Anomalous Quantum Diffusion in Order-Disorder Separated Double Quantum Ring

Santanu K. Maiti†,‡,∗

†Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
‡Department of Physics, Narasinha Dutt College, 129, Belilious Road,
Howrah-711 101, India

Abstract

A novel feature for control of carrier mobility is explored in an order-
disorder separated double quantum ring, where the two rings thread different
magnetic fluxes. Here we use simple tight-binding formulation to describe the
system. In our model, the two rings are connected through a single bond and
one of the rings is subjected to impurity, keeping the other ring as impurity
free. In the strong impurity regime, the electron diffusion length increases
with the increase of the impurity strength, while it decreases in the weak
impurity regime. This phenomenon is completely opposite to that of a
conventional disordered double quantum ring, where the electron diffusion
length always decreases with the increase of the disorder strength.

PACS No.: 73.23.Ra; 73.23.-b; 73.63.-b

Keywords: Persistent current; Drude weight; Double quantum ring; Impurity.

∗Corresponding Author: Santanu K. Maiti

Electronic mail<EMAIL_ADDRESS>
##  1 Introduction

Over the last few decades, the physics at sub-micron length scale provides
enormous evaluation both in terms of our understanding of basic physics as
well as in terms of the development of revolutionary technologies. In this
length scale, the so-called mesoscopic or nanoscopic regime, several
characteristic quantum length scales for the electrons such as system size and
phase coherence length or elastic mean free path and phase coherence length
are comparable. Due to the dominance of the quantum effects in the
mesoscopic/nanoscopic regime, intense research in this field has revolved its
richness. The most significant issue is probably the persistent currents in
small normal metal rings. In thermodynamic equilibrium, a small metallic ring
threaded by magnetic flux $\phi$ supports a current that does not decay
dissipatively even at non-zero temperature. It is the well-known phenomenon of
persistent current in mesoscopic normal metal rings which is a purely quantum
mechanical effect and gives an obvious demonstration of the Aharonov-Bohm
effect.1 The possibility of persistent current was predicted in the very early
days of quantum mechanics by Hund,2 but their experimental evidences came much
later only after realization of the mesoscopic systems. In $1983$, Büttiker et
al.3 predicted theoretically that persistent current can exist in mesoscopic
normal metal rings threaded by a magnetic flux $\phi$, even in the presence of
impurity. In a pioneering experiment, Levy et al.4 first gave the experimental
evidence of persistent current in the mesoscopic normal metal ring, and later,
the existence of the persistent current was further confirmed by several
experiments.5-8 Though the phenomenon of persistent current has been addressed
quite extensively over the last twenty years both theoretically9-27 as well as
experimentally,4-8 but yet we cannot resolve the controversy between the
theory and experiment. The main controversies come in the determinations of
(a) the current amplitude, (b) flux-quantum periodicities, (c) low-field
magnetic susceptibilities, etc. In recent works,24-26 we have pointed out that
the higher order hopping integrals, in addition to the nearest-neighbor
hopping integral, have a significant role to enhance the current amplitude
(even an order of magnitude). In other recent work,27 we have focused that the
low-field magnetic susceptibility can be predicted exactly only for the one-
channel systems with fixed number of electrons, while for all other cases it
becomes random. To grasp the experimental behavior of the persistent current,
one has to focus attention on the interplay of quantum phase coherence,
disorder and electron-electron correlation and this is a highly complex
problem.

Using the advanced molecular beam epitaxial growth technique, one can easily
fabricate a quantum system where the impurities are located only in some
particular region of the system, keeping the other region free from any
impurity. This is completely opposite from a conventional disordered system,
where the disorders are given uniformly throughout the system. Traditional
wisdom is that, the larger the disorder stronger the localization.28 However,
some recent experimental studies29-31 as well as theoretical
investigations32-35 on these special class of systems where the disorders are
not distributed uniformly, have yielded completely different behavior which
predicts that the electron diffusion length decreases in the weak disorder
regime, while it increases in the strong disorder regime. Motivated with these
results, in this article, we focus our attention in an order-disorder
separated double quantum ring system. To reveal the variation of the electron
diffusion length in such a particular system, here we study the behavior of
persistent current and Drude weight and our results may illuminate some of the
unusual experimental results for such diverse transport property. The
parameter Drude weight $D$ characterizes the conducting nature of the system
as originally introduced by Kohn.36 In our present model, two mesoscopic
rings, threaded by different magnetic fluxes, are connected by a single bond
and impurities are given in any one of these two rings, while the other ring
becomes impurity free. For this order-disorder separated double quantum ring,
we observe an anomalous behavior of electron mobility in which the electron
diffusion length increases with the increase of the impurity strength in the
strong impurity regime, while the diffusion length decreases in the weak
impurity regime. This phenomenon is completely opposite to that of a
conventional disordered double quantum ring, in which the electron diffusion
length always decreases with the increase of the disorder strength.

In what follows, we describe the model and the method in Section $2$. Section
$3$ contains the significant results and the discussion, and finally, we
summarize our results in Section $4$.

##  2 The model and the method

The schematic representation of a double quantum ring is shown in Fig. 1 where
the two rings, threaded by different magnetic fluxes, are connected by a
single bond. In the non-interacting picture, the system is usually modeled by
a single-band tight-binding Hamiltonian,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i}\epsilon_{i}^{I}c_{i}^{\dagger}c_{i}+v_{I}\sum_{<ij>}\left[e^{i\theta_{I}}c_{i}^{\dagger}c_{j}+e^{-i\theta_{I}}c_{j}^{\dagger}c_{i}\right]$
(1) $\displaystyle+$
$\displaystyle\sum_{k}\epsilon_{k}^{II}c_{k}^{\dagger}c_{k}+v_{II}\sum_{<kl>}\left[e^{i\theta_{II}}c_{k}^{\dagger}c_{l}+e^{-i\theta_{II}}c_{l}^{\dagger}c_{k}\right]$
$\displaystyle+$ $\displaystyle
v_{\alpha\beta}\left[c_{\alpha}^{\dagger}c_{\beta}+c_{\beta}^{\dagger}c_{\alpha}\right]$

Here $\epsilon_{i}^{I}$’s ($\epsilon_{i}^{II}$’s) are the site energies in the
ring $I$ (ring $II$), $c_{i}^{\dagger}$ ($c_{k}^{\dagger}$) is the creation
operator of an electron at site $i$ ($k$) of the ring $I$ (ring $II$) and
$c_{i}$ ($c_{k}$) is the annihilation operator of an electron at site $i$
($k$)

Figure 1: Schematic view of a double quantum ring in which the two rings
thread magnetic fluxes $\phi_{I}$ and $\phi_{II}$ respectively. These two
rings are connected through the lattice sites $\alpha$ and $\beta$. The filled
circles correspond to the position of the atomic sites (for color
illustration, see the web version).

of the ring $I$ (ring $II$), $v_{I}$ ($v_{II}$) is the hopping strength
between nearest-neighbor sites in the ring $I$ (ring $II$), and
$v_{\alpha\beta}$ gives the hopping strength between these two rings. In this
expression, $\theta_{I}=2\pi\phi_{I}/N_{I}$ and
$\theta_{II}=2\pi\phi_{II}/N_{II}$ are the phase factors due to the fluxes
$\phi_{I}$ and $\phi_{II}$ (measured in units of $\phi_{0}=ch/e$, the
elementary flux quantum), respectively where $N_{I}$ and $N_{II}$ correspond
to the total number of atomic sites in the ring $I$ and ring $II$,
respectively. In order to introduce the impurities in the system, we choose
the site energies ($\epsilon_{i}$’s, omitting the ring index in the
superscript) from the relation: $\epsilon_{i}=W\cos(i\lambda\pi)$, where $W$
is the strength of the disorder and $\lambda$ is an irrational number, and as
a typical example we take it as the golden mean $\left(1+\sqrt{5}\right)/2$.
Setting $\lambda=0$, we get back the pure system with identical site potential
$W$. The idea of considering such an incommensurate potential is that, for
such a correlated disorder we do not require any configuration averaging and
therefore the numerical calculations can be done in the low cost of time. Now
to achieve the order-disorder separated double quantum ring, we introduce the
correlated disorder in any one of the rings, keeping the other one as impurity
free.

At absolute zero temperature, the persistent currents in the two rings can be
calculated from the expressions,

$\displaystyle
I(\phi_{I})=-\frac{\partial{E(\phi_{I},\phi_{II})}}{\partial{\phi_{I}}}$ (2)
$\displaystyle
I(\phi_{II})=-\frac{\partial{E(\phi_{I},\phi_{II})}}{\partial{\phi_{II}}}$ (3)

where, $I(\phi_{I})$ and $I(\phi_{II})$ correspond to the currents in the ring
$I$ and ring $II$, respectively and $E(\phi_{I},\phi_{II})$ represents the
ground state energy of the complete system. We evaluate this energy exactly to
understand unambiguously the anomalous behavior of persistent current, and
this is achieved by exact diagonalization of the tight-binding Hamiltonian Eq.
(1).

Now the response of the double quantum ring system to a uniform time-dependent
electric field can be determined in terms of the Drude weight $D$,37-38 a
closely related parameter that characterizes the conducting nature of the
system as originally noted by Kohn.36 The Drude weights for the two rings can
be calculated through the relations,39

$\displaystyle
D_{I}=\left.\frac{N_{I}}{4\pi^{2}}\left(\frac{\partial{{}^{2}E(\phi_{I},\phi_{II})}}{\partial{\phi_{I}}^{2}}\right)\right|_{\phi_{I}\rightarrow
0,\phi_{II}\rightarrow 0}$ (4) $\displaystyle
D_{II}=\left.\frac{N_{II}}{4\pi^{2}}\left(\frac{\partial{{}^{2}E(\phi_{I},\phi_{II})}}{\partial{\phi_{II}}^{2}}\right)\right|_{\phi_{I}\rightarrow
0,\phi_{II}\rightarrow 0}$ (5)

where $D_{I}$ and $D_{II}$ represent the Drude weights for the ring $I$ and
ring $II$ respectively.

Our main aim in this article is the determination of the conducting properties
of an order-disorder separated double quantum ring, which can be computed
through the parameters $D_{I}$ and $D_{II}$. From these parameters we can
clearly describe the mobility of the charge carriers in the system and
accordingly, the variation of the electron diffusion length might be expected.

##  3 Results and discussion

In the order-disorder separated double quantum ring, we introduce the
correlated disorder in ring $I$ (for the sake of simplicity), keeping the ring
$II$ as impurity free. Throughout the numerical computations, we take the
values of the different parameters as: $v=-1$, $v_{\alpha\beta}=-1$ and for
the sake of simplicity, we use the units where $c=1$, $e=1$ and $h=1$. During
these calculations, we fix the chemical potential ($\mu$) for all the systems
to a constant value $0$. The main focus of this article is to describe how the
disordered states affect the ordered states in the order-disorder separated
system. Since for such a system the impurities are introduced in the ring $I$,
we evaluate the conducting properties of the double quantum ring by measuring
the Drude weight $D_{II}$. This actually provides the response of the ordered
states in presence of the disordered states.

Figure 2: Drude weight ($D_{II}$) of the ring $II$ as a function of the
disorder strength ($W$) for the systems with $N_{I}=30$, $N_{II}=30$ and the
fixed chemical potential $\mu=0$. The red and the blue lines correspond to the
order-disorder separated and the complete disordered double quantum ring
systems respectively (for color illustration, see the web version).

Otherwise, if we measure the parameter $D_{I}$, then we will get the trivial
result as obtained in a traditional disordered system since the response of
the disordered states will not be changed by coupling these states with the
ordered states.

In Fig. 2, we show the variation of the Drude weight $D_{II}$ as a function of
the disorder strength $W$ for some double quantum rings, where we choose
$N_{I}=30$ and $N_{II}=30$. The chemical potential $\mu$ is fixed to $0$. The
red and the blue curves represent the results for the order-disorder separated
and the complete disordered double quantum rings, respectively. From the
results it is observed that, in the complete disordered double quantum ring
the Drude weight sharply decreases with the increase of the disorder strength
and eventually it drops to zero. Therefore, we can say that for such a system
the electron diffusion length as well as the electron mobility decreases
sharply with the disorder strength. Such a behavior can be well understood
from the theory of Anderson localization, where we get more localization with
the increase of the disorder strength.28 The anomalous behavior is observed
when the impurities are given only in any one of the two rings, keeping the
other one as impurity free i.e., for the order-disorder separated system. Our
results predict that the Drude weight initially decreases with the increase of
the disorder strength, but after reaching to a minimum it again increases with
the strength of the disorder. Such a phenomenon is completely opposite to that
of the traditional disordered system and can be justified in the following
way. For the order-disorder separated double quantum ring, the energy spectra
of the disordered ring are gradually separated from the energy spectra of the
ordered ring with the increase of the disorder strength $W$. Therefore, the
influence of random scattering in the ordered ring due to the strong
localization in the disordered ring decreases. It has been examined that the
energy spectrum of the order-disorder separated double quantum ring with large
disorder contains localized tail states with much small and central states
with much large values of localization length, contributed approximately by
disordered and ordered rings, respectively. Hence the central states gradually
separated from the tail states and delocalized with the increase of the
strength of the disorder. Thus we see that, for the coupled order-disorder
separated double quantum ring, the coupling between the localized states with
the extended states is strongly influenced by the strength of the disorder,
and this coupling is inversely proportional to the disorder strength $W$.
Accordingly, in the limit of weak disorder the coupling effect is
significantly high, while the coupling effect becomes very weak in the strong
disorder regime. Hence, in the limit of weak disorder the electron transport
is strongly influenced by the impurities at the disordered ring such that the
electron states are scattered more and therefore the electron diffusion length
decreases which manifests the lesser electron mobility. On the other hand, for
the stronger disorder limit the extended states are weakly influenced by the
disordered ring and the coupling effect gradually decreases with the increase
of the disorder strength which provides the larger electron mobility in the
strong disorder limit. This reveals that the electron diffusion length
increases in this limit. For large enough impurity strength, the extended
states are almost unaffected by the impurities at the disordered ring and in
that case the electrons are carried only by these extended states in the
ordered ring which is the trivial limit. So the novel phenomenon will be
observed only in the intermediate limit of $W$.

In order to emphasize the dependence of the electron mobility on the system
size, here we focus our attention on the results those are plotted in Fig. 3.
In this figure, we display the Drude weight for some

Figure 3: Drude weight ($D_{II}$) of the ring $II$ as a function of the
disorder strength ($W$) for the systems with $N_{I}=50$, $N_{II}=50$ and the
chemical potential $\mu=0$. The red and the blue lines correspond to the
order-disorder separated and the complete disordered double quantum ring
systems respectively (for color illustration, see the web version).

typical double quantum rings, where we fix $N_{I}=50$ and $N_{II}=50$. Similar
to the previous systems, here we also take $\mu=0$ for these systems. The red
and the blue lines correspond to the same meaning as in Fig. 2. From this
figure (Fig. 3) it is also observed that, the Drude weight in the order-
disorder separated double quantum ring decreases with the increase of the
disorder strength $W$ in the weak disorder regime, while it increases with the
strength $W$ in the strong disorder regime. On the other hand, the Drude
weight always decreases with the strength of the disorder for the complete
disordered system, as expected. Though the results plotted in Fig. 3 seem to
be quite similar in nature with the results those are described in Fig. 2, but
the significant point is that, the typical magnitude of the Drude weight
strongly depends on the size of both these two rings which manifest the finite
quantum size effects. Now the other significant factor that raises to our mind
is the existence of the location of the minimum in the Drude weight versus
disorder curves of the order-disorder separated double quantum rings. This
minimum can be implemented as follows. The carrier mobility in the order-
disorder separated double quantum ring is controlled by the two competing
mechanisms. One is the random scattering in the ordered ring due to the
localization in the disordered ring which tends to decrease the carrier
mobility, and the other one is the vanishing influence of random scattering in
the ordered ring due to the strong localization in the disordered ring which
provides the enhancement of the carrier mobility. Depending on the ratio of
the total number of atomic sites in the disordered ring to the total number of
atomic sites in the ordered ring, the vanishing effect of random scattering
from the ordered states dominates over the non-vanishing effect of random
scattering from these states for a particular disorder strength $(W=W_{c})$,
which provides the location of the minimum in the Drude weight versus disorder
curve.

##  4 Concluding remarks

In conclusion, we have established a novel feature for control of the electron
diffusion length in an order-disorder separated double quantum ring in which
the two rings thread different magnetic fluxes. From our study it has been
observed that, in the order-disorder separated double quantum ring, the
electron diffusion length increases with the increase of the disorder strength
in the strong disorder regime, while it decreases in the weak disorder regime.
Such a peculiar behavior is completely opposite to that of the conventional
disordered systems, where the electron diffusion length always decreases with
the increase of the disorder strength. Lastly, we have noticed that, both the
electron mobility and the location of the minimum in the Drude weight versus
disorder curve strongly depend on the size of both the two rings which
manifest the finite quantum size effects. Our theoretical results in this
article might be helpful to illuminate some of the unusual experimental
phenomena which have been observed in the order-disorder separated quantum
systems.29-31

Throughout our study, we have ignored the effect of the electron-electron
(e-e) correlation since the inclusion of the e-e correlation will not provide
any new significant result in our present investigations.

Acknowledgment

I acknowledge with deep sense of gratitude the illuminating comments and
suggestions I have received from Prof. S. Sil during the calculations.

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# Avoiding Inference Heuristics in Few-shot Prompt-based Finetuning

Prasetya Ajie Utama†‡ , Nafise Sadat Moosavi‡, Victor Sanh♣, Iryna Gurevych‡

†Research Training Group AIPHES
‡UKP Lab, Technische Universität Darmstadt
♣Hugging Face, Brooklyn, USA
‡https://www.ukp.tu-darmstadt.de
<EMAIL_ADDRESS>
###### Abstract

Recent prompt-based approaches allow pretrained language models to achieve
strong performances on few-shot finetuning by reformulating downstream tasks
as a language modeling problem. In this work, we demonstrate that, despite its
advantages on low data regimes, finetuned prompt-based models for sentence
pair classification tasks still suffer from a common pitfall of adopting
inference heuristics based on lexical overlap, e.g., models incorrectly
assuming a sentence pair is of the same meaning because they consist of the
same set of words. Interestingly, we find that this particular inference
heuristic is significantly less present in the zero-shot evaluation of the
prompt-based model, indicating how finetuning can be destructive to useful
knowledge learned during the pretraining. We then show that adding a
regularization that preserves pretraining weights is effective in mitigating
this destructive tendency of few-shot finetuning. Our evaluation on three
datasets demonstrates promising improvements on the three corresponding
challenge datasets used to diagnose the inference heuristics.111The code is
available at https://github.com/UKPLab/emnlp2021-prompt-ft-heuristics

##  1 Introduction

Prompt-based finetuning has emerged as a promising paradigm to adapt
Pretrained Language Models (PLM) for downstream tasks with limited number of
labeled examples Schick and Schütze (2021a); Radford et al. (2019). This
approach reformulates downstream task instances as a language modeling
input,222E.g., appending a cloze prompt “It was [MASK]” to a sentiment
prediction input sentence “Delicious food!”, and obtaining the sentiment label
by comparing the probabilities assigned to the words “great” and “terrible”.
allowing PLMs to make non-trivial task-specific predictions even in zero-shot
settings. This in turn, provides a good initialization point for data
efficient finetuning Gao et al. (2021), resulting in a strong advantage on low
data regimes where the standard finetuning paradigm struggles. However, the
success of this prompting approach has only been shown using common held-out
evaluations, which often conceal certain undesirable behaviors of models Niven
and Kao (2019).

One such behavior commonly reported in downstream models is characterized by
their preference to use surface features over general linguistic information
Warstadt et al. (2020). In the Natural Language Inference (NLI) task, McCoy et
al. (2019) documented that models preferentially use the lexical overlap
feature between sentence pairs to blindly predict that one sentence entails
the other. Despite models’ high in-distribution performance, they often fail
on counterexamples of this inference heuristic, e.g., they predict that “the
cat chased the mouse” entails “the mouse chased the cat”.

At the same time, there is a mounting evidence that pre-training on large text
corpora extracts rich linguistic information Hewitt and Manning (2019); Tenney
et al. (2019). However, based on recent studies, standard finetuned models
often overlook this information in the presence of lexical overlap Nie et al.
(2019); Dasgupta et al. (2018). We therefore question whether direct
adaptation of PLMs using prompts can better transfer the use of this
information during finetuning. We investigate this question by systematically
studying the heuristics in a prompt-based model finetuned across three
datasets with varying data regimes. Our intriguing results reveal that: (i)
zero-shot prompt-based models are more robust to using the lexical overlap
heuristic during inference, indicated by their high performance on the
corresponding challenge datasets; (ii) however, prompt-based finetuned models
quickly adopt this heuristic as they learn from more labeled data, which is
indicated by gradual degradation of the performance in challenge datasets.

We then show that regularizing prompt-based finetuning, by penalizing the
learning from updating the weights too far from their original pretrained
values, is an effective approach to improve the in-distribution performance on
target datasets, while mitigating the adoption of inference heuristics.
Overall, our work suggests that while prompt-based finetuning has gained
impressive results on standard benchmarks, it can has a negative impact
regarding inference heuristics, which in turn suggests the importance of a
more thorough evaluation setup to ensure meaningful progress.

##  2 Inference Heuristics in Prompt-based Finetuning

### Prompt-based PLM Finetuning

In this work, we focus on sentence pairs classification tasks, where the goal
is to predict semantic relation $y$ of an input pair $x=(s_{1},s_{2})$. In a
standard finetuning setting, $s_{1}$ and $s_{2}$ are concatenated along with a
special token [CLS], whose embedding is used as an input to a newly
initialized classifier head.

The prompt-based approach, on the other hand, reformulates pair $x$ as a
masked language model input using a pre-defined template and word-to-label
mapping. For instance, Schick and Schütze (2021a) formulate a natural language
inference instance $(s_{1},s_{2},y)$ as:

$\texttt{[CLS]}s_{1}\texttt{?}\texttt{[MASK]}\textrm{, }s_{2}\texttt{[SEP]}$

with the following mapping for the masked token: “yes”$\rightarrow$
“entailment”, “maybe”$\rightarrow$“neutral”, and “no” $\rightarrow$
“contradiction”. The probabilities assigned by the PLM to the label words at
the [MASK] token can then be directly used to make task-specific predictions,
allowing PLM to perform in a zero-shot setting. Following Gao et al. (2021),
we further finetune the prompt-based model on the available labeled examples
for each task. Note that this procedure finetunes only the existing pre-
trained weights, and does not introduce new parameters.

Original Input
---
Premise |  The actors that danced saw the author.
Hypothesis |  The actors saw the author.
Label |  entailment (support)
Premise |  The managers near the scientist resigned.
Hypothesis |  The scientist resigned.
Label |  non-entailment (against)
Reformulated Input
Premise |  The actors that danced saw the author? [MASK], the actors saw the author.
Label word |  Yes
Premise |  The managers near the scientist resigned? [MASK], the scientist resigned.
Label word |  No / Maybe
Table 1: Top: input examples of the NLI task that support or are against the
lexical overlap heuristics. Bottom: reformulated NLI instances as masked
language model inputs with the expected label words.

### Task and Datasets

We evaluate on three English language datasets included in the GLUE benchmark
Wang et al. (2018) for which there are challenge datasets to evaluate the
lexical overlap heuristic: MNLI Williams et al. (2018), SNLI Bowman et al.
(2015), and Quora Question Pairs (QQP). In MNLI and SNLI, the task is to
determine whether premise sentence $s_{1}$ entails, contradicts, or is neutral
to the hypothesis sentence $s_{2}$. In QQP, $s_{1}$ and $s_{2}$ are a pair of
questions that are labeled as either duplicate or non-duplicate.

Researchers constructed corresponding challenge sets for the above datasets,
which are designed to contain examples that are against the heuristics, i.e.,
the examples exhibit word overlap between the two input sentences but are
labeled as non-entailment for NLI or non-duplicate for QQP. We evaluate each
few-shot model against its corresponding challenge dataset. Namely, we
evaluate models trained on MNLI against entailment and non-entailment subsets
of the HANS dataset McCoy et al. (2019), which are further categorized into
lexical overlap (lex.), subsequence (subseq.), and constituent (const.)
subsets; SNLI models against the long and short subsets of the Scramble Test
challenge set Dasgupta et al. (2018); and QQP models against the PAWS dataset
Zhang et al. (2019).333See appendix A for details of HANS, PAWS, and Scramble
Test test sets. We illustrate challenge datasets examples and their
reformulation as prompts in Table 1.

Figure 1: In-distribution (bold) vs. challenge datasets (italic) evaluation
results of prompt-based finetuning across different data size $\mathbf{K}$ (x
axis), where $\mathbf{K}=0$ indicates zero-shot evaluation. In all challenge
sets, the overall zero-shot performance (both blue and green plots) degrades
as the model is finetuned using more data.

### Model and Finetuning

Our training and standard evaluation setup closely follow Gao et al. (2021),
which measure finetuning performances across five different randomly sampled
training data of size $\mathbf{K}$ to account for finetuning instability on
small datasets Dodge et al. (2020); Mosbach et al. (2021). We perform five
data subsampling for each dataset and each data size $\mathbf{K}$, where
$\mathbf{K}\in\\{16,32,64,128,256,512\\}$. Note that $\mathbf{K}$ indicates
the number of examples per label. We use the original development sets of each
training dataset for testing the in-distribution performance. We perform all
experiments using the RoBERTa-large model Liu et al. (2019b).

### Inference heuristics across data regimes

We show the results of the prompt-based finetuning across different
$\mathbf{K}$ in Figure 1. For the in-distribution evaluations (leftmost of
each plot), the prompt-based models finetuned on MNLI, SNLI, and QQP improve
rapidly with more training data before saturating at $\mathbf{K}=512$. In
contrast to the in-distribution results, we observe a different trajectory of
performance on the three challenge datasets. On the Scramble and HANS sets,
prompt-based models show non-trivial zero-shot performance ($\mathbf{K}=0$)
that is above its in-distribution counterpart. However, as more data is
available, the models exhibit stronger indication of adopting heuristics.
Namely, the performance on examples subset that support the heuristics
increases, while the performance on cases that are against heuristics
decreases. This pattern is most pronounced on the lexical overlap subset of
HANS, where the median accuracy on non-entailment subset drops to below 10%
while the entailment performance reaches 100%. The results suggest that few-
shot finetuning can be destructive against the initial ability of prompt-based
classifier to ignore surface features like lexical overlap. Finetuning appears
to over-adjust model parameters to the small target data, which contain very
few to no counter-examples to the heuristics Min et al. (2020); Lovering et
al. (2021).

##  3 Avoiding Inference Heuristics

Here we look to mitigate the adverse impact of finetuning by viewing the issue
as an instance of catastrophic forgetting French (1999), which is
characterized by the loss of performance on the original dataset after
subsequent finetuning on new data. We then propose a regularized prompt-based
finetuning based on the Elastic Weight Consolidation (EWC) method Kirkpatrick
et al. (2017), which penalizes updates on weights crucial for the original
zero-shot performance. EWC identifies these weights using empirical Fisher
matrix Martens (2020), which requires samples of the original dataset. To omit
the need of accessing the pretraining data, we follow Chen et al. (2020) that
assume stronger independence between the Fisher information and the
corresponding weights. The penalty term is now akin to the L2 loss between
updated weights $\theta_{i}$ and the original weights $\theta_{i}^{*}$,
resulting in the following overall loss:

$\mathbf{L}_{rFT}=\alpha\mathbf{L}_{FT}+(1-\alpha)\frac{\lambda}{2}\sum_{i}(\theta_{i}-\theta_{i}^{*})^{2}$

where $\mathbf{L}_{FT}$ is a standard cross entropy, $\lambda$ is a quadratic
penalty coefficient, and $\alpha$ is a coefficient to linearly combine the two
terms. We use the RecAdam implementation Chen et al. (2020) for this loss,
which also applies an annealing mechanism to gradually upweight the standard
loss $\mathbf{L}_{FT}$ toward the end of training.444See Appendix A for
implementation details.

|  MNLI (acc.) |  QQP (F1) |  SNLI (acc.)
---|---|---|---
| In-dist. | HANS | avg. | In-dist. | PAWS | avg. | In-dist. | Scramble | avg.
Prompt-based
zero-shot #0 | 51.1 | 62.6 | 56.8 | 35.4 | 51.8 | 43.6 | 49.7 | 64.7 | 57.2
FT #512 | 84.3 | 54.8 | 69.5 | 82.1 | 29.6 | 55.8 | 88.1 | 50.1 | 69.1
rFT #512 | 82.7 | 60.2 | 71.5 | 81.5 | 37.1 | 59.3 | 87.6 | 55.4 | 71.5
FT-fix18 #512 | 76.5 | 61.6 | 69.1 | 78.6 | 35.6 | 57.1 | 84.5 | 45.3 | 64.9
FT-fix12 #512 | 83.5 | 54.3 | 68.9 | 81.9 | 35.3 | 57.1 | 87.1 | 50.5 | 68.8
FT-fix6 #512 | 84.2 | 52.9 | 68.5 | 82.1 | 32.7 | 57.4 | 87.9 | 50.1 | 68.9
Classifier head
FT #512 | 81.4 | 52.6 | 67.0 | 80.9 | 26.8 | 53.8 | 86.5 | 49.8 | 68.1
Table 2: Results of different strategies for finetuning prompt-based model
(using #$k$ examples). Models are evaluated against the in-distribution set
and corresponding challenge sets. The zero-shot row indicates prompting
results before finetuning. The avg columns report the average score on in-
distribution and challenge datasets.

### Baselines

We compare regularized finetuning with another method that also minimally
update the pretraining weights. We consider simple weight fixing of the first
$n$ layers of the pretrained model, where the $n$ layers are frozen (including
the token embeddings) and only the weights of upper layers and LM head are
updated throughout the finetuning. In the evaluation, we use
$n\in\\{6,12,18\\}$. We refer to these baselines as FT-fix$n$.

### Results

We evaluate all the considered finetuning strategies by taking their median
performance after finetuning on 512 examples (for each label) and compare them
with the original zero-shot performance. We report the results on Table 2,
which also include the results of standard classifier head finetuning (last
row). We observe the following: (1) Freezing the layers has mixed challenge
set results, e.g., FT-fix18 improves over vanilla prompt-based finetuning on
HANS and PAWS, but degrades Scramble and all in-distribution performances; (2)
The L2 regularization strategy, rFT, achieves consistent improvements on the
challenge sets while only costs small drop on the corresponding in-
distribution performance, e.g., +6pp, +8pp, and +5pp on HANS, PAWS, and
Scramble, respectively; (3) Although vanilla prompt-based finetuning performs
relatively poorly, it still has an advantage over standard classifier head
finetuning by +2.5pp, +2.0pp, and +1.0pp on the average scores of each in-
distribution and challenge dataset pair.

Figure 2: Relative difference between median accuracy of prompt-based
finetuning across data regimes (y axis) with and without regularization on
MNLI and HANS.

Additionally, Figure 2 shows rFT’s improvement over vanilla prompt-based
finetuning across data regimes on MNLI and HANS. We observe that the advantage
of rFT is the strongest on the lexical overlap subset, which initially shows
the highest zero-shot performance. The results also suggest that the benefit
of rFT peaks at mid data regimes (e.g., $\mathbf{K}=32$), before saturating
when training data size is increased further. We also note that our results
are consistent when we evaluate alternative prompt templates, or finetune for
varying number of epochs.555See Appendix B for the detailed results. The
latter indicates that the adoption of inference heuristics is more likely
attributed to the amount of training examples rather than the number of
learning steps.

##  4 Related Work

### Inference Heuristics

Our work relates to a large body of literature on the problem of “bias” in the
training datasets and the ramifications to the resulting models across various
language understanding tasks Niven and Kao (2019); Poliak et al. (2018);
Tsuchiya (2018); Gururangan et al. (2020). Previous work shows that the
artifacts of data annotations result in spurious surface cues, which gives
away the labels, allowing models to perform well without properly learning the
intended task. For instance, models are shown to adopt heuristics based on the
presence of certain indicative words or phrases in tasks such as reading
comprehension Kaushik and Lipton (2018), story cloze completion Schwartz et
al. (2017); Cai et al. (2017), fact verification Schuster et al. (2019),
argumentation mining Niven and Kao (2019), and natural language inference
Gururangan et al. (2020). Heuristics in models are often investigated using
constructed “challenge datasets” consisting of counter-examples to the
spurious cues, which mostly result in incorrect predictions Jia and Liang
(2017); Glockner et al. (2018); Naik et al. (2018); McCoy et al. (2019).
Although the problem has been extensively studied, most works focus on models
that are trained in standard settings where larger training datasets are
available. Our work provides new insights in inference heuristics in models
that are trained in zero- and few-shot settings.

### Heuristics Mitigation

Significant prior work attempt to mitigate the heuristics in models by
improving the training dataset. Zellers et al. (2019); Sakaguchi et al. (2020)
propose to reduce artifacts in the training data by using adversarial
filtering methods; Nie et al. (2020); Kaushik et al. (2020) aim at a similar
improvement via iterative data collection using human-in-the-loop; Min et al.
(2020); Schuster et al. (2021); Liu et al. (2019a); Rozen et al. (2019)
augment the training dataset with adversarial instances; and Moosavi et al.
(2020a) augment each training instances with their semantic roles information.
Complementary to this, recent work introduces various learning algorithms to
avoid adopting heuristics including by re-weighting He et al. (2019); Karimi
Mahabadi et al. (2020); Clark et al. (2020) or regularizing the confidence
Utama et al. (2020a); Du et al. (2021) on the training instances which exhibit
certain biases. The type of bias can be identified automatically Yaghoobzadeh
et al. (2021); Utama et al. (2020b); Sanh et al. (2021); Clark et al. (2020)
or by hand-crafted models designed based on prior knowledge about the bias.
Our finding suggests that prompted zero-shot models are less reliant on
heuristics when tested against examples containing the cues, and preserving
this learned behavior is crucial to obtain more robust finetuned models.

### Efficiency and Robustness

Prompting formulation enables language models to learn efficiently from a
small number of training examples, which in turn reduces the computational
cost for training Le Scao and Rush (2021). The efficiency benefit from
prompting is very relevant to the larger efforts towards sustainable and green
NLP models Moosavi et al. (2020b); Schwartz et al. (2020a) which encompass a
flurry of techniques including knowledge distillation Hinton et al. (2015);
Sanh et al. (2019), pruning Han et al. (2015), quantization Jacob et al.
(2018), and early exiting Schwartz et al. (2020b); Xin et al. (2020).
Recently, Hooker et al. (2020) show that methods improving compute and memory
efficiency using pruning and quantization may be at odds with robustness and
fairness. They report that while performance on standard test sets is largely
unchanged, the performance of efficient models on certain underrepresented
subsets of the data is disproportionately reduced, suggesting the importance
of a more comprehensive evaluation to estimate overall changes in performance.

##  5 Conclusion

Our experiments shed light on the negative impact of low resource finetuning
to the models’ overall performance that is previously obscured by standard
evaluation setup. The results indicate that while finetuning helps prompt-
based models to rapidly gain the in-distribution improvement as more labeled
data are available, it also gradually increases models’ reliance on surface
heuristics, which we show to be less present in the zero-shot evaluation. We
further demonstrate that applying regularization that preserves pretrained
weights during finetuning mitigates the adoption of heuristics while also
maintains high in-distribution performances.

## Acknowledgement

We thank Michael Bugert, Tim Baumgärtner, Jan Buchman, and the anonymous
reviewers for their constructive feedback. This work is funded by the German
Research Foundation through the research training group AIPHES (GRK 1994/1)
and by the German Federal Ministry of Education and Research and the Hessen
State Ministry for Higher Education, Research and the Arts within their joint
support of the National Research Center for Applied Cybersecurity ATHENE.

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##  Appendix A Experimental Details

### Manual templates and mapping

We use the following prompt templates and word-to-label mapping for the three
tasks we evaluate on:

Template | Label Words
---|---
MNLI (manual): entailment, neutral, contradiction
$s_{1}\texttt{?}\texttt{[MASK]}\textrm{, }s_{2}$ | Yes, Maybe, No
SNLI (manual): entailment, neutral, contradiction
$s_{1}\texttt{?}\texttt{[MASK]}\textrm{, }s_{2}$ | Yes, Maybe, No
QQP (manual): duplicate, non-duplicate
$s_{1}\texttt{[MASK]}\textrm{, }s_{2}$ | Yes, No
MNLI (auto): entailment, neutral, contradiction
$s_{1}\texttt{.}\texttt{[MASK]}\textrm{, you are right , }s_{2}$ | Fine, Plus, Otherwise
$s_{1}\texttt{.}\texttt{[MASK]}\textrm{, you're right , }s_{2}$ | There, Plus, Otherwise
$s_{1}\texttt{.}\texttt{[MASK]}\textrm{ ! }s_{2}$ | Meaning, Plus, Otherwise
Table 3: Templates and label words used to finetune and evaluate on MNLI,
SNLI, and QQP.

The last 3 rows are automatically generated templates and label words that are
shown by Gao et al. (2021) to improve the few-shot finetuning further. Note
that we use the corresponding task’s template when evaluating on the challenge
datasets.

### Challenge datasets

We provide examples from each challenge datasets considered in our evaluation
to illustrate sentence pairs that support or are against the heuristics. Table
4 shows examples for HANS, PAWS, and Scramble Test. Following McCoy et al.
(2019), we obtain the probability for the non-entailment label by summing the
probabilities assigned by models trained on MNLI to the neutral and
contradiction labels. We use the same-type subset of Scramble Test Dasgupta et
al. (2018) which contain examples of both entailment (support) and
contradiction (against) relations.

HANS McCoy et al. (2019)
---
premise |  The artists avoided the senators that thanked the tourists.
hypothesis |  The artists avoided the senators.
label |  entailment (support)
premise |  The managers near the scientist resigned.
hypothesis |  The scientist resigned.
label |  non-entailment (against)
PAWS Zhang et al. (2019)
S1 |  What are the driving rules in Georgia versus Mississippi?
S2 |  What are the driving rules in Mississippi versus Georgia?
label |  duplicate (support)
S1 |  Who pays for Hillary’s campaigning for Obama?
S2 |  Who pays for Obama’s campaigning for Hillary?
label |  non-duplicate (against)
Scramble Test Dasgupta et al. (2018)
premise |  The woman is more cheerful than the man.
hypothesis |  The woman is more cheerful than the man.
label |  entailment (support)
premise |  The woman is more cheerful than the man.
hypothesis |  The man is more cheerful than the woman.
label |  contradiction (against)
Table 4: Sampled examples from each of the challenge datasets we used for
evaluation.

### HANS details

HANS dataset is designed based on the insight that the word overlapping
between premise and hypothesis in NLI datasets is spuriously correlated with
the entailment label. HANS consists of examples in which relying to this
correlation leads to incorrect label, i.e., hypotheses are not entailed by
their word-overlapping premises. HANS is split into three test cases: (a)
Lexical overlap (e.g., “The doctor was paid by the actor” $\rightarrow$ “The
doctor paid the actor”), (b) Subsequence (e.g., “The doctor near the actor
danced” $\rightarrow$ “The actor danced”), and (c) Constituent (e.g., “If the
artist slept, the actor ran” $\rightarrow$ “The artist slept”). Each subset
contains both entailment and non-entailment examples that always exhibit word
overlap.

### Hyperparameters

Following Schick and Schütze (2021b, a), we use a fixed set of hyperparameters
for all finetuning: learning rate of $1e^{-5}$, batch size of 8, and maximum
length size of 256.

### Regularization implementation

We use the RecAdam implementation by Chen et al. (2020) with the following
hyperparameters. We set the quadratic penalty $\lambda$ to $5000$, and the
linear combination factor $\alpha$ is set dynamically throughout the training
according to a sigmoid function schedule, where $\alpha$ at step $t$ is
defined as:

$\alpha=s(t)=\frac{1}{1+\texttt{exp}(-k\cdot(t-t_{0}))}$

where parameter $k$ regulates the rate of the sigmoid, and $t_{0}$ sets the
point where $s(t)$ goes above $0.5$. We set $k$ to $0.01$ and $t_{0}$ to $0.6$
of the total training steps.

Figure 3: Results of prompt-based finetuning with varying number of epochs and
fixed amount of training examples. Top: finetuning on 32 examples per label
for epochs ranging from 10 to 50. Bottom: finetuning on 512 examples per label
for 1 to 9 epochs. Both results show an immediate drop of non-entailment HANS
performances which later stagnate even after more training steps.

Figure 4: Results of non-prompt finetuning.

##  Appendix B Additional Results

### Standard CLS finetuning

Previously, Gao et al. (2021) reported that the performance of standard non-
prompt finetuning with additional classifier head (CLS) can converge to that
of prompt-based counterpart after certain amount of data, e.g., 512. It is
then interesting to compare both finetuning paradigm in terms of their
heuristics-related behavior. Figure 4 shows the results of standard finetuning
using standard classifier head across varying data regimes on MNLI and the 3
subsets of HANS. We observe high instability of the results when only small
amount of data is available (e.g., $\mathbf{K}=64$). The learning trajectories
are consistent across the HANS subsets, i.e., they start making random
predictions on lower data regime and immediately adopt heuristics by
predicting almost all examples exhibiting lexical overlap as entailment. We
observe that standard prompt-based finetuning still performs better than CLS
finetuning, indicating that prompt-based approach provides good initialization
to mitigate heuristics, and employing regularization during finetuning can
improve the challenge datasets (out-of-distribution) performance further.

|  MNLI (acc.)
---|---
| IN | HANS
manual | 51.1 | 62.6
manual Ft-#512 | 84.3 | 54.8
template-1 | 46.3 | 62.0
template-1 Ft-#512 | 84.2 | 53.2
template-2 | 49.9 | 61.3
template-2 Ft-#512 | 83.9 | 52.7
template-3 | 44.5 | 61.7
template-3 Ft-#512 | 84.4 | 56.0
Table 5: Evaluation results of different MNLI templates provided by Gao et al. (2021). Models are evaluated against both the in-distribution (IN) set and corresponding challenge set of MNLI. |  MNLI (acc.) |  QQP (F1) |  SNLI (acc.)
---|---|---|---
| In. | HANS | avg. | In. | PAWS | avg. | In. | Scramble | avg.
zero-shot RoBERTa-large | 51.1 | 62.6 | 56.8 | 35.4 | 51.8 | 43.6 | 49.7 | 64.7 | 57.2
FT #512 RoBERTa-large | 84.3 | 54.8 | 69.5 | 82.1 | 29.6 | 55.8 | 88.1 | 50.1 | 69.1
zero-shot RoBERTa-base | 48.2 | 58.1 | 53.15 | 37.3 | 41.5 | 39.4 | 48.8 | 56.4 | 52.6
FT #512 RoBERTa-base | 74.4 | 49.9 | 62.15 | 79.0 | 26.9 | 52.9 | 83.7 | 48.5 | 66.1
zero-shot BERT-large-uncased | 45.3 | 55.4 | 50.4 | 34.7 | 33.4 | 34.0 | 41.5 | 54.8 | 48.1
FT #512 BERT-large-uncased | 70.9 | 50.0 | 60.4 | 77.3 | 26.3 | 51.8 | 79.9 | 49.5 | 64.7
zero-shot BERT-base-uncased | 43.5 | 55.9 | 49.7 | 40.7 | 50.8 | 45.8 | 38.7 | 49.9 | 44.3
FT #512 BERT-base-uncased | 63.2 | 50.1 | 56.65 | 73.9 | 29.1 | 51.5 | 74.5 | 42.6 | 58.5
Table 6: Evaluation results of different pretrained language models. Models
are evaluated against both the in-distribution (In.) set and corresponding
challenge set.

Figure 5: Relative difference between median accuracy of prompt-based
finetuning across data regimes (y axis) with and without regularization on QQP
/ PAWS and SNLI / Scramble Test.

### Impact of prompt templates

A growing number of work propose varying prompt generation strategies to push
be benefits of prompt-based predictions Gao et al. (2021); Schick et al.
(2020). We therefore questions whether different choices of templates would
affect the model’s behavior related to lexical overlap. We evaluate the 3 top-
performing templates for MNLI that are obtained automatically by Gao et al.
(2021) and show the results in Table 5. We observe similar behavior from the
resulting models over the manual prompt counterpart, achieving HANS average
accuracy of around 62% and below 55% on zero-shot and finetuning with 512
examples.

### Impact of learning steps

We investigate the degradation of the challenge datasets performance as the
function of the number of training data available during finetuning. However,
adding more training examples while fixing the number of epochs introduces a
confound factor to our finding, which is the number of learning steps to the
model’s weights. To factor out the number of steps, we perform similar
evaluation with a fixed amount of training data and varying number of training
epochs. On 32 examples per label, we finetune for 10, 20, 30, 40, and 50
epochs. Additionally, we finetune on 512 examples for 1 until 10 epochs to see
if the difference in learning steps results in different behavior. We plot the
results in Figure 3. We observe that both finetuning settings result in
similar trajectories, i.e., models start to adopt heuristics immediately in
early epochs and later stagnate even with increasing number of learning steps.
For instance, finetuning on 32 examples for the same number of training steps
as in 512 examples finetuning for 1 epoch still result in higher overall HANS
performance. We conclude that the number of finetuning data plays a more
significant role over the number of training steps. Intuitively, larger
training data is more likely to contain more examples that disproportionately
support the heuristics; e.g. NLI pairs with lexical overlap are rarely of non-
entailment relation McCoy et al. (2019).

### Regularization across data regimes

Figure 5 shows the results improvement of L2 weight regularization over
vanilla prompt-based finetuning on QQP and SNLI. Similar to results in
MNLI/HANS, the improvements are highest on mid data regimes, e.g., 32 examples
per label.

### Impact of pretrained model

In addition to evaluating RoBERTa-large, we also evaluate on other commonly
used pretrained language models based on transformers such as RoBERTa-base,
BERT-base-uncased, and BERT-large-uncased. The results are shown in Table 6.
We observe similar pattern across PLMs, i.e., improved in-distribution scores
come at the cost of the degradation in the corresponding challenge datasets.