Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: This paper studies how to mitigate the gradient bias of multiobjective learning. They use the idea of momentum-based gradient bias correction to substitute the gradient when finding the common gradient.

I have had some earlier attempts on this problem but didn't end up very successful and I am very excited that someone also works on this problem. However, I find some potential mistakes in a very core Lemma (Lemma 4) used in this paper and thus I doubt the correctness of the result in this paper.
strength_and_weaknesses: Strength:
This is an open and important problem in MOO.

Questions:

I am *pretty* unsure about the correctness of Lemma 4 which indicates the Lipschitz continuity of the lambda. Dontchev & Rockafellar, 2009, Theorem 2F7 does not imply this Lemma.

Using the notation in Dontchev & Rockafellar, 2009, Theorem 2F7, that result holds locally (given $x$, we have a $\mu$) but when we approach the Pareto stationary point ($x$ changes), that $\mu$ might change and we cannot have a global bound of the $\mu$. In other words, the Lipschitz constant of $\lambda$ should actually increase (to infinite) when we are closer to the stationary.

The lambda can be viewed as a solution map of a dual problem of a quadratic programming problem and its Lipschitzness is a very classic problem. To my best knowledge, all of the existing estimations of the Lipschitz are local, not global. Some literatures [1,2,3]


[1] On the continuity of the minimum in Parametric quadratic programs.
[2] On the Lipschitz Behavior of Optimal Solutions in Parametric Problems of Quadratic Optimization and Linear Complementarity
[3] Sufficient Conditions for the Lipschitz Continuity of QP-based Multi-Objective Control of Humanoid Robots
clarity,_quality,_novelty_and_reproducibility: Clarity: good
Quality: unsure due to the potential issue
Novelty: unsure due to the potential issue
summary_of_the_review: My main concern is the correct of a core Lemma. My current rating is mainly due to that issue if this can be addressed during the rebuttal period, I am happy to raise the score to 6 or 8. If it can't be addressed, this paper is not ready to publish.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: This paper studied the multi-objective optimization problem. Here, it reformulated it as a bilevel optimization problem and then leveraged the tools in bilevel optimization to establish the convergence rate. Since the main proof is adapted from bilevel optimization, the novelty is incremental. 
strength_and_weaknesses: Strength:
1. The problem studied is important.
2. The writing is easy to follow. 

Weakness:
1. It is not clear why multi-objective optimization problems should be formulated as bilevel optimization problems. This critical point should be discussed clearly. At present, it is a weird combination of those two problems. 
2. The theoretical analysis is mainly adapted from the bilevel optimization literature. This makes the contribution incremental. The authors should highlight the challenges and the different parts compared with existing bilevel optimization literature.
3. This paper assumes that the loss function is Lipschitz continuous. With this strong assumption, existing bilevel optimization literature can achieve the convergence rate $O(K^{-1/2})$, which is much stronger than the convergence rate in this paper. 
4. In lemma 2, the number of objectives $M$ should depend on the number of iterations. This is definitely NOT feasible in real-world applications. 
clarity,_quality,_novelty_and_reproducibility: Clarity: good
Quality: Neutral
Novelty: good
Reproducibility: N/A
summary_of_the_review: This paper studied the multi-objective optimization problem. Here, it reformulated it as a bilevel optimization problem and then leveraged the tools in bilevel optimization to establish the convergence rate. Since the main proof is adapted from bilevel optimization, the novelty is good.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: The author(s) studied the multi-objective optmization problem and proposed a new algorithm called MoCo that provably converges to a Pareto stationary point in the stochastic gradient setting. Experiments on both toy and real data are conducted to support the proposed method.
strength_and_weaknesses: Strengths:
- The paper is well-motivated and well-written, the story line and analysis is easy to follow for experienced readers;
- The MoCo algorithm proposed in this work is the fisrt stochastic multi-gradient method that can converge to Pareto stationary point under mild condition, which is a decent contribution to the field.

Weakness:
- I did not find any obvious weakness of the paper. The notation is a little bit heavy, maybe it is better to add a notation subsection.
clarity,_quality,_novelty_and_reproducibility: The paper is well-written; the result is novel to my knowledge, reproducbility is unclear to me.
summary_of_the_review: Overall, this paper makes a decent contribution to the field; the author(s) give the first stochastic multi-gradient method can converges under mild conditions. I did not carefully check the proofs, the paper is above the bar of acceptence if other reviewers all agree that the proof is correct.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: The paper is proposing an analyzing a stochastic multi-objective optimization method. Existing multi-objective optimizers are typically posed as solving a meta-optimization problem using gradients. However, these gradients are stochastic resulting a bias in the final solution. The proposed method uses an adaptive approach to track optimization solution through time effectively alleviating bias. Method is further fully analyzed in convex and non-convex setting. Moreover, it is evaluated in various interesting applications including supervised learning and RL. Hence, it is a solid work on an important problem. Moreover, three experts reviewed the paper independently and raised various issues which are all addressed by the authors during the discussion phase. In their final scores, all reviewers agree on the quality and acceptance.
justification_for_why_not_lower_score: The paper technically sounds and solving an important problem. Moreover, the contributions are both theoretical and empirical. Hence, it is clearly a large interest within the community. I believe there is some significant knowledge for majority of the audience, specifically:
- ML Practitioners: The resulting algorithm is effective in various practical problems including scene understanding and reinforcement learning both are active research sub-fields.
- Theory: The theoretical advancement is non-straightforward and relates to both online learning and optimization theory.
- Multi-objective optimization/Multi-task learning: This is the core subfield where the paper lies. It would have a major impact on this core sub-field.
Decision: Accept: notable-top-5%