# Perfect QCD – a new Universal approach to soft QCD

P. Christiansen  Division of Particle Physics, Lund University, Sweden

###### Abstract

The ideas presented in this proceeding aims to be a first step towards a
description of hadronic collisions where all soft processes are fundamentally
strongly coupled and the same Universal strongly coupled physics drives both
initial and final-state interactions.
As it is not currently possible to derive such a picture from first
principles, instead, an attempt to generalize the perfect liquid observation
to a “perfect QCD” guiding principle is presented, focusing on implications
for particle production in small systems. The first steps towards a
microscopic model is taken by arguing that “perfect QCD” suggests that the
screening in the initial state is so large that multi-parton interactions are
of little or no importance. Instead, a target and projectile remnant is
coherently excited and particle production is mainly driven by radiation in a
qualitative similar manner as $e^{+}e^{-}\rightarrow q\bar{q}$.
Finally, some of the possible implications of this “excited remnant model” are
presented. It is argued that the time ordering of soft and hard physics can
explain the absence of jet quenching in small systems and that the coherence
scale of the projectile and target provides insights into what small systems
will exhibit flow.

##  1 Introduction

The goal of this proceeding for the Winter Workshop 2022 is to present a new
picture for hadronic collisions. To be precise, the focus in this paper is
only on non-diffractive inelastic collisions and only the soft
physics111Meaning that momentum transfers are small so that perturbative
calculations are inaccurate. As, for example, the Pythia generator [1, 2] for
proton-proton collisions treats all interactions perturbatively, this is not a
unique definition but part of the motivation for exploring a completely
different picture in this paper., which is expected to be responsible for bulk
particle production. When hadronic collisions are mentioned in the following
it always refers to this type of collision unless another type is explicitly
mentioned.
The motivation for doing this is the observation of several phenomena in small
systems222Small systems are taken to mean proton-proton, proton-nuclei and
ultra-peripheral nuclei-nuclei collisions. that has traditionally been
associated with the formation of a quark-gluon plasma (QGP) in large systems,
see, e.g., Refs. [3, 4] for an overview. These new phenomena can all be
explained by the presence of large final-state interactions in small system
and many excellent ideas have been presented for describing this with weakly
coupled physics, see e.g., [5], but what seems to the author to be a
fundamental flaw in these models is that a weakly coupled interaction leads to
a non-vanishing mean free path so that the QGP-like effects will build up as
the system grows and first dominate at a certain system size [5]. This means
that QGP-like effects do not in a natural way extend down to the smallest
systems, even if there is no indication in data of an onset [3, 4]. At the
same time, a non-vanishing mean free path will introduce diffusion and
dissipation effects that will supposedly modify the initial-state
correlations, which the author is unaware of experimental evidence for, see
e.g. C. A. Pruneau’s contribution to these proceedings [6].

Figure 1: Illustration of how the initial soft scatterings are described in
different models/pictures. Top: in Pythia, soft and hard interactions are
modeled in the same way as leading-order perturbative processes and _multiple_
interactions occur in most pp collisions. The strings forming between color
charges are not shown. Bottom: in the “perfect QCD” picture a remnant of each
nucleon is excited as a whole, as if there was only a _single_ interaction,
and the picture is therefore denoted the “excited remnant model”.

In this paper, the decision has been to take a fresh look at things from the
perspective offered by the new measurements and try to bring forth a picture
that is fundamentally strongly coupled with a vanishing mean free path so that
large final-state effects are present in all systems and do not introduce
diffusion or dissipation (are essentially reversible) thereby hopefully
preserving correlations such as those introduced by string breakings or
similar processes. In traditional pictures, “soft” can have two very different
meanings:

  1. 1.

The extrapolation from high-momentum transfers to low momentum transfer, e.g.,
using leading-order perturbative cross sections even for situations where
next-to-leading order correlations are large

  2. 2.

Phenomenological physics such as the Lund string model [7]

The approach in this paper is to claim that point 1 does not work, meaning
that next-to-leading order corrections distorts the leading-order picture, and
the proposal is instead that “perfect QCD” is a Universal version of point 2
and can provide guidance in that way. This means that any time where soft is
mentioned in the text one should in principle be able to apply the “perfect
QCD” principle. To help convince the reader that this leads to fundamentally
different physics from that found in existing models, one of the main findings
will be already discussed here and illustrated in Fig. 1. In pp event
generators, such as Pythia [1, 2], one typically treats the initial stages of
pp collisions as two interacting parton gases where the scattering of each
parton-parton interaction is motivated by perturbative (weakly coupled) QCD,
Fig. 1 top. In the Color-Glass Condensate (CGC) model, not shown, one instead
considers it as a weakly coupled interaction between dense gluon fields [8]
that produce longitudinal Glasma tubes, Fig. 1. In both models the collision
can involve one or more interactions and _the number of interactions is the
main driving mechanism of the final-state multiplicity_. In the picture
motivated in this paper, one considers a strongly coupled scenario where the
color field of each projectile parton is neutralized by the target partons. It
is argued that this results instead in that the remnant of the projectile and
the target is coherently excited, corresponding essentially to a _single_ soft
interaction. This gives rise to two semi-independent color fields, Fig. 1
bottom, which would mean that most of the particle production is driven by
final-state radiation from the colored target and projectile remnants, similar
to $e^{+}e^{-}\rightarrow q\bar{q}$.

Concretely, the idea of this paper is to extend the experimental observation
that the QGP behaves like a perfect liquid to a “perfect QCD” principle that
can guide our understanding of particle production in general. The goal is not
to come up with a full model, but to demonstrate that it is possible using the
proposed “perfect QCD” principle to obtain surprising insights into particle
production where the physics and the explanations for observed phenomena are
very different from those found in existing models, such as Pythia and the
CGC.

##  2 Perfect QCD

One of the most remarkable discoveries of the heavy-ion program at RHIC and
LHC is that the Quark-Gluon Plasma (QGP) behaves as a perfect liquid [9, 10,
11, 12, 13, 14, 15]. The shear-viscosity-to-entropy density ($\eta/s$) is as
low as possible [16]. This means that the build up of flow is almost
deterministic, which has enabled the precise measurement of fluctuations in
the initial distribution of matter, e.g., Refs. [17, 18]. At the Winter
Workshop it was further shown how the same minimal $\eta/s$ is also obtained
when analyzing balance functions and momentum correlations, see C. A.
Pruneau’s contribution to these proceedings [6].

The perfect nature of the liquid seems to indicate that it is very fundamental
and since it is observed in all hadronic collisional systems (pp, p–Pb, and
Pb–Pb collisions), see for example Refs. [19, 20] for small systems, one could
hope that it provides a deep insight into QCD.

Based on the characteristics of the perfect liquid it is proposed that
“perfect QCD” has to have the following two characteristics:

  * •

Strongly interacting

  * •

Minimal entropy production

The minimal entropy production comes from the observation that the
hydrodynamic description of the QGP is as close to ideal (reversible) as it
can be and means that dissipation and diffusion can play no significant role
in the description of the system.

##  3 The Perfect QCD Picture of Particle Production

It might seem impossible to derive a microscopic picture from a strongly
interacting soft QCD model because one looses the perturbative guidance but
the surprise is that the proposed picture is extremely simple. The “perfect
QCD” principle dictates that the entropy production during the initial
collisions should be as small as possible, yet strongly interacting, and this
suggests that all that happens is the exchange of a single soft gluon so that
only color and essentially no momentum is exchanged. As the interacting
hadrons are of course made up of partons, this would require that the
screening in the initial state is so strong for the initial interactions that
the soft parton-parton (and/or CGC equivalent) interactions are suppressed to
a degree where they can be neglected. One will of course have parton-parton
interactions for very large momentum transfers but they are not of interest
here where the focus is on bulk production.

Let us first treat the rest of the collision, ignoring possible radiation,
using the Lund string model [7], which, as it is derived from the confining
long-range part of the QCD potential, is a strongly coupled model. In the Lund
string model, strings will form between colors and anti-colors that eventually
breaks, producing hadrons uniformly in rapidity. In this case, two strings
will form as the gluon carries both a color and anti-color. Let us assume that
all the energy of each proton is carried by the color and the anti-color
systems. If both have half the energy, the total string length will be
${\approx}4(y_{\rm beam}-\log{2})$ while if one color (or anti-color) has all
the energy one can supposedly form a string of length $2y_{\rm beam}$ (this
must be the minimal length for the color field to stretch between the target
and projectile). As the average number of particles produced a by a string is
proportional to the string length [7], the “perfect QCD” principle tells us
that nature will take the 2nd solution. This means that instead of having two
remnants with a similar amount of energy, one will have a “valence”-like
remnant with almost all the energy and a “sea”-like remnant with almost no
energy. This is reminiscent of the BGK picture [21], and so it is naturally to
propose that the “valence” remnant in one proton is color-coupled to the “sea”
remnant in the other proton, and vice versa, so that one in some sense has two
semi-independent systems carrying approximately half the total initial energy
each.
Let us finally try to give a partonic picture of how the “perfect QCD” picture
can be understood. As the two nucleon penetrate at high energy the partons
inside them are interacting strongly but the claim is that they interact in a
way that screens the partonic interactions. However, this screening can only
happen in a certain regime. If $x$ denotes the usual four momentum fraction
then one can maximally “organize” the nucleon into $n\approx 1/x$
constituents. Screening will be impossible when the four momentum transfer,
$Q^{2}$, is very large because one can resolve individual partons (the hard
scattering limit), or when $x$ is large so that the number of constituents is
small. The latter argument is why nucleon remnants will be excited as a whole.

In the current picture, $\text{d}N_{\text{ch}}/\text{d}\eta$ at $\eta=0$ would
be independent of $\sqrt{s}$ as all the energy will go to extend the strings
in rapidity. What has been ignored is radiation: the color charge carrying
most of the energy is, as QCD is strongly interacting, very likely to emit
soft or collinear radiation. How to calculate this radiation is not trivial,
but one can at least note that one qualitatively get a system very similar to
what one has for $e^{+}e^{-}\rightarrow q\bar{q}$ (denoted $e^{+}e^{-}$ in the
following). Comparing particle production in $e^{+}e^{-}$ collisions to that
of pp collisions, one finds that the former produces _more_ particles on the
average [11]. The common understanding is that it is possible for part of the
proton to escape as a color neutral object, taking away around 50% of the
energy [22]. Based on the observed particle production in $e^{+}e^{-}$, it is
concluded that there is no fundamental reason one should not be able to create
the observed particle production via radiation also in pp, pA, and AA
collisions.

To recap, the general microscopic “perfect QCD” picture of pp, pA, and AA
collisions will be that the soft initial interactions will excite a remnant of
each nucleon in a “projectile” coherently and that the main particle
production at high energy collisions is driven by final-state radiation. For
this reason the picture will be denoted the “excited remnant model”. This
might sound like the Dual Parton Model but it is important to note that the
Dual Parton Model contains MPIs [23].

In the limit that particle production is dominated by radiation, the color-
connections to the “sea” systems in the “target” can be ignored and one can
therefore factorize the soft particle production into $N_{\text{part}}$ semi-
independent terms. Semi-independent, because there must be some dependence on
the nucleon-nucleon impact parameter to explain the slightly increased
particle production per participant in AA collisions.

###  3.1 An Illustration of Particle Production in pp Collisions

Figure 2: $\text{d}N_{\text{ch}}/\text{d}\eta$ measured in $\sqrt{s}=200$
GeV/$c$ pp collisions by UA5 for NSD events and for events with different
final-state charged particle multiplicities, $n$. The data have been read off
from the published figures [24]. As the figure is just meant to illustrate a
trend, the statistical uncertainties have not been included for clarity.

The main goal here is to discuss small systems. In these systems, e.g., pp
collisions, the full “perfect QCD” picture of a collision is:

  1. 1.

the initial interactions produce up to three semi-independent systems:

     * •

coherently excited target and projectile remnants

     * •

possible color-neutral target and projectile remnants that act as spectators
(escape with energy along beam direction)

     * •

possible hard parton-parton scatterings

  2. 2.

the excited remnants radiate gluons

  3. 3.

the color fields decay into partons

  4. 4.

final-state partonic interactions: flow, strangeness enhancement

  5. 5.

hadronization

  6. 6.

possible final-state hadronic rescattering

One could in principle try to implement a generator along these lines but the
goal here is to illustrate the picture using UA5 data [24]. Fig. 2 shows the
$\text{d}N_{\text{ch}}/\text{d}\eta$ measured by UA5 for NSD events as well as
for multiplicity selected events. In low-multiplicity events,
$\text{d}N_{\text{ch}}/\text{d}\eta$ is flat as one would expect for a single
long string. As the multiplicity grows, one observes a narrowing of
$\text{d}N_{\text{ch}}/\text{d}\eta$, which in the perfect QCD picture should
be caused by the radiation adding shorter and shorter (less energetic)
strings. In this way the “excited remnant model” is at least qualitatively
consistent with the observed trends by UA5.

##  4 Insights and Predictions for Small Systems

In this section, the hope is to demonstrate for the reader that the perfect-
QCD picture of particle production can provide many new insights and
predictions.

###  4.1 A Simple Explanation for the Absence of Jet Quenching in Small
Systems

One can immediately notice that, if the time scales involved with the hard
interactions are shorter than the formation time for step 2 (“the excited
remnants radiate gluons”) as one would imagine from the scales of the momentum
transfers involved, then one can understand why there is no jet quenching in
small systems even if there is a relation between flow and jet quenching in a
large system. The medium simply has not been produced yet when the jet
propagates. This seems very attractive to the author as this is in line with
experimental findings, see, for example, Ref. [25], and it is hard to explain
in most existing models.

###  4.2 Flow in pp Collisions $\gg$ Flow in $\mathbf{e^{+}e^{-}}$ Collisions

It should be clear from the way the “excited remnant model” work that it
“postdicts” that the particle production in pp collisions and $e^{+}e^{-}$
collisions should be very similar because in this model, and unlike
traditional MPI-based models, the growth with $\sqrt{s}$ is in both cases
driven by radiation. Indeed this surprising similarity have been noted and
discussed much in the past by experimental collaborations [26, 11], even it
was never theoretically understood.

It can therefore be surprising that while one observes strong flow in pp
collisions, one does not observe it for $e^{+}e^{-}$ collisions [27]. However,
there could be a simple explanation for that. As the “excited remnant model”
postulates that for each nucleon a single “valence” remnant is excited as a
whole, then it is clear that the radiation in step 2 will have to have very
low transverse momentum, $p_{\text{T}}<1/R$, where $R$ is the size of the
excited remnant. As the $p_{\text{T}}$ is so low, the color fields will have
to stack and so one will naturally get a quite dense system of parallel color
fields with a large energy density. In the “perfect QCD” picture these color
fields will be strongly interacting and so they will immediately start to
build up collective flow. This makes a big difference when comparing to
$e^{+}e^{-}$, where all the energy is located with a single parton and so the
radiated gluons can and will typically have very large $p_{\text{T}}$. This
means that most energy will be radiated away from the initial color field and
so there is little time where system is dense and can build up collective
flow.

###  4.3 How to Control Flow in Ultra-Small Systems

In the previous subsection it was argued that for small systems, the size of
the excited remnant determines the flow that can be built up in the final
system. This then is naturally in line with the observation of flow in Ultra-
Peripheral Collisions (UPCs), where the photon field of one nuclei interacts
with the other nuclei, because in this case the photon field has a long
wavelength since it is emitted coherently by the protons in the nuclei. Recall
that photons can interact as a “hadronic” system by fluctuating into a
$q\bar{q}$ pair, which will have a size that reflects the photon four momentum
($Q^{2}$). ATLAS has observed flow in UPC Pb–Pb events [28] and CMS has
reported non-zero $v_{2}\\{2\\}$ in p–Pb events [29], which is in line with
the ideas presented here.

By going to electron-proton or electron-ion collisions one can in principle
measure the wavelength of the photon from the change in electron four
momentum. One can in this way select different sizes of the excited remnants
and if the picture is true, control the $p_{\text{T}}$ radiation and switch on
(low $Q^{2}$) and off (high $Q^{2}$) flow. ZEUS and H1 has reanalyzed old data
both for low and high $Q^{2}$ but neither ZEUS [30, 31] nor H1 [32] observes
any signatures of collective flow. This clearly goes against the ideas
presented here. However, it seems that if there is flow in UPCs at LHC then
there would also likely be flow in low $Q^{2}$ $e$p collisions at HERA and
vice verse. On the other hand, one knows that flow in small systems is very
hard to detect. Looking from the outside, it would be good if one could
resolve the situation so that one is as certain as possible that similar
procedures have been used before one concludes too strongly on the current
results.

##  5 Conclusions

An attempt to generalize the perfect-liquid nature from flow to particle
production has been presented. The “perfect QCD” principle has been proposed
to be a Universal principle for soft QCD that applies both in the initial and
final state of hadronic collisions. Using the idea of minimal entropy
production, a microscopic picture, the “excited remnant model”, has been
presented. In the microscopic picture, the screening as the two hadronic
systems penetrate is so large that subcollisions between constituents does not
occur, in contrast to most existing pictures, e.g., MPI and CGC based ones.

No attempt has been done to prove the “perfect QCD” principle in this paper
but several surprising insights have been provided, such as simple arguments
for why jet quenching is absent in small systems and which collisional systems
will exhibit flow. The hope is that the principle can be used to provide novel
insights into a wide range of topics, for example, jet quenching in large
systems and the relation between diffractive and non-diffractive physics.

##  6 Acknowledgements

The author would like to thank Adrian Nassirpour for many valuable comments on
earlier versions of similar manuscripts.

##  7 References

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# DualMix: Unleashing the Potential of Data Augmentation for Online Class-
Incremental Learning

Yunfeng Fan1, Wenchao Xu1, Haozhao Wang2,, Jiaqi Zhu3, Junxiao Wang4,5, and
Song Guo1
1The Hong Kong Polytechnic University, 2Huazhong University of Science and
Technology
3School of Automation, Beijing Institute of Technology, 4KAUST, 5SDAIA-KAUST
AI Center
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author

###### Abstract

Online Class-Incremental (OCI) learning has sparked new approaches to expand
the previously trained model knowledge from sequentially arriving data streams
with new classes. Unfortunately, OCI learning can suffer from catastrophic
forgetting (CF) as the decision boundaries for old classes can become
inaccurate when perturbated by new ones. Existing literature have applied the
data augmentation (DA) to alleviate the model forgetting, while the role of DA
in OCI has not been well understood so far. In this paper, we theoretically
show that augmented samples with lower correlation to the original data are
more effective in preventing forgetting. However, aggressive augmentation may
also reduce the consistency between data and corresponding labels, which
motivates us to exploit proper DA to boost the OCI performance and prevent the
CF problem. We propose the Enhanced Mixup (EnMix) method that mixes the
augmented samples and their labels simultaneously, which is shown to enhance
the sample diversity while maintaining strong consistency with corresponding
labels. Further, to solve the class imbalance problem, we design an Adaptive
Mixup (AdpMix) method to calibrate the decision boundaries by mixing samples
from both old and new classes and dynamically adjusting the label mixing
ratio. Our approach is demonstrated to be effective on several benchmark
datasets through extensive experiments, and it is shown to be compatible with
other replay-based techniques.

##  1 Introduction

Deep learning (DL) has made remarkable achievements by imitating human
intelligence to mine the knowledge from carefully gathered datasets. Further
inspired by human learning process, continual learning (CL), also known as
incremental learning, has taken a next step to broaden the model knowledge
from a number of sequentially arrived tasks [25, 34, 36]. However, while
learning new tasks, CL can suffer from catastrophic forgetting (CF) that the
model’s classification capability can drop severely on old tasks [29, 12, 20].
The CF phenomenon can be significant especially under more realistic settings,
i.e., the online class-incremental (OCI) learning, where new classes are
continually arriving with the data stream and every batch samples can be
observed only once [16, 28].

Figure 1: (a) Normal data augmentation increases the diversity of limited
samples (within the opaque border), but excessive augmentation can produce
samples that are unrelated to the labels (outside of the transparent border).
To achieve more robust augmentation, we mix the augmented samples from various
classes. (b) In OCI, imbalanced class distribution leads to a biased decision
boundary that favors the old classes stored in memory. To overcome this, we
mix the samples from both old and new classes and adjust the label mixing
ratio to push the decision boundary away and make it more adaptive.

Replay-based methods [6, 27, 33] have been shown to be effective against CF
for OCI by storing past task samples in memory buffer and replaying them while
training for new tasks. Recent research [45, 46, 27, 15] has aimed to
alleviate CF by performing data augmentation (DA) [37, 35] to both previously
buffered and newly arrived samples. These methods focus on complex loss design
coupled with contrastive or self-supervised learning (SSL) [42, 30] techniques
to replace the less effective cross-entropy (CE) loss with advanced loss
functions such as InfoNCE loss [17] or mutual information [13]. However, the
role of DA in OCI still remains unclear and requires exploitation. Different
with literature status quo, in this paper, we will consider the input side of
OCI, exploring what constitutes a good DA in OCI and how to apply it
effectively?

Intuitively, DA could create samples with extended diversity, which is shown
to be beneficial to representation learning [31, 11, 41] and can potentially
prevent CF in OCI. However, counterintuitively, [44] reveals that directly
applying DA, without multiple iterations [5] on experience replay (ER) [10],
leads to worse performance than ER without augmentation. Such anomaly inspires
us to compare different DA methods for OCI. As shown in Figure 2, DA does not
always harm OCI performance, but selecting the appropriate augmentation method
and strength111For the RandomResizedCrop operator with the scale range
$\left[a,b\right]$, its strength is defined as
$\left(1-a\right)+\left(1-b\right)$. is critical. The results indicate that
aggressive augmentation is more likely to perform well in OCI, whereas
excessively strong DA can lead to worse performance (Crop-1.3 is worse than
Crop-0.8).

To explain the aforementioned phenomenon, we conducted a theoretical analysis
on the relationship between DA and the forgetting in OCI. Our analysis
suggests that the augmented samples from the memory buffer should have low
covariance on the model’s mean cross entropy compared to the original samples.
However, excessive augmentation with low covariance may generate new samples
that deviate from the ground-truth labels, leading to the introduction of
erroneous information. To address this issue, we propose the Enhanced Mixup
(EnMix), which applies mixup [43] on augmented samples from memory to ensure
stronger DA while still maintaining high consistency with their labels, as
shown in Figure 1. Additionally, we observed that the decision boundaries
between old and new classes are biased towards old classes due to class
imbalance [19, 8] in OCI. To address this, we propose Adaptive Mixup (AdpMix)
to adjust the decision boundaries between old and new classes according to the
weight imbalance from the classifier. The two methods we propose are
collectively referred to as DualMix.

Figure 2: Average accuracy on different augmentation strategies with the
baseline ER. “Identity”: copy the original samples as augmentation; “Flip”:
random horizontal flip; “Crop-xx”: random resized crop, number indicates
strength. “Identity” is slightly weaker than ER with no augmentation. “Crop”
is usually better and different strengths correspond to different promotions.
All experiment were performed on CIFAR-100 and memory size is 1k.

To sum up, this paper makes the following contributions.

  * •

We provide a theoretical explanation that low covariance between DA samples
and original samples in terms of mean cross entropy is advantageous in OCI. In
addition, we introduce EnMix, which enhances standard DA by generating
stronger and more dependable samples.

  * •

The AdpMix method is proposed to strike a balance between the decision
boundaries of old and new classes by dynamically adjusting the mixing ratio of
labels.

  * •

Our proposed augmentation methods, which only adjust the input side, are shown
to outperform existing methods on several benchmark datasets and be compatible
with other replay-based techniques, as demonstrated by extensive empirical
results.

##  2 Related works

###  2.1 Replay-based online continual learning

In the online continual learning (OCL) [7, 24] setting, data arrives in small
batches sequentially, and previous batches from current and previous tasks
cannot be reused, presenting the challenge of efficient single-pass learning.
OCL can be classified into two types based on whether the new task contains
new classes: online class-incremental (OCI) and online domain-incremental
(ODI) [2, 14]. This work mainly focuses on the more challenging OCI setting,
which relies heavily on replay-based methods. Chaudhry _et al_. [10] proposed
experience replay (ER) to store a subset of data from previously seen tasks in
memory and replay them during training on new tasks. Variants of ER have been
developed to optimize sample selection and representation learning strategies.
A-GEM [9] constrains the gradient from memory samples to prevent the average
loss from previous tasks from increasing. MIR [3] compares the loss increments
of memory samples after updating the model based on current batch data to
perform sample retrieval. GSS [4] stores samples with more diversity in
gradient directions in memory. ASER [33] scores each sample by the Sharply
value according to its ability to preserve the latent decision boundaries of
previously seen classes. Although these methods aim to prevent forgetting by
storing and revisiting representative memory samples, they do not address the
severe class imbalance problem in replay-based OCI. GDumb [32] was developed
specifically to address class imbalance problem by greedily keeping the number
of samples from each class balanced in memory and training the model with only
memory data. However, it excessively reduces the usage of new task data,
resulting in insufficient feature exploitation. In this work, we propose two
simple yet efficient methods to prevent forgetting and alleviate class
imbalance simultaneously. Our methods are specifically designed for the OCI
setting and only adjust the input side. We demonstrate the superiority of our
methods on multiple benchmark datasets.

###  2.2 Data augmentation and continual learning

To alleviate the CF phenomenon, various recent studies [27, 45, 15] have
focused on learning representations with appropriate features. For instance,
Mai _et al_. [27] proposed SCR, which utilizes contrastive learning to
encourage clustering samples from the same class, and replaces the linear
classifier with the Nearest-Class-Mean classifier to address the problem of
imbalanced weights. Gu _et al_. [13] analyzed representation learning with
mutual information and proposed DVC to retain more information from previous
tasks, the similar way with [15]. Zhu _et al_. [45] employed data mixture [43]
to generate samples with virtual new classes and was also inspired by SSL to
augment classes by rotating training samples to promote learning of holistic
features [46]. Although these methods use DA to generate diverse samples,
their purpose is to satisfy the requirements for contrastive learning or SSL,
and they have not examined the role of DA in CL. Moreover, these methods
necessitate changes in the model structure (such as additional classifier
heads for new classes) during training. In this paper, we investigate the role
of DA in replay-based OCI with theoretical explanations, and propose two DA
methods that solely regulate the input side.

##  3 Method

###  3.1 Insights derived from theoretical exposition

In this paper, we study a supervised OCI learning setting as in [26, 5, 23].
Consider a data stream $D=\left\\{D_{1},D_{2},...,D_{N}\right\\}$ and
$D_{i}=\left(X_{i},Y_{i}\right)$ is the data for task $i$. $X_{i}$ and $Y_{i}$
represent the samples and corresponding labels in task $i$. $Y_{i}\cap
Y_{j}=\emptyset$ for $i\neq j$. In the training phase, the data $D_{i}$ from
the task $i$ can be seen only once, which means only one epoch for training is
allowed. The model can be divided into feature extractor $f_{\theta}$, with
parameters $\theta$, and a linear classifier (single-head for all classes)
with weights $\left\\{\mathbf{w}_{c}\right\\}_{c=1}^{C}$ ($C$ is the class
number for all tasks, and we omit the bias for simplification). We denote
$\mathbf{h}_{i}=f_{\theta}\left(x_{i}\right)\in\mathcal{R}^{d}$ as the
extracted features of sample $x_{i}$.
$F_{\theta}\left(x_{i}\right)=\sigma\left(\mathbf{w}^{T}f_{\theta}\left(x_{i,j}\right)\right)\in\mathcal{R}^{C}$
is the output probabilities of sample $x_{i}$. $\sigma\left(\cdot\right)$ is
the softmax operation.

When the model is trained on task $t$, according to [13], the empirical risk
which the model aims to minimize is defined as

$\begin{split}R_{t}\left(F\right)\overset{def}{=}&\underset{\left(x,y\right)\sim
D_{t}}{\mathrm{E}}\left[L\left(y,F\left(x\right)\right)\right]+\\\
&\beta\lambda\underset{\left(x,y\right)\sim
D_{t-1}^{\mathcal{M}}}{\mathrm{E}}\left[L\left(y,F\left(x\right)\right)\right]\end{split}$
(1)

where $D_{t-1}^{\mathcal{M}}$ is the fixed-size memory after trained on task
$t-1$. We omit $\theta$ in $F_{\theta}$ for simplicity.
$\lambda:=\frac{\left|D_{t}\right|}{\left|D^{\mathcal{M}}\right|}$ and
$\beta:=1/\left(1+\frac{2\left|D_{t}\right|}{\left|D_{\left[1,t\right]}\right|}\right)$.
$D_{\left[1,t\right]}$ is all the seen data
$\left\\{D_{1},D_{2},...,D_{t}\right\\}$. If we apply a random transform $g\in
G$ on memory samples, the expansion memory data is
$D^{\mathcal{M}g}=\left\\{x,g\left(x\right)|x\in D^{\mathcal{M}}\right\\}$.
The risk with DA of the second term in Equation 1, indicating the accquired
knowledge from past tasks, is as:

$R_{t}^{\mathcal{M}g}\left(F\right)=\beta\lambda\underset{\left(x,y\right)\sim
D_{t-1}^{\mathcal{M}g}}{\mathrm{E}}\left[L\left(y,F\left(x\right)\right)\right]$
(2)

The objective of CL is to minimize the empirical risk over all the tasks seen
so far, so the empirical risk from task 1 to $t-1$ without forgetting should
be

$R_{t-1}^{obj}\left(F\right)\overset{def}{=}\underset{\left(x,y\right)\sim
D_{\left[1,t-1\right]}}{\mathrm{E}}\left[L\left(y,F\left(x\right)\right)\right]$
(3)

Therefore, we define the forgetting gap as:

$FG^{\mathcal{M}g}=\underset{g}{\mathrm{E}}\left[\left(R_{t}^{\mathcal{M}g}\left(F\right)-R_{t-1}^{obj}\left(F\right)\right)^{2}\right]$
(4)

We further define the mean covariance of sample’s CE loss in dataset
$D_{t-1}^{\mathcal{M}g}$ as

$CO^{\mathcal{M}g}=\underset{x_{i},x_{j}\sim
D_{t-1}^{\mathcal{M}g}}{\mathrm{E}}\left[\mathrm{Cov}\left[q\left(x_{i}\right),q\left(x_{j}\right)\right]\right]$
(5)

where $\mathrm{Cov}\left[\cdot,\cdot\right]$ stands for the covariance. $q$ is
defined as
$q\left(x_{i}\right)=-{y_{i}}^{T}\log\left(F\left(x_{i}\right)\right)$.

Figure 2 shows that if DA could generate more diverse information, then the
forgetting problem can be alleviated accordingly. Inspired from [40], we give
the rigorous analysis and prove it in the Appendix.

Figure 3: The workflow for integrating EnMix and AdpMix techniques into the
learning process of OCI.

Proposition 1. Given a model trained on sequential data
$\left\\{D_{1},D_{2},...,D_{t-1}\right\\}$ with memory buffer
$D^{\mathcal{M}}_{t-1}$, there comes a new task with $D_{t}$. Considering the
data transforms $g_{1}\in G$ and $g_{2}\in G$, they are applied on memory
samples to obtain $D_{t-1}^{\mathcal{M}g_{1}}$ and
$D_{t-1}^{\mathcal{M}g_{2}}$. If the mean covariance of CE loss in
$D_{t}^{\mathcal{M}g_{1}}$ is lower than that in $D_{t}^{\mathcal{M}g_{2}}$,
i.e., $CO^{\mathcal{M}g_{1}}<CO^{\mathcal{M}g_{2}}$, model will suffer less
forgetting on previous tasks, $FG^{\mathcal{M}g_{1}}<FG^{\mathcal{M}g_{2}}$.

This proposition shows that DA should reduce the correlation between the
augmented and the original samples. Since the covariance is determined by the
model output and less forgetting can be achieved if model changes slightly
after the new task learning, we calculate the variance of the output
probability based on old model as an indicator for evaluating the strength of
DA, inspired from [40]:

$\bar{m}=\frac{1}{C_{old}}\sum_{i\in\left[C_{old}\right]}{\left(\mathrm{Var}_{\mathcal{M}g}\left(\mathbf{u}\right)\right)}^{\frac{1}{2}},\mathbf{u}=\frac{1}{\left|\mathcal{M}g\right|}\sum_{x_{i}\in\mathcal{M}g}{F\left(x_{i}\right)}$
(6)

See Section 4.5 for the experimental results.

###  3.2 Enhanced mixup augmentation

As described in Section 3.1, we should conduct a strong DA in the memory for
OCI to alleviate forgetting. However, when the DA is too strong, i.e., the
output probability of augmented view is far from that of the original sample,
the augmented view is likely to keep weak correlation with the label, which
will destroy the supervised learning (e.g. “Crop-1.3” with stronger DA but
poor performance than “Crop-0.8” in Figure 2). How can we further reduce the
correlation between the augmented views and the original samples without
destroying the correspondence with their labels?

In order to solve the problem, we propose a simple yet effective method,
enhanced mixup (EnMix), based on intensive DA with reliable label correlation.
Given a data transform $g\in G$ and a sample $x$, the augmented view will be
$\tilde{x}=g\left(x\right)$. EnMix constructs virtual samples by mixing the
augmented views from different original samples and also linearly
interpolating their labels:

$\displaystyle\tilde{x}^{e}=\mu\tilde{x}_{i}+\left(1-\mu\right)\tilde{x}_{j}$
(7) $\displaystyle\tilde{y}^{e}=\mu y_{i}+\left(1-\mu\right)y_{j}$

where $\mu\in\left[0,1\right]$ and
$\mu\sim\mathrm{Beta}\left(\alpha,\alpha\right)$,
$\alpha\in\left(0,\infty\right)$. Due to mixing with other samples (including
other classes) in memory, the model output probability of the mixed samples
should be dissimilar with the original samples’. What counts is that the label
is also mixed, building correlation between the enhanced samples with
different labels. Through this way, we not only reduce the correlation between
the augmented views and original samples, but also preserve the consistency
between views and labels. The workflow about how to mix is shown as Figure 3.

###  3.3 Adaptive mixup for balance

In the above sections, we try to produce new samples through DA, enriching the
diversity of previous task data in memory and alleviating the CF problem.
However, according to previous works [26], the CF phenomenon emerges not only
because of the forgetting about previous knowledge, but also due to the
inherent class imbalance property in replay-based OCI. The number of samples
from new classes is generally greater than the number of samples from old
classes in memory, and this imbalance intensifies as the number of tasks
increases.

The class imbalance eventually results in severely biased decision boundary
towards the old classes. In order to verify it, we denote the
misclassification ratio following [26]: $er\left(n,o\right)$ denotes the ratio
of new class test samples misclassified as old classes to the total number of
misclassified new class samples. Same notation rule is applied for
$er\left(o,n\right)$. As shown in Figure 4, the samples from old classes can
be incorrectly classified to new classes with high probability and the
probability of samples from new classes being misclassified to old is
relatively small, indicating that the decision boundaries between old and new
classes are severely uneven. Moreover, as the memory size decreases, this
imbalance is further aggravated.

Figure 4: The misclassification ratios during the training process. The
results were acquired from CIFAR-100 and trained by ER with different memory
size.

In order to adjust the decision boundaries, DA can also be a straightforward
method. Performing DA on the memory expands the sample diversity, and also
balances the number to a small extent. However, the randomness of DA still
limits its effectiveness on OCI. For the purpose of adjusting the decision
boundary more directly, we induce DA to generate samples near the decision
boundaries by mixing samples from old and new classes. In order to push the
decision boundary farther away from old classes , we adaptively adjust the
mixing ratio on labels by AdpMix:

$\displaystyle x^{a}=\mu_{x}x_{i}+\left(1-\mu_{x}\right)x_{j}$ (8)
$\displaystyle y^{e}=\mu_{y}y_{i}+\left(1-\mu_{y}\right)y_{j}$

where $x_{i}$ and $x_{j}$ are from old classes (memory data) and new classes
(current task data). Equation 8 is similar to Equation 7. The difference are
two-fold: 1) EnMix mixes the augmented samples from memory and AdpMix mixes
the original data from memory and imminent task dataset. 2) EnMix use the same
ratio $\mu$ for raw data and label while the ratio $\mu_{y}$ for label is
different from the ratio $\mu_{x}$ for data in AdpMix. The main idea is that
when the decision boundary is too closer to old classes, the mixing ratio
$\mu_{y}$ should be also more biased towards old classes, i.e., giving
$\mu_{y}$ a larger value relative to $\mu_{x}$.

The main reason for class imbalance is from the quantity problem, but in OCI,
we cannot know the number of new task samples in advance, making it
inappropriate to derive the value of $\mu$ based on the quantity gap. In
addition, CL retains the knowledge stored in memory by learning from previous
tasks. Therefore, it is not reliable to evaluate the imbalance only by the
sample size. A recent work [1] revealed that the class imbalance results in a
biased weights $\mathbf{w}$ of linear classifier. Therefore, we use the
weights $\mathbf{w}$ to design the adjustment scheme of $\mu_{y}$:

$\begin{split}\mu_{y}=\left\\{\begin{matrix}\min\left(\mu_{x}+\delta\frac{\left\|\mathbf{w}_{new}\right\|}{\left\|\mathbf{w}_{old}\right\|},1\right),&\frac{\left\|\mathbf{w}_{new}\right\|}{\left\|\mathbf{w}_{old}\right\|}>\kappa,\mu_{x}>\tau\,\,\\\
\mu_{x},&others\\\ \end{matrix}\right.\,\,\end{split}$ (9)

where $old$ or $new$ denotes an index for old or new class respectively.
$\delta$, $\kappa$ and $\tau$ are the hyper-parameters. $\kappa$ is used to
confirm the occurrence of class imbalance. $\delta$ and $\tau$ control the
degree of the adaptive adjustment. When the decision boundary deviation
occurs, we push it away from old classes. The magnitude of the adjustment is
calculated according to the relative L-2 norm value of weights.

We rewrite the output probability of class $c$ to show how AdpMix can adjust
the updating of weights $\mathbf{w}$:

$p_{i,c}=\frac{\exp\left(\mathbf{w}_{c}^{T}\mathbf{h}_{i}\right)}{\sum\nolimits_{j=1}^{C}{\exp\left(\mathbf{w}_{c}^{T}\mathbf{h}_{j}\right)}}$
(10)

For simplicity, we assume that the buffer is not updated during the training
of current task $t$, that is, the previous tasks data and current task data
are clearly divided into $D^{\mathcal{M}}$ and $D_{t}$. The CE loss is
calculated as:

$L=-\sum_{x_{i}\in D_{t}\cup
D^{\mathcal{M}}}{\sum_{c=1}^{C}{\mathcal{I}\left\\{y_{i}=c\right\\}\log
p_{i,c}}}$ (11)

where $\mathcal{I}\left\\{\cdot\right\\}$ is the indication function.
According to gradient calculation of CE loss, we can obtain the update formula
of $\mathbf{w}_{c}$ as:

$\begin{split}\mathbf{w}_{c}=\mathbf{w}_{c}+\underset{current\,\,task\,\,data}{\underbrace{\eta\sum_{x_{i}\in
D_{t},y_{i}=c}{\left(1-p_{i,c}\right)\mathbf{h}_{i}}-\eta\sum_{x_{i}\in
D_{t},y_{i}\neq c}{p_{i,c}\mathbf{h}_{i}}}}\\\
+\underset{memory\,\,data}{\underbrace{\eta\sum_{x_{i}\in
D^{\mathcal{M}},y_{i}=c}{\left(1-p_{i,c}\right)\mathbf{h}_{i}}-\eta\sum_{x_{i}\in
D^{\mathcal{M}},y_{i}\neq c}{p_{i,c}\mathbf{h}_{i}}}}\end{split}$ (12)

Method | CIFAR-100 | Mini-ImageNet | Tiny-ImageNet
---|---|---|---
Finetune | 3.4 ± 0.4 | 3.2 ± 0.2 | 2.4 ± 0.5
iid offline | 48.8 ± 2.1 | 52.1 ± 1.0 | 39.4 ± 2.2
Memory Size | M=1K | M=2K | M=5K | M=1K | M=2K | M=5K | M=1K | M=2K | M=5K
A-GEM | 3.2 ± 0.3 | 3.4 ± 0.2 | 3.2 ± 0.3 | 2.9 ± 0.3 | 3.2 ± 0.3 | 3.7 ± 0.3 | 2.6 ± 0.2 | 2.5 ± 0.2 | 2.6 ± 0.3
SCR | 14.3 ± 0.8 | 16.6 ± 0.5 | 17.5 ± 0.4 | 13.1 ± 0.4 | 14.8 ± 0.6 | 15.9 ± 0.6 | 6.2 ± 0.7 | 7.8 ± 0.4 | 9.4 ± 0.6
DVC | 16.3 ± 0.7 | 17.8 ± 0.9 | 20.2 ± 1.5 | 14.0 ± 0.8 | 16.4 ± 2.0 | 16.8 ± 1.1 | 6.8 ± 1.2 | 8.4 ± 1.4 | 12.0 ± 1.2
ER | 8.4 ± 0.5 | 12.7 ± 0.8 | 18.0 ± 1.1 | 7.8 ± 0.7 | 11.9 ± 1.2 | 15.0 ± 2.6 | 4.6 ± 0.4 | 6.3 ± 0.6 | 10.4 ± 0.8
ER+DualMix | 17.5 ± 1.0 | 20.3 ± 0.9 | 22.5 ± 1.2 | 13.2 ± 0.8 | 16.7 ± 0.8 | 17.9 ± 1.1 | 6.9 ± 0.9 | 9.4 ± 1.4 | 13.6 ± 1.1
GSS | 7.9 ± 0.6 | 11.1 ± 0.9 | 15.1 ± 0.9 | 7.7 ± 0.9 | 11.5 ± 0.9 | 12.8 ± 1.0 | 4.1 ± 0.4 | 5.7 ± 0.7 | 9.7 ± 1.0
GSS+DualMix | 9.4 ± 0.8 | 13.4 ± 1.1 | 15.7 ± 1.5 | 9.5 ± 1.2 | 12.3 ± 1.5 | 13.7 ± 0.9 | 5.5 ± 0.8 | 6.4 ± 1.0 | 10.9 ± 0.9
MIR | 8.7 ± 0.4 | 13.2 ± 0.9 | 18.6 ± 1.2 | 8.0 ± 0.6 | 12.8 ± 1.1 | 16.5 ± 2.2 | 4.2 ± 0.5 | 6.0 ± 0.5 | 11.3 ± 1.0
MIR+DualMix | 17.8 ± 0.8 | 17.2 ± 2.3 | 19.2 ± 2.8 | 11.1 ± 1.3 | 14.8 ± 1.8 | 17.1 ± 2.7 | 7.1 ± 1.2 | 9.9 ± 0.8 | 13.9 ± 0.4
ASER | 13.1 ± 0.7 | 16.4 ± 0.6 | 21.0± 0.8 | 8.9 ± 1.3 | 11.7 ± 0.9 | 16.3 ± 2.1 | 7.1 ± 0.3 | 8.8 ± 0.9 | 11.6 ± 0.7
ASER+DualMix | 13.8 ± 0.8 | 17.9 ± 2.4 | 21.7± 1.5 | 9.6 ± 0.9 | 13.1 ± 1.8 | 16.9 ± 1.9 | 6.3 ± 0.5 | 9.5 ± 0.6 | 12.2 ± 1.0
Table 1: Average Accuracy (end of training, higher is better) on three
benchmarks with 1K, 2K and 5K memory. DualMix boosts the performance of ER-
based baselines, making them even better than pervious SOTA (DVC) in most
cases.

Because the activation function for the feature extractor [18] is usually
ReLU, $\mathbf{h}_{i}$ is always positive. Due to the number of samples in
$D_{t}$ is usually far more than that of each class samples in
$D^{\mathcal{M}}$, the gain or reduction for $\mathbf{w}_{c}$ is mainly
affected by the current task $D_{t}$. We can see that the weights of class $c$
from old tasks are extremely weakened as no sample with class $c$ in $D_{t}$
(the first term in current task data equals to 0), while the weights of class
from current task are going to be effectively increased. When we apply AdpMix
as Equation 8, Equation 12 will be changed to (the memory data term is omitted
because of lesser influence):

$\begin{split}\mathbf{w}_{c}=\mathbf{w}_{c}+\eta\sum_{x_{i}\in
D_{t},y_{i}=c}{\left(\mu_{y}-p_{i,c}\right)\mathbf{h}_{i}}-\eta\sum_{x_{i}\in
D_{t},y_{i}\neq c}{p_{i,c}\mathbf{h}_{i}}\end{split}$ (13)

When $c$ belongs to the current task, the ratio $\mu_{y}$ reduces the
excessive growth of the corresponding weights. When $c$ is a old class, the
mixture operation makes the gain not equal to 0. And we further balance the
weights by adaptively adjusting $\mu_{y}$. This also shows that it is
reasonable to adjust the mixing ratio according to the weights of the
classifier.

##  4 Evaluation

###  4.1 Datasets

Split CIFAR-100 is constructed by splitting the CIFAR-100 dataset [21] into 20
tasks with no overlapping classes between each task. Every task is randomly
assigned 5 classes data. The image size is $3\times 32\times 32$. There are a
total 3,000 images in each class, which are divided into 2,500 training
samples and 500 test samples.

Split Mini-ImageNet splits the Mini-ImageNet dataset [38], containing 100
classes, into 20 disjoint tasks as in [26]. Each task includes 5 random
classes, and every class consists of 500 $3\times 84\times 84$ images for
training and 100 images for testing.

Split Tiny-ImageNet is used to verify algorithms effectiveness in more complex
scenarios. We split it Tiny-ImageNet [22] into 20 disjoint tasks and each task
contains 10 classes. Each class contains 500 $3\times 64\times 64$ images for
training and 100 images for testing.

###  4.2 Baselines and metrics

The baselines we compare are the methods mentioned in Section 2: A-GEM, and
ER, GSS, MIR, ASER, focusing on sample selection, and SCR and DVC, two
algorithms with standard DA. We apply our DualMix on the four ER-based methods
and compare them with A-GEM, SCR and DVC. Finetune and iif offline are used as
the lower and upper bounds as the same setting in [26].

We use two standard metrics in the continual learning to measure performance:
Average Accuracy and Average Forgetting. Let $a_{i,j}$ be the accuracy of the
model on testing set of task $j$ after trained from task 1 to task $i$.
$f_{i,j}$ represents how much the model has forgot about task $j$ after being
trained on task $i$. The two metrics are defined as:

$\mathrm{Average}\,\,\mathrm{Accuracy}\left(A_{i}\right)=\frac{1}{i}\sum_{j=1}^{i}{a_{i,j}}$
(14)
$\begin{split}\mathrm{Average}\,\,\mathrm{Forgetting}\left(F_{i}\right)=\frac{1}{i-1}\sum_{j=1}^{i-1}{f_{i,j}}\\\
\mathrm{where}\,\,f_{k,j}=\underset{l\in\left\\{1,\cdots,k-1\right\\}}{\max}\left(a_{l,j}\right)-a_{k,j},\forall
j<k\end{split}$ (15)

Dataset | CIFAR-100 | Mini-ImageNet | Tiny-ImageNet
---|---|---|---
Memory Size | M=1K | M=2K | M=5K | M=1K | M=2K | M=5K | M=1K | M=2K | M=5K
A-GEM | 57.4 ± 0.8 | 58.0 ± 1.1 | 57.3 ± 1.2 | 52.3 ± 1.2 | 52.1 ± 1.3 | 52.5 ± 1.5 | 44.4 ± 1.0 | 43.8 ± 1.1 | 44.5 ± 0.6
ER | 52.1 ± 1.5 | 47.9 ± 1.1 | 44.4 ± 1.1 | 46.8 ± 1.8 | 43.6 ± 1.8 | 42.1 ± 1.6 | 45.6 ± 1.5 | 43.0 ± 1.4 | 39.1 ± 1.6
GSS | 51.1 ± 1.1 | 47.5 ± 1.1 | 44.5 ± 1.8 | 47.4 ± 1.8 | 41.6 ± 1.4 | 39.4 ± 1.3 | 45.1 ± 1.0 | 42.9 ± 1.2 | 41.1 ± 1.4
MIR | 50.2 ± 1.0 | 42.9 ± 1.7 | 40.6 ± 1.5 | 44.9 ± 1.1 | 39.8 ± 1.6 | 37.4 ± 2.1 | 44.9 ± 2.0 | 40.9 ± 1.8 | 36.7 ± 1.9
ASER | 52.1 ± 1.0 | 46.6 ± 0.8 | 37.9 ± 1.4 | 48.1 ± 1.4 | 44.8 ± 1.3 | 38.7 ± 2.2 | 46.4 ± 0.8 | 43.1 ± 0.6 | 39.7 ± 0.4
DVC | 40.2 ± 1.0 | 38.8 ± 1.1 | 37.2 ± 1.4 | 40.0 ± 1.0 | 37.0 ± 1.7 | 36.6 ± 2.4 | 40.5 ± 1.6 | 36.7 ± 1.7 | 33.2 ± 3.1
ER+DualMix | 36.7 ± 1.4 | 31.8 ± 1.7 | 32.8 ± 1.1 | 35.7 ± 1.1 | 31.2 ± 1.7 | 30.4 ± 2.1 | 29.5 ± 2.1 | 22.7 ± 2.0 | 26.5 ± 1.9
Table 2: Average Forgetting (end of training, lower is better) on three
benchmarks with 1K, 2K and 5K memory. DualMix is only applied on ER, which is
enough to demonstrate our good performance.

###  4.3 Implementation details

We use a reduced ResNet-18 for all datasets as in [26, 13]. A single-head is
used for all classes. The normal DA used here is a combination of four
augmentation methods as used in [13, 27]: random crop, horizontal flip, color
jittering and grayscale. We use Stochastic Gradient Descent (SGD) to optimize
the model and set the learning rate to 0.1, batch size to 10. $\alpha$ is set
to 0.2. $\kappa$, $\tau$ and $\delta$ are set to 2.0, 0.5 and 0.05
respectively. EnMix and AdpMix are both combined with standard DA. All the
experimental results we present are averages of 10 runs, performed on one
NVIDIA GeForce RTX 3090 GPU.

###  4.4 Comparative performance evaluation

Accuracy performance. The accuracy results are illustrated in Table 1. We
apply our DualMix strategy on four ER-based methods, ER, GSS, MIR and ASER,
which optimize the sample selection for memory in OCI. According to the
results, DualMix can improve the ER-based methods significantly in the three
datasets, showing that our mixing method is effective on a variety of sample
distributions. As we can see, our method performs better in relatively small
memory (double the accuracy on CIFAR-100 with ER and MIR), due to the severe
lack of data diversity in such setting. In Mini-ImageNet, our method can also
achieve 69.2% and 40.3% performance gain with 1K and 2K memory respectively.
Interestingly, compared with ER, the gains on other ER-based methods are
relatively small. Intuitively, the combination of better methods should lead
to better performance. However, our method increases the performance on ASER
and GSS with a limited degree. On the one hand, this may be due to the
inherently advanced performance of these method, making it difficult to keep
improving. On the other hand, sample selection strategy produces a
distribution that is quite different from the original data, resulting in
biased augmentation. In contrast, the samples obtained by ER through reservoir
sampling [39] are more uniform. Combined with DA, there is a greater potential
to obtain more diverse and adequate samples. In addition to effectively
improving the performance of some existing methods, our strategy also
demonstrates superiority over state-of-the-art methods, SCR and DVC. For a
fair comparison, the memory batch size of SCR is the same with our DualMix and
DVC. It can be seen that our method performs remarkably better than DVC in
most cases, despite the fact that DVC and SCR still require new loss functions
to enhance feature exploration about old and current tasks. These results
suggest that DA plays an important role in OCI, and its potential has not been
fully tapped in previous approaches.

Forgetting rate. Table 2 shows the Average Forgetting by the end of training.
We apply DualMix on ER to compare with other baselines in CIFAR-100, Mini-
ImageNet and Tiny-ImageNet. We don’t illustrate the Average Forgetting results
of SCR because it performs poorly on tasks due to limited memory batch size.
Our method can achieve the lowest forgetting on the three benchmark datasets
with different memory sizes. In CIFAR-100, our method can achieve 8.7% $\sim$
18.0% reduction on Average Forgetting compared with the strongest baseline
DVC. This number in Tiny-ImageNet can even reach 38.1% with 2K memory, opening
up a huge gap with other methods.

Figure 5: The mean variance calculated by Equation 6 on CIFAR-100 with
different DA methods. “Combine” means the combination of “Crop”, “Flip”,
“ColorJitter” and “GrayScale” and EnMix is performed based on “Combine” to get
stronger DA.

###  4.5 Ablation study

Effectiveness of each component. We investigate the effectiveness of each
component of our method. As shown in Table 3, a strong DA can already make
improvement compared with the baseline. Applied the stronger EnMix, further
great improvement occurs on Average Accuracy and Average Forgetting in all
cases. In addition, AdpMix also improves the performance of the model in
various experiments, which is even more remarkable than EnMix. After combining
all the components, our method achieves the best performance especially when
the memory is small. This is an intuitive phenomenon, which highlights the
importance of data augmentation to enrich diversity and alleviate the class
imbalance. The results indicate that each of our components is essential.

Method |  M=1k (AA$\uparrow$/AF$\downarrow$) |  M=2k (AA$\uparrow$/AF$\downarrow$) |  M=5k (AA$\uparrow$/AF$\downarrow$)
---|---|---|---
ER |  8.4 ± 0.5 / 52.1 ± 1.5 |  12.7 ± 0.8 / 47.9 ± 1.1 |  18.1 ± 1.1 / 44.4 ± 1.1
ER+DA |  14.6 ± 1.6 / 48.0 ± 2.4 |  15.5 ± 2.0 / 48.5 ± 2.6 |  18.5 ± 1.4 / 43.2 ± 1.9
ER+EnMix |  15.7 ± 1.1 / 42.7 ± 1.6 |  17.2 ± 4.0 / 43.4 ± 3.9 |  18.7 ± 1.4 / 43.8 ± 2.3
ER+AdpMix |  16.6 ± 1.2 / 38.7 ± 1.8 |  18.9 ± 1.8 / 33.5 ± 3.8 |  21.3 ± 2.2 / 33.3 ± 2.4
ER+DualMix |  17.5 ± 1.0 / 36.7 ± 1.4 |  20.3 ± 0.9 / 31.8 ± 1.7 |  22.5 ± 1.2 / 32.8 ± 1.1
Table 3: Comparison with different augmentation strategies on CIFAR-100 with
1K, 2K and 5K memory. AA and AF denote Average Accuracy and Average Forgetting
respectively. Every component is essential in our method and their combination
achieves the best performance.

Empirical results about EnMix and correlation. To verify our proposition 1, we
use the mean variance of the output probability based on old models, as
described in Equation 6, to demonstrate the effect of EnMix. In Figure 5, we
plot the scatters of $\bar{m}$ and the final Average Accuracy on CIFAR-100
with various DA strategies. We can see that there exists an obviously negative
correlation between covariance and accuracy. “Flip”, “Colorjitter” and
“GrayScale” are with relatively weak strength, so the augmentation samples
have higher correlation with original samples, which results in weaker
improvement. The strength of “Crop” and their combination is stronger and our
EnMix further promotes them, achieving better performance.

Correct decision boundary with AdpMix. As we discussed in 3.1, there exists
the serious biased decision boundary between old and new classes. Therefore,
we propose the AdpMix to push the decision boundary away from samples of old
classes. As shown in Figure 6, we plot the error ratio of $er\left(n,o\right)$
and $er\left(o,n\right)$ when we apply AdpMix on ER. Compared with the results
in Figure 4, $er\left(o,n\right)$ decreases notably and the gap between
$er\left(o,n\right)$ and $er\left(n,o\right)$ is almost wiped out. This shows
that our method does correct the previous decision boundary.

Running time comparison. Our method only perform mixing augmentation to enrich
samples, which adds a certain amount of training time. However, the increased
time cost of our method is pretty small than that of some other sample
selection or contrastive-relevant methods. We can see from Figure 7 that ER
and A-GEM, which use the reservoir method to update and retrieve memory, have
the shortest training time. Various sample selection strategies are applied in
other approaches, significantly increasing running time especially SCR, DVC
and GSS. Our method only augments memory data, and the increased running time
is less and within a controllable range.

Figure 6: The misclassification ratios during the training process on
CIFAR-100 by ER+AdpMix with different memory sizes. The biased decision
boundary is greatly alleviated. Figure 7: Running time comparison with a 2K
memory buffer and trained on CIFAR-100.

##  5 Discussion

Continual learning can falls into the poor performance because of catastrophic
forgetting. We revisit the data augmentation strategy in online class-
incremental learning, a challenging and more realistic setting. We prove that
aggressive augmentation which generates samples with low correlation with
original samples are beneficial to forgetting avoidance. However, label
consistency occurs when the standard augmentation is too strong. We propose
the Enhanced Mixup (EnMix), which mixes the augmented samples and their labels
based on standard DA, resulting in increased sample diversity and consistency
with labels. Further, to address the class imbalance in replay-based OCI, we
introduce the Adaptive Mixup (AdpMix) to mix samples from old and new classes,
which can recalibrate the biased decision boundary. Our method can be directly
combined with existing methods and boost them effectively. The practice in
this paper shows that the CF in CL can be greatly alleviated merely through
appropriate DA strategy.

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