For the convenience of the reader, we summarize a list of notations blow. 1. null G In Appendix B.1, we present a general statement of Theorem 3.1 (a) along with its proof. Theorem 3.1 (a) states the order recovery guarantee for a specified parameter We summarize the bounds for (I) and (II) in Lemma B.1 and Lemma B.2, which can be found in Collecting the results in Lemma B.1 and Lemma B.2 and reorganizing the terms in the inequalities, we have the following conclusion. We now state the proof of this Lemma. Then we bound the first term using the concentration bound on Chi-squared random variables. For the non-identifiable models, we can use Lemma H.1 in a similar way to obtain that with probability We now state the proof of this Lemma.

Adam is a widely used optimizer in neural network training due to its adaptive learning rate. However, because different data samples influence model updates to varying degrees, treating them equally can lead to inefficient convergence. To address this, a prior work proposed adapting the sampling distribution using a bandit framework to select samples adaptively. While promising, the bandit-based variant of Adam suffers from limited theoretical guarantees. In this paper, we introduce Adam with Combinatorial Bandit Sampling (AdamCB), which integrates combinatorial bandit techniques into Adam to resolve these issues. AdamCB is able to fully utilize feedback from multiple samples at once, enhancing both theoretical guarantees and practical performance. Our regret analysis shows that AdamCB achieves faster convergence than Adam-based methods including the previous bandit-based variant. Numerical experiments demonstrate that AdamCB consistently outperforms existing methods.

Controlling payoffs in repeated games is one of the important topics in control theory of multi-agent systems. Recently proposed zero-determinant strategies enable players to unilaterally enforce linear relations between payoffs. Furthermore, based on the mathematics of zero-determinant strategies, regional payoff control, in which payoffs are enforced into some feasible regions, has been discovered in social dilemma situations. More recently, theory of payoff control was extended to multichannel games, where players parallelly interact with each other in multiple channels. However, the existence of payoff-controlling strategies in multichannel games seems to require the existence of payoff-controlling strategies in some channels, and properties of zero-determinant strategies specific to multichannel games are still not clear. In this paper, we elucidate properties of zero-determinant strategies in multichannel games. First, we relate the existence condition of zero-determinant strategies in multichannel games to that of zero-determinant strategies in each channel. We then show that the existence of zero-determinant strategies in multichannel games requires the existence of zero-determinant strategies in some channels. This result implies that the existence of zero-determinant strategies in multichannel games is tightly restricted by structure of games played in each channel.

We use the following numerical experiment to further illustrate our finite-time bounds on the convergence of double Q-learning. In such an experiment, the optimal Q-function can be explicitly calculated and thus the learning errors can be tracked. We choose γ = 0 .8,α We prove Lemma 1 by induction. First, it is easy to justify that the initial case is satisfied, i.e., In this appendix, we will provide a detailed proof of Theorem 1.

In contrast to the independent sampling assumed in classical matrix completion literature, the observed entries, which arise from past matching data, are constrained by matching capacity. This matching-induced dependence poses new challenges for both estimation and inference in the matrix completion framework. We propose a non-convex algorithm based on Grassmannian gradient descent and establish near-optimal entrywise convergence rates for three canonical mechanisms, i.e., one-to-one matching, one-to-many matching with one-sided random arrival, and two-sided random arrival. To facilitate valid uncertainty quantification and hypothesis testing on matching decisions, we further develop a general debiasing and projection framework for arbitrary linear forms of the reward matrix, deriving asymptotic normality with finite-sample guarantees under matching-induced dependent sampling. Our empirical experiments demonstrate that the proposed approach provides accurate estimation, valid confidence intervals, and efficient evaluation of matching policies.

Recent research has observed that in machine learning optimization, gradient descent (GD) often operates at the edge of stability (EoS) [Cohen et al., 2021],

We summarize the results in Figure S1-S3. Figure S1 presents the results for ridge regression. Lastly, Figure S3 shows the iteration plot for the real data example. We omit their proofs as all of them can be found in standard convex analysis textbooks [see, e.g., Boyd and V andenberghe, 2004]. This completes the proof of Lemma S1.