Title: On the Stability and Generalization of Meta-Learning: the Impact of Inner-Levels
Venue: NeurIPS 2025
Accepted: yes
Keywords: Stability and Generalization

Meta-Learning
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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.\n\nQUESTIONS:\nI 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.\n\nLIMITATIONS:\nThe work is purely theoretical in nature, and does not pose any negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-23"}

comments: 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.

STRENGTHS AND WEAKNESSES:
Pros:
- Different from previous work, this work dives into the impact of inner-loop optimizations on the generalization in meta-learning algorithms.
- This work achieves some interesting insights from the perspective of theory.
- The paper starts from an interesting phenomenon of the generalization error and then develops many valuable insights regarding meta-learning, which is impressive.
- The results reported in this paper are good.

Cons:
From my perspective, I do not think this work has obvious drawbacks. I like this work.

QUESTIONS:
- Could you provide some explanations regarding Eq. (2)? I do not really understand the formulation.
- 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
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-28

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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).\n\nSTRENGTHS AND WEAKNESSES:\nTo 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.\n\nQUESTIONS:\n- 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\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n(+) \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.\n\nQUESTIONS:\nSee above.\n\nLIMITATIONS:\nSee above.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}