies the following controllable guarantee on the FDR-E, i.e.,

̸

/a80 /a80 where the inner and outer probabilities are taken over ( x , y , e, v ) ∼ D and Z ∼ D n , respectively, and ( ˆ S, ˆ U ) := A SGen Semi ( Z ) . Here, δ := δ W + δ S + δ E is a desired confidence level, where δ W is for the upper bounds on w SL and w SSL, δ S is for (C) in (2) and the NER, and δ E is for the FER and FNER.

Here, A SGen Semi is controllable in the sense that it upper-bounds the FDR-E of a learned selective generator to a desired level ε S or at least to a minimum achievable level ˆ U with confidence δ .

5 Experiments

We demonstrate the efficacy of our methods in controlling the FDR-E on pre-trained GLMs under various setups. We use two GLMs, GPT-3.5-Turbo and Alpaca-7B, alongside the Natural Questions (NQ) dataset to annotate entailment labels for question-answer pairs. Details on model configurations, datasets, and additional experimental results can be found in Section A.3 and Appendix K.

Methods. We consider two heuristic semi-supervised algorithms, SGen H-Semi PL and SGen H-Semi PFL (Algorithm 9) and an unsupervised learning algorithm [9] SGenEM (Algorithm 10) as baselines to show the efficacy of our certified semi-supervised method SGen Semi (Algorithm 7). SGen H-Semi PL and SGen H-Semi PFL exploit the unlabeled data by pseudo-labeling textual entailment based on a threshold as a hyperparameter without any guarantee on mislabeling error. SGen H-Semi PFL additionally filters out

a pseudo-labeled sample if its entailment score is below a specific threshold. SGenEM is a certified unsupervised method that takes the EM metric for measuring the correctness. We also report results on SGen Semi NoMS (Algorithm 5) for two different scoring functions , f M 1 and f M 2 , used in SGen Semi . SGen Semi NoMS is a certified semi-supervised learning algorithm using a single-threshold indicator function given a scoring function. We also take SGen Sup (Algorithm 8) as a baseline, since it is a direct modification of [9] to the language generation problem.

Scoring Functions. We use the conditional probability of an answer as f M 1 and the self-consistency score [19] as f M 2 , since our goal is to generate the sequence which is not only logically consistent to the true answer but also linguistically correct.

Control Parameters. To control an FDR-E, we use two user-specified parameters ( ε, δ ) , where we use (0 . 25 , 0 . 02) unless specified. For our methods ( i.e., SGen Semi , SGen Semi NoMS , and SGen Semi-Sup NoMS ), we have five control parameters ( ε S , δ S , δ E , δ W ), where we maps as follows: ε S = ε , δ S = ( δ -δ W ) / 2 , δ E = ( δ -δ W ) / 2 , δ W = 10 -5 . For other methods without using entailment sets, Algorithm 8, Algorithm 9, and Algorithm 10, we use ε and δ accordingly. Additionally, we use Q = 5 for Algorithm 3.

FDR-E Guarantee and Efficiency. As can be seen in Table 1, our method SGen Semi can overall achieve desired FDR-E guarantees with better efficiency compared to baselines. Depending on the quality of scoring functions ( e.g., f M 1 ), our variation SGen Semi NoMS may not find a selective generator that satisfies a desired FDR-E (denoted in the underlined FDR-E). The heuristic methods, i.e., SGen H-Semi PL and SGen H-Semi PFL , do not provide theoretical guarantees on FDR-E. In Figure 1 and Table 2, we can correctly predict even with the complicated answers, e.g., which have many equivalent words, because we do not rely on the EM metric. We conducted 100 random experiments for each method to show how well FDR-E is bounded under a desired FDR-E. As shown by the green boxes In Figure 4, which are successfully bounded under ε S = 0 . 25 , we can see that the FDR-E for a learned selective generator is well controlled below ε S under the test environment. Among the certified methods with theoretical guarantees, results appear to align well with the expected theoretical basis.

Why Entailment Labels. As expected and can be seen in Table 3 by comparing SGenEM and SGen Sup , a metric like EM cannot measure correctness correctly. Unlike classification, generative tasks can have infinite number of true answers so it is not likely to have exact match. Instead, entailment labels provide semantic correctness, so SGen Sup can perform better and more efficient than SGenEM .

SGen Semi

NoMS

10k

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on GPT-3.5-turbo

Why Semi-Supervised Learning. We observe that our semi-supervised learning for selective generation is effective. In particular, the fully supervised methods in Table 3 achieves the efficiency of 0 . 7535 and 0 . 2959 for GPT-3.5 and Alpaca-7B, respectively, with the entire labeled samples Z E (when they satisfy a ε -FDR-E guarantee). Compared to these, the proposed semi-supervised method SGen Semi Table 1 achieves the efficiency of 0 . 7334 and 0 . 3173 for GPT-3.5 and Alpaca-7B, respectively, by only using 75% of labeled examples. Additionally, we observe that more unlabeled samples are beneficial to achieving better efficiency as can be seen in Figure 3. This implies that if we can approximate the entailment set well and the size of Z U is enough, we can enjoy our certified pseudo-entailment labeling by the semi-supervised learning even with small Z E .

Why Neuro-Selection. It is hard to manually find a well calibrated scoring function. But, given multiple scoring functions, a neuro-selection function learns to choose right scoring functions that achieves a desired FDR-E and maximizes selection efficiency. This is empiricially validated in Table 1, as SGen Semi is better on average efficiency.

6 Conclusion

Wepropose selective generation, a generalized version of [9] for GLMs to handle semantic correctness between two structured answers. To this end, we leverage logical entailment to define a new entailment-based FDR (FDR-E) metric. As obtaining entailment labels are expensive, we propose novel semi-supervised learning for selective generation by using entailment sets as a pseudo-labeling function. To enhance the low selective efficiency due to inefficient scoring functions, we propose neuro-selection functions for effectively optimizing scoring functions for better selective efficiency and the FDR-E guarantee. The efficacy of our proposed algorithms SGen Semi and SGen Sup are theoretically and empirically justified.

Limitations. Our algorithm needs the i.i.d. assumption for a correctness guarantee, which can be violated in practical situations. We leverage expensive entailment labels, where the labels are obtained by considering logical entailment between a true answer and a generated answer. This limitation is partially mitigated by proposing the semi-supervised method to propagate entailment-labeled samples to samples without entailment labels. Also, our results show the empirical FDR-E is not much closely bounded under ε , especially on Alpaca7B, which implies that we may need a tighter FDR-E bound.

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A Discussion

A.1 Conformal Prediction

Conformal prediction [10] provides a promising way to quantify uncertainty of a model with a correctness guarantee under minimal assumptions. Here, we consider PAC prediction sets [11], an interpretation of tolerance region [31] and training-conditional inductive conformal prediction [20] in the lens of PAC learning theory [32] ( i.e., learning a 'good' function within a function family from data). This interpretation inspires us to generalize selective generation for GLMs via neural selection functions.

Conformal Set Model. We consider a conformal (prediction) set model ˆ C : X → 2 Y that measures the uncertainty of a target model; in conformal prediction, this model is specifically called a scoring function f : X ×Y → R ≥ 0 that measures the conformity (or likelihood) of x for being y with respect to f ; thus, f ( x , y ) is called a conformity score . In particular, we consider scalar parameterization of a conformal set [11, 33] as follows: C ( x ) := { y ∈ Y | f ( x , y ) ≥ τ } , where τ ∈ R ≥ 0 is a scalar parameter.

Conformal Sets and Uncertainty. The output of the conformal set model is a set of labels, which naturally represents the uncertainty of a scoring function on an example via the size of a conformal set. In particular, if the scoring function f is unsure on its prediction on x (due to uncertainty on a label distribution of x , i.e., aleatoric uncertainty, and due to uncertainty in the modeling of f , i.e., epistemic uncertainty), the conformal set is larger than it is when the scoring function is sure on its prediction.

To be precise, we consider a true conformal set C ∗ ( x ) := { y ∈ Y | f ( x , y ) ≥ f ( x , y ∗ ) } , where y ∗ is the true label of x . In particular, the true conformal set is a minimal set that contains a true label and labels with larger scores than the true label score; thus, the size of the true conformal set intuitively measures the uncertainty of a scoring function on the given example, i.e., the scoring function's possibilities on making wrong predictions, instead of the true prediction.

The true conformal set clearly captures the uncertainty, but the true label is unknown in inference time. Thus, the true conformal set is approximated via scalar parameterization [11, 33] as follows:

where τ ∈ R ≥ 0 is a scalar parameter.

Correctness. As we desire to construct a conformal set close to the true conformal set, we define the correctness of the conformal set based on its similarity to the true one. In particular, we wish to have the smallest C ( x ) such that C ∗ ( x ) ⊆ C ( x ) , or equivalently C ( x ) needs to have the smallest τ while y ∈ C ( x ) . This correctness definition is realized into two ways: a coverage guarantee [10] or a PAC guarantee [20].

Assumption. We assume that samples are independent and identically distributed (i.i.d.), i.e., the i.i.d. assumption. In particular, all samples for testing and learning prediction sets are independently drawn from the same but known distribution D .

PAC guarantee. Under the i.i.d. assumption, we learn a conformal set ˆ C that includes the most true labels ( approximately correct ). In particular, this means that the miscoverage of ˆ C is less than a desired level ε ∈ (0 , 1) , i.e., R MC ( ˆ C ) := /a80 { y / ∈ ˆ C ( x ) } ≤ ε, where the probability is taken over i.i.d. samples ( x , y ) ∼ D . This risk on micoverage can be generalized to be the risk on indicator loss, R 01 ( ˆ C ) := ❊ D ℓ 01 ( ˆ C, x , y ) . Here, the conformal set ˆ C is learned from a randomly drawn calibration set, so we desire to construct ˆ C that has a desired error for the most of random calibration sets ( probably approximately correct ), i.e., /a80 {R 01 ( ˆ C ) ≤ ε } ≥ 1 -δ, where δ ∈ (0 , 1) is a desired confidence level and the probability is taken over n i.i.d. calibration samples Z ∼ D n , used to learn ˆ C .

Algorithm. The PAC conformal prediction set method [11, 34] considers the following algorithm A Binom : ( X × Y ) ∗ → H to learn a conformal set model ˆ C , parameterized by ˆ τ , where H is a finely-discretized R ≥ 0 :

where k τ := ∑ n i =1 ℓ 01 ( ˆ C, x i , y i ) . Here, U Binom is a binomial tail bound, i.e., /a80 {R 01 ( C ) ≤ U Binom ( k τ ; n, δ ) } ≥ 1 -δ for any C , where U Binom ( k ; n, δ ) := inf { θ ∈ [0 , 1] | F ( k ; n, θ ) ≤ δ }∪{ 1 } and F ( k ; n, θ ) is a cumulative distribution function (CDF) of a binomial distribution with n trials and success probability θ . This algorithm is PAC.

Here, we slightly generalize the known PAC guarantee to hold for any risk with indicator loss. See Appendix F for a proof. Note that the PAC guarantee generally holds only if an enough number of samples is provided (when we know a function family including a true function). However, we consider PAC algorithms that hold for any number of samples due to the structural property of prediction sets, i.e., a prediction set is always correct if τ = 0 (thus ˆ C ( x ) = Y ), regardless of the sample size. In other words, if the calibration samples are not sufficient, the prediction set is constructed to return Y to satisfy the PAC guarantee.

Theorem 2. ([11, 20, 34]) The algorithm A Binom is PAC, i.e., for any f , ε ∈ (0 , 1) , δ ∈ (0 , 1) , and n ∈ Z ≥ 0 , we have /a80 {R 01 ( ˆ C ) ≤ ε } ≥ 1 -δ , where the probability is taken over i.i.d. labeled examples Z ∼ D n , and ˆ C = A Binom ( Z ) .

A.2 Sample Space Decomposition

Given the generator G and the entailment set function ˆ E , the sample space Ω := X ×Y ×E ×V can be partitioned as follows:

Here, the short-hands are defined as follows:

True discovery rate (TDR): P (Ω E true TD )

True entailment rate (TER): P (Ω ˆ E TE )

False discovery rate (FDR): P (Ω E true FD )

False non-entailment rate (FNER): P (Ω ˆ E FNE )

False entailment rate (FER): P (Ω ˆ E FER )

True non-entailment rate (TNER): P (Ω ˆ E TNE )

A.3 Experiment Setup

A.3.1 Computing Environment

Our system environment consists of 4 NVIDIA A100 80GB with 128 CPUs.

A.3.2 Models and Datasets

We use two large language models (LLMs), GPT-3.5-Turbo and Alpaca-7B , for language generation. We use deberta-v2-xxlarge-mnli as our entailment model.

For each GLM to annotate entailment labels for each question, answer, and generated answer pair, we annotate entailment labels. Specifically, we consider the open-ended QA task, where the model is prompted to generate the answer in a declarative form given a question. To validate our method and its theoretical guarantee on controlling FDR-E, we create a dataset on textual entailment using the Natural Questions (NQ) dataset [17] for each GLM. Based on the transformation method by [38] that converts the question and answer pair in QA dataset into a declarative form, we manually labeled textual entailment by letting the generated sequence as the premise and the reference answer in declarative form as the hypothesis. Similar work can be found in [39], but they label the textual entailment based on the extractive answer from the model. Approximately 7.3k (7,374) and 4.6k (4,595) samples are labeled for Alpaca-7B and GPT-3.5-Turbo , respectively, and both are split into calibration and test data at an 8:2 ratio. For semi-supervised learning algorithms that exploit unlabeled data (Algorithm 7, Algorithm 9), at most 27k and 10k unlabeled samples are used to train a selective generator, varying its size. Besides, semi-supervised learning algorithms use only 75% of the labeled calibration data compared to what is used by supervised methods (Algorithm 8, Algorithm 10).

B Semi-supervised Selective Generation Algorithms (Certified)

Algorithm 1 Entailment Set Learning with a False Entailment Rate (FER) Guarantee

Algorithm 2 U SSL Computation (for Single ε E )

1: procedure COMPUTEU SSL( f E , Z E , Z U , δ S , ε E , δ E ) 2: τ E ← ES ( f E , Z E , ε E , δ E / 2) 3: ℓ ← ∑ ( x , y ,e ) ∈ Z E ✶ ( e = 1 , f E ( G ( x ) , y ) < τ E ) 4: k ← ∑ ( x , y ) ∈ Z U ✶ ( f E ( G ( x ) , y ) < τ E ) 5: U SSL ← ε E -L Binom ( ℓ ; | Z E | , δ E / 2) + U Binom ( k, | Z U | , δ S / 2) 6: return U SSL

Algorithm 3 Optimal U SSL Search

1: procedure COMPUTEU OPT ( f E , Z E , Z U , δ S , Q , δ E )

SSL 2: Z E ← SORT f E ( Z E ) ( ▷ ) In an increasing order of f E ( y i , G ( x i )) 3: ( i, i ) ← (1 , | Z E | ) 4: ε max ← ∑ ( x , y ,e ) ∈ Z E ✶ ( e = 0) / | Z E | 5: H E ←{ ε 1 = ε max , . . . , ε Q = 1 / | Q | ε max } 6: U OPT SSL ←∞ 7: for i in { 1 , . . . , Q } do 8: U ( i ) SSL ← ComputeU SSL ( f E , Z E , Z U , δ S /Q,ε i , δ E /Q ) 9: if U ( i ) SSL ≤ U OPT SSL then 10: U OPT SSL ← U ( i ) SSL 11: return U OPT SSL

̸

C Supervised Selective Generation Algorithms (Certified)

D Semi-supervised Selective Generation Algorithms (Heuristic)

E Unsupervised Selective Generation Algorithms (Certified)

̸

F Proof of Theorem 2

Let C τ be a prediction set C with a parameter τ , H ε := { τ ∈ H | R 01 ( C τ ) > ε } , and τ ∗ := inf H ε , where H is finely-discretized non-negative real values. Then, we have

/a80 where the last equality in (11) holds as ✶ ( y / ∈ C τ ( x )) and U B are non-decreasing in τ ( i.e., Lemma 2 in [34]) and the last inequality in (12) is due to the property of the binomial tail bound U Binom.

G Proof of Lemma 2

Since (E) in (2) is decomposed into three terms in Lemma 1, we first find upper bounds on each of the terms and take the union bound as follows. This will return a single upper bound on (E) in (2), which we denote U SSL.

FER Bound. First, recall that

R FER ( ˆ E ) := /a80 D ˆ S { e = 0 ∧ G ( x ) ∈ ˆ E ( y ) } . Learning ˆ E via A FER is equivalent to the PAC prediction set learning algorithm that considers the optimization problem in (10), where the indicator loss is ℓ 01 ( ˆ E, x , y , e ) := ✶ ( e = 0 ∧ G ( x ) ∈ ˆ E ( y )) and the target model is the entailment scoring function f E . Therefore, by Theorem 2, for any n E := | Z E | , we have

Note that (13) holds as the PAC guarantee for conformal prediction holds for any number of samples.

The same bound holds with respect to Z . Specifically, letting ℓ FER ( Z E , Z U ) := ✶ ( R FER ( ˆ E ) ≤ ε E ) , we have

where the second equality holds due to the i.i.d. assumption on the calibration data and the inequality holds due to (14).

FNER Bound. Recall

Since our goal is to upper-bound -R FNER ( ˆ E ) , we consider a lower bound R FNER ( ˆ E ) as follows for any n E := | Z E | :

where the inequality holds due to the binomial tail bound. The same bound holds when the probability is taken over Z . First, let

Then,

where the second equality holds due to the i.i.d. assumption and the inequality holds due to (16). NER Bound. Recall

Then, we upper bound R NER ( ˆ E ) as follows for any n U := | Z U | :

where the inequality holds due to the binomial tail bound. Again, the same bound holds when the probability is taken over Z . First, let

Then,

where the inequality holds due to (18).

Finally, taking the union bound of (15), (17), and (19) completes the proof.

H Proof of Lemma 3

Let U ( i ) SSL be U SSL for the i -th candidate of ε E in Algorithm 3. Due to Lemma 2, the following holds:

/a80 /a80 Since U OPT SSL = min i ∈ [ Q ] U ( i ) SSL , we have

where the second inequality is due to a union bound. This completes the proof.

I Proof of Theorem 1

Let H be the calibration set-dependent hypothesis space of selective generators, where n H := |H| is always calibration set independent. Letting U ( i ) be the FDR-E bound computed given the i -th selective generator S i in H , we first describe how to derive an upper bound of the FDR-E for a given hypothesis S i .

Since an upper bound of (E) in (2) is proved in Lemma 3, the remaining parts are (i) to derive upper bounds on the others and (ii) to take the union bound. For proportions of the visibility of textual entailment labels, i.e., (B) and (D) in (2), and the FDR-E for the supervised case only using entailment-labeled examples, i.e., (C) in (2), the followings hold due to the binomial tail bound:

where ˆ Z E and ˆ Z U are defined same as Lemma 2 does, and ˆ Z e =0 E := { ( x , y , e ) ∈ ˆ Z E | e = 0 } . Note that the binomial tail bound is applied to filtered sets by the given selective generator ( e.g., ˆ Z E ), but we can use the same bound for the non-filtered set Z , by using the same marginalization technique over the size of a filtered set, as in, e.g., (15).

Thus, by taking the union bound along with Lemma 3 when δ ′ E = δ E and δ ′ S = δ S / 2 ,

/a80 Z { R E ( S i ) ≤ U ( i ) } ≥ 1 -( δ E + δ S + δ W ) / |H| , (20) where U i := w ( i ) SL U ( i ) SL + w ( i ) SSL U OPT ( i ) SSL is the computed FDR-E bound a given selective generator S i . Here, U OPT ( i ) SSL refers to the smallest FDR-E bound of (E) in (2) given the i -th selective generator.

Since (20) holds for all S i ∈ H , and the final bound ˆ U is chosen among them, this completes the proof by taking an union bound, i.e.,

J Proof of Lemma 4

We say f M is perfectly calibrated with respect to D , G , E true if

The true discovery rate with respect to E true conditioned on f M ( x , G ( x )) ≥ τ S , i.e., 1 -FDR-E, is as follows:

where and (22) holds as f M is perfectly calibrated, i.e., (21).

Letting h ( t ) := P { f M ( x , G ( x )) = t } , H ( t ) := ∫ 1 t h ( t ′ ) dt ′ , i ( t ) := t P { f M ( x , G ( x )) = t } , and I ( t ) := ∫ 1 t i ( t ′ ) dt ′ , since we have τ S ≤ ∫ 1 τ S t P { f M ( x ,G ( x ))= t } dt ∫ 1 τ S P { f M ( x ,G ( x ))= t } dt ≤ 1 , the following holds:

Therefore,

This completes the proof.

Note that the classification problem can be reduced from the special case, i.e., E true ( y ) := E EM ( y ) , where Y := W and E EM ( y ) := { y } = arg max w ∈W P ( Y = w | X = x ) .

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K Additional Experiments

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Neuro-Selective Entailing-Generation (NSEGen), a novel approach to enhance the trustworthiness of generative language models. The method extends selective classification to language generation tasks, utilizing textual entailment to measure semantic correctness between generated and true answers. NSEGen employs a semi-supervised method that leverages both labeled and unlabeled data by learning an entailment set, addressing the challenge of expensive entailment labeling. The authors introduce neuro-selection functions to optimize feature selection for minimizing false discovery rate and provide theoretical guarantees on false discovery rate control. The approach aims to control the entailment-based false discovery rate while maximizing selection efficiency.\n\nSTRENGTHS:\n- The paper introduces a new framework for improving GLM trustworthiness that addresses limitations of existing methods. This is particularly important because it tackles the metric misalignment issue in GLMs, where conventional metrics like exact match fail to capture semantic correctness. By introducing selective generation and leveraging textual entailment, the authors provide a more nuanced and accurate way to evaluate language model outputs.\n- The approach is supported by thorough theoretical analysis and guarantees. The authors provide a detailed proof for the correctness guarantee of their algorithm (Theorem 1) and establish conditions for achieving monotonicity in precision (Lemma 3).\n\nWEAKNESSES:\n- The method introduces several new components and parameters, which may make it challenging to implement and tune. For instance, the algorithm involves learning an entailment set, designing neuro-selection functions, and tuning multiple parameters. This complexity could make it difficult for practitioners to adopt the method.\n- Experiments are conducted only on Natural Question with two models (GPT-3.5 and Alpaca-7B). While these experiments do demonstrate the method's effectiveness, they leave open questions about its generalizability. Especially if the method claims to adapt uncertainty learning methods to generation tasks. It would be valuable to see how the method performs on a wider range of language generation tasks (e.g., summarization, translation, or open-ended text generation and so on).\n\nQUESTIONS:\nN/A\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates the trustworthiness of generative language models in critical decision-making systems, identifying deficiencies in current uncertainty learning methods, such as selective classification and conformal prediction, which fail to address metric misalignment in GLMs. The authors propose an entailment-based false discovery rate  by leveraging logical entailment to define the relationship between true and generated answers. They introduce a supervised selective generation approach using entailment labels to control the FDR and a semi-supervised method to mitigate the high cost of obtaining these labels. This method employs an entailment set for pseudo-labeling and neuro-selection functions to enhance data space learning, reducing FDR. The neuro-selective entailing-generation algorithm  is theoretically validated, meeting the PAC guarantee on the desired FDR. NSeGen demonstrates improved efficiency and reliability in achieving a desired FDR level compared to baseline models, highlighting its practical applications.\n\nSTRENGTHS:\nThe work works to provide good grounding of both the problem and proposed path to the solution. With the proposed path, the authors work well to highlight the deficiencies both with current approaches, as well as their chosen approach and then incorporate fixes that make the deficiencies less of a problem. The work to do psuedo labelling and also deal with entailment is important as well as showing the theoretical guarantees.\n\nWEAKNESSES:\n1. There is a bit of repetitiveness in the Experiment section when describing the GLMs and Datasets. \n2. It should be made clear that your approach deals with the generation results from the underlying model. For example, when reading table 1 earlier I thought you were comparing Alpaca to your method, but later got it as you have different ways to deal with the selective generation. \n3. It would be great to get a practical understanding of how much data would be needed to train a good enough selective algorithm. This is important especially in low resource scenarios (both language and application areas) as the space of application of where such an algorithm such as yours might be used is highly likely in places were there is not much data. So ablation studies on sizes of the training set would have been great.\n\nQUESTIONS:\nSee my weaknesses. esp. \n\n3. It would be great to get a practical understanding of how much data would be needed to train a good enough selective algorithm. This is important especially in low resource scenarios (both language and application areas) as the space of application of where such an algorithm such as yours might be used is highly likely in places were there is not much data. So ablation\n\nLIMITATIONS:\nThe work covers their limitations well and this was very appreciated. A practical guide for researchers might have been great in terms of knowing how much data is needed to help the psuedo labelling.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work presents a method for selective generation from language models for question answering. Their approach functions as a secondary decision function on top of an existing language model, determining whether to accept the language model's generation or to abstain. Their method is approach is based on constructing an entailment set of correct answers to a given question. The entailment set is defined as all answers that imply or textually entail the ground truth answer. They then design their secondary decision function such that it satisfies bounds on the correctness of accepted, generated answers based on whether it is within the constructed entailment set, or the answer not being in the ground truth entailment set. The authors then demonstrate that their method satisfies correctness guarantees based on conformal prediction, and demonstrate the efficacy of using their entailment set approach to identifying correctness empircally through experiments on open-domain question answering (NaturalQuestions) and with two language models (Alapaca7b and GPT3.5)\n\nSTRENGTHS:\nThe goal of this work, selective generation for language models to prevent hallucinations and erroneous outputs is sounds and well motivated.\n\nThis work provides theoretical guarantees on the efficacy of the selective generation approach, builds upon insights from prior works on the efficacy of using textual entailment to aid in calibrating QA predictions.\n\nWEAKNESSES:\nRelying on single-directional textual entailment as a method for determining correctness of an answer is susceptible to accepting generations that produce that untrue fact or hallucinations in addition to the correct answer. [1] is a very relevant work (that was not cited) that similarly uses entailment to determine equivalence between two answers; however, in their work they rely on bi-directional entailment to ensure answer equivalence.\n\nThe baselines used in this work lack clear descriptions, and are not in line with current simple and effective methods for selective generation and calibration of language models [1, 2, 3]. [2] and [3] both do not require additional entailment training data and [1] takes a related approach using entailment to determine answer equivalence among a set of generated answers. [1] also does not require domain-specific entailment data.\n\n[1] Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation\nLorenz Kuhn, Yarin Gal, Sebastian Farquhar\nICLR 2023\n\n[2] Self-consistency improves chain of thought reasoning in language models.\nX. Wang, J. Wei, D. Schuurmans, Q. Le, E. Chi, S. Narang, A. Chowdhery, and D. Zhou.\nICLR 2023 \n\n[3] Language Models (Mostly) Know What They Know\nSaurav Kadavath, Tom Conerly, Amanda Askell, Tom Henighan, Dawn Drain, Ethan Perez, Nicholas Schiefer, Zac Hatfield-Dodds, Nova DasSarma, Eli Tran-Johnson, Scott Johnston, Sheer El-Showk, Andy Jones, Nelson Elhage, Tristan Hume, Anna Chen, Yuntao Bai, Sam Bowman, Stanislav Fort, Deep Ganguli, Danny Hernandez, Josh Jacobson, Jackson Kernion, Shauna Kravec, Liane Lovitt, Kamal Ndousse, Catherine Olsson, Sam Ringer, Dario Amodei, Tom Brown, Jack Clark, Nicholas Joseph, Ben Mann, Sam McCandlish, Chris Olah, Jared Kaplan\nArXiv 2022\n\nQUESTIONS:\nThe presentation of the experimental details could benefit from additional explanation. Clearly defining what the baselines are and how their implemented (not just the algorithms in the appendix) would greatly improve readability. This applies to the experimental results and metrics as well.\n\nEvaluations could be designed in a way that (1) more clearly describe each methods efficacy and performance in the end task (QA in this case) and (2) allow for direct comparison against other methods also designed for calibration and selective generation.\n\nLIMITATIONS:\nThe authors address that they do not report statistical significance.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper looks at a selective generative language system, meaning one which can produce a I-Don't-Know label rather than an answer, and calibrating it such that some guarantees on the precision of the system can be made. \nThe paper expands upon citation [1] on selective generation, to improve the efficiency by not requiring an exact match metric on the answer in order to be able to say whether it is correct or not. This is important for language generation tasks where there are endless possibly valid answers. It does this via the use of entailment measures, ie saying whether a candidate question+answer pair have positive/negative or neutral entailment. \nThe paper reports theoretical analysis of when the proposed method can be learnt within given error margins, doing so under reasonable assumptions of the data being IID. This may not truly be the case for say actual question-answer datasets, but is a required assumption for the theoretical statements that are presented. \nThe paper also presents empirical results on a question-answer dataset (Natural Questions), comparing against other selective generation methods with exact matching (rather than entailment), and with entailment sets (from labels or inferred). \nThe proposed method which learns the selective threshold is shown to have the best false-discovery-rate (ie the inverse of precision on the generated answer).\n\nSTRENGTHS:\nThe requirement of having labels for answer correctness is clearly a large limiting factor in language generation tasks given the large number of possible correct answers. Relaxing this via entailment is therefore well motivated. \nThorough theoretical analysis given, albeit under\n\nWEAKNESSES:\n* Say more about what language problems this could and could not be applied to. The key appears to be in being able to have an accurate entailment function. For how many types of language-generation problems is such obtainable? \n\n* Some data on rates of IDK generation on the test set would be useful to see. \n\n* Minor suggestion: an algorithm box showing the steps for the whole proposed approach would be helpful for the reader. \n\n* The paper would benefit from an editor / proofreading. It doesn't impact understanding, but there are several small typos (lines 34, 69, 367 as some examples)\n\nQUESTIONS:\n* In figure 2, are the SG-EM and SG-ES systems scoring 0? I'm not familiar with the prior literature, but with this paper being strongly influenced by [1] where SG-EM is from, it seems surprising these 2 baselines are so poor.\n\nLIMITATIONS:\n* The theoretical analysis relies on the IID assumption, which won't hold on real world language datasets. This is noted by the authors.\n* Not clear how many language problems this is applicable to.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-15"}