We thank the reviewers for their time, effort, and helpful feedback. We address individual comments below. The training loss in all of our examples can be written in the form of lines 17-18. We will include these additional references and add a proof to the appendix. Consider equation (6) when the assumptions of Section 2.1 apply, i.e., F (x) is a quadratic Our test can be considered as a more general version of testing whether the loss has reached a constant value.

Assumptions of fixed gradient noise and particular form of preconditioner (R2).

The k-sparse parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the k-sparse parity problem with sign stochastic gradient descent, a variant of stochastic gradient descent (SGD) on two-layer fullyconnected neural networks.

We study the evaluation of a policy under best-and worst-case perturbations to a Markov decision process (MDP), using transition observations from the original MDP, whether they are generated under the same or a different policy. This is an important problem when there is the possibility of a shift between historical and future environments, e.g.

We consider the problem of learning Bayesian networks optimally, when subject to background knowledge in the form of ancestral constraints. Our approach is based on a recently proposed framework for optimal structure learning based on non-decomposable scores, which is general enough to accommodate ancestral constraints. The proposed framework exploits oracles for learning structures using decomposable scores, which cannot accommodate ancestral constraints since they are non-decomposable. We show how to empower these oracles by passing them decomposable constraints that they can handle, which are inferred from ancestral constraints that they cannot handle. Empirically, we demonstrate that our approach can be orders-of-magnitude more efficient than alternative frameworks, such as those based on integer linear programming.

We study the problem of learning a series of tasks in a fully online Meta-Learning setting. The goal is to exploit similarities among the tasks to incrementally adapt an inner online algorithm in order to incur a low averaged cumulative error over the tasks. We focus on a family of inner algorithms based on a parametrized variant of online Mirror Descent. The inner algorithm is incrementally adapted by an online Mirror Descent meta-algorithm using the corresponding within-task minimum regularized empirical risk as the meta-loss. In order to keep the process fully online, we approximate the meta-subgradients by the online inner algorithm. An upper bound on the approximation error allows us to derive a cumulative error bound for the proposed method. Our analysis can also be converted to the statistical setting by online-to-batch arguments. We instantiate two examples of the framework in which the meta-parameter is either a common bias vector or feature map. Finally, preliminary numerical experiments confirm our theoretical findings.

We thank the reviewers for their valuable comments. We reply to each outstanding point below. A. One starting point in this direction would be to R. Related work in supplementary. A. We agree, we will move the Thanks also for the additional reference, which we will add to the paper. REVIEWER 2. R. I expected more meta-learning insights. A. The proposed framework provides us with a flexible meta-learning Furher examples in our framework are e.g.

In many scientific settings there is a need for adaptive experimental design to guide the process of identifying regions of the search space that contain as many true positives as possible subject to a low rate of false discoveries (i.e.

We now dive into these ideas more carefully and address specific comments by the reviewers. Reviewer #1: Thank you for the encouraging review. The proof relies heavily on the sampling scheme and the choice of estimators. Reviewer #3: Thank you for your comments. This also relates to your concerns in points 3 and 4. Bounds in statistical learning theory based on VC (local) dimensions They have received less attention in the bandit and active learning literature.

We provide a sample image 2 and its debate process.