__ Summary and Contributions__: In this paper, the authors address the problem of identifying the optimal mixture of distributions over multiple training sets, that has maximal performance on a validation set. The difficulty are that the distributions of data might differ between each set, which is compounded by the presence of unobserved variables. The authors introduce a new algorithm,Mix&Match, based on hierarchical bandits and stochastic gradient descent, that solves this problem, and prove an upper bound on the simple regret that it incurs. Mix&Match is then empirically evaluated and compared to existing methods.

__ Strengths__: - The problem of identifying the best mixture in the presence of unobserved variables is new (as far as I know) and is interesting.
- The upper bound on the regret of the SGD (Theorem 2/5) is an interesting contribution
- The solution proposed by the authors is insightful
- Overall the paper is well written and pleasant to read

__ Weaknesses__: - The hypotheses required by Mix & Match are very strong. Proposition 1 assume thats there exists a mixture of the training distribution that perfectly matches the validation distribution. This seems very limiting to me, in particular as it requires the different training set to be correctly split -- which is impossible to verify in practice, since there are unobserved variables.
- Similarly, the clustering argument of the authors (line 184 - 186) is unclear to me due to the unobserved variables.
- In the experiment section, Mix& Match appears to frequently outperform the Genie algorithm, depending on the SGD budget. But Genie is an "oracle" method, that has access to the true mixture, so how is it possible ?
- Figure 1a is very difficult to read

__ Correctness__: All the claims appear to be correct, up to many typos that makes the statements difficult to follow. For instance, Theorem 2 appears to prove a lower bound on the regret instead of an upper bound, and the definition of \beta-smooth in the appendix is false.

__ Clarity__: The paper is well written, but there are multiple errors/ typos in the statements.

__ Relation to Prior Work__: The relationship between this work and the literature are clearly discussed.

__ Reproducibility__: Yes

__ Additional Feedback__: [After Rebuttal] Thank you for the clarifications. The authors adressed some of my concerns, however I think that the argument of "simply adding more distributions and re-run" does not fully address my main concern as this approach drastically increases the model space. Therefore I choose to not increase my score.

__ Summary and Contributions__: This paper considers the problem of dealing with the covariate shift in machine learning. The setup is that we are given a set of training data distributions and a validation data distribution. The key assumption is that the validation dataset is a covariate-shifted version that takes the form of a particular mixture of train data distribution. This is defined in a way that allows unobserved features. This setting is more general and flexible than previously proposed ones because those can fail in this setup. The authors propose an algorithm that combines SGD with tree search, and prove its convergence.

__ Strengths__: The problem of covariate shift (or distributional robustness) is important, and the authors provide one method with the computational complexity in mind. I find it relevant and important, and the solution seems novel. The new h.p. bound on SGD is interesting technically.

__ Weaknesses__: The particular formulation of forming a tree search may not be the best thing to do, but it seems quite competitive among available algorithms as of now.

__ Correctness__: Yes

__ Clarity__: Yes. I found numerous discussions to support the claim, defend the weakness, and distill the contribution.

__ Relation to Prior Work__: yes.

__ Reproducibility__: Yes

__ Additional Feedback__: (after the rebuttal)
I've read the rebuttal and keep the same score. The authors responded that "The emphasis in Theorem 2 is on the scaling with respect to d0 and G∗", but it's not quite clear in the current form with G undefined, etc. I suggest that the authors find ways to show distill the dependence on d_0 and G^*, and connect it to the exponentially decaying term in d_0^2 to make the point clear.
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The paper is well-written. I found the problem setting quite interesting. The h.p. bound on the SGD convergence where the dependence on the stochastic gradient norm only at the optimal solution is interesting (though I am not expert in these algorithms).
Q: Could there be a way to form some sort of alternating minimization between w and alpha? If theoretical statements on this are hard, at least can we see how it works empirically?
The form of Theorem 2: I would like to see a simpler and more modular form of the statement. First, I would not directly use \Lambda here so that the contribution on SGD itself is decoupled from the overall computational budget which is needed only for the covariate shift problem here. Could the authors restate it so we can clearly see the dependence on \kappa, \delta (the failure rate of the probabilistic statement "w.p. >= 1-\dt"), t, ||w_0 - w^*||, and \mathcal{G}_* ?
comments
• Abstract: To me, it was hard to identify the 'interesting part' from the abstract. I would mention that the presence of unobserved feature is something that was not dealt with before, but more general and cannot be handle from existing work.
• L515: "Definition Definition 1"
• L521: isn't the inequality the other direction..?
• Isn't the strong convexity assumption too strong, given that we assume those datasets are all from a different distribution? In the linear model with squared loss, the data must span the whole space. We also must know \mu to tune the learning rate, and the convergence is slow when \mu is small. When dealing with one dataset, we can preprocess it well, but for multiple datasets, I am not sure if we can do it. Please comment on this.
• Theorem 2: It seems to me that the dependence on large \kappa means faster convergence at first sight, because the convergence scales like poly(1/E)? Please specify the dependence on \kappa.
• Smoothness of G(): I was expecting to see something that bounds || \nabla G(\alpha) - \nabla(\alpha') || to check the smoothness. But I can only find something about |G(\alpha) - G(\alpha')|, which is more like Lipschitzness. Could the authors explain more on this?
• L260: "that that"
• L266: I am not sure if the explanation here suffices to explain why the error can decay exponentially in h+1. Explain a bit more please.
• Theorem 2: Is G(.,.) explained somewhere? Note that G() was used before, so it seems like a notation clash.
• Algorithm 1: Was T_t defined before? Also, I don't know where this variable t came from. Is this some sort of iteration number?
• L251: doesn't this comment conflict with the one in L289 (and the choice of \lambda(h) = \lambda in the appendix)
• I was surprised to see the naïve implementation of mix&match is not shown. How does it work and why was it excluded? If there is a reason why it did not work well, what's the authors' guess? I think this is something worth being discussed in the main text.

__ Summary and Contributions__: Given access to K datasets with distributions $(p_i)_{i=1}^K$, the authors consider the problem of finding the best mixture of these distributions, such that a learning algorithm trained on the mixture distribution performs well on a target distribution (p^{(te)}).
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AFTER FEEDBACK: I thank the authors for their clarifications regarding the requirement on the validation set size.
Based on the authors' feedback and the comments of other reviewers I am happy to upgrade my score to 7.
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Under the assumption that the target distribution can be written as a mixture of the K given distributions, the authors frame the problem as a noisy black-box optimization problem over the (K-1) dimensional simplex. For every value of a mixture vector $\alpha$ in the simplex, the proposed approach runs SGD for a certain number of steps.
Under strong convexity and smoothness assumptions on the loss function, the authors show that the optimization objective function is Lipschitz continuous. Next, the authors derive a bound on the suboptimality of a solution after $t$ steps of running SGD. These two results together allow the authors to use the ideas from Lipschitz black-box function optimization algorithms, such as HOO of Bubeck et al. 2011, to design an algorithm to find the best mixture vector given a total computational budget of $\Lambda$ SGD steps.
The authors derive a bound on the Simple regret of the proposed algorithm, and also demonstrate its good performance over various baselines in several empirical tasks.

__ Strengths__: The key technical contribution of this work, in my opinion, is to set up the task of estimating the best mixture vector $\alpha^*$ as that of optimizing a Lipschitz continuous black-box function which can be accessed via 'noisy' observations, where the amount of 'noise' can be controlled (i.e., by varying the number of SGD steps). This involves two results:
1. The first result (Theorem~1 in Section~5 and Theorem~4 and Corollary~2 in Appendix~C) demonstrate the Lipschitz continuity of the optimal weights (w^*(\alpha)) and the optimization objective function $G(\alpha)$.
2. An improved high probability bound on the potential function $\|w_t - w^*\|_2^2$ after $t$ steps of SGD.
I feel that the proofs of these two results are the main technical contributions of this work. Once these results have been established the design of the algorithm (Mix&Match) and its analysis follow along somewhat standard lines.
Besides the theoretical contributions, the authors also demonstrate the good performance of the proposed algorithm empirically on two datasets.

__ Weaknesses__: The setup of the problem seems a bit contrived to me. The authors assume that the size of the validation set is smaller than the computational budget and hence it is infeasible to perform SGD directly on the validation set. At the same time, they also assume that the validation set is large enough that the $F^{(te)}(.)$ value can be obtained accurately *uniformly for all w* trained after at least one SGD step.
Since most non-trivial ML models have very large parameter size (i.e., w lies in high dimensions) and the fact that the SGD guarantees are dimension-free, it seems to me that the number of samples required to ensure the accurate oracle access to $F^{(te)}$ might be larger than those required to train a model using SGD.
It would be very helpful if the authors justify the above two seemingly contradictory assumptions on the size of the validation dataset, in case I have misinterpreted them.

__ Correctness__: I checked the proof of the continuity results (Theorem~4 and Corollary~2 in Appendix~C) and they look correct to me. I did not have the time to go through the other proofs in details.
The methodology used in the experiments look sound to me. The authors describe the details of all the baseline methods used, and report their performance along with the appropriate error bars.

__ Clarity__: Yes.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: Minor Comments:
Is there a typo in the statement of Theorem~2? Shouldn't it be $\|w_{t+1} - w^*\|_2^2$ instead of $\|w_{t+1}-w_0\|_2^2$?
In Assumption~1, is F^{(\alpha)} assumed to be a strongly-convex function for all possible choices of (p_i)_{i=1}^K? It would be helpful if the authors include some examples to demonstrate the gain in generality achieved by making this assumption, in comparison to assuming that the individual loss functions $f(\cdot, z)$ is convex.

__ Summary and Contributions__: The paper proposes a new algorithm for one particular setup of transfer learning: the training data is partitioned into N distributions and there is a small validation set of the target distribution; the goal is to do well on the validation set assuming its distribution is well modeled as an unknown mixture of the training distributions. The method proposed in this paper does provably almost as well as any mixture (it presents a regret bound). The method works by using a hierarchical partition of the simplex of distributions over the datasets and using warm-started SGD on mixture distributions to evaluate candidate distributions in a tree search algorithm.

__ Strengths__: At first value the result is surprisingly powerful; there are many transfer learning settings which are approximately solvable as mixtures of known distributions and it's rare to find mathematical guarantees of this strength.
The algorithm itself is fairly simple; it repeatedly picks a node in the search tree to explore and explores it by doing a small number of SGD updates and computing the validation loss of those models.
In the experiments the proposed algorithm performs in practice in line with the theoretical expectations, coming close to the oracle performance when the distribution is known and outperforming reasonable baselines when the distribution isn't known.

__ Weaknesses__: The way the paper is written somewhat hides the exponential complexity of the algorithm as a function of the number of training distributions. This is hidden AFAICT in the near-optimality dimension of the loss function over mixtures. The paper also glosses over the procedure for bisecting the simplex, which is a key part of the algorithm since the bisection has to preserve the metric very well for the results to be practically useful (or there will be a lot of backtracking during the search).
This lack of clarity about the algorithm would make it quite hard to reproduce the results without going over the code. The code was provided, however.
The experimental results are somewhat suspicious. Specifically, in figure 1a (though the "1" is not present in the text) we see in much of the budget space the non-optimal strategy "OnlyCT" which trains on data from only one domain outperforming the strategy "Genie" which is the oracle strategy with knowledge of the true data distribution (and similarly you can see one mixmatch variant outperforming genie). The text makes no attempt to explain this counterintuitive result; my best guess is that the baselines were not properly tuned.
Not tuning the baselines in figure 1a makes me not trust the experimental results in figure 1a and also in figure 1b since it's not clear whether those trend lines would revert if proper experimental practices were adopted.

__ Correctness__: As far as I can tell the paper is correct, though I did not thoroughly verify the proofs as they run through many pages of the supplemental material. The overall gist of the results is believable, though.
I wish the paper was a little more clear about the limits of the applicability of this method. Things like what are the conditions on the data distribution for the performance with a reasonable budget to be good? Why would a practitioner not always divide a training dataset of N examples into N independent domains for maximum flexibility?

__ Clarity__: The paper is generally well written.

__ Relation to Prior Work__: No issues here.

__ Reproducibility__: No

__ Additional Feedback__: