Exploring Example Influence in Continual Learning
Authors: Qing Sun
Fan Lyu
Fanhua Shang
Wei Feng
Liang Wan
Venue: NeurIPS 2022 Accept (2022)
Keywords: Continual Learning

Abstract: Continual Learning (CL) sequentially learns new tasks like human beings, with the goal to achieve better Stability (S, remembering past tasks) and Plasticity (P, adapting to new tasks). Due to the fact that past training data is not available, it is valuable to explore the influence difference on S and P among training examples, which may improve the learning pattern towards better SP. Inspired by Influence Function (IF), we first study example influence via adding perturbation to example weight and computing the influence derivation. To avoid the storage and calculation burden of Hessian inverse in neural networks, we propose a simple yet effective MetaSP algorithm to simulate the two key steps in the computation of IF and obtain the S- and P-aware example influence. Moreover, we propose to fuse two kinds of example influence by solving a dual-objective optimization problem, and obtain a fused influence towards SP Pareto optimality. The fused influence can be used to control the update of model and optimize the storage of rehearsal. Empirical results show that our algorithm significantly outperforms state-of-the-art methods on both task- and class-incremental benchmark CL datasets.

Exploring Example Influence in Continual Learning
Qing Sun", Fan Lyu", Fanhua Shang, Wei Feng, Liang Wan College of Intelligence and Computing, Tianjin University {sssunqing, fanlyu, fhshang, wfeng, @ ju. edu. https://githubb com//SSmmminng/Eaaalllee Inf Influence_CC CL
Abstract
Continual Learning (CL) sequentially learns new tasks like human beings, with the goal to achieve better Stability (S, remembering past tasks) and Plasticity (P, adapting to new tasks). Due to the fact that past training data is not available, it is valuable to explore the influence difference S and P among training examples, which may improve the learning pattern towards better SP. Inspired by Influence Function (IF), we first study example influence via adding perturbation to example weight and computing the influence derivation. To avoid the storage and calculation propose simple yet effective
burden of Hessian inverse in neural networks, MetaSP algorithm to simulate the two key steps in the computation of and obtain the S- and P-aware example influence. Moreover, we propose to fuse two kinds of example influence by solving a dual-objective optimization problem, and obtain fused influence towards SP Pareto optimality. The fused influence be used to control the update of model and optimize the storage of rehearsal. Empirical results show that algorithm significantly outperforms state-of-the-ar methods on both task- and class-incremental benchmark CL. datasets.
Introduction
By mimicking human-lite learning, Continual Learning (CL) aims to enable model to continuously learn from novel knowledge (new tasks, new classes, etc.) in a sequential order. The major challenge in CL. is to harness catastrophic forgetting and knowledge transition, namely the Stability- dilemma, with Stability (S) showing the ability to prevent performance drops for old tasks and Plasticity (P) referring if the new task can be learned rapidly and unimpededly. Intuitively speaking, a robust CL system should achieve outstanding S and P through sequential learning.
The sequential paradigm means CL does not access past training data. Comparing to traditional machine learning, the training data in CL is thus more precious. It valuable to explore the influence difference on S and P among training examples. Following the accredited influence chain "Data- Model-Perf ormance". exploring this difference is equivalent tracing from performance back to example difference. With appropriate control, this may improve the learning pattern towards better SP. On top of this, the goal of this paper explore the reasonable influence from each training example to SP, and apply the example influence to Cl. training.
To understand example influence, one classic successful technique is the Influence Function (IF) 20 which leverages the derivation chain rule from a test objective to training examples. However, directly applying the chain rule leads computing the inverse of Hessian with the complexity of O )²) (n is the number of examples and is parameter size). which is computationally intensive and may run out-of- oe--ory in neural networks. In this paper, we propose novel meta-learning algorithm, called MetaSP, to compute example influence via simulating IF. We design based on the sidearsal-based
"Co-first authors.
1 CCoresponding author.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Old Task:
New
Negalive SP
Positive
Upiiaee Mixi
Ca Dog
Positive
Bird VS Fish
Update by IF
IF= 0.02
0.03
0.08
0.11
0.15
0.17
0.21
0.25
Modil
Modil Rg
Negative P
Positive
Wooe
Chily P
Pareto Fusion
2 Selection For Rehesrssl
Store
Meuocy Butfer
Stae
F Fusion
Negative
Only 5
Parete Front
Drop
Better
Wore
61) Influence A cquirement
(b) Influence Fusion
(c) Influence Utilization
Training examples have different influences on Stability and Plasticity. Given an old task with classes "cat" and "dog" and a new task with classes "Bird" and "Fish", we compute the influence on and for each example. Then, we fuse the two kinds of influence towards SP Pareto front. We also show that example influence can be used to adjust model update and optimize rehearsal selection.
Figure I:
CL framework, which avoids forgetting via retraining part of old data. First, a pseudo update is held with example-level perturbations. Then, two validation sets sampled from seen data are used to compute the gradients on example perturbations. The gradients are regarded the example influence on S and P. As shown in Fig. Ta), examples can be distinguished by the value of influence on S and P.
To leverage the two independent kinds of influence in CL, need to take full account of the influence on both S and P. However, the influence on S and P may interfere with each other, which leads us to make trade-off. This be seen as a Dual-Objective Optimization (DOO) problem, which aims to find solutions not dominated (no other better solution) by any other one, i.e. Pareto optimal solutions (8) We say the solutions the example influence SP. Following the gradient-based MGDA algorithm [12., we obtain the fused example influence on SP by meeting the Karush- Kuhn- Tucker (KKT) condition, illustrated in Figure [b).
Finally, we show that the fused influence can used to control the update model and optimize the storage of rehearsal in Figure Mc). On one hand, the fused influence can be directly used to control the magnitude of training loss for each example. On the other hand, under a fixed memory budget, the fused influence can be used to select appropriate examples storing and dropping, which keeps the rehearsal memory always have larger positive influence on SP.
In summary, our contributions are four-fold: 1) Inspired by the influence function, we study CL. from the perspective of example difference and propose MetaSP to compute the example influence on S and P. 2) We propose to trade off S and P influence via solving DOO problem and fuse them towards SP Pareto optimal. 3) We leverage the fused influence to control model update and optimize the storage of rehearsal. 4) The verification contribution: by considering the example influence, in our experiments on both task- and class-incremental CL. better and more stable P can observed.
2 Related Work
Continual Learning. Due to many researchers' efforts, lots of methods for CL have been proposed, which can be classified into three categories. The regularization-baser methods [19, Ii [0] are based on regularizing the parameters corresponding to the old tasks and penalizing the feature drift. 26 generate task-specific parameter expansion or sub-
The parameter isolation based methods [4.
branch. Rehearsal-basee based methods (29 7 22
28 3 25 tackle the challenge of SP dilemma by retaining a subset of old tasks in a stored memory buffer with bounded resources. Although the existing methods apply themselves to achieve better SP, they fail to explore what contributes to the Stability and Plasticity inside the training data. In this work, we explore the problem in the perspective of example difference, where we argue that each example contributes differently to the SP. We focus our work the rehearsal-bee -baed CL framework in order to omit the divergence between models, while evaluating the old data's 's influence simultaneously.
Example Influence. In recent years, as the impressive Interpretable Machine Learning (IML) 27 develops, people realize the importance of exploring the nature of data-driven machine learning. Examples are different, even they belong to the same distribution. Because of such difference, the example contributes differently to the learning pattern. In other words, the influence acquired in advance from different training examples significantly improve the CL training. Some studies

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propose a similar idea and use the influences to reweight or dropout the training data B3 [6]. In contrast to complicated model design, a model-agnostic algorithm estimates the training example influence via computing the derivation from test loss to training data weight. One typical example method is the Influence Function [20], which leverages pure second-order derivation (Hessian) with the chain rule. In this paper, to avoid the expensive computation of Hessian inverse, we design meta
learning [18] based method, which can be used to control the training.
3 Demystifying Example Influence on SP
3.1 Preliminary: Rehearsal-based CL
Given T different tasks w.r.t. datasets (D1. ... Dr., Continual Learning (CL) seeks to learn them in
sequence. For the 1-th dataset (task), D; = {(x, Mi ) )}== is split into training set D and test
set De, where N, is the number of examples. At any time, CL aims at learning a multi-askkmmulti-lasss
predictor to predict tasks/classes that have been learned (say iask-incremental and glass-incremeet
CL). To suppress the catastrophic forgetting, the rehearsal-bas bbased CL small size memory buffer M₁ sampled from D for each task (i.e. M < (Dim
[30 22 32
6 [6] builds
At training
phase, the data in the whole memory M UR will be retrained together with the current tasks.
Accordingly, mini-batch training step of task in rehearsal-based CL is denoted as
min 0.
E(Botd U Brew, 0t), Botd C C and Brew C DiD,
(1)
where / is the empirical loss. 0, is the trainable parameters at task / and is updated from scratch.
3.2 Example Influence on Stability and Plasticity
Definition (Stability and Plasticity) Suppose the parameter of a model is initialized to Og At the training on the 1-th task, given test sets of an old task Diat (I: < 1) and the current task Dist, the Stability S and Plasticity P. can be evaluated by:
- ==
P. P(= 11)))
(.1, pDD (0-1),
where (D D2) represents the performance (accuracy in classification) of Dj conditioned to the model O training D2, p(D(0) denotes the performance of 'D tested on the model O.
The S of a task is evaluated by the performance difference on the test set after training on any later tasks, which is also known Forgetting [6]. The P of task is defined as the ability to integrate new knowledge, which regarded as the test performance of this task. As many existing CL. methods demonstrate, the SP inevitably interferes mutually.
Definition 2 (Example Influence on SP) At the training on the t-th task, with a sampled example ..tr Dem, the example influence from znt to Stability St. and Plasticity P. for < t can be evaluated by the gap from deleting it then retraining the model:
a 01,1, a 0,-1, Dirm /221m), Ls(D)at x _
(D _nm) p(D 0. Dirm) p(D) 0.-1, Dium / tm
where Dim /2 denotes the dataset Di without the training example trm
However, deleting every example compute full influences is impractical due to the highly computa- tional cost. Instead, the performance change can be indicated by the loss change, which leads to derivable way to approximate the influence:
Is(Di x (mm) del d
tm)) del (l()))t) de
Ip(D)st
de
where is the weight perturbation to the training example and def means define. This influence can be computed by the Influence Function (20) that will be introduced in the next section.

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Taskt- 1
Taskt
Tast+11
Continual Learning
Vpr 8Ex)
[(ver B
(1
Imput
arg min 6(B, (7
Ōg.
Traizing Data
Ded(Vou. (ks)
Pass
I(Vold. B)
Penudo
DIT Vintad
(a) Influrnce A equinement in Mini-Batte
Select top
Better
infflsecc
Ingar
(100.0)-1"A00.0))
Clusner
Select by top E00
M
Worse SP influence
Treining Data
Select by top ELU
New Task Dalaset
Store
Drop
(b) Model update with example influence
(c) Rehearsel selection with example influence
Evaluating and making use example influence in mini- batch Continual Learning. (a) At each iteration in CL training, MetaSP updates in pseudo and use two validation sets representing old tasks and new task obtain the example influence on S and The two kinds of influence are fused
Figure 2:
towards a Pareto optimal. (b) The computed influence can be directly used to update CL model and (c) select examples for rehearsal storing and dropping.
Meta Learning on Stability and Plasticity
4.1 Influence Function for SP
mini-batch, B, from the training data is sampled, and the normal model update is
i
arg min & (B, 0).
(3)
In Influence Function (IF) a small weight perturbation added the training example tm
arg min e (B, + el(a. O), tm E
(4)
0 We easily promote this to the batc-
θg arg min (B, 0)
EE L(B,()
(5)
where L denotes the loss vector for mini-batch and E || denotes the perturbation on each example in it. It is easy to know that the example influence / (DDist BB) is reflected in the derivative
Vgl(Dist Og..) B- By the chain rule, the example influence in IF can be computed by
I(Dist: B) def VEl(Dist 0g.0)) E Vo H V L(B.O).
(6)
where = V8((B 0) is Hessian. Unfortunately, the inverse of Hessian requires the complexity O(IBlq² + )2) and huge storage for neural networks (maybe off-or-- which challenging for efficient training.
In Eq. (6). have I(D B)
B) and find the loss will get larger if T(Dist = > O, which means the negative influence on the test set Dist Similarly, (Dist xx")) < means the positive influence
""""""""""""""""""
the test set Diat, Fortunately, the second-order derivation in IF is not necessary under the popular meta learning paradigm such as [18], instead we can easily get the derivation like IF through a one-step pseudo update. In the following, we will introduce simple yet effective meta-based method, named MetaSP, to simulate IF at each step with a two-level optimization avoid computing Hessian inverse.
4.2 Simulating IF for SP
Based on the meta learning paradigm, we transform the example influence computation into solving meta gradient descent problem, named MetaSP. For each training step in rehearsal-bar based CL,

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Algorithm 1: Computation of Example Influence (MetaSP)
Input: Bold. Brew; Vold. Vnew :
Training batches, Validation batches // Pareto example influence SP // Pseudo update // Cradient from old val loss // Gradient from new val loss // Optimal fusion ayper-paramette // Influence fusion
Output: r²;
ŌE, = arg ming ((Bala U Bnew: 0) + E L(Boid Bnew; 0):
I(Vaa, B) = VgL(Vola, 0K,B);
I[vew: B)
Vge(Vaew Bg,B):
7" < Eq.
5 - 7" [ old: B) + (1 - 7²) I(Venv B):
we have two mini-batches data Bold and Bnew in respect to old and new tasks. Our goal is to obtain the influence on S and P from every example in Bold U Bnew- Note that both S-aware and P-aware influence are applied to every example regardless of old or new tasks. That is, the contribution of an example is not deterministic. Data of old tasks may also affect the new task in positive, and versa. In rehearsal-based CL, we turn to computing the derivations V Vge(Vola 0)
B== for example influence.
To compute the derivation, as shown in Fig. 2(a,, our MetaSP has two key steps:
(1) Pseudo update. This step simulate Eq. in IF via pseudo update
Eg arg min
{(Bold U Bew. θ) + E L(Bold U Bnew, 0),
(7)
where L denotes the loss vector for mini-batch combining both old and new tasks.
(2) Compute example influence. This step computes example influence on S and P for all training examples as simulating Eq. (6). Based the pseudo updated model in Eq. ( we compute S- and P-aware example influence via two validation sets Vata and Vnew. Noteworthily, because the test set Dist is unavailable at training phase, we use two dynamic validation sets Votd and View to act as the alternative in the CL training process. One is sampled from the memory buffer (Vola) representing the old tasks, and the other is from the seen training data representing the new task (Vnow). With E initialized to O, the two kinds of example influence are computed as
I(Vola B) Vgl(Vold: ŌE,B), I(Vnow, B3 Vge((Vnew Ōg,B).
Generally, each elements in two influence vectors I(Val, B) and [(V B) represents the example influence on S and P. Similar to IF, elements with positive value mean negative influence while elements with negative value mean positive influence.
5 Using Influence for Continual Learning
5.1 Before Using: Influence for SP Pareto Optimality
As shown in Eq. ( the example influence equal the derivation from validation loss of old and new tasks to the perturbations E. However, the two kinds of influence are independent and interfere with each other. That is, using only one of them may fail the other performance. We prefer to find a solution that makes a trade-off between the influence on both S and P. Thus, we integrate the two influence I(Vola B) and I(Voew, B) into a DOO problem with two gradients from different objectives.
{((Vas Ōr, E(Vnews ŌF,.)
min E
The goal of Problem is to obtain a fused way that satisfies the SP Pareto optimality.
Definition (SP Pareto Optimality)
1. (Pareto Dominate) Let E,, Eg be two solutions for Problem (9) E, is said to dominate Es (E, x
E,)) if and only if E(V, ŌE.,.) I E(V, θg.,B), YV E (Vold, Vnew ) and E(V, Qg...) < E(V, Bg,
EV (Vola, Vnow
2. (SP Pareto Optimal) E is called SP Pareto optimal if no other solution can have better values in
E(Vota, θg,B) and E(Vnew. θg,8).
5

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Algorithm 2: Using Example Influence in Rehearsal-bare based Continual Learning.
Input: Initialized Og. Learning rate a, Training set (DI ... ;DT). Memory M
Output: 0r:
// Final model
for task do
T==rninewwwaa Dim M) (Alg- 306
for i 1:
M do
C2.C2,.... Clm t K-Means(Dff"):
7
Pop the bottom of M, Push the top of C, to M:
Rank C, with (z)), E,;
8
Rank M with (() M:
9 end
end
Inspired by the Soltiple-Gradiens Descent Algorithm (MGDA) [2] we transform Problem min-norm problem. Specifically, according to the KKT conditions [1], have
9 to
= arg min ||> Vgee(Vo ÕR,S) + (1 Vge(Vnew 0 9.b,0575,
(10)
Referring to the study from Sener et al. 34 the optimal y" is easily computed as
(VEllVnew ÕF,B) Vel(VVoaa ÕF,B) VVl(( ÕE.5) . .0.
.1).
min
max
(11)
VVE 0g.8) Vgg(Vons θE,S) (e..s)ll 113
Thus, the SP Pareto influence the training batch can be computed by
I' I(Vala B) + --) I(V B).
(12)
This process can be seen in Fig. Za). Different from the S-aware and P-aware influence, the integrated influence consider the Pareto optimum both S and P. i.e., reducing the negative influence on and keeping the positive influence on both S and P. Then we will introduce how to leverage example influence in CL training, our algorithm can be seen in Alg.
5.2 Model Update Using Example Influence
With the computed example influence in each mini-batch, we can easily control the model update of this mini-bac batch to adjust the training towards an ensemble positive direction. Given parameter f from the previous iteration the step size a, the model can be updated in traditional a Ve (E(B, 0)), where = Bou Bnew. By regularizing the update with the example influence I* we have
A Algorithm 3: Training New Task
Input: Initialized 0,, Training set Dim
Memory M,, Learning rate
Output: Trained 0,
for
:IEE NUM do Beew ~ Dim: if l 1 then
SGD as 0"
2
0, = O: a Vol(Bnew. O.):
else
0- - Vo (((B,)) (-I") L(B,
Bold ~ M. Vold ~ M. Vnew ~ Dim
(13)
I METASP(BBBB. Brew, Vola. Vnew)
MetaSP offers regularized updates every step for rehearsal-basse -based CL, which leads the CL training better SP but with only the complex- 10
0, = Og a Ve(e(Bols U Bnew O,) +( +(-I)) L(BaM U Bnew. O,));
end
ity of O + vg) (v denotes the validation II end size) compared with that of IF, O(IB)22 y²).
We show this application in Fig. bb. By updating like the above equation, we can make use of the influence of each example to large extent. In this way, some useless examples are restrained and some positive examples are emphasized, which may improve the acquisition of new knowledge and the maintenance of old knowledge simultaneously.
5.3 Rehearsal Selection Using Example Influence
Rehearsal fixed budget needs to consider storing and dropping to keep the memory M having the core set of all old tasks. In tradition, storing and dropping are both based on randomly example

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is ranked by its SP influence value, and the most positive influence on both SP will be selected to selection, which ignores the influence difference on SP from each example. Given influence I* (x) representing contributions from example x to SP, we further design to use it to improve the rehearsal strategy under fixed memory budget. The above example influence on S and P is computed in mini-batch level, we can promote to the whole dataset according to the law of large numbers, and the influence value for the example a: is the value of expectation over batches, i.e., (x))
The fixed-size memory divided averagely by the seen task number. After task finishes its training, we conduct our influence-aware rehearsal selection strategy as shown in Fig. 2c. For storing, we first cluster all training data into ? [M] groups using K-means to diversify the store data. Each group
store. For dropping, rank again the memory buffer via their influence value, and drop the most negative [M example. In this way, M always stores diverse examples with positive SP influence.
Experiments
6.1 Datasets and implementation details
We use three commonly used benchmarks for evaluation: 1) Split CIFAR-10 337 consists of 5 tasks, with 2 distinct classes each and 5000 exemplars per class, deriving from the CIFAR-10 dataset; 2) Split CIFAR-100 (3 splits the original CIFAR- 100 dataset into 10 disjoint subsets, each of which is considered separate task with 10 classes; 3) Split Mini-Imagenet (5 is subset of 100 classes from ImageNet [9], rescaled to 32 x 32. Each class has 600 samples, randomly subdivided into training (80% and test sets (20%). Mini-Imagenet dataset is equally divided into 5 disjoint tasks,
We employ ResNet- 18 17 as the backbone which is trained from scratch. We use Stochastic Gradient Descent (SGD) optimizer and set the batch size 32 unchanged in order to guarantee an equal number of updates. Also, the rehearsal batch sampled from memory buffer is set to 32. We construct the SP validation sets in MetaSP by randomly sampling 10% of the seen data and 10% of the memory buffer at each training step. We set other hyper-settings following ER tricks [4), including 50 total epochs and seyper-paramtters All results are averaged over fixed seeds for fairness.
To better evaluate the CL. process, we suggest evaluating SP with four metrics as follows. We use the sign function 1()) to represent if the prediction of model is equal to the ground truth. 1) First Accu- racy (A1 = + E. Ex.ep 1yy,, Or (z:))): For each task, when is first trained done, we evaluate its testing performance immediately, which indicates the Plasticity, ie.. the capability of learning new knowledge. 2) Final Accuracy (A + E, Ex,epy (1, (r;))) This metric is the final performance for each task, which indicates Stability, i.e., the capability of suppressing catastrophic for-
getting. 3) Mean Average Accuracy (Am + E, (t Exso Ex,Epy 1 ( Or (zz))) ): This metric computes along CL process, indicating the SP performance after each task trained done. 4) Backward Transfer (BWT- = - Σi Σ ((,y))D (1, (x)( (v, (x())) = (Ax AA)): This metric is the performance drop from first to final accuracy of each task.
6.2 Main Comparison Results
We compare our method against 8 rehearsal- based methods (including GDUMB 288, GEM (2). AGEM (6), HAL (5) GSS (2]. MIR D. GMED (33 and ER (7).. What's more, we also provide lower bound that train new data directly without any forgetting avoidance strategy (Fine-tune) and an upper bound that is given by all task data through joint training (Joint).
In Table we show the quantitative results of all compared methods and the proposed MetaSP in insseinnreeeetal and ies-inncemeetal settings. First of all, by controlling the training according to the influence on SP, the proposed MetaSP outperforms other methods on all metrics. With the memory buffer size growth, all the rehearsal- of MetaSP are more obvious. In terms of the First Accuracy A₁, indicating the ability to learn new tasks, our method outperforms most of the other methods with little numerical advantage. In terms of the Final Accuracy Accs which is used to measure the forgetting, we have an obvious improvement of an average of 3.17 for inasemaacrem setting and averagely 1.77 for task-incremental setting w.r.t. the second best result. This shows MetaSP can significantly keep stable learning of the new task while suppressing the catastrophic forgetting. It is because although the new tasks may have
CL get better performance, while the advantages
7

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Table 1: Comparisons on three datasets, averaged across 5 runs (See std. in the Appendix). Red and blue values mean the best our methods and the compared methods. indicates that our method is significantly better than the compared method (paired t-tests at 95% significance level).
CIEARIO (Clas increment)
CIFARIO (Tisk increment)
Method
bu Tuleer size
500
bulfer size 500
hufler size 500
butter size 500
DWT
BWI
DWT
Firetune
19.06
Join
01.70
Pinctune
05:27
Jain
95.16
GIUMMI
50.02
14:27
GEM
93,50
37.51
55.43*
-70.45
92.70
36.95
57.30
-69.76
56. 62+
92.09e
-9.00
96.73e
90.12
92. 93
-7.88
AGEM
20.00
-05.65
-95.60
06.71
53
91.30
24.45
46.31
91.91i+
27.90
49.05
11.41
-53.75
-10.39
92.00
51.841
MIR
96,70
38.53+
56.96*
-72.72+
96.85
02.65.
50.90
-67.50
56.7Ce
53.33 50.50
90.87
-10.33
96.73
90.05
91.99
-7.62
GSS
96.53
35.80
64.33
-70.90+
96.55
41.
11.996
58.16+
24
56. -5.
85.05
90..80
10.63+
96.57
50..38
52. 15w
-7.75
GMB3
53
96.63
38, 12+
58,
58.922
-73.16+
96.05
02.08
022,00
-66.21
550
7m
88, 91
91
201
-9.70
96.72
59.72+
52. 11e
-8.75
FR
96.73
34.15 10w
53.72
-78.15
06.74
40.45m
67.60+
70.36
06 90
076
01 12e
9.85
96.79
90.60
92. 78-
-7.74
49.10
7.30
RehSel
96.85
43.76
63.69
-66.36
96.81
50.10
68.28
-58.38
97.11
89.91
92.66
-8.99
97.30
91.41
50.28
CIFARIO0 (Class increment)
CIFARIOO (Tank increment)
Method
buffer size 500
buffee size 1000
butfer size 500
butfer size 1000
A
Ave
An
BWT
A
Ax
Aa
BWT
4
Ase
An
BWT
A
Ale
Ax
BWT
Finetune
9.14
Joint
71:25
Finetune
30.85
Joint
91.03
GDUMB 23
11.11
15.75
36.401
435
GEM 122
85.28
15.91
29.
29.38
-77.07
84.28
22.79
34.00
-68.32
85.53+
68, 8.55
68. :0
-18.72
65.21=
73.71
72.50
-12
AGEM
6
86.97
9.33
124.00>
-85.18
85.06
9.27m
24.07
-84.88
85.977
55.254
08 08.233
334.10
80.06+
05.96
660.96
-33.01
HAL
67.33e
8:23
-65.70e
68.06
24.71-
-63.800
67.64+
44.55e
50.77
-25.1
68.624
50.07
84.01
-20.61
MIR
87.38
13:45+
28
-82.09
17.56+
87.50
71.20+
GSS
86.0
14.00
80.02
86.31
17.87
31.82+
76.04
10.00
(6.93)
9G.44e
71.05
71.00
GMED
87.18
14.56+
-80.65
87.20
18.67
60m
76 23
87 17.300
66
20.53
87.40
73.91
78.36
-15.
ER 7
87.23
13.70
28. 884
-11.5
87.33
17.58
32. 45
-77.32
57.29
06. 822
67.500
22.73
97.40
71.74
71.00
-17.00
Dan
S3.13
18.00
70.35
87.58
24.78
46,20
030.76
30.994
70.00
74:07
21.01
88.34
75.32
78.00
15.14
Ours+RehSel
87.81
19.28
39.23
-76.13
87.55
25.72
45.48
-68.69
88. 55
70.81
74.34
-19.74
89.03
76.14
78.27
-14.32
Mini-Imagenne (Class Increnannt
Imml--mageeett (Tack Incresecntt
buffer
butfer
A
Ave
Ap
RWT
Ag
BWT
Agp
Ag
RWT
RWT
Finetune
11.12
44.39
Finetune
23.46
62.20
GDUMB
7.15
AGEM
50.06
10.00
22. 20w
49.22
50.03
10.69e
22.25e
-10.16
50.00e
30.65
50.03e
18.75
128. 122
MIR
61.44
11.07
23.65
-50.00
51.25
11:32
24.00e
-49.52
51.47
229 10
35.201
27.05
51.31
31.33l
37.24
-24 20
GSS
51.63
11:00
23.62
-50.06+
51.35
11:42+
24.05
-19.91
51 11.61+
25. 67.
35.222
-25.71
51.40
31.75
37:23
-24 56.
GMED
33
61.21
11:00
24.47e
-50.25
50.87
11.73+
25.50
48.99
51 13.250m
40.47e
37.66
26.02e
51.00
32. 32.88
38 63.666
600
ER
51.68
11:00+
23.71
-50.80
51.41
11.35
24.08
-50.08+
51 70
55,99.
-28.
51.55
31.50
37.36
-21 95
100.0
Ours-RehSel
51.81
12.74
25.43
-48.84
50.96
14.54
28.44
-45.52
54.73
34.36
40.48
-21.70
51.47
37.30
42.19
-17,83
2015
Posttive on
Influence suutibes
CANA
Negativa
24
3514
Positive
Negative
Positive SP
Negative IP oz SP
100%
max
Inffeence proportio
00%
80%
I1 52
mih 5
80%
1F on P
mut
42%-
18 P0
20%
m0n
0%
4- maa min
(a) Total 500 memory data
(b) Total 10000 new data per tisk
Figure 3: Top: Statistics of examples with positive and negative influence on S, P, and SP. Bottom:
We divide all example influences equally into groups, and count the number in each range.
larger gradient to dominant the update for all rehearsal-based CL, our method improves the example with positive effective and restrain the negative-impact example. In terms of the Mean Average Accuracy Am. which evaluates the SP throughout the whole CL process, our method shows its significant superiority with an average improvement of over 4.44 and 1. 1.24 w.r.t the second best results in caaaseinccrmettlll and inss-inncreeetal settings. The complete results with std. can viewed in the Appendix. Moreover, with the proposed rehearsal selection strategy (Ours+RehSel), we have our Axe improved, which means the selected example according to their influence has a clear ability for reducing catastrophic forgetting. With our Rehearsal Selection (RehSel) strategy, we have an improvement of 0.77 on Acc, but A₁ and Am have uncertain performance. This means better memory may bring in worse task conflict.
8

---

6.3 Analysis of Dataset Influence on SP
In Fig. 3 we count the example with positive/negative influences on old task (S), new task (P), and total SP in Split-CIFRR-10 10. At each task after task 2, we have 500 fixed-size memory and 10,000 new task data. We first find that most data of old tasks has a positive influence on S and a negative influence on P. while most data of new tasks has a positive influence on P and a negative influence on S. Even so, some data in both new and old tasks has the opposite influence. Then, for the total SP influence, most of memory data has positive influence. In contrast, examples of new tasks have near equal number of positive and negative SP influence. Thus, by clustering and storing examples via higher influence to rehearsal memory buffer, the old knowledge can be kept. By dividing all example influences equally into 5 groups from the minimum to the maximum, we find that most examples have mid-level influence, and server as the main part of the dataset. Also, the numbers of examples with large positive and negative influence are small, which means unique examples are few in the dataset. The observations suggest the example difference should be used to improve model training.
6.4 Analysis on SP Pareto Optimum
In this paper, we propose convert the S-aware and P-aware influence fusion into a DOO problem and use the MGDA guarantee the fused solution is an SP Pareto optimum. As shown in Fig. 4 we show the comparison of the First Accuracy and Final Accuracy coordinate visualization for all compared methods. We also evaluate with only stability-aware (Ours only S) and with only Plasticity-awar ence. Obviously, with only one kind of influence, our method can already get better SP than other methods. The integration of two kinds of influence yield an even more balanced SP. On the other hand, the existing methods cannot approach the SP Pareto front well.
AGEM
ER
GEM
(Ours only P) influ-
MIR GMED
OuisodyyPP
Ours
SP Pardo Front
97
90
54
20
92
Figure 4: SP Pareto front.
6.5 Training Time
We list the training time of one-step update and total update overhead for all compared methods for Split CIFAR-I0 dataset. In one-step update, we evaluate all methods with a batch on one update. Our method takes more time than other methods except for GEM, because of the pseudo update, backward on perturbation and influence fusion. To guarantee the efficiency, we utilize our proposed method only in the last 5 epochs among the total, and the rest are naive fine-tuning (See details in the Appendix). The results show the strategy is as fast as other light- methods but achieve huge improvement on SP. We also use this setting for the comparison in Table
Table 2: Comparison of training time [s] on CIFAR- 10.
Method
ER
GSS
AGEM
MIR
GMED
GEM
MetaSP
One-Siep
0.013
0.015
0.029
0,043
0.077
0.093
0.290
0.250
Total
2685
2672
3812
7223
24768
5898
7 Conclusion
In this paper, we proposed explore the example influence on Ktability-Plasticity (SP) dilemma in rehearsal-baad -baed continual learning. To achieve that, we evaluated the example influence via small perturbation instead of the computationally expensive Hessian-like influence function and proposed simple yet effective MetaSP algorithm. At each iteration in CL training, MetaSP builds a pseudo update and obtains the S- and P-aware example influence in batch level. Then, the two kinds of influence are combined via an SP Pareto optimal factor and can support the regular model update. Moreover, the example influence can be used optimize rehearsal selection. The experimental results on three popular CL datasets verified the effectiveness of the proposed method. We list the limitation of the proposed method, (1) The proposed method relies on rehearsal selection, which may affect privacy and extra storage is needed. (2) The proposed method is not fast enough for online continual learning. In most situations, however, we can leverage our training tricks reduce the time. (3) Our method is limited in the extremely small memory size. Large memory size means better remembering and an accurate validation set. The proposed method does not perform well when the memory size is extremely small.

---

Acknowledgement
This work is financially supported in part by the National Key Research and Development Pro- gram of China under Grant (No. 2019YFC1520904) and the Natural Science Foundation of China (Nos. 62072334, 62276182, 61876220). The authors would like thank constructive and valuable suggestions for this paper from the experienced reviewers and AE,
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12


Reviews:
[{"review_id": "vlt0HF_VcSr", "reviewer": "Reviewer_Mnn3", "summary": "The method augments rehearsal-based methods for continual learning. At the heart of the method is a measurement of the influence of each example on the stability and the plasticity of the algorithm. First, the authors introduce a method for estimating said influence. Second, they present a method for using the influence to regularise the parameter updates. Third, they show how to use the influence in order to select which examples of past tasks to store in memory.", "questions": "Do you anticipate that your method would work on sequences of tasks with different input domains? For instance, if you had FMNIST classification mixed with CIFAR10.", "limitations": "I did not see a discussion on the limitations of the method. I’d be curious to better understand the number of tasks the method can handle before it breaks, possibly as a function of the memory size.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper introduced an interesting direction of combining research on Example Influence with Continual Learning. It appears that the methods they introduce are novel and well motivated. Finally, the experiments show good improvement over the many rehearsal-based baselines.\n\nHowever, I feel that the paper’s presentation can benefit from another iteration. There are many sentences which need to be improved and that at the moment hinder the readability of the paper. E.g. lines 44, 52, 190, and 203. Moreover, I was not left with the impression that the presentation of the background material, in particular on influence functions, was satisfactory. For instance, I am confused by eq. (6) and the use of iff. \nApart from the presentation, it’d be good if the experiments also introduced baselines which represent other CL directions, s.a. regularisation-based methods.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "s2x1OqNETKI", "reviewer": "Reviewer_MauF", "summary": "The paper investigates the question of how important each example is for plasticity and stability in continual learning. The authors motivate their approach from the perspective of the Influence Function which was previously used as an explainability method. However, since computing the actual Influence Function requires computing an inverse of the Hessian, the authors decide to instead propose MetaSP, a two-level optimization technique, which aims to approximate the Influence Function. Equipped with this, one can find a loss balancing factor $\\gamma$ which is Pareto optimal with respect to stability and plasticity, and then use this balanced loss for training as well as choosing which examples to keep in the buffer. Finally, the authors show an empirical evaluation on multiple datasets and perform additional analysis of the method.", "questions": "I would like to ask the authors about the issues mentioned in the \"weaknesses\" section above. In particular, I'd like to highlight he first one:\ndo you have theoretical or empirical proof of how well your approach approximates the results obtained from the Importance Function?", "limitations": "I don't think a discussion on potential negative societal impact is needed. \nThe authors do not list the limitations of their method and in my opinion, a section like that should be added. Just to name a few: \n- The method relies on experience replay, so it cannot be applied in settings where data privacy is crucial.\n- The proposed approach is rather slow (2x slower than ER even with the \"use MetaSP only in the last 5-epoch\" trick), so it might be an issue in some settings.\n- The method was implemented only for classification tasks. How hard would it be to extend it to other problems? ", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The idea of disentangling plasticity and stability through IF-like approach and then using this information to get better performance seems novel and intriguing.\n- The empirical evaluation is quite thorough and the proposed method performs well. In particular, I appreciate including the training time as the algorithm seems to be computationally complex. \n- The ablation studies are useful and give a deeper insight into the method. It's interesting to see how the number of examples with a positive impact on plasticity and stability changes with time. \n\nWeaknesses:\n- The major weakness of the paper from my perspective is that in the end, I'm not sure how well the Importance Function motivation connects with the final MetaSP algorithm. Intuitively, these two seem connected, but a theoretical or empirical argument confirming that would be very useful. In particular, do you know how well the MetaSP approach approximates the gradient from Equation (6)? Can we show their equivalence, either theoretically or by running some experiments on toy problems? The authors hint at this kind of result in Section 4.1, but no hard proof is given.\n- The fact that the algorithm is actually applied only on the last 5 epochs out of 50 should be highlighted. This is only mentioned in a single subsection near the end of the paper but seems quite crucial. Have you checked how well the other methods perform in this scenario? For example, what happens if we use fine-tuning for 45 epochs and then ER for 5 epochs? \n- The presentation could be improved, as the paper is hard to read at times. There are some minor mistakes (see the Minor comments section below) and in general I think the paper could be more self-contained. In particular, checking the previous works was necessary for me to understand Section 5.1 (SP Pareto Optimal), as the authors do not list the KKT conditions and do not really explain what the result of Eq (10) is. Grammar could also be improved, but for the most part, the mistakes do not make it harder to understand the paper so this is not an important issue.\n- I'm a bit skeptical whether K-Means on the input space learns anything more than just basic background colors. Have you checked how the examples get clustered? What happens if you do not perform clustering at all (i.e. 1 cluster) or just cluster the examples randomly?\n- The paper does not list limitations although in the checklist the authors claim that it does.\n\n\nMinor comments:\nLine 22: \"robust CL system should achieve outstanding S while keeping a stable P\" - that statement might be a bit controversial as it depends on your desiderata - you might want a model that is plastic but can remember the past moderately well.\nLine 96: What do you mean by performance? You are using the $p$ symbol which suggests probability, but that's a bit confusing.\nLine 120: \"Hessian [...] assumed as positive definite\" - that's a pretty strong assumption if we're not close to the minimum (i.e. during the training).\nLine 121: \"Always out-of-memory\" - that's a bit informal, given enough storage or a small number of parameters I think this operation should be tractable. Of course, I agree that in most cases this statement is true, but I would suggest using more precise wording.\nLine 168: might be useful to state those KKT conditions\nLine 225: I would suggest, when possible, reusing metrics from previous CL papers, as introducing new metrics make it difficult to compare between papers.\nFigure 3: At what point is each of these plots generated? I.e. these metrics are dependent on the parameters $\\theta$, so which $\\theta$ are we using? The one from the end of the current task?\nFigure 4: I don't quite understand this figure. Why are there three bars per task and why there are two separate plots? \nFigure 5: How was the Pareto front plotted?\nTable 2: I assume you mean \"training time\" rather than \"implementation time\"?\n\n\n**EDIT:** Authors' responses and improvements made to the paper convinced me to increase my score to Accept.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "mDYZ2tlZUvI", "reviewer": "Reviewer_Qv4L", "summary": "The authors investigate the stability dilemma through the lens of the training data and the selection for the replay buffer.\nThey evaluate the influence of data points on stability and plasticity using small perturbation to data instead of computing hessian (for computation sake).\nThe authors use the stability and plasticity influence to optimize the gradient update and the sample selection.\nThe algorithm is evaluated on CIFAR10, CIFAR100, and Mini-Imagenet and compared with several state-of-the-art algorithms.\n\nOverall I like the paper and the idea. I think, however, that the authors could realize some improvements to make the paper clearer and better.", "questions": "Table 1: \"ours\" method use random sampling selection? Maybe comparing the sampling selection to another example from the literature would make sense to assess the method better. \n(cf. eg \"An Investigation of Replay-based Approaches for Continual Learning\" Bagus et al)\n\nWhat does that mean that a sample increases plasticity or stability? Beyond the definition provided in section 3, I miss the high-level idea....\n\nl266 \"We divide all example influence equally to 5 groups from the minimum to the maximum.\" how does this division is realized exactly?\n\nl269 \"The two observations suggest we should consider the example difference to improve model training.\" but you do not do it, right? So what is the point of this figure in the paper?\n\nFig.3: left is the evaluation of the data stored in the memory, and the left is the new data? I think the legend could be more detailed to make it clear from the beginning.\n\nTab 1: code color is not defined.\n\nIs the data used for validation in the memory buffer used at some point for training?\n\nTable 2: how many seeds lead to those results?\n\nl.69 \"4) By considering the example influence, in our experiments on both task- and class-incremental, better S and more stable P can be observed\" I do not understand what this contribution is...\n\n\n********\nSome Comments:\n**************\n\n\"KKT\" is never defined.\n\nI would move the task-incremental results in the appendix (task incremental is not very classical to study when using replay, but this is not problematic) and add std of results in the main body.\n\nIt would be better to group figures 3 and 4 or to modify figure 4, such as the legend is not in figure 3.", "limitations": "It would be nice to see in the table which results are statistically significant.\n\nThe legend of several figures could be improved.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\nThe paper is well written and organized, with a detailed description of the method.\nSample selection is a crucial problem to study to improve replay methods.\nthe method is interesting and looks well-performing.\n\nWeaknesses:\nThe definition of the metrics is not very clear. \"Finished Accuracy\" seems to be the classical \"Final Accuracy\". \"First Accuracy\" and its relationship with plasticity is not very clear.\nIs it the average best performance realized on each task?\nMaybe, \"backward transfer\" from \"Gradient Episodic Memory for Continual Learning\" Lopez-Paz et al would be more precise in measuring stability.\nAnd \"Final performance\" would measure the mixture between S and P. I am not sure about what \"Mean Average Accuracy\" brings here.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept