On the Stability and Generalization of Meta-Learning: the Impact of Inner-Levels
Authors: Wenjun Ding
Jingling Liu
Lixing Chen
Xiu Su
Tao Sun
Fan Wu
Zhe Qu
Venue: NeurIPS 2025 poster ()
Keywords: Stability and Generalization
Meta-Learning
Primary area: theory

Abstract: Meta-learning has achieved significant advancements, with generalization emerging as a key metric for evaluating meta-learning algorithms. While recent studies have mainly focused on training strategies, data-split methods, and tightening generalization bounds, they often ignore the impact of inner-levels on generalization. To bridge this gap, this paper focuses on several prominent meta-learning algorithms and establishes two generalization analytical frameworks for them based on their inner-processes: the Gradient Descent Framework (GDF) and the Proximal Descent Framework (PDF). Within these frameworks, we introduce two novel algorithmic stability definitions and derive the corresponding generalization bounds. Our findings reveal a trade-off of inner-levels under GDF, whereas PDF exhibits a beneficial relationship. Moreover, we highlight the critical role of the meta-objective function in minimizing generalization error. Inspired by this, we propose a new, simplified meta-objective function definition to enhance generalization performance. Many real-world experiments support our findings and show the improvement of the new meta-objective function.

On the Stability and Generalization of Meta-Learning: the Impact of Inner-Levels
Wenjun Ding Jingling Liu!.", Lixing Chen Xiu Su Tao Sun Fan Wu', Zhe Qu School of Computer Science and Engineering, Central South University 2Xiangjaan Laboratory
Sccoool of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University
SSanggaai Key Laboratory of Integrated Administration Technologies for Information Security
BBgg Data Institute, Central South University
*College of Computer, National University of Defense Technology {234712244) inglingling xiusuliuus1. 1994, zue_euueeeeeeeeeee Qucccuu edu. cn, exchen@sjtan juu edu. en, suntao. saltfisshoutlooo com
Abstract
Meta-learning has achieved significant advancements, with generalization emerging key metric for evaluating meta-leaming algorithms. While recent studies have mainly focused on training strategies, data-split methods, and tightening general- ization bounds, they often ignore the impact of inner-levels generalization. To bridge this gap, this paper focuses on several prominent meta-learning algorithms and establishes generalization analytical frameworks for them based their inner-processed the Gradient Descent Framework (GDF) and the Proximal Descent Framework (PDF). Within these frameworks, we introduce two novel algorithmic stability definitions and derive the corresponding generalization bounds. Our find- ings reveal a trade-off of inner-levels under GDF, whereas PDF exhibits a beneficial relationship. Moreover, we highlight the critical role of the meta-objective function in minimizing generalization error. Inspired by this, we propose new, simplified meta-objective function definition to enhance generalization performance. Many real-world experiments support our findings and show the improvement of the new meta-objective function.
Introduction
Meta-learning has been proven to be a powerful paradigm for extracting well-generalization from previous tasks and quickly learning tasks (I) It has received increasing attention in many machine learning applications such as few-shot learning [2), robust learning (3), and natural language processing [4] The key idea of meta-learning
improve the learning ability of agents through learning-to-leat process. In recent years, optimization-base meta-learning algorithms have emerged popular approach [5-8] These studies formulate the problem as a bi-level optimization problem and have demonstrated impressive performance across various domains, drawing significant attention from the research community. particular, the outer-level, trains meta-learner to extract task-shared knowledge from meta- tratt-t tasks. At the inner-level, a basic model, which is initialized using the meta- parameters,,, adapts each task by taking Q inner-level gradient updates, where the Q times inner-level update is commonly called an inner-process 5 1
Despite the remarkable success of meta-learning, its theoretical understanding of generalization remains largely unexplored. Recent studies have primarily focused on analyzing the effect of training
"Correspodiing authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Table 1: The summary of main meta-learning algorithms
Frame.
Algorithm
Tnner-Level Process
Convex
Non-Comve
MAML
E (u D.)
FO-MAML
aug minaa (2 2 Veifut D...
y yiiie
o ( (1 So 88 (F(nttr)
Meu-SGDD G
2 107 w13)
IMAML 20
PDF
Meta Mini
oygmin Di Alooy w/p
of
of ا (F()))))))))))) ih)
PO-MuML
strategy [12 13,, data splitting methods 14 [5] on generalization error or on advancing tighter
generalization bounds [7), while overlooking the impact of inner-levels generalization, i.e., the relationship between the generalization error and the number of inner-levels Q.
In particular, there are two main inner-processes frameworks in current meta-learning algorithms, as shown in [ One is the Gradient Descent-based Framework (GDF) (5 6 [8], where the key idea is to measure the closeness of the initial prior hypothesis to the target optimal hypothesis by the number of gradient descent steps. However, this approach incurs high computational costs due to the need for second-orde derivatives and requires careful tuning of multiple hyper- parameetrss [19]. To address these limitations, another inner-prove process framework PDF, which is based on proximal descent, has been developed [20 7 [ 2n. It only depends the solution to the inner optimization and not the path taken by the inner optimization algorithm.
To validate the impact of Q on the two frameworks, we conducted two simple experiments on the Omniglot dataset [22] using MAML [5] and eralization error (Test Loss Training Loss) first decreases but then increases as the number of inner-levels Q grows in MAMI, while always decreases in MittaMMMnibatchPro.. This different behav- ior, attributed to the distinct inner processes of the two frameworks, motivates the need for a deeper analysis to help the design of Q for improved generalization performance.
7 examples. In Figure 1 the gen-
Conernteatton
Epocis.
To analyze the relationship between inner-levels Q and generaliza- tion error under these two frameworks: GDF and PDF, this paper leverages the algorithmic stability to characterize the generalization of algorithm [16] 23) which measures sensitivity to perturbations in the training dataset. To the best of our knowledge, this is the first study to investigate the influence of inner-levels on the generalization of meta-learning under two frameworks. Our findings offer valuable insights for developing efficient meta-learning algorithms. The main contributions can be summarized as follows:
(a) MAML.
O-28
Groeriieaton
6.2
0.1
(1) We summarize six mainstream meta-learning algorithms and extract their structural features. Based on their mner--poeessses we classify these algorithms into two frameworks: GDF and PDF, and develop two definitions of on-average stability, respectively. Accordingly, we establish a quantitative relationship between inner- levels and the generalization error in convex and non-convex settings.
Epacis,
(b) Meta-MinniattchPro.
Figure 1: Effect of Q on Om- niglot dataset.
(2) Our results reveal the influence of the inner-levels Q on general- ization error. In particular, we identify a trade-off relationship GDF, whereas PDF demonstrates beneficial relationship in its generalization bound. The primary reason for this difference lies in the term introduced by the inner-process For example, in convex setting. the term for GDF, O T ) increases with Q, whereas the term for PDF, O ((,))) decreases with Q. These findings help to design a more efficient inner-process of leta-learning.
(3) Based on the generalization results of GDF and PDF, we further derive the generalization bounds for six meta-learning algorithms and analyze their implications. In general, note that the meta-objective F(w) plays a crucial role in reducing the generalization bound. Motivated by this, we propose a new meta-objective Fnew (w) and prove (new(w) S F(wr), thereby enhancing generalization performance. Extensive experiments confirm the efficiency of the proposed objective.

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2 Related Work
Algorithm Stability. Algorithmic stability is critical for learnability [24]. There are two main approaches to investigating the stability- based generalization bound of meta-learning: (1) The first approach, introduced by (25], focuses on deriving generalization bounds for the transfer error, based on the assumption of independent task environments. This line of research has been further developed [3] 17 27 28). In this approach, the generalization error is defined as the transfer eттоr minus the training eiror. (2) The second approach, by quantifying the test error, successfully eliminates the assumption of independent task environments and provides tighter generalization bounds compared to the first approach [16]. Additionally, [23] re-examined the generalization performance of meta-learning from a bi-level perspective, i.e., the inner-level and outer-level, and concluded that inverted regularization at the inner level helps to reduce the generalization bound. Differently, we make a further development in the second group by re- examining the generalization ability from two main structures at inner-level and reveal the two distinct impacts of inner-levels. Notably, the algorithm stability of prior work is limited in one-step MAML, highlighting the need for novel algorithm stability definition.
in several influential works [22 26
Meta-Learning. Recent advances in meta-learning have spurred significant progress. From the optimization perspective, 229 30] analyzed the convergence rate and computational complexity of ANII., while 31 investigated the convergence properties with a single adaptation step. Extending this, developed a theoretical framework for MAMI. with multiple inner-levels. [1I introduced a novel analysis of first-order meta-leaminn algorithms. To improve the in-context learning performance of pre-trained models, several mncta-rraining-basse
32
approaches have been proposed. For example, MetalCT 333 uses BinaryCLFs and LAMA datasets to create tasks, while pre-pending human- generated instructions to each task. In contrast, MetalCL [4] leverages wide variety of disjoint tasks to meta-train large language models. [35 36) demonstrates that meta-learning can aid in cross- task generalization for prompt tuning. Moreover, MAML-en-LLM, (37) is capable of learning truly generalizable parameters that not only perform well across disjoint tasks but also adapt effectively to unseen tasks.
3 Preliminaries
In this paper, each data point : = (x., y) E Z consists of an input E X' and its corresponding label y E y/. We assume access m tasks, denoted by TT,--.; Tm with the data for each task Ti, i E [m]] generated from unknown distribution P. We use the loss function & gd x - IR+ to evaluate the performance a model parameterized by w E W, where W is a closed subset of Rd. The population loss corresponding for task T, is defined L. == Ez-P, [[(w, 2)}. In addition, we use the notation (w) denote the empirical loss for task dataset D,, i.c., [, D,) == क. ExED, E(w, 2), where |D:] is the size of D; The goal of meta-learning is to learn good meta-parameters w that perform wells across different tasks, which can be written as follows:
:
min F(w) WEW
m Σ R(ww),
where (w( is used to denote the performance of on task Tr after undergoing multiple (or one) gradient updates to adapt the parameters to the specific task. At a high level, a.k.a., the outer-level, the learner needs to figure out useful meta-information that can generalize across tasks. At the inner-level, the learner needs to find task-specific parameters WT, that perform well on individual tasks after undergoing an inner-process.
3.1 The GDF Framework
From the inner-level perspective, two distinct inner-processes have primarily been developed, giving rise to two dominant meta-learning frameworks, as illustrated Table I One prominent meta- learning framework, called the Gradient Descent Framework (GDF), is exemplified by MAML (3), FO-MAML [5), and Meta-SGD 6). In particular, the inner-process finds the task specific optimal hypothesis by solving the following optimization problem:
Σ Vii(uy D.), wT, ww + 2a |WT, - w w112
min BFT,
(2)
3

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Algorithm
GDF and PDF
1: The set of datasets S {S outer iterations T, inner-levels Q, regulation X.
2: Choose arbitrary initial point b E W;
3: for t 7 --1 do
4: Randomly choose the task é.
5: Inner-Level: w Wt
6: for q 0, 1, ,..,, Q do
7:
w// whr,. V ( St))
8: w//gt11 = w/y.gg a avVK aVKK(wu'iss St))
9: end for
10: wL+1 w² 11. V www.(ww 5
w++1 wt TTd(w' w'y,,0
end for
13: w" and Th = T+T Eizovd w';
where Q is the number of inner-levels, i.e., gradient descent steps in GDF and a the inner step size. In fact, by taking the derivative on WT, and making the gradient equal to (2) can be written as WT, = w w--a Eg (W, D.), which reveals that task-specific parameters are iteratively learned by minimizing the empirical loss based using gradient descent. Specifically, in GDF, without loss of generality, we mainly follow MAML [ and summarize the training procedure in Algorithm [ For each task Ti, we group the samples into two distinct sets among n training samples: a support set S. of size nt for meta-train at inner-level and a query set S of size n for meta-validation at outer-level. Then, for each task Tú, we have one corresponding training set S, == 1 } in GDF Referring back to ( the formal definition of the meta-loss of task T is:
Fw := Ep, [. (w, D.),
(3)
Because it is difficult to obtain the 3 in practical applications, we usually use the empirical loss fw S:) = L(w (w, S, Erest ((w), (w, 2) to approximate it.
3.2 The PDF Framework
Another widely used framework of meta-learning, known as the Proximal Descent Framework (PDF), is based on a "proximal" descent, wherein task-specific parameters are iteratively learned by minimizing the empirical loss and an &g regularizer. This framework is exemplified by iMAML 200 Meta-MinibatchPooo 77, Fo-MuML I1. In this framework, the inner-level optimization problem
formulated as follows:
min (WT., D.) + W w
where 2 is a regularization constant. In PDF, without loss of generality, we mainly follow Meta-Miiibbtthhros 77]. In particular, the task-specific training set S; of size n is used directly for meta-train inner-level, without the data into two distinct sets. Then, we present the following formulation of the meta-loss for task T, corresponding to the PDF:
Fi(w): = min [BANP, e(WT,, 2) 2 1| -wll) Bosp:ex(W,.) wT;
(5)
and the empirical loss is Fw S,) = minwy; 1 Exes, f(WT. 2) + I1l w|l } = Ex(w. S.).
Note that we use Eg-p, & (w, 2) and (ww S,) forlicity without impacting our analysis.
For a practical PDF-hased algorithm, it's difficult to get an exact solution of (w, Si). thereby we usually turn to get the inexact solution, K(w, St) E.es, ((WT,, 2) + AlW w|l2. In- stead of solving (T). we solve its sample average surrogate problem as arg min F(w,S) == 1 Σ"h -== Fw,,, St), in which each F; is calculated by its corresponding empirical loss.

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Specifically, by comparing 2 and 4 we observe that GDF only uses the first-order in- formation of (W(TT, D.). leading to inexact solution. In contrast, PDF focuses optimizing L, D,) = VV, (WT, D,), WT; w) (v>L, D,), (wy, w) w)s²) ( D:), (WT, w))") +... and thus implicitly leveraging higher-order information. This finding suggests that fewer inner-levels Q may be more suitable for GDF, while more inner-levels Q are better suited for PDF.
3.3 Stability and Generalization
Test error is generally considered the most critical metric for evaluating the performance of meta- learning algorithms. To control it, most studies focus two key perspectives: generalization error and optimization error [38/4]] Specifically, let == {Si Im be the concatenation of all tasks data sets. Given a randomized optimization algorithm A that acts on the dataset S and produces an output A(S), the generalization error is formally defined 'gen = [. ((A(S)) F(A(S), S)] and the optimization error is Copt == EAA. [. (F(((S)) - miny F(., S)
Then the test error of
A can be decomposed into three distinct terms:
EAS F(A(S)) - min F
Egen Eopt + Ess min F., S)
min F
(6)
50
32 33 have shown that Caps will converge to 0 as the number of outer iterations T increases, given that the loss function E(w, 2) satisfies certain assumptions, and the third term non-positive. As such, analyzing Egen to improve the performance of the test error is more crucial. Note that [16 23 only provide Egen of GDF in the strongly-conver setting. To fill the gap, in this paper, we establish comprehensive theoretical analysis of two meta-learning frameworks in convex and non-convex settings. Before presenting our results, we state the following definition and assumptions widely used in generalization analysis [43 [6] (2).
Definition We say a function f(w) is X-strongly convex if Vwi, w2, ((w1) 2
((w2)
(Vl(wz), wi w2) + lll w2ll2. If X then we say f(w) convex.
Assumption 1. We assume Z is polish space (i.e., complete, separable, and metric) and Fz is the Borel p-algebra over Z. Moreover, for any i, Pi is a non-atomic probability distribution over Z.
F2), i.e., P.(=) = 0 for every
Assumption 2. For any 2, the function f(. 2) is twice continuously differentiable. Furthermore, we assume it satisfies the following properties for any w, u E Rdd
(i) The loss is G-Lipschitz over Rd, i.e., [1((w, 2) E(u, 2) < Gllw ul|:
(ii) The loss is L-smooth over Rdd, ie., [V Vl(u, 2| < Till - ull:
(iii) The second derivative is Lipschitz continuous with constant p over Rd,, i.e. [[ 72E(u, 2)ll I p|ww- -ull;
(iv) The third derivative is Lipschitz continuous with constant is over Rdd iee. || { < x||lw- - ull-
Assumption 3. For any E [m] and any w IRd, the stochastic gradients F.(w,, S) have bounded variance, i.e, E P.(w, (w, S) FF(w) <².
Stability-based generalization error analysis has been widely used characterize the generalization properties for optimization algorithms such as stochastic gradient [43), adversarial training [44), and federated learning [45 46). Next, we will establish two different stability definitions and then leverage them to prove the generalization bound for GDF and PDF, respectively.
4 Theoretical Analysis
Under stability analysis, [16] improves the previous generalization error bound in the homogeneous case and provides total variation-base based analysis in the heterogencous case. In addition, [23].
5

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focusing on the mnner-proess, demonstrates that introducing inverted regularization at the inner-level can enhance generalization performance. However, their analyses are limited GDF with one inner-level update (Q 1), which ignores the effect of multiple inner-levels (Q > 1). To fully characterize the effect of multiple inner-level updates, a different stability definition is required.
Definition 2. (on-average stability for GDF). A randomized algorithm called e-on-average stable if the following condition holds for any i E [m]] Take the dataset S which is the same as S, except that Si and Sty differ from S and S by replacing the k-th and j-th data points, respectively, where k E [n'],j E [nd*]. Then, we have: [aX ,,,,,,,,,,,,, [[(WT, (ws, Si,,), 2 - f(WT, (ug, (ik 2.. S.
with output ws is
Different with the stability definition for stochastic gradient descent in [47 48 which considers only single-level structure, our definitions for meta-learning mean any perturbation of samples across levels cannot lead to a big change of the model trained by an algorithm in expectation. The main reason that we are interested in the stability of an algorithm is its connection with generalization error. Next, we formalize this connection for GDF and show that whether an Algorithm A is e-on-average stable, then its generalization eπror is bounded above by
Theorem 1. Consider the populction risk F(w) and empirical risk F(w). The corresponding F((w) and (((w)) are approached by GDF in Algorithm / randomized GDF-based algorithm, then Egen S EA,S F(ws)) F(ws, 5) S.
Under Assumption and Definition i
Building on the GDF analysis, we now extend this concept to the PDF setting, where the stability definition needs to account for differences in training procedures.
Definition (on-average stability for PDF). A randomized algorithm A with output ws is called e-on-average stable if the following condition holds for any i E [m]: Take the dataset S which is the same as S, except that Stg, differ from S; by replacing the j-th data point. Then, for any ā E 2, we have: maxje]m] ES..A.,,, [ (ws, ²..) - (wg; T..s)] S e, where lx is defined in (5).
Under this definition, we can establish connection between generalization and stability. Although similar to Theorem i.e., Ggen S A,S [(ws)) F(ws, S)] I c, the proof of this is significantly different from it. In addition, F(w) and F(w) in PDF also differ from those in GDF, as composed of different (w) and (w)) respectively.
Based on the above theorems, we will present the generalization bounds under convex and non-convex.
4.1 Generalization bounds of GDF
Theorem 2. Let the outer-level step size and inner-level step size be chosen as т S Liq and a S 1 respectively. Under Assumptions 3 the generalization error Egen of GDF can be bounded by:
En-o T-1 ni(1 0-)--1 (6QG + Q2a2Gpp))
1 V m
F(ww)) min F + LoO² ΣT
mn
where LQ = apQ(1 + + can obtain a concise result as follows:
+ a1+
If we further let a S 1 and in n be fixed, we La
TQ mnn
1 V F((w))
Egen S
0
min F +T W
Remark 1. The first observation from the above results is that the generalization bound worsens as increases, primarily due to the first term, O TQ)). Similarly, [15] provides comparable bound, O(VT+ TQ), using the information-theoretio analysis. However, by revisiting ①I and 3 we observe that F(w) influenced by w D;). Consequently, after Q steps of gradient descent, the divergence between F(w²) and miny F decreases, which in turn positively impacts the generalization bound. In summary, our findings indicate a trade-off in selecting the number of inner steps Q which also explains the phenomenon in Figure Ia
Remark 2. In previous work, (9) also provides similar trade-off relationship, O (1=) +F(
miny F), with < 1. However, under the choice - Φ.. the first term simplifies to (E2),

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effectively mitigating the trade off relationship. In contrast, our result demonstrates greater robustness to the choice of a. In fact, we can consider a special case understant the robutst relationship. For sufficiently large Q, a S or yields negligibly small learning rate, implying the inner update will be close to initialisation " which means the model fails to learn task-specific information effectively, leading to larger generalization error. Furthermore, unlike the generalization bound
O mm provided by [T6], our generalization bound grows with due to our consideration of general convex setting. In contrast, their analysis assumes strongly convex setting, which is consistent with 43
Theorem 3. Let the outer-level step size be chosen as ttt & satisfy min Lo 4(2Loin(T))
and the inner-level step size be chosen S QL. Under Assumptions 7
3 and further assume that
Es,A(F(wss) & F(w"), the generalization error 'gen of GDF can be bounded by:
ET
2Q
(F( 1705
m
1+ Q LQ = 3 + (1+ L PQ = 3p(l+a))+aL)2( 1) + (1
where Q2
nic.
L and min Bslll (w
(e(F(ww) minw F))}
Remark 3. Note that we find that the trade-off relationship also exists in the non-convex setting, which similar to Theorem Specifically, on the one hand, we observe that Egen increases with & = nar. On the other hand, Egen decreases with F(w"), where F(w") itself decrease with Q. In addition, y also characterizes the effect of Egen through the term po(co vc(F(w9) minw F)). Here, PQ grows with Q, while F(w0)= miny F decreases with Q. Importantly, inappropriate choice of the inner step size a can exacerbate the generalization error. Even when the standard loss f(w, 2) is L-smoothness and has p-Lipschitz hessian, the compositional loss E(w (w, S²") 2) may become LQ -mssohhthssss and pq-Lipschitz, with LQ and pQ decreasing with Q. However, under our principled choice of a QI we can ensure Lo S (1) and pQ ((). thereby mitigating these negative effects.
Q
4.2 Generalization bounds of PDF
Theorem 4. Let the outer-level step size be chosen 1r2 S L. the inner-level step size be fixed and C being constant. Under Assumptions 2 bounded by:
the generalization error ggn of PDF can be
Σ
T-1 2 mCQ + mm
F(ww)) - min + W
Loo²
Σ
T-1 t=
Ti
2
where LQ X+L If we further let mp as follows:
Nb
n
1
be constant,
obtain more concise result
Lo
Egen 5 50
T 1 mCQ
F((w)) min F + m
W
T Remark 4. An immediate conclusion from the first term, T. in the above result is that the generalization error PDF decreases as the adaption steps Q increase. Additionally, the second term, m (F(w0) - miny F + T)). also benefits from Q for similar reasons discussed in Remark It is worth mentioning that, in strong-conver setting, the generalization error does not grow with the number of iterations T' [4]. However, the objective in (5) that we are minimizing is actually strongly convex w.r. WT; (due to the strongly-convex regularizer, 3 BllWT, - w|l2 and convex function ((. 2)), but only convex w.r.t w [49 50 40). As result, generalization bound still grows with T. Furthermore, if we let Q n* the generalization error of GDF becomes O I
m / F (w)) minw F + T), which greater than the generalization error of PDF, O mC F 0 minw F + T).
m
Theorem 5. Let the outer-level step size be chosen as satisfy S min 1/ 12Lg inner-level step size be fixed and C > being constant. Under Assumptions and further
7

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Table 2: Summary of our results.
Frame.
Algurihm
Convex
Non-comve
MAML
O1 fel1 o2)0
o0 war (P(
POMAML
O (0-T sor ta i) (P(W
Meta: SGD
0 Wil 6
O 1
+1 az
a (Flu")
IMAML
O (a 2
(1 sueteoy (F(w"1))
Men-MiniaachPuux
Of [0 + )) (FW")))
Fo-MaML
O1) P Jomnl S22)))
O 25 (1+ Bel (F(
assume that A S F(w"), the generalization error Egen of PDF can be bounded by:
o
m x 2 (1'((( (F'(w))))(t)) (.x)
where LQ = X+I 7 min Esll + po + ve(F(w") minw F})}, DQ
AL
1 + 2o and PQ =
Remark 5. Similarly, we get a lower generalization error since Q = 1 + पम and F(w") gets AL
smaller when Q increases. In addition, observe that LQ X+ and pQ p are independent of Q, because PDF eliminates the compositional structure of E(WT, (w. S!"), 2) in GDF. Furthermore, set Q n²" we can get $ 2 in GDF, whereas $ 1 Cn in PDF, which indicate that the generalization error of PDF is lower than GDF.
Remark 6. Based on Theorems
we present the generalization bounds for the corresponding algo-
Here, the term ((F(w")) = v F(wx)) minw F + Loo' decreases as Q increases. Specifically, Meta-SGD redefines as learnable parameter, denoted by ãt. As discussed in Remarks 2 the choice of the inner step size critical, where a S Qr. is necessary to mitigate potential adverse effects caused by the inner process. In addition, our results reveal that algorithms based on the GDF framework exhibit a trade-off between the inner process and generalization error, as shown in Theorems 2 contrast, the PDF framework avoids the adverse effects of the adaptation process by implicitly leveraging higher-order information. It is important to emphasize that our analysis focuses on the relationship between Q and Egen- However, directly comparing Égen across algorithms remains challenging due to the influence of algorithm-specific terms, such as LQ and PQ-
rithms, summarized in Table
2 Ei= vi
4.3 New Optimization Objective
Recalling from the Theorems 2 and the results Table 2 we can observe that F(w") plays a critical role in reducing the generalization bound, due to the terms, F(w") minyy F and F (""")..
To reduce the impact of F(w"), note that F(w") = 1 m Σ Fi(w)), then let us consider a simple
[0, 1) and in S m with the losses sorted by (F)))) in ascending order, we can obtain that Σ ()(w))(()) < Σm Fe(w0) where (i) is derive Emm- F. (1000000) Motivated by
relaxation technique: given a constant
2
mapping function associating the index with the original loss F.(wO). Based on this, F(w²)- - Em ( > 1+0
m 1 (1)))))00) + 1- >m
this result, propose the following optimization objective:
Fnew (w)
1 1+ B
m Σ ==1 F((
+
1 1-3 3
Fw) P00
(7)
m
1=m++1
where m = (1+ 8) + (1- (m i)) is to ensure the lower bound derivation. A clear advantage of our new optimization objective is its ability to achieve smaller initial value, Fnew (w²), thereby reducing the generalization error. In addition, this new optimization objective is designed to easily integrate with all types of algorithms in both the GDF and PDF frameworks without requiring modification of their training procedures. During the training phase, we usually select batch B with size B. After performing the inner-process (,( ), E [B], at the outer-level and sort the losses in ascending order, yielding (Cr(w)) () (1). Then, based on the sorted losses, we then use weighting factor B to adjust the weight of each
each task, we evaluate their task losses
8

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12
64-
1.0
25.
Innir Stepp
Step.
(a) Convex GDF
(b) Non-Convex GDF
(c) Convex PDF
(d) Non-Convex PDF
Figure 2: Relationship between generalization errors and Q.
25
35
0.
7.0
64
(a) Convex GDF.
(b) Non-Convex GDF.
(c) Convex PDF.
(d) Non-Convex PDF,
Figure 3: Convergence results with or without our new objective.
task when updating the meta-model w'. Importantly, our strategy achieves a dynamical fairness for each task without compromising convergence performance. The design most similar our new optimization goal is [51], however, their goal is to maximize the meta-loss as much as possible to obtain task-robust meta- peta-parameters, whereas our goal is reduce the original meta-loss, and our approach is easier to practice.
Experiment
Few-shot regression. This problem focuses on approximating family of sine functions represented as [(x) = a sin(ter), The task distribution, denoted as P, corresponds to the joint distribution p(a, b), where 50 is the amplitude and b ~ U00, π| is the phase. Both the training and test tasks are randomly generated from P = p(a, b). Following (7), we use an MLP network with Mean square error loss function. The generalization error is assessed as the difference between training and test errors. For each training task, we set the number of support samples and query samples to n² and n = 5, respectively, while for each test task, use n²" 5, n = 15.
Few-shot classification. We follow the standard experimental setup described 22 using the real-world Omniglot dataset, which comprises 1,623 characters from 50 different alphabets, with each character having 20 instances drawn by different individuals. We employ 3 x CNN to align with (5) [1], and use the Cross-Entropy Loss as the loss function. For each task, the number of support samples and query samples to n² and ss = 5, respectively. For each test task, we use n² ===11 n t = 15. In addition, each task is formulated as a 5-way classification problem.
We fix the training task number m 100 and generate 10000 new tasks at test time by using the standard library [52). To ensure a fair comparison. We report the average generalization error during the last 10 iterations of GDF and PDF under convex and non-convex settings, As shown in Figure 2 GDF's generalization ability benefits considerably from smaller inner-levels Q, but deteriorates when Q becomes larger. In contrast, PDF's generalization improves with increasing Q. These observations align with our findings in Table Under our new optimization Objective, the generalization error of different algorithms can effectively reduced. Furthermore, we also compare the convergence performance of GDF and PDF with or without our new objective within the same number of epochs. In particular, we set Q = for the convex setting and Q = 10 for the non-convex setting. Figure 3 the results indicate that new optimization objective leads to lower test errors for most algorithms.

---

Conclusion
In this paper, we introduce two theoretical frameworks, GDF and PDF. They are summarized by several popular meta-learning algorithms based on inner processes. Within these frameworks, we derive generalization upper bounds for both convex and non-convex settings, which offer detailed understanding of how the number of inner-levels influences the generalization error of GDF and PDF. Our analysis reveals trade-off' relationship induced by the inner-level in GDE, while PDF demonstrates a more favorable relationship that improves generalization. Building on these insights, we propose a novel meta-objective designed to significantly reduce generalization error. Extensive experiments validate the effectiveness of our findings and the proposed objective. We believe that the insights, the proof techniques, and the new meta-objective can inspire further research and open new directions in meta-learning and related areas.
Acknowledgment
This work was supported by the Project of Xiangjiang Laboratory (24XJJCYJ01003) the National Nat- ural Science Foundation of China (62302525, 62202293, 62303306, 62302524, 62372472. 62306109 62320106006), the Hunan Province Natural Science Foundation of China (2024JJ6527, 2024JJ6531, 2024JJ4068), and the Guizhou Provincial Science and Technology Projects (025222)), It also utilized computing resources at Central South University's ' High Performance Computing Center of Central South University.
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A Useful Lemmas
Lemma 1. Let o be convex and L-soooth L-smoothh function. Then, for n t. we have
Il(w 17 Vo(w)) - (u - ---((V) & w| - u/].
for any and
Proof.
Since o is L-smooth and convex, we know that
(Vó(w) Vo(u), u) 2 _ [V(ww vo(u)l
Using this fact,
[I( VVo(w) - ( (/o
- -
-|| [ u|| 1 [
Vo(u), u)
S w u|| 2 + 1 - - [V((u))
when 1) S 7.
Lemma 2. 43
Let @ be X-strongly convex and L-smooth function. Then, for any 8 S X+I.
2
have
L8A (Vo(u) - (o - (vo()) & (1
X ll wlll
for any and
Lemma Let fx(w) = min, (f(u) + 1 - ull3), if f G-Lipschitz, then we have || [V((w)] S 2G.
Proof. Define (u)) & f(u) + 2110 w/ and n" arg min, Φ(u). Now, observe that
X ~1 f|lw-u' u²11
0 s d(w) o(u") f(w) f(u")
2
Thus,we have
X l n² 1² 1 fw)) f(")(())))))));{))) = -|| --
2
where the last inequality follows from the fact that f is G-Lipschitz. Thus, get ||w =u'l < Since [[)]] = >l[w- w²|l. This together with the above bound gives the desired result.
2C
6
B Stability and Generalization of MAML
B.1 Proof of Theorem 1
To show the claim, it just suffices show that for any we have
EAS Fi(ws) Fi(ws, .:) S
Take the dataset S which the same as S, except that S.C. and 5/g differ from S and S in most one data point, respectively. In particular,
S {
2,ntr
ir ), S = (- .... >
S = f Th ... 2tr = 1. 5ts = {2/. ... 2ig )
Under Assumption population risk by
we could assume zng is different with zne Then, we relate empirical risk and
E. (i(wss, S;)]
Ess E.s (wT, (ws, Str), =1
ns
j=1
(8)
Lu
1
E
A. see 1e
3tt 21s
18
20

---

Moreover, we have
t5
EAss(Pi(wss)
1
E S
S S.
[e (((T. (ws, (1,))
nt
Putting S and together, we have
EAS F(ws) F(ws,
1
E
(W. (1))) - -
n
A.S,
5tr
Se.
Then obtain the desired result.
B.2
Lemmas
In the following proofs, for simplicity, we use [(WT,a)= L. Str), ((WT..)) ((wT.g:)) 2) without other explnation, where q E [Q].
Lemma 4, For any i E [m]] = O,... Q - 1 and w, u E Rd, if is a convex function, we have | "T:9+1 ||w ull
Proof. Based on the updates that WT.g+1 = WT.. og V (W(T. and uT.001 = uT.9 ag (Li(u, we obtain
|| WT.g+1ll -
0g (W.T.))))) - - (UT.q - on VC,
|| UT
where we use the Lemma] 1 in the first inequality. Unrolling it over g from 0 to q + 1, we obtain
||T - uT. S || ull
Lemma 5. Suppose the conditions in Assumtpion 2 are satisfied. Then, if S 3. and E is convex function, we have
[[((w,, (w, St 2) E(WT, St",, 2)1l S Gl|w all
Proof. For any w, E W. note that
[1((1, (w, x) E(WT, (t",,, SG l (w, wT, (u, Stri
= =G a V WWW. 11 St
(WT,Q-1 V VL Str)) S(
SG <G|WT,QQ11 -
<G| -
where we use Lemma I in the second inequality and Lemma in the last inequality.
Lemma 6. For any i [m], q 0, -.... and w. u IRdd, we have
|WT UT. S (1 +1+ (1+aL)++i) aaa)+llll -|u-ll u/l
Proof. Based on the updates that WTig+1 WT..q ag V C. and uT.991 = uT..9 ag VLi(ur.)) obtain
|| WTig+1ll
= =I[ - 0g (W.T,g)) - (UT. or a (u,.a)))
< |T uT.7 [.]||WTT,. uT.aal
= (1 + aL)]| uT
where we use the Assumption 2 in the first inequality. Unrolling it over g from to q + 1, we obtain
||0 uT. I aL) ul]
21

---

Lemma 7. Suppose the conditions in Assumtpion 2 are satisfied. Then, with a S QL non-convex function,, we have
1 and & is
(w, E(WT,, (u, s cGllw ull
Proof. For any w, u E W, note that
[[([T,, (w, ), E(WT, (u, S,"), 2)1l SG||uu, (w, (w) (w,
(T, S.
=C - -ttpa///.t.aaaaaaiaaaaaaaaaaaaa-
<G + -
SG + 1
SeG||w u|]
where we use Assumption Zin the first and second inequality, Lemmao6in the third inequality. S QL' we have the last inequality.
[.
Lemma 8. For any i [m], q = O,... Q and w E Rd, we have
Proof. Recalling the update rule WT.g++ WT,, 0g V[.(w..))) Using Assumption we have
| VV )l
V (W - V(((w...)) + V (W
(W VZ. )1 +
(wT:
T.llTa.tt1 WT..gl + VVL
-||WTi -VV.. - WT. [V (1+ (L+
where we use Assumption in the second inequality. Then, unrolling the above inequality over from to q + 1, we get
V s (
Lemma 9, Suppose that Assumption hold and the function F((w) defined in Eq 3. Then, for any w, E IR4, if f is convex, have
(w) V LQllw ull
where LQ ===[pppQ100 app((++a))0-1 A with < 1
22

---

Proof. Similar to the proof of Lemma
9 we have
Q-1
Q-1 -
-|1
II
(I
7=0
q=0
Q-1
Q-1
(I a
av- (I-aaV"((vr,)))V(oorr...
9=0
Q-1
L. (T
()
II a (VV,a
/=0
9-1
a
V
9=0
Q-1
(T
) [ -
22
4=0
Q-1
Q-1
vV2
2,( a)
(1
+ (1+ ((L)))) v
Q=0
Q-1
Q-1 ( + (1+ L. - wn.ool
2. (WT..))
4=0
-=0
Q-1
Q-1 [I
V2 (WT.
(1 a/|,
q=0
-
(10)
where in the last inequality, Assumption 2 and Lemma 4 inequality. Specifically, we have
We next upper-bound A in the above
Q-2 A(Q 1) V [ (1tp::/ww_-----T---///////----00
Q-2
4=0
q=U
Q-2 [I V [ - -
Q-2
+
q=0
q=0
Q-2 <
q=0
Q-2
Q-2
+1
I|
2 2, (WT.))
(,
[I a 2
Q=0
9=0
<(1+aL)
( - V
-
Q-2
Q-2
+(1+aL)
[T 4/2 (W(T.)))
II
aV -V
0
0
5(1 -(+o0))9-ool10000011-1-
u +1+ 2)
+ ( (1
23

---

where we use Assumption in the second inequality and Lemma 4 in the last inequality. Telescoping
the above inequality over from 1 to Q 1 and noting A()) - app w ull. have
AQQ - 1) 5(1 (1+ a1) appQ - -
ull
- 1 + ull
(11)
SapQ(1 |
Substituting
11 into (10) and let LQ apQ a Q-1 + (1+ get
[[V Vii(u)ll & LQll --w-uj ull,
which completes the proof.
Lemma 10. Suppose that Assumption w, u E TRd, if f is non-convex, have
hold and the function Fe(w)) defined in Eg 3). Then, for any
[VF Vf((w) s LQllw- ull
where LQ 3p(1+01)2000 + (1+ a
Proof. Similar the proof of Lemma 9 have
|| (w) (uull)l
Q-1
Q-1 a + (1 + un.ell
(1 (wWTT.))
<=0
q=0
Q-1
Q-1
II 9=0
(I V E, (WT.g)
(I (v., (1 ull-
-=0
M(Q11)
(12)
where in the last inequality, we use Assumption and Lemma We next upper-bound A in the above inequality which is also similar to the proof of (1I). Specifically, we have
A(Q - 1) (0) + ap + (L)1-=)))0- ull
Q-2
5(1+ + =(1+aL)0-applu
Σ
ap( ab a0)
Q-2
(13)
(1 + aap
(1 (1+) -
1=0
(1 (1+al) ap f +aL) (1 11) 1 ull
3p(1 + aL)
|| ull
L
where I in the last inequality. Substituting (13) into 12 and let Lo 30(1+aZ)2001)
(1+ L, we get
[[VF(()) V.((u)) < Loll u/l,
which completes the proof.
Lemma 11. Suppose that Assumption w, u E IRd, if & is convex, have
hold and the function F((w) defined in Eq 3) Then, for any
2
F S nellw- mellw-ni ull
where PQ = (1 + 2aL22-2111 [apQ 2p Qap + Gax + Ga?
24

---

Proof. For simplicity, we denote 144 = - av (WT,))
[V2F(w) v2R(())l
Q-1
Q-1
OJa
II
(1 V ((W
II Ji
v((WT,))
Ow
1=9-1
Q-1
UJa
(I
V Li ( ( 2
II Ji (=++1
v((un.Q)
du
Q-1
Q-1
<||
(
a
V
(T- a "
k00
9
Q-1
Q-1
Q-1
OJg
JJg
J1
II Ji
V((WT.1))
Σ
J
II Ji
Ve(uT..))
dw
Ou
4=0
I=0
l=q+1
9=0
1=0
l=9+1
(14)
For the first term, we have
Q-1
Q-1
[I - ov (
a (
q=0
g=0
Q-1
Q-1
a
(I V [
q=0
4=0
Q-1
Q-1
-
0=0
0=0
Q-1
Q-1
(15)
2
>
4=0
Q-1
+
(1 (1-
(I - v
Q-1
Q-1
<
II
(T oV
E. (WT,g))
II
(T (V (1 2I)22 p|[w - u|l
0=0
9=0
MQ-1)
25

---

where we use Assumption 2 and Lemma 4 in the last inequality. We next upper-bound A(Q 1) in the above inequality. Specifically, we have
Q-1 (I a (
Q-1 - IU-
A(Q -)
II
9=0
9=0
Q-1 (I a
Q-1 II
( a V L (uT,,)))
q=0
[I a V (WT.
(I v L.
(I
V2 غ, (UT.
q=0
Q-1
Q-1
Q-1
+
II
(I - a v 2,
II q=0
[I a V2 (wT
Il
(I - a V2 (uT,
q=0
q=0
Q-1
Q-1
II (T 2V2LL 9=0
II (7 - (uT,,)))
9=0
S(1 + aT) Q al. |0 ull.
(16)
where we use (II in the last inequality. Putting 16 into
15
we have
Q-1
Q-1
II (T - LV 222" ((
II (T v (
(17)
y=0
9=0
< [ + a + a + +11 at) pl ||l - ull
To bound the second term in 14), we have
JJg
1
JJ q
II J
v((TT.))
II Ji
Vi(un.Q)l
Ow
du
1=0
1=9+1
1=0
1=
Q-1 II JI
Q-1
JJu
JJg
J Ow
V((WTTQ)
J1
II Ji
V((uT.Q)
ow
1=0
1=++1
9=0
1=0
1=g+1
J1
JJg Ow
Q-1
II JI
ve(uT.q)
Q-1
Σ
g-11
Ji
JJg
Q-1
II
Ji
Ve(UT.2))l
Ou
2=0
1=0
1=9+1
4=0
V==
1=9+1
Q-1
JJg
J
II 1=++1
J
VI(TTTT)) - Ve(UT.Q)
Ow
1=0
JJg
Q-1 II 1=++1
Q-1
0Jg
Q-1
J1
J1
J1
II J1
|||
aw
Iu
0=0
q=0
1=0
imq+1
DJg
Q-1
Q-1
9--1
OJg
Q-1 II
OJg
Q-1
II Jr
11|
- w|l +
Σ
||
Jr
JI
II V=
Jr
II 1=9+1
Jr
||
Ow
Iw
Ou
0=
1=9-1
7=0
1=0
1=9+1
Q-1
DJg
Q Q-1 Σ
DJg
0Jg
(1+ (+)
||||
w|l +
(1 (1+aL) aL)
Q Q-1G1
G
||-
Dw
dw
Ou
9=0
q=0
(18)
Firstly, we have
0.g
9-1
=|la ((WT, 2
II (1 av ((WT.())
8w
(19)
1=
5(1 + aa)--100
26

---

where we use Assumption
Secondly, we have
DJg dw
0Jg
1= V ( =11=
a -
(1 aV
a V ((UT:
du
I=
q-1
4-1
a
3 D (T
((wT.))) vV 2( (TTT..))
2
WT TT,1 )
1=0
1=0
a aV a. L(un.g)
a
V
((ur
1m0
1=0
9-1
9-1 1
Sa(1 + +ab)"
v L(uTr.
appl
II
(1 - a ((w...))
((ur.g)))
1=0
1=0
San(1 a ul + aL) |
(20)
Putting 20 and
19
into 18
we have
Q-1
Q-1
OJg
J1
II Ji
V((UT.Q) -
Σ
J1
II JI
Ve(uT.Q)l
Ow
Su
q=0
1=0
1=9-1
1=0
1=g+1
<Q
[ap(1 + aa))
Gan + + Gappppp' +aI)
1|
ull.
(21)
Putting
and
21
into
14
we get
[[) F3
(00 ( ||
ull
+Q a =.)) 2) + Gan(1 + 2 Go p²Q2 + aT))2(0-1) 11] ||0- ul
-(1 + aa))
lapQ + 2p + Qap + Gan + Go pQ] [ ull
=pqlw w - u|l,
where we denote PQ (1 + 2L) [apQ + 2p Qap Gak + Go Q²Q]
Lemma 12. Suppose that Assumption hold and the function F:w)) defined in Eq 3 Then, for any w, E Rdd, if E is non-convex, we have
|V² - F²F((u)]l S SPQll pQllW- wollwwol ull
3pl 221
(1 + aL) 30 p+ + + aL)22 (1
where PQ
Proof. Similar to the proof of [4
we have
Q-1
Q-1 a (T
( <1
II
(1
a
Q-1
9
JJg
Q-1
Jk
II Jk
V
Ow
0=
k=0
k=
Q-1
Σ
Jk
II
Jk
v((u))))
Ou
k==
k=
(22)
27

---

For the first term, we have
Q-1
Q-1 V(
II (T - alT-aV-L (V
-=0
q=0
Q-1
Q-1 a ((.,
a 22 (w
4=0
4=0
Q-1
(23)
(
| -
4=0
Q-1
Q-1
V )
V
II (T Lv (1 + aL)3 ull
where we use Assumption 2 and Lemma 6 in the last inequality. We next upper-bound A(Q - 1) in the above inequality. Similar to the proof of (16), have
3p(1 + 2L)
A(Q 1)
|| - ull
(24)
L
Putting 24 into 23
we have
Q-1
Q Q-1 II 9=0
(1
(1
(
9=0
(25)
Tpp1 + aL)(10-11
+ (1 + aL)
3Q
|| ull
L
To bound the second term, similar to the proof of 18
we have
OJ
Q-1
OJg
Q-1
Js
II
Jk
ve(ws,))
Jk
πI
Ju
v((u.)))
Zw
Ou
k=
k=
7=0
k=U
k=
Q-1
ÖJg
Jk
II k=0+1
Jk
|||
VI(w()) vl(uiq))
Ow
A=
JJg
Q-1
Jk
J
3Jg
II J
IIIe((u)))l
dw
Du
k=
9=0
k=
Q-1
OJg
Jk
II
Jk
[[11
+
L w
ull
aw
k==
A==-1
Q-1
9-1 1
0Jg
Q-1
Jk
Jk
J
Jk
||G
Ow
Iu
q=0
k=
k=
k=
k=
Q-1
AJg
Q-1
OJg
JJg
+
aL+
22-1 L
|||| - ull +
Σ (1 +
Gil
||-
Dw
Dw
ou
-=0
9=0
(26)
where we use Assumption
and Lemma
in the second inequality. First, we have
0J vu
g-1 2 L.
||a 3 L(wig)
(27)
k=0
5(1 + aa)
28

---

Secondly, we have
DJg
0Jg
9-1
9-1
-1laalaV (ww.g)
a V2 -
a
V2 I4 (wc.)))
du
k=
< Ll
a LLLL
a V2 L ())l
km
km0
+ +|aaV>L(ui,))
-1-a a [V -a av L(un))
a 2 Lu (ui
km0
9-1
Sa(1+ (o(++ - v ((uia)))) || | apll
II
(1 - a Li(w,a))
II (1
a vv2 Li (u(i,)))
k=U
k=0
Sax(1 2 u|| + 3ap² (1 + aL)2(Q-1)
| ull
(28)
Putting (0
and 19
into 18).
we have
Q
aJ
Q-1
Q-1 Σ
OJ,
Q-1
Jk
II Jk
ve(wi)))
-
Jk
II Jk
V((un.))
Ow
Ou
k=
k=0
k=
3Gap (1+ (1 aaa0011-11)
SQ
ap(1 + a
| - uul-
L
(29)
Putting
15
and
29
into
we get
3pp1 + aL))(Q-1)
[v)) v P(()) <
+ (1 + al)33
1/0 ull
ann(1 + aL) lw u/l 3ap + 21))((-11)
|| u//.
L
=PQ ww u/|
where we denote PQ = +o + +11 a1+ + +an(1 + aL)2
As shown in Algotihm [ in the outter-level, We are also performing SGD by considering S meta-sample (which is equivalent to 2), then we can obtain the following Lemmas.
Lemma 13. (Lemma SSpposs that Assumption and Assumption hold and F T. smoothness. Then, for any W. we have
T-1 Es t=
T-1 Σ
- min F + Loo W
T-1
T-1
nel mell F Stll 2
Th
F(wt+1)=
2 +
Σ
om-
t=
Lemma 14. (Lemma 6 in Lett Ge(w) : -- n1 FF S.) and assume that the loss function Fi), is LQ Eq-smooth and that its Hessian is Lipsscchh Then,
G.(ws.) G. (w
< (1+ ncEt)
||ws Wggs, Al]-
where & = /|V S,) + ng
Σ 8, V W s/>
+ 62
Σ AVF s.)| Further-
| Be (W S.)||. Furthermore,
more, we have Es.s(E]] = Es,s V [wo, (Wp, S:) + PQBBs. for any t [T].
Es.s (S, S)] & Es.s
v2 F S.)
2pQ v
(F(wo) min F)c(1 + In())) W
+ +op2
(V2cLe dl1 + +n((T)))
29

---

Lemma 15. (Lemma 5 in [47])Suppose that Assumption 2 hold, then every to E 20. T}
we have that
(s( (w3, (t),, E(WT, (wg (1"), - eGEs,s.alw3 -o:lww''' = = -s, (F(wS)) 1o
m
B.3 Proof of Theorem 2 convex)
Let's consider two parallel processes of generating iterates (w*) and (wi} by using datasets S and S, respectively. We use the tilde superscript to refer to the second process throughout the proof. By Lemma [9] we know E is Lg-smooth and convex function. Hence, for given time index with probability 1 L. the task Ty is selected, where By using Lemma have
|| - 1++1| =11 =1((0 - 1/1 V.VViww (T, , (wi 7/ - ViWW,; (w', ("", ,,
1 n1
Σ
(1 ni VE(WT, (w', ,5"), - (wi - 7 V WT, (wi, ( ,)'),
(30)
Next, for given time index t, with probability 1 the task Ti is selected. In this case, we have
E| Bl|10/ 2+1 - 10 16 =B] (ne
V C (W'T (w) (4)) - (w² 1)t V (WT, (10 5,
Bl| (w-
Ne V WT: (w" ,,
(w² n VL WT, (w) 5:h
+ El
V [( (T) (1"" ),t STa) n V S[(w7: ( 3,r) STa)
7: v E W (w', S: z11) (w' n7 v 2 (W (w (,0) 2b11
ns
ESt
B
VL WT, (w. S (tt) VŻ (wt 5 St+)
Σ
E w ny Ve(WT, (x))),,,,,) (Ttt - ny NeVVeWT (WT,. ( ,*))
ns
*EST
27p
B
(31)
where the last inequality follows that 3 and St are sampled from the same distribution, then E|| V[ (wr $ E|| V wT, (w¹ 8t)), E|| (w)]. Note that
E (w - 97 (w S!:h) - (w² T/ Ve(WT. (wt 1,,,, ,,))) <Ell( (w' 7e V VE ,,²,),)))))))) - nu V.VVew (w 2())))
(wi-
(32)
V (w (ū St9). Vi(wT, (w0,, 3, |
Let us bound the two terms on the RHS of 32 separately. First, similar to how we derived 30 we could bound the first term by
Bll(w' Toe Fe(wTT('', ,(,)),))))))))
(w² n Ve(WT ("'") 28)) E wn T6|l
30

---

To bound the second term on the RHS of 32 we consider two parallel processes of generating iterates { and { (T,g) J by using datasets S/" and S respectively. Note that
E
V U (w' S,)),
Ve(WT, (w. (T' 5/0) |
Q-1
Q-1
-BI
II (I
Ti.g S: V[
II
aI- Vil 21)
(I -
q=0
4=0
Q-1
Q-1
<BI
II (1
2. 11 T:gg1 S:1)
II T
a
1 T,:q: S V(
q=0
Q-1
Q-1
B|
II
(I - a
(w
22
1Q01
q=0
q=U
Q-1
<B||
II
[I
S
2/ا
[ 21) ||
q=0
+B|
II
(T
V
(T
v2
2 =t!
<(1
aL)
T. )
ve(wi 2
Q-1
(I a
2 .. (t
(I a a
5t1))l 1.0"
q=0
4=0
V(Q)
(33)
where we use Assumption in the last inequality. Hence, what remains to bound the two terms in 33 To do so, notice that
E| V(
-V
21) SLE
TT w
-LE||
wh. @ - avi(w) VV 11: 5tr)
"iQ-1 - a V. 5to
1-11"
Q-1
<|EI|
W W Q - av
Str)
"iq-1
V 511
LE
V C 5hh a
( St'
2aLG
11- -1l ntr
2(Q 1)oLG
SLE||
T
aV sP)] [ avī(w' 35t1 (()) || ||
nr
2(Q 1)LLG
=aLE
VQ (t) vi(w". Š,-)
nt
2QaLG
(34)
31

---

where we use Lemma 1 and Assumption in in the third inequality. Next,
Q-1
Q-1 II (1 ( aI- StT)
((Q) =1 II
(1 - V²
((
q=0
9=0
Q-2
V 2 ",,
Q=0
Q-2
II [I
9=0
Q-2
(I a 2 L(
17
""
q=0
Q-2
(I a- (1 31
q=0
Q-2
2 - 3t))
a 2 2
q=0
Q-2 (I a
-
q=0
Q-2
=| II
2 2 w T., S
q=0
+ - +VQ0-N)I4-AA2(((00111111011
aL)V(Q <(1+ (4 (1) 1)
where we use Assumption in the last inequality. To bound the above inequality, we first consider the first term.
(1+
-
Sa(1 aa(1+
2 1.0 11
V2
1 St VV "1 (( 42 V Str
21.
Sa(1 aLL0+
p pl Q- w
4 +
n²
2L pll - aVi(wt Str) - (w) - (( nt
=a(1+ + Q
p|| Q- ال + 2[T+ Gap)
Sa(1 + +LL)--1
n¹"
Sa(1 + QLL
Ml| T - avā (pT) - T avV 5 ||
2(I + Gap(Q 22)
nt
04p Vi("", ,, - Vi(Te' ,,t0l + 2(L Gap(( - 2))
Sa(1 + +T.)
nr
Sa(1 + 1-11+a0)--1200 2(L + Gap(Q -1) 1))
fr
where we use Assumption 2 to derive the inequalities from the second to the last step. Putting it in to (Q) and Unrolling it, noting that V(1) " SSF)
a 2 L(w 3 < 20L then
32

---

have
(1 [+ab))0-1 2a(L + GaplQ - k))
v(Q) 5(1 =(L+a()0--VV0
n
20L(1 + aL)
Q-1
(1 ab))--1 20(L + Gapk)
(35)
n1
n1
k=1
+ - 2 Gapk)
n*
By plugging (34)
and (35)
into (33)
then we have
2(1 aL)QaLG
2aG Gask))
E| E] (T) (5)))
88,), Vi(wT (w),
s1 (1+
n
n²
$ (1 (1+ (6QG Q a
n
(36)
where we using aL < in the second inequality. Substituting
36 and
32) into
we have
++1 - - <BI|ww - W + nr + aL) (6QG + Q2 p) 2n B (w")))
n/
Combing the above two cases, we obtain
1 Ell'" -w'l
E | ++1 2 5(1 - Bllw w
- E
1
1
(6QG 0- Q' G2 p) + 2 + B.BBVP. F. (w')ll
+ m
(1 + aL) Q-
nt
m
ا + 17. (1 a (6QG + Q p) = Ellww
21
e| V (w")
+
mn
m
Unrolling it and noting that wm10 - 1 0. we have
T-1
(1 + ab)o-11 (6QG + P22
21 E v V ( S:)I
Blllw" w
tnntr
t=
ne + a (6QG + Q'a2G')) p) +
LQ T-1
F(w")
min F + W
Σ nh
mnn
2
t=
where we use Lemma have
13 in the last inequality. Now we are ready to conclude. For any i E [m]], we
Egen I Blle((UT, (w (,)) l(WT, ",r),)))))3) ,,,,
< GE GE|lw - wF|
which completes the proof.
B.4 Proof of Theorem 3
In this section, we establish stability results that do not rely on convexity, and consider two cases. For the first case, using Lemma 10 and Lemma [4] we have
=1 =1 nn NovV(wrp (WWTT (w', ST), (w' The V. (wi ( St))) || ++
(37)
<(1 + +noolll 16/l,
where
the Σ v w
min miq, & with &
v F(wo, (w, St)l
E F
PQ
3p 2 1)
(1 +L) + ax + aL) +
3ap
(1
Lo
30 20 1)
L
33

---

(1 L.
Next, for the second case, similar the proof of (3I. we have
Bl
w-+1|
Ell(w*
- 71 Ve(WT, S 2())))))))))) - -(w² - )x²)
to
=ES!
B (w)ll
(38)
Note that
E (w' ni Ve WT, (w' -1)), - (w' Vl Ve((T. (w² ),
EB| (w' ne V WT, (w" , St"), =1)) (w' 1/2 Ve(WT: (T: (wi S ;5"))))))) ||
(39)
+ B (ū. S 2 V WT, (w' )
Let us bound the two terms on the RHS of separately. First, similar to how we bound could bound the first term by
37 we
Ell (w' Ve(TT, (w", S,)), (w' T/t V ( ("), S (1 = w'Il
To bound the second term on the RHS of (39). similar to the proof of (33 we have
E| V
(T', S.h ) - Ve(wT, , ,,,,=
<(1
aL+
E T. Ve(w )|"
Q-1
Q-1
(40)
GE
(I - aV
L( Q. 5T))
II
(1 - av 3r1)I
4=0
9=0
V(Q)
where we use Assumption in the last inequality. Hence, what remains is to bound the two terms in To do so, notice that
40
El T.:
Ve(W (1))))))))))
=LB]|
wh.q-1
aV AVL(W,,Q-i 11, ST:
W
1Q-1 a 1 ||
<LB||
W 110-- aV 10 Q- ST
W
1-Q11
av
Strr)
+ LE +LEB
11 845 a
1:: S
2xLG
<T(1 + t..
//// 1 nt
2(Q 1)(1+ -
|/:) <1(1+ + <L(1+aL)9E] 1 - avī ,
16 aVV (3)) +
nt
=al(1 + QL.) S v 3t0) + 2( 2(Q - 1)(1 + aL)R-LaLG
2Q 2Q(1+ (L)1+aL)aaG
tt
(41)
Next, we have
V(Q) <(1 + (/aa
(42)
To bound the above inequality, we first consider the first term,
(1 + +1 -aa St a(1 - Gap(Q 1))
n
34

---

Putting it in to V(Q) and Unrolling it, noting that V(1) = ||a ||av2Σ(w', . - 2 8 -
2ad
nir then we have
R-1 v(Q) 5(1 (1+a1))-1v(1))
+al))--1 2a(L + GaplQ - k))
n
2aL(1+aL)0-
(1 +ab]Q-1 20(L + Gapk)
(43)
n
n
k=1
Q-1
(1 a1+0-)0-11 20(L + Gapk)
nt
k=0
By plugging and (22
into 40) then we have
2(1 + aaL+aL]aaG
2aG(L + Gapk)
(1 V| (T' 2)))) Ve(WT, (w², 2 (1+
n
n²
$ (1 (1+aL) (6QG +
n
(44)
where we using aL < in the second inequality. Substituting 44 and
(39h
into
38
we obtain
Bl|w**1 " <(1 no E| wu' - = + 71 aL) aL) (6QG + Q²
2m B
(F("))l
n²
From Lemma
we can know E V F, (w') < eG. Then combing the above two cases, we obtain
_ (1 ne |B| w w
Bl ++1 w/ S(
(1 + no0) w"
m
m
+ - m/o (1 + aT.)2(9911) (6QG + Qaa p) + 2n.eG
n
<exp(m El/ - w1 + n m
where we use + x S exp(x) and a S QT. the second inequality, and we denote
-= 2eG (1 + aL))(Q-1) (6QG+Q'a²""*) Following the same proof technique in 47 (Eq (23) in
Theorem 4), we can easily get
w
Σ
exp(2c
1
2c4
k
ml
1=
1=++1
T 2c
exp(2cyln(
In
1=
2c (T
Σ t 2000-11
m
t=to
2c T ,207
$ 20y m
to
and
EllE(WT (wT ",,,), 2) - E(WT, (w. S,,),, 2) 5
cGa T 207
+
r
to
(45)
7m
where (F(s))
Omin Lq. Es
[|] V
(w
S|)
+ pQ(co
+
Vc(F(w²) minw F)))). Next, let b = 20y Then, setting
to - 20eG4
T
r
minimizes 45 Plugging to back we get that (44)
equals to
(2ceG4) T&E (rT) +
m
This completes the proof.
35

---

C Stability and Generalization of PDF
C.1 Proof of stability of PDF
To show the claim, it just suffices to show that for any i, we have
EAS [((ws)) Fi Si) S.
Take the dataset 560 which the same as S, except that S differ from S; in at most one data point. In partieular,
S = { ... ) S; { Z.g. .... Zi,n
Then, relate empirical risk and population risk by
Es. (
Es Es
j=1
(46)
Es A. ex ex Zig).
Moreover, have
1 n
EA.s (F.s(F((ws))
Es A, (x
(47)
T1 j=1
Putting 446 and 47
together, we have
m
n
EAS F(ws) Fwws, 0.)
s
E...A.,,, (x(ws, Zng) - Ex Ex(wg: F1.3)
m
n
S.
Then we obtain the desired result.
C.2 Lemmas
Lemma 16. Assume that P is differentiable and wiy is the unique minimizer of (( ) || w|
Then the gradiens of (w) = [(w+) + 21100 - w|l2 is given by VL, A(w - w7)
Proof. Since L is differentiable, from the first-order optimality condition we know that
vū(wT,) + A(w w) =
From the chain rule we have
Vix(w) = OWT:
v((wF) + A (T- (T- (Owi
(ow7
)) (w (w-w7)
Ow
Ow
:WT: )T (VL(w>) Xw--wi) wf)) Iw
>(w -w+)
>(w wir)
Lemma 17. Suppose Assumption 2 hold. Then if X > L, (x(,2)) = =(w7,=) + + A1l w|l2
soosmmotthssss with T.Q X+I and p-lipcchitz Hessian with respect to w, where w argmin ((WT. + 2 I[WT. w|?
Proof. From the first-order optimiality condition we know that V((wū,) ) + A( - w) = 0.
36

---

Therefore, we can further obtain
Twl
Dwot dv
(((
1) =
dw
and
(wit
Pug
+X x Pu Ow
(((
+ [2((wit]
0.
8w
Ow?
This implies
ow X(V2 + dw
Pwj (v2 + - (((((()))))))))
Owi 2 dw
Ow2
From Lemma, we have V(x(w) >(w- wj,)). Therefore, we can further have
Dw
Vla(w) = X(I
X(1 - (VV
dw
and
(+) = -X S
(V +1) 3 (w;;)
Ow
Note that ((WT,) is L-smooth and p-Hessian Lipschitz with respect to WT. Then it yields
AL
\ X+
[
sp
C.3 Proof of Theorem 4
To facilitate the analysis of stability, we rewrite R(wT, w. S))
[(WT,S) + 2 1uo
will; (wr, S) miney (Z(wT, 5) + 2 |uu =1 From Lemma 17 we can know Lx is T.g-smoohh. Hence, by using Lemmal T for the first case that task T, is selected, we have
E| Bl|w5* w =E]((w no Vi w0, w²; S,) (w' n (
a. (())),
(w' (T)) <E]((w' V (w,, St)) - (w' V
23,, E V w 7, 0. w/ ;,) Vix( S,)Il
(48)
EIl(ww n Ex (w", St)) (w' ne V2x (w' S)))
AB w3,0 / (w')l
<E|lw² - wt + 2m Al| 1 - w (w')
where (w) arg mitlor Σ(WT, S) + 1lW wll. where we use Lemma in the second inequality. In the second case, we have
Ellw0- w =E] (wo 1y VK (. w.) (w' m VK w/; ;S))
<B]l(w - T/ ViVi(uus w0, ,S:) (wi T) V 33))
(49)
+ 2E VKC Q: : v
For the first term in 44 we have
Bll (w - 7/ VL, (w" (S:) - (w' VLx ("', ,3)))l
<B|| (w- 17t VZ,
(w w0 St)) (wi ne VZx (w), S:)) + V S.) VÊx(w',
3.)1
<Ellwo 1011- 2m E B. F (w", S:)ll
(50)
For the second term in 50 we have
V ), w/ - (mx(wxx)) AB| =
(51)
37

---

Putting 150 and into
49 then we have
E
w
E [w w'
|| + 2n ,.)ll + 2T (w')' (52)
Combining
48 and
522
we obtain
1
E
Ellw+t -011
1| 5(1
)B
w' || + m
Blll' w0|l
2 V FF"",, s3 \ll + 2n - (ww))]
m
m
2n EE| V F( (w', 8t)ll +
211
=B
=B||w*
r5'l +
BBlli w (n'')l
m
m
Unrolling it and noting that ||000 - 11 = 0, we have
T-1
217e
2n
Blllw" T
w
S
E1|
w/ w (""( +
E E i (w'
m
m
1=0
t=
Loo T-1
2m
El| w/
wt (w") |
F(wo) min F + W
m/8
m
m
where we use Lemma and Lemma 17 in the second inequality, and we denote LQ we are ready to conclude. By using Lemma A we have
AL Now X+L
Bl((x(WT, 2) (2T, 21 $2GIE|lw - IT
T-1 2nd
LO
T-1
Ell Bllwi, - (0)(""))) (w0))
F(wo) mmin F +
m
W
2
1=0
Then we completes the proof.
C.4
Proof of Theorem
5
For the non-convex case, under Lemma 17. for the first case we have
Ellw't
w =B][ (w' 7/p V. (1 or w²; S5)) (wt 1/e VR (w T"; S1
<B][(w' - m/ v (w"" {j)) (wt n VLx ( S;))I
+ 2 Fe w g. w²: S, V.x(w", (S)]l-
(53)
=E]((w' - 7/6 VZ, (w- 8,)) - (w* - n V Ex (w' ,S)))
+ 22 w (w') (w')ll-
<(1 + (Bll))lllu w² 2m N|w w (w²))
where
Φi
mi with & = | Sill + & (VP(())
E (VF ||. (w)) arg min E(WT, 9) + 2 |lw - w|l. In the second case, we
have
El|w011 - /1 -E]l (w² TT WKK(Ku (wt Teip : - (w' - 77 VKK w²; w²;
<E] - T/t VL, (wo". SS:) - (w'- mp VZ2((i (wi 5.))
+ 2 VK w/y .. w²; (.) V[x("', S:)Il
(54)
<B| (w0' 1/1 v Ex( (wo', S.)) - (w' m1 Vix((x (wi (3)))
+ 2m 1110000 - w (w Ill
For the first term in 554)
Ell(w' - 7/ V Lox w/ st )) - (wi m) v Σx (w' 3))1
<Ell(w' - 7/e LZ, (w' xx)) -
- t.. -wi-
V (w ))Il + + (w2(1")))
38

---

Putting it into 54 then we have
E w
S (1 mese WB| 2 ( + =,,(w')) (55)
we can know E (w=")) S G. Then combing two cases, we obtain
E| 1t 1
<(1 - [1 no)
T0ll
+ m
- (1 + nT) w'll 1
2n.BI :)ll 2n,Billu (17((("))
m
1
2G
<(1 + (10.))BllIu ui|l + m
2nd ( (w' )11 Cº
(cxp(n -w1l= 21/ +>
m
where we use the typical convergence rate of GD results in the second inequality, 1 + x I exp(x) in
the third inequality. Additionally, we denote & = G +
(++GG
Following the same proof technique
C4
in [7 (Eq (23) in Theorem 4), we can easily get
T
T
1
2c
exp(2c7
k
mt
t= +1
1=++1
T 2c
exp(20>1n(
mt
+=
T
2c
(m)
t 200-1
1=1o+
2c
T
207
2ey
to
and
(l (w" (, ElWT, (wi" 5, 2)
eG4
T 207
+
to
(56)
mm
tn
m
where
Es ((s)))
O Es Fw.
S|D + plco
Ve(F(ww) minw F)))). Next, let b = 20y Then, setting
2cGB
to r
r TiE
minimizes (56). Plugging to back we get that
44
equals to
1
(2cG+) +ें T
This completes the proof.
D Results of Table 2
In the subsequent proofs other algorithms, we provide only the proof of generalization bound under the assumptions of the Lm-sthhtsssss constant and Hessian Lipschitz continuity, which can be established by referring our previous proof. We first present their corrosponding algorithm.
D.I Algorithms
39

---

Algorithm 2 MAML
Require: The set of datasets S {S with S, {S]", (.r). outer iterations T, adaptation steps
Q.
Require: Choose arbitrary initial point w" E W;
1: for t= 0 to 7 1 do
2: Randomly choose the task i.
3: Inner-Level: w/, = W/
4: for ! - 0, ...,, Q
5: WT T,+++1 = wT, T,99
a (( V.V.(WT... Tiip S::);
6: end for
7: Outer-level: WT: w
8: m++1 = w' 1/: V w Li (T, S:a))
9: end for
10: n and TH == T+I Et=o w";
1
Algorithm 3 FOMAMI
Require: The set of datasets {5:) with S; (S 51² outer iterations T, adaptation steps
Q.
Require: Choose arbitrary initial point w E W;
1: for = to T 1 do
2: Randomly choose the task
3: Inner-Level: w
We
4: for 0, ....,
1,.,, Q 1 do
5: h.g+1 w/y, v L( (( St:);
6: end for
7: Outer-level: WT, w²y
8: w*+1 w¹ 17c wT,, Li (W,, S
9: end for
10: T and T. = T+1 Eizo w":
Algorithm 4 MetaSGD
Require: The set of datasets S {S with S, {S) S!²),, outer iterations T, adaptation steps
Q.
Require: Choose arbitrary initial point w0 E W;
1: for = T do
2: Randomly choose the task i.
3: Inner-Level: w = Wl
4: for 1 0, 1,.., Q do
T. = w/ - a VL (
5:
6: end for
7: Outer-level: WTT w².
8: wl+1 w¹ 17; V N L. (WT,, S7.)
9: a+1 = -a 1/. V L, (WT, 5115
10: end for
11: of and -- := T Σi= wh:
40

---

Algorithm 5 iMAML
Require:
The set of datasets S { with S, {S]", (. outer iterations T,, adaptation steps Qregularization constant
Require: Choose arbitrary initial point w" E W;
1: for t = 0 to T 1 do
2: Randomly choose the task i.
3: Inner-Level: w/ = W1
4: for ! - 0, 1, ...., Q
5: "T,++1 = wT T,99
a Ex (wt, Tr,0" S;
6: end for
7: Outer-level: WT: = TT..0
8: wLt1 = w' 11(11 + V wy, -,: VV wx, (wT,,
9: end for w and TH T+I Σ w":
Algorithm 6 Meta-Minibatchlrox
Require: The set of datasets S = {S:)m outter iterations T, adpation steps Q, regularization constant A.
Require: Choose arbitrary initial point wn E W;
for t 0 to T do
2: Randomly choose the task i.
3: Inner-Level: w7.00 0 Wt
4: for q= 0, .,.,, Q do
5: TT.0+1 w4., a v (( Si);
6: end for
7: Outer-level: WT, = w
8: w(+1_ w' ( w0t - WT,)
9: end for
10: w" and TH Et= wt:
Algorithm 7 FoMuML
Require: The set of datasets S [5 with S; {S s1h), outer iterations T, adaptation steps
Q. regularization constant X.
Require: Choose arbitrary initial point w E W;
1: for /= = 0 to T 1 do
2: Randomly choose the task i.
3: Inner-Level: w Wl
4: for 1 0, 1, Q 1 do
5: T.,++1 w4
V Ls (w.r.gr S11);
6: end for
7: Outer-level: WT, w²,00
8: w w' 17. V WT. Li Li(WT,, STo)
9: end for
10: w and .F T+1 Σl= w":
D.2 (cover)
This proof is similar the proof of Theorem 2 we have
||
w++||
=|| ww'
VnVā(wy (TT, (w', S. ,,"), (w² -1, V.. W.(WT, (wi )
1
(w ne Ve (w', ,,,), - (1²² - 77 Ve (WT, (w¹, ( )1)
1/00"
(57)
41

---

Next, for given time index t. with probability
1. the task Ti is selected. In this case, we have mi
B| Bl|w - 1 =B|l (w T/ V ( WT: (w" (t)) (w' 17, VV(WT, (ūt (++),,
< (w² T E WT: ,w', ,, - (T²² - In ' ( (T6,, , ((a))
Ellm V [( WT; 10 S S S1,)) no VV
U (wi,, (,,), Sta)
1
B|| ww' 8/e V L WT: (w', 2())) - (i n/ V W ( ,, e11)]
ts
EST
WT, (w) , St)) Vi (WT,, , ,,tt)) sSai
+ T
Ell Ell(w nu Ve(WT, (w', , - (i²² nn VE(WT, (w', ,, 21))
ts
ZEST
+ (w))l.
(58)
where the last inequality follows that Si and S are sampled from the same distribution, then E| ([VV[( (w', ( Sit) = El V[ ("", S!*) = (w)]]. Note that
E (w" - 7) V (w", 5:hh ) 21) - (w' -n Ve (w . ,,)) (wi - no Ve (wT, (w,, """),,"))))) + nh V (6 S 2*² - VE(WT, (w', ("",""")) >")
<BI| (w" - n: Ve (w' (("),
(59)
Let us bound the two terms on the RHS of 59 separately. First, similar to how we derived 57 we could bound the first term by
Bll - n/ ( (w", ST)) ))
2 T7 VE(W,, S."), 2*)] - E w² 77'1 ||-
To bound the second term on the RHS of (5 we consider two parallel processes of generating iterates {wi4 } and { by using datasets S/r and 5 respectively. Note that
SIIVEWTT (w', """),, ;²)),= Ve(WT, ("",,,)); =,"),"
VE (w, Q. )0 Ve( qQ اا
SLE| 7 /
=LE||
1: |T-)
W.
iQQ-1 v VZ wt.
1211 (( Str) ||
<LE]
W 11 a VL(
11: (t-)
a
+ LEI.a
Villa i 5::T) V
1: St511l
(60)
2aLG
SLEI (illl QQQ-1 0//2 || + ntt
2(Q - 11)aTG
SLEI|
T - VV sstT ,5t) - [w - a VV ,,))) ||+
nte
Str - v L(wi, (5t5 + 2(Q - 1)aLG
n"
2QaLG
nt
Substituting (60) and
59 into 58
we have
2QaLG
Ellw*t! +1 w +1
E Iw' w0ll + m
ntr + 2 FV(('))
Combing the above two cases, we obtain
-
2QaLG nt
Ellwo*
'1 <(1 +1
=Ell - w¹1l + - Ellw" - w0ll + -m
+ E VF(""')
m
m
2QaLG
2
=Ellw x' + m.
E| F((("')l
mnt
m
42

---

Unrolling it and noting that || - 1 O, we have
2QaLG
2n
Elll"
w" 110
5 71 mn
E V F (w', St)ll
m
T
2QaLG
LO00 T-1
+
F(w*) - min F +
Σ 11:
mn
m
W
1=
where we use Lemma 13 in the last inequality. which completes the proof.
D.3 FOMAMI.non-coomee
This proof is similar to the proof of Theorem B we have
|| w (Iw nn, V WT, (w' ".),),))) - (w' n1 V
(w {.
(61)
S(1 + = T011,
where
6F
min & with & = F(wo, Sill +
Pu 2
| BVF -
PQ
El= 8 V (w))
2
Next, for the second case, similar the proof of 58 we have
Bl|w** w
1
Σ
Ell(w' ma VVVeWr S," 21)), - -(w' m Ve(WT , ,3")) S,)),
ts
EES!"
2m
Bl (w)l
(62)
Note that
E (w' n V ( WT, (w', ,s!"), (w' ne Ve(w. (it,, ,3,),, 2,2),))))) < [wl - n Ve (WT, (w= Str) 21)) (wi 7 Ve (T (w² str 2(*))(
(63)
+ VE (UT, (ū'
hh V (WT, (w', ), ||
Let us bound the terms the RHS of 63 separately. First, similar to how we bound 66 could bound the first term by
El| (w"
7/s Ve (w (w", ST",, (w' n (0 2())) S (1 + no W'll
To bound the second term on the RHS of (63) similar to the proof of (60). we have
2Q + aLL
E Ve(w WTT (w", )*))
Ve(w,, (w
(64)
n
Substituting 64 and (63 into 622 we obtain
<11 + T El| 5e 2Q(1 + aa))aLG
E El +1
2T T Fr
n2
From Lemma we can know F Fo (w])] < eG. Then combing the above two cases, we obtain
w +1 +++ <(1 (1 - no - - + (1 + [) E01l
2Q(1 + Q 2g.eG
mn
Th
(elp( Wi|l
where we use + x I exp(x) and - Qu in the second inequality, and we denote + 2eG
2Q(1+at)90I.G Using the same technique in Theorem
Then complete the proof.
43

---

D.4 (coNE(convex))
Since our goal is to demonstrate the extensibility of our framework analysis, here we consider the exact version of the solved iMAML algorithm, and of course we believe that this can be equally generalized
to the iMAML algorithm with an enror term. Let f*+ argmin wy, {[(WT, st + a W! - w' 1
we have
|| x
11 =11 =1(w²' m/
( (w', (3)) (w' T7 VVVi(w ( (w), S
1
Σ I1ww~ no Ve((w, (w', (5)) - (w² 7/ ( (w) STJ) 21)1
n ts
100
(65)
Next, for given time index t. with probability 1.2 the task Ti is selected. In this case, we have
Bllw**+1 - ""+ =BIl (w* Ti V Vv(wi, (w (w" ,,"),
(w' m/ Vā(w,, (.")", 3+")
<B|| (w' T7 Vi (w) (w", S.)) (()) - (w - mn (
5th 51 nt VL W 555 ("², 3,t)), S.:) E (w' - Ti v 2 w (w', S:r )())) - (w² Tn V 2 (wi, (w 3, )|
Ellne V
(w (w
W (w), 344)) Vi(Wi 3, 1
n
E| w mp Ve(wT, (w', ,.""), - (w" - m V Ve(ui ("')"")))) )""))*)))
=ES!
+ Bmh (w)
(66)
where the last inequality follows that S and S/ are sampled from the same distribution, then [I (wi , = E (w (wr, St = V Note that
E (w' - nu V ( (w 5th 21) - (w² m VeVe(wi (w ,,"))
<BI| (w* - n V (w77 (w' ( 2) (w² n Ve( Ww (w²
(67)
+ B (w7, (ät ","")," 2h - (w ("",)")) ".
Let us bound the two terms on the RHS of 67 separately. First, similar to how we derived 665 we could bound the first term by
Bll(w' - The Ve( (w S ) 2))
(wt Ty Velu st)), s El w - w0|l
To bound the second term on the RHS of (67 note that
ElIVE(wt, (w' 24h) - ve(Wi,
,
=B]((T 1 vi )t -1 VVe(u,, =0x) (I + v2 V' ((-) - -V =
<BI[(I + π 42 V2L ) -Velun )00) (1 + x
v2 E( ( ' ((W,,, ||
+ +E] + 1
1 1 C( wT, St (17 1 - (1 + v2 [(w 3, V ||] (68)
<B][(I+ x v2 E (
|1I =² V |
+ + V s - (1 1 V - ||[[ =
For the first term in 68 since the inner-optimization problem is in À strongly-conver setting, then by using the standard stability result in 43), we have
1 VVL[(ug El([I+ + A
2LG Antrr
- 2ا
(69)
44

---

For the second term in 68 since S." different with S at most one point, then we have
B +
2 L (w )) (I+ A
vz V= 5: - VIIV 2| -
2LG
(70)
An
By plugging (69) and (70) into 68)) then we have
Ve(WT, 5,t), (11) 2L GG) And
E VE WT (w. s, ²*)
(71)
Substituting 71 and 67
into (66). we have
< w - =0'l + 2L + +G)
E w
w
+ B1 VF:(w')]
An
Combing the above two cases, we obtain
_ |R| 10 w0ll
Bl|w0/+1 - |1 S(1
R w w'
m
L + G)
2TT E V (ww))l
+ m Antr
+
m
2mp LG G +
2n
=Bl - T|ll +
E (w'))l
Amn
Unrolling it and noting that Im10
1 O, we have
2m LGG + G Amn
2n E V (w',
BlllwT
n 11 S
(w', ,S1l
m
2mp LGG GG) Amn
1
F(w") min F + LQ02 T-1
Σ 1/2
+
1=0
where we use Lemma 13 in the last incquality. which completes the proof.
D.5 (MAML(non-convex))
This proof is similar to the proof of Theorem 3 we have
|| w = [1( n v [ (((wr, .. - (w² no V. St (72)
$(1 + 71000ll w'1l
where
Φt min(Lq, Ee} with & =
F(2,, +
po 2
| (VF(wus)
+
P2 2
El=l 3, V (wo
Next, for the second case, similar the proof of (66). we have
Ellw 1+1 V 1
E (w'
m/
ts
SEST
(73)
(w" 1/t VEV(WTTT (w) - 20) 27, E
Note that
E (w² N/ V
VeWW WT, (w' , ,((),))1)) - (w' m) V (W' (w ,,,),,
< Ell((w - n. V WT, (w", Sth =))) (wi 7/e V (((WT. (T), (wi STT) 2))
(74)
+ BnB/VE((TT (6' S ) 2T - V4(wu, (w. ,,)),, ||
Let us bound the two terms on the RHS of 74 separately. First, similar to how we bound could bound the first term by
72 we
E|| (w" mu Ve(T (wi, 2())) (wi The V ((),)))))))) (1 + nl a
45

---

To bound the second term on the RHS of (4). similar to the proof of 68 under the X - L strongly-convex setting, we have
Ve(wT ST), = 2 +G)
[ (w" ;
(75)
() L)n
Substituting (5 and 774 into 73 we obtain
2T GG) (A - L)ntr
E m w (1 + 7e E w + nt
+ Bn VP
From Lemma 7
we can know E V F. (w')l S cG. Then combing the above two cases, obtain
(1 (1 + FFl - -0 + = (1 + mese Ell - T-
(A L)mm*r + 2n.eG
2L + G)
% +
Sexp(mo FBIl W
m
where use 1 + x S exp(x) and < QL in the second inequality, and denote + 2eG 2L(G*+G) Using the same technique Theorem (A-T)ntt"
Then we complete the proof.
D.6 Fo-MuML(convex)
Fo-MuML is more like a application of FoMAML in PDF In particular, in the ourter-level, We no longer derive the derivative of w, take the derivative of WT, Then we have
|| w ||
n/
vV ( S.)).
(w T7. V..ā (wT . Sf.)1
1
Σ
11 - n VE VE(TT,, ,))), (w² n VE (T² S z )
1000
(76)
Next, for given time index I, with probability 1. the task T: is selected. In this case, we have
B|| - -1+1||
= Ell (w" V WT, (w",, S,
(wr (((TT, (,)))))))
< [l| (w' T7e V (TT (w', ,, (w ne V.
+ Bllm.
V Vi(TT. 51 5:0 n. VV([,, ).1) Sta)
1
(w n v 2 W (T. ST² (*)) (w' n Vi(WT.,( ,5))),
n
S
E
(WT.Q (w' (,t)
" S;
Ell (w - nh Ve(wT. " - (T² - m V (,) );)),,)))
ZEST
2m 2
(77)
where the last inequality follows that 3/- and S are sampled from the same distribution, then E [ (w*, S - (wi, Str), St*) [ Note that
E| 1w - n7c V [ (w' Shh) 21)) (w' n Vl TT ,,)))
< (w' - n V ( (w: ,"))," (w' n (eVV(WTT.(( S(T),
(78)
+ E (w. ) Ve(WT, (ū 3r ||
Let us bound the two terms on the RHS of 78 separately. First, similar to how we derived (6).. we could bound the first term by [ll(ww n/ Ve(Wn.Q(wi) Sth 2 ))
T n V 2)),) s El|w"* w
46

---

To bound the second term on the RHS of 778 we consider two parallel processes of generating iterates {w and ( J by using datasets Spr and Str respectively. Note that
B Vl]VE(WT, (i' (",,,,),,")) - Ve(WT, ST), )| =E V T Q.QQ - Ve( (2 21
LLE 1021
(79)
2G
Antrr
Substituting 79 and 78 into 77) have
2G
E | ++
a++1
E
Iwo
will || + 1tt
+ =
Ant
Combing the above two cases, obtain
1 778
1
1
20
E w ++1 w(+1l || <(1
E
w +
E
1/0 w' + - 1/!
+
m
m
An
m
2G
2T
Ellw
T
+ 1/2
E
V (w')l-
Amm
Unrolling it and noting that wm100 - -1
0. we have
T
2G
2n
Elllw*
E V (w',
Th
Xmn
m
2G
LQo2
F(w0) min F +
n
Amntr
m
t=
1=u
where we use Lemma 13 in the last inequality. which completes the proof.
D.7 FnMuML(nnnnccovee)
This proof is similar the proof of Theorem have
|| w+ (ww n/. S((WT, (w', (.)), - (w' no VWVWV[(WTTT (wi S.))s)9)))
(80)
5(1 + n T61l,
Li with & F Sill + Pa E BVF
where
EE 3 (
Next, for the second case, similar to the proof of 77 we have
Ellw ++1
w+11| < ts
Σ
Ell
(w'
m7 Ve(wT, S(Th),
=ES
(77 1 V WT, (W) S:0 20) 27p E (
Note that
E (w' n we.Ve((s (w' S )11)) - (wi Ve (WT (To² )
< Ell (w' ni V WT, (w" Strh 23 (wi 7 V WT, (w/ 21
(81)
+ n. El V (w 5:h 2 V WT, (W', ,,
Let us bound the two terms on the RHS of 81 separately. First, similar to how we bound 80 we could bound the first term by
E|l (w" 7/ Ve (WT (w", StD), 2 =2) - (w' - 7) (WT, (0" ST), 2)) S (1 + Vr T'|l.
To bound the second term on the RHS of (81). similar to the proof of 779 we have
20
Ve(WT, (w), S,), Ve(w (w', 3 z |0 S
(82)
( - L)n
47

---

Substituting 82
and 8 into 81 we obtain
2G
Bll
++1 <(1 + me Bllw we'l + 11
(A L)n't + 21/ B. B (f
From Lemma 7 we can know EE] V F [w')] S eG. Then combing the above two cases, we obtain
E w ++ w
11 S(1
m (1 no):)llll will m (1 + nooo)Elluu T01l
(A L)mner + 21yreG
202
m
<exp(Th<>)BBI1) w
m
where we use + x S exp(x) and S Q in the second inequality, and we denote d = 2eG - (A-L)m Using the same technique in Theorem
20
Then we complete the proof.
48


Reviews:
[{"review_id": "S9eMwg8cZB", "reviewer": "Reviewer_9dZJ", "summary": "This paper investigates the impact of inner-level updates on the stability and generalization of meta-learning algorithms. The study classifies prominent meta-learning algorithms into two frameworks: the Gradient Descent Framework (GDF) and the Proximal Descent Framework (PDF), based on their inner-processes. It introduces novel definitions of algorithmic stability for both frameworks and derives generalization bounds, revealing a trade-off between inner-levels and generalization in GDF, while PDF exhibits a beneficial relationship. The meta-objective function's critical role in minimizing generalization error is highlighted, leading to the proposal of a simplified meta-objective function to enhance performance. Experiments on datasets like Omniglot validate that GDF's generalization error first decreases then increases with inner-levels, whereas PDF's error consistently decreases, supporting the theoretical findings and the effectiveness of the new objective.", "questions": "See above.", "limitations": "See above.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "(+) \n1. This paper considers an important topic in meta-learning, that is, the generalization performances. The authors use algorithmic stability to deal with the problem, which is algorithm-dependent bound. \n2. This paper considers a group of meta-learning methods (gradient descent framework, and proximal descent framework) instead of a single method, which is general. This point is new to me. \n3. The authors provide rigorous theoretical analysis on generalization. While I did not check all the details, I believe that the arguments here are mostly correct. \n4. The authors further provide empirical evidence to validate their theoretical findings.\n\n(-)\n1. The technical novelty seems limited, and the comparison between the upper bounds (instead of the upper and lower bound) stops me from giving a higher score. We know that due to technical limits, we may have to compare between the upper bounds. But this still harms the scope of this paper. \n2. The authors may need to add experiments acorss different datasets to validate the generalizability of findings.\n\n\nOverall, I still lean to recommand accpetance, given these limitations.", "was_revised": true, "final_justification": "After reading the authors' responses, I keep my score of 4 unchanged. \nThis score is mostly due to the fact that the authors consider a group of meta-learning methods (instead of a single method), and this is general. \nBesides, the theoretical part seems correct to me. \n\nI did not give a higher score, since the theoretical contributions still seem limited to me. But this is just a minor problem, and I still vote for an acceptance.", "rebuttal": "We thank for your valuable comments and will reply them as the following.\n\n**Question 1:** In generalization analysis, upper bounds are particularly valuable because they characterize the worst-case performance of an algorithm, thereby enhancing its reliability in real-world applications. Especially in stability-based analyses, upper bounds provide deeper insights into the intrinsic properties of learning algorithms. An algorithm $\\mathcal{A}$ is said to be $\\epsilon\\_{\\text{gen}}$-stable if the change in its output loss, when applied to two neighboring datasets that differ by a single sample, is bounded by $\\epsilon\\_{\\text{gen}}$. By deriving an upper bound on this stability, we are able to systematically trace how differences between neighboring datasets propagate through both the inner and outer learning levels. This enables our bound to precisely reflect the influence of the inner-level optimization. Moreover, our work makes a significant technical contribution: to the best of our knowledge, this is the first study to establish generalization bounds for meta-learning under two scalable algorithmic frameworks. In addition, we introduce two novel definitions of stability, which serve as the theoretical foundation for our analysis. We also recognize that the lower bound is valuable for the next generation of deductions, and will consider exploring this area of work in the future.\n\n**Question 2:** Thanks for you suggestions and we additionally provide two experiments results in the convex setting and non-convex setting to verify our theoretical results again.\n\n**Few-shot regression on the new synthetic dataset:** Unlike the experiments conducted in our paper that the tasks are created randomly as random sinusoid function, we recreated tasks randomly as either random sinusoid functions, or random linear functions. The amplitude of the sinusoids varies within $[0.1,5.0]$ and the phase within $[0,\\pi]$. The slope and intercept of the lines vary in $[-3.0, 3.0]$. The inputs are sampled uniformly in $[-5.0,5.0]$. We also use an MLP network with Means square error loss function. The generalization error is assessed as the difference between training and test errors. For each training task, we set the number of support samples and query samples to $n^{\\text{tr}}=5$ and $n^{\\text{ts}}=5$, respectively, while for each test task, we use $n^{\\text{tr}}=5$,$n^{\\text{ts}}=15$. Then we report our results as follows:\n\n| Sinusoid & lines | MAML   | MetaSGD | FOMAML | MetaProx | iMAML  | FOMuML |\n| ---------------- | ------ | ------- | ------ | -------- | ------ | ------ |\n| Q=2              | 0.7916 | 0.8682  | 0.6884 | 0.3330   | 0.3803 | 0.9063 |\n| Q=5              | 0.6393 | 0.6731  | 0.3196 | 0.1507   | 0.3569 | 0.6033 |\n| Q=10             | 0.4863 | 0.4296  | 0.2034 | 0.1040   | 0.3422 | 0.4674 |\n| Q=15             | 0.5356 | 0.5179  | 0.2209 | 0.0233   | 0.1222 | 0.2212 |\n\n| Sinusoid & lines | MAML(Ours)   | MetaSGD(Ours) | FOMAML(Ours) | MetaProx(Ours) | iMAML(Ours)  | FOMuML(Ours) |\n| ---------------- | ------ | ------- | ------ | -------- | ------ | ------ |\n| Q=2              | 0.7624 | 0.8484  | 0.6720 | 0.3190   | 0.3562 | 0.7381 |\n| Q=5              | 0.5964 | 0.6497  | 0.3071 | 0.1263   | 0.3247 | 0.5041 |\n| Q=10             | 0.4681 | 0.4118  | 0.1732 | 0.0594   | 0.2524 | 0.3338 |\n| Q=15             | 0.5144 | 0.4921  | 0.2018 | 0.0162   | 0.0864 | 0.1583 |\n\n**Few-shot classification on CIFARFS:** We conduct the experiment by using the real-world CIFARFS dataset, which is also a few-shot classification taks. It splits the 100 classes from CIFAR-100 into 64, 16 and 20 classes for training, validation, and test respectively. Each class contains 600 images of size 32×32×3. We employ a 3×3 CNN to align support samples and query samples to $n^{\\text{tr}}=1$ and $n^{\\text{ts}}=5$, respectively. For each test task, we use $n^{\\text{tr}}=1$ and $n^{\\text{ts}}=15$. Then we report our results as follows:\n\n| CIFARFS | MAML   | MetaSGD | FOMAML | MetaProx | iMAML  | FOMuML |\n| ------- | ------ | ------- | ------ | -------- | ------ | ------ |\n| Q=2     | 0.5989 | 0.5944  | 0.5865 | 0.6536   | 0.8020 | 0.5577 |\n| Q=5     | 0.4022 | 0.4423  | 0.4277 | 0.4636   | 0.7188 | 0.3894 |\n| Q=10    | 0.3677 | 0.2673  | 0.4709 | 0.3721   | 0.4105 | 0.2588 |\n| Q=15    | 0.3951 | 0.4798  | 0.4510 | 0.1360   | 0.2794 | 0.2478 |\n\n| CIFARFS | MAML(Ours)   | MetaSGD(Ours) | FOMAML(Ours) | MetaProx(Ours) | iMAML(Ours)  | FOMuML(Ours) |\n| ------- | ------ | ------- | ------ | -------- | ------ | ------ |\n| Q=2     | 0.5819 | 0.5855  | 0.5434 | 0.6196   | 0.5796 | 0.5430 |\n| Q=5     | 0.3884 | 0.4035  | 0.4137 | 0.3211   | 0.4458 | 0.3675 |\n| Q=10    | 0.3411 | 0.2556  | 0.4352 | 0.2748   | 0.3628 | 0.2473 |\n| Q=15    | 0.3861 | 0.4340  | 0.4493 | 0.0834   | 0.1342 | 0.2310  |\n\nAs shown in the table above, a trade-off relationship exists among the GDF algorithms, i.e., MAML, MetaSGD, and FOMAML, while a beneficial relationship is observed among the PDF algorithms, including MetaProx, iMAML, and FOMuML. Furthermore, we present experiments with our new objective, demonstrating that it improves the generalization error across the six algorithms."}, {"review_id": "zq1c19PmyZ", "reviewer": "Reviewer_3fQk", "summary": "This paper considers two settings - the gradient descent framework (GDF) and proximal descent framework (PDF) - for a number of common meta-learning strategies. It discusses two types of common metrics, the generalisation error and optimisation error, whcih constitute the test error. Given convergence of the optimisation error, the paper produces theory about how the generalisation error changes for the GDF and PDF. It subsequently provides a new meta-objective function, informed by the theory, and a small number of experiments demonstrating that their method has better optimisation characteristics (i.e., better generalisation behaviour than more standard meta-objectives).", "questions": "- This work focuses on a subset of meta-learning. Does this theory generalise beyond these MAML-oriented meta-learning approaches?\n- Can you explain for my sake what $Q$ is actually referring to?\n- How does the tightness and derivation of your bound compare to the others derived in theory that are referenced here?\n- Where does the intuition of the new meta-loss come from?\n\nBelow I provide some examples of typos:\nLine 17 - well-generalisation is confusing\n26 - Q inner-level gradient updating\n45 - increases with the number of inner-levels Q grows\n111 wells -> well\n145 - forlicity\n329 - 'We' shouldn't be capitalised. Neither should Objective on 333", "limitations": "N/A", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "To preface, I am an empirical researcher and so while I did my best to follow the theory I admit that a times certain bounds and literature were unfamiliar to me. For this reason, I submit my review with slightly lower confidence.\n\nStrengths:\n- Throughout the paper, the work is constantly compared and contrasted with preexisting literature. As such, I felt it was very well framed with regards to the surrounding field.\n- For a theoretically-focused paper, I felt their was still a reasonable empirical comparison and demonstration that their theoretical bounds transferred to practical tests.\n- The paper had a natural narrative, following from introducing the problem setting -> introducing types of errors -> discussing how these errors affect generalisation -> providing theoretical bounds -> showing these bounds echo what happens in practice.\n- From what I could follow, the paper is rigorous in its theorems and provides corresponding proofs in the supplementary material. For the sake of final publications, I think it would be good to provide these proofs in the appendix (so that they can all be found in the same PDF).\n\nWeaknesses:\n- There were a number of typos that I spotted (discussed below).\n- I felt like the paper could have introduced a number of concepts in a background section before jumping in. For instance, what is $Q$ - initially I had assumed this was the number of inner-level gradient updates for e.g., MAML, but the way it is framed in the paper has me guessing if this is the case.\n- I think this may be more common in theoretical papers, but it felt like a lot of important concepts were quoted by reference without actually saying what they were. It would be good for some of these to be directly discussed in the main test.\n- Occasionally, I found it difficult to see where leaps in theory came from. This is partly a result of my naivete of theoretical meta-learning, as well as the fact that the proofs were held in a separate PDF.\n- There is no access to the code, and while in the questionaire it states that significance in experiments was provided, this does not seem to be the case. It is not completely obvious to me that the new proposed objective (which basically just weights larger losses more) actually improves performance, given most lines are practically on top of each other and there are no error bars.\n- I cna't quite follow the logic from theory to the new loss. To me the new loss seems to sort of come out of nowhere, and also doesn't seem to have the framing for GDF vs PDF and why this is useful.", "was_revised": true, "final_justification": "I feel the theory is good, and htat the empirical side of the paper leaves some to be desired. I also find that there are a number of clarity issues. However, I think the contribution of the paper outweighs these issues, and thus maintain a score of 4 - borderline acceptance.", "rebuttal": "Thank you for your valuable suggestions, we will reply these as follows.\n\n**Weakness 1:** We appreciate the reviewer for pointing out the typographical errors, which we will promptly correct.\n\n**Weakness 2 and Question 2:** To further clarify the meaning of $Q$, we can use MAML as an example. MAML utilizes available training data from multiple tasks to derive a meta-initialization that performs well after a few updates during testing, tailored to a new task. Unlike traditional supervised learning, where the goal is to find a model that generalizes well to a new task without any adaptation, MAML focuses on finding an initial model that can efficiently learn a new task with only a small amount of labeled data by performing $Q$ gradient updates. Therefore, in this paper, our framework requires undergoing $Q$ gradient updates at the inner level to adapt to a small amount of data. We will add some relevant concepts explanation based on your advice.\n\n**Weakness 3:** We are sorry to make you confused and will explain them in details. Our work primarily focuses on the generalization error in meta-learning, which quantifies the performance gap between training and testing phases. The upper bound of this generalization error characterizes the worst-case performance at test time, and thus, optimizing this upper bound is critical for ensuring the real-world applicability of meta-learning algorithms. Moreover, stability is a powerful analytical tool in generalization theory. It helps us understand which parameters affect generalization and how they influence it. Crucially, the definition of stability plays a central role in this analysis, as it directly determines the tightness and relevance of the resulting bounds. Therefore, we introduce two novel stability definitions for our frameworks.\n\n**Weakness 4:** We further elaborate on our results and the proof approach. Our work focuses on the generalization error in meta-learning, and stability proves to be a powerful tool for deriving the generalization error of algorithms. The first step involves introducing two novel stability definitions for our frameworks, namely GDF and PDF. Based on these definitions, we derive Theorem 1, which states:  $\\mathbb{E}\\_{\\mathcal{A}, \\mathcal{S}}\\left[F(w\\_{\\mathcal{S}})-\\widehat{F}(w\\_{\\mathcal{S}},\\mathcal{S}\\_{i})\\right] \\leq \\frac{1}{m} \\sum\\_{i=1}^m \\frac1{n^{\\rm{ts}}} \\sum\\_{j=1}^{n^{\\rm{ts}}} \\mathbb{E}\\_{\\mathcal{A}, \\mathcal{S}, \\widetilde{S}\\_{i,k}^{\\rm{tr}}, \\tilde{z}\\_{i,j}^{\\rm{ts}}}[\\ell(w\\_{\\mathcal{T}\\_i}(w\\_{\\mathcal{S}}, \\widetilde{S}\\_{i,k}^{\\rm{tr}}), \\widetilde{z}\\_{i,j}^{\\rm{ts}})-  \\ell\\left(w\\_{\\mathcal{T}\\_i}(w\\_{\\widetilde{\\mathcal{S}}}, \\widetilde{S}\\_{i,k}^{\\rm{tr}}), \\widetilde{z}\\_{i,j}^{\\rm{ts}}\\right)] \\leq  \\epsilon.$ This provides a bound for the generalization error, $\\epsilon\\_{\\rm{gen}}$. It is important to note that $w\\_{\\mathcal{S}}$ and $w\\_{\\widetilde{{\\mathcal{S}}}}$ are two different outputs of a randomized algorithm $\\mathcal{A}$, which correspond to two neighboring datasets, $\\mathcal{S}$ and $\\widetilde{{\\mathcal{S}}}$, differing by a single data point. Therefore, in our proof, the key idea is to bound the difference between $\\ell(w\\_{\\mathcal{T}\\_i}(w\\_{\\mathcal{S}}, \\widetilde{S}\\_{i,k}^{\\rm{tr}}), \\widetilde{z}\\_{i,j}^{\\rm{ts}})-  \\ell\\left(w\\_{\\mathcal{T}\\_i}(w\\_{\\widetilde{\\mathcal{S}}}, \\widetilde{S}\\_{i,k}^{\\rm{tr}}), \\widetilde{z}\\_{i,j}^{\\rm{ts}}\\right)$. Under our assumption that the loss function is $G$-Lipschitz over $\\mathbb{R}^d$, i.e., $| \\ell(w,z) - \\ell(u,z)|\\leq G|w-u|$, we aim to bound the difference between $w\\_{\\mathcal{T}\\_i}(w\\_{\\mathcal{S}}, \\widetilde{S}\\_{i,k}^{\\rm{tr}})$ and $w\\_{\\mathcal{T}\\_i}(w\\_{\\widetilde{\\mathcal{S}}}, \\widetilde{S}\\_{i,k}^{\\rm{tr}})$. This difference is inherently dependent on the updating rule used by different algorithms.\n\n**Weakness 5:** We have submitted our code in the supplementary materials. You can check them by downloading the supplementary materials.\n\n**Weakness 6 and Question 4:** Our results indicate that the objective function $F(w^0)$ plays a crucial role in reducing the generalization bound. For instance, in the non-convex setting, the generalization bound of GDF is $\\mathcal{O}((\\frac{1+\\frac{1}{c\\gamma}}{m}\\Phi_Q)^{\\frac{1}{c\\gamma}}(F(w^0)T)^{\\frac{c\\gamma}{1+c\\gamma}})$ and the generalization bound of PDF is $\\mathcal{O}((\\frac{1+\\frac{1}{c\\gamma}}{m}\\Phi_Q)^{\\frac{1}{c\\gamma}}(F(w^0) T)^{\\frac{c\\gamma}{1+c\\gamma}})$. We can observe that size of $F(w^0)$ is important when $T$ is sufficiently large. \n\nTo reduce the term $F(w^0)$, we begin with the definition of $F(w)=\\frac{1}{m}\\sum_{i=1}^m F_i(w)$. Let us consider a simple relaxation technique: given a constant $\\beta\\in[0,1)$ and $\\hat{m}\\leq \\frac{m}{2}$ with the losses sorted by $\\{F_i(w^0)\\}$ in  ascending order, we can obtain that $\\sum_{i=1}^{\\hat{m}}F_i(w^0)^{\\psi{(i)}} \\leq \\sum_{i=\\hat{m}+1}^{m}F_i(w^0)^{\\psi{(i)}}$, where $\\psi{(i)}$ is a mapping function associating the index with the original loss $F_i(w^0)$. Based on this, we can derive that $F(w^0) = \\frac{1}{m}\\sum_{i=1}^mF_i(w^0)^{\\psi{(i)}} \\geq \\frac{1+\\beta}{m}\\sum_{i=1}^{\\hat{m}}F_i(w^0)^{\\psi{(i)}} + \\frac{1-\\beta}{m} \\sum_{i=\\hat{m}+1}^{m} F_i(w^0)^{\\psi{(i)}}$. Motivated by this result, we propose the following optimization objective:\n$$F_{\\rm{new}}(w)=\\frac{1+\\beta}{\\tilde{m}}\\sum\\nolimits_{i=1}^{m}F_i(w)^{\\psi{(i)}} \n    + \\frac{1-\\beta}{\\tilde{m}}\\sum\\nolimits_{i=\\hat{m}+1}^{m}F_i(w)^{\\psi_{(i)}},\n$$ where $\\tilde{m}=(1+\\beta)\\hat{m}+(1-\\beta)(m-\\hat{m})$ is to ensure the lower bound derivation.\n\n**Question 1:** Our works mainly focus on the optimization-oriented approaches, which is a main learning field for meta-learning. Therefore, we summarize the common characteristics of most optimization-oriented algorithms and classify them into GDF and PDF according to their different inner training methods. To validate the scalability of our theoretical frameworks, we derived six popular algorithms, five of which are MAML-oriented (except MetaProx). In fact, our frameworks can also be extended to other meta-learning approaches, e.g., MetaOptNet [1] and Hyper-representation[2].\n\n**Question 3:** Since our work primarily focuses on generalization error, while other studies emphasize transfer error, it is difficult to make a fair comparison regarding the tightness of the bounds. Generalization error measures the discrepancy between training tasks and test tasks, making it a key metric for evaluating the practical capability of meta-learning algorithms. Although some previous works [3][4] have analyzed the generalization error of MAML by using stability, their analyses are limited to the strongly convex setting. In contrast, our study extends the analysis to the more challenging and practically relevant convex and non-convex settings. The most close relevant results to our theorem 2 is theorem 1 in their work [5]. Under the convex-setting, if we let $T=m$, then our theorem 2 yields a bound $\\mathcal{O}(\\frac{Q}{n}+\\frac{\\sqrt{F(w^0)-\\min_{\\mathcal{W}}F}}{m})$. In contrast, the bound in [5] is $\\mathcal{O}(\\frac{1}{n}+F(w^T)-\\min_{\\mathcal{W}}F))$. We can observe that our second term $\\frac{\\sqrt{F(w^0)-\\min_{\\mathcal{W}}F}}{m})$ is significantly lower than the corresponding  $F(w^T)-\\min_{\\mathcal{W}}F$ in [5]. Although our first term $\\frac{Q}{n}$ is looser than theirs $\\frac{1}{n}$, we remain the trade-off relationship about $Q$. \n\n[1] Lee, Kwonjoon, et al. \"Meta-learning with differentiable convex optimization.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2019.\n\n[2] Franceschi, Luca, et al. \"Bilevel programming for hyperparameter optimization and meta-learning.\" International conference on machine learning. PMLR, 2018.\n\n[3] Fallah, Alireza, Aryan Mokhtari, and Asuman Ozdaglar. \"Generalization of model-agnostic meta-learning algorithms: Recurring and unseen tasks.\" Advances in Neural Information Processing Systems 34 (2021): 5469-5480.\n\n[4] Wang, Lianzhe, et al. \"Improving generalization of meta-learning with inverted regularization at inner-level.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.\n\n[5] Zhou, Pan, et al. \"Task similarity aware meta learning: Theory-inspired improvement on MAML.\" Uncertainty in artificial intelligence. PMLR, 2021."}, {"review_id": "xMYPsfWbFi", "reviewer": "Reviewer_wmex", "summary": "This paper mainly focuses on exploring meta-learning algorithms. As previous works usually pay more attention to training strategies and data, the impact of inner-loop optimization on generalization is ignored. Thus, in this paper, the authors explore both gradient descent and proximal descent frameworks to obtain insights regarding meta-learning paradigm.", "questions": "- Could you provide some explanations regarding Eq. (2)? I do not really understand the formulation.\n- What is the $\\mathcal{K}$ in Algorithm 1? Although it is explained in the later part, it would be better to define it before.", "limitations": "N/A", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n- Different from previous work, this work dives into the impact of inner-loop optimizations on the generalization in meta-learning algorithms.\n- This work achieves some interesting insights from the perspective of theory.\n- The paper starts from an interesting phenomenon of the generalization error and then develops many valuable insights regarding meta-learning, which is impressive.\n- The results reported in this paper are good.\n\nCons:\nFrom my perspective, I do not think this work has obvious drawbacks. I like this work.", "was_revised": true, "final_justification": "I will maintain my score regarding this paper since this paper is good and my concerns have been well addressed.", "rebuttal": "We thank for all your valuable comments. According to the questions, we will reply these individually as the following.\n\n**Question 1:** In the inner-level process of GDF, the learner $w$ needs to find task-specific parameters $w\\_{\\tau_i}$ after undergoing $Q$ times gradient updates. This process can be formulated as $w\\_{\\tau\\_i}=w-\\alpha\\sum\\_{q=0}^{Q-1}\\nabla \\widehat{\\mathcal{L}}\\_i(w\\_{\\tau\\_i}^q,\\mathcal{D}\\_i)$. To emphasize its difference of inner-levels from PDF, we give the optimization objective of inner-process, i.e., Eq(2)=$\\min\\_{w\\_{\\mathcal{T}\\_{i}}}\\left\\langle\\sum\\nolimits\\_{q=0}^{Q-1}\\nabla \\widehat{\\mathcal{L}}\\_{i}(w\\_{\\mathcal{T}\\_{i}}^{q},\\mathcal{D}\\_i),w\\_{\\mathcal{T}\\_{i}}-w\\right\\rangle+\\frac{1}{2\\alpha}\\|w\\_{\\mathcal{T}\\_{i}}-w\\|\\_{2}^{2}$. By taking the derivative on $w\\_{\\tau_i}$ and making the gradient equal to $0$, Eq(2) can be written as $w\\_{\\tau_i}=w-\\alpha\\sum\\_{q=0}^{Q-1}\\nabla \\widehat{\\mathcal{L}}\\_i (w\\_{\\tau_i}^q,\\mathcal{D}\\_i)$..\n\n**Question 2:** As we illustrated in section 3.2, PDF aims to obtain an exact solution of its empirical meta-loss $\\widehat{L}\\_{\\lambda}(w, S\\_i)=\\min\\_{w\\_{\\mathcal{T}\\_i}}\\{\\frac{1}{n}\\sum\\_{z\\in S\\_i} \\ell(w\\_{\\mathcal{T}\\_i} , z)+\\frac\\lambda2\\|w\\_{\\mathcal{T}\\_i}-w\\|^2 \\}$ which is difficult in practical algorithms. Therefore, we often solve an approximation, i.e., $\\widehat{\\mathcal{K}}(w, S\\_i)=\\frac{1}{n}\\sum\\_{z\\in S\\_i} \\ell(w\\_{\\mathcal{T}\\_i}, z)+\\frac\\lambda2\\|w\\_{\\mathcal{T}_i}-w\\|^2$ in Algorithm 1."}, {"review_id": "lxiDyl6v5O", "reviewer": "Reviewer_hYHQ", "summary": "This paper provides new algorithmic-stability based generalisation bounds for meta-learning focussing on the impact of inner level optimization. The authors consider two broad classes of inner-optimisation frameworks: gradient descent framework (GDF), which includes classical MAML algorithm,  and proximal descent framework (PDF) that includes the iMAML and Meta-Minibatch Pros algorithms. Leveraging on-average algorithmic stability framework, the authors derive new meta-generalisation bounds that reveals the influence of the number $Q$ of inner-level iterations. For the GDF framework, authors report a trade-off relationship with respect to the choice of $Q$. The authors finally also propose a meta-objective with improved generalization performance.", "questions": "I think the following points need clarification:\n\n1. Authors use Fig.1 to motivate the problem setting. However, their claim regarding MAML -- that generalization error first decreases with Q and then increases -- is not convincing from Fig.1(a). This will require more plots corresponding to increasing values of Q>20.\n\n2. In Section 4, the authors present Theorem 2 and Theorem 3. The assumptions in both theorems are the same. Its hard to see the difference in the setting (strongly convex/non-convex) considered. Better statement of the theorems are required. Along this line, even the statement of Theorem 1 is incomplete with no reference to $\\epsilon$.\n\n\n3. Following Theorem 2 and Remark 2, it seems like the proposed bound in Theorem 2 yields looser bound  than the one in reference [9] in the paper. The authors claim that setting $\\alpha \\leq 1/(QL)$ simplifies the first term in the bound of [9] to scale as $O(1/n)$, mitigating the dependence on $Q$. However, under the same $\\alpha$, the first term in Theorem 2 still has a linear dependence on $Q$. Note that for sufficiently large $Q$, $\\alpha \\leq 1/(QL)$ yields negligibly small learning rate, implying that the inner update will be close to the initialisation $w^t_{\\mathcal{T}_i,0}$ for each task. Isn't it then reasonable to assume that no dependence on $Q$ in this regime is better? \n\n4. Additionally, the term $F(w^0)-\\min_wF(w)$ accounts for the difference between the meta-objective $F(\\cdot)$ evaluated at initialisation $w_0$ and that evaluated with respect to the best initialisation $w^* =\\arg \\min_w F(w)$. It is not clear how increasing $Q$ can decrease this difference, as this difference depend on the proximity of $w^0$ to $w^*$.  Of course, this is true if the term was $F(w^T)-\\min_w F(w)$, the difference between the meta-learned initialisation and the best initialisation possible.  I believe reference [9] has the latter term in their bound. The above issue also arises  Theorem 3, 4 and 5. This concern could also be attributed to the lack of clear definitions in the paper.  In this sense, I am also unsure of the relevance of Section 4.3. \n\n5. More discussion on how the proposed bounds in Theorems 4 and 5 for PDF framework compare to other existing bounds are required.", "limitations": "The work is purely theoretical in nature, and does not pose any negative societal impact.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The paper belongs to a recent series of works that have explored the generalization analysis of meta-learning algorithms. The general classification of the existing frameworks into GDF and PDF is interesting, and enables authors to simultaneously investigate multiple existing algorithms. \n\nHowever, the paper is not easy to read and lack clarity in many sections. There is an overload of notation especially in Section 3, which makes it difficult to follow. I believe a thorough re-writing is required in Section 3. For instance, in Section 3.2, authors define $\\hat{F_i}(w,S_i)$ and relate it to $\\hat{\\mathcal{L}}_{\\lambda}(w,S_i)$, which is not used anywhere else . Later on, they introduce $\\hat{\\mathcal{K}}(w,S_i)$, whose relationship with $\\hat{F}_i(w,S_i)$ requires clarification. In Algorithm 1, line 7, what is the loss function $\\mathcal{L}$? Shouldnt this be $\\mathcal{L}_i$?\n\nSignificance and Originality: The investigation of the impact of inner-level optimization on the generalization of meta-learning algorithms is not new. For instance, reference [1] has already studied this in the GDF framework and has reported a similar trade-off relationship with respect to the number Q of inner level iterations. The authors in this work have gone beyond the GDF and have analysed PDF under both convex and non-convex assumptions on the loss function.\n\nWeaknesses: Please find the weaknesses under the Questions Tab.\n\n\nReference:\n[1] Zhou, P., Zou, Y., Yuan, X.T., Feng, J., Xiong, C. and Hoi, S., 2021, December. Task similarity aware meta learning: Theory-inspired improvement on MAML. In Uncertainty in artificial intelligence (pp. 23-33). PMLR.", "was_revised": true, "final_justification": "Following the rebuttal, the authors have clarified my main concerns. The notational discrepancies were corrected, and additional experimental results are included. Given the strong theoretical contributions of the paper, I am happy to increase my score to 4: Borderline acceptance.", "rebuttal": "We thank for all your valuable comments. According to the weaknesses and questions, we will reply these individually as the following.\n\n**Weakness 1:** Although we acknowledge certain typographical errors pointed out by the reviewer, we believe we have clarified the meaning of all symbols used in our paper. Regarding the terms $\\hat{F}\\_i(w, S\\_i)$ and $\\hat{L}\\_{\\lambda}(w, S\\_i)$ and their relationship, we have explained them in lines 144-145. In particular, $\\hat{F}\\_i(w, S\\_i)$ represents the empirical loss, i.e., the training loss, which has been explained in lines 132-133. We use $\\hat{L}\\_{\\lambda}(w, S\\_i)$ to reflect the effect of $\\lambda$. If we did not do so, the definition of $F$ would be confusing. As for the term $\\hat{\\mathcal{K}}(w, S\\_i)$, we have addressed it in lines 146-149: \"For a practical PDF-based algorithm, it's difficult to obtain an exact solution of $\\widehat{\\mathcal{L}}\\_{\\lambda}(w, S\\_i)$, so we typically resort to an inexact solution, $\\widehat{\\mathcal{K}}(w, S\\_i) = \\frac{1}{n} \\sum\\_{z \\in S\\_i} \\ell(w\\_{\\mathcal{T}\\_i}, z) + \\frac{\\lambda}{2} \\| w\\_{\\mathcal{T}\\_i} - w \\|^2$.\"\n\n**Weakness 2 and Question 3:** We would like to emphasize that our work significantly extends the contributions of [1]. While [1] investigates the trade-off in MAML, their analysis is conducted within a relatively narrow scope and lacks a general or extensible framework. In contrast, we introduce two flexible and broadly applicable frameworks—GDF and PDF—which are designed to accommodate a wide range of meta-learning algorithms. To illustrate the generality and practicality of these frameworks, we derive generalization bounds for six different algorithms. Moreover, our GDF framework provides a more robust and interpretable characterization of the trade-off. Specifically, when comparing our Theorem 2 with the results of [1], let us consider the case where $T=m$. Under this, our theorem 2 yields a bound $\\mathcal{O}(\\frac{Q}{n}+\\frac{\\sqrt{F(w^0)-\\min\\_{\\mathcal{W}}F}}{m})$ whereas the bound in [1] is $\\mathcal{O}(\\frac{1}{n}+F(w^T)-\\min\\_{\\mathcal{W}}F))$. It is evident that our second term $\\frac{\\sqrt{F(w^0)-\\min\\_{\\mathcal{W}}F}}{m})$ is significantly lower than the corresponding  $F(w^T)-\\min\\_{\\mathcal{W}}F$ in [1]. Although our first term is looser than theirs $\\frac{1}{n}$, we argue that it is intuitive outcome.. In particular, this statement, \"For sufficiently large $Q$, $\\alpha\\leq \\frac{1}{QL}$ yields negligibly small learning rate, implying the inner update will be close to initialisation $w\\_{\\tau\\_i,0}^t$\" reinforces the validity of our bound. Intuitively, if the inner-loop update is too close to the initialization, the model fails to learn task-specific information effectively, leading to larger generalization error—similar to the behavior observed when $Q=1$. However, the results in [1] ca not explain the trade-off relationship either is sufficiently small or large $Q$ when $\\alpha\\leq \\frac{1}{QL}$, which means their results is not robust. \n\n**Question 1:** The core idea of MAML involves applying a few gradient descent steps on a small number of training samples to adapt the meta-model to test tasks. Therefore, We use Fig. 1 to illustrate an interesting phenomenon, i.e., a larger number of $Q$ is not suitable for MAML. To address the reviewers' concerns, we also provide further experiment for MAML when $Q>20$. The experiment is also conducted on the Omniglot dataset. We employ a $3\\times 3$ CNN and use the Cross-Entropy Loss as the loss function. For each training task, the number of support samples and query samples to $n^{\\rm{tr}}=1$ and $n^{\\rm{ts}}=5$, respectively. For each test task, we use $n^{\\rm{tr}}=1, n^{\\rm{ts}}=15$. In addition, each task is formulated as a $5$-way classification problem.\n\n| Q=2    | Q=5    | Q=10   | Q=20   | Q=30   | Q=40   | Q=50   |\n| ------ | ------ | ------ | ------ | ------ | ------ | ------ |\n| 0.6429 | 0.2868 | 0.2300 | 0.2417 | 0.2423 | 0.2541 | 0.2622 |\n\n**Question 2:** In theoretical analysis, two identical assumptions do not necessarily imply that the difficulty and process of derivation under the two settings is the same, especially in stability analysis. In fact, the condition of some parameter is stricter in non-convex. For instance, under convex setting, the step size $\\eta\\leq\\frac{1}{L\\_Q}$ with $L\\_Q = \\alpha\\rho Q(1+\\alpha L)^{Q-1} + (1+\\alpha L)^{Q}L$. In contrast, under non-convex setting, the step size $\\eta\\_t \\leq \\frac{c}{t}$ and $c\\leq\\min\\{\\frac{1}{L_Q},\\frac{1}{4(2L\\_Q\\rm{ln}(T))^2}\\}$ with $L\\_Q= \\frac{3\\rho(1+\\alpha L)^{2(Q-1)}}{L} + (1+\\alpha L)^{Q}L$.\n\n**Question 4:** Referring to the Eq (1) and Eq(2) in this paper, we can get $F(w)=\\frac{1}{m}\\sum\\_{i=1}^{m} F\\_i(w)=\\frac{1}{m}\\sum\\_{i=1}^{m} \\mathbb{E}\\_{\\mathcal{D}\\_i} \\mathbb{E}\\_{z\\in\\mathcal{P}\\_i}[\\ell(w\\_{\\mathcal{T}\\_{i}}^Q(w, \\mathcal{D}\\_i),z]$. Then we can find that $F_i(w)$ is benefited by $Q$ due to $w\\_{\\tau\\_i}^Q(w, \\mathcal{D}\\_i)=w-\\alpha\\sum\\_{q=0}^{Q-1}\\nabla \\widehat{\\mathcal{L}}\\_i( w\\_{\\tau\\_i}^q,\\mathcal{D}\\_i)$. Therefore, both $F(w^T)-\\min\\_{\\mathcal{W}}F$ and $F(w^0)-\\min\\_{\\mathcal{W}}F$ can be improved by increasing $Q$. The main difference is that $F(w^T)-\\min\\_{\\mathcal{W}}F$ can also be improved by the number of outer-loop iterations $T$.\n\n**Question 5:** Since our work primarily focuses on generalization error, while other studies emphasize transfer error, it is difficult to make a fair comparison regarding the tightness of the bounds. Generalization error measures the discrepancy between training tasks and test tasks, making it a key metric for evaluating the practical capability of meta-learning algorithms.\n\n[1] Zhou, P., Zou, Y., Yuan, X.T., Feng, J., Xiong, C. and Hoi, S., 2021, December. Task similarity aware meta learning: Theory-inspired improvement on MAML. In Uncertainty in artificial intelligence (pp. 23-33). PMLR."}]

Author rebuttal:

Meta-review: This paper theoretically evaluates how the inner update in meta-learning affects generalization performance. This work centers on two representative approaches to the inner update, the gradient descent framework (GDF) and the proximal descent framework (PDF). From the derived theory, the authors show that the trade-off present in the GDF does not arise in PDF. As a consequence, they propose a new reweighting-style meta-objective and demonstrate its empirical effectiveness.

Multiple reviewers positively noted the theoretical significance of analyzing the impact of inner-loop optimization on generalization within two unified frameworks (GDF and PDF) and of providing stability-based generalization bounds. The analysis is sound, and by focusing on GDF and PDF the theory applies to a broad range of existing meta-learning algorithms, which is valuable. The consistency between the empirical results and the theory is also appreciated. Consequently, the proposed loss function is recognized as a contribution to the community.

Reviewers pointed out that the statistical significance of the experiments was not fully validated; in response, the authors committed to multi-seed runs and adding error bars. Several reviewers also found the logical development not always clear; the authors addressed this reasonably in the rebuttal and they promised the update of the manuscript. Although some questioned the novelty of studying the effect of inner-loop optimization on generalization, the authors argued that their analysis of GDF and PDF, which identifies the trade-offs over a wide range of existing methods, is valuable. A remaining concern is that the theoretical analysis provides only upper bounds, but this is a very common approach in generalization-error analysis and is not considered a critical flaw.

Taken together, the paper offers a novel theoretical analysis and a valuable examination of how inner updates influence generalization from the perspectives of GDF and PDF. It introduces a new loss function grounded in solid theory, and its performance is supported by experiments. Since the authors have responded reasonably to the main concerns, I recommend acceptance.
Decision: Accept (poster)