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: The problem of learning problem-specific learning schedules for non-negative matrix factorization is studied. Some theoretical upper bounds for the optimal learning rate schedule are studied, and a reinforcement learning based approach is implemented and empirically studied.
strengths_and_weaknesses: Strengths:
- Important problem with lots of applications
- Great approach to try to obtain theoretical bounds, coupled with experiments

Weaknesses:
- Theoretical bounds need more precision
- Nice empirical demonstration, but more details about the experiment could help reproducibility
suggested_changes: Lemmas/Theorems should be stated precisely:
- Lemma A.1: statement should include convex w.r.t. which argument(s) since F is a function with two arguments.
- Lemma A.2: again argument, and also Lipschitz constant should come in lemma statement. But why is L even constant (an assumption?).

Section 2: Assumptions for any theorems should be clearly stated in the main body.

Experiments:
- How are the sparse matrices initialized?
- Details of training the RL agent are missing.
- Nice to haves: link to code; more discussion on curves in Appendix B.

Typo:
section ?? in last line Pg 1.
review_title: null

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: The work studies finding the optimal learning rate when alternating gradient descent is used to solve the non-negative matrix factorization problem. Mathematically the authors show a range of the learning rate lead to the algorithm convergence. The authors also propose a RL based solution to tune the learning rate and conduct experiments to visualize its performance.
strengths_and_weaknesses: Using learning-based methods to select a good learning rate sequence is an interesting and well-motivated problem. However, the work is still in a very early stage. The theoretical result presented in Theorem A.3 follows from simple Lipschitz continuity and PL condition and is quite elementary. The experimental setup and results are not well explained. I believe major work needs to be performed before this paper can become a good contribution to the community.
suggested_changes: 1. What exactly is the loss function $F$? Is it $F(W,H)=||V-WH||_F^2$? If so, we should be able to compute the exact gradient of $F$ with respect to the two factors, which means that we can perform a deterministic gradient descent algorithm. The paper claims that SGD is performed, but I do not see how the stochasticity or noise arises in the gradient.

2. How the PL condition is established needs to be clearly explained. $F^*$ is probably the optimal function value of $\min_{W,H}F(W,H)$ but is not defined.

3. In RL application paper, it is critical to clearly define the state space, action space, and reward function. Is the action space continuous in this problem or is it discretized? The description of the reward function sounds like rewards are only given at the end of the episode, which may be very sparse. To improve RL training, I suggest designing a denser reward signal.

4. The takeaway from Figure 1 is not clear. Why is the performance of the RL policy fluctuating?

5. The operator P is undefined in the second paragraph of the introduction, which I believe is the projection to $\mathbb{R}_{+}^{m\times r}$.
review_title: null

Meta-review: metareview: Strengths:
- interesting and well-motivated problem
- great principled attempt at the problem from both theoretical and empirical angles

Drawbacks:
- experimental setup and results are not well explained
- some theoretical results need more clarification (see reviewer comments)

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