Paper:
We are interested in the optimization of a high-dimensional function when only function evaluations are possible. Although this derivative-free setting arises in many applications, existing methods suffer from high sample complexity since their sample complexity depend on problem dimensionality, in contrast to the dimensionality-independent rates of first-order methods. The recent success of deep learning methods suggests that many data modalities lie on low-dimensional manifolds that can be represented by deep nonlinear models. Based on this observation, we consider derivative-free optimization of functions defined on lowdimensional manifolds. We develop an online learning approach that learns this manifold while performing the optimization. In other words, we jointly learn the manifold and optimize the function. Our analysis suggests that the proposed method significantly reduces sample complexity. We empirically evaluate the presented method on continuous optimization benchmarks and high-dimensional continuous control problems. Our method achieves significantly lower sample complexity than Augmented Random Search and other derivative-free optimization algorithms.

Review:
Contributions: 
	-Authors have proposed a methodology to optimise high dimensional functions in a derivative-free setup by reducing the sample complexity by simultaneously learning and optimising the low dimensional manifolds for the given high dimensional problem. 
Although, performing dimensionality reduction to learn the low dimensional manifolds is popular in the research community, the extensions made and the approach authors have considered seems to be novel.
Comments:
	
	- Authors have talked about the utilization of domain knowledge on the geometry of the problem. How feasible is to expect the availability of the domain knowledge? Authors have not discussed the downsides of the proposed method if the domain knowledge is not available, and a possible strategy to overcome the same.
	- Authors have said that they are specifically interested in random search methods. Is there any motivating reason to stick to the random search methods? Why not consider other sample efficient search methods? 
	“…….random search scale linearly with the dimensions”, why one should not consider other sample efficient methods that grow sub-linearly as against random search? 
Srinivas, N., Krause, A., Kakade, S. M., and Seeger, M. Gaussian process optimization in the bandit setting: No regret and experimental design. International Conference on Machine Learning, 2010
	- Please derive Lemma 1 in the appendix for the sake of completeness.
	- I am missing a discussion about manifold parameters like “λ” in the important equations. 
	- Authors have made a strong claim that neural networks can easily fit any training data, but it may be not be true for many datasets.
	 - Authors have claimed that they have fast and no-regret learning by selecting mixing weight β=1/d. Author might want to discuss more on this as this is an important metric.
	 - “ …. total time spent on learning the manifold is negligible…. ” – any supporting results for this claim.
	 - “…….communication cost from d+2k to d+2k+kd… ” – curious to know if there is any metric like wall-clock time to talk about the optimisation time.
	- Authors have restricted the comparisons to only three important methods, but it is always comparing with other baselines in the same line. Authors should consider Bayesian optimisation as it is a good candidate for the performance comparison, even though the researchers are interested only in random search methods (just like CMA–ES).
Kirschner, Johannes, Mojmír Mutný, Nicole Hiller, Rasmus Ischebeck, and Andreas Krause. "Adaptive and Safe Bayesian Optimization in High Dimensions via One-Dimensional Subspaces." arXiv preprint arXiv:1902.03229 (2019).
	- It is seen from the results that the proposed method is not performing better for low dimensional problem like “Swimmer” function. But according to the initial claim, method was supposed to work better in low dimensional problems. Is it because of the fact that the problem space is not drawn from high dimensional data distributions? 
	- “…..improvement is significant for high dimensional problems” – It will be better if the authors compare their proposed method with some more derivative-free optimisers that are proven to be good in high dimensions (like high dimension Bayesian optimisation).
	-  “The no-learning baseline outperforms random search ……….” – this statement is not very clear, does it mean like the proposed method works only when the problem is reduced from higher dimensions to lower dimensions and not on the lower dimensional problem itself?
	- “Performance profiles represent how frequently a method is within the distance T of optimality” – Any thumb rule considered for the choice of T?. Can we think of any relation with standard metrics like simple regret or cumulative regret that are used to measure the optimisation performance?
	- “Although BO methods typically do not scale…… ” – Authors have made a strong assumption here. In the literature, we see active research in the context of high dimensional optimisation.
Rana, Santu, Cheng Li, Sunil Gupta, Vu Nguyen, and Svetha Venkatesh. "High dimensional Bayesian optimization with elastic gaussian process." In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2883-2891. JMLR. org, 2017.
	
Minor Issues:
	“Improve the sample complexity” may not convey the meaning very clearly to the readers, something like “Improve the sample efficiency” or “Reduce the sample complexity” would add more clarity.
	Inconsistency in the terms used in Proposition 1 and Proposition 2. What does “I” signify in the formula? Was that supposed to be “t”?
	Even though the constants are mentioned in the appendix, it is always better to mention the constants used in the algorithm like “α“ as step size for quick understanding.
	“Follow The Regularized Leader (FTRL)” is more appropriate than “follow the regularized leader (FTRL)”
	“noise table in pre-processing” – Should it mean something relevant to the paper? 
	“We use widely used…..” – may be consider rephrasing the sentence here
	“treshold” – Typo in Table 1
	Y – Axis in Figure 2 is missing
	Appendix B :
	“We also perform grid search …. “ would look better
	MuJoCo Experiments – is the parameter space continuous and what is the search space considered for n, α and δ. Do we deal with smaller search spaces in every problem? Any other way of searching the parameter space to further improve the efficiency?