We consider the problem of prediction with expert advice for ``easy'' sequences. We show that a variant of NormalHedge enjoys a second-order $ε$-quantile regret bound of $O\big(\sqrt{V_T \log(V_T/ε)}\big) $ when $V_T > \log N$, where $V_T$ is the cumulative second moment of instantaneous per-expert regret averaged with respect to a natural distribution determined by the algorithm. The algorithm is motivated by a continuous time limit using Stochastic Differential Equations. The discrete time analysis uses self-concordance techniques.

Varying domains and biased datasets can lead to differences between the training and the target distributions, known as covariate shift. Current approaches for alleviating this often rely on estimating the ratio of training and target probability density functions. These techniques require parameter tuning and can be unstable across different datasets. We present a new method for handling covariate shift using the empirical cumulative distribution function estimates of the target distribution by a rigorous generalization of a recent idea proposed by Vapnik and Izmailov. Further, we show experimentally that our method is more robust in its predictions, is not reliant on parameter tuning and shows similar classification performance compared to the current state-of-the-art techniques on synthetic and real datasets.

A central problem in machine learning and statistics is to model joint densities of random variables from data. Copulas are joint cumulative distribution functions with uniform marginal distributions and are used to capture interdependencies in isolation from marginals. Copulas are widely used within statistics, but have not gained traction in the context of modern deep learning. In this paper, we introduce ACNet, a novel differentiable neural network architecture that enforces structural properties and enables one to learn an important class of copulas--Archimedean Copulas. Unlike Generative Adversarial Networks, Variational Autoencoders, or Normalizing Flow methods, which learn either densities or the generative process directly, ACNet learns a generator of the copula, which implicitly defines the cumulative distribution function of a joint distribution. We give a probabilistic interpretation of the network parameters of ACNet and use this to derive a simple but efficient sampling algorithm for the learned copula. Our experiments show that ACNet is able to both approximate common Archimedean Copulas and generate new copulas which may provide better fits to data.

Conformal prediction provides a pivotal and flexible technique for uncertainty quantification by constructing prediction sets with a predefined coverage rate. Many online conformal prediction methods have been developed to address data distribution shifts in fully adversarial environments, resulting in overly conservative prediction sets. We propose Conformal Optimistic Prediction (COP), an online conformal prediction algorithm incorporating underlying data pattern into the update rule. Through estimated cumulative distribution function of non-conformity scores, COP produces tighter prediction sets when predictable pattern exists, while retaining valid coverage guarantees even when estimates are inaccurate. We establish a joint bound on coverage and regret, which further confirms the validity of our approach. We also prove that COP achieves distribution-free, finite-sample coverage under arbitrary learning rates and can converge when scores are $i.i.d.$. The experimental results also show that COP can achieve valid coverage and construct shorter prediction intervals than other baselines.

The theory describes four physical zones (or territories) defined by growing distances around each person, as can be seen in Figure 3 (top left). With those hidden unwritten rules for spaces around a person, only socially close people are welcome within the intimate zone, while generally close people can enter the personal zone, followed by generally familiar people who are allowed in the social space. Otherwise, general public are only permitted within the public space. The concept of group proxemics has been investigated in literature with most attention being paid to detailing the classical proxemics theory. For instance, the authors of [14] explored proxemics and their impact on shape of group formation, the authors of [2] explored proxemics dispersion as average distances people maintain between each other as they walk in group, and in [18], [19] focus was given to studying the effect of proxemics on crowd and its traffic flow dynamics. Within robot-human interactions, the authors of [20]-[22] studied appropriate (safety, comfort, acceptability, etc) distance robots are expected to maintain from people (as individuals). It could be noticed that proxemics are structured around interactions between individuals and details are specified in terms of social relationships between them. In what follows, we explore the situation when an individual is part of a bigger and more complex social entity such as a group. We study the nature of such interactions and and explore associated proxemics.

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Traditional machine learning approaches assume that training data and target data are drawn from the same distribution.

Extinction times in resampling processes are fundamental yet often intractable, as previous formulas scale as $2^M$ with the number of states $M$ present in the initial probability distribution. We solve this by treating multinomial updates as independent square-root diffusions of zero drift, yielding a closed-form law for the first-extinction time. We prove that the mean coincides exactly with the Wright-Fisher result of Baxter et al., thereby replacing exponential-cost evaluations with a linear-cost expression, and we validate this result through extensive simulations. Finally, we demonstrate predictive power for model collapse in a simple self-training setup: the onset of collapse coincides with the resampling-driven first-extinction time computed from the model's initial stationary distribution. These results hint to a unified view of resampling extinction dynamics.

The Appendix is organized as follows: In Appendix A, we state the symbols and notation used in this paper. In Appendix B, we provide the proofs and related lemmas of Theorem 1. In Appendix C, we provide the proofs of Theorem 2. In Appendix D, we provide the proofs and related lemmas of Theorem 3. In Appendix F, we discuss several limitations of this work. Finally, in Appendix G, we discuss the societal impact of this paper. In the paper, vectors are indicated with bold small letters, matrices with bold capital letters.