Russia's Defence Ministry said air defences shot down 105 Ukrainian drones over Russian regions, including 35 over the Moscow region, after the ministry said a day earlier that it had downed more than 300 Ukrainian drones. Kherson Governor Oleksandr Prokudin said one person was killed in a Russian artillery attack on the region. H said over the past day, 35 areas in Kherson, including Kherson city, came under artillery shelling and air attacks, wounding 11 people. Ukrainian President Zelenskyy said the "most intense situation" is in the Donetsk region, and the army is continuing "active operations in the Kursk and Belgorod regions". Russia's Defence Ministry said air defences shot down 105 Ukrainian drones over Russian regions, including 35 over the Moscow region, after the ministry said a day earlier that it had downed more than 300 Ukrainian drones.

Bayesian optimization (BO) is a popular method for efficiently inferring optima of an expensive black-box function via a sequence of queries. Existing informationtheoretic BO procedures aim to make queries that most reduce the uncertainty about optima, where the uncertainty is captured by Shannon entropy. However, an optimal measure of uncertainty would, ideally, factor in how we intend to use the inferred quantity in some downstream procedure. In this paper, we instead consider a generalization of Shannon entropy from work in statistical decision theory [13, 39], which contains a broad class of uncertainty measures parameterized by a problem-specific loss function corresponding to a downstream task. We first show that special cases of this entropy lead to popular acquisition functions used in BO procedures such as knowledge gradient, expected improvement, and entropy search. We then show how alternative choices for the loss yield a flexible family of acquisition functions that can be customized for use in novel optimization settings.

Sequences of events, such as crashes in the stock market or outages in a network, contain strong temporal dependencies, whose understanding is crucial to react to and influence future events. In this paper, we study the problem of discovering the underlying causal structure from event sequences. To this end, we introduce a new causal model, where individual events of the cause trigger events of the effect with dynamic delays. We show that in contrast to existing methods based on Granger causality, our model is identifiable for both instant and delayed effects. We base our approach on the Algorithmic Markov Condition, by which we identify the true causal network as the one that minimizes the Kolmogorov complexity. As the Kolmogorov complexity is not computable, we instantiate our model using Minimum Description Length and show that the resulting score identifies the causal direction.

We present theoretical results to show that K-priors with limited memory can achieve low gradientreconstruction error. We will discuss the optimal K-prior which can theoretically achieve perfect reconstruction error. Note that the prior is difficult to realize in practice since it requires all past training-data inputs X. Our goal here is to establish a theoretical limit, not to give practical choices. Our key idea is to choose a few input locations that provide a good representation of the training-data inputs X.

Real-world classification problems must contend with domain shift, the (potential) mismatch between the domain where a model is deployed and the domain(s) where the training data was gathered. Methods to handle such problems must specify what structure is common between the domains and what varies. A natural assumption is that causal (structural) relationships are invariant in all domains. Then, it is tempting to learn a predictor for label Y that depends only on its causal parents. However, many real-world problems are "anti-causal" in the sense that Y is a cause of the covariates X--in this case, Y has no causal parents and the naive causal invariance is useless.

International series in pure and applied mathematics.

In this section, we briefly introduce the four kinds of emebddings consists the fusion embedding. The goal of position embedding module is to calibrate the position of each time point in the sequence so that the self-attention mechanism can recognize the relative positions between different time points in the input sequence. We design the token embedding module in order to enrich the features of each time point by fusion of other features from the adjacent time points within a certain interval. The role of spatial embedding is to locate and encode the spatial locations of different nodes, by which each node at different location possesses a unique spatial embedding. Thus, it enabling the model to identify nodes in different spatial and temporal planes after the dimensionality is compressed in the later computation.

We provide more details of training the teacher network in Section A, more experimental results on synthetic functions in Section B, and the hyperparameter settings for benchmark datasets in Section C. Here, we omit the iteration subscript t for simplicity. To solve Eq. (10), we obtain the hypergradient regarding to and backpropagate it to = {W 2 R As shown in Algorithm 1, we train the teacher network one step when each time it is called by an underperforming student model, where the step refers to one iteration on synthetic functions and one epoch of the validation set on benchmark datasets in the experiment. In Section 4.1, we have shown the experimental results of HPM on two population synthetic functions, i.e., the Branin and Hartmann6D functions. In the following, we will provide more details about synthetic functions and the implementation, as well as more results on the other two functions. We used the Branin and Hartmann6D functions in Section 4.1.

We thank all reviewers for their time and constructive comments. We first address concerns that were brought up by multiple reviewers. NMODE is more sample efficient than other methods (Appendix C.2, first paragraph), so for density estimation The quantifier for Prop 5.1 should be "for some"; this will be fixed. Note that for small dimensions (e.g. Riemannian metric, and are thus Riemannian.

Generative Adversarial Networks (GANs) are a widely-used tool for generative modeling of complex data. Despite their empirical success, the training of GANs is not fully understood due to the min-max optimization of the generator and discriminator. This paper analyzes these joint dynamics when the true samples as well as the generated samples are discrete, finite sets, and the discriminator is kernel-based. A simple yet expressive framework for analyzing training called the Isolated Points Model is introduced. In the proposed model, the distance between true samples greatly exceeds the kernel width, so each generated point is influenced by at most one true point. Our model enables precise characterization of the conditions for convergence, both to good and bad minima. In particular, the analysis explains two common failure modes: (i) an approximate mode collapse and (ii) divergence. Numerical simulations are provided that predictably replicate these behaviors.