This paper studies the online learning (stochastic and full-information) problem of bidding in multi commodity first price auctions. The paper introduces a polynomial time algorithm that achieves a regret of \sqrt{T log(T)} that has a near optimal dependence on T. The main challenge that the paper has to deal with is to find a computationally efficient algorithm for computing the best biding strategy given a known distribution.The authors first demonstrate that natural approaches for solving this problem exactly are not computationally efficient (this is not a formal np-hardness proof). Then, they provide a FPTAS for solving the problem using dynamic programming. Once they have a FPTAS for the offline problem, their results hold for the stochastic online setting using existing reductions. I haven't carefully looked in to the details of their analysis of the dynamic programming, but I think the effectiveness of it here is interesting and surprising -- specially given that the variation of this problem for the second price auctions is hard to approximate.

The paper concerns online linear optimization where at each trial, the player, prior to prediction, receives a hint about the loss function. The hint has a form of a unit vector which is weakly correlated with the loss vector (its angle's cosine with loss vector is at least alpha). The paper shows that: - When the set of feasible actions is strongly convex, there exists an algorithm which gets logarithmic regret (in T). The algorithm is obtained by a reduction to the online learning problem with exp-concave losses. The bound is unimprovable in general, as shown in the Lower Bounds section.

A) My main concern with this paper is with respect to the main results (Theorems 2 and 3). It seems the authors have not put sufficient care to the fact that \partial \hat{Q} in Algorithm 2 is a biased estimator of the true gradient \partial Q. Also, \hat{Q} defined in Line 189 depends on \hat{\pi} which is an estimate of \pi. Thus, a probabilistic proof would require to look at a conditional probability of the estimation of Q depending on the estimation of \pi. B) Regardless of the above, the final high probability statement in Theorems 2 and 3, seem to be missing the union bound of the error probability in Assumption 1.

This can be done in a batch or in an online fashion from the data. The authors provide an online approach for doing this and prove that it can achieve a regret O(1/sqrt(T)) as a function of data size T. 1. I believe the paper is not written very well. For example, the underlying idea of inverse optimization is not clearly explained as in main references such as [1]. Moreover, the problem has been posed in a quite general setup but at the end the authors have focused on the convex and strongly convex setup where they have used the result already in the literature with some minor modifications.

This paper studies online convex optimization in dynamic environments, where one wishes to control the regret with respect to sequences of comparators, as opposed to a static comparator. The complexity of a comparator sequence is measured by its path length P_T, and the goal is to obtain an optimal regret bound simultaneously for all path lengths. A first bound in this setting was established in the pioneering paper [1], which showed that online gradient descent (OGD, projected on the convex compact domain) with step-size 1/sqrt(T) achieves an at most T {1/2} (1 P_T) regret for all comparator sequences. However, there is a gap between this upper bound and the (T (1 P_T)) {1/2} lower bound on worst-case regret established in Theorem 2 of the present paper, which is the first lower bound for this problem. On the other hand, if the path length P_T one wishes to compare against is known in advance, optimally tuning the step-size of OGD with respect to P_T in the OGD regret bound yields optimal (T (1 P_T)) {1/2} regret for this path length.

We introduce an efficient algorithmic framework for model selection in online learning, also known as parameter-free online learning. Departing from previous work, which has focused on highly structured function classes such as nested balls in Hilbert space, we propose a generic meta-algorithm framework that achieves online model selection oracle inequalities under minimal structural assumptions. We give the first computationally efficient parameter-free algorithms that work in arbitrary Banach spaces under mild smoothness assumptions; previous results applied only to Hilbert spaces.

We develop a nonparametric and online learning algorithm that estimates the triggering functions of a multivariate Hawkes process (MHP).

We study online learning with the general notion of transductive regret, that is regret with modification rules applying to expert sequences (as opposed to single experts) that are representable by weighted finite-state transducers. We show how transductive regret generalizes existing notions of regret, including: (1) external regret; (2) internal regret; (3) swap regret; and (4) conditional swap regret. We present a general and efficient online learning algorithm for minimizing transductive regret. We further extend that to design efficient algorithms for the time-selection and sleeping expert settings. A by-product of our study is an algorithm for swap regret, which, under mild assumptions, is more efficient than existing ones, and a substantially more efficient algorithm for time selection swap regret.

Most online optimization algorithms focus on one of two things: performing well in adversarial settings by adapting to unknown data parameters (such as Lipschitz constants), typically achieving O( T) regret, or performing well in stochastic settings where they can leverage some structure in the losses (such as strong convexity), typically achieving O(log(T)) regret. Algorithms that focus on the former problem hitherto achieved O( T) in the stochastic setting rather than O(log(T)).

In this paper, we propose a quasi-Newton method for solving smooth and monotone nonlinear equations, including unconstrained minimization and minimax optimization as special cases. For the strongly monotone setting, we establish two global convergence bounds: (i) a linear convergence rate that matches the rate of the celebrated extragradient method, and (ii) an explicit global superlinear convergence rate that provably surpasses the linear convergence rate after at most ${O}(d)$ iterations, where $d$ is the problem's dimension. In addition, for the case where the operator is only monotone, we prove a global convergence rate of ${O}(\min\{{1}/{k},{\sqrt{d}}/{k^{1.25}}\})$ in terms of the duality gap. This matches the rate of the extragradient method when $k = {O}(d^2)$ and is faster when $k = \Omega(d^2)$. These results are the first global convergence results to demonstrate a provable advantage of a quasi-Newton method over the extragradient method, without querying the Jacobian of the operator. Unlike classical quasi-Newton methods, we achieve this by using the hybrid proximal extragradient framework and a novel online learning approach for updating the Jacobian approximation matrices. Specifically, guided by the convergence analysis, we formulate the Jacobian approximation update as an online convex optimization problem over non-symmetric matrices, relating the regret of the online problem to the convergence rate of our method. To facilitate efficient implementation, we further develop a tailored online learning algorithm based on an approximate separation oracle, which preserves structures such as symmetry and sparsity in the Jacobian matrices.