# Quantifying Prediction Consistency
Under Model Multiplicity in Tabular LLMs

Faisal Hamman1 Pasan Dissanayake1 Saumitra Mishra2 Freddy Lecue2 Sanghamitra
Dutta1
1University of Maryland, College Park 2JPMorgan AI Research
{fhamman, pasand<EMAIL_ADDRESS>
{saumitra.mishra<EMAIL_ADDRESS>
###### Abstract

Fine-tuning large language models (LLMs) on limited tabular data for
classification tasks can lead to _fine-tuning multiplicity_ , where equally
well-performing models make conflicting predictions on the same inputs due to
variations in the training process (i.e., seed, random weight initialization,
retraining on additional or deleted samples). This raises critical concerns
about the robustness and reliability of Tabular LLMs, particularly when
deployed for high-stakes decision-making, such as finance, hiring, education,
healthcare, etc. This work formalizes the challenge of fine-tuning
multiplicity in Tabular LLMs and proposes a novel metric to quantify the
robustness of individual predictions without expensive model retraining. Our
metric quantifies a prediction’s stability by analyzing (sampling) the model’s
local behavior around the input in the embedding space. Interestingly, we show
that sampling in the local neighborhood can be leveraged to provide
probabilistic robustness guarantees against a broad class of fine-tuned
models. By leveraging Bernstein’s Inequality, we show that predictions with
sufficiently high robustness (as defined by our measure) will remain
consistent with high probability. We also provide empirical evaluation on
real-world datasets to support our theoretical results. Our work highlights
the importance of addressing fine-tuning instabilities to enable trustworthy
deployment of LLMs in high-stakes and safety-critical applications.

##  1 Introduction

Large language models are generating significant interest in high-stakes
applications, e.g., finance, healthcare, etc., particularly in few-shot
classification scenarios on tabular datasets. Recent findings demonstrate that
these models perform commendably in such scenarios, especially when very
little training data is available, due to their transfer learning abilities
(Hegselmann et al., 2023; Dinh et al., 2022; Yin et al., 2020; Yan et al.,
2024; Wang et al., 2023). However, models in these settings are often fine-
tuned from large pretrained models with millions or billions of parameters on
small proprietary datasets (Hu et al., 2021; Liu et al., 2022). This paucity
of training data, combined with large parameter spaces, risks instability
across different fine-tuned variants of the pre-trained model.

We introduce the the notion of _fine-tuning multiplicity_ as to the phenomenon
where multiple competing models, fine-tuned from the same pre-trained LLM but
under slightly varying conditions (such as different random seeds or minor
changes in the training data), exhibit comparable performance (e.g., in terms
of accuracy) yet generate conflicting predictions for the same inputs. This
concept is closely related to predictive multiplicity, often referred to as
the Rashomon effect in the context of neural networks (Marx et al., 2020;
Breiman, 2003; Hsu and Calmon, 2022). This is particularly alarming for high-
stakes applications, such as finance (Yin et al., 2023), healthcare (Wang et
al., 2024; Chen et al., 2023; Kim et al., 2024), where arbitrary predictions
can have significant consequences.

The fine-tuned models for high-stakes decisions might have to be updated due
to various reasons, e.g., to retrain on additional data points to improve
performance (Wu et al., 2024), or even removing datapoints for privacy. For
instance, regulatory frameworks like the GDPR (Voigt, 2017) introduces the
right to be forgotten which necessitates the removal of an individual’s data
upon request, potentially leading to model updates. These updates could, in
turn, impact the validity of previously issued predictions. The phenomenon of
fine-tuning multiplicity not only raises concerns about model reliability but
also paves the way for fairwashing and explanation bias (Black et al., 2022;
Sokol et al., 2023) making quantifying robustness against fine-tuning
multiplicity an important problem.

Since methods to measure multiplicity in classical ML are impractical for LLMs
due to the computational challenge of retraining several fine-tuned models, we
propose a novel metric to quantify the robustness of individual predictions
without requiring expensive retraining of multiple models. Our metric, termed
_consistency_ , leverage the model’s local behavior around each input sample
within the embedding space to estimate the prediction’s susceptibility to
multiplicity. Interestingly, by analyzing this local neighborhood, we can
derive probabilistic guarantees on the robustness of predictions with high
consistency scores under a broad range of potential fine-tuned model updates.

Our contribution is summarized as follows:

  * •

Model multiplicity in fine-tuned tabular LLMs. We show that model multiplicity
is a critical issue in the context of fine-tuning large language models (LLMs)
for tabular data tasks. Multiple fine-tuned models may perform equally well,
yet assign conflicting predictions to the same input due to random training
initialization, we term this fine-tuning multiplicity. To evaluate the extent
of multiplicity, we finetune several models and evaluate the _Arbitrariness_ ,
_Discrepancy_ , _Pairwise Disagreement_ , _Prediction Variance_ , and
_Prediction Range_ (see Definitions 1,2,3,4, and 5 in Section 2). This
multiplicity issue raises concerns about the robustness and reliability of
fine-tuned LLMs, which are increasingly being deployed in high-stakes
decision-making scenarios.

  * •

A measure to quantify prediction robustness amidst fine-tuning multiplicity.
We introduce a novel metric, termed consistency (see Definition 6), to
quantify the robustness of model predictions in the face of fine-tuning
multiplicity, without retraining several models. Given an input
$x\in\mathcal{X}$ and model $f(\cdot)\in[0,1]$, our robustness measure is
given as: $S_{k,\sigma}(x,f)=\frac{1}{k}\sum_{x_{i}\in
N_{x,k}}(f(x_{i})-|f(x)-f(x_{i})|)$, where $N_{x,k}$ is a set of $k$ points
sampled independently from a distribution over a hypersphere of radius
$\sigma$ centered at $x$. Consistency leverages the neighbors of a given
sample (in the embedding space) to inform the reliability of its prediction.
The first term essentially captures the mean value of the model output in a
region around it. The second term captures the local average variability of
the model output around it (lower variability is expected to be more
reliable).

  * •

Probabilistic guarantees on consistency over a broad range of fine-tuned
models. Our main contribution is to provide a theoretical guarantee (see
Theorem 1) that predictions with sufficiently high consistency (as defined by
our measure) will remain consistent with high probability over a broad range
of fine-tuned models. To achieve this guarantee, we characterize the behavior
and statistical properties of a broad class of fine-tuned models (see
Assumption 1; Stochastic Fine-Tuned Model Class). Our results leverage
Bernstein’s Inequality (see Lemma 2) to establish rigorous concentration
bounds on the Neighborhood Divergence (see Lemma 1), which are then used to
prove our theoretical guarantee.

  * •

Empirical validation. In our empirical analysis, we evaluate the extent of
fine-tuning multiplicity and validate our proposed consistency measure
$S_{k,\sigma}(x,f)$ on the Diabetes, German Credit, and Adult datasets (Kahn,
; Hofmann, 1994; Becker and Kohavi, 1996). We employ the BigScience T0
encoder-decoder model (Sanh et al., 2021), fine-tuned via the T-Few recipe
(Liu et al., 2022), and LORA (Hu et al., 2021). We measure Arbitrariness,
Discrepancy, Pairwise Disagreement, and Prediction Variance (see Section 2) to
evaluate the extent of fine-tuning multiplicity. We study how effectively our
consistency measure $S_{k,\sigma}(x,f),$ (as measured only on model
$f(\cdot)$) captures multiplicity of predictions over a broad range of fine-
tuned models.

###  1.1 Related Works

LLM in tabular predictions. The application of LLMs to tabular data is a
growing area of research, demonstrating significant performance due to the
transfer learning capabilities (Yin et al., 2020; Li et al., 2020; Narayan et
al., 2022; Borisov et al., 2022; Bertsimas et al., 2022; Onishi et al., 2023;
Zhang et al., 2023; Wang et al., 2023; Sui et al., 2024; Yan et al., 2024;
Yang et al., 2024). Yin et al. (2020); Jaitly et al. (2023) incorporated
column data types into the serialized strings, adding a layer of descriptive
context to the data. Dinh et al. (2022) proposes LIFT, a method for adapting
LLMs to non-language classification and regression tasks without changing the
model architecture or loss function. Hegselmann et al. (2023) investigates the
use of LLMs for zero-shot and few-shot classification of tabular data and
finds that this method outperforms previous deep-learning-based approaches and
is competitive with traditional baselines like gradient-boosted trees. Wang et
al. (2024) presents MediTab, a method that uses LLMs to combine different
medical datasets, significantly improving predictions for patient and trial
outcomes. Tabular LLMs have also been applied in other high-stakes domains
(Chen et al., 2023; Kim et al., 2024; Li et al., 2023; Yin et al., 2023). Yin
et al. (2023) presents FinPT an LLM based approach to financial risk
prediction. We refer to Fang et al. (2024) for a more detailed survey on LLMs
on Tabular Data.

Figure 1: (a) illustrates the process of fine-tuning LLMs for Tabular data
using few labeled examples (Hegselmann et al., 2023; Dinh et al., 2022). (b)
demonstrates the concept of finetuning multiplicity. Models fine-tuned from
the same pre-trained LLM under slightly varying conditions, such as different
random seeds, can exhibit comparable performance metrics but may yield
conflicting predictions for the same input. (c) introduces our proposed
consistency measure designed to quantify the robustness of individual
predictions without requiring the retraining of multiple models. By sampling
points in a bounded neighborhood around a given input in the embedding space,
the consistency measure $S_{k,\sigma}(x,f)$ informs a prediction’s
susceptibility to multiplicity.

Model multiplicity in machine learning. Breiman (2003) introduced the idea
that machine learning models can differ significantly while achieving similar
average performance, known as the Rashomon effect. Marx et al. (2020)
highlighted the prevalence of arbitrary decisions in simple classification
problems, coining this phenomenon predictive multiplicity. (Creel and Hellman,
2022) discuss the harms of predictive multiplicity and arbitrary decisions.
Efforts to leverage model multiplicity beneficially while addressing its
implications have been explored by (Black et al., 2022; Fisher et al., 2019;
Xin et al., 2022; Coston et al., 2021). The effect of model multiplicity in
fairness (Sokol et al., 2022) and explainability are examined by Hamman et al.
(2023); Black et al. (2021); Dutta et al. (2022); Pawelczyk et al. (2020).
Watson-Daniels et al. (2023); Hsu and Calmon (2022) offered a framework for
measuring predictive multiplicity in classical ML models, however this
involves retraining several models. Model multiplicity has not been
extensively studied in tabular LLMs. The closest work is by (Gomez et al.,
2024), which investigates prediction arbitrariness for online content
moderation. We leverage the rich embedding space of LLMs to quantify proneness
to multiplicity without the need for expensive retraining, as fine-tuning LLMs
is computationally expensive.

###  1.2 Preliminaries

We consider a classification task for a tabular dataset
$D=\\{(x_{i},y_{i})\\}_{i=1}^{n}$, where each $x_{i}$ is a $d$-dimensional
feature vector (rows of a tabular input), and each label $y_{i}$ is binary,
$y_{i}\in\\{0,1\\}$. We study an $n$-shot classification problem by fine-
tuning a pre-trained model on $n$ examples from a training set. This fine-
tuning process aims to adapt the pre-trained model to effectively predict new,
unseen data points by learning from a limited number of training examples.

Serialization of Tabular Data for LLMs: To effectively apply LLMs to tabular
data, it is crucial to transform the data into a natural text format. This
process, known as serialization, involves converting the table rows into a
text string that includes both the column names and their corresponding values
(Yin et al., 2020; Jaitly et al., 2023; Hegselmann et al., 2023; Dinh et al.,
2022). The resultant serialized string is combined with a task-specific prompt
to form the input for the LLM. There have been various proposed methods for
serialization, and this is still a topic of active research Hegselmann et al.
(2023); Jaitly et al. (2023). Among the serializations we have examined are:
list template (A list of column names and feature values), and text template (
“ _The <column name> is <value>._”). LLMs can be adapted for classification
tasks by training them on serialized tabular data. This training involves
using the natural-language outputs of the LLM, mapped to valid classes in the
target space, as part of a fine-tuning process (see Figure 1).

To clarify, table values are serialized into $\text{serialize}(x)$ and then
transformed into a format understandable by the LLM,
$\text{tokenize}(\text{serialize}(x))$, which is some embedding. Since these
transformations are one-to-one mappings, we denote the embedded form of $x$ as
$x\in\mathcal{X}$ to represent $x$ in the embedding space. This allows us to
simplify the notation and directly use $x$ to refer to the table values in the
embedding space.

Goals. Our primary goal is to define a measure of predictive consistency to
fine-tuning multiplicity that does not require retraining multiple models
since this is computational expensive for LLMs. We aim to theoretically
motivate this measure and provide theoretical guarantees on its effectiveness
in ensuring consistent predictions across a broad range of fine-tuned models
(see Figure 1). Additionally, we intend to evaluate this measure against
multiplicity metrics by fine-tuning several models to empirically validate its
robustness and reliability.

##  2 Model Multiplicity in Fine-Tuned Tabular LLMs

Let $f(\cdot):\mathcal{X}\to[0,1]$ denote an LLM that performs binary
classification, outputting the probability distribution over the classes. We
let $\mathcal{F}$ denote a broad class of fine-tuned models that are equally
well-performing (i.e., a set of competing models as measured by the accuracy),
i.e, $\mathcal{F_{\delta}}=\\{f:err(f)\leq err(f_{0})+\delta\\}$ where
$err(f_{0})=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}[\hat{f}_{0}(x_{i})\neq y_{i}]$
for a reference model $f_{0}$ (with satisfactory accuracy) a dataset with $n$
examples. Here, $\hat{f}(x)=\mathbb{I}[f(x)\geq 0.5]$ denotes the predicted
labels. This is a set of models that perform just as well as the baseline
classifier where $\delta\in(0,1)$ is the error tolerance (Marx et al., 2020).
The appropriate choice of $\delta$ is application-dependent.

Fine-tuning multiplicity. We introduce the notion of fine-tuning multiplicity
as to the phenomenon observed where multiple competing models, fine-tuned from
the same pre-trained LLM but under slightly varying conditions (such as
different random seeds or minor changes in the training data), exhibit
comparable performance yet generate conflicting predictions for the same
inputs (e.g., model in $\mathcal{F}_{\delta}$). This concept is closely
related to predictive multiplicity, often referred to as the Rashomon effect
in the context of neural networks (Marx et al., 2020; Breiman, 2003; Hsu and
Calmon, 2022). However, this issue becomes more pronounced with LLMs because
these models are typically fine-tuned on small datasets from large
foundational models (to leverage their transfer learning ability), which
possess millions or billions of parameters. This paucity of training data,
combined with large parameter spaces, risks instability across different fine-
tuned variants of the model (see Figure 1).

###  2.1 Evaluating Fine-tuning Multiplicity

To effectively evaluate the extent of fine-tuning multiplicity, we introduce
specific empirical metrics that assess how predictions may vary across
different versions of fine-tuned models. These can only be evaluated when we
access to several fine-tuned models in the competing set.

######  Definition 1 (Arbitrariness (Gomez et al., 2024)).

Arbitrariness over set $\mathcal{F}_{\delta}$ measures the extent of
conflicting predictions across the model space for a given set of inputs
$\\{x_{1},\ldots,x_{n}\\}$. It is defined as:

$A_{\delta}=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}[\exists
f,f^{\prime}\in\mathcal{F}_{\delta},:\hat{f}(x_{i})\neq\hat{f}^{\prime}(x_{i})]$
(1)

Arbitrariness generalizes the _Ambiguity_ measure from Marx et al. (2020).
While _Ambiguity_ computes the fraction of points where at least one model in
$\mathcal{F}_{\delta}$ disagrees with a reference model, arbitrariness
measures the percentage of points that receive conflicting predictions from
any two models within the set $\mathcal{F}_{\delta}$. Arbitrariness can be
defined on an input, i.e., $A(x_{i})=\mathbb{I}[\exists
f,f^{\prime}\in\mathcal{F}_{\delta},:\hat{f}(x_{i})\neq\hat{f}^{\prime}(x_{i})]$

######  Definition 2 (Discrepancy).

Discrepancy quantifies the maximum proportion of conflicting predictions
between the reference model and any competing model in the set. It is defined
as:

$D_{\delta}(f_{0}):=\max_{f\in\mathcal{F}_{\delta}}\left(\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}[\hat{f}(x_{i})\neq\hat{f}_{0}(x_{i})]\right)$
(2)

Discrepancy measures the maximum number of predictions that could change if a
reference model is replaced with a competing model. This means that, in
practice, altering multiple predictions requires that all conflicting
predictions come from a single competing model.

######  Definition 3 (Pairwise Disagreement (Black et al., 2022)).

Pairwise Disagreement assesses the variability among models by measuring the
proportion of instances where pairs of models within the competing set
disagree:

$\text{PD}_{\delta}(x):=\frac{1}{|\mathcal{F}_{\delta}|(|\mathcal{F}_{\delta}|-1)}\sum_{f_{i},f_{j}\in\mathcal{F}_{\delta},f_{i}\neq
f_{j}}\mathbb{I}[\hat{f}_{i}(x)\neq\hat{f}_{j}(x)]$ (3)

Most existing measures of multiplicity focus on predicted labels. We propose
more nuanced measures that leverages the predicted probabilities of model
outputs:

######  Definition 4 (Prediction Variance).

Prediction Variance $PD$ measures the variability of the model outputs for a
given input $x$ across different models within the set $\mathcal{F}_{\delta}$.
It is defined as:

$\text{PV}_{\delta}(x):=\frac{1}{|\mathcal{F}_{\delta}|}\sum_{f\in\mathcal{F}_{\delta}}\bigg{(}f(x)-\frac{1}{|\mathcal{F}_{\delta}|}\sum_{f\in\mathcal{F}_{\delta}}f(x)\bigg{)}^{2}$
(4)

Prediction Variance captures the variability of predictions for a given input
across the competing set of models (higher $PV$ indicates more multiplicity).
Unlike threshold-dependent measures, $PV$ does not rely on accept/reject
thresholds. This means low $PV$ centered around the $0.5$ threshold may
indicate high multiplicity when using arbitrariness, discrepancy, or pairwise
disagreement while these other measures may overlook high $PV$ centered on one
side of the decision boundary. We also introduce a similar measure that
captures the maximum disparity in predictions:

######  Definition 5 (Prediction Range).

Prediction Range (PR) measures the range of model outputs for a given input
$x$ across different models in the set $\mathcal{F}_{\delta}$. It is defined
as:

$\text{PR}_{\delta}(x):=\max_{f\in\mathcal{F}_{\delta}}f(x)-\min_{f\in\mathcal{F}_{\delta}}f(x)$
(5)

######  Remark 1.

Given the infeasibility of computing the exact size of
$|\mathcal{F}_{\delta}|$ due to its potentially vast model space, we employ an
expensive sampling approach, i.e., re-fine-tuning on various seeds. We select
a finite number of models from $\mathcal{F}_{\delta}$ for practical
evaluation, allowing us to evaluate the multiplicity metrics. It is very
computationally expensive to fine-tune several models to evaluate
multiplicity. This motivates the need for a measure to quantify consistency
given one fine-tuned model.

##  3 Quantifying Prediction Consistency amidst Fine-tuning Multiplicity

Our objective is to develop a measure denoted as $S_{k,\sigma}(x,f)$, for an
input $x$ and a given fine-tuned model $f$, that quantifies its robustness of
predictions to a broad class of fine-tuned models. Ideally, we desire that the
measure $S_{k,\sigma}(x,f)$ should be high if the input $x$ is consistent
across this broad class of fine-tuned models (see Figure 1).

###  3.1 Proposed Consistency Measure

Prediction probabilities as a measure of consistency. Using the prediction
probabilities of model $f(\cdot)$ as a measure of prediction certainty can
offer insights into the model’s confidence in its predictions in choosing a
certain class. While a high prediction probability $f(\cdot)$ might suggest a
strong confidence in a particular class, we show that relying solely on $f(x)$
for assessing the robustness of predictions against model multiplicity is
insufficient (see Table 4, Figure 2, i.e., samples with high $f(x)$ or
confidence can still be susceptible to multiplicity). Therefore, we propose
leveraging the local neighborhood around the input $x$ in the embedding space.
This motivates us to derive a theoretical measure of consistency.

######  Definition 6 (Consistency).

The consistency of a given prediction $f(x)\in[0,1]$ is defined as follows:

$S_{k,\sigma}(x,f)=\frac{1}{k}\sum_{x_{i}\in
N_{x,k}}(f(x_{i})-|f(x)-f(x_{i})|),$ (6)

where $N_{x,k}$ is a set of $k$ points sampled independently from a
distribution over a hypersphere of radius $\sigma$ centered at $x$, i.e.,
$N_{x,k}=\\{x_{1},x_{2},\ldots,x_{k}\\}\subset
B(x,\sigma)=\\{x^{\prime}\in\mathcal{X}:\|x^{\prime}-x\|_{2}<\sigma\\}$.

######  Remark 2.

Our consistency measure is fundamentally tied to the confidence in predicting
a specific class. The concept can be seamlessly applied by considering the
logits (or softmax outputs) for predicting any given class. This approach can
also be extended to multi-class classification by providing logits for each
class, thereby maintaining the measure’s applicability across various
classification tasks.

###  3.2 Theoretical Guarantees on Consistency

Here, we present theoretical insights that motivate and provide guarantees for
our proposed robustness measure $S_{k,\sigma}(x,f)$, ensuring consistent
predictions across a broad class of fine-tuned models. We represent the class
of fine-tuned models by a stochastic (random) function $F$, such that
$F\in\mathcal{F}$. We denote two random models, $F$ and $F^{\prime}$, both of
which are independently and identically distributed within $\mathcal{F}$. For
clarity, we use capital letters (e.g., $F,F^{\prime},X_{i},Z$) to denote
random variables, while lowercase letters (e.g., $x_{i},f,\epsilon$) indicate
specific realizations.

In our framework, we define a set of assumptions that delineates the behavior
of a broad class of finetuned models and the statistical properties of their
predictions.

######  Assumption 1 (Stochastic Fine-Tuned Model Class).

We define the _stochastic divergence_ between predictions of two random
models, $F$ and $F^{\prime}$ as:

$Z_{i}:=F^{\prime}(X_{i})-F(X_{i})-|F(X_{i})-F(x)|+|F^{\prime}(X_{i})-F^{\prime}(x)|$

with $Z=\frac{1}{k}\sum_{i=1}^{k}Z_{i}$ where $X_{i}$’s are random points
sampled independently from a distribution over a hypersphere $B(x,\sigma)$. We
assume:

  * •

$F(X)$ and $F^{\prime}(X)$ are independent and identically distributed given
an input $X=x$.

  * •

$\mathrm{Var}\left[{Z_{i}|F^{\prime}=f^{\prime},F=f}\right]\leq\beta$ for all
$f,f^{\prime}\in\mathcal{F}$.

Intuition: The variable $Z$ captures the neighborhood stochastic divergence
between predictions of two independently fine-tuned models $F$ and
$F^{\prime}$. This captures both the difference in predictions and variability
around a given point $x$. The first assumptions ensure that $F$ and
$F^{\prime}$ provide an unbiased estimate of the prediction for $x$. The
assumption on the variance of $Z_{i}$ indicates that the variance of the
stochastic neighborhood divergence within a $\sigma$-Ball of a sample between
any two models’ predictions is controlled. The parameter $\beta$ essentially
captures the similarity of the models within the local neighborhood of a
sample. This concept is also somewhat analogous to the Lipschitz constant of a
general function, which bounds how much the function’s output can change
relative to changes in its input. However, in this context, the $\beta$-bound
reflects an average behavior of the models’ predictions within the local
neighborhood. It does not strictly enforce a uniform Lipschitz constant,
especially considering that transformer models are not typically Lipschitz
continuous (Kim et al., 2021).

######  Theorem 1 (Probabilistic Guarantee on Consistency).

Given a data point $x$, a random model $F^{\prime}$ and consistency measure
$S_{k,\sigma}(x,F^{\prime})$. Then under Assumption 1, and
$|\mathbb{E}\left[{Z_{i}|F^{\prime}=f^{\prime},F=f}\right]|\leq\epsilon^{\prime}$,
a prediction over a broad class of fine-tuned models satisfies:

$\Pr\left(F(x)\geq S_{k,\sigma}(x,F^{\prime})-\epsilon\right)\geq
1-\exp\left(\frac{-k\epsilon^{2}}{8\beta+\frac{16}{3}\epsilon}\right),$ (7)

for all $\epsilon>2\epsilon^{\prime}$, The probability is over the stochastic
models $F$ and $F^{\prime}$, and the random perturbations $X_{i}$’s are random
points sampled independently from a distribution over a hypersphere
$B(x,\sigma)$.

#### Consistency guarantee interpretation.

Essentially, our consistency measure $S(x,F^{\prime})$ provides a
probabilistic guarantee that if a sample $x$ has a sufficiently high
consistency score with respect to a random model $F^{\prime}$, then the
prediction of another random model $F$ from the same broad class of fine-tuned
models will be at least $S(x,F^{\prime})-\epsilon$ with high probability. For
example, if $S(x,F^{\prime})=0.8$, we can be confident that $F(x)$ will be at
least $0.8-\epsilon$ with _high_ probability (i.e, prediction will remain on
the accepted side). This implies that high consistency scores are indicative
of robust predictions across different fine-tuned models. Conversely, a low
consistency score does not provide significant information about the
prediction’s behavior, as it does not guarantee a lower bound on the
prediction. For $F(x)\geq S(x,F^{\prime})-\epsilon$ to hold with high
probability, a large $k$ is needed, ideally $k\gg\beta$. This implies that
when $\beta$ is large then more samples are needed. The complete proof of
Theorem 1 is provided in Appendix B. Here, we include a proof sketch.

###### Proof Sketch:.

From Assumption 1, $F$ and $F^{\prime}$ are identically distributed given
$X_{i}$, hence
$\mathbb{E}[F^{\prime}(X_{i})|X_{i}]=\mathbb{E}[F(X_{i})|X_{i}]$ and
$\mathbb{E}[|F^{\prime}(X_{i})-F^{\prime}(x)||X_{i}]=\mathbb{E}[|F(X_{i})-F(x)||X_{i}]$.
The terms in $\mathbb{E}\left[{Z}\right]$ cancel each other out, resulting in
$\mathbb{E}[Z]=0$.

The next step of the proof leverages the Bernstein’s inequality (see Lemma 2)
to provide a bound on the stochastic neighborhood divergence (see Lemma 1).

######  Lemma 1 (Neighborhood Divergence Bound).

Given the neighborhood discrepancy
$Z=\frac{1}{k}\sum_{i=1}^{k}(F^{\prime}(X_{i})-F(X_{i}))$, under Assumption 1,
for any $\tilde{\epsilon}>\epsilon^{\prime}>0$, we have:

$\Pr(Z\geq\epsilon^{\prime}+\tilde{\epsilon})\leq\exp\left(\frac{-k(\tilde{\epsilon}+\epsilon^{\prime})^{2}}{8\beta+\frac{16}{3}(\tilde{\epsilon}+\epsilon^{\prime})}\right).$
(8)

######  Lemma 2 (Bernstein Inequality).

For a given random variable $X_{i}$ such that $\Pr(|X_{i}|\leq c)=1$ , and
$\beta=\frac{1}{k}\sum_{i=1}^{k}\mathrm{Var}\left[{X_{i}}\right]$ then, for
any $\varepsilon>0$,

$\Pr\left(\bigg{|}\frac{1}{k}\sum_{i=1}^{k}X_{i}-\mathbb{E}(X_{i})\bigg{|}>\varepsilon\right)\leq
2\exp\left(\frac{-k\varepsilon^{2}}{2\beta+\frac{2c\varepsilon}{3}}\right),$
(9)

See Sridharan (2002) for detailed proof of Bernstein’s Inequality. The final
steps of the proof leverages the reverse triangle inequality so show:
$F(x)\geq\frac{1}{k}\sum_{i=1}^{k}(F(X_{i})-|F(X_{i})-F(x)|)$. Combining that
along with Lemma 1 derives our consistency measure and guarantees.∎

#### Properties of consistency.

While our measure was theoretically motivated, it also exhibits intuitive
properties. Our consistency measure $S_{k,\sigma}(x,f)$ leverages the model’s
predictions within a local neighborhood around an input in the embedding
space. It considers two key factors: (i) the average prediction across $k$
samples drawn from a $\sigma$-Ball around $x$, and (ii) the absolute
variability of these predictions measured by $|f(x)-f(x_{i})|$. A high
$S_{k,\sigma}(x,f)$ score indicates the model makes consistent predictions
with low variability in $x$’s local neighborhood, hence predictions are more
robust to minor perturbations. Conversely, a low score implies the prediction
is sensitive to local variations and hence less reliable under model updates.
The idea is that sampling informs the behaviour of the model in a local
neighborhood. By leveraging the average predictions and their variability,
consistency is more comprehensive than solely using prediction probabilities,
which may not capture local consistencies.

##  4 Empirical Validation

In this section, we experiment across different datasets to (i) quantify the
prevalence of fine-tuning multiplicity in Tabular LLMs, and (ii) validate the
effectiveness of our proposed measure in quantifying the consistency of
predictions over a broad range of fine-tuned models.

Datasets and Serialization. Our experiments utilize the Diabetes (Kahn, ),
German Credit (Hofmann, 1994), and Adult datasets (Becker and Kohavi, 1996),
serialized using the “Text Template” method (Hegselmann et al., 2023; Dinh et
al., 2022) where each tabular entry is converted into a natural language
format by stating “ _The <column name> is <value>_.” This approach helps align
the inputs with the training distribution of LLMs, enhancing their performance
in both zero-shot and few-shot scenarios.

Models and Fine-tuning Methods. We utilize the BigScience T0 encoder-decoder
model as our pretrained LLM for tabular tasks (Sanh et al., 2021). This model
was trained across a diverse array of task-specific prompts incorporating
numerous datasets, making it well-suited for our experiments in the few-shot
settings. For fine-tuning, we adopt the T-Few recipe (Liu et al., 2022), known
for its performance in few-shot settings, and LORA (Hu et al., 2021), a
parameter-efficient method that constrains weight matrix updates to be low-
rank. Detailed setup can be found in Appendix C

Evaluating Extent of Fine-tuning Multiplicity. We measure the extent of fine-
tuning multiplicity across the various datasets and fine-tuning methods, we
use the following multiplicity evaluation metrics (introduced in Section 2):
(1) Arbitrariness- extent of conflicting predictions across fine-tuned models
(see Definition 1), (2) Discrepancy- the maximum proportion of conflicting
predictions between the baseline classifier and any competing model (see
Definition 2), (3) Average Pairwise Disagreement- the proportion of instances
where pairs of models disagree (see Definition 3), (4) Average Prediction
Variance- variability of model outputs over different fine-tuned models
averaged over a set of inputs (see Definition 4), and (5) Average Prediction
Range- range of model outputs over different fine-tuned models averaged over a
set of test inputs (see Definition 5). To evaluate these multiplicity metrics
across our datasets, we fine-tune 40 models on Tfew recipe and LORA using
different random seeds and test on a sample set. Our results for Tfew are
summarized in Table 1. See Table 3 in Appendix C for fine-tuning using LORA.

Table 1:  Evaluated Multiplicity for Different Datasets and Number of Shots. Evaluated on $40$ fine-tuned models on T-Few recipe using different random seeds. Multiplicity observed in predictions across different fine-tuned model, even when models exhibit similar accuracy (in this setting $\delta=0.02$). Fine-tuning using LORA achieves results in the same ballpark (see LORA Table 3 in Appendix C) Dataset | No. | Multiplicity Evaluation Metrics
---|---|---
Shots | Arbitrariness | Discrepancy | Avg. Pairwise | Avg. Pred. | Avg. Pred. | Avg. Model
|  |  |  | Disagreement | Variance | Range | Accuracy
Adult | 64 | 10% | 9% | 7% | 0.01 | 0.10 | 83%
128 | 10% | 7% | 8% | 0.01 | 0.10 | 84%
512 | 11% | 8% | 7% | 0.01 | 0.12 | 85%
German | 64 | 18% | 10% | 6% | 0.01 | 0.20 | 71%
128 | 17% | 11% | 6% | 0.01 | 0.16 | 71%
512 | 23% | 12% | 7% | 0.02 | 0.23 | 72%
Diabetes | 64 | 29% | 18% | 10% | 0.04 | 0.31 | 71%
128 | 13% | 17% | 11% | 0.03 | 0.13 | 72%
512 | 16% | 16% | 10% | 0.02 | 0.18 | 78%

Comparing Consistency Measure to Evaluated Multiplicity. We compare our
measure of consistency $S_{k,\sigma}(x,f)$ (measured on just one model) with
the prediction probability $f(x)$ in informing the multiplicity (evaluated on
several trained models). This comparison is made using the Spearman
correlation coefficient (see Definition 7), a nonparametric measure that
evaluates the rank correlation between two variables. Specifically, we
calculate the Spearman correlation between the consistency $S_{k,\sigma}(x,f)$
and the multiplicity, e.g., Prediction Variance ($PV(x)$) across all test
samples. This analysis provides insights into consistency’s utility in
indicating the presence of fine-tuning multiplicity. In Figure 2, we visually
illustrate the evaluated multiplicity versus our consistency measure for the
128-shot setting on the Adult dataset (see Figure 3 and Figure 4 in Appendix C
for Diabetes and German Credit datasets). In Table 4, we report the absolute
Spearman correlation between the consistency measure and the various
multiplicity evaluation metrics for $128$ shots on the Adult, German Credit,
and Diabetes datasets. See detailed Table with $64$ and $512$ shot cases in
Appendix C.

Hyperparameter Selection. Based on our theoretical guarantees, choosing a
larger $k$ is advantageous as it ensures the guarantee holds with high
probability $k$. However, this increases the computational cost of model
inference (forward pass). In our experiments, we used $k=30$, the maximum
number that fits in one inference pass through the GPU, yielding good results.
For $\sigma$, we sampled points from a truncated Gaussian distribution with a
variance of $0.01$. This choice performed well across all our experiments. To
guide the choice of $\sigma$, one could consider the spread of training
samples. The choice of $\delta$ in the competing set $\mathcal{F}_{\delta}$ is
application-dependent. For our experiments, we use $\delta=0.02$, which
corresponds to a $2\%$ margin of accuracy deviation. Evaluating multiplicity
by refining several models is computationally expensive, which motivated this
work. Ideally, sampling more models provides a better evaluation of
multiplicity. However, due to computational constraints, we used $40$ models.

Figure 2: Evaluated multiplicity (assessed on $40$ retrained models) versus our consistency measure (evaluated on one model) for the 128-shot setting on the Adult dataset. The plots demonstrate that high consistency values correspond to low multiplicity across various multiplicity evaluation metrics. Also, observe that high predicted probability values (i.e., high prediction confidence) does not imply low multiplicity. Our consistency measure provides better insight into the multiplicity of predictions compared to the predicted probabilities alone. See Appendix C for visualizations on other Datasets. Table 2:  This table reports the absolute Spearman correlation between the consistency measure and various multiplicity evaluation metrics for $128$ shots on the Adult, German Credit, and Diabetes datasets. In most cases, the consistency measure $S_{k,\sigma}(x,f)$ shows a higher correlation with these multiplicity measures compared to predicted probabilities, indicating that the consistency measure $S_{k,\sigma}(x,f)$ better informs about the multiplicity than predicted probabilities $f(x)$ do. See full Table with $64$ and $512$ shot cases in Appendix C. Dataset | Number of Shots | Measure | Arbitrariness | Pairwise Disagreement | Prediction Variance | Prediction Range
---|---|---|---|---|---|---
Adult | 128 | Consistency | 0.80 | 0.96 | 0.84 | 0.91
Pred. Prob. | 0.67 | 0.62 | 0.30 | 0.54
German | 128 | Consistency | 0.54 | 0.54 | 0.87 | 0.87
Pred. Prob. | 0.57 | 0.57 | 0.86 | 0.86
Diabetes | 128 | Consistency | 0.92 | 0.95 | 0.93 | 0.95
Pred. Prob. | 0.88 | 0.93 | 0.93 | 0.95

##  5 Discussions

Our multiplicity evaluation metrics, summarized in Table 1, reveal significant
variability in model predictions across different fine-tuned variants, even
when they exhibit similar accuracy. This multiplicity is not captured by
merely examining predicted probabilities, as predictions with high confidence
can still be susceptible to multiplicity (see Figure 2). Our consistency
measure, $S_{k,\sigma}(x,f)$, was compared with prediction probabilities
$f(x)$. The results, presented in Table 4, demonstrate that our consistency
measure consistently shows mainly higher correlation with multiplicity metrics
across the datasets compared to prediction probabilities. This indicates that
$S_{k,\sigma}(x,f)$ is more informative than the predictive probabilities at
informing the multiplicity.

Marx et al. (2020) argue for the necessity of measuring and reporting
multiplicity to better inform predictions. Traditional methods to measure
multiplicity in classical ML are impractical for LLMs due to the computational
challenge of retraining several fine-tuned models (Marx et al., 2020; Hsu and
Calmon, 2022; Watson-Daniels et al., 2023). Our proposed measure, which
requires only the given model and leverages the embedding space to inform
multiplicity, addresses this issue. This approach reduces the complexity from
retraining and inference to just inference, making it more feasible to apply
in practice. Although, from our theoretical guarantee, a large $k$ (number of
sampled points) might be needed for accurate consistency estimation
(particularly when $\beta$ is large), it remains computationally more
efficient than retraining multiple models. Our work provides practitioners
with meaningful information about the multiplicity of predictions, which may
lead them to carefully evaluate which predictions to trust and which to treat
with caution. Our research has significant implications in several high-stakes
applications, e.g., hiring, finance, education, etc., where inconsistent
predictions can lead to distrust. A limitation of our work is that while we
inform about fine-tuning multiplicity for a given sample, we do not resolve
it. Future work could focus on developing methods to mitigate fine-tuning
multiplicity, ensuring more consistent model predictions.

## Acknowledgements

This paper was prepared for informational purposes in part by the CDAO group
of JPMorgan Chase & Co and its affiliates (“J.P. Morgan”) and is not a product
of the Research Department of J.P. Morgan. J.P. Morgan makes no representation
and warranty whatsoever and disclaims all liability, for the completeness,
accuracy or reliability of the information contained herein. This document is
not intended as investment research or investment advice, or a recommendation,
offer or solicitation for the purchase or sale of any security, financial
instrument, financial product or service, or to be used in any way for
evaluating the merits of participating in any transaction, and shall not
constitute a solicitation under any jurisdiction or to any person, if such
solicitation under such jurisdiction or to such person would be unlawful.

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##  Appendix A Relevant Definition

######  Definition 7 (Spearman’s Correlation).

Spearman’s correlation, $\text{Spearman}(X,Y)$, measures the strength and
direction of the monotonic relationship between two ranked variables. It is
calculated as the Pearson correlation coefficient between the ranked
variables.

For $n$ pairs of observations $(X_{i},Y_{i})$, Spearman’s rank correlation is
given by:

$\text{Spearman}(X,Y)=1-\frac{6\sum_{i=1}^{n}d_{i}^{2}}{n(n^{2}-1)},$

where $d_{i}$ is the difference between the ranks of $X_{i}$ and $Y_{i}$.
Alternatively, in terms of covariance:

$\text{Spearman}(X,Y)=\frac{\text{cov}(\text{rank}(X),\text{rank}(Y))}{\sigma_{\text{rank}(X)}\sigma_{\text{rank}(Y)}},$

where $\sigma$ denotes standard deviation. $Spearman(X,Y)$ ranges from $-1$
(perfect negative monotonic relationship) to $1$ (perfect positive monotonic
relationship), with $0$ indicating no monotonic relationship.

##  Appendix B Proof of Theoretical Guarantee

See 1

###### Proof.

To prove Theorem 1, we begin with Lemma 1.

Assume the fine-tuned models $F$ belong to a discrete class of random
variables. A specific model realization is represented as $f_{i}$ for
$i=1,2,\ldots,|\mathcal{F}_{\delta}|$, with the complete set denoted by
$\mathcal{F}=\\{f_{1},f_{2},\ldots,f_{|\mathcal{F}|}\\}$. Each model $f_{i}$
is selected with probability $p_{i}$, where
$\sum_{i=1}^{|\mathcal{F}_{\delta}|}p_{i}=1$.

See 1

We show that $\mathbb{E}\left[{Z}\right]=0$:

$\displaystyle\mathbb{E}[Z]$
$\displaystyle\overset{(a)}{=}\mathbb{E}_{X_{i}}\bigg{[}\mathbb{E}_{F|X_{i}}\bigg{[}\frac{1}{k}\sum_{i=1}^{k}(F^{\prime}(X_{i})-F(X_{i})-|F(X_{i})-F(x)|+|F^{\prime}(X_{i})-F^{\prime}(x)|)\bigg{]}\bigg{]}$
(10)
$\displaystyle\overset{(b)}{=}\frac{1}{k}\sum_{i=1}^{k}\mathbb{E}_{X_{i}}[\mathbb{E}_{F|X_{i}}[(F^{\prime}(X_{i})-F(X_{i})-|F(X_{i})-F(x)|+|F^{\prime}(X_{i})-F^{\prime}(x)|)]]$
(11)
$\displaystyle\overset{(c)}{=}\frac{1}{k}\sum_{i=1}^{k}\mathbb{E}_{X_{i}}\big{[}\mathbb{E}[F^{\prime}(X_{i})|X_{i}]-\mathbb{E}[F(X_{i})|X_{i}]-\mathbb{E}[|F(X_{i})-F(x)||X_{i}]$
(12)
$\displaystyle+\mathbb{E}[|F^{\prime}(X_{i})-F^{\prime}(x)||X_{i}]\big{]}$
(13)
$\displaystyle\overset{(d)}{=}\frac{1}{k}\sum_{i=1}^{k}\mathbb{E}_{X_{i}}\big{[}\mathbb{E}[F(X_{i})|X_{i}]-\mathbb{E}[F(X_{i})|X_{i}]-\mathbb{E}[|F(X_{i})-F(x)||X_{i}]$
(14) $\displaystyle+\mathbb{E}[|F(X_{i})-F(x)||X_{i}]\big{]}=0$ (15)

Here (a) holds from applying the law of total expectation. (b) Distributing
the expectation over the summation. (c) Applying the linearity of expectations
inside the inner expectation. (d) From Assumption 1, $F$ and $F^{\prime}$ are
identically distributed given $X_{i}$, hence
$\mathbb{E}[F^{\prime}(X_{i})|X_{i}]=\mathbb{E}[F(X_{i})|X_{i}]$ and
$\mathbb{E}[|F^{\prime}(X_{i})-F^{\prime}(x)||X_{i}]=\mathbb{E}[|F(X_{i})-F(x)||X_{i}]$.
The terms cancel each other out, resulting in $\mathbb{E}[Z]=0$. The rest of
the proof leverages Bernstien’s Inequality:

See 2

Observe that
$|Z_{i}|=|F^{\prime}(X_{i})-F(X_{i})-|F(X_{i})-F(x)|+|F^{\prime}(X_{i})-F^{\prime}(x)||\leq
2$. Hence, we have:

$\Pr\left(|Z-\mathbb{E}[Z|F^{\prime}=f^{\prime},F=f]|\geq\tilde{\epsilon}\mid
F^{\prime}=f^{\prime},F=f\right)\leq
2\exp\left(-\frac{k\tilde{\epsilon}^{2}}{2\beta+\frac{4}{3}\tilde{\epsilon}}\right)$

where
$\frac{1}{k}\sum_{i=1}^{k}\mathrm{Var}\left[{Z_{i}|F^{\prime}=f^{\prime},F=f}\right]\leq\beta$
from Assumption 1.

Given
$|\mathbb{E}\left[{Z|F^{\prime}=f^{\prime},F=f}\right]-\mathbb{E}\left[{Z}\right]|<\epsilon^{\prime}$
and $\mathbb{E}\left[{Z}\right]=0$,

we have
$-\epsilon^{\prime}<\mathbb{E}\left[{Z|F^{\prime}=f^{\prime},F=f}\right]<\epsilon^{\prime}$
$\forall f,f^{\prime}$. Now observe that:

$\displaystyle\Pr(Z\geq\epsilon^{\prime}+\tilde{\epsilon}|F^{\prime}=f^{\prime},F=f)$
$\displaystyle{\stackrel{{\scriptstyle\text{(a)}}}{{\leq}}}\Pr(Z\geq\mathbb{E}\left[{Z|F^{\prime}=f^{\prime},F=f}\right]+\tilde{\epsilon}|F^{\prime}=f^{\prime},F=f)$
(16)
$\displaystyle\leq\exp\left(\frac{-k\tilde{\epsilon}^{2}}{2\beta+\frac{4}{3}\tilde{\epsilon}}\right).$
(17)

Here, (a) holds since
$\mathbb{E}\left[{Z|F^{\prime}=f^{\prime},F=f}\right]<\epsilon^{\prime}$. The
event on the left is a subset of that on the right. Therefore, the probability
of the event {$Z\geq\epsilon^{\prime}+\tilde{\epsilon}$} occurring cannot be
more than the probability of the event
{$Z\geq\mathbb{E}\left[{Z|F^{\prime}=f^{\prime},F=f}\right]+\tilde{\epsilon}$}
occurring.

$\displaystyle\Pr(Z\geq\epsilon^{\prime}+\tilde{\epsilon})$
$\displaystyle{\stackrel{{\scriptstyle\text{(b)}}}{{=}}}\sum_{i,j}\Pr(Z\geq\epsilon^{\prime}+\tilde{\epsilon}|F^{\prime}=f_{i},F=f_{j})\Pr(F^{\prime}=f_{i},F=f_{j})$
(18)
$\displaystyle{\stackrel{{\scriptstyle\text{(c)}}}{{\leq}}}\exp\left(\frac{-k\tilde{\epsilon}^{2}}{2\beta+\frac{4}{3}\tilde{\epsilon}}\right)\sum_{i,j}\Pr(F^{\prime}=f_{i},F=f_{j})$
(19)
$\displaystyle=\exp\left(\frac{-k\tilde{\epsilon}^{2}}{2\beta+\frac{4}{3}\tilde{\epsilon}}\right)$
(20)
$\displaystyle{\stackrel{{\scriptstyle\text{(d)}}}{{\leq}}}\exp\left(\frac{-k(\tilde{\epsilon}+\epsilon^{\prime})^{2}}{8\beta+\frac{16}{3}(\tilde{\epsilon}+\epsilon^{\prime})}\right)$
(21)

Here, (b) holds from the law of total probability. Next, (c) follows from
Equation (17). Finally, (d) holds from using the inequality
$4\tilde{\epsilon}^{2}>(\tilde{\epsilon}+\epsilon^{\prime})^{2}$ which holds
for $\tilde{\epsilon}>\epsilon^{\prime}>0$ at the numerator and
$\tilde{\epsilon}\leq\tilde{\epsilon}+\epsilon^{\prime}$ at the denominator.
Setting $\epsilon=\tilde{\epsilon}+\epsilon^{\prime}$.

We have:

$\Pr\bigg{(}\frac{1}{k}\sum_{i=1}^{k}F(X_{i})\geq\frac{1}{k}\sum_{i=1}^{k}\big{(}F^{\prime}(X_{i})-|F^{\prime}(X_{i})-F^{\prime}(x)|+|F(X_{i})-F(x)|\big{)}-\epsilon\bigg{)}\geq
1-\exp\left(\frac{-k\epsilon^{2}}{8\beta+\frac{16}{3}\epsilon}\right).$ (22)

Observe that $F(x)\geq F(x_{i})-|F(x_{i})-F(x)|$. This is a applies directly
from the reverse triangle inequality, i.e., for any real numbers $a$ and $b$,
we have:$|a|\geq|b|-|a-b|.$

Hence,

$F(x)\geq\frac{1}{k}\sum_{i=1}^{k}(F(X_{i})-|F(X_{i})-F(x)|)$ (23)

Therefore, plugging (23) into (22), we have:

$\displaystyle\Pr\big{(}F(x)\geq\frac{1}{k}\sum_{i=1}^{k}(F^{\prime}(X_{i})-|F^{\prime}(X_{i})-F^{\prime}(x)|+|F(X_{i})-F(x)|-|F(X_{i})-F(x)|-\epsilon)\big{)}$
(24)
$\displaystyle=\Pr\big{(}F(x)\geq\frac{1}{k}\sum_{i=1}^{k}(F^{\prime}(X_{i})-|F^{\prime}(X_{i})-F^{\prime}(x)|)-\epsilon\big{)}\geq
1-\exp\left(\frac{-k\epsilon^{2}}{8\beta+\frac{16}{3}\epsilon}\right).$ (25)

Given
$S_{k,\sigma}(x,F^{\prime})=\frac{1}{k}\sum_{i=1}^{k}(F(X_{i})-|F^{\prime}(x)-F^{\prime}(X_{i})|)$,
we have:

$\displaystyle\Pr\big{(}F(x)\geq S_{k,\sigma}(x,F^{\prime})-\epsilon\big{)}$
$\displaystyle\geq
1-\exp\left(\frac{-k\epsilon^{2}}{8\beta+\frac{16}{3}\epsilon}\right).$ (26)

∎

##  Appendix C  Appendix to Experiment Section

Figure 3: Evaluated multiplicity (assessed on $40$ retrained models) versus
our consistency measure (evaluated on one model) for the 512-shot setting on
the Diabetes dataset. The plots demonstrate that high consistency values
correspond to low multiplicity across various multiplicity evaluation metrics.
Predictive probabilities not providing any providing any useful insight about
multiplicity. Figure 4: Evaluated multiplicity (assessed on $40$ retrained
models) versus our consistency measure (evaluated on one model) for the
512-shot setting on the German Credit dataset. The plots demonstrate that high
consistency values correspond to low multiplicity across various multiplicity
evaluation metrics. In this setting Prediction probability is performing
competitively. But generally consistency measure provides better insight into
the multiplicity of predictions compared to the predicted probabilities.

###  C.1 Dataset Details

####  Adult Dataset

The Adult dataset [Becker and Kohavi, 1996], also known as the "Census Income"
dataset, is used for predicting whether an individual earns more than $50,000
annually based on various demographic attributes. It consists of 48,842
instances with 14 attributes, including age, work class, education, marital
status, occupation, relationship, race, sex, capital gain, capital loss, hours
per week, and native country. The dataset is commonly used in classification
tasks.

####  German Credit Dataset

The German Credit dataset [Hofmann, 1994] is used for credit risk evaluation.
It consists of $1,000$instances with $20$ attributes, which include personal
information, credit history, and loan attributes. The target variable
indicates whether the credit is good or bad. This dataset is often used for
binary classification problems and helps in understanding the factors
affecting creditworthiness. The dataset is commonly used in classification
tasks.

#### Diabetes Dataset

The Diabetes dataset Kahn  is used for predicting the onset of diabetes based
on diagnostic measurements. It contains 768 instances with 8 attributes,
including the number of pregnancies, glucose concentration, blood pressure,
skin thickness, insulin level, body mass index (BMI), diabetes pedigree
function, and age. The target variable indicates whether the individual has
diabetes. The dataset is commonly used in classification tasks.

###  C.2 Experimental Setup

Our experiments were conducted using the BigScience T0 model fine-tuned on
three datasets: German Credit, Diabetes, and Adult Income. We explored the
performance and robustness of the fine-tuned models in few-shot scenarios. The
number of shots was set to $64$,$128$, and $512$ for each dataset. To evaluate
model multiplicity and consistency, we fine-tuned $40$ models with different
random seeds for each dataset and recorded their predictions. The training
process involved setting the batch size to 2. The learning rate was set to
0.003. For each dataset, we determined the number of training steps adaptively
based on the number of shots, ensuring sufficient iterations for model
convergence. Specifically, for the number of shots-shot setting, the training
steps were calculated as $20\times(\text{number of shots}/\text{batch size})$.
All experiments were performed on 2 NVIDIA RTX A4500 GPU, utilizing CUDA for
efficient computation. The evaluation of model performance and consistency was
conducted at regular intervals, with an evaluation epoch interval set to 30.
To ensure reproducibility and robustness of the results, different random
seeds were used for each fine-tuning iteration. For fine-tuning with LORA we
use a rank of 4.

Table 3: Multiplicity Evaluation Metrics for Different Datasets and Number of Shots. Evaluated on $40$ fine-tuned models on LORA using different random seeds. Multiplicity observed in predictions across different fine-tuned model, even when models exhibit similar accuracy (in this setting $\delta=0.02$). Dataset | No. | Multiplicity Evaluation Metrics
---|---|---
Shots | Arbitrariness | Discrepancy | Avg. Pairwise | Avg. Pred. | Avg. Pred. | Avg. Model
|  |  |  | Disagreement | Variance | Range | Accuracy
Adult | 64 | 11% | 6% | 9% | 0.01 | 0.11 | 83%
128 | 10% | 9% | 6% | 0.01 | 0.10 | 84%
512 | 11% | 3% | 10% | 0.01 | 0.12 | 85%
German | 64 | 19% | 10% | 6% | 0.04 | 0.40 | 70%
128 | 17% | 11% | 6% | 0.01 | 0.16 | 71%
512 | 21% | 14% | 8% | 0.03 | 0.26 | 72%
Diabetes | 64 | 20% | 13% | 11% | 0.04 | 0.21 | 70%
128 | 16% | 14% | 11% | 0.08 | 0.14 | 73%
512 | 19% | 13% | 11% | 0.04 | 0.17 | 76%
Table 4: This table reports the Spearman correlation between the consistency measure and various multiplicity evaluation metrics for different numbers of shots on the Adult, German Credit, and Diabetes datasets. In most cases, the consistency measure $S_{k,\sigma}(x,f)$ shows a higher correlation with these multiplicity measures compared to predicted probabilities, indicating that the consistency measure $S_{k,\sigma}(x,f)$ better informs about the multiplicity than predicted probabilities $f(x)$ do. Dataset | Number of Shots | Measure | Arbitrariness | Pairwise Disagreement | Prediction Variance | Prediction Range
---|---|---|---|---|---|---
Adult | 64 | Consistency | 0.95 | 0.90 | 0.91 | 0.89
Pred. Prob. | 0.67 | 0.66 | 0.50 | 0.62
128 | Consistency | 0.80 | 0.96 | 0.84 | 0.91
Pred. Prob. | 0.67 | 0.62 | 0.30 | 0.54
512 | Consistency | 0.90 | 0.86 | 0.93 | 0.92
Pred. Prob. | 0.70 | 0.69 | 0.56 | 0.72
German Credit | 64 | Consistency | 0.95 | 0.95 | 0.98 | 0.84
Pred. Prob. | 0.99 | 0.99 | 0.80 | 0.79
128 | Consistency | 0.54 | 0.54 | 0.87 | 0.87
Pred. Prob. | 0.57 | 0.57 | 0.86 | 0.86
512 | Consistency | 0.59 | 0.60 | 0.87 | 0.86
Pred. Prob. | 0.54 | 0.56 | 0.83 | 0.82
Diabetes | 64 | Consistency | 0.45 | 0.51 | 0.31 | 0.23
Pred. Prob. | 0.03 | 0.38 | 0.04 | 0.08
128 | Consistency | 0.92 | 0.95 | 0.93 | 0.95
Pred. Prob. | 0.88 | 0.93 | 0.93 | 0.95
512 | Consistency | 0.80 | 0.89 | 0.74 | 0.68
Pred. Prob. | 0.21 | 0.23 | 0.24 | 0.30