#  HOP to the Next Tasks and Domains for Continual Learning in NLP

Umberto Michieli, Mete Ozay

###### Abstract

Continual Learning (CL) aims to learn a sequence of problems (_i.e_., tasks
and domains) by transferring knowledge acquired on previous problems, whilst
avoiding forgetting of past ones. Different from previous approaches which
focused on CL for one NLP task or domain in a specific use-case, in this
paper, we address a more general CL setting to learn from a sequence of
problems in a unique framework. Our method, HOP, permits to hop across tasks
and domains by addressing the CL problem along three directions: (i) we employ
a set of adapters to generalize a large pre-trained model to unseen problems,
(ii) we compute high-order moments over the distribution of embedded
representations to distinguish independent and correlated statistics across
different tasks and domains, (iii) we process this enriched information with
auxiliary heads specialized for each end problem. Extensive experimental
campaign on 4 NLP applications, 5 benchmarks and 2 CL setups demonstrates the
effectiveness of our HOP.

## Introduction

Current practice to obtain a deep learning model to perform a specific
assignment is to train the model on a specific dataset for that particular
assignment (Chen and Liu 2018). However, this paradigm has several inherent
limitations. For example, as models get larger, training from scratch requires
a larger amount of expensive labeled data and computation time, which can be
reduced by knowledge transfer (KT) from a pre-trained model on a base domain.
This problem is generally addressed via adapting (_e.g_., through fine-tuning,
FT) large pre-trained language models (_e.g_., BERT, Devlin et al. 2019) to
various downstream NLP applications, such as Text Classification (TC, Wang et
al. 2023a; Wu et al. 2023; Ke, Xu, and Liu 2021; Hu et al. 2021), Natural
Language Inference (NLI, Pfeiffer et al. 2020), Document (Ke et al. 2020) or
Aspect (Ke, Xu, and Liu 2021; Zhou et al. 2021) Sentiment Classification (DSC
and ASC). In our work, we employ the Adapter-BERT (Houlsby et al. 2019) model.
FT a large pre-trained model reaches state-of-the-art results on NLP
benchmarks with a static distribution. However, if a stream of problems111For
the sake of clarity, we refer to problems as either tasks or domains
experienced by the CL method over time. are presented sequentially, the
naïvely FT model faces catastrophic forgetting (CF, McCloskey and Cohen 1989)
of previous knowledge due to the non-stationary data distribution, and cannot
make use of past knowledge to improve capability on subsequent problems
(forward KT, Lopez-Paz and Ranzato 2017) or vice-versa (backward KT). In
particular, high CF and low KT hinder performance of CL for NLP, as several
NLP applications share similar knowledge that can be exploited to achieve
higher accuracy on future/previous problems, without accuracy degradation.
Indeed, ideally, learning a sequence of problems should allow multiple
problems to support each other via KT (Rusu et al. 2016; Ke, Liu, and Huang
2020). To address such issues towards more versatile NLP models, we focus on
CL to tune a pre-trained Adapter-BERT for a stream of problems.

In CL, a model learns a sequence of problems incrementally. After each
incremental learning (IL) stage is completed, its training data is typically
discarded (Chen and Liu 2018). Three main families of CL setups can be
identified (Van de Ven and Tolias 2019): namely, Task-IL (TIL), Domain-IL
(DIL), and Class-IL (CIL). TIL builds one model for each task (_e.g_., to
classify sentiment in products’ reviews). At test time, task identifier
specifies the proper model for each input sample. This could significantly
increase the number of parameters; however, in our case, most of the
parameters are shared across problems and only a few parameters are problem-
specific. DIL is similar to TIL, however, builds a single head for each domain
as classes are shared across domains. In DIL, no identifier is required at
test time and subsequent problems present data from different domains (_e.g_.,
reviews from online commerce, or from movie critique, _etc_.). In CIL, non-
overlapping classes are learned progressively. Opposed from traditional CL
approaches used in Computer Vision (CV), most of the NLP problems are
formulated as either TIL or DIL (Biesialska, Biesialska, and Costa-jussà 2020;
Ke and Liu 2022; Ke et al. 2021b; Sun, Ho, and Lee 2019) and, to the best of
our knowledge, no prior work has addressed them both jointly.

Differently from concurrent CL NLP approaches, in this paper, we evaluate
models on both TIL and DIL in a unified framework which employs parameter-
efficient transfer learning strategies to adapt the models to each end
problem: (i) as in current state-of-the-art approaches (Ke, Xu, and Liu 2021;
Ke, Liu, and Huang 2020), we use Adapter-BERT with a separate set of adapters
tuned for each problem; (ii) we discard the [CLS] token, which we show being
counterproductive for CL, and rather compute high-order statistical measures
over the distribution of extracted features (_i.e_., tokens representing text
embeddings); (iii) we use an MLP head specialized for each problem to process
and combine such information. With a slight abuse of notation, we call our
method HOP, from High-Order Pooling, that is the most distinctive component of
our method.

Our HOP extracts multiple cues from the limited samples drawn from non-
stationary distributions while preserving previous knowledge. HOP accurately
models the variable distribution of problems since input-level distribution
shift is reflected into feature-level distribution shift. Indeed, we show that
this variation is not properly captured by the single [CLS] token. We present
an extensive validation on 2 CL setups (DIL and TIL) and 5 benchmarks
outperforming current state-of-the-art on accuracy, KT, CF, and runtime.

## Related Work

For a wide survey of the current state-of-the-art, we refer to recent CL
reviews (De Lange et al. 2021; Lesort et al. 2020; Michieli, Toldo, and
Zanuttigh 2022; Biesialska, Biesialska, and Costa-jussà 2020; Ke and Liu
2022).

Traditional CL methods focused on image classification and can be grouped
according to the proposed technique. (1) Regularization-based methods are
generally based on knowledge distillation (Li and Hoiem 2017; Jung et al.
2016; Michieli and Zanuttigh 2019) or on importance score for each parameter
to compute a penalty term in the optimization to reduce weight deviation while
learning new problems (Kirkpatrick et al. 2017; Zeng et al. 2019; Nguyen et
al. 2018; Zenke, Poole, and Ganguli 2017; Ahn et al. 2019). (2) Parameter-
isolation approaches dedicate a set of parameters to each problem to reduce
forgetting when learning subsequent problems. Parameters can be either masked
out (Serra et al. 2018; Mallya and Lazebnik 2018; Mallya, Davis, and Lazebnik
2018; Wang et al. 2023b), frozen (Rusu et al. 2016; Xu and Zhu 2018; Michieli
and Zanuttigh 2021), or new branches are grown over time (Rusu et al. 2016; Xu
and Zhu 2018). (3) Replay-based methods either retain an exemplar set of
previously seen data (Rebuffi et al. 2017; Isele and Cosgun 2018; Lopez-Paz
and Ranzato 2017; Chaudhry et al. 2018) or generated pseudo-samples (Shin et
al. 2017; Maracani et al. 2021) to reduce CF and promote KT.

CL in NLP is in rapid expansion due to its great importance. Early works
tackled lifelong learning (with no knowledge preservation, hence no CF) for
sentiment analysis (Carlson et al. 2010; Silver, Yang, and Li 2013; Ruvolo and
Eaton 2013; Chen, Ma, and Liu 2015; Wang et al. 2019; Qin, Hu, and Liu 2020;
Wang et al. 2018). Recent works have dealt with CF in many applications:
sentiment analysis (Lv et al. 2019; Ke et al. 2021b; Ke, Xu, and Liu 2021),
dialogue systems (Shen, Zeng, and Jin 2019; Madotto et al. 2020; Qian, Wei,
and Yu 2021; Chien and Chen 2021), language modeling (Sun, Ho, and Lee 2019;
Chuang, Su, and Chen 2020) and learning (Li et al. 2019), cross-lingual
modeling (Liu et al. 2020), sentence embedding (Liu, Ungar, and Sedoc 2019),
machine translation (Khayrallah et al. 2018; Zhan et al. 2021), question
answering (Greco et al. 2019), named entity recognition (Monaikul et al.
2021).

Most of the previous literature focuses on the simpler TIL setup. SRK (Lv et
al. 2019) and KAN (Ke et al. 2020) tackled DSC via recurrent architectures.
They are mainly conceived for KT, hence they suffer from CF and cannot be
easily extended to BERT. B-CL (Ke, Xu, and Liu 2021) is the first CL framework
for ASC: it employs Adapter-BERT and is based on capsule network and dynamic
routing, bringing only limited KT. CAT (Ke, Liu, and Huang 2020) works on
mixed sequences of similar and dissimilar problems, and can transfer knowledge
among similar problems. Snapshot (Wang et al. 2023a) regularizes training with
adapters learned over previous problems. Parallel to CL, AdapterFusion
(Pfeiffer et al. 2021) uses a two-stage method to learn the adapters of
Adapter-BERT to improve multi-task learning, hence it has no CF. CTR (Ke et
al. 2021a) extends the adapters concept to the idea of CL plugins to adapt
BERT to each problem, and it is the state-of-the-art in TIL.

Recently, DIL has gained attention. LAMOL (Sun, Ho, and Lee 2019) uses a
language model (_i.e_., GPT-2 Radford et al. 2019) to solve sequential
problems and to generate training pseudo-samples against CF; CLASSIC (Ke et
al. 2021b) uses contrastive learning to promote KT.

Pooling in NLP has been recently studied to improve accuracy (Wu et al. 2020;
Ács, Kádár, and Kornai 2021; Zhao et al. 2022). In particular, Wu et al. 2020
propose an attentive pooling scheme with learnable norm to extract accurate
text representations in different problems, motivated by 3 observations: (i)
different contexts have different informativeness for learning text
representations (_e.g_., they might be important to determine sentiment
polarity, however, probably less relevant for TC); (ii) different problems
have different characteristics; (iii) popular pooling methods (_e.g_., MAX or
AVG) may over-emphasize some concepts and disregard other useful contextual
information. To summarize, some problem specific words or sentences contain
information regarding output class in various ways. Our work is motivated by
these results. However, such pooling schemes cannot be applied to CL. To cope
with this, our HOP computes multiple statistical moments from the encoded text
to capture evolution of different statistics of the input.

## Problem Formulation

CL learns a sequence of problems ${t\in\\{1,\dots,T\\}}$. Each problem $t$ has
its test data $\hat{\mathcal{S}}_{t}$ and training data
${{\mathcal{S}}_{t}=\\{(x_{t}^{k},y_{t}^{k})\\}_{k=1}^{N_{t}0}}$, where
${x_{t}^{k}\in\mathcal{X}_{t}}$ is a training sample with label
$y_{t}^{k}\in\mathcal{Y}_{t}$ (_i.e_., supervised problems). Then, the CL goal
is to minimize the empirical loss $\mathcal{L}$ over all seen problems. At
problem $T$, we aim at training models $f_{t},\forall t$, parameterized by
$\theta$ (_i.e_., $\hat{y}_{t}^{k}=f_{t}(x_{t}^{k};\theta)$), which minimize
the loss

$\sum_{t=1}^{T}l_{t},\quad\mathrm{with}\ \
l_{t}\\!=\\!\frac{1}{N_{t}}\sum_{k=1}^{N_{t}}\mathcal{L}(\hat{y}_{t}^{k},y_{t}^{k}).$
(1)

However, Eq. (1) cannot be minimized since no (in case of replay-free CL
methods) or limited (in case of replay CL methods) access to previous data is
guaranteed. In the most challenging replay-free setup, we can minimize the
empirical loss on the current problem $T$ only, _i.e_., $l_{T}$.

Therefore, CL methods try to approximate Eq. (1) in different ways (_e.g_.,
via regularization, replay, _etc_.). Instead, we extract high-order statistics
from the input dataset and we process this additional information via an
auxiliary problem-specific multi-layer perceptron (MLP) to adapt the current
model to the current problem.

Depending on the properties of $(\mathcal{X}_{t},\mathcal{Y}_{t})$ $\forall
t$, we can identify TIL, DIL and CIL. In CIL, models are progressively trained
with new classes, and it has been less attractive for NLP applications as the
number of classes is generally determined a priori (Ke et al. 2021b).
Therefore, we address both TIL and DIL in a unified framework of CL for NLP.

Figure 1: The proposed HOP framework. During the incremental step $T$, only
orange modules are trained, while gray and green modules are frozen. Figure 2:
Accuracy matrix showing the main CL metrics used in this work.
$\hat{\mathcal{S}}_{t}$ and ${\mathcal{S}}_{t}$ are the testing and training
datasets at step $t$.

## Our Proposed Framework: HOP

Our framework employs parameter-efficient learning strategies to adapt models
to each end problem, namely: (i) adapter modules, (ii) computation of high-
order moments, and (iii) a specialized MLP for each end problem. In the
experimental validation, we apply HOP to BERT-based models due to its superior
performance in NLP, however, it can be seamlessly applied to other
architectures as well.

(i) Following recent CL approaches for NLP (Ke, Xu, and Liu 2021; Ke, Liu, and
Huang 2020), our best architecture relies on Adapter-BERT, with a separate set
of adapters tuned for each problem. Adding adapters to BERT is a highly
parameter-efficient transfer learning paradigm: in CL, this means that
subsequent problems have separate adapters (which are small in size). An
adapter layer is a tunable 2-layer fully-connected network, which adapts the
pre-trained model to the end problem at hand. In this way, there is no need
for a separate BERT model fine-tuned on each problem, which is extremely
parameter-inefficient if many problems are learned in sequence.

(ii) HOP computes high-order statistical moments from the distribution of
extracted tokens to capture most of the information from the input sequence.
Current approaches (Ke, Xu, and Liu 2021; Ke et al. 2021a) design their CL
systems relying on the [CLS] token embeddings. More formally, we consider
$f_{t}$ composed of a tokenizer $\mathcal{T}$ and a classifier $\mathcal{C}$
by $f_{t}=\mathcal{C}\circ\mathcal{T}$ to recognize $N_{C}$ classes ($\circ$
represents function composition). Traditionally, $\mathcal{T}$ includes a
reduction function $\mathcal{R}$ as the last layer to summarize the whole
input sequence into one element. Therefore, we can write
$\mathcal{T}=\mathcal{R}\circ\mathcal{T}^{\prime}$, with
$\mathcal{T}^{\prime}$ being the tokenizer without the final reduction
function. $\mathcal{R}$ is often identified by the [CLS] token or by AVG
pooling. However, different problems usually have different peculiar patterns
in the input samples, and the output should be an explicit function of the
whole, non-reduced, embedding sequence. Therefore, we compute high-order
central moments from the input sequence and concatenate ($\mathrm{concat}$)
them. We define the reduction function by

$\mathcal{R}=\mathrm{concat}(m_{1},m_{2},\dots,m_{p}),$ (2)

where $p$ is the order of considered moments, that is, $m_{1}$ is the first
moment (_i.e_., AVG), $m_{2}$ is the second moment (_i.e_., the variance),
_etc_. (Papoulis and U. Pillai 2002; Michieli, Parada, and Ozay 2023; Michieli
and Ozay 2023; Michieli et al. 2024). Such moments are computed over the
distribution of tokens identified by the unreduced tokenizer
$\mathcal{T}^{\prime}:x_{t}^{k}\mapsto{h_{t,d}^{k}}$ where $d$ denotes the
dimension of the embedded sequence and each $h_{t,d}^{k}\in\mathbb{R}^{Q}$
with $Q$ channels.
This step may recall the statistical moments of tokens identified by standard
N-gram models (Cortes and Mohri 2004).

(iii) We process and combine embeddings computed by $\mathcal{T}$ with an
auxiliary MLP head specialized for each problem. We replace the usual linear
layer constituting $\mathcal{C}$ with an MLP. The MLP head increases the
adaptation capacity to process the high order information while being highly
parameter efficient. We design the MLP as a 2-layer network consisting of
$p\cdot Q$ and $N_{C}$ neurons at each layer, respectively.

Overall, our HOP can extract richer information from the samples drawn from
the non-stationary input sequence distributions while preserving previous
knowledge. Therefore, our method can hop across the distributions of
subsequent tasks and domains, since input-level distribution shift is
reflected into a feature-level distribution shift via the embedding tokenizer.
Our framework is applicable to both TIL and DIL setups: in TIL, one head per
task adapts models to each of the tasks separately; in DIL, one single head is
incrementally adapted to the varying domains.

HOP alleviates CF thanks to (i) frozen shared backbone to extract shared rich
features, and (ii) adapters tuned for each problem. It promotes KT via (i)
initialization of adapters to the last achieved ones, therefore bringing
onward the previous information; (ii) modeling tailed distributions of tokens
encountered in subsequent CL problems with different moments. Finally, MLP
heads improve plasticity for new tasks.

An overview of our framework is given in Figure 1, where we depict the modular
property of HOP, so it can be plugged on top of other CL methods. Compared to
competing methods, HOP only brings a minimal computation and complexity
footprint. Detailed analyses are provided in Sec. Experimental Results.

The main novelty of our HOP are: (i) we are the first to address tailed
distribution of features and its relation to CL; (ii) we revisit high-order
feature statistics (inspired from N-grams co-occurrence statistics) to deep
NLP architectures; (iii) we exploit them in CL for the first time; (iv) we use
adapters for CL extending concurrent works to jointly tackle TIL/DIL; (v) we
include specialized MLP heads to improve plasticity in learning new concepts.
Overall, we propose a simple yet effective CL baseline that outperforms
complex architectures in all experiments, while requiring significantly lower
training time. Additionally, it can be seamlessly applied on top of
competitors.

## Experimental Setup

Architectures. We evaluate HOP on 3 BERT-based architectures (Devlin et al.
2019). Previous works (Ke et al. 2021a; Ke, Xu, and Liu 2021) have shown that
naïvely fine-tuning BERT increases CF, hence we focus on more versatile ways
to learn new concepts limiting CF. BF + Lin consists in a Frozen BERT (BF)
with a trainable linear layer on top. BF + CNN consists in a BF with a
trainable CNN TC network (Kim 2014) on top. Adapter-BERT trains only the
adapter blocks built of 2 linear layers each with $2000$ neurons.

Datasets. We consider 4 applications, unifying the setups proposed by (Ke et
al. 2021b, a; Asghar et al. 2020). (1) ASC classifies a review sentence on
either positive, negative or neutral aspect-level sentiments. We use 19
datasets (_i.e_., reviews of 19 products) taken from four sources: 5 products
from HL5Domains (Hu and Liu 2004), 3 products from Liu3Domains (Liu et al.
2015), 9 products from Ding9Domains (Ding, Liu, and Yu 2008), and 2 products
from SemEval14 Task 4 (Pontiki et al. 2014). We applied the same data
filtering of previous works for fair comparison (Ke et al. 2021a). (2) DSC
classifies product reviews into either positive or negative opinion classes,
using TC formulation of Devlin et al. 2019. We use 10 DSC datasets (_i.e_.,
reviews of 10 products, Ke et al. 2020). We consider both a small training
version of 100 positive and 100 negative reviews per problem, and the full
training version of 2500 positive and 2500 negative reviews per problem.
Validation and test sets are fixed, and each consists of 250 reviews per each
class. The first experiment is arguably more useful in practice because
labeling a large number of examples is costly, thus, ablation is carried out
on this split. (3) TC classifies text into 20 classes using 20News data (Lang
1995). We split documents into 10 problems with 2 classes per problem (in DIL,
$N_{C}$ is supposed known a priori). Classes are variegate and share little
knowledge, hence forgetting is the main issue. (4) We target NLI for sentence
understanding using the MultiNLI dataset (Williams, Nangia, and Bowman 2018)
which is one of the largest corpus of its kind. Sentences are classified into:
entailment, neutral and contradiction. We split data in 5 problems, each
belonging to a specific domain (fiction, telephone, _etc_., Asghar et al.
2020).

Baselines. As the first baseline, we consider a separate model learned for
each problem independently, which we call SDL (standalone). No KT or CF occurs
here. Second, we compare against FT which simply optimizes the model over the
sequence of problems. Third, we examine 13 CL competitors. Among them, some
approaches have been proposed for CL in NLP and have been briefly described in
Sec. Related Work. Additionally, we adapted CL methods proposed in CV for our
NLP applications. For TIL-based works: UCL (Ahn et al. 2019) proposes
uncertainty-based regularization via a Bayesian online learning framework; HAT
(Serra et al. 2018) focuses on employing embeddings preserving information of
previous problems while learning new ones. CIL approaches can be adapted to
TIL by training the head of the specific problem and considering predictions
of the specific head during testing. Among them, we used (i) regularization-
based methods, _e.g_., EWC (Kirkpatrick et al. 2017), OWM (Zeng et al. 2019),
and L2 (Kirkpatrick et al. 2017), and (ii) replay-based methods, _e.g_., the
efficient A-GEM (Chaudhry et al. 2018), and DER++ (Buzzega et al. 2020) for
pseudo replay. We note that a single upper bound for accuracy does not exist,
since different methods have a different number of parameters, as shown in
Table 3.

Hyperparameters. We employ the same scenarios as current state-of-the-art
approaches (Ke et al. 2021a). We follow the CL evaluation of (De Lange et al.
2021): after training on one problem is completed, the respective training
data is no longer accessible; all hyperparameters are chosen according to the
performance on the validation set; after all problems are learned, testing is
carried out on the test set. We report results averaged over $5$ random seeds
(_i.e_., different ordering of problems) and we report the mean (the standard
deviation is negligible - lower than $0.1$ in all cases). All baseline
approaches use the [CLS] token as the reduction function of the tokenizer. We
show that this is a major limitation, concurrently hinted also by (Mirzadeh et
al. 2022). However, (Mirzadeh et al. 2022) address the issue only marginally,
while we propose a simple and effective framework to overcome it. The only
hyperparameter specific to our framework is $p$, which is set according to the
best validation results. Empirically, $p=3$ provides the best results and
represents a good compromise with additional computational complexity.

Table 1: $mAcc$ and $MF1$ over 5 benchmark datasets on both TIL and DIL
setups. We evaluate 3 network architectures based on BERT and 14 baselines.
Results are color-coded according to the column-wise value and best results
are in bold.

| CL Method | ASC | DSC (small) | DSC (full) | 20News | NLI
---|---|---|---|---|---|---
TIL | DIL | TIL | DIL | TIL | DIL | TIL | DIL | TIL | DIL
$\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$
BF+Lin | SDL | 56.32 | 27.8 | 58.48 | 35.21 | 48.97 | 42.21 | 57.87 | 56.97 | 82.32 | 80.11 | 77.45 | 76.94 | 95.31 | 95.30 | 54.96 | 54.31 | 48.69 | 43.78 | 43.24 | 41.55
2-22 | FT | 61.18 | 32.00 | 68.42 | 49.97 | 60.41 | 52.36 | 66.12 | 65.06 | 70.53 | 63.62 | 75.07 | 74.19 | 66.30 | 65.59 | 55.85 | 54.88 | 42.53 | 37.09 | 41.08 | 38.17
| HOP (ours) | 86.38 | 78.64 | 87.54 | 81.79 | 78.48 | 75.31 | 81.96 | 80.93 | 84.59 | 83.18 | 85.63 | 84.65 | 96.45 | 96.45 | 68.45 | 66.16 | 61.05 | 60.88 | 59.24 | 58.82
Bert (frozen) + CNN | SDL | 78.14 | 58.13 | 78.14 | 58.13 | 73.88 | 67.97 | 73.88 | 67.97 | 85.34 | 80.17 | 85.34 | 80.17 | 96.49 | 96.48 | 96.49 | 96.48 | 68.21 | 67.49 | 68.21 | 67.49
2-22 | FT | 85.51 | 76.64 | 86.85 | 78.73 | 83.12 | 79.23 | 85.66 | 84.87 | 61.88 | 45.79 | 85.54 | 84.59 | 83.28 | 81.81 | 64.45 | 58.68 | 71.98 | 71.70 | 71.70 | 71.47
| L2 | 56.04 | 38.40 | 86.31 | 77.90 | 59.17 | 48.39 | 82.29 | 81.32 | 69.80 | 62.63 | 64.92 | 61.06 | 72.14 | 65.39 | 62.50 | 58.85 | 56.80 | 56.49 | 68.71 | 68.55
| A-GEM | 86.06 | 78.44 | 79.74 | 71.78 | 59.33 | 45.94 | 80.23 | 79.27 | 70.67 | 61.77 | 87.53 | 86.61 | 93.31 | 92.95 | 58.50 | 46.86 | 71.95 | 71.89 | 72.49 | 72.34
| DER++ | 84.27 | 75.08 | 87.53 | 80.09 | 72.29 | 66.28 | 83.26 | 82.29 | 86.70 | 85.46 | 86.87 | 85.99 | 60.44 | 49.67 | 54.20 | 39.22 | 70.48 | 69.98 | 71.84 | 71.69
| EWC | 86.37 | 74.52 | 86.60 | 78.31 | 82.38 | 78.41 | 81.88 | 80.99 | 72.77 | 65.76 | 85.78 | 84.65 | 80.26 | 78.60 | 75.40 | 71.98 | 68.85 | 68.18 | 71.10 | 70.90
| OWM | 87.02 | 79.31 | 86.11 | 76.65 | 58.07 | 42.63 | 80.25 | 79.45 | 86.30 | 85.36 | 82.31 | 81.44 | 84.54 | 82.73 | 50.00 | 33.33 | 72.23 | 71.96 | 65.97 | 65.81
| UCL | 83.89 | 74.82 | 85.38 | 76.90 | 80.12 | 74.13 | 84.39 | 83.59 | 74.76 | 69.48 | 89.00 | 88.27 | 94.65 | 94.63 | 72.20 | 67.15 | 71.61 | 71.46 | 72.66 | 72.50
| HAT | 86.74 | 78.16 | 84.73 | 76.49 | 79.48 | 72.78 | 79.98 | 79.10 | 87.29 | 86.14 | 88.07 | 87.26 | 93.51 | 92.93 | 53.80 | 42.56 | 69.78 | 69.70 | 71.06 | 70.93
| CAT | 83.68 | 68.64 | 87.32 | 81.42 | 67.41 | 56.22 | 71.03 | 69.40 | 87.34 | 86.51 | 83.56 | 82.74 | 95.17 | 95.16 | 51.15 | 44.11 | 72.39 | 72.08 | 73.21 | 73.10
| KAN | 85.49 | 77.38 | 83.20 | 73.52 | 77.27 | 72.34 | 68.02 | 66.13 | 82.32 | 81.23 | 86.10 | 85.30 | 73.07 | 69.97 | 64.95 | 58.84 | 72.77 | 72.72 | 73.97 | 73.86
| SRK | 84.76 | 78.52 | 83.91 | 74.38 | 78.58 | 76.03 | 75.58 | 74.64 | 83.99 | 82.66 | 88.14 | 87.45 | 79.64 | 77.89 | 57.80 | 51.26 | 69.84 | 69.79 | 70.69 | 70.54
| HOP (ours) | 87.51 | 80.45 | 88.16 | 82.76 | 83.79 | 81.45 | 85.74 | 84.98 | 87.98 | 86.91 | 89.30 | 88.54 | 95.23 | 95.20 | 74.32 | 72.10 | 74.49 | 74.34 | 76.59 | 76.45
Adapter-BERT | SDL | 85.96 | 78.07 | 85.96 | 78.07 | 76.31 | 71.04 | 76.31 | 71.04 | 88.30 | 87.31 | 88.30 | 87.31 | 96.20 | 96.19 | 96.20 | 96.19 | 72.54 | 71.26 | 72.54 | 71.26
2-22 | FT | 54.03 | 44.81 | 86.67 | 78.04 | 55.19 | 35.28 | 82.22 | 81.32 | 64.94 | 63.40 | 88.77 | 88.21 | 68.29 | 61.70 | 63.10 | 57.87 | 69.31 | 67.80 | 81.63 | 81.59
| L2 | 63.97 | 52.43 | 75.64 | 59.78 | 70.87 | 69.11 | 83.78 | 82.86 | 73.03 | 71.50 | 89.02 | 88.49 | 69.56 | 65.50 | 63.85 | 61.29 | 77.16 | 76.85 | 78.81 | 78.91
| A-GEM | 45.88 | 28.21 | 86.78 | 77.84 | 59.35 | 54.20 | 86.25 | 85.49 | 71.22 | 69.94 | 86.54 | 56.12 | 60.29 | 50.40 | 62.24 | 60.30 | 78.84 | 78.04 | 77.45 | 77.36
| DER++ | 47.63 | 35.54 | 88.59 | 79.85 | 63.11 | 61.96 | 84.00 | 83.16 | 59.67 | 57.82 | 87.12 | 86.43 | 58.95 | 49.58 | 60.18 | 57.89 | 74.12 | 73.78 | 79.84 | 79.56
| EWC | 56.30 | 49.58 | 88.05 | 78.75 | 58.34 | 42.85 | 87.29 | 86.34 | 62.69 | 61.51 | 88.58 | 87.94 | 61.86 | 53.94 | 49.95 | 36.86 | 75.72 | 75.38 | 80.63 | 80.59
| OWM | 72.99 | 66.51 | 87.66 | 78.82 | 73.97 | 71.96 | 77.32 | 76.68 | 85.46 | 84.57 | 85.78 | 85.45 | 71.10 | 66.25 | 60.23 | 57.94 | 70.45 | 68.24 | 77.65 | 76.89
| UCL | 64.46 | 36.64 | 71.23 | 39.61 | 48.36 | 32.07 | 56.53 | 52.96 | 57.06 | 55.86 | 88.18 | 87.54 | 51.75 | 36.06 | 50.25 | 36.78 | 76.54 | 76.43 | 80.36 | 80.40
| HAT | 86.14 | 78.52 | 88.23 | 79.19 | 80.83 | 78.41 | 86.22 | 85.44 | 88.00 | 87.26 | 86.52 | 85.78 | 95.22 | 95.21 | 63.30 | 60.44 | 71.51 | 68.83 | 81.02 | 80.92
| B-CL | 88.29 | 81.40 | 89.83 | 84.22 | 84.34 | 83.12 | 85.92 | 85.11 | 79.76 | 76.51 | 88.12 | 87.48 | 95.07 | 95.04 | 64.50 | 61.87 | 72.92 | 72.71 | 81.23 | 81.02
| CTR | 89.47 | 83.62 | 89.13 | 83.52 | 83.96 | 83.00 | 86.01 | 85.16 | 89.31 | 88.75 | 88.36 | 87.89 | 95.25 | 95.23 | 65.76 | 63.04 | 75.95 | 75.46 | 80.78 | 80.64
| HOP (ours) | 89.84 | 85.06 | 89.87 | 84.12 | 85.63 | 84.18 | 87.84 | 86.96 | 90.08 | 89.44 | 89.93 | 89.41 | 95.30 | 95.29 | 72.30 | 69.67 | 81.75 | 81.66 | 82.46 | 82.22

Metrics. We compute both mean accuracy (mAcc, $\uparrow$) and macro-F1 (MF1,
$\uparrow$), to reduce biases in accuracy originating from imbalanced classes.
To fully characterize the different approaches, we report a wide range of
forgetting and transfer metrics computed from the accuracy matrix (Mai et al.
2022), as illustrated in Figure 2. Namely, we report: backward transfer
($BwT$, $\uparrow$), which tracks the influence that learning a new problem
has on the preceding problems’ performance, to measure stability; forward
transfer ($FwT$, $\uparrow$) measures the positive influence of learning a
problem on future problems’ performance; forgetting ($Forg$, $\downarrow$)
averages the difference of class-wise accuracy achieved at the last step and
the best class-wise accuracy achieved previously; plasticity ($Pla$,
$\uparrow$) averages the accuracy achieved on each problem evaluated right
after learning that problem. Additionally, we report the number of overall
(#OP, $\downarrow$) and of trainable parameters (#TP, $\downarrow$), and the
computation time ($\downarrow$, in minutes).

## Experimental Results

Table 2: Aggregate results from Tab. 1. First (second) vertical group of 2
columns: results averaged over the 5 benchmarks for TIL (DIL). Last vertical
group: averaged over both benchmarks and CL setups.

| CL Method | Avg Benchmarks
---|---|---
TIL | DIL | Avg Setups
$\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$
BF+Lin | SDL | 66.32 | 57.84 | 58.40 | 53.00 | 62.36 | 55.42
2-8 | FT | 60.19 | 50.13 | 61.31 | 56.45 | 60.75 | 53.29
| HOP (ours) | 81.39 | 78.89 | 76.56 | 74.47 | 78.98 | 76.68
Bert (frozen) + CNN | SDL | 80.41 | 74.05 | 80.41 | 74.05 | 80.41 | 74.05
2-8 | FT | 77.15 | 71.03 | 78.84 | 75.67 | 78.00 | 73.35
| L2 | 62.79 | 54.26 | 72.95 | 69.54 | 67.87 | 61.90
| A-GEM | 76.26 | 70.20 | 75.70 | 71.37 | 75.98 | 70.79
| DER++ | 74.84 | 69.29 | 76.74 | 71.86 | 75.79 | 70.58
| EWC | 78.13 | 73.09 | 80.15 | 77.37 | 79.14 | 75.23
| OWM | 77.63 | 72.40 | 72.93 | 67.34 | 75.28 | 69.87
| UCL | 81.01 | 76.90 | 80.73 | 77.68 | 80.87 | 77.29
| HAT | 83.36 | 79.94 | 75.53 | 71.27 | 79.44 | 75.61
| CAT | 81.20 | 75.72 | 73.25 | 70.15 | 77.23 | 72.94
| KAN | 78.18 | 74.73 | 75.25 | 71.53 | 76.72 | 73.13
| SRK | 79.36 | 76.98 | 75.22 | 71.65 | 77.29 | 74.32
| HOP (ours) | 85.80 | 83.67 | 82.82 | 80.97 | 84.31 | 82.32
Adapter-BERT | SDL | 83.86 | 80.77 | 83.86 | 80.77 | 83.86 | 80.77
2-8 | FT | 62.35 | 54.60 | 80.48 | 77.41 | 71.42 | 66.00
| L2 | 70.92 | 67.08 | 78.22 | 74.27 | 74.57 | 70.67
| A-GEM | 63.12 | 56.16 | 79.85 | 71.42 | 71.48 | 63.79
| DER++ | 60.70 | 55.74 | 79.95 | 77.38 | 70.32 | 66.56
| EWC | 62.98 | 56.65 | 78.90 | 74.10 | 70.94 | 65.37
| OWM | 74.79 | 71.51 | 77.73 | 75.16 | 76.26 | 73.33
| UCL | 59.63 | 47.41 | 69.31 | 59.46 | 64.47 | 53.44
| HAT | 84.34 | 81.65 | 81.06 | 78.35 | 82.70 | 80.00
| B-CL | 84.08 | 81.76 | 81.92 | 79.94 | 83.00 | 80.85
| CTR | 86.79 | 85.21 | 82.01 | 80.05 | 84.40 | 82.63
| HOP (ours) | 88.52 | 87.13 | 84.48 | 82.48 | 86.50 | 84.80

Main Results. We evaluate methods on 5 benchmarks (ASC, DSC small, DSC full,
20News, NLI) targeting 4 applications (ASC, DSC, TC, NLI) in 2 CL setups (DIL
and TIL) using 3 network architectures based on BERT. Table 1 shows that HOP
clearly outperforms or achieves comparable results to baseline competitors in
every scenario.

In the first block, we evaluate them on BF + Lin. Due to the low accuracy of
this architecture, we compare our framework only against SDL and FT. HOP
outperforms both by a large margin in every case. In the second block, we use
BF + CNN (Kim 2014). Here, we report comparison against several approaches (we
note that B-CL and CTR cannot be employed with a CNN head). Finally, the best
results are achieved on Adapter-BERT reported in the third block (we remark
that CAT, KAN, SRK cannot work with adapters). We observe that $mAcc$ and
$MF1$ generally show consensus in identifying the best methods. Also, results
are higher in DIL since a single head can transfer knowledge more easily.

SDL outperforms some approaches, due to increased model size for adaptation to
end problems. However, it builds a model for each problem independently using
a separate network, therefore, it does not handle CF or KT. On the other hand:
FT, regularization-based approaches (such as EWC, OWM, and L2) and replay-
based approaches (such as A-GEM and DER++) perform generally better in BF+CNN
than in Adapter-BERT, due to the fewer parameters used to update models and
apply regularization on them.

KAN and HAT require problem identifier, and suffer from CF in TIL. We extended
them to DIL by using the last model, which, however, shows low results in DIL.
Similarly, also CAT (which extends HAT), SRK and UCL cannot achieve
competitive results. Approaches specifically designed for CL in NLP (_i.e_.,
B-CL and CTR) show clear improvements compared to the others. B-CL and CTR
have been mainly designed for TIL: they achieve competitive results in TIL
setup, however they fail when employed in DIL. HOP outperforms or it is
comparable to the current state-of-the-art competitors in every scenario, and
it can deal both with large scale data (_e.g_., DSC full) and with limited
data (_e.g_., DSC small) in both TIL and DIL. We confirm these findings by
looking at the aggregate results reported in Table 2. The results show that
HOP robustly outperforms all baseline competitors in both TIL and DIL (first
and second vertical groups). Also, CIL-based methods are inadequate for TIL
and DIL in NLP. Finally, the last vertical block provides a further comparison
aggregated across all benchmarks and CL setups, and is helpful to grasp an
overall sense of the results. In general, the best performing frameworks for
BF+CNN are HOP, UCL and HAT; while for Adapter-BERT, they are HOP, CTR, B-CL,
and HAT.

Table 3: Collection of metrics on the DSC small dataset for every competing
approach on both TIL and DIL setups.

| TIL | DIL |  |  |
---|---|---|---|---|---
| $\mathbf{mAcc}\uparrow$ | $\mathbf{MF1}\uparrow$ | $\mathbf{BwT}\uparrow$ | $\mathbf{FwT}\uparrow$ | $\mathbf{Forg}\downarrow$ | $\mathbf{Pla}\uparrow$ | $\mathbf{mAcc}\uparrow$ | $\mathbf{MF1}\uparrow$ | $\mathbf{BwT}\uparrow$ | $\mathbf{FwT}\uparrow$ | $\mathbf{Forg}\downarrow$ | $\mathbf{Pla}\uparrow$ |  #OP $\downarrow$ |  #TP $\downarrow$ |  Time $\downarrow$
FT | 55.19 | 35.28 | 59.59 | 45.50 | 28.79 | 74.71 | 82.22 | 81.32 | 83.17 | 79.24 | 5.56 | 82.99 | 183.3 | 73.8 | 1.2
L2 | 70.87 | 69.11 | 71.48 | 57.46 | 13.60 | 82.86 | 83.78 | 82.86 | 83.48 | 81.22 | 1.44 | 82.99 | 183.3 | 73.8 | 1.4
A-GEM | 59.35 | 54.20 | 56.77 | 46.14 | 26.05 | 84.24 | 86.25 | 85.49 | 84.92 | 83.03 | 0.90 | 84.80 | 183.3 | 73.8 | 8.7
DER++ | 63.11 | 61.96 | 50.92 | 40.26 | 22.76 | 83.31 | 84.00 | 83.16 | 85.06 | 80.97 | 3.46 | 84.15 | 183.3 | 73.8 | 1.2
EWC | 58.34 | 42.85 | 54.15 | 43.10 | 25.65 | 79.13 | 87.29 | 86.34 | 86.05 | 82.25 | 0.83 | 83.19 | 183.3 | 73.8 | 1.3
OWM | 73.97 | 71.96 | 72.19 | 58.32 | 12.40 | 83.39 | 77.32 | 76.68 | 73.78 | 54.12 | 6.76 | 79.87 | 184.4 | 74.8 | 1.4
UCL | 48.36 | 32.07 | 51.19 | 49.21 | 7.48 | 52.12 | 56.53 | 52.96 | 51.07 | 43.16 | 7.08 | 57.71 | 183.4 | 73.9 | 1.3
HAT | 80.83 | 78.41 | 75.97 | 58.15 | 6.50 | 83.12 | 86.22 | 85.44 | 85.25 | 80.57 | 1.57 | 84.26 | 184.0 | 74.5 | 1.6
B-CL | 84.34 | 83.12 | 84.76 | 48.40 | 0.67 | 81.98 | 85.92 | 85.11 | 85.18 | 78.73 | 1.34 | 85.92 | 220.2 | 110.7 | 4.8
CTR | 83.96 | 83.00 | 84.01 | 47.95 | 0.28 | 82.86 | 86.01 | 85.16 | 85.26 | 78.81 | 1.56 | 85.89 | 186.6 | 77.1 | 30.8
HOP (ours) | 85.63 | 84.18 | 84.39 | 45.32 | 0.57 | 82.76 | 87.84 | 86.96 | 85.11 | 82.33 | 0.39 | 84.41 | 189.2 | 79.7 | 1.3
B-CL + HOP (ours) | 86.93 | 86.25 | 86.84 | 51.47 | 0.24 | 86.53 | 87.08 | 86.30 | 86.47 | 84.49 | 0.88 | 87.22 | 244.8 | 134.3 | 4.9
CTR + HOP (ours) | 87.08 | 86.32 | 86.67 | 57.04 | 0.49 | 86.33 | 86.54 | 85.73 | 85.73 | 81.72 | 1.44 | 86.26 | 210.2 | 100.8 | 32.7

TIL | DIL
---|---
|  |  |

Figure 3: Per-problem accuracy ($mAcc_{t}$ and $mAcc_{t,\leq T}$) on the DSC
small dataset for both TIL and DIL setups.

CF and KT. We report additional metrics to evaluate the intrinsic CF and KT
properties of CL models in Table 3 for both TIL and DIL in the DSC small
dataset. Most regularization- and replay-based approaches designed for image
classification (first group of eight rows) are inadequate to address CL in
NLP. These methods show low accuracy due to high forgetting and low KT ($BwT$
and $FwT$), despite having good plasticity ($Pla$) to learn representations
for a new problem. Methods designed for CL in NLP (second group of three
rows), instead, can effectively increase accuracy ($mAcc$ and $MF1$) by
increasing KT, reducing $Forg$ whilst maintaining $Pla$. Compared to
competitors, our HOP can find a better balance between CF and KT. In both TIL
and DIL, modeling high order statistics using HOP leads to increased $mAcc$
and $MF1$ by reducing $Forg$, although showing comparable or more conservative
results in terms of KT properties ($BwT$, $FwT$ and $Pla$ of HOP are not
always maximized). Overall, our framework achieves a better trade-off and
outperforms methods proposed specifically for TIL and for DIL.

Table 4: Combination of HOP with competing CL methods on the DSC small
dataset.

HOP + | TIL | DIL |
---|---|---|---
$\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$ | Time [min]
FT | 85.63 (+55.2%) | 84.18 (+138.6%) |  87.94 (+7.0%) |  87.04 (+7.0%) | 1.3 (+8.3%)
L2 | 82.57 (+16.5%) | 80.02 (+15.8%) | 85.60 (+2.2%) | 84.84 (+2.4%) | 1.4 (+0.0%)
A-GEM | 86.87 (+46.4%) | 85.96 (+58.6%) | 86.91 (+0.8 %) | 86.08 (+0.7%) | 8.8 (+1.1%)
DER++ | 85.49 (+35.5%) | 83.97 (+35.5%) | 87.00 (+3.6%) | 86.05 (+3.5%) | 1.5 (+25.0%)
EWC | 84.43 (+44.7%) | 83.30 (+94.4%) | 86.91 (-0.4%) | 86.14 (-0.2 %) | 1.4 (+7.7%)
B-CL | 86.93 (+3.1%) | 86.25 (+3.8%) | 87.08 (+1.4%) | 86.30 (+1.4%) | 4.9 (+2.1%)
CTR |  87.08 (+3.7%) |  86.32 (+4.0%) | 86.54 (+0.6%) | 85.73 (+0.7%) | 32.7 (+6.2%)

HOP Improves Other CL Methods. To ensure that HOP is beneficial to CL in NLP
applications, we include it in competing CL methods and report the results in
Table 4 for DSC small in both TIL and DIL. Comparing the results against Table
1 (gains are reported within brackets in Table 4 for convenience) emerges
clear how HOP improves CL methods almost every time, only exception for EWC
DIL. In some cases, we observe a large gain up to about $140\%$. The gain is
experienced in both TIL and DIL, the former being more largely improved by our
HOP. The integrated methods robustly outperform the original methods along all
the evaluated metrics. Remarkably, also current state-of-the-art approaches as
B-CL and CTR are significantly improved by our framework.

Per-Problem Acc is shown in Figure 3. Acc evolution over problem is measured
by $mAcc_{t}$ (_i.e_., per-problem accuracy averaged over all problems) and
$mAcc_{t,\leq T}$ (_i.e_., per-problem accuracy averaged over the problems
seen so far). FT exhibits a clear performance drop due to CF and inability to
perform KT. Methods designed for CL in NLP show an almost perfect
monotonically increasing behaviour of $mAcc_{t,\leq T}$, since they are
capable of learning new problems (high plasticity) without forgetting previous
ones.

Efficiency. HOP only adds a small increase in parameters and computation time.
We observe in Table 3 that adapters of Adapter-BERT account only for about
$40\%$ (73.8M) of the total number of parameters (183.3M), while clearly
outperforming architectures with only a linear or convolution-based trainable
head (see Table 1). Compared to FT, HOP introduces just about $3\%$ more total
parameters, increasing the average running training time per problem by about
$8\%$ ($1.2$ to $1.3$ min). We confirm in Table 4 that HOP only adds a minimal
increase in computation time when added on top of existing CL methods. On
average, HOP increases the mean running time per problem by just $7.2\%$.

We further highlight the effectiveness of HOP in Figure 4. Our approach is
more computationally efficient than existing methods while outperforming them
in terms of accuracy. In particular, HOP is much faster than the main
competitors being about $24\times$ faster than CTR, $4\times$ faster than
B-CL.

Figure 4: mAcc vs. training time per problem on the TIL setup. Optimal results are in the top-left corner. Table 5: Ablation results on DSC small for different pooling schemes and HOP with different values of $R$. | TIL | DIL
---|---|---
| $\mathbf{mAcc}$ | $\mathbf{MF1}$ | $\mathbf{mAcc}$ | $\mathbf{MF1}$
[CLS] (Devlin et al. 2019) | 55.19 | 35.28 | 82.22 | 81.32
AVG (LeCun et al. 1998) | 81.44 | 80.53 | 82.94 | 81.68
MAX (Riesenhuber and Poggio 1999) | 79.49 | 78.48 | 82.63 | 81.26
AVGMAX (Monteiro, Alam, and Falk 2020) | 81.51 | 80.47 | 83.34 | 82.27
TSDP (Wang et al. 2021) | 77.52 | 76.30 | 78.91 | 77.50
iSQRT-COV (Li et al. 2018) | 81.47 | 80.59 | 79.62 | 78.06
HOP $p=2$ (ours) | 83.52 | 82.53 | 86.91 | 86.12
HOP $p=3$ (ours) | 85.63 | 84.18 | 87.84 | 86.96
HOP $p=4$ (ours) | 84.47 | 83.58 | 86.65 | 85.81
HOP with $m_{1}=$ [CLS] (ours) | 85.47 | 84.61 | 87.50 | 86.70

Other Pooling Schemes and Order of HOP. Next, we observe in Table 5 how
popular pooling schemes underperform our solution. As a baseline, [CLS] token
is used for the final classification (Devlin et al. 2019): its low results are
due to the variable distribution of tokens over training. AVG pooling (LeCun
et al. 1998) already shows remarkable improvements especially in handling the
problem variability in TIL. MAX pooling (Riesenhuber and Poggio 1999) has a
slightly worse effect than AVG. Concatenating AVG and MAX (AVGMAX, Monteiro,
Alam, and Falk 2020) improves compared to using single clues alone. Employment
of second order statistics of tokens alone, _i.e_. either standard deviation
(TSDP, Wang et al. 2021) or covariance (iSQRT-COV, Li et al. 2018), improves
on TIL but not on DIL compared to the baseline. HOP with $p=2$ improves
results compared to AVGMAX whilst using the same number of statistical
measures from the distribution of tokens. We observe that the best results are
obtained for HOP with $p=3$. Intuitively, features of different problems have
similar distribution of first moments, while higher moments are discriminative
for the specific problems. We compute the average Wasserstein distance between
distributions of features of different problems to quantify this effect. The
mean distance of first moments ($0.15\pm 0.03$) is considerably lower than the
mean distance of second ($0.36\pm 0.05$) and third moments ($0.26\pm 0.07$),
indicating that problems are more entangled in the feature space of first
moment (lower distance) than of second-third moments. Thus, second-third
moments improve accuracy. On the other hand, moments $>3$ show lower distance
(_e.g_., the average distance of fourth moments is $0.04\pm 0.01$) and yield
lower results (Table 5). Therefore, they have been ignored. HOP with $m_{1}=$
[CLS] concatenates the [CLS] token with high order statistics and shows
results similar to our framework, suggesting that the [CLS] token can be used
in conjunction with high order statistics in par with AVG. In other words, in
HOP, $m_{1}$ can be either AVG or [CLS].

## Conclusion

We proposed HOP, which, to our knowledge, is the first CL method for both TIL
and DIL in various NLP applications (ASC, DSC, NLI, TC). HOP is a novel
approach to adapt a pre-trained NLP model for CL. HOP relies on adapter
modules and auxiliary MLPs specialized for each problem. Then, it computes
high order moments of embedded tokens to extract rich sentence-wide
information, opposed to relying on a single token for classification (_e.g_.,
[CLS]), which fails to adapt to dynamic non-stationary input distributions.
HOP encourages KT among problems and protects problem-specific knowledge
reducing CF. Experiments show that HOP sets new state-of-the-art results on
the most widely used CL NLP scenarios. At the same time, HOP only adds a
minimal computation footprint.

## References

  * Ács, Kádár, and Kornai (2021) Ács, J.; Kádár, A.; and Kornai, A. 2021.  Subword Pooling Makes a Difference.  In _Conference of the European Chapter of the Association for Computational Linguistics (EACL)_. Association for Computational Linguistics.
  * Ahn et al. (2019) Ahn, H.; Cha, S.; Lee, D.; and Moon, T. 2019.  Uncertainty-based continual learning with adaptive regularization.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 32\.
  * Asghar et al. (2020) Asghar, N.; Mou, L.; Selby, K. A.; Pantasdo, K. D.; Poupart, P.; and Jiang, X. 2020\.  Progressive Memory Banks for Incremental Domain Adaptation.  In _International Conference on Learning Representations (ICLR)_.
  * Biesialska, Biesialska, and Costa-jussà (2020) Biesialska, M.; Biesialska, K.; and Costa-jussà, M. R. 2020.  Continual Lifelong Learning in Natural Language Processing: A Survey.  In _International Conference on Computational Linguistics (COLING)_ , 6523–6541.
  * Buzzega et al. (2020) Buzzega, P.; Boschini, M.; Porrello, A.; Abati, D.; and Calderara, S. 2020.  Dark experience for general continual learning: a strong, simple baseline.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 33: 15920–15930.
  * Carlson et al. (2010) Carlson, A.; Betteridge, J.; Kisiel, B.; Settles, B.; Hruschka, E. R.; and Mitchell, T. M. 2010.  Toward an architecture for never-ending language learning.  In _AAAI Conference on Artificial Intelligence_.
  * Chaudhry et al. (2018) Chaudhry, A.; Ranzato, M.; Rohrbach, M.; and Elhoseiny, M. 2018.  Efficient Lifelong Learning with A-GEM.  In _International Conference on Learning Representations (ICLR)_.
  * Chen and Liu (2018) Chen, Z.; and Liu, B. 2018.  Lifelong machine learning.  _Synthesis Lectures on Artificial Intelligence and Machine Learning_ , 12(3): 1–207.
  * Chen, Ma, and Liu (2015) Chen, Z.; Ma, N.; and Liu, B. 2015.  Lifelong Learning for Sentiment Classification.  In _Association for Computational Linguistics and International Joint Conference on Natural Language Processing_ , 750–756.
  * Chien and Chen (2021) Chien, J.-T.; and Chen, Y.-H. 2021.  Continuous-time attention for sequential learning.  _AAAI Conference on Artificial Intelligence_ , 35(8): 7116–7124.
  * Chuang, Su, and Chen (2020) Chuang, Y.-S.; Su, S.-Y.; and Chen, Y.-N. 2020.  Lifelong Language Knowledge Distillation.  In _Conference on Empirical Methods in Natural Language Processing (EMNLP)_.
  * Cortes and Mohri (2004) Cortes, C.; and Mohri, M. 2004.  Distribution kernels based on moments of counts.  In _International Conference on Machine Learning (ICML)_ , 25.
  * De Lange et al. (2021) De Lange, M.; Aljundi, R.; Masana, M.; Parisot, S.; Jia, X.; Leonardis, A.; Slabaugh, G.; and Tuytelaars, T. 2021.  A continual learning survey: Defying forgetting in classification tasks.  _IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI)_ , 44(7): 3366–3385.
  * Devlin et al. (2019) Devlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K. 2019.  BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding.  In _Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , 4171–4186.
  * Ding, Liu, and Yu (2008) Ding, X.; Liu, B.; and Yu, P. S. 2008.  A holistic lexicon-based approach to opinion mining.  In _International Conference on Web Search and Data Mining_ , 231–240.
  * Greco et al. (2019) Greco, C.; Plank, B.; Fernández, R.; and Bernardi, R. 2019.  Psycholinguistics Meets Continual Learning: Measuring Catastrophic Forgetting in Visual Question Answering.  In _Annual Meeting of the Association for Computational Linguistics_ , 3601–3605.
  * Houlsby et al. (2019) Houlsby, N.; Giurgiu, A.; Jastrzebski, S.; Morrone, B.; De Laroussilhe, Q.; Gesmundo, A.; Attariyan, M.; and Gelly, S. 2019.  Parameter-efficient transfer learning for NLP.  In _Proceeding of the International Conference on Machine Learning (ICML)_ , 2790–2799. PMLR.
  * Hu and Liu (2004) Hu, M.; and Liu, B. 2004.  Mining and summarizing customer reviews.  In _ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , 168–177.
  * Hu et al. (2021) Hu, W.; Qin, Q.; Wang, M.; Ma, J.; and Liu, B. 2021.  Continual learning by using information of each class holistically.  _AAAI Conference on Artificial Intelligence_ , 35(9).
  * Isele and Cosgun (2018) Isele, D.; and Cosgun, A. 2018.  Selective experience replay for lifelong learning.  In _AAAI Conference on Artificial Intelligence_ , volume 32.
  * Jung et al. (2016) Jung, H.; Ju, J.; Jung, M.; and Kim, J. 2016.  Less-forgetting learning in deep neural networks.  _arXiv:1607.00122_.
  * Ke and Liu (2022) Ke, Z.; and Liu, B. 2022.  Continual Learning of Natural Language Processing Tasks: A Survey.  _arXiv:2211.12701_.
  * Ke, Liu, and Huang (2020) Ke, Z.; Liu, B.; and Huang, X. 2020.  Continual learning of a mixed sequence of similar and dissimilar tasks.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 33\.
  * Ke et al. (2021a) Ke, Z.; Liu, B.; Ma, N.; Xu, H.; and Shu, L. 2021a.  Achieving Forgetting Prevention and Knowledge Transfer in Continual Learning.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 34: 22443–22456.
  * Ke et al. (2020) Ke, Z.; Liu, B.; Wang, H.; and Shu, L. 2020.  Continual learning with knowledge transfer for sentiment classification.  In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , 683–698. Springer.
  * Ke et al. (2021b) Ke, Z.; Liu, B.; Xu, H.; and Shu, L. 2021b.  CLASSIC: Continual and Contrastive Learning of Aspect Sentiment Classification Tasks.  In _Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , 6871–6883.
  * Ke, Xu, and Liu (2021) Ke, Z.; Xu, H.; and Liu, B. 2021.  Adapting BERT for Continual Learning of a Sequence of Aspect Sentiment Classification Tasks.  _Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , 4746–4755.
  * Khayrallah et al. (2018) Khayrallah, H.; Thompson, B.; Duh, K.; and Koehn, P. 2018.  Regularized Training Objective for Continued Training for Domain Adaptation in Neural Machine Translation.  In _Workshop on Neural Machine Translation and Generation_ , 36–44. Melbourne, Australia: Association for Computational Linguistics.
  * Kim (2014) Kim, Y. 2014.  Convolutional Neural Networks for Sentence Classification.  In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , 1746–1751. Doha, Qatar: Association for Computational Linguistics.
  * Kirkpatrick et al. (2017) Kirkpatrick, J.; Pascanu, R.; Rabinowitz, N.; Veness, J.; Desjardins, G.; Rusu, A. A.; Milan, K.; Quan, J.; Ramalho, T.; Grabska-Barwinska, A.; et al. 2017.  Overcoming catastrophic forgetting in neural networks.  _Proceedings of the National Academy of Sciences (PNAS)_ , 114(13): 3521–3526.
  * Lang (1995) Lang, K. 1995.  Newsweeder: Learning to filter netnews.  In _Machine Learning Proceedings_ , 331–339. Elsevier.
  * LeCun et al. (1998) LeCun, Y.; Bottou, L.; Bengio, Y.; and Haffner, P. 1998.  Gradient-based learning applied to document recognition.  _Proc. of the IEEE_ , 86(11): 2278–2324.
  * Lesort et al. (2020) Lesort, T.; Lomonaco, V.; Stoian, A.; Maltoni, D.; Filliat, D.; and Díaz-Rodríguez, N. 2020.  Continual learning for robotics: Definition, framework, learning strategies, opportunities and challenges.  _Information Fusion_ , 58: 52–68.
  * Li et al. (2018) Li, P.; Xie, J.; Wang, Q.; and Gao, Z. 2018.  Towards faster training of global covariance pooling networks by iterative matrix square root normalization.  In _Conference on Computer Vision and Pattern Recognition (CVPR)_ , 947–955.
  * Li et al. (2019) Li, Y.; Zhao, L.; Church, K.; and Elhoseiny, M. 2019.  Compositional Language Continual Learning.  In _International Conference on Learning Representations (ICLR)_.
  * Li and Hoiem (2017) Li, Z.; and Hoiem, D. 2017.  Learning without forgetting.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 40(12): 2935–2947.
  * Liu et al. (2015) Liu, Q.; Gao, Z.; Liu, B.; and Zhang, Y. 2015.  Automated rule selection for aspect extraction in opinion mining.  In _International Joint Conference on Artificial Intelligence (IJCAI)_.
  * Liu, Ungar, and Sedoc (2019) Liu, T.; Ungar, L.; and Sedoc, J. 2019.  Continual Learning for Sentence Representations Using Conceptors.  In _Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , 3274–3279.
  * Liu et al. (2020) Liu, Z.; Winata, G. I.; Madotto, A.; and Fung, P. 2020.  Exploring fine-tuning techniques for pre-trained cross-lingual models via continual learning.  _arXiv:2004.14218_.
  * Lopez-Paz and Ranzato (2017) Lopez-Paz, D.; and Ranzato, M. 2017.  Gradient episodic memory for continual learning.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 30\.
  * Lv et al. (2019) Lv, G.; Wang, S.; Liu, B.; Chen, E.; and Zhang, K. 2019.  Sentiment classification by leveraging the shared knowledge from a sequence of domains.  In _International Conference on Database Systems for Advanced Applications_ , 795–811. Springer.
  * Madotto et al. (2020) Madotto, A.; Lin, Z.; Zhou, Z.; Moon, S.; Crook, P.; Liu, B.; Yu, Z.; Cho, E.; and Wang, Z. 2020.  Continual learning in task-oriented dialogue systems.  _arXiv:2012.15504_.
  * Mai et al. (2022) Mai, Z.; Li, R.; Jeong, J.; Quispe, D.; Kim, H.; and Sanner, S. 2022.  Online continual learning in image classification: An empirical survey.  _Neurocomputing_ , 469: 28–51.
  * Mallya, Davis, and Lazebnik (2018) Mallya, A.; Davis, D.; and Lazebnik, S. 2018.  Piggyback: Adapting a single network to multiple tasks by learning to mask weights.  In _European Conference on Computer Vision (ECCV)_ , 67–82.
  * Mallya and Lazebnik (2018) Mallya, A.; and Lazebnik, S. 2018.  Packnet: Adding multiple tasks to a single network by iterative pruning.  In _Conference on Computer Vision and Pattern Recognition (CVPR)_ , 7765–7773.
  * Maracani et al. (2021) Maracani, A.; Michieli, U.; Toldo, M.; and Zanuttigh, P. 2021.  Recall: Replay-based continual learning in semantic segmentation.  In _International Conference on Computer Vision (ICCV)_ , 7026–7035.
  * McCloskey and Cohen (1989) McCloskey, M.; and Cohen, N. J. 1989.  Catastrophic interference in connectionist networks: The sequential learning problem.  In _Psychology of Learning and Motivation_ , volume 24, 109–165. Elsevier.
  * Michieli et al. (2024) Michieli, U.; Moon, J.; Kim, D.; and Ozay, M. 2024.  Object-Conditioned Bag of Instances for Few-Shot Personalized Instance Recognition.  In _IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_.
  * Michieli and Ozay (2023) Michieli, U.; and Ozay, M. 2023.  Online Continual Learning for Robust Indoor Object Recognition.  _International Conference on Intelligent Robotics_.
  * Michieli, Parada, and Ozay (2023) Michieli, U.; Parada, P. P.; and Ozay, M. 2023.  Online Continual Learning in Keyword Spotting for Low-Resource Devices via Pooling High-Order Temporal Statistics.  _INTERSPEECH_.
  * Michieli, Toldo, and Zanuttigh (2022) Michieli, U.; Toldo, M.; and Zanuttigh, P. 2022.  Domain adaptation and continual learning in semantic segmentation.  In Davies, E.; and Turk, M. A., eds., _Advanced Methods and Deep Learning in Computer Vision_ , Computer Vision and Pattern Recognition, 275–303. Academic Press.  ISBN 978-0-12-822109-9.
  * Michieli and Zanuttigh (2019) Michieli, U.; and Zanuttigh, P. 2019.  Incremental learning techniques for semantic segmentation.  In _International Conference on Computer Vision Workshops (ICCVW)_ , 0–0.
  * Michieli and Zanuttigh (2021) Michieli, U.; and Zanuttigh, P. 2021.  Knowledge distillation for incremental learning in semantic segmentation.  _Computer Vision and Image Understanding (CVIU)_ , 205: 103167.
  * Mirzadeh et al. (2022) Mirzadeh, S. I.; Chaudhry, A.; Yin, D.; Nguyen, T.; Pascanu, R.; Gorur, D.; and Farajtabar, M. 2022.  Architecture matters in continual learning.  _arXiv:2202.00275_.
  * Monaikul et al. (2021) Monaikul, N.; Castellucci, G.; Filice, S.; and Rokhlenko, O. 2021.  Continual learning for named entity recognition.  _AAAI Conference on Artificial Intelligence_ , 35(15): 13570–13577.
  * Monteiro, Alam, and Falk (2020) Monteiro, J.; Alam, M. J.; and Falk, T. 2020.  On the performance of time-pooling strategies for end-to-end spoken language identification.  In _Conference on Language Resources and Evaluation Conference (LREC)_ , 3566–3572.
  * Nguyen et al. (2018) Nguyen, C. V.; Li, Y.; Bui, T. D.; and Turner, R. E. 2018.  Variational Continual Learning.  In _International Conference on Learning Representations (ICLR)_.
  * Papoulis and U. Pillai (2002) Papoulis, A.; and U. Pillai, S. 2002.  _Probability, random variables and stochastic processes_.
  * Pfeiffer et al. (2021) Pfeiffer, J.; Kamath, A.; Rücklé, A.; Cho, K.; and Gurevych, I. 2021.  AdapterFusion: Non-destructive task composition for transfer learning.  In _Conference of the European Chapter of the Association for Computational Linguistics (EACL)_ , 487–503. Association for Computational Linguistics (ACL).
  * Pfeiffer et al. (2020) Pfeiffer, J.; Vulić, I.; Gurevych, I.; and Ruder, S. 2020.  MAD-X: An Adapter-Based Framework for Multi-Task Cross-Lingual Transfer.  In _Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , 7654–7673.
  * Pontiki et al. (2014) Pontiki, M.; Galanis, D.; Pavlopoulos, J.; Papageorgiou, H.; Androutsopoulos, I.; and Manandhar, S. 2014.  SemEval-2014 Task 4: Aspect Based Sentiment Analysis.  In _PInternational Workshop on Semantic Evaluation (SemEval 2014)_ , 27–35. Dublin, Ireland: Association for Computational Linguistics.
  * Qian, Wei, and Yu (2021) Qian, K.; Wei, W.; and Yu, Z. 2021.  A student-teacher architecture for dialog domain adaptation under the meta-learning setting.  _AAAI Conference on Artificial Intelligence_ , 35(15): 13692–13700.
  * Qin, Hu, and Liu (2020) Qin, Q.; Hu, W.; and Liu, B. 2020.  Using the past knowledge to improve sentiment classification.  In _Findings of the Association for Computational Linguistics: Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , 1124–1133.
  * Radford et al. (2019) Radford, A.; Wu, J.; Child, R.; Luan, D.; Amodei, D.; Sutskever, I.; et al. 2019\.  Language models are unsupeke2022continualpretrainingrvised multitask learners.  _OpenAI blog_ , 1(8): 9.
  * Rebuffi et al. (2017) Rebuffi, S.-A.; Kolesnikov, A.; Sperl, G.; and Lampert, C. H. 2017.  icarl: Incremental classifier and representation learning.  In _Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2001–2010.
  * Riesenhuber and Poggio (1999) Riesenhuber, M.; and Poggio, T. 1999.  Hierarchical models of object recognition in cortex.  _Nature Neural Networks (NN)_ , 2(11): 1019–1025.
  * Rusu et al. (2016) Rusu, A. A.; Rabinowitz, N. C.; Desjardins, G.; Soyer, H.; Kirkpatrick, J.; Kavukcuoglu, K.; Pascanu, R.; and Hadsell, R. 2016.  Progressive neural networks.  _arXiv:1606.04671_.
  * Ruvolo and Eaton (2013) Ruvolo, P.; and Eaton, E. 2013.  ELLA: An efficient lifelong learning algorithm.  In _International Conference on Machine Learning (ICML)_ , 507–515. PMLR.
  * Serra et al. (2018) Serra, J.; Suris, D.; Miron, M.; and Karatzoglou, A. 2018.  Overcoming catastrophic forgetting with hard attention to the task.  In _International Conference on Machine Learning (ICML)_ , 4548–4557. PMLR.
  * Shen, Zeng, and Jin (2019) Shen, Y.; Zeng, X.; and Jin, H. 2019.  A progressive model to enable continual learning for semantic slot filling.  In _Conference on Empirical Methods in Natural Language Processing and International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , 1279–1284.
  * Shin et al. (2017) Shin, H.; Lee, J. K.; Kim, J.; and Kim, J. 2017.  Continual learning with deep generative replay.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 30\.
  * Silver, Yang, and Li (2013) Silver, D. L.; Yang, Q.; and Li, L. 2013.  Lifelong machine learning systems: Beyond learning algorithms.  In _AAAI spring symposium series_.
  * Sun, Ho, and Lee (2019) Sun, F.-K.; Ho, C.-H.; and Lee, H.-Y. 2019.  LAMOL: LAnguage MOdeling for Lifelong Language Learning.  In _International Conference on Learning Representations (ICLR)_.
  * Van de Ven and Tolias (2019) Van de Ven, G. M.; and Tolias, A. S. 2019.  Three scenarios for continual learning.  _arXiv:1904.07734_.
  * Wang et al. (2019) Wang, H.; Liu, B.; Wang, S.; Ma, N.; and Yang, Y. 2019.  Forward and backward knowledge transfer for sentiment classification.  In _Asian Conference on Machine Learning (ACML)_ , 457–472. PMLR.
  * Wang et al. (2023a) Wang, J.; Dong, D.; Shou, L.; Chen, K.; and Chen, G. 2023a.  Effective Continual Learning for Text Classification with Lightweight Snapshots.  _AAAI Conference on Artificial Intelligence_ , 37(8): 10122–10130.
  * Wang et al. (2018) Wang, S.; Lv, G.; Mazumder, S.; Fei, G.; and Liu, B. 2018.  Lifelong learning memory networks for aspect sentiment classification.  In _IEEE International Conference on Big Data (Big Data)_ , 861–870. IEEE.
  * Wang et al. (2021) Wang, S.; Yang, Y.; Qian, Y.; and Yu, K. 2021.  Revisiting the statistics pooling layer in deep speaker embedding learning.  In _International Symposium on Chinese Spoken Language Processing (ISCSLP)_ , 1–5. IEEE.
  * Wang et al. (2023b) Wang, Y.; Ma, Z.; Huang, Z.; Wang, Y.; Su, Z.; and Hong, X. 2023b.  Isolation and impartial aggregation: A paradigm of incremental learning without interference.  _AAAI Conference on Artificial Intelligence_ , 37(8): 10209–10217.
  * Williams, Nangia, and Bowman (2018) Williams, A.; Nangia, N.; and Bowman, S. R. 2018.  A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference.  In _Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_. Association for Computational Linguistics.
  * Wu et al. (2020) Wu, C.; Wu, F.; Qi, T.; Cui, X.; and Huang, Y. 2020.  Attentive pooling with learnable norms for text representation.  In _Annual Meeting of the Association for Computational Linguistics_ , 2961–2970.
  * Wu et al. (2023) Wu, T.; Liu, Q.; Cao, Y.; Huang, Y.; Wu, X.-M.; and Ding, J. 2023.  Continual Graph Convolutional Network for Text Classification.  _AAAI Conference on Artificial Intelligence_.
  * Xu and Zhu (2018) Xu, J.; and Zhu, Z. 2018.  Reinforced continual learning.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 31\.
  * Zeng et al. (2019) Zeng, G.; Chen, Y.; Cui, B.; and Yu, S. 2019.  Continual learning of context-dependent processing in neural networks.  _Nature Machine Intelligence_ , 1(8): 364–372.
  * Zenke, Poole, and Ganguli (2017) Zenke, F.; Poole, B.; and Ganguli, S. 2017.  Continual learning through synaptic intelligence.  In _International Conference on Machine Learning (ICML)_ , 3987–3995. PMLR.
  * Zhan et al. (2021) Zhan, R.; Liu, X.; Wong, D. F.; and Chao, L. S. 2021.  Meta-curriculum learning for domain adaptation in neural machine translation.  _AAAI Conference on Artificial Intelligence_ , 35(16): 14310–14318.
  * Zhao et al. (2022) Zhao, S.; You, F.; Chang, W.; Zhang, T.; and Hu, M. 2022.  Augment BERT with average pooling layer for Chinese summary generation.  _Journal of Intelligent & Fuzzy Systems_, 1–10.
  * Zhou et al. (2021) Zhou, Y.; Zhu, F.; Song, P.; Han, J.; Guo, T.; and Hu, S. 2021.  An adaptive hybrid framework for cross-domain aspect-based sentiment analysis.  _AAAI Conference on Artificial Intelligence_ , 35(16): 14630–14637.


# Implications of non-extensivity on Gamow theory

H<EMAIL_ADDRESS>M. Javaherian, E. Namvar, A. H. Ziaie
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
University of Maragheh, P.O. Box 55136-553, Maragheh, Iran

###### Abstract

Relying on the quantum tunneling concept and the Maxwell-Boltzmann-Gibbs
statistics, Gamow shows that the star burning process happens at temperatures
comparable to a critical value, called the Gamow temperature (T) and less than
the prediction of the classical framework. In order to highlight the role of
the equipartition theorem in the Gamow argument, a thermal length scale is
defined and thereinafter, the effects of non-extensivity on the Gamow
temperature have been investigated focusing on the Tsallis and Kaniadakis
statistics. The results attest that while the Gamow temperature decreases in
the framework of the Kaniadakis statistics, it can be bigger or smaller than T
when the Tsallis statistics is employed.

##  I Introduction

In the framework of the Maxwell-Boltzmann-Gibbs statistics, also known as the
Gibbs statistics, consider a gas with temperature $T_{g}$, including particles
with mass $m$ and mean velocity $v$. In this manner, using the equipartition
theorem, one finds

$\displaystyle\frac{1}{2}mv^{2}=\frac{3}{2}K_{B}T_{g}.$ (1)

Here, $K_{B}$ is the Boltzmann constant. For a pair of particles (called the
$i$-th and $j$-th particles) with atomic numbers $Z_{i}$ and $Z_{j}$,
respectively, located at a distance $r_{0}$ from each other, the Kinetic
energy lets particles overcome the Coulomb barrier meaning that nuclear fusion
begins, and consequently, a star is born if $\frac{3}{2}K_{B}T_{g}\geq
U(r_{0})=\frac{Z_{i}Z_{j}e^{2}}{4\pi\epsilon_{0}r_{0}}$ leading to 1

$\displaystyle
T_{g}\geq\frac{Z_{i}Z_{j}e^{2}}{6\pi\epsilon_{0}K_{B}r_{0}}\simeq 1\cdot
1\times 10^{10}\frac{Z_{i}Z_{j}}{r_{0}}\equiv T_{0},$ (2)

for the gas temperature. On the other hand, for the temperature of the gas, we
also have 1

$\displaystyle\mathcal{T}\approx 4\times
10^{6}(\frac{M}{M_{\odot}})(\frac{R_{\odot}}{R}),$ (3)

in which $M$ ($M_{\odot}$) and $R$ ($R_{\odot}$) denote the mass and radius of
the gas (Sun), respectively. As an example, consider the Sun for which we have
$\mathcal{T}\ll T_{0}$ meaning that the Sun should not burn 1 . Therefore,
nuclear fusion cannot be launched in gasses whose temperature ($T$) is lower
than $T_{0}$ (i.e., $T<T_{0}$) 1 .

Thanks to the scorching Sun, the above argument becomes questionable. Indeed,
Gamow was the first one who was able to find a proper answer by proposing a
mechanism: quantum tunneling 1 . Based on this theory, if the particles become
close to each other by their de Broglie wavelength
($r_{0}\simeq\frac{\hbar}{p}\equiv\lambda$), then they overcome the Coulomb
barrier. In this manner, the corresponding de Broglie wavelength of particles
can be calculated as

$\displaystyle\lambda=\frac{2\pi\epsilon_{0}\hbar^{2}}{mZ_{i}Z_{j}e^{2}},$ (4)

where $p=mv$ is considered. We have replaced $r_{0}$ with
$\lambda\equiv\frac{\hbar}{p}$ along with using

$\displaystyle\frac{p^{2}}{2m}=\frac{Z_{i}Z_{j}e^{2}}{4\pi\epsilon_{0}\lambda}.$
(5)

Now, with the help of $\frac{3}{2}K_{B}T_{g}\geq U(r_{0})$, one reaches 1

$\displaystyle
T_{g}\geq\frac{Z_{i}Z_{j}e^{2}}{6\pi\epsilon_{0}K_{B}\lambda}\simeq 9\cdot
6\times 10^{6}Z_{i}^{2}Z_{j}^{2}(\frac{m}{\frac{1}{2}})\equiv\texttt{T},$ (6)

instead of Eq. (2) meaning that nuclear fusion can be started in gases whose
temperature is comparable with T (the Gamow temperature) not $T_{0}$ 1 .
Moreover, using $\frac{3}{2}K_{B}T_{g}\geq U(r_{0})$, one gets

$\displaystyle\lambda\geq\frac{Z_{i}Z_{j}e^{2}}{6\pi\epsilon_{0}K_{B}T_{g}}\equiv
r_{0}^{T}.$ (7)

Now, bearing the equal signs within Eqs. (1) and (5) in mind, we can finally
deduce that the minimum requirement for quantum tunneling in a gas with
temperature $T_{g}\geq\texttt{T}$ is $\lambda=r_{0}^{T}$. Hence, the
equipartition theorem has a vital role i.e., if it changes then both Eq. (6)
and $r_{0}^{T}$ would change.

Although extensivity is the backbone of the Gibbs statistics, there are
various arguments in favor of the non-extensivity, especially in relativistic
systems and those that involve long-range interactions kanient ; kani0 ; rev ;
sal ; kani1 ; kani2 . The Tsallis and Kaniadakis ($\kappa$) statistics are two
of the most famous and widely used generalized statistics kanient ; rev ; sal
; kani1 ; kani2  which propose generalized versions of the equipartition
theorem plaTsal ; kani2 ; nunes . Motivated by various reasons such as the
long-range nature of gravity, and the probable relationship between quantum
aspects of gravity and the non-extensivity nunes ; epl ; shab , these
statistics have been employed leading to notable outcomes in $i$) describing
dark energy nunes ; epjc , MOND theory mond , $ii$) studying the Jeans
instability jtsal ; jk ; jk1 ; morad , and also $iii$) stellar sciences st1 ;
st4 ; st2 ; st3 ; st5 .

Therefore, finding the Gamow temperature in the Tsallis and Kaniadakis
statistics is an important task that also helps to get a better understanding
of the non-extensivity, and correspondingly, the non-extensive aspects of
gravity. To achieve this goal, we focus on the Tsallis and Kaniadakis
statistics in the next section, and a summary will be presented at the end.

##  II Generalized statistics and the Gamow temperature

###  II.1 Tsallis framework

The Tsallis entropy content of a statistical distribution with $W$ states
while the $i$-th state happens with probability $P_{i}$ is defined as sal

$\displaystyle S^{T}_{q}=\frac{1}{1-q}\sum_{i=1}^{W}(P_{i}^{q}-P_{i}),$ (8)

where $q$ is a free parameter calculated by other parts of physics or matching
with experiments rev ; sal . The Gibbs entropy is recovered at $q\rightarrow
1$; in fact, each sample has its own $q$ rev ; sal . For a three dimensional
particle, the ordinary thermal energy ($\frac{3}{2}K_{B}T$) is modified as
$\frac{3}{5-3q}K_{B}T$ meaning that Eq. (1) changes as

$\displaystyle\frac{1}{2}mv^{2}=\frac{3}{5-3q}K_{B}T_{g},$ (9)

where $0\leq q<\frac{5}{3}$ nunes . Now, simple calculations lead to

$\displaystyle
T_{g}\geq\frac{5-3q}{2}\texttt{T}\equiv\texttt{T}_{q}\Rightarrow
0<\texttt{T}_{q}\leq 2\cdot 5~{}\texttt{T},$
$\displaystyle\lambda_{q}=\frac{2\pi\epsilon_{0}\hbar^{2}}{mZ_{i}Z_{j}e^{2}}=\lambda,$
(10)

in which the subscript $q$ is used to distinguish the previous results with
those of the Tsallis statistics. Moreover, solving
$\frac{3}{5-3q}K_{B}T_{g}=U(r_{0})$, one reaches the Tsallis thermal length
scale
$r_{0q}^{T}(\equiv\frac{(5-3q)Z_{i}Z_{j}e^{2}}{12\pi\epsilon_{0}K_{B}T_{g}}=\frac{5-3q}{2}r_{0}^{T})$
that meets the condition $r_{0q}^{T}(q\rightarrow 1)\rightarrow r_{0}^{T}$.
Therefore, quantum tunneling can happen at temperature comparable to
$\texttt{T}_{q}$, and in this manner, we should have at least
$r_{0q}^{T}=\lambda_{q}$.

###  II.2 The $\kappa$ statistics

In this framework, entropy is given by kani1

$\displaystyle
S_{\kappa}=-\sum_{i=1}^{W}\frac{P_{i}^{1+\kappa}-P_{i}^{1-\kappa}}{2\kappa}=$
(11)
$\displaystyle\frac{1}{2}\bigg{(}\frac{\sum_{i=1}^{W}(P_{i}^{1-\kappa}-P_{i})}{\kappa}+\frac{\sum_{i=1}^{W}(P_{i}^{1+\kappa}-P_{i})}{-\kappa}\bigg{)},$

leading to epjc

$\displaystyle S_{\kappa}=\frac{S_{1+\kappa}^{T}+S_{1-\kappa}^{T}}{2},$ (12)

which clearly testifies that the Gibbs entropy is achieved for $\kappa=0$
kanient . Indeed, $\kappa$ is an unknown free parameter estimated by
observations that varies from case to case kanient . Moreover, the
equipartition theorem changes kani2 ; nunes , and thus, Eq. (1) takes the form

$\displaystyle\frac{p^{2}}{2m}=\frac{3}{2}\gamma_{\kappa}K_{B}T_{g},$ (13)

in which

$\displaystyle\gamma_{\kappa}=\frac{(1+\frac{\kappa}{2})\Gamma(\frac{1}{2\kappa}-\frac{3}{4})\Gamma(\frac{1}{2\kappa}+\frac{1}{4})}{2\kappa(1+\frac{3\kappa}{2})\Gamma(\frac{1}{2\kappa}+\frac{3}{4})\Gamma(\frac{1}{2\kappa}-\frac{1}{4})},$
(14)

where $0\leq\kappa<\frac{2}{3}$ and $\Gamma(n)$ denote the Gamma function
nunes . Moreover, $\gamma_{\kappa}$ diverges for $\kappa=\frac{2}{3}$ and the
ordinary equipartition theorem ($\frac{3}{2}K_{B}T$), and thus Eq. (1) are
recovered when $\kappa=0$ leading to $\gamma_{\kappa}=1$ nunes .

Finally, it is a matter of calculation to find the Kaniadakis counterpart of
Eq. (II.1) and the Kaniadakis length scale as

$\displaystyle
T_{g}\geq\frac{\texttt{T}}{\gamma_{\kappa}}\equiv\texttt{T}_{\kappa},$ (15)
$\displaystyle\lambda_{\kappa}=\frac{2\pi\epsilon_{0}\hbar^{2}}{mZ_{i}Z_{j}e^{2}}=\lambda,$

and

$\displaystyle r_{0\kappa}^{T}=\frac{r_{0}^{T}}{\gamma_{\kappa}},$ (16)

respectively. Since $1\leq\gamma_{\kappa}$ nunes , the conditions
$\texttt{T}_{\kappa}\leq\texttt{T}$ and $r_{0\kappa}^{T}\leq r_{0}^{T}$ are
obtained as the allowed intervals for $\texttt{T}_{\kappa}$ and
$r_{0\kappa}^{T}$.

##  III Conclusion

Reviewing the Gamow theory shows the role of equipartition theorem in more
clarification via defining a thermal length scale ($r_{0}^{T}$). It was
deduced that the nuclear fusion would occur in a gas whose temperature is
comparable to the Gamow temperature (T) if the minimum requirement
$\lambda=r_{0}^{T}$ is satisfied. Moreover, equipped with the fact that
generalized statistics modifies the equipartition theorem and motivated by
their considerable achievements in various setups nunes ; epl ; shab ; epjc ;
mond ; jtsal ; jk ; jk1 ; morad ; st1 ; st4 ; st2 ; st3 ; st5 , we studied the
effects of non-extensivity on the Gamow temperature. The results indicate that
the Gamow temperature (T) decreases in the Kaniadakis statistics
($\texttt{T}_{\kappa}\leq\texttt{T}$), and in the Tsallis statistics, it can
be smaller or bigger than T (i.e., $0<\texttt{T}_{q}\leq 2\cdot
5~{}\texttt{T}$), depending on the value of $q$. The same result applies to
the corresponding thermal length scales.

Finally, if stars obey the Tsallis (Kaniadakis) statistics, then by using
$\mathcal{T}=\texttt{T}_{q}$ ($\mathcal{T}=\texttt{T}_{\kappa}$), one can find
the value of $q$ ($\kappa$) corresponding to each star. Therefore, the upper
and lower bounds on the $q$ ($\kappa$) parameter for nuclear fusion process
occurring can be found in the coldest and hottest stars.

## References

  * (1) D. Prialnik, An Introduction to the Theory of Stellar Structure and Evolution, (Cambridge University Press, 2000).
  * (2) G. Kaniadakis, Phys. Rev. E 72, 036108 (2005).
  * (3) G. Kaniadakis, Entropy. 15, 3983 (2013).
  * (4) J. M. Amigo, S. G. Balogh, S. Hernández, Entropy. 20, 813 (2018).
  * (5) C. Tsallis, J. Stat. Phys. 52, 479 (1988).
  * (6) G. Kaniadakis, Physica A 296, 405 (2001).
  * (7) G. Kaniadakis, Phys. Rev. E 66, 056125 (2002).
  * (8) A. R. Plastino, J. A. S. Lima, Phys. Lett. A 260, 46 (1999).
  * (9) R. C. Nunes, E. M. Barboza Jr., E. M. C. Abreu, J. A. Neto, JCAP 08, 051 (2016).
  * (10) H. Moradpour, C. Corda, A. H. Ziaie, S. Ghaffari, EPL 127(6), 60006 (2019).
  * (11) H. Shababi, K. Ourabah, Eur. Phys. J. Plus 135, 697 (2020).
  * (12) H. Moradpour, A. Ziaie, M. K. Zangeneh, Eur. Phys. J. C 80, 8 (2020).
  * (13) H. Moradpour, A. Sheykhi, C. Corda, I. G. Salako, Phys. Lett. B 783, 82 (2018).
  * (14) D. Jiulin, Phys. Lett. A 320, 347 (2004).
  * (15) E. M. C. Abreu et al., EPL 114, 55001 (2016).
  * (16) W. H. Yang et al., Chin. Phys. B 29, 110401 (2020).
  * (17) H. Moradpour, A. H. Ziaie, S. Ghaffari, F. Feleppa, MNRAS 488(1), L69 (2019).
  * (18) A. R. Plastino, A. Plastino, Physica A 174, 384 (1993).
  * (19) G. Kaniadakis, A. Lavagno, P. Quarati, Phys.Lett. B 369, 308 (1996).
  * (20) M. Sakagami, A. Taruya, Continuum Mech. Thermodyn. 16, 279 (2004).
  * (21) B. B. Soares, J. C. Carvalho, J. D. do Nascimento Jr., J. R. De Medeiros, Physica A 364, 413 (2006).
  * (22) A. Esquivel, A. Lazarian, ApJ 710, 125 (2010).