# DialoGLUE: A Natural Language Understanding Benchmark for Task-Oriented
Dialogue

Shikib Mehri♠ Mihail Eric♢ Dilek Hakkani-Tur♢
♠Language Technologies Institute, Carnegie Mellon University
♢Amazon Alexa AI
<EMAIL_ADDRESS>{mihaeric<EMAIL_ADDRESS>This work was done while
the first author was an intern at Amazon.

###### Abstract

A long-standing goal of task-oriented dialogue research is the ability to
flexibly adapt dialogue models to new domains. To progress research in this
direction, we introduce DialoGLUE (Dialogue Language Understanding
Evaluation), a public benchmark consisting of 7 task-oriented dialogue
datasets covering 4 distinct natural language understanding tasks, designed to
encourage dialogue research in representation-based transfer, domain
adaptation, and sample-efficient task learning. We release several strong
baseline models, demonstrating performance improvements over a vanilla BERT
architecture and state-of-the-art results on 5 out of 7 tasks, by pre-training
on a large open-domain dialogue corpus and task-adaptive self-supervised
training. Through the DialoGLUE benchmark, the baseline methods, and our
evaluation scripts, we hope to facilitate progress towards the goal of
developing more general task-oriented dialogue
models.111https://evalai.cloudcv.org/web/
challenges/challenge-page/708/

##  1 Introduction

One of the ultimate goals of task-oriented conversational systems is the
ability to flexibly bootstrap new dialogue functionalities across diverse
domains of user interest. For example, once we have successfully built a
dialogue assistant that can handle restaurant booking queries, we would
ideally like that assistant to quickly start servicing hotel reservation
queries without too much additional manual effort. Unfortunately in the modern
conversational assistant ecosystem, the work required to scale up
functionalities is often linear with respect to the number of desired domains.

For modelling improvements to claim meaningful progress towards generality,
the improvements must extend beyond a single dataset and instead hold across
several different dialogue tasks and corpora. We argue that one of the core
roadblocks in progressing the generality of conversational systems toward this
desired state is a lack of standardization in both datasets and evaluation
used by the community. These problems have been noted more broadly in the
natural language understanding field, inspiring numerous efforts to propose
unified benchmarks spanning multiple downstream tasks across different corpora
with consolidated evaluation procedures Wang et al. (2018, 2019); McCann et
al. (2018).

While today there is a reasonable abundance of available corpora for building
data-driven dialogue systems Serban et al. (2018), little work has been done
to bring together the efforts of researchers to reflect the properties we care
about in our systems: statistical learning that is data-efficient and robustly
transfers across domains and tasks.

In this work we propose DialoGLUE, a public benchmark consisting of 7 diverse
task-oriented spoken-language datasets across 4 distinct natural language
understanding tasks including intent prediction, slot-filling, semantic
parsing, and dialogue state tracking. Many of these datasets include multiple
system functionalities and in total, the DialoGLUE benchmark covers over 40
different domains. Our benchmark is designed to encourage dialogue research in
representation-based transfer, domain adaptation, and sample-efficient task
learning. Furthermore, it consists entirely of previously-published datasets
that have reported results, as these resources have been vetted by the broader
community to be sufficiently difficult and interesting.

As part of DialoGLUE, we also release evaluation scripts and competitive BERT-
based baselines on the downstream tasks. We introduce ConvBERT, a BERT-base
model which has been trained on a large open-domain dialogue corpus. In
combination with task-adaptive pre-training and multi-tasking, ConvBERT
matches or exceeds state-of-the-art results on five of the seven DialoGLUE
datasets. Most notably, we attain a +2.98 improvement in the joint goal
accuracy over the best dialogue state tracking models on the MultiWOZ corpus.
While our baselines demonstrate the efficacy of task-adaptive finetuning in
transferring across datasets, there is still plenty of headroom in the
aggregated benchmark scores encouraging further research.

In summary, the contributions of this work are four-fold: (1) a challenging
task-oriented dialogue benchmark consisting of 7 distinct datasets across 4
domains, (2) standardized evaluation measures for reporting results (3)
competitive baselines across the tasks, and (4) a public leaderboard for
tracking scientific progress on the benchmark.

The DialoGLUE leaderboard and evaluation scripts are hosted using the EvalAI
platform (Yadav et al., 2019). Our code and baseline models are open-
sourced.222https://github.com/alexa/DialoGLUE/

##  2 Related Work

###  2.1 NLP Benchmarks

In part, the development of unified benchmarks have helped drive progress
towards more general models of language. The GLUE (Wang et al., 2018) and
SuperGLUE benchmarks (Wang et al., 2019) have provided a consistent benchmark
that allows pre-trained language models (Devlin et al., 2018; Radford et al.,
2018) to be evaluated on a variety of tasks.

While GLUE and SuperGLUE are concentrated on language understanding tasks,
decaNLP (McCann et al., 2018) consists of a broader set of NLP tasks including
question answering, machine translation and summarization.

Within conversational systems, the recently proposed Dialogue Dodecathlon is a
benchmark for knowledge-grounded, situated, and multi-modal dialogue
generation consisting of several open-domain dialogue datasets Shuster et al.
(2019). In contrast to the Dialogue Dodecathlon, DialoGLUE focuses on task-
oriented dialogue and language understanding.

DialoGLUE is inspired by GLUE and SuperGLUE, however our efforts differ in
that we aim to produce a benchmark for natural language understanding in the
context of task-oriented dialogue. Task-oriented dialogue poses a unique set
of challenges; dialogue is intrinsically goal-driven, multi-turn and often
informal/noisy (Henderson et al., 2019; Zhang et al., 2019b). Through
DialoGLUE, we hope to provide a benchmark for assessing models that aim to
tackle these unique challenges across several tasks within task-oriented
dialogue.

###  2.2 Pre-trained Models

Large-scale pre-training has attained significant performance gains across
many tasks within NLP (Devlin et al., 2018; Radford et al., 2018). Through
self-supervised pre-training on large natural language corpora, these models
gain generalized language understanding capabilities that transfer effectively
to downstream tasks (Wang et al., 2018), including intent prediction (Chen et
al., 2019; Castellucci et al., 2019) and dialogue state tracking (Heck et al.,
2020). However, recent work has demonstrated that leveraging dialog-specific
pre-trained models, such as ConveRT (Henderson et al., 2019; Casanueva et al.,
2020) obtains better results.

Large-scale pre-training on open-domain dialogue has demonstrated surprising
generality, with models like DialoGPT (Zhang et al., 2019b), Meena (Adiwardana
et al., 2020) and Blender (Roller et al., 2020) achieving response generation
performance competitive with humans in certain settings. ConveRT has
demonstrated that pre-training on open-domain dialogue transfers to task-
oriented dialogue, with significant performance improvements over BERT on both
intent prediction (Casanueva et al., 2020) and slot filling (Coope et al.,
2020). Inspired by these results, we fine-tune BERT (Devlin et al., 2018) on a
large open-domain dialogue corpus prior to fine-tuning on the downstream tasks
of DialoGLUE.

###  2.3 Task-Adaptive Training

Though large-scale pre-training results in strong performance when
transferring to downstream tasks, performing self-supervised training on a
target dataset allows the model to better adapt to the dataset prior to fine-
tuning (Mehri et al., 2019; Gururangan et al., 2020). Since our target
datasets differ from the pre-training data, we hypothesize that task-adaptive
training will allow pre-trained models to better adapt to task-oriented
dialogue.

##  3 DialoGLUE Tasks

The DialoGLUE benchmark consists of 7 different datasets spanning 4 different
tasks: intent prediction, slot filling, semantic parsing and dialogue state
tracking. These datasets all share the common goal of language understanding
in the context of dialogue, making them suitable for DialoGLUE.

###  3.1 Intent Prediction

Intent prediction is the task of extracting meaning from a natural language
utterance in order to understand the user’s goal (Hemphill et al., 1990;
Coucke et al., 2018). We leverage three different datasets for the task of
intent prediction, all of which span several domains and consist of many
different intents.

banking77 (Casanueva et al., 2020) contains 13,083 utterances related to
banking with 77 different fine-grained intents. Despite only consisting of a
single domain, this dataset is challenging as it requires fine-grained
differentiation betweeen very similar intents like card payment wrong exchange
rate and wrong exchange rate for cash withdrawal.

clinc150 (Larson et al., 2019) contains 23,700 utterances spanning 10 domains
(e.g., travel, kitchen/dining, utility, small talk, etc.) and 150 different
intent classes. This dataset also consists of out of scope utterances, which
do not belong to any of the other intents and must therefore be classified as
out of scope.

hwu64 (Liu et al., 2019) includes 25,716 utterances for 64 intents spanning 21
domains (e.g., alarm, music, IoT, news, calendar, etc.). The domains and
intents of this dataset are similar to ones we expect users to ask a virtual
assistant (e.g., Alexa, Google Assistant, Siri).

Casanueva et al. (2020) forego a validation set when using these datasets for
intent prediction and instead only use a training and testing set. We instead
designate a portion of the training set to be the validation set.

###  3.2 Slot Filling

Slot filling, a vital component of task-oriented dialogue systems, is the task
of identifying values for pre-defined attributes in a natural language
utterance. Following the set up of Coope et al. (2020), we include two slot
filling datasets in DialoGLUE.

restaurant8k (Coope et al., 2020) comprises of 8,198 utterances from a
commercial restaurant booking system and consists of 5 slots (date, time,
people, first name, last name).

DSTC8 (Rastogi et al., 2019) consists of slot annotations extracted by Coope
et al. (2020) spanning 4 domains (buses, events, homes, rental cars) for a
total of 5,569 utterances.

For both these datasets, the value for a particular slot will always be a
contiguous span of the utterance. For some utterances, the expected slot
(e.g., “people”, “time”) is provided. This allows an otherwise ambiguous
utterance like “four” to be interpreted either as “four people” or “four
o’clock”.

###  3.3 Semantic Parsing

TOP (Gupta et al., 2018) is a dataset of 44k utterances wherein each utterance
is annotated with a hierarchical semantic representation. Each utterance has a
top-level intent, which serves as the root of the tree. Every word of the
utterance is a leaf node of the tree. The sub-trees correspond to different
slots and intents, with the dataset having an average tree depth of 2.54. TOP
covers two different domains, navigation (directions, distance, traffic) and
events.

###  3.4 Dialogue State Tracking

MultiWOZ (Budzianowski et al., 2018; Eric et al., 2019) is a multi-domain
dialogue dataset that contains 7 domains including restaurants, hotels and
attractions. We use MultiWOZ for dialogue state tracking, the task of
interpreting the user utterances throughout a dialogue in order to maintain a
state representation of their requests. Dialogue state tracking is an
important component of task-oriented dialogue systems; in order to fulfill a
user request, it is necessary to track the user’s goals over the course of
multiple turns.

##  4 Methods

In this section, we describe several methods employed for the DialoGLUE tasks.
We begin by describing the architectures for the 4 different tasks, all of
which are built around an underlying BERT-like model. We then discuss
ConvBERT, a BERT model that was fine-tuned on a large open-domain dialogue
corpus. Finally, we describe our use of task-adaptive masked language
modelling, which allows us to better adapt our pre-trained models to the
DialoGLUE tasks.

###  4.1 Architectures

Intent Prediction: We fine-tune a BERT model to encode an utterance and
predict its intent. Specifically, we use the pooled representation output by a
BERT model and pass it through a linear layer to predict the intent class.

Slot Filling: For slot filling, we represent the problem as IOB tagging
(Ramshaw and Marcus, 1999) wherein every token in the utterance is labeled as
either being the beginning of a slot value (B-), inside a slot value (I-) or
not belonging to a slot value (O). We use BERT to produce a latent
representation of each token, which is passed through a linear layer that
predicts the appropriate tag (e.g., “B-time”, “I-people”).

Semantic Parsing: We transform the hierarchical representations of the TOP
dataset into (i) a top-level intent for the utterance which corresponds to the
root of the tree and (ii) a label for each word of the utterance which is the
path from the root to each leaf node (which is always a word). Given this, we
train a model to simultaneously predict the top-level intent for the utterance
using the BERT pooled representation and the labels for each word using BERT’s
latent representation of each word.

Dialogue State Tracking: Our state tracking architecture is inspired by the
TripPy of Heck et al. (2020) which uses an underlying BERT model and a triple
copy strategy to perform state tracking. The TripPy model uses (i) span
prediction and a copy mechanism to extract values from a user utterance, (ii)
a copy mechanism over concepts mentioned by the system utterance and (iii) a
copy mechanism over the DS memory, the existing dialogue state.

These architectures are held consistent throughout our experiments. Given a
BERT-like encoder, we can plug it into all of the aforementioned architectures
and evaluate its performance on all of the DialoGLUE tasks. This allows us to
assess the quality of the underlying encoders, with confidence that
performance improvements come from the improved representational power of the
encoders.

###  4.2 ConvBERT

Though pre-trained models (e.g., BERT) have exhibited strong language
understanding capabilities, recent work has suggested that they may be
insufficient for modelling dialogue, due to dialogue’s intrinsically goal-
driven, linguistically diverse, multi-turn and often informal/noisy nature
(Henderson et al., 2019; Zhang et al., 2019b). The unique challenges of
modelling dialogue have been addressed by training on large amounts of
conversational data, from online forums. We extend these efforts by fine-
tuning BERT on a large open-domain dialogue corpus consisting of nearly 700
million conversations to produce ConvBERT.

By training ConvBERT with large amounts of open-domain dialogue, we
hypothesize that it is better able to produce semantically meaningful latent
representations of utterances and multi-turn dialogues than a BERT model.
Specifically, we fine-tune an uncased BERT-base for 4 epochs using a masked
language modelling objective. Here our input representation is the last 3
turns of dialogue context followed by the [SEP] token and then the dialogue
response. We truncate the entire input to have sequence length 72, and we
train using the Adam optimizer with an initial learning rate of $3\cdot
10^{-4}$.

###  4.3 Task-Adaptive Training

Task-adaptive training is the process of adapting a pre-trained model to a
specific task or domain, by performing self-supervised training prior to fine-
tuning on the downstream task. This has been shown to help with domain
adaptation (Mehri et al., 2019; Gururangan et al., 2020). To adapt BERT-based
encoders to the various DialoGLUE tasks, we leverage task-adaptive training.
Specifically, we perform self-supervised training with the masked language
modelling (MLM) objective on each dataset. We explore both (i) pre-training
with MLM prior to fine-tuning on the specific task and (ii) multi-tasking by
simultaneously performing self-supervised MLM and fine-tuning on the task. An
example experimental setting is as follows: (1) start with the pre-trained
ConvBERT model, (2) do MLM pre-training on the utterances of HWU, (3) fine-
tune on HWU using the intent prediction architecture and simultaneously
perform self-supervised MLM training on the utterances of HWU.

To further study the benefits of self-supervised training, we fine-tune both
BERT and ConvBERT with masked language modelling over all the DialoGLUE
datasets. In this manner we aim to adapt the two pre-trained models to task-
oriented dialogue through self-supervised training.

##  5 Experiments

Model | Average | banking77 | hwu64 | clinc150 | restaurant8k | DSTC8 | TOP | MultiWOZ
---|---|---|---|---|---|---|---|---
BERT | 86.08 | 93.02 | 89.87 | 95.93 | 95.53 | 90.05 | 81.90 | 56.30
\+ Pre | 86.18 | 92.34 | 91.82 | 96.27 | 95.78 | 89.48 | 81.54 | 56.07
\+ Multi | 85.97 | 92.27 | 90.99 | 96.22 | 95.61 | 89.93 | 81.46 | 55.30
\+ Pre, Multi | 85.92 | 93.20 | 90.99 | 95.67 | 95.04 | 89.96 | 82.08 | 55.06
ConvBERT | 86.01 | 92.95 | 90.43 | 97.07 | 95.90 | 87.58 | 82.13 | 56.00
\+ Pre | 86.19 | 93.25 | 92.84 | 97.09 | 95.33 | 87.02 | 82.00 | 55.67
\+ Multi | 85.97 | 93.20 | 91.36 | 97.09 | 95.39 | 90.02 | 82.63 | 56.48
\+ Pre, Multi | 86.89 | 93.44 | 92.38 | 97.11 | 95.44 | 91.20 | 82.08 | 56.56
BERT-DG | 86.11 | 91.75 | 90.89 | 95.98 | 95.23 | 90.24 | 81.16 | 57.54
\+ Pre | 86.16 | 92.01 | 91.26 | 96.20 | 94.61 | 89.79 | 81.29 | 58.00
\+ Multi | 86.38 | 92.53 | 90.61 | 95.89 | 95.44 | 90.81 | 81.04 | 58.34
\+ Pre, Multi | 86.18 | 92.57 | 91.26 | 96.22 | 95.11 | 88.69 | 80.89 | 58.53
ConvBERT-DG | 82.9 | 93.21 | 91.64 | 96.96 | 93.44 | 74.54 | 72.22 | 58.57
\+ Pre | 84.1 | 93.05 | 92.94 | 97.11 | 95.38 | 90.88 | 60.68 | 58.65
\+ Multi | 82.78 | 93.02 | 91.73 | 97.13 | 95.93 | 88.97 | 53.97 | 58.70
\+ Pre, Multi | 85.34 | 92.99 | 91.82 | 97.11 | 94.34 | 86.49 | 76.36 | 58.29
Table 1: Full data experiments on DialoGLUE. The average score on the
DialoGLUE benchmark is shown in the leftmost column.

###  5.1 Experimental Setup

The experiments are carried out with four BERT-like models: (1) BERT-base, (2)
ConvBERT which is BERT trained on open-domain dialogues, (3) BERT-DG which is
BERT trained on the full DialoGLUE data in a self-supervised manner with
masked language modelling and (4) ConvBERT-DG which is ConvBERT trained on the
full DialoGLUE data in a self-supervised manner.

We carry out experiments with each of these four models in four different
settings: (1) directly fine-tuning on the target task, (2) pre-training with
MLM on the target dataset prior to fine-tuning, (3) multi-tasking with MLM on
the target dataset during fine-tuning and (4) both pre-training and multi-
tasking with MLM.

We perform self-supervised MLM pre-training for 3 epochs prior to fine-tuning.
Fine-tuning on a target task is carried out until the performance on the
validation set does not improve for 10 epochs. When multi-tasking, we
alternate between fine-tuning on the target task and self-supervised training
with MLM after every epoch.

To assess the effectiveness of our pre-trained models for few-shot learning,
we carry out few-shot experiments. In such experiments, self-supervised
training is performed only on the few-shot data, which is 10% of the full
data. The MLM pre-training and multi-tasking is performed with only the few-
shot versions of each dataset. However, both BERT-DG and ConvBERT-DG are
trained with the full DialoGLUE data, albeit in a self-supervised manner,
meaning that they see more in-domain data than either BERT or ConvBERT in the
few-shot experiments. For all the few-shot experiments, we train five times
with different random seeds and report the average performance across the five
runs.

banking77 (accuracy) ✓
---
USE (Casanueva et al., 2020) | 92.81
ConveRT (Casanueva et al., 2020) | 93.01
USE + ConveRT (Casanueva et al., 2020) | 93.36
ConvBERT \+ Pre + Multi | 93.44
hwu64 (accuracy) ✓
USE (Casanueva et al., 2020) | 91.25
ConveRT (Casanueva et al., 2020) | 91.24
USE + ConveRT (Casanueva et al., 2020) | 92.62
ConvBERT-DG \+ Pre | 92.94
clinc150 (accuracy) ✓
USE (Casanueva et al., 2020) | 95.06
ConveRT (Casanueva et al., 2020) | 97.16
USE + ConveRT (Casanueva et al., 2020) | 97.16
ConvBERT-DG \+ Multi | 97.13
restaurant8k (F-1) ✓
Span-BERT (Coope et al., 2020) | 93.00
V-CNN-CRF (Coope et al., 2020) | 94.00
Span-ConveRT (Coope et al., 2020) | 96.00
ConvBERT-DG \+ Multi | 95.93
DSTC8 (F-1)
Span-BERT (Coope et al., 2020) | 91.50
V-CNN-CRF (Coope et al., 2020) | 91.25
Span-ConveRT (Coope et al., 2020) | 94.00
ConvBERT \+ Pre + Multi | 91.20
TOP (Exact Match)
RNNG (Gupta et al., 2018) | 78.51
SR + ELMo (Einolghozati et al., 2019) | 87.25
Seq2Seq-PTR (Rongali et al., 2020) | 86.67
ConvBERT \+ Multi | 82.63
MultiWOZ (Joint Goal Accuracy) ✓
DST-Picklist (Zhang et al., 2019a) | 53.30
TripPy (Heck et al., 2020) | 55.30
SimpleTOD (Hosseini-Asl et al., 2020) | 55.72
ConvBERT-DG \+ Multi | 58.70
Table 2: Comparison to prior work on all seven datasets. We match or exceed
state-of-the-art results on five out of seven datasets (marked with
checkmarks), with significant improvements (+3) on the MultiWOZ corpus.

###  5.2 Evaluation

Our evaluation metrics are consistent with prior work on these datasets. For
intent prediction (banking77, clinc150, hwu64) we use accuracy. For the slot
filling tasks (restaurant8k, DSTC8), we use macro-averaged F-1 score as
defined by Coope et al. (2020). We use exact-match for TOP, which measures how
often we exactly reconstruct the hierarchical semantic representation. For
MultiWOZ, we use joint goal accuracy.

Model | Average | banking77 | hwu64 | clinc150 | restaurant8k | DSTC8 | TOP | MultiWOZ
---|---|---|---|---|---|---|---|---
BERT | 66.07 | 79.87 | 81.69 | 89.52 | 87.28 | 45.05 | 74.38 | 4.69
\+ Pre | 66.57 | 80.72 | 83.05 | 89.73 | 86.37 | 47.17 | 74.41 | 4.55
\+ Multi | 66.11 | 79.89 | 82.32 | 89.69 | 87.53 | 44.92 | 74.45 | 3.95
\+ Pre, Multi | 66.87 | 81.49 | 82.70 | 90.53 | 86.34 | 48.55 | 74.17 | 4.29
ConvBERT | 68.03 | 83.63 | 83.77 | 92.10 | 86.90 | 49.08 | 74.86 | 5.90
\+ Pre | 67.36 | 83.68 | 83.77 | 92.10 | 86.90 | 45.20 | 74.92 | 5.09
\+ Multi | 68.16 | 83.15 | 82.32 | 92.33 | 86.71 | 50.49 | 75.21 | 5.48
\+ Pre, Multi | 68.22 | 83.99 | 84.52 | 92.75 | 86.17 | 48.40 | 78.84 | 6.87
BERT-DG | 72.70 | 81.47 | 83.23 | 90.57 | 85.31 | 43.85 | 74.80 | 49.70
\+ Pre | 72.80 | 81.79 | 83.74 | 90.44 | 86.66 | 43.45 | 74.34 | 49.40
\+ Multi | 73.00 | 81.60 | 83.18 | 90.43 | 86.48 | 44.86 | 74.79 | 49.67
\+ Pre, Multi | 72.90 | 81.08 | 83.40 | 90.09 | 86.26 | 46.32 | 73.56 | 49.86
ConvBERT-DG | 73.75 | 84.42 | 85.17 | 92.87 | 87.65 | 41.94 | 75.27 | 48.94
\+ Pre | 74.10 | 84.74 | 85.63 | 93.16 | 86.95 | 43.61 | 75.32 | 49.26
\+ Multi | 74.35 | 84.09 | 85.74 | 93.14 | 87.48 | 45.31 | 75.37 | 49.35
\+ Pre, Multi | 73.80 | 85.06 | 85.69 | 93.06 | 87.58 | 44.36 | 72.01 | 48.89
Table 3: Few-shot data experiments on DialoGLUE. The values in this table are
averaged across five runs, with different random seeds.

###  5.3 Results

The results of the full data experiments are shown in Table 1. We attain a
performance gain over the vanilla BERT model (Devlin et al., 2018) across all
seven datasets. These results highlight the efficacy of both the ConvBERT
model and the task-adaptive self-supervised training. Across four datasets,
the best results are attained by ConvBERT with both MLM pre-training and
multi-tasking. We compare to prior work in Table 2, wherein we demonstrate
that ConvBERT in combination with task-adaptive training, matches or exceeds
state-of-the-art performance across five out of seven of the datasets. We
attain a +2.98 improvement in the joint goal accuracy on the dialogue state
tracking task of MultiWOZ. These strong results, which hold true across
several datasets, suggest that large-scale pre-training on open-domain
dialogue data in combination with task-adaptive self-supervised training
transfers effectively to several task-oriented dialogue tasks.

When looking at the aggregate performance across all the DialoGLUE tasks,
neither ConvBERT nor task-adaptive training attain improvements over BERT.
However by combining these two approaches, there is a +0.81 improvement in the
average score. This suggests that through large-scale pre-training on open-
domain dialogue, ConvBERT learns skills that are valuable to DialoGLUE,
however it is only through task-adaptive training that these skills are
transferred effectively to the downstream tasks.

A noteworthy outcome of these experiments is the fact that the BERT model with
task-adaptive self-supervised training sometimes outperforms ConvBERT. This
indicates that in certain settings, it is more beneficial to perform self-
supervised training on the the downstream dataset rather than a much larger
dialogue dataset.

Performing self-supervised training across the combination of the DialoGLUE
datasets gives mixed results. ConvBERT-DG attains a significant performance
gain on MultiWOZ, suggesting that self-supervised training on other task-
oriented dialogue corpora helps significantly in modelling MultiWOZ dialogues.
Across other datasets, it is only marginally better than the ConvBERT model
and sometimes worse. Aside from the unique case of MultiWOZ, it appears that
self-supervised training with additional task-oriented dialogue data, beyond
just the dataset in question, does not provide significant improvements. For
two datasets, DSTC8 and TOP, there is a decrease in performance which may be
indicative of catastrophic forgetting. Namely, the ConvBERT-DG model may have
lost the language understanding capabilities captured by the ConvBERT model
through the additional self-supervised training, and only partially recovers
this through the task-specific self-supervised training. Future work should
explore better mechanisms for performing self-supervised training across the
combination of the DialoGLUE datasets, as well as multi-tasking across the
seven tasks.

While our models achieve state-of-the-art performance across five of the seven
tasks, they underperform on TOP and DSTC8. On the TOP dataset, the best models
use sophisticated architectures which have been tailored to the task of
semantic parsing (Einolghozati et al., 2019; Rongali et al., 2020). With the
DialoGLUE benchmark, our objective is to improve the underlying language
encoders in a manner that results in consistent performance gains across all
of the tasks. We are more concerned with the aggregate improvement across the
DialoGLUE benchmark, rather than the performance on a single task. As such, we
try to avoid complex task-specific architectures when simpler models achieve
competitive results.

The results of the few-shot experiments are shown in Table 3. The few-shot
experiments are particularly important for assessing the generalizability of
the methods and their ability to transfer to downstream tasks. In low data
environments, self-supervised training on the entirety of the DialoGLUE
datasets results in performance gains – with BERT-DG and ConvBERT-DG doing
better than BERT and ConvBERT respectively. However, this is not entirely
surprising as these models are exposed to more utterances from every dataset,
albeit without any of the labels.

Most significantly, on MultiWOZ we see a 40 percent difference between BERT-DG
and ConvBERT-DG over BERT and ConvBERT. For state tracking in particular, it
appears that seeing additional dialogue data in a self-supervised setting,
results in significant improvements. This may suggest that dialogue state
tracking is more dependent on having semantically meaningful representations
of dialogue.

Self-supervised training on the same dataset also helps significantly in few-
shot environments. Across almost every dataset, the best result is obtained
through some form of task-adaptive MLM training. Especially in settings with
fewer training examples, adapting the pre-trained models to the domains of the
dataset is necessary for good performance on the downstream task.

ConvBERT is also far more effective in the few-shot experiments, than it was
in the full data experiments with a +1.96 point improvement in the aggregate
score over BERT. While the full datasets may be sufficient to effectively
transfer BERT to task-oriented dialogue, with only 10% of the data, the
benefits of the large-scale open-domain pre-training are far clearer.

##  6 Conclusion

To facilitate research into producing generalizable models of dialogue, we
introduce DialoGLUE, benchmark for natural language understanding in the
context of task-oriented dialogue. We experiment with several baseline methods
for the DialoGLUE benchmark, demonstrating the efficacy of large-scale pre-
training on open-domain dialogue and task-adaptive self-supervised training.

To improve performance on DialoGLUE, future work should explore: (1) Large-
scale pre-training that attains generalized language understanding
capabilities which transfer effectively to task-oriented dialogue. (2)
Mechanisms of adapting pre-trained models to specific tasks, beyond the task-
adaptive masked language modelling we explore. In particular, we believe there
to be potential in extending our preliminary exploration of self-supervised
training on the combination of all the DialoGLUE datasets. (3) Multi-tasking
across the seven datasets of DialoGLUE, as a means of transferring skills
across the datasets.

The results of our experiments demonstrate that there is significant room for
improvement on DialoGLUE, particularly in the few-shot settings. The DialoGLUE
benchmark is hosted publicly and we invite the research community to submit to
the leaderboard.333https://evalai.cloudcv.org/web/
challenges/challenge-page/708/

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∎

11institutetext: Yuri Heymann 22institutetext: Georgia Institute of
Technology, Atlanta, GA 30332, USA
22email<EMAIL_ADDRESS>
_Present address:_ 3 rue Chandieu, 1202 Geneva, Switzerland

# A lower bound for the modulus of the Dirichlet eta function when
$\Re(s)\geq\frac{1}{2}$ using 2-D principal component analysis

Yuri Heymann

(Received: date / Accepted: date)

###### Abstract

The present manuscript aims to derive an expression for the lower bound of the
modulus of the Dirichlet eta function on vertical lines $\Re(s)=\alpha$. An
approach based on a two-dimensional principal component analysis matching the
dimensionality of the complex plane, which is built on a parametric
ellipsoidal shape, has been undertaken to achieve this result. This lower
bound, which is expressed as $\forall s\in\,\mathbb{C}$ s.t.
$\Re(s)\in\mathcal{P}$, $|\eta(s)|\geq|1-\frac{\sqrt{2}}{2^{\alpha}}|$, where
$\eta$ is the Dirichlet eta function, has implications for the Riemann
hypothesis as $|\eta(s)|>0$ for any $s\in\,\mathbb{C}$ s.t.
$\Re(s)\in\mathcal{P}$, where $\mathcal{P}$ is a partition spanning one half
of the critical strip, on a side of the critical line $\Re(s)=1/2$ upon a
variable.

###### Keywords:

Dirichlet eta function, PCA, Analytic continuation

##  1 Introduction

The Dirichlet eta function is an alternating series related to the Riemann
zeta function of interest in the field of number theory for the study of the
distribution of primes Riemann1859 . Both series are tied together on a two-
by-two relationship expressed as $\eta(s)=\left(1-2^{1-s}\right)\,\zeta(s)$
where $s$ is a complex number. The location of the so-called non-trivial zeros
of the Riemann zeta function in the critical strip $\Re(s)\in\,]0,1[$ is key
in the prime-number theory. For example, the Riemann-von Mangoldt explicit
formula, which is an asymptotic expansion of the prime-counting function,
involves a sum over the non-trivial zeros of the Riemann zeta function
Mangoldt . The Riemann hypothesis, which is focused on the domain of existence
of the zeros in the critical strip, has implications for the accurate estimate
of the error involved in the prime-number theorem and a variety of conjectures
such as the Lindelöf hypothesis Edwards , conjectures about short intervals
containing primes Dudek , Montgomery’s pair correlation conjecture Montgomery
, the inverse spectral problem for fractal strings Lapidus , etc. Moreover,
variants of the Riemann hypothesis falling under the generalized Riemann
hypothesis in the study of modular L-functions Milinovich  are core for many
fundamental results in number theory and related fields such as the theory of
computational complexity. For instance, the asymptotic behavior of the number
of primes less than $x$ described in the prime-number theorem,
$\pi(x)\sim\frac{x}{\ln(x)}$, provides a smooth transition of time complexity
as $x$ approaches infinity. As such, the time complexity of the prime-counting
function using the $x/(\ln x)$ approximation is of order $\mathcal{O}(M(n)\log
n)$, where $n$ is the number of digits of $x$ and $M(n)$ is the time
complexity for multiplying two n-digit numbers. This figure is based on the
time complexity to compute the natural logarithm using the arithmetic-
geometric mean approach, which is of the same order as above but where $n$
represents the number of digits of precision at which the natural logarithm is
to be evaluated.

The below definitions provided in the current manuscript are supplementary to
standard definitions when referring to reals, complex numbers and holomorphic
functions. Any real number can be expressed either as an integer, a rational
or an irrational number. A complex number is the composite of a real and
imaginary number, forming a 2-D surface, where the pure imaginary axis is
represented by the letter $i$ such that $i^{2}=-1$. The neutral element and
index $i$ form a complementary basis spanning a vector space. The Dirichlet
eta function is a holomorphic function having for domain a subset of the
complex plane where reals are positive denoted $\mathbb{C}^{+}$, which
arguments are sent to the codomain belonging to $\mathbb{C}$. Any holomorphic
function is characterized by its modulus, a variable having for support an
axial vertex orthonormal to the complex plane forming a hologram comprised of
five semi-surfaces and a hemisphere, which is akin to a polyhedron of four
vertices. The conformal way to describe space in geometrical terms is the
orthogonal system, which consists of eight windows delimited by the axes of
the Cartesian coordinates, between the three even surfaces of the Euclidean
space adjacent to each other. The hexagon known today as a flat shape refers
to the six directions orthonormal to the faces of a cube embedded within the
Cardinal mund, which led to the definition of Tensors, to describe the three
orthogonal directions associated with each face of the cube.

As a reminder, the definition of the Riemann zeta function and its analytic
continuation to the critical strip are displayed below. The Riemann zeta
function is commonly expressed as follows:

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\,,$ (1)

where $s$ is a complex number and $\Re(s)>1$ by convergence.

The standard approach for the analytic continuation of the Riemann zeta
function to the critical strip $\Re(s)\in\,]0,1[$ is performed with the
multiplication of $\zeta(s)$ with the function
$\left(1-\frac{2}{2^{s}}\right)$, leading to the Dirichlet eta function. By
definition, we have:

$\eta(s)=\left(1-\frac{2}{2^{s}}\right)\zeta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{s}}\,,$
(2)

where $\Re(s)>0$ and $\eta$ is the Dirichlet eta function. By continuity as
$s$ approaches one, $\eta(1)=\ln(2)$.

The function $\left(1-\frac{2}{2^{s}}\right)$ has an infinity of zeros on the
line $\Re(s)=1$ given by $s_{k}=1+\frac{2k\pi i}{\ln 2}$ where
$k\in\,\mathbb{Z}^{*}$. As
$\left(1-\frac{2}{2^{s}}\right)=2\times\left(2^{\alpha-1}\,e^{i\,\beta\ln
2}-1\right)/\,{2^{\alpha}\,e^{i\,\beta\ln 2}}$, the factor
$\left(1-\frac{2}{2^{s}}\right)$ has no poles nor zeros in the critical strip
$\Re(s)\in\,]0,1[$. We can infer that the Dirichlet eta function can be used
as a proxy of the Riemann zeta function for zero finding in the critical strip
$\Re(s)\in\,]0,1[$.

From the above, the Dirichlet eta function is expressed as:

$\eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\,e^{-i\,\beta\ln(n)}}{n^{\alpha}}\,,$
(3)

where $s=\alpha+i\,\beta$ is a complex number, $\alpha$ and $\beta$ are real
numbers.

We have $\frac{1}{n^{s}}=\frac{1}{n^{\alpha}\exp(\beta i\ln
n)}=\frac{1}{n^{\alpha}\left(\cos(\beta\ln n)+i\sin(\beta\ln n)\right)}$. We
then multiply both the numerator and denominator by
$\cos(\beta\ln(n))-i\sin(\beta\ln(n))$. After several simplifications,
$\eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\left[\cos(\beta\ln
n)-i\sin(\beta\ln n)\right]}{n^{\alpha}}$.

In the remaining portion of the manuscript, the Riemann zeta function is
referring to its formal definition and analytic continuation by congruence.

##  2 Mathematicals

###  2.1 Elementary propositions

Proposition 1 Given $z_{1}$ and $z_{2}$ two complex numbers, we have:

$\Big{|}z_{1}+z_{2}\,\Big{|}\geq\Big{|}|z_{1}|-|z_{2}|\,\,\Big{|}\,,$ (4)

where $|z|$ denotes the modulus of the complex number $z$. As a reverse
triangle inequality, (4) can be extended to any normed vector space.

###### Proof

Let us write the complex numbers in polar form and set
$z_{1}=r_{1}{e}^{i\,\theta_{1}}$ and $z_{2}=r_{2}{e}^{i\,\theta_{2}}$, where
their respective modulus and the constituents of their imaginary arguments,
$\theta_{1}$ and $\theta_{2}$, are reals. We get:

$\begin{split}&\Big{|}z_{1}+z_{2}\Big{|}=\Big{|}r_{1}{e}^{i\,\theta_{1}}+r_{2}{e}^{i\,\theta_{2}}\Big{|}\\\
&=\sqrt{\left(r_{1}\cos(\theta_{1})+r_{2}\cos(\theta_{2})\right)^{2}+\left(r_{1}\sin(\theta_{1})+r_{2}\sin(\theta_{2})\right)^{2}}\\\
&=\sqrt{r_{1}^{2}+r_{2}^{2}+2r_{1}r_{2}\cos(\theta_{1}+\theta_{2})}\,.\end{split}$
(5)

The trigonometric identities $2\cos(a)\cos(b)=\cos(a-b)+\cos(a+b)$ and
$2\sin(a)\sin(b)=\cos(a-b)-\cos(a+b)$ were used in (5), see Stegun  formulas
4.3.31 and 4.3.32, page 72. Eq. (4) follows from (5) and (6) below:

$\begin{split}\Big{|}|z_{1}|-|z_{2}|\Big{|}&=\Big{|}r_{1}+r_{2}{e}^{i\,\pi}\Big{|}\\\
&=\sqrt{r_{1}^{2}+r_{2}^{2}-2r_{1}2_{2}}\,.\end{split}$ (6)

Proposition 2 Let us consider an ellipse
$(x_{t},y_{t})=\left[a\cos(t),b\sin(t)\right]$ where $a$ and $b$ are two
positive reals corresponding to the lengths of the semi-major and semi-minor
axes of the ellipse ($a\geq b$) and $t\in[0,2\pi]$ is the angle between the
x-axis and the vector $(x_{t},y_{t})$.

Let us set $t$ such that the semi-major axis of the ellipse is aligned with
the x-axis, which is the angle maximizing the objective function defined as
the modulus of $(x_{t},y_{t})$. When $|(x_{t},y_{t})|$ is maximized, we have:

$|(x_{t},y_{t})|=x_{t}+y_{t}=a\,.$ (7)

Note that by maximizing $x_{t}+y_{t}$, we would get
$|(x_{t},y_{t})|<x_{t}+y_{t}$, as the expression $x_{t}+y_{t}$ is maximized
when $t=\arctan(b/a)$, leading to
$\text{max}(x_{t}+y_{t})=\sqrt{a^{2}+b^{2}}$.

###### Proof

Principal Component Analysis (PCA), is a statistical method for reducing the
dimensionality of a variable space by representing it with a few orthogonal
variables capturing most of the variability of an observable. In the current
context, a two-dimensional principal component analysis built on a parametric
ellipsoidal shape is introduced to match the dimensionality of the complex
plane. An ellipse centered at the origin of the coordinate system can be
parametrized as follows: $(x_{t},y_{t})=\left[a\cos(t),b\sin(t)\right]$ where
$a$ and $b$ are positive real numbers corresponding to the lengths of the
semi-major and semi-minor axes of the ellipse ($a\geq b$) and $t\in\mathbb{R}$
is the angle between the x-axis and the vector $(x_{t},y_{t})$. As the
directions of the x and y-axes are orthogonal, the objective function is
maximized with respect to $t$ when the major axis is aligned with the x-axis.
When $|(x_{t},y_{t})|$ is at its maximum value, we get
$|(x_{t},y_{t})|=x_{t}+y_{t}=a$ where $a$ is the length of the semi-major
axis. The modulus of $(x_{t},y_{t})$ is as follows:
$|(x_{t},y_{t})|=\sqrt{a^{2}\cos^{2}(t)+b^{2}\sin^{2}(t)}\leq a,\forall
t\in\mathbb{R}$. By analogy with principal component analysis, $x_{t}$
represents the first principal component and $y_{t}$ the second principal
component. Say $x_{t}$ and $y_{t}$ were not orthogonal, then there would be a
phase shift $\varphi$ between the components, i.e. $x_{t}=a\cos(t)$ and
$y_{t}=b\sin(t+\varphi)$.

Proposition 3 Given a vector $V_{\mathcal{E}}=[u(\mathcal{E}),v(\mathcal{E})]$
defined in a bidimensional vector space, where $u(\mathcal{E})$ and
$v(\mathcal{E})$ are two real-valued functions say on
$\mathbb{R}^{\mathcal{\nu}}\rightarrow\mathbb{R}$, where $\nu$ represents the
degrees of freedom of the system. The reference of a point in such system, is
described by the set
$\mathcal{E}=\\{\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{\nu}\\}$
representing a multidimensional coordinate sytem. Thus, we have:

$|V_{\mathcal{E}}|=u(\mathcal{E})+v(\mathcal{E})\,,$ (8)

if only $u(\mathcal{E})\,v(\mathcal{E})=0$ and
$u(\mathcal{E})+v(\mathcal{E})\geq 0$.

Given a basis set $\\{e_{1},e_{2}\\}$ of the above-mentioned vector space,
where $|e_{i}|=1$ for $i=1,2$, proposition 3 is true if only the inner product
across basis elements is equal to zero, i.e. $e_{1}\cdot e_{2}=0$, meaning
that the basis elements are disentangled from each other. We say that
$\\{e_{1},e_{2}\\}$ is an orthonormal basis. This condition is also necessary
for $\textit{propositions 2}\,and\,\textit{4}$ to be true. The above vector
space is commonly referred to as the Euclidean space in Cartesian coordinates.

###### Proof

By the square rule, we have $\left(u+v\right)^{2}=u^{2}+v^{2}+2\,u\,v$. The
modulus of a vector $V$ as defined in a orthonormal basis is
$|V|=\sqrt{u^{2}+v^{2}}$, leading to $|V|=|u+v|$ if only $u\,v=0$.

Proposition 4 Given a circle of radius $r\in\,\mathbb{R}^{+}$ parametrized as
follows: $(x_{t},y_{t})=[r\,\cos(t),r\,\sin(t)]$ where $t$ is a real variable
in $[0,2\pi]$, we construct a function $f(t)=a\cos(t)+b\sin(t+\varphi)$ where
$a$ and $b$ are two positive reals and $\varphi$ a real variable which can be
positive or negative such that:

$r\,\cos(t)+r\,\sin(t)=a\cos(t)+b\sin(t+\varphi)\,.$ (9)

Let us say $u_{t}=a\cos(t)$ and $v_{t}=b\sin(t+\varphi)$.

When $a\geq b$, the first component $u_{t}$ is the one which carries most of
the variance of $f(t)$ and we have:

$|(x_{t},y_{t})|\leq\text{max}(v_{t})\leq\text{max}(u_{t})\,,$ (10)

where $t\in\,[0,2\pi]$, $\text{max}(u_{t})$ is the maximum value of $u_{t}$
and $\text{max}(v_{t})$ is the maximum value of $v_{t}$.

When $a\leq b$, the component $v_{t}$ carries most of the variance of $f(t)$
and we have:

$\text{max}(u_{t})\leq|(x_{t},y_{t})|\leq\text{max}(v_{t})\,,$ (11)

where $t\in\,[0,2\pi]$.

When the functions $u_{t}$ and $v_{t}$ are orthogonal i.e. $\varphi=0$, we
have $r=a=b$.

###### Proof

We have
$(r+\delta)\cos(t)+(r-\delta_{2})\sin(t)=r\cos(t)+r\sin(t)+\delta\cos(t)-\delta_{2}\sin(t)$.
Hence, we want that $\delta\cos(t)-\delta_{2}\sin(t)=0$. Thus,
$\delta_{2}=\delta\cot(t)$. Therefore we get:
$r\cos(t)+r\sin(t)=(r+\delta)\cos(t)+\left(r-\delta\cot(t)\right)\sin(t)$. As
$r\sin(t)-\delta\cos(t)=\sqrt{r^{2}+\delta^{2}}\sin(t+\varphi)$ where
$\varphi=-\arctan{\delta/r}$, we get $\forall t\in\,[0,2\pi]$,
$r\cos(t)+r\sin(t)=(r+\delta)\cos(t)+\sqrt{r^{2}+\delta^{2}}\sin(t+\varphi)$.
We set $a=r+\delta$ and $b=\sqrt{r^{2}+\delta^{2}}$ where $-r\leq\delta\leq
r$, leading to (9). If $\delta\geq 0$, we have
$r\leq\sqrt{r^{2}+\delta^{2}}\leq r+\delta$, leading to (10). If $\delta\leq
0$, we have $r+\delta\leq r\leq\sqrt{r^{2}+\delta^{2}}$, leading to (11). As
$\text{max}\\{r\cos(t)+r\sin(t)\\}=\sqrt{2}\,r$ which occurs when
$t=\frac{\pi}{4}$, we have $\delta\in[-r,r]$ for any $r\geq 0$.

Proposition 5 Given an alternating series $S_{m}$ constructed on a sequence
$\\{a_{n}\\}$ monotonically decreasing with respect to its index $n\geq
m\in\,\mathbb{N}$ where $a_{n}>0$ and $\lim_{n\to\infty}a_{n}=0$, defined such
that:

$S_{m}=\sum_{n=m}^{\infty}(-1)^{n+1}\,a_{n}\,,$ (12)

where $m$ is an integer, we get the below upper bound inequality on the
absolute value of the series:

$\left|\sum_{n=m}^{\infty}(-1)^{n+1}\,a_{n}\right|\leq a_{m}\,,$ (13)

where $a_{m}$ represents a radius in Leibniz’s notations.

###### Proof

Given the Leibniz’s rule, the series $S_{m}$ is convergent as $\\{a_{n}\\}$ is
monotonically decreasing and $\lim_{n\to\infty}a_{n}=0$. Let us define the
series $L=\sum_{n=1}^{\infty}(-1)^{n+1}a_{n}$ and its partial sum
$S^{k}=\sum_{n=1}^{k}(-1)^{n+1}a_{n}$. The odd partial sums decrease as
$S^{2(m+1)+1}=S^{2m+1}-a_{2m+2}+a_{2m+3}\leq S^{2m+1}$. The even partial sums
increase as $S^{2(m+1)}=S^{2m}+a_{2m+1}-a_{2m+2}\geq S^{2m}$. As the odd and
even partial sums converge to the same value, we have $S^{2m}\leq L\leq
S^{2m+1}$ for any finite $m\in\,\mathbb{N^{*}}$.

When $m$ is odd:

$S_{m}=a_{m}+\sum_{n=m+1}^{\infty}(-1)^{n}\,a_{n}=a_{m}+(L-S^{m})\leq
a_{m}\,.$ (14)

When $m$ is even:

$S_{m}=-a_{m}+\sum_{n=m+1}^{\infty}(-1)^{n}\,a_{n}=-a_{m}+(L-S^{m})\geq-
a_{m}\,.$ (15)

Leading to:

$|S_{m}|\leq a_{m}\,,$ (16)

where $m$ is a natural number, finite in $\mathbb{N^{*}}$ by definition.

###  2.2 The lower bound of the Dirichlet eta modulus as a floor function

In complex notations, the point $s$ is expressed as $s=\alpha+i\,\beta$ where
$\alpha$ and $\beta$ are reals in their corresponding basis.

Note the zeros of the Dirichlet eta function are also the zeros of its complex
conjugate. For convenience, let us introduce the conjugate of the Dirichlet
eta function, which is expressed as follows:

$\widebar{\eta}(s)=\sum\limits_{n=1}^{\infty}(-1)^{n+1}\frac{e^{i\,\beta\ln
n}}{n^{\alpha}}\,,$ (17)

where $\Re(s)>0$. By applying proposition 1 to (17), we get:

$\begin{split}\left|\widebar{\eta}(s)\right|&\geq\left|1-\left[\sum\limits_{n=2}^{\infty}(-1)^{n+1}\frac{e^{i\,\beta\ln
n}}{n^{\alpha}}\right]\right|\,,\end{split}$ (18)

where $[z]$ denotes the norm of the complex number $z$. The square brackets
are used as smart delimiters for operator precedence.

With respect to the expression
$\left|\sum\limits_{n=2}^{\infty}(-1)^{n+1}\frac{e^{i\,\beta\ln
n}}{n^{\alpha}}\right|$, its decomposition into sub-components
$u_{n}=\frac{(-1)^{n+1}}{n^{\alpha}}e^{i\beta\ln n}$ is a vector
representation where $\beta\ln n+(n+1)\pi$ is the angle between the real axis
and the orientation of the vector itself and $\frac{1}{n^{\alpha}}$ its
modulus. The idea is to apply a rotation by the angle $\theta$ to all the
component vectors simultaneously, resulting in a rotation of the vector of
their sum. The resulting vector after rotation $\theta$ expressed in complex
notations is
$v_{\theta,\beta}=\sum\limits_{n=2}^{\infty}\frac{(-1)^{n+1}}{n^{\alpha}}e^{i(\,\beta\ln
n+\theta)}$, where $\theta$ and $\beta$ are real numbers.

Let us introduce the objective function $w$, defined as the sum of the real
and imaginary parts of $v_{\theta,\beta}$, i.e. $w=v_{x}+v_{y}$ where
$v_{x}=\Re(v_{\theta,\beta})$ and $v_{y}=\Im(v_{\theta,\beta})$. We get:

$\begin{split}w=&\sum\limits_{n=2}^{\infty}\frac{(-1)^{n+1}}{n^{\alpha}}\left(\cos(\beta\ln
n+\theta)+\sin(\beta\ln n+\theta)\right)\\\
&=\sum\limits_{n=2}^{\infty}(-1)^{n+1}\frac{\sqrt{2}}{n^{\alpha}}\cos\left(\beta\ln
n+\theta-\frac{\pi}{4}\right)\,.\end{split}$ (19)

The trigonometric identity
$\cos(x)+\sin(x)=\sqrt{2}\cos\left(x-\frac{\pi}{4}\right)$ which follows from
$\cos(a)\cos(b)+\sin(a)\sin(b)=\cos(a-b)$ with $b=\frac{\pi}{4}$ is used in
(19), see Stegun  formulas 4.3.31 and 4.3.32, page 72. The finite sum of a
subset of the elements of the second line of (19) from 2 to $n\in\mathbb{N}$
is further referred to as the $w$-series.

In the remainder of the manuscript, orthogonality between functions is defined
in terms of the inner product. While orthogonality between vectors is defined
in terms of the scalar product between such pair of vectors, for functions we
usually define an inner product. Let us say we have two real-valued functions
$f$ and $g$, which are squared-$Lebesgue$ integrable on a segment $[a,b]$ and
where the inner product between $f$ and $g$ is given by:

$\langle f,g\rangle=\int_{a}^{b}f(x)\,g(x)dx\,.$ (20)

The functions $f$ and $g$ are squared-$Lebesgue$ integrable, meaning such
functions can be normalized, i.e. the squared norm as defined by $\langle
f,f\rangle$ is finite. For sinusoidal functions such as sine and cosine, it is
common to say $[a,b]=[0,2\,\pi]$, which interval corresponds to one period.
The condition for the functions $f$ and $g$ to be orthogonal is that the inner
product as defined in (20) is equal to zero.

The objective function $w$ was constructed by adding together the real and
imaginary parts of $v_{\theta,\beta}$. We note that the real and imaginary
parts of $v_{\theta,\beta}$ are orthogonal due to the Euler formula. Hence,
$\Re(v_{\theta,\beta})$ is maximized in absolute value when
$\Im(v_{\theta,\beta})=0$, which occurs for example when $\beta=0$. As
$w_{\theta=\frac{\pi}{4},\beta}=\sqrt{2}\,\Re(v_{\theta=0,\beta})$, the
maximum value of the objective function $w$ provided that
$\theta=\frac{\pi}{4}$ occurs when $\beta=0$. We get:

$\textit{max}_{\star}\\{|w|\\}=\left|\sum\limits_{n=2}^{\infty}(-1)^{n+1}\frac{\sqrt{2}}{n^{\alpha}}\right|\,,$
(21)

which is also the maximum value of the objective function $w$, given that
$\beta=0$. Although there could be other values of the tuple $(\theta,\beta)$
leading to larger values of $\textit{max}\\{|w|\\}$, a particularity of the
expression $\textit{max}_{\star}\\{|w|\\}$ as defined above is to be equal to
$\sqrt{2}\left|v_{\theta=\frac{\pi}{4},\beta=0}\right|$.

By applying proposition 5 to (21), we get
$\textit{max}_{\star}\\{|w|\\}\leq\frac{\sqrt{2}}{2^{\alpha}}$, leading to the
below inequality:

$\left|v_{\theta=\frac{\pi}{4},\beta=0}\right|\leq\frac{1}{2^{\alpha}}<\frac{\sqrt{2}}{2^{\alpha}}\,,$
(22)

which is less than the maximum of $|v_{\theta,\beta}|$, while the inner
components of $v_{\theta,\beta}$ collinear as $\beta=0$.

We proceed with the decomposition of the objective function $w$ into say
principal components $w_{1}$ and $w_{2}$, expressed as follows:

$w_{1}=-\frac{\sqrt{2}}{2^{\alpha}}\cos\left(\beta\ln(2)+\theta-\frac{\pi}{4}\right)\,,$
(23)

and

$w_{2}=\sum\limits_{n=3}^{\infty}(-1)^{n+1}\frac{\sqrt{2}}{n^{\alpha}}\cos\left(\beta\ln
n+\theta-\frac{\pi}{4}\right)\,,$ (24)

where most of the variance of the $w$-series comes from the leading component.

By construction $w$ is the sum of the real and imaginary parts of
$v_{\theta,\beta}$ which are orthogonal functions. Let us say $v_{\theta}$ is
the parametric notation of $v_{\theta,\beta}$ for a given $\beta$ value. We
note that for any given $\beta$ value, the complex number $v_{\theta}$
describes a circle in the complex plane, centered in the origin. Hence, in
symbolic notations, $w$ can be written as $w=r\cos(t)+r\sin(t)$. The
components $w_{1}$ and $w_{2}$ can be expressed as $w_{1}=a\cos(t)$ and
$w_{2}=b\sin(t+\varphi)$ where $t$ is in $[0,2\pi]$. As we suppose that
$w_{1}$ carries most of the variance of $w$ (i.e. $a\geq b$), the modulus
$|v_{\theta}|$ is smaller or equal to the maximum value of $|w_{1}|$, by
proposition 4. Yet, $|v_{\theta}|$ is equal to $\text{max}\\{w_{1}\\}$, if
$w_{1}$ and $w_{2}$ are orthogonal and $t=0$. By proposition 2, at its maximum
value $|(w_{1},w_{2})|=w_{1}+w_{2}$ when $\varphi=0$ and $t=0$, which in light
of the above, is also equal to the maximum of $|v_{\theta}|$. As the inner
product of $w_{1}$ and $w_{2}$ does not depend on $\theta$, orthogonality
between $w_{1}$ and $w_{2}$ is determined by $\beta$ values. We then apply a
rotation by an angle $\theta$ to maximize the objective function
$|(w_{1},w_{2})|$. We consider two scenarios respectively, whether $w_{1}$ or
$w_{2}$ is the leading component of the $w$-series.

When $w_{1}$ is the leading component:

In this scenario, if we suppose that $w_{1}$ and $w_{2}$ are orthogonal, at
the maximum value of $|(w_{1},w_{2})|$, $w_{1}=\frac{\sqrt{2}}{2^{\alpha}}$
and $w_{2}=0$. Hence, we get
$\textit{max}\\{|(w_{1},w_{2})|\\}=\frac{\sqrt{2}}{2^{\alpha}}$. If we suppose
that $w_{1}$ and $w_{2}$ are not orthogonal, by proposition 4 we would get
$\left|\sum_{n=2}^{\infty}(-1)^{n+1}\frac{e^{i\,\beta\ln
n}}{n^{\alpha}}\right|<\frac{\sqrt{2}}{2^{\alpha}}$, and $|\eta(s)|$ would be
strictly larger than zero when $\alpha=1/2$ in (26). This would imply that the
Dirichlet eta function does not have zeros on the critical line $\Re(s)=1/2$,
which is known to be false. Hence, we can say that when $w_{1}$ is the leading
component, the functions $w_{1}$ and $w_{2}$ are orthogonal at the maximum
value of $|v_{\theta}|$. Thus, we have:

$\forall s\in\,\mathbb{C^{+}}\text{ s.t. }w_{1}\text{ is leading
component},\,\left|\sum_{n=2}^{\infty}(-1)^{n+1}\frac{e^{i\,\beta\ln
n}}{n^{\alpha}}\right|\leq\frac{\sqrt{2}}{2^{\alpha}}\,.$ (25)

Say for $\Re(s)=\alpha\geq\frac{1}{2}$, (18) and (25) imply that:

$|\eta(s)|\geq 1-\frac{\sqrt{2}}{2^{\alpha}}\,.$ (26)

When $w_{2}$ is the leading component:

In this scenario, we fall on eq. (11) of proposition 4, leading to:

$\forall s\in\,\mathbb{C^{+}}\text{ s.t. }w_{2}\text{ is leading
component},\,\left|\sum_{n=2}^{\infty}(-1)^{n+1}\frac{e^{i\,\beta\ln
n}}{n^{\alpha}}\right|\geq\frac{\sqrt{2}}{2^{\alpha}}\,.$ (27)

Say for $\Re(s)=\alpha\leq\frac{1}{2}$, (18) and (27) imply that:

$|\eta(s)|\geq\frac{\sqrt{2}}{2^{\alpha}}-1\,.$ (28)

By the Riemann zeta functional, if $|\eta(s)|>0$ on either side of the
critical strip, then the same applies to the other side. Therefore, the
relations (26) and (28) belonging to two complementary scenerios, which are
mutually exclusive, is necessary and sufficient for our purpose.

If we would have chosen a decomposition of $w$ introducing multiple terms of
the $w$-series as first principal component, say
$w_{1}=\frac{\sqrt{2}}{2^{\alpha}}\cos(\beta\,\ln
2+\theta-\frac{\pi}{4})+\frac{\sqrt{2}}{4^{\alpha}}\cos(\beta\,\ln
4+\theta-\frac{\pi}{4})$, $|w_{1}|$ is maximized when $\beta=0$ and
$\theta=\frac{\pi}{4}$. Hence
$\left|v_{\theta=\frac{\pi}{4},\beta=0}\right|\leq\frac{\sqrt{2}}{2^{\alpha}}+\frac{\sqrt{2}}{4^{\alpha}}$
which introduces a bias in (22). Furthermore, the floor function of the
modulus of the Dirichlet eta function on vertical lines $\Re(s)=\alpha$ does
not depend on $\beta$, which is reflected by the linear relationship between
$\theta$ and $\beta$ in the cosine argument of the first principal component
as a single term of $w$-series. This is no longer true when introducing
several terms of the $w$-series as first principal component. In such
scenarii, neither $w_{1}$ nor $w_{2}$ is a principal component. A special case
of polynomial made up of a subset of terms of the $w$-series which does not
depend on $\beta$ occurs, if there exists such a polynomial which is equal to
zero for any $\beta$. For such a polynomial to be a first principal component
involves that the subsequent components are also equal to zero, leading to the
degenerate case $|v_{\theta}|=0$. This occurs when $\alpha$ tends to infinity,
leading to $w=0$. The combination of multiple terms of the $w$-series as first
principal component involves that $w_{1}$ is a function composed of terms of
the form $a_{n}=\pm\frac{1}{n^{\alpha}}\cos(\beta\ln(n)+\theta)$, where $n$ is
the index of the corresponding term in $v_{\theta}$. Due to the multiplicity
of bivariate collinear arguments in the cosine functions, which comovements
are not parallel across the index $n$, there is no bijection between $\theta$
and $\beta$, i.e. a one-degree of freedom relationship, such that all cosine
arguments $\beta\ln n+\theta$ of the $w_{1}$ component are decoupled from
$\beta$. As aforementioned, the lower bound of the modulus of the Dirichlet
eta function obtained from (18) needs to be decoupled from $\beta$, to be a
floor function on vertical lines $\Re(s)=\alpha$. Moreover, the principal
components involved in modulus maximization need to be disentangled for PCA to
be applicable. As a simple rule, one degree of freedom is needed for every
additional principal component, when matching the dimensionality of the
variable space in the parametric ellipsoidal model. Hence, the approach
consistsing of using a countable set of terms of the $w$-series of
multiplicity larger than one, as first principal component is not suitable to
the problem addressed in the present text.

A further observation is that in the first scenario, when $w_{1}$ is the
leading component, the second term of the Dirichlet eta function is orthogonal
to the vector comprised of the remaining terms of the series of index larger
than $2$ at zeros of the function, by the Hadamar product. Furthermore, the
angle between the axis defined by the second component of the Dirichlet eta
function and the vector comprised of the remaining components of index larger
than $2$ is in the range $[\frac{\pi}{2},\frac{3\pi}{2}]$.

The below are based on the premise that the Dirichlet eta function can be used
as a proxy of the Riemann zeta function for zero finding in the critical
strip, and interpretations about the lower bound of the modulus of the
Dirichlet eta function as a floor function. The surface spanned by the modulus
of the Dirichlet eta function is a continuum resulting from the application of
a real-valued function over the dimensions of the complex plane, which is a
planar representation where the reals form an ad-dextram vertice contiguous
with the imaginary axis, and where the square of imaginary numbers are
subtracted from zero. The modulus of the Dirichlet eta function is a
holographic function sending a complex number into a real number, whereas the
floor function is a projection of the former onto the real axis.

When $\alpha=1/2$, $|\eta(s)|\geq 0$ means the Dirichlet eta function can have
some zeros on the critical line $\Re(s)=1/2$, which is known to be true
Titchmarsh , p. 256. $\forall\alpha\in\,]\frac{1}{2},1[,|\eta(s)|>0$ means the
Dirichlet eta function has no zeros in the strip $\Re(s)\in\,]\frac{1}{2},1[$.
As the Dirichlet eta function and the Riemann zeta function share the same
zeros in the critical strip, involves the Riemann zeta function has no zeros
in the strips, $\Re(s)\in\,]\frac{1}{2},1[$. As an excerpt from Heymann , i.e.
Given $s$ a complex number and $\bar{s}$ its complex conjugate, if $s$ is a
zero of the Riemann zeta function in the strip $\Re(s)\in\,]0,\,1[$, then we
have: $\zeta(s)=\zeta(1-\bar{s})$. Accordingly, there are no zeros in the
strip $\Re(s)\in\,]0,\frac{1}{2}[$. It follows that the Dirichlet eta function
does not have any zeros on either parts of the critical strip, i.e.
$\Re(s)\in\,]0,1/2[$ and $]1/2,1[$, which is a prerequisite to say that the
non-trivial zeros of the Riemann zeta function lie on the critical line
$\Re(s)=1/2$.

##  3 Conclusion

The lower bound of the modulus of the Dirichlet eta function derived in the
present manuscript from 2-D principal component analysis is $\forall
s\in\mathbb{C}$ s.t. $\Re(s)\in\mathcal{P}$,
$|\eta(s)|\geq|1-\frac{\sqrt{2}}{2^{\alpha}}|$, where $\mathcal{P}$ is a
partition spanning one half of the critcal strip, on a side of the critical
line depending upon a variable, and where $\eta$ is the Dirichlet eta
function. As a proxy of the Riemann zeta function for zero finding in the
critical strip $\Re(s)\in\,]0,1[$, the floor function of the modulus of the
Dirichlet eta function involves the Riemann zeta function does not have zeros
on both sides of the critical strip. Further observations are made, with
respect to the orthogonality between the second component of the Dirichlet eta
function and the vector comprised of the remaining elements of the series of
index larger than $2$ at the so-called zeros of the Dirichlet eta function.

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