We study online convex optimization in a setting where the learner seeks to minimize the sum of a per-round hitting cost and a movement cost which is incurred when changing decisions between rounds.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. It presents a slight modification of the NAC algorithm, where the original algorithm is a special case which is called forgetful NAC. The authors show that forget full Nac and optimistic policy iteration are equivalent. The authors also present a non-optimality result for soft-greedy Gibbs distribution, I.e., the optimal solution is not a fixed point of the policy iteration algorithm. I liked the unified view on both type of algorithms.

The authors propose an efficient scheme to solve LP relaxations of combinatorial optimization problems. Their contribution is a novel scheme and analysis that takes into account the original goal of constructing an integral feasible solution from the relaxed solution. They prove that an approximate solution is sufficient to construct an integral alpha-approximate solution to the vertex cover problem. They also prove a convergence result for an algorithm solving a suitably constructed QP approximation to a general standard form LP problem. The proposed method is evaluated experimentally on a number of combinatorial optimization problems and shown to be competitive with Cplex, a state-of-the-art LP solver.

Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning.

The Frank-Wolfe algorithm is a classic method for constrained optimization problems.

The Frank-Wolfe algorithm is a classic method for constrained optimization problems.

We thank the reviewers for their constructive feedback on our paper. We especially appreciate our reviewers' conviction Reviewers 1 and 2 found some of our explanations of ASA form and DPP difficult to follow. We will also explain the motivation for our ruleset (reviewer 1's guess is essentially correct). This is what we meant by our vague phrasing "jointly DCP ... [with] one We will separately explain how to reduce certain expressions in which parameters are multiplied together ( e.g., We will clarify this point. In the revision, we will make sure to clearly explain this.

We propose new continuous-time formulations for first-order stochastic optimization algorithms such as mini-batch gradient descent and variance-reduced methods.

Here we introduce a few more notations. In this section we provide the details of the subroutines used in the training algorithm for Frugal ML . There are 3 steps for solving problem 3.3. T o solve Problem 3.2, let us first denote C.1 Helpful Lemmas W e first provide some useful lemmas throughout this section. Lemma 4. Suppose the linear optimization problem max Lemma 5. Let F (w) be the optimal value of the linear optimization problem max Thus, the objective value must be smaller than the optimal one, i.e., Given the expected accuracy and cost provided by Lemma 2, the problem 3.1 becomes Let us first consider the expected accuracy .

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