B.1 Proof of Theorem 1 The proof of Theorem 1 is based on the following two lemmas. Moreover, when there is a tie (i.e., the set Proof of Lemma 2. Recall that the Wasserstein metric of order 1 is defined as W ( P,P For the sake of completeness, we extend our algorithm to non-few-training-sample setting. The depth of the shaded area shows the level of samples entropy. The entropy of a sample is defined as follows. As a simple example, for Bernoulli random variable (which can represent, e.g., the outcome for flipping a coin with bias Now we use this entropy to define the "uncertainty" associated with each training points. Figure 6 reveals that the most informative samples usually lie in between categories.

We propose a two-step method PLNet to estimate the precision matrix.

Newton method of Polyak and Nesterov (2006) and of regularized Newton method of Mishchenko (2021) and Doikov and Nesterov (2021), b) we prove a local quadratic rate, which matches the best-known local rate of second-order methods, and c) our stepsize formula is simple, explicit, and does not require solving any subproblem.

Having a mixed-integer linear program (MIP) or a nonlinear program (NLP) in the second stage further aggravates the intractability, even when specialized algorithms that exploit problem structure are employed.

Theorem A.1 (Deterministic scaling limit of stochastic processes) . The reader interested in the proof is referred to the supplementary materials of [21, 31]. Although the theorem wasn't originally proven in the A.1 corresponds to 1 /δt, where δt is defined in Theorem 2.1. Before proving this proposition, we begin with a small lemma: Lemma B.2. We are now in a position to show Theorem B.1: 16 Proof.

Linear programs are often specified by hand, using prior knowledge of relevant costs and constraints.

Motivated by sequential budgeted allocation problems, we investigate online matching problems where connections between vertices are not i.i.d., but they have fixed

Solving quadratic programs (QPs) efficiently is critical to applications in finance, robotic control and operations research.

This will be proved as follows. Hence, the upper bound of condition (b) is satisfied. Now we introduce the Fano's inequality. We will use the version provided by [24] in our proofs. We firstly consider entrywise KL-divergence.