Venue: ICLR 2023 TinyPapers

Reviews:

final_score: rating: Great Start (GS): a submission which meets some of the reviewing criteria but has room for improvement
review_content: summary_of_contributions: This paper studies the convergence of the last iterate of SAM, rather than the time average convergence. The results show that the gradient of the deterministic version of SAM converges to zero as $t$ approaches infinity.
strengths_and_weaknesses: # Strengths
- The paper studies an important question regarding the convergence of the last iterate of SAM.
- The paper provides a discussion of the assumptions made.

# Weaknesses
- The paper only examines the convergence of the deterministic version of SAM and does not provide information on the **convergence rate**.
-  The paper does not investigate the convergence of the stochastic version of SAM, which may limit the practical applicability of the theory.
- The clarity of the paper is acceptable. However, there are some informal expressions used throughout the text. See the suggestions below.
-  While the proofs appear correct, there are some redundant steps on page 6.

suggested_changes: - The writing could be improved. Some expressions are informal. For example, the sentence "The proof of Corollary 2.1 is straightforward by the Lemma" could be revised to "The proof of Corollary 2.1 follows easily from Lemma 2." The expression "we can get the theorem 1" could be revised to  "we can obtain Theorem 1".
- The intuition behind assumption (A.4) could be discussed with more detail.
review_title: null

final_score: rating: High Potential (HP): a submission which meets the reviewing criteria and has potential to make an impact on the field
review_content: summary_of_contributions: Sharpness-aware optimization, where flatness of the minima is encouraged while optimizing any (potentially non-convex) function, is studied w.r.t. convergence of a certain algorithm. In particular, the two-step update rule of Foret et al. (2021) is analyzed, and last iterate is shown to converge a.s.
strengths_and_weaknesses: Strengths:
- Paper is very well-written. Sufficient amount of detail and context is provided to gauge impact and correctness.
- The writing looks well-aware of the contributions and limitations, and presents them upfront.
- The results seem correct as far as I checked.
- All relevant details are in the appendix.

Weaknesses:
- Some minor writing edits would be good (see suggestions).
suggested_changes: - It is a bit unclear from the write-up where "SAM" is used to refer to the optimization problem or the particular algorithm (two-step update rule of Foret et al., 2021) for it. I recommend reading through various usage of the term to try to make it clearer to the reader to avoid potential confusion/ambiguity in some places.
- Adding 1-2 lines on key insights in proving Theorem 1 would be a nice addition.

Optional minor writing suggestions:
- flatter it shapes --> flatter it is shaped
- typo in footnote 3: iterature --> literature
- in the main body replace "_Proof._" by "_Proof sketch._", and add □ ("tombstone"/"qed") symbol at the end of proofs.

Optional things to think about:
- Does this have connections to adversarially robust training under L2-ball perturbations? It seems flatter minima should be more robust.
review_title: null

Meta-review: metareview: Pros
- The paper is reasonably clear, correct and all proof details as well as assumptions needed are included (therefore claims are verifiable).

Cons
- As the reviewers note, there are some limits to the usefulness of the setup considered in the paper. It is still a step in the right direction, but comments from the reviewers can be useful for identifying useful future extensions of the work.

confidence: 4: The area chair is confident but not absolutely certain
Decision: Invite to present