# Parameter-Efficient Abstractive Question Answering
over Tables or Text

Vaishali Pal1 Evangelos Kanoulas2 Maarten de Rijke2
1Discovery Lab, University of Amsterdam 2University of Amsterdam
v.pal, e.kanoulas<EMAIL_ADDRESS>
###### Abstract

A long-term ambition of information seeking question answering (QA) systems is
to reason over multi-modal contexts and generate natural answers to user
queries. Today, memory intensive pre-trained language models are adapted to
downstream tasks such as QA by fine-tuning the model on QA data in a specific
modality like unstructured text or structured tables. To avoid training such
memory-hungry models while utilizing a uniform architecture for each modality,
parameter-efficient adapters add and train small task-specific bottle-neck
layers between transformer layers. In this work, we study parameter-efficient
abstractive QA in encoder-decoder models over structured tabular data and
unstructured textual data using only 1.5% additional parameters for each
modality. We also ablate over adapter layers in both encoder and decoder
modules to study the efficiency-performance trade-off and demonstrate that
reducing additional trainable parameters down to 0.7%–1.0% leads to comparable
results. Our models out-perform current state-of-the-art models on tabular QA
datasets such as Tablesum and FeTaQA, and achieve comparable performance on a
textual QA dataset such as NarrativeQA using significantly less trainable
parameters than fine-tuning.

##  1 Introduction

Information seeking systems over diverse contexts require model capabilities
to reason over unstructured and structured

Figure 1: Parameter-efficient transfer learning using modality-specific
(table/text) adapters for Abstractive Question Answering

data such as free-form text, tables, and images (Agrawal et al., 2016;
Vakulenko et al., 2019; Hudson and Manning, 2019; Zhang et al., 2020; Zhu et
al., 2021; Deldjoo et al., 2021). Such systems might have the additional
requirement of generating natural language responses if deployed as task-
oriented conversational agents (Wen et al., 2015; Carnegie and Oh, 2000;
Rambow et al., 2001; Ratnaparkhi, 2002). Recent work on open-domain QA
predominately addresses these challenges by fine-tuning massive pre-trained
language models on different modalities such as tables and text (Yin et al.,
2020; Herzig et al., 2020, 2021; Katsis et al., 2021; Nan et al., 2021).
However, each model trained on a specific input type is incompatible with
other modalities and requires modality-specific fine-tuning. For example, in
tabular QA (Herzig et al., 2020), the structure of the table is learnt by
training additional position embeddings (row and column identifiers) to
identify which row and column a table cell belongs to. This renders such
modality specific models incompatible with free-form text-based models. Multi-
modal models (Zhu et al., 2021) can reason over both tables and text by
concatenating the textual context and the flattened table, leading to longer
input sequences and limiting the length of the context that can be encoded.

To address these challenges, we study parameter-efficient transfer learning
for abstractive QA over tables and over text. We are motivated to use adapter-
layers that inject small bottle-neck layers between frozen pre-trained
transformer layers as they achieve comparable performance to fine-tuning on a
variety of tasks such as multi-lingual translation (Pfeiffer et al., 2020;
Philip et al., 2020; Guo et al., 2020), classification (Houlsby et al.,
2019a), text-to-text generation (Lin et al., 2020), domain-adaptation in
dialogue state tracking, and response generation (Hung et al., 2021).

Ablation studies on adapter layers (Rucklé et al., 2020) on masked language
models such as BERT-base and RoBERTa over the GLUE benchmark demonstrate that
removing beginning adapter layers leads to a minimal drop in performance.
Extending adapter layer ablation over separate encoder and decoder modules is
non-trivial as the conventional approach of sequential pruning of layers does
not extend to consecutive encoder and decoder modules. Our work explores the
interaction of adapter layers from both modules in the context of abstractive
QA.

Lin et al. (2020) explore the impact of the adapter bottle-neck dimension for
various language generation tasks over an auto-regressive model such as GPT-2
(Radford et al., 2019). They do not study tabular data nor ablate adapter
layers, which is crucial in understanding impact of individual adapters in
sequential transformer module architectures such as encoder-decoder. Our
analysis is complementary to (Lin et al., 2020) as we ablate adapter layers to
study parameter-performance trade-off whereas they only focus on adapter
bottle-neck size. Also, we generalize beyond the text-to-text setting and
explore language generation from structured or unstructured input such as
tables and text. This introduces domain-shift in both the _task_ and
_structure_ of the downstream data.

We propose a system, named Parameter, Efficient, Abstractive Question
Answering (PeaQA), shown in Figure 1, which learns to reason over unstructured
and structured input using a _shared_ pre-trained language model and modality-
specific adapter layers. We automatically transform hierarchical tables to
regular tables to have a uniform representation without breaking associations
between table cells. In addition, we extend the study of ablating adapter
layers over both encoder and decoder modules.

Our main contributions are summarized as:

  1. (1)

We perform parameter-efficient abstractive question answering over multi-modal
context using only additional 1.5% of trainable parameters for each modality.
Our adapter-tuned model outperforms existing work by a large margin on tabular
QA datasets and achieves comparable performance on a textual QA dataset.

  2. (2)

We study tabular QA as a new modality that introduces massive input domain
shift to pre-trained language models. We propose a 2-step transformation of
hierarchical tables to sequences, which produces a uniform representation to
be used by a single, shared pre-trained language model and modality-specific
adapter layers. To the best of our knowledge, this is the first work that
explores tabular QA question answering in a parameter-efficient manner.

  3. (3)

We ablate adapter layers in both encoder and decoder modules to study their
impact and show that beginning layers from both encoder and decoder can be
eliminated without significant drop in performance. We also demonstrate that
last encoder adapter layers are indispensable and have greater contribution
than decoder layers at the same level.

##  2 Related Work

Tabular question answering. Tabular QA systems aim to answer questions from
structured tables, which can be regular or hierarchical. Hierarchical tables
can have header cells and body cells spanning across multiple rows and columns
(Cheng et al., 2021). In most tabular QA systems (Herzig et al., 2020; Zhu et
al., 2021; Katsis et al., 2021), the structure of the table is encoded in the
embedding layer of large language models by introducing table specific
position information such as row id and column id. Concurrent to our work,
abstractive QA over tables (Nan et al., 2021; Cheng et al., 2021) poses
additional challenges of generating natural answers by reasoning and
aggregating discontinuous facts from the table.

Textual question answering. Question answering over text measures a system’s
ability to comprehend free-form text in the user question and context
passage(s) and predict an answer. The answer predicted can be extractive in
nature, where the system identifies short text spans in the context passage to
answer the user query (Lee et al., 2016; Seo et al., 2016; Rajpurkar et al.,
2016; Pearce et al., 2021), or it can be abstractive, where it is required to
generate a free-form answer (Yin et al., 2016; Mitra, 2017; Bauer et al.,
2018; Reddy et al., 2019).

Transfer learning. Transfer learning techniques such as fine-tuning pre-
trained models for downstream tasks, require a new set of parameters to be
learnt for each new task. To avoid such memory intensive transfer learning
methods, adapters have been proposed as a parameter-efficient method of
adapting to new domains (Houlsby et al., 2019b; Pfeiffer et al., 2020).
Adapters have been extended to language generation in a variety of generative
tasks such as translation, summarization, multi-turn dialogue, and task-
oriented natural language generation (Lin et al., 2020).

Our work combines all the aforementioned aspects to generate abstractive
answers from _both_ tables and text with only $0.7\%$–1.0$\%$ trainable
parameters without compromising performance.

##  3 Model

We focus on encoder-decoder models for the task of abstractive question
answering. We use a BART (Lewis et al., 2019) encoder-decoder architecture
which comprises of a bidirectional encoder and an auto-regressive decoder. The
input sequence consists of the question, the context title and context
sequence preceded with prompts indicating the beginning of the each sub-
sequence. Formally, the input sequence is represented as ${<}$_question_ ${>}$
$q_{0}$ $q_{1}$ … $q_{m}$ ${<}$_title_ ${>}$ $t_{1}$ $t_{2}$ … $t_{p}$
${<}$_context_ ${>}$ $c_{0}$ $c_{1}$ … $c_{n}$, where $q_{i}$ is the $i$-th
question token, $t_{j}$ is the $j$-th title token, and $c_{k}$ is the $k$-th
context token. The context can either be a text passage or a flattened table.
The parameters of the pre-trained BART model are frozen during training.
Modality specific adapter layers added to the model are trained on either
tabular context or textual context to generate natural answers.

##  4 Textual Question Answering

To study multi-modal abstractive QA, we first focus on free-form text as
context to the system. We train adapter layers for textual context on the
NarrativeQA dataset (Kočiský et al., 2018). NarrativeQA is a complex
abstractive question answering dataset over stories. The dataset contains
$32,747$ samples in the training set, $3,461$ samples in the validation set,
and $10,557$ samples in the test set. For our task, we have selected the input
context passage to be the human annotated _summary_ of each sample which is
the Wikipedia page summary of the story and represented as a paragraph. The
input to the model is the _question_ , _title_ and _summary_ of each passage
and the target is the abstractive answer.

##  5 Tabular Question Answering

We study tabular QA as a new modality which introduces massive input domain
shift to pre-trained language models. Tables enforce structural constraints in
their representation which is incompatible with the expected input format of
pre-trained language models. To achieve our goal of parameter efficiency by
utilizing a uniform pre-trained language model, we only train table specific
adapter layers while keeping the pre-trained model frozen. However, this
necessitates a uniform input representation for both tables and text. An
additional challenge is introduced to maintain uniformity across different
table types (regular, hierarchical).

For our task, we explore $2$ tabular QA datasets, namely, Tablesum (Zhang et
al., 2020) and FeTaQA (Nan et al., 2021). Tablesum consists of 200 unique
Wikipedia tables over which questions and abstractive answers are manually
annotated; 40% of the samples are questions over hierarchical tables but the
tables in their released data are missing information in the hierarchical
cells and their work do not handle hierarchies. We address this issue by
extracting the wikitables from the respective Wikepedia pages and release a
clean version of the dataset.111The cleaned data and code can be found at
https://github.com/kolk/Pea-QA

FeTaQA (Nan et al., 2021) is a larger abstractive tabular QA dataset
consisting of question and free-form answers over $10,330$ regular tables. The
dataset consists of $7,326$ samples in the training set, $1,001$ in the
validation set, and $2,003$ in the test set. FeTaQA consists of human-
annotated answers containing explanations involving entities and relations.

###  5.1 Table Representation

For our work, we choose to represent all tables uniformly in a two-step
process:  (1) Transformation of a hierarchical table into a regular table; and
(2) Linearization of a regular table into a flattened sequence which can be
encoded with a language model.

(a) A multi-span table represented as a regular table.

(b) Linearize regular table to a sequence of _key_ :_value_ pairs.

Figure 2: Table representation.

Linearize hierarchical table headers. Hierarchical table headers are
linearized into a single row of headers by the following process. A header
cell spanning multiple columns is duplicated and split into multiple cells.
Next, the cell values over which this header spans are concatenated with the
entire split. Repeating this process over all header rows flattens the
hierarchical header into a sequential one. We depict this process in Figure
2(a), which yields a linear header $a(d)$, $a(d)$, $b$, $e(f)$.

Linearizing table body. Multi-span table body cells are parsed differently
than headers. Each table body cell is replicated with one or multiple header
cells depending on its span across columns. Cells that span across multiple
rows are replicated with all the spanned rows. This process leads to a regular
table. We flatten the regular table in row-major form, concatenating rows
sequentially. Each row is a sequence of _(key, value)_ pairs where a key is a
column header and the value is the cell value of that column as depicted in
Figure 2(b).

##  6 Experimental Setup

We seek to answer the following research questions with our experiments:
(RQ1) How does adapter-tuning perform compared to fine-tuning in the context
of multi-modal input?  (RQ2) Do all adapter layers across the encoder and
decoder contribute equally to performance across tasks/modalities?

###  6.1 Fine-Tuning

We perform all our experiments on the _large_ variant of BART model. We fine-
tune the BART-large model over the 3 datasets as the state-of-the-art fine-
tuned models utilize different architectures for different datasets making
comparison with adapter-tuning difficult. We treat our fine-tuned BART models
on the 3 datasets as baselines. We sweep learning rates from {$8e^{-4}$,
$6e^{-4}$, $3e^{-4}$, $1e^{-4}$, $5e^{-5}$, $4^{e}{-5}$, $3e^{-5}$, $2e^{-5}$,
$1e^{-5}$} and select the best performing learning rate for each dataset. We
select $4e^{-5}$ for fine-tuning on Tablesum, $8e^{-4}$ on FeTaQA datasets and
$2e^{-5}$ to fine-tune NarrativeQA. We use a batch size of $4$ and gradient
accumulation of $8$ to emulate an effective batch size of $32$. The maximum
target sequence length is set to $200$ for tabular QA datasets and to $100$
for the textual QA dataset. On the Tablesum dataset, we follow 5-fold cross
validation as described in the original work to evaluate our models. On FeTaQA
and NarrativeQA, we utilize the test split for evaluating our models. We train
the model on each dataset for $15$ epochs and evaluate on Rouge-2, Rouge-L and
sacreBLEU metrics.

###  6.2 Adapter-Tuning

We perform adapter-tuning as a parameter-efficient alternative to adapt BART-
large model to the abstractive question answering task across different
modalities. We first freeze all layers of the pre-trained BART-large model
which was trained on text reconstruction as mentioned in the original BART
paper Lewis et al. (2019). We add bottle-neck adapter layers from the Houlsby
adapter configuration (Houlsby et al., 2019a) which are trained to adapt to
the downstream abstractive question answering task and also to modality
specific input context. Each adapter layer has a bottle-neck embedding size of
$64$. As mentioned in Section 6.1, we sweep learning rates and select the best
performing learning rate for each dataset. We select $6e^{-4}$ for the tabular
QA datasets Tablesum and FeTaQA, and select $1e^{-1}$ to train the textual QA
dataset NarrativeQA. We use the same batch size and maximum target sequence
length as fine-tuning for effective comparison. A summary of hyper-parameters
are mentioned in Table 1.

Dataset | Params | ATune | FTune
---|---|---|---
All | scheduler | linear | linear
batch size | 32 | 32
| seed | 6 | 6
| max epochs | 15 | 15
Tablesum | learning rate | 6e-4 | 4e-5
input length | 200 | 200
FeTaQA | learning rate | 6e-4 | 8e-4
input length | 100 | 100
NarrativeQA | learning rate | 1e-4 | 2e-5
input length | 50 | 50
Table 1: Hyper-parameters for training. ATune indicates Adapter-tuning, FTune
indicates Fine-tuning, All indicates all 3 datasets.

Dataset | Model | Training | Rouge-1 | Rouge-2 | Rouge-L | BLEU
---|---|---|---|---|---|---
Tablesum (Zhang et al., 2020) | GPT2 | fine-tune | $0.272$ | $0.073$ | $0.200$ | $5.35$
T5 | $0.362$ | $0.143$ | $0.276$ | 10.43
Ours (Pea-QA) | fine-tune(Baseline) | 0.400 | 0.186 | 0.316 | $6.30$
Adapter-tune | $0.393$ | 0.186 | $0.312$ | $6.75$
FeTaQA (Nan et al., 2021) | T5-small | fine-tune | $0.550$ | $0.330$ | $0.470$ | $21.60$
T5-base | $0.610$ | $0.390$ | $0.510$ | $28.14$
T5-large | $0.630$ | $0.414$ | $0.530$ | $30.54$
Ours (Pea-QA) | fine-tune(Baseline) | $0.632$ | $0.415$ | $0.534$ | $30.81$
Adapter-tune | 0.651 | 0.436 | 0.553 | 33.45
NarrativeQA (Kočiský et al., 2018) |  MasqueMasqueMasq (Nishida et al., 2019) | fine-tune | – | – | 0.547 | –
Ours (Pea-QA) | fine-tune(Baseline) | $0.518$ | $0.268$ | $0.515$ | $21.07$
Adapter-tune | $0.510$ | $0.270$ | $0.500$ | $20.08$

Table 2: Results: Scores obtained on the Tablesum, FeTaQA and NarrativeQA
datasets.

###  6.3 Ablation Study: Adapter Pruning

Adapter-layer pruning has been explored on the GLUE benchmark in (Rucklé et
al., 2020), which demonstrates that removing adapter layers from the beginning
of BERT-base and RoBERTa models leads to minimal performance drop. We extend
adapter layer ablation to encoder-decoder architectures and hypothesize that
this phenomenon should be observed on both the encoder and decoder modules.
However, it is non-trivial how the adapter-layers in the encoder and decoder
interact with each other and contribute to performance. Previous studies
(Rucklé et al., 2020) on adapter ablation prune consecutive adapter layers in
masked language models. This approach does not extend directly to sequential
modules of encoder-decoder where intra-module adapters not only contribute to
their respective objective of encoding and decoding but also contributes to
inter-module interaction and performance. To measure the impact of the adapter
layers in different modules, we perform adapter ablation in both the encoder
and decoder. First, we uniformly remove adapter layers from both encoder and
decoder modules starting from the beginning layers of both modules and finally
deleting all layers. This leads to 12 experiments corresponding to eliminating
12 encoder and 12 decoder adapter layers. To study interaction across inter-
module adapters at different levels, we conduct 36 experiments of different
configurations of adapter elimination from the last 6 levels of encoder and
decoder. We analyze the performance by each configuration in Section 7.3.

Question: What and when were Akhila Kishore’s first two films?
---
Target: akhila kishore made her debut in the kannada film padhe padhe (2013),
and appeared in kathai thiraikathai vasanam iyakkam (2014).
Table: Year Film Role Language 2013 Padhe Padhe Kanchana Kannada 2014 Kathai
Thiraikathai Vasanam Iyakkam Daksha Tamil 2015 Inimey Ippadithaan Akhila Tamil
… … …
Adaper-tune: akhila kishore made her debut in the kannada film padhe padhe
(2013) and kathai thiraikathai vasanam iyakkam (2014).
Fine-tune: kathai thiraikathai vasanam iyakkam (2014) and inimey ippadithaan
(2015) were kannada films.
Question: Who is the starring actor of Aastik?
Target: aastik is a 1956 hindi film starring shahu modak, paro devi and
meenakshi.
Table: Title Director Cast … … … Aastik S. P. Kalla Shahu Modak, Paro Devi,
Meenakshi, B. M. Vyas, Praveen Paul Alam Ara Nanubhai Vakil Daljeet, Chitra,
Tiwari, Niranjan Sharma, Minu Mumtaz,… … … …
Adaper-tune: aastik is a 1956 bollywood film starring shahu modak.
Fine-tune: a directed by s. p. kalla.
Question: What were the three films directed by Yakub and when were they
released?
Target: yakub directed three films: sagar ka sher in 1937, uski tamanna in
1939, and, in 1949, aiye.
Table: Year Film Director … … … 1937 Sagar Ka Sher (Lion of Sagar) Yakub … … …
1939 Uski Tamanna (Her Last Desire) Yakub … … … 1949 Aiye Yakub … … …
Adaper-tune: yakub directed three films: sagar ka sher (lion of sagar) in
1937, uski tamanna (her last desire) in 1939 and aiye in 1949.
Fine-tune: y directed by yakub.
Table 3: Samples where adapter-tune outperforms fine-tune

Question: how many times was ed sheeran listed as the performer? Targets: • Ed
Sheeran was listed as a performer twice in the table documenting the top hits
of 2014 in Sweden. Other English-Language top performers included Bruce
Springsteen, Sam Smith, and Coldplay, implying that English-Language music has
significant success in Sweden. • According to the table, in 2014, Ed Sheeran
was only listed as the performer one time. It was for the song that he
performed that is called ""I See Fire"", which was out in January and February
of 2014 Table:  Week Date Song title Performer Album title Performer 1 3 Jan
2014 "Timber" Pitbull feat. … True Avicii 2 10 Jan 2014 3 17 Jan 2014 "I See
Fire" Ed Sheeran High Hopes Bruce … 4 24 Jan 2014 5 31 Jan 2014 True Avicii 6
7 Feb 2014 Christer Sjögren sjunger Sinatra Christer Sjögren 7 14 Feb 2014 8
21 Feb 2014 True Avicii 9 28 Feb 2014 … … … … 31 31 July 2014 Prayer in C…
Lilly Wood &… X Ed Sheeran 32 7 Aug 2014 Honky Tonk Rebels Lasse Stefanz … … …
… 42 16 Oct 2014 "The Days" Avicii X Ed Sheeran 43 23 Oct 2014 44 30 Oct 2014
Songs for Daddy Jill Johnson … … … … Adaper-tune: I found a table of the year
2014. It shows the week of the week, the song title, the artist, the title of
the song, the album title, and the performer. There are 11 times that Ed
Sheeran was listed as the performer in the year. The song title is "Timber"
and the album is True. Fine-tune: I found a table of Ed Sheeran’s year in
2014. He was listed as the performer 14 times in the year 2014. The first time
he was listed was on 3 January 2014 with the song "Timber" and the last time
was on 4 April 2014 with "I See Fire".

Table 4: Example from the Tablesum dataset. Adapter-tune |  #Trainable parameters
---|---
Encoder adapters removed |  Decoder adapters removed
– | – |  $6,343,680$ $(1.56\%)$
0–2 | 12–14 |  $4,757,760$ $(1.17\%)$
0–4 | 12–16 |  $3,700,480$ $(0.91\%)$
0–6 | 12–18 |  $2,643,200$ $(0.65\%)$
0–8 | 12–20 |  $1,585,920$ $(0.39\%)$
0–10 | 12–22 |  $528,640$ $(0.13\%)$
0–11 | 12–22 |  $264,320$ $(0.07\%)$
fine-tune |  $406,291,456$ $(100\%)$
Table 5: Trainable parameters in the encoder and decoder. Encoder adapter
layers are numbered from 0–11 and decoder adapter layers are numbered from
12–22. $x$–$y$ implies all adapter layers from $x$ to $y$ inclusive.

(a) FeTaQA Rouge-L scores

(b) Tablesum Rouge-L scores

(c) NarrativeQA Rouge-L scores

(d) FeTaQA sacreBLEU scores

(e) Tablesum sacreBLEU scores

(f) NarrativeQA sacreBLEU scores

Figure 3: Adapter layer ablation scores. The X-axis represents range of
encoder adapter layers deleted, the Y-Axis represents range of decoder adapter
layers deleted. $x$-$y$ implies all adapter layers from $x$ to $y$ inclusive.
There are 36 model ablation configurations displayed. The ablation starts from
$0$ to $6$ encoder adapter layers removal and 12 to 18 decoder adapter layer
removal represented by the bottom left cell ((0–6), (12–18)) and progressively
increases deletion of encoder adapter layers along the X-axis and decoder
adapter layers along the Y-axis.

##  7 Results

We compare the results of our baseline fine-tuned models with the state-of-
the-art fine-tuned models in Section 7.1. We address (RQ1) “How does adapter-
tuning perform compared to fine-tuning in the context of multi-modal input?”
in Section 7.2 and (RQ2) “Do all adapter layers across the encoder and decoder
contribute equally to performance across tasks/modalities?” in 7.3.

###  7.1 Fine-Tuned Models

We study the results of our baseline fine-tuned models with the state-of-the-
art fine-tuned models for the 3 datasets. The results of the experiments are
shown in Table 2. We observe that for the Tablesum dataset, our fine-tuned
model outperform the best state-of-art T5 model on Rouge-1 by $3.8\%$ ,
Rouge-2 by $4.3\%$ and Rouge-L score by $4\%$. This can be attributed to fine-
tuning our model on the clean version of the dataset. Our fine-tuned models
perform comparably to the state-of-the-art T5-large on FeTaQA dataset, i.e,
$0.2\%$ on Rouge-1, $0.01\%$ higher on Rouge-2, and $0.04\%$ higher on
Rouge-L. Our fine-tuning results on NarrativeQA are lower than state-of-the-
art models trained with sophisticated reasoning architecture. The focus of
this work was primarily on comparing fine-tuning and adapter-tuning and hence
we leave explicit reasoning as part of future work.

###  7.2 Adapter-Tuned Models

We address (RQ1) by comparing the performance of adapter-tuned models to our
baseline fine-tuned models. For Tablesum, as observed in Table 2 fine-
tuning(baseline) marginally outperforms adapter-tuning with $0.7\%$ higher
Rouge-1 and $0.4\%$ higher Rouge-L scores while having the same Rouge-2 score.
For FeTaQA, adapter-tune shows a larger performance gain with $1.9\%$ on
Rouge-1 and Rouge-L and $2.1\%$ on Rouge-2 compared to fine-tuning. The
insignificant gains of fine-tuning over adapter-tuning in tabular QA can be
attributed to catastrophic forgetting (French, 1999; Kirkpatrick et al., 2017;
Chen et al., 2020) induced by differences in the distribution of downstream
tabular data format from the original text data format of pre-training.

To explore this phenomenon further, we analyse examples from FeTaQA dataset in
Table 3 where adapter-tuning outperforms fine-tuning. We observe that the
fine-tuned model is unable to disambiguate surface-form similarities from the
column semantics in the first example. The intended semantics of the named-
entity _Akhila Kishore_ in the question is _Actor_. While the surface-form is
similar to the column value _Akhila_ , the intended semantics is that of the
column header _Role_. The fine-tuned model wrongly predicts the second and
third row of the tabular context as correct grounding of information while
adapter-tuning is able to disambiguate and predicts information from the first
2 rows as answer. We observe that the fine-tuned model also predicts
information from the wrong column _Director_ instead of _Cast_ in the second
example. Adapter-tune correctly identifies the column but partially generates
the required information in the prediction. The third example depicts both
non-factual and non-fluent prediction by the fine-tuned model.

We demonstrate an example of a hierarchical table of Tablesum in Table 4. The
question requires aggregation on the table cells and has various summary-like
targets associated with it. The hierarchical table mentions _Ed Sheeran_ $3$
times, but the actual number of occurrence is $10$ times, from _Week 3_ to
_Week 9_ , _Week 31_ and from _Week 42_ to _Week 43_. Our table transformation
process handles this to produce a regular table with $10$ cells containing _Ed
Sheeran_ as value. The models can simply aggregate over the mentions. As shown
in Table 4, both models generates long answers summarizing information from
the context table. However, as the models do not explicitly handle cell
aggregation, we observe factual mistakes in both adapter-tuned and fine-tuned
models. The models find Tablesum samples challenging even though the generated
language is fluent and readable.

For textual QA, on the NarrativeQA dataset, adapter-tuning performs comparable
to fine-tuning with the adapter-tuned model achieving $0.8\%$ lower Rouge-1,
$1.8\%$ higher Rouge-2 and $1.5\%$ lower Rouge-L scores than fine-tuning.

We conclude that adapter-tuning performs better than fine-tuning for out-of-
domain tabular data and comparable performance on in-domain text.

Figure 4: Adapter layer ablation Rouge2 F-scores. The X-axis depicts encoder-
adapter layers (0–11) and decoder adapter layers (12–23) deleted
progressively. Each $(x-y)\atop(r-s)$ represents F-score with encoder layers
$p$ to $q$ deleted and decoder layers $r$ to $s$ deleted. Figure 5: Adapter
layer ablation Rouge-L scores. The X-axis depicts encoder-adapter layers
(0–11) and decoder adapter layers (12–23) deleted progressively. Each
$(x-y)\atop(r-s)$ represents F-score with encoder layers $p$ to $q$ deleted
and decoder layers $r$ to $s$ deleted. Figure 6: Adapter layer ablation
sacreBLEU F-scores. The X-axis depicts encoder-adapter layers (0–11) and
decoder adapter layers (12–23) deleted progressively. Each $(x-y)\atop(r-s)$
represents F-score with encoder layers $p$ to $q$ deleted and decoder layers
$r$ to $s$ deleted.

###  7.3 Ablation of adapter layers

We study (RQ2) by ablating adapter layers in both the encoder and decoder
modules. We uniformly eliminate successive adapter layers from both encoder
and decoder starting from the first layer in both modules and finally deleting
all layers. This leads to 12 experiments corresponding to 12 encoder and 12
decoder adapter layers. We number the encoder adapter layers from 0–11 and the
decoder adapter layers from 12–23. We measure the performance of the models
using Rouge-2, Rouge-L222https://pypi.org/project/rouge-score/ and
sacreBLEU333https://github.com/mjpost/sacreBLEU scores. The F-scores for each
dataset (NarrativeQA, Tablesum, FeTaQA) are shown in Figure 4, 5 and 6,
respectively. We observe that as more adapter layers are eliminated, the
performance drops across all datasets. However, the performance drop is
minimal until the last adapter layers are also deleted. The inflection point
varies across dataset but is limited to the last $2$ layers of the encoder and
decoder. For the NarrativeQA dataset, this point is when all layers till the
second last adapter layer from both the encoder and decoder are deleted. For
the FeTaQA and Tablesum datasets, the performance drops sharply only when the
last encoder and decoder layers are removed.

To analyze contribution of the $i$-th adapter layer of encoder and decoder to
performance, we perform ablation of adapter layers (0–6), (0–7), …, (0–11)
from encoder and adapter layers (12–18), (12–19), …, (12–23) from decoder
(decoder layers are numbered 12–23). This leads to $36$ configurations where a
configuration ($p$–$q$, $r$–$s$) represents removal of all encoder adapters
from $p$-th to $q$-th layer and all decoder adapters from $r$-th to $s$-th.
The results are shown in Figure 3. We observe that performance remains
comparable as we progressively eliminate adapter layers from encoder and
decoder until the last layers. The performance drops steeply when we remove
the last encoder and decoder adapter layers depicted towards the top-right
corner of RougeL scores in Figures 3(a), 3(b), and 3(c) and BLEU scores in
Figures 3(d), 3(e), and 3(f). This implies that last adapter layers learns
most of the domain information.

We also observe that the last encoder and decoder layers contribute
differently to performance. Removing the last encoder layer (column 0–11)
leads to substantial score drop across all decoder layers. This indicates that
the last encoder layer is indispensable. Keeping only the last decoder adapter
(row 12–23) is comparable to keeping last two last encoder layers (column
0–10). We also observe that retaining just the last $50\%$ of adapter layers
from both encoder and decoder increases parameter efficiency by $0.7\%$
parameters as summarized in Table 5 without significant compromise to
performance.

##  8 Conclusion

We are the first to study parameter-efficient transfer learning over tables
and text for abstractive question answering using adapters. We demonstrate
that parameter efficient adapter-tuning outperforms fine-tuning on out-of-
domain tabular data and achieves comparable results on in-domain textual data.

We propose a transformation from hierarchical tables to regular ones and
further into a sequential form compatible with pre-trained model. We extend an
existing ablation study of adapter layers to encoder-decoder setting and
demonstrate that adapter layers from the end of the encoder is indispensable
to encoding modality specific information than decoder adapter layers at the
same level.

Our results are useful for exploring scalability of QA models in memory
constrained situations with comparable performance while scaling across
modalities using light-weight adapters.

One of the limitations of our work is that our models do not explicitly reason
and aggregate over the table cells. This might lead to fluent but factually
incorrect answers on challenging Tablesum dataset. Addressing this limitation
is left as future work.

##  9 Acknowledgements

We would like to thank Elsevier for their support throughout this project and
funding this work. This work was also supported by the NWO Innovational
Research Incentives Scheme Vidi (016.Vidi.189.039), the NWO Smart Culture -
Big Data / Digital Humanities (314-99-301), the H2020-EU.3.4. - SOCIETAL
CHALLENGES - Smart, Green And Integrated Transport (814961). All content
represents the opinion of the authors, which is not necessarily shared or
endorsed by their respective employers and/or sponsors.

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## APPENDICES

We provide further details on statistics of the datasets used (Appendix A) and
on the Rouge-2 scores for an encoder-decoder adapter layer ablation study
(Appendix B).

##  Appendix A Dataset Statistics

Statistics of the three datasets, i.e., Tablesum, FeTaQA and NarrativeQA are
listed in Table 6. Tablesum has the longest answer length. The answers are
summary-like, often, describing aspects of the table contents. The FeTaQA
dataset contains answers of mostly single sentences and targeted towards
specific facts asked in the question. The NarrativeQA dataset focuses on
questions from stories. The answer lengths vary from single words to long
sentences. For the tabularQA dataset, Tablesum contains larger tables than the
FeTaQA dataset even though it is limited to $200$ unique tables over which
questions are asked. The FeTaQA dataset’s tables contain more columns on
average than Tablesum.

Tablesum
---
Domain | Open
Modality | Table
Table-type | Regular
Training samples | 798
Validation samples | 200
Test samples | –
Max question length | 114
Max target length | $1,579$
Max table row | 155
Max table column | 8
FeTaQA
Domain | Open
Modality | Table
Table-type | Hybrid
Training samples | $7,326$
Validation samples | $1,001$
Test samples | $2,003$
Train max question length | 165
Train max target length | 338
Train max table rows | 34
Train max table columns | 30
Val max question length | 182
Val target length | 325
Val max table rows | 34
Val max table columns | 22
Test max question length | 193
Test max target length | 295
Test max table lows | 34
Test max table columns | 22
NarrativeQA
Domain | Stories
Modality | Text
Training samples | $65,494$
Validation samples | $6,922$
Test samples | $21,114$
Train max question length | 175
Train max target length | 171
Train max context length | $6,045$
Val max question length | 158
Val target length | 187
Val max context length | $6,033$
Test max question length | $1,220$
Test target length | 224
Test max context length | $6,090$
Table 6: Dataset Statistics

##  Appendix B Encoder-Decoder Adapter Layer Ablation Rouge-2 Scores

Ablation results (Rouge-2 F-scores) of $36$ configurations of adapter layers
deleted from the later half of the encoder and decoder. Deleting the last
encoder adapter layers leads to massive drop in performance as observed in the
last three columns of Figures 7(a), 7(b) and 7(c). However, deleting the last
decoder adapter layers results in better performance in comparison to the
encoder layers at the same level as observed from the top 3 rows.

(a) FeTaQA Rouge-L scores

(b) Tablesum Rouge-L scores

(c) NarrativeQA Rouge-L scores

Figure 7: Adapter layer Rouge-2 ablation scores. The X-axis represents range
of encoder adapter layers deleted, the Y-Axis represents range of decoder
adapter layers deleted. $x$-$y$ implies all adapter layers from $x$ to $y$
inclusive. There are 36 model ablation configurations displayed. The ablation
starts from $0$ to $6$ encoder adapter layers removal and 12 to 18 decoder
adapter layer removal represented by the bottom left cell ((0–6), (12–18)) and
progressively increases deletion of encoder adapter layers along the X-axis
and decoder adapter layers along the Y-axis.

  *[QA]: question answering


# Walk a Mile in Their Shoes: a New Fairness Criterion for Machine Learning

Norman Matloff
<EMAIL_ADDRESS>
Department of Computer Science
University of California, Davis
Davis, CA 95616, USA

###### Abstract

The old empathetic adage, “Walk a mile in their shoes,” asks that one imagine
the difficulties others may face. This suggests a new ML counterfactual
fairness criterion, based on a group level: How would members of a
nonprotected group fare if their group were subject to conditions in some
protected group? Instead of asking what sentence would a particular Caucasian
convict receive if he were Black, take that notion to entire groups; e.g. how
would the average sentence for all White convicts change if they were Black,
but with their same White characteristics, e.g. same number of prior
convictions? We frame the problem and study it empirically, for different
datasets. Our approach also is a solution to the problem of covariate
correlation with sensitive attributes.

##  1 Introduction

The issue of fairness in machine learning (ML) is the subject of increasing
concern, with many interesting algorithms proposed, both for fair prediction
and for assessing fairness. As always, though, “The devil is in the details.”
Exactly what do we mean by “fair”? Many criteria for fairness have been
proposed. See for instance [3] [35] [4] [26] [6] for overviews, some of them
written from a critical point of view.

Much work has been done on fair ML from counterfactual points of view. What
sentence would this Caucasian convict receive if he had been Black but
otherwise with the same relevant characteristics? See [21] for a comprehensive
review. In this paper, a novel approach to the counterfactual is proposed.

###  1.1 Relation to the Legal Realm

Having served as an expert witness in a number of discrimination cases in
litigation (e.g. [7]), I view an important point in choosing fairness criteria
to be consideration of legal issues, including legal standards for evidence
but even more important, development of methodology that is easily understood
by judges and juries.

[39], [17] and [2] give detailed analyses of how fairness criteria developed
in the ML literature may or may not be consistent with US federal statutes and
case law. The legal situation in the European Union is apparently less
precisely formulated at this point [36], but will certainly evolve in the near
future; this may produce constraints. See [32] and [33] for interesting South
and East Asian perspectives.

A key point regarding statistical analyses presented in litigation is the
Daubert standard [24]) [12] [16]. The vast majority of fair ML methods do not
account for sampling variability, i.e. do not include mechanisms for
significance testing and confidence intervals.111The authors in [14] speculate
that they are the first to have developed such mechanisms; a literature search
seems to confirm this. Accordingly, most published methods may be vulnerable
to challenge in court.

Anohter key point in litigation is that the statistical analysis presented be
intuitively clear and reasonable to judges and juries ([34]). Indeed, in the
case of judges, the US Federal Judicial Center, a federal agency chaired by
the Chief Justice of the Supreme Court, found the issue of statistical
literacy pressing enough to commission a guide to statistical methods for
judges [11]. This was viewed as essential even though presumably most judges
had some exposure to the subject in school; that will not be the case for
some, likely most, jurors. It is thus imperative that fair ML methodology be
easily grasped on an intuitive level, a major goal of this paper.

There is also concern among many regarding intersectionality, the pernicious
interaction between the separate discriminatory actions against different
protected groups [9], [10]. As explained in [9],

> Intersectionality is a lens for examining societal unfairness which
> originally arose from the observation that sexism and racism have
> intertwined effects, in that the harm done to Black women by these two
> phenomena is more than the sum of the parts

Methodology aimed at fairness must take intersectionality into account. Linear
models of effects of race, gender and aga, for instance, may require inclusion
of interaction terms.

###  1.2 Contributions of This Paper

Although much of the presentation here will be motivated by legal aspects, the
need for fair ML goes far beyond the legal realm. A firm or government agency,
for instance, may be concerned that its AI tool may be effective at addressing
some objective function, but at the same time may be unfair to some important
groups.

Indeed, no single fairness criterion is best for all possible applications.
Thus it is desirable for ML analysts to have available a variety of fairness
criteria to choose from.

This paper proposes a new fairness criterion, motivated specifically by these
desiderata:

  * (a)

A criterion should be easily relatable to not only ML analysts, but also to
non-STEM academics, the press, participants in litigation, and the general
public.

  * (b)

As such, a criterion should be kept simple, describable easily in plain
language.

  * (c)

Conditional distributions can be confusing. Fairness criteria with lesser
explicit involvement with conditional quantities may be preferred.

  * (d)

Again in the interest of simplicity, a criterion should not consist of many
numbers; it should consist of a single number for each protected (base group,
counterfactual group) pair.

  * (e)

A criterion should include a mechanism for generating associated standard
errors, thus enabling statistical inference.

  * (f)

A criterion should be easily adaptable to intersectionality analyses.

  * (g)

A criterion should properly account for covariates that may be related both to
the outcome of intereset and to status as a member of a protected group.

Concerning that last point, let $X$ and $S$ denote our feature vector and
indicator of protected status, respectively. Let $Y$ be the outcome of
interest, and let $\widehat{Y}$ denote the predicted value of $Y$, given $X$.
Ideally we wish that $\widehat{Y}$ is independent of $S$, or nearly so. This
goal is obstructed by the fact that there may be features in $X$ that are
correlated with $S$, so that the model makes some use of $S$ after all. Let’s
call this the X,S correlation problem.

###  1.3 Organization of This Paper

As seen in the paper’s title and abstract, the proposed method is inspired by
the old empathetic adage, “Walk a mile in their shoes,” which asks that one
imagine the difficulties others may face. Accordingly the proposed fairness
criterion will be called Walk a Mile (WaM).

WaM will be motivated in Section 2, then described in detail in Section 5. In
Section 6, we illustrate the criterion on various real datasets. Section 7
will be devoted to mathematical derivations and empirical methods to assess
the statistical accuracy of WaM. Extensions of WaM, such as for false-positive
rates, will be discussed in Section 8. Finally, Section 10 consists of
conclusions and discussion of future work.

##  2 Introducing WaM

WaM takes the counterfactual view of a protected group as a whole. As a quick
example, consider some data from the 2000 US census (similar to the Adult
dataset, another common example in fair ML studies). WaM would yield
statements like,

> Hey, you men out there! Your mean income in this study was $63554.04. But if
> you were women of the same occupation, age and education, your mean would be
> only $53829.78.

Or, in COMPAS, an algorithm for assisting judges in sentencing of defendants:

> Take note, Caucasian defendants! Your mean decile risk score is 3.67.
> (Smaller values are better.) But if you were Black while having the same
> characteristics as yours, your mean would be 4.19.

This directly addresses the X,S correlation problem, in a novel way. Unlike
methods that attempt to remove the influence of $S$ in $X$, we adjust for that
influence.

In the employment example above, in which $S$ is gender, the occupation
variable in $X$ is strongly related to $S$. This variable codes for one of six
computer-professional occupations in the study, and is strongly gendered:

| A | B | C | D | E | F
---|---|---|---|---|---|---
men | 0.20 | 0.22 | 0.34 | 0.02 | 0.04 | 0.17
women | 0.31 | 0.23 | 0.33 | 0.04 | 0.03 | 0.06

Men are only 2/3 as likely as women to be in occupation A, while the male rate
for occupation F is nearly triple that of women.

And of course there is considerable variation in pay across occupations in the
study:

| A | B | C | D | E | F
---|---|---|---|---|---|---
mean $ | 50396 | 51374 | 53640 | 67019 | 68798 | 69494

WaM adjusts for this by equalizing the distribution of $X$: How would men fare
with their distribution of $X$ if they were women?

##  3 Why WaM?

Other than the desideratum regarding standard errors, most of the multitude of
existing fairness criteria satisfy at least several of the desiderata listed
above. Yet, not all the desiderata are important in all application settings.
With that in mind, let us discuss some of the desiderata,

###  3.1 Use of Conditional Quantities

Desideratum (c) states,

> Conditional distributions can be confusing. To the degree possible, fairness
> criteria that do not explicitly involve conditional quantities are
> preferred.

This one may be surprising, as a number of fairness criteria are based on
explicit conditional quantites, such as the true positive rate (TPR,
conditional probability) and regression functions (conditional mean) [3].

Many such criteria are used successfully in practice, such as in [19]. There a
credit-equity measure is proposed,

$SP=\frac{1}{2}|(FPR_{x_{\alpha}=1}-FPR_{x_{\alpha}=0})+(FNR_{x_{\alpha}=1}-FNR_{x_{\alpha}=0})|$
(1)

This may be effective in a corporate internal introspective document, but
difficult to comprehend by most jurors,

It is well known that conditional quantities are subject to frequent
misunderstanding [30]. Such confusion is not limited to classroom settings,
and indeed arises often in ordinary discourse. The authors in [20] cite a
number of misleading newspaper headlines, as well as court caess. After the
Covid-19 pandemic emerged, for instance, it was common to see news article
with titles such as “Coronavirus Testing: What Is a False Positive rate?”
([25]), and there was much confusion over the efficacy of policy ([27]).

Careful presentation, say in describing data to a jury, may succeed in
ameliorating such problems, but it is desirable to have available criteria
that less explicitly cite conditional quantities. Though WaM technically does
involve conditional quantities, it does so in an implicit manner that is
easier for non-STEM people to grasp.

It should be noted in passing that if a fairness analysis involves ranks,
other major problems arise [22], as two individuals may have roughly equal
traits yet be far apart in rankings.

###  3.2 Dimensionality of the Fairness Criterion

Desideratum (d) says,

> Again in the interest of simplicity, a criterion should not consist of many
> numbers; it should consist of a single number for each protected (base
> group, counterfactual group) pair.

Many methods involve setting up multiple scenarios of real or hypothetical
individual cases, with a typical scenario being say, “Consder a 36-year-old
single woman applying for credit, with a current bank balance of $1200, a
yearly income of $63,250” and so on. Again, for an internal, introspective
corporate analysis, this finely detailed approach may be informative, indeed
necessary. But in many settings, again such as litigation, it is best to keep
to a single summary number. WaM satisfies this crtierion that our fairness
measure consists of a single number, as opposed to measures involving
pointwise counterfactual fairness, such as [21].

Computing a single summary number is especially useful in situations in which
multiple policies, multiple ML prediction methods and so on are being compared
over multiple protected groups. If possible, having just a single number for
each case in the comparison is desirable.

On the other hand, however, note Einstein’s famous quote, “Everything should
be made as simple as possible, but not simpler.” Some settings are
fundamentally complex, and it will be seen below that WaM can help uncover
more complex fairness problems.

###  3.3 Ability of the Consumer (e.g. Juror) to Empathize

It is well known in legal circles that empathy is crucial in judges and
jurors. As pointed out in [23], this is especially important when, for
instance, jurors and a defendant are of a different race. Presumably this
holds for gender as well.

The proposed method’s name, Walk a Mile, is motivated by empathetic aims,
arising from the saying, “Walk a mile in their shoes.” [28] recounts the
history of this phrase, which goes back at least as early as the Cherokee
tribe of Native Americans.

WaM is aimed at capturing this spirit, helping the intended recipient of the
presentation, say a jury, relate emotionally. The hypothetical example given
earlier in this paper,

> Hey, you men out there! Your mean income in this study was $63554.04. But if
> you were women of the same occupation, age and education, your mean would be
> only $53829.78.

should be something that strikes a chord with either men or women. Men, for
instance, may be somewhat detached when presented with evidence of
discrimination against women, but may better relate to comparisons such as the
above, which bring in their own gender on a groupwide level.

###  3.4 Amenability to Intersectionality Issues

The analyst may find that accounting for intersectional issues may complicate
implementation for some types of fairness criteria. But intersectionality
flows naturally from the definition of WaM, with no special extensions or
special cases required. An example will be presented shortly.

###  3.5 Further Comments

It is not claimed here that WaM uniquely satisfies the desiderata (again,
ignoring the standard errors issue). Yet, different application settings may
have different goals, and the novel method presented here should be very
effective in a number of settings. The WaM approach to measuring fairness
should have simple, immediate relevance in many discussion venues, such as
policymaking, litigation, social science research and so on. It thus should be
part of any analyst’s toolkit.

##  4 Notation and Background

As above, let $X$ be a vector of features, such as credit history and jobs,
and let $S$ indicate membership in a sensitive group, such as a racial
minority.

Let $Y$ denote the score value of interest—say a credit rating or a sentence
given to a convicted criminal. Or, the “score” may be a direct outcome, such
as income in our example above. Note that though one could distinguish between
an outcome $Y$ and a decision $D$ as in [26], this is not done here, in the
interests of simplicity; indeed, for convenience let us refer to $Y$ as a
“policy,” whether a formal one such as the decile risk score, or a virtual one
imposed by society such as via discriminatory wages. For binary $Y$, we assume
coding as a dummy/one-hot variable, coded 0,1.

Note that while $S$ will typically be a scalar, such as gender or race, it
could be an intersectional vector, consisting of gender and race. In the
vector case, assume for notational convenience that $S$ has been recoded to
scalar form, in which $S$ is categorical, with $s$ values. If $S$ is
continuous, such as age, assume that it has been discretized to $s$ values.

Though WaM may remind some of covariate shift methods, such as [29], [40], its
goals and operation are actually opposite to those methods. They are concerned
with adapting a fitted ML model in the face of future distributional shifts in
data to be predicted. In WaM, we utilize the fact that the distribution of
$X|S$ varies with $S$, as a means of measuring unfairness..

One more point to get started: Let $U$ and $V$ be any random variables for
which the quantities below exist. Here and below we will be using the Law of
Total Expectation,

$EV=E[E(V|U)]$ (2)

and the Law of Total Variance (“Pythagorean Theorem” for variance) [31],

$Var(V)=E[Var(V|U)]+Var[E(V|U)]$ (3)

##  5 Details of the Method

Now, in the context of the probabilistic generative process of our data, let
$X_{i},~{}i=1,...,s$ denote a random vector of length $p$ having the
distribution of $X|S=i$. Then define

$\mu_{i}(t)=E(Y|X_{i}=t),~{}i=1,...,s$ (4)

Here are the quantities that WaM estimates,

$\nu_{ij}=E[\mu_{j}(X_{i})],~{}i,j=1,...,s$ (5)

Then $\nu_{ij}$ is the mean outcome that members of group $i$ would have if
they were subjected to the policies used for group $j$.

In the case $i=j$, we simply have

$\nu_{ii}=E[\mu_{i}(X_{i})]=E[(E(Y|X)~{}|~{}S_{i}]=E(Y|S=i)$ (6)

using the Tower Theorem for conditional expectations ([18]). That is,
$\nu_{ii}$ is the mean (non-counterfactual) outcome in group $i$.

These quantities $\nu_{ij}$ must be estimated from our data. Let $X_{ik}$
denote the $k^{th}$ realization of $X_{i}$ in group $i$, $k=1,...,n_{i}$, and
let $Y_{ik}$ denote the associated $Y$ value. In other words, considering only
the rows $R_{i}$ of our training data for which $S=i$, then $X_{ik}$ is the
value of $X$ in the $k^{th}$ such row, and $Y_{ik}$ is the value of $Y$ in
that row. Let $N_{i}$ denote the number of such rows, with $n$ rows in all.

We fit one’s favorite predictive algorithm, say linear regression or random
forests, taking the training set to be $R_{i}$, yielding the estimated
regression function for group $i$, $\widehat{\mu}_{i}(t)$. We can then form
the counterfactual outcomes at the individual level,

$\widehat{Y}_{jk}=\widehat{\mu}_{i}(X_{jk})$ (7)

Our WaM estimates are then

$\widehat{\nu}_{ij}=\sum_{k=1}^{N_{i}}\widehat{\mu}_{j}(X_{ik})$ (8)

For the purpose of forming $\widehat{\mu}_{i}(t)$., linear or logistic
regression, k-nearest neighbors, gradient boosting,and random forests all work
in a straightforward manner. Neural networks are also fine for continuous $Y$;
in the case of categorical $Y$, the final-stage activation function must be
chosen appropriately to produce class probabilities.

SVM also is fine for the categorical $Y$ case, provided calibration methods
are used on the output scores [13]. For continuous $Y$, various forms of
support vector regression are available, even for advanced forms such as
quantile regression [15].

It is important to note that unlike many ML fairness criteria, WaM has no
problem with unbalanced data, since $\widehat{\mu}_{i}(t)$ is estimated
separately for each $i$. However, one does of course need the generative
process underlying $(X_{ik},Y_{ik})$ to be that of $E(Y|X_{i})$.

Other than of course requiring that the expectations exist, there are really
no statistical assumptions underlying WaM.

##  6 Empirical Evaluation

Here we illustrate WaM via various datasets, using both parametric regression
methods, either linear or logistic, and random forests or k-Nearest Neigbhors.
The software is available at https://github.com/matloff/WAMfair.

The examples are presented solely for illustrating the workings of the method.
No feature engineering is performed, and default values are used for
hyperparameters.

###  6.1 Census Data

This is a dataset from the 2000 US Census, focused on Silicon Valley.
Variables are age, education, occupation (among six programmer and engineer
job categories), gender, wage income, and number of weeks worked.

First, a look at gender (example from above), with a linear model:

| men cf.. | women cf.
---|---|---
men, act. | 63554.04 | 53829.78
women, act. | 58646.87 | 50403.48

We use “act.” and “cf.” to denote the actual and counterfactual cases. Women
as a group had a mean wage of about $50,403, but if they had been men with the
same characteristics, their mean would have been $58,647.

The random forest analysis is similar:

| men cf. | women cf.
---|---|---
men act. | 63554.04 | 53728.65
women act. | 59464.30 | 50403.48

(The diagonal elements do not change in this second analysis, due to (6).)

###  6.2 Employment Race Test Data

This data is from [5]. Re’sume’s were sent to random employers, with the same
qualifications but some with “Black-sounding” given names, such as Lakisha.
$Y$ is an indicator variable recording whether the employer called the
applicant after receiving the re’sume’. A number of variables were used, such
as education, years of experience, military background and so on.

Only the variables ”education”, ”ofjobs”, ”yearsexp”, ”honors”, ”volunteer”,
”military”, ”empholes”, ”workinschool” , ”email”, ”computerskills”, ”race”,
”call” and ”adid” were used.

Here is the random forests analysis:

act | bcf | wcf
---|---|---
b | 0.06 | 0.05
w | 0.03 | 0.10

We see that “Caucasian” applicants had a 10% callback rate, but would have had
only a 3% rate if they had been Black with the same qualifications.
Interestingly, though “Black” applicants did not seem to benefit from having
“White” names.

As noted, WaM automatically includes intersectional cases, no extension or
special cases involved. Again with random forests:

| fbcf | mbcf | fwcf | mwcf
---|---|---|---|---
fbact | 0.07 | 0.06 | 0.05 | 0.10
mbact | 0.03 | 0.10 | 0.05 | 0.11
fwact | 0.02 | 0.05 | 0.06 | 0.04
mwact | 0.02 | 0.05 | 0.03 | 0.09

Here ’fb’ means female Black, ’mb’ means male Black, and so on. So, Black
women would get a boost from being White men, with their actual 7% callback
rate increasing to 10%, but not, for instance, from being Black men. The
latter, on the other hand, would benefit by being White men, but not much by
being White women.

The latter result, of course, may depend on the job type, and a more segmented
analysis is needed, together with domain expertise, but WaM does seem to have
uncovered some interesting, complex relationships.

###  6.3 COMPAS

We used the variables ’age’, ’juv_fel_count’, ’decile_score’,
’juv_misd_count’, ’juv_other_count’, ’priors_count’, ’sex’ and ’race’. Due to
insufficient data, the Asian and Native American instances were changed to
Other.

Using a linear regression model to predict decile score, we have

| African-American.cf | Caucasian.cf | Hispanic.cf | Other.cf
---|---|---|---|---
African-American.act | 5.28 | 4.78 | 4.41 | 4.13
Caucasian.act | 4.19 | 3.67 | 3.20 | 2.94
Hispanic.act | 4.36 | 3.77 | 3.32 | 3.07
Other.act | 4.36 | 3.76 | 3.31 | 3.05

We showed two of these numbers in Section 1 above, with possible implications
that the COMPAS system rates Black defendants unfairly highly. Various other
numbers in the table suggest that as well.

The random forest analysis is similar:

| African-American.cf | Caucasian.cf | Hispanic.cf | Other.cf
---|---|---|---|---
African-American.act | 5.28 | 4.73 | 4.28 | 4.16
Caucasian.act | 4.15 | 3.67 | 3.20 | 3.00
Hispanic.act | 4.27 | 3.74 | 3.32 | 3.07
Other.act | 4.26 | 3.70 | 3.22 | 3.05

What about gender?

| Female | Male
---|---|---
Female | 4.05 | 3.60
Male | 5.04 | 4.50

The COMPAS algorithm seems to have a substantial bias against women. Their
mean decile score was 4.05, but would have been only 3.60 had they been men
with the same characteristics.

###  6.4 German Credit Data

This is another commonly used example in ML fairness research, though one of
the aspects considered below is less common. Let’s start with the standard
aspect, investigating possible gender bias. Using a logistic model for a
‘Good’ credit rating, we have

| female.cf | male.cf
---|---|---
female.act | 0.72 | 0.64
male.act | 0.70 | 0.65

Women in general had a 72% rate of being assigned a Good rating, but if they
were men with the same characteristics, only 64% would get this score. There
may be bias against men here. The results using random forests are similar:

| female | male
---|---|---
female | 0.72 | 0.66
male | 0.72 | 0.65

Let’s take a look at (discretized) age, using random forests:

| (18.9,37.7].cf | (37.7,56.3].cf | (56.3,75.1].cf
---|---|---|---
(18.9,37.7].act | 0.68 | 0.75 | 0.70
(37.7,56.3].act | 0.74 | 0.75 | 0.71
(56.3,75.1].act | 0.68 | 0.74 | 0.72

Possibly a slight bias in favor of the middle-aged consumers, less favorable
to the young and old.

Is there discrimination against foreign workers in the credit realm? Again,
running an analysis with random forests, we have

| not for.cf | for.cf
---|---|---
not for.act | 0.89 | 0.74
for.act | 0.84 | 0.69

About 69% of foreign workers have Good credit, but if they were citizens with
the same characteristics, 84% would be judged good credit risks. So, there may
indeed be some discrimination here.

##  7 Assessment of Sampling Variance

Claims of discrimination are sensitive. As such, there must be some indication
of the accuracy of the fairness criterion used, in the form of statistical
standard errors (“error bars”). This is vital in academia, the corporate world
and especially in litigation, where, as previously noted, case law requires
it.

Let’s recall the notation for our data: Define $Y$, $X$ and $X_{i}$ as before.
Denote the $k^{th}$ data point in group $i$ by $X_{ik}$ and $Y_{ik}$,
respectively. Data points are i.i.d. across $k$ for fixed $i$, and independent
across $i$. Let $n_{i}$ denote the number of data points in group $i$. Vectors
are written as columns.

###  7.1 Exact Expression for Variance in the Linear Case

For notational convenience, it is assumed here that all data has been
centered, thus having mean 0. This eliminates the intercept term for the
coefficent vector, and the need to have a 1s column in the design matrix.

Here we have

$E(Y|X_{i})=\beta_{i}^{T}{X}_{i}$ (9)

where $\beta_{i}=(\beta_{i1},\beta_{i2},...,\beta_{ip})^{T}$ and T denotes
matrix transpose.

In this setting,

$\nu_{ij}=E(\beta_{j}^{T}{X}_{i})=\beta_{j}^{T}E({X}_{i})$ (10)

Using OLS, we obtain an unbiased linear estimate $\widehat{\beta_{j}}$ of
$\beta_{j}$. Denote its covariance matrix by $Cov(\widehat{\beta_{j}})$
(computed via the standard formula and available for instance in R via the
vcov() function). We are not assuming normal $Y_{ij}$, but under our i.i.d.
conditions $\widehat{\beta_{j}}$ will be asymptotically normal.

Then our estimated $\nu_{ij}$ is

$\widehat{\nu}_{ij}=\widehat{\beta}_{j}^{~{}T}{A}_{i}$ (11)

where our sample estimate of $E{X}_{i}$ is

${A}_{i}=\frac{1}{n_{i}}\sum_{k=1}^{n_{i}}{X}_{ik}$ (12)

Then conditioning on $A_{i}$ and using (3), we have

$\displaystyle Var\left(\widehat{\beta_{j}}^{T}{A}_{i}\right)$
$\displaystyle=$ $\displaystyle Var\left(A_{i}^{T}\widehat{\beta}_{j}\right)$
(13) $\displaystyle=$ $\displaystyle
E\left[Var(A_{i}^{T}\widehat{\beta}_{j}|A_{i})\right]+Var\left[E(A_{i}^{T}\widehat{\beta}_{j})|A_{i}\right]$
(14) $\displaystyle=$ $\displaystyle
E\left[A_{i}^{T}Cov(\widehat{\beta}_{j})A_{i}]\right]+Var\left[A_{i}^{T}\beta_{j}\right]$
(15) $\displaystyle=$ $\displaystyle
E\left[A_{i}^{T}Cov(\widehat{\beta}_{j})A_{i}]\right]+\beta_{j}^{T}Cov(A_{j})\beta_{j}$
(16) $\displaystyle=$ $\displaystyle
E\left[A_{i}^{T}Cov(\widehat{\beta}_{j})A_{i}]\right]+\frac{1}{n_{i}}\beta_{j}^{T}Cov(X_{i})\beta_{j}$
(17)

where $Cov(A_{i})$ and $Cov(X_{i})$ are the covariance matrices of $A_{i}$ and
$X_{i}$, respectively. Hence we have the exact expression for variance of WaM.

These quantities can then easily be estimated directly from their sample
analogs. In fact, our estimate for (17) is simply

${A}_{i}^{T}\widehat{Cov}(\widehat{\beta}_{j})A_{i}]+\frac{1}{n_{i}}\widehat{\beta}_{j}^{T}\widehat{Cov}(X_{i})\widehat{\beta}_{j}$
(18)

where $\widehat{Cov}$ denotes covariance matrices estimated from the data.

Note that in (11) we have the product of a quantity $\widehat{\beta}_{i}$ that
converges in distribution (to a multivariate Gaussian distribution with mean
$\beta_{i}$) and a quantity $A_{j}$ that converges in probability (to
$EX_{i}$). In other words, $\widehat{\nu}_{ij}$ too is asymptotically normal
([18]), with standard error given by the square root of (18), and confidence
intervals can be formed.

###  7.2 The Bootstrap

For nonparametric models, say random forests, one may apply bootstrap methods
([37]). One does resampling of the data many time, generating many values of
$\widehat{\nu}_{ij}$. Our new point estimate is the average of these values,
and the standard error is their standard deviation.222The software computes
only bootstrap results, not the formula derived above for the linear case.

###  7.3 Examples

We earlier broke down the COMPAS data by gender, finding an apparent bias
against women. Here are the standard errors, based on 100 resamplings:

act | cf | myGrpEst | theirGrpEst | bias | biasSE
---|---|---|---|---|---
Female | Male | 4.11 | 3.65 | 0.46 | 0.05
Male | Female | 4.60 | 5.16 | -0.56 | 0.08

So for instance an approximate 95% confidence interval for that 0.46 figure
would be $0.46\pm 1.96\times 0.05$, or about 0.36 to 0.56. The bias is real.

As another example, consider the Employment Race Test Data, Section 6.2. Again
with 100 resamples, the bootstrap results were

act | cf | myGrpEst | myGrpSE | theirGrpEst | theirGrpSE | bias | biasSE
---|---|---|---|---|---|---|---
b | w | 0.06 | 0.01 | 0.06 | 0.01 | 0.00 | 0.01
w | b | 0.10 | 0.01 | 0.04 | 0.00 | 0.06 | 0.01

So there is a genuine difference in callback ratings. If the White applicants
were to apply under “Black” names with the same employment-relevant
characteristics, their chances of being called back by employers would only be
about 40% as high.

##  8 Extensions

The WaM approach can be extended, applying it to various common fairness
criteria. For instance, we could address questions such as:

> Consider loan applications, predicting default, with the criterion false
> positive rate. If Caucasian applicants were to have their present
> qualifications but were Black, what would their false positive rate be?

Assume and the classification rule (if distributions were fully known)

$\widehat{Y}=\begin{cases}1,&\mu_{j}(X_{i})\geq 0.5\\\
0,&\mu_{j}(X_{i})<0.5\\\ \end{cases}$ (19)

Let’s use $\gamma_{ij}$ to denote the false positive rate that those in group
$i$ would experience if they were subjected to the treatment of those in group
$j$. The quantities being estimated are

$\gamma_{ij}=\frac{E\boldsymbol{\\{}1_{[0.5,1]}\boldsymbol{[}\mu_{j}(X_{i})~{}\cdot~{}(1-\mu_{j}(X_{i}))\boldsymbol{]}\boldsymbol{\\}}}{E\left[1_{[0.5,1]}[\mu_{j}(X_{i})]\right]}$
(20)

These are then estimated by their sample analogs.

For instance, consider a Dutch census dataset ([38]). The outcome of interest
is whether a respondent’s occupation is considered prestigious, using various
demographic, economic and household characteristics. Here are the results
using a k-Nearest Neighbors analysis:

| male c.f. | female c.f. | myGrpEst | theirGrpEst | bias | biasSE
---|---|---|---|---|---|---
male | 1 | 2 | 0.19 | 0.22 | -0.03 | 0.01
female | 2 | 1 | 0.21 | 0.22 | -0.01 | 0.01

For example, the FP rate for men was about 19%, but would be 22% if they were
women with the same characteristics.

##  9 Accounting for Model Edge Bias

Fit of one’s regression model is important for WaM. Standard fit assessment
techniques may be used, but here we discuss some issues concerning bias at the
edges of one’s dataset.

To motivate this discussion, think of a simple setting in which we use
k-Nearest Neighbors to predict human weight from height. Consider a data point
in which, say, the person is unusually tall. Then his/her neighbors are likely
to be shorter than this person, thus lighter. In other words, the prediction
on this data point may be biased downward.

The situation with random forests is similar. The leaf in a decision tree into
which this hypothetical tall person falls is also a neighborhood, and again,
the other points in this leaf will likely be shorter/lighter.

But remedies have been developed for both k-NN [8] and random forests [1], in
the form of calculating a linear fit within a neighborhood or leaf. These will
soon be incorporated in our WaM software.

##  10 Conclusions and Future Work

We have introduced a new approach to fairness in machine learning, a group-
counterfactual measure we call WaM. It serves as a simple measure of fairness
that can be easily grasped by nontechnical people, and it is a solution to the
X,S correlation problem.

We believe WaM opens the door to extensive further research. We mention two
possible areas:

  * •

Extension to other fairness and predictive power measures: In Section 8, we
extended WaM to False Positive Rates. There are various other rates that may
be of interest in WaM’s group-counterfactual approach. AUC may also be
extended to WaM.

  * •

As explained earlier, any report of a fairness measure in ML should be
accompanied by a standard error. This is especially important in litigation,
but also vital in any context in which the report will be used in decision
making.

The bootstrap solution we presented here is convenient, but is time consuming.
Moreover, it cannot easily be used in pre-planning, e.g. of sample size,
needed in a study. For such purposes, explicit formulas for estimator variance
such as (17) can be helpful, and worthy of further study. Extension of (17) to
the generalized linear model, notably the logistic and Poisson regression
cases, should be pursued.

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