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Identifying latent variables and causal structures from observational data is essential to many real-world applications involving biological data, medical data, and unstructured data such as images and languages. However, this task can be highly challenging, especially when observed variables are generated by causally related latent variables and the relationships are nonlinear. In this work, we investigate the identification problem for nonlinear latent hierarchical causal models in which observed variables are generated by a set of causally related latent variables, and some latent variables may not have observed children. We show that the identifiability of both causal structure and latent variables can be achieved under mild assumptions: on causal structures, we allow for the existence of multiple paths between any pair of variables in the graph, which relaxes latent tree assumptions in prior work; on structural functions, we do not make parametric assumptions, thus permitting general nonlinearity and multi-dimensional continuous variables. Specifically, we first develop a basic identification criterion in the form of novel identifiability guarantees for an elementary latent variable model. Leveraging this criterion, we show that both causal structures and latent variables of the hierarchical model can be identified asymptotically by explicitly constructing an estimation procedure. To the best of our knowledge, our work is the first to establish identifiability guarantees for both causal structures and latent variables in nonlinear latent hierarchical models.

We consider the problem of testing the fit of a discrete sample of items from many categories to the uniform distribution over the categories. As a class of alternative hypotheses, we consider the removal of an $\ell_p$ ball of radius $\epsilon$ around the uniform rate sequence for $p \leq 2$. We deliver a sharp characterization of the asymptotic minimax risk when $\epsilon \to 0$ as the number of samples and number of dimensions go to infinity, for testing based on the occurrences' histogram (number of absent categories, singletons, collisions, ...). For example, for $p=1$ and in the limit of a small expected number of samples $n$ compared to the number of categories $N$ (aka "sub-linear" regime), the minimax risk $R^*_\epsilon$ asymptotes to $2 \bar{\Phi}\left(n \epsilon^2/\sqrt{8N}\right) $, with $\bar{\Phi}(x)$ the normal survival function. Empirical studies over a range of problem parameters show that this estimate is accurate in finite samples, and that our test is significantly better than the chisquared test or a test that only uses collisions. Our analysis is based on the asymptotic normality of histogram ordinates, the equivalence between the minimax setting to a Bayesian one, and the reduction of a multi-dimensional optimization problem to a one-dimensional problem.

In this paper, we investigate the two-dimensional extension of a recently introduced set of shallow water models based on a regularized moment expansion of the incompressible Navier-Stokes equations \cite{kowalski2017moment,koellermeier2020analysis}. We show the rotational invariance of the proposed moment models with two different approaches. The first proof involves the split of the coefficient matrix into the conservative and non-conservative parts and prove the rotational invariance for each part, while the second one relies on the special block structure of the coefficient matrices. With the aid of rotational invariance, the analysis of the hyperbolicity for the moment model in 2D is reduced to the real diagonalizability of the coefficient matrix in 1D. Then we prove the real diagonalizability by deriving the analytical form of the characteristic polynomial. Furthermore, we extend the model to include a more general class of closure relations than the original model and establish that this set of general closure relations retain both rotational invariance and hyperbolicity.

This paper investigates the efficiency of the K-fold cross-validation (CV) procedure and a debiased version thereof as a means of estimating the generalization risk of a learning algorithm. We work under the general assumption of uniform algorithmic stability. We show that the K-fold risk estimate may not be consistent under such general stability assumptions, by constructing non vanishing lower bounds on the error in realistic contexts such as regularized empirical risk minimisation and stochastic gradient descent. We thus advocate the use of a debiased version of the K-fold and prove an error bound with exponential tail decay regarding this version. Our result is applicable to the large class of uniformly stable algorithms, contrarily to earlier works focusing on specific tasks such as density estimation. We illustrate the relevance of the debiased K-fold CV on a simple model selection problem and demonstrate empirically the usefulness of the promoted approach on real world classification and regression datasets.

In this paper, we study the problems of detection and recovery of hidden submatrices with elevated means inside a large Gaussian random matrix. We consider two different structures for the planted submatrices. In the first model, the planted matrices are disjoint, and their row and column indices can be arbitrary. Inspired by scientific applications, the second model restricts the row and column indices to be consecutive. In the detection problem, under the null hypothesis, the observed matrix is a realization of independent and identically distributed standard normal entries. Under the alternative, there exists a set of hidden submatrices with elevated means inside the same standard normal matrix. Recovery refers to the task of locating the hidden submatrices. For both problems, and for both models, we characterize the statistical and computational barriers by deriving information-theoretic lower bounds, designing and analyzing algorithms matching those bounds, and proving computational lower bounds based on the low-degree polynomials conjecture. In particular, we show that the space of the model parameters (i.e., number of planted submatrices, their dimensions, and elevated mean) can be partitioned into three regions: the impossible regime, where all algorithms fail; the hard regime, where while detection or recovery are statistically possible, we give some evidence that polynomial-time algorithm do not exist; and finally the easy regime, where polynomial-time algorithms exist.

Ensuring the security of networked systems is a significant problem, considering the susceptibility of modern infrastructures and technologies to adversarial interference. A central component of this problem is how defensive resources should be allocated to mitigate the severity of potential attacks on the system. In this paper, we consider this in the context of a General Lotto game, where a defender and attacker deploys resources on the nodes of a network, and the objective is to secure as many links as possible. The defender secures a link only if it out-competes the attacker on both of its associated nodes. For bipartite networks, we completely characterize equilibrium payoffs and strategies for both the defender and attacker. Surprisingly, the resulting payoffs are the same for any bipartite graph. On arbitrary network structures, we provide lower and upper bounds on the defender's max-min value. Notably, the equilibrium payoff from bipartite networks serves as the lower bound. These results suggest that more connected networks are easier to defend against attacks. We confirm these findings with simulations that compute deterministic allocation strategies on large random networks. This also highlights the importance of randomization in the equilibrium strategies.

The multi-armed bandit (MAB) model is one of the most classical models to study decision-making in an uncertain environment. In this model, a player chooses one of $K$ possible arms of a bandit machine to play at each time step, where the corresponding arm returns a random reward to the player, potentially from a specific unknown distribution. The target of the player is to collect as many rewards as possible during the process. Despite its simplicity, the MAB model offers an excellent playground for studying the trade-off between exploration versus exploitation and designing effective algorithms for sequential decision-making under uncertainty. Although many asymptotically optimal algorithms have been established, the finite-time behaviors of the stochastic dynamics of the MAB model appear much more challenging to analyze, due to the intertwine between the decision-making and the rewards being collected. In this paper, we employ techniques in statistical physics to analyze the MAB model, which facilitates the characterization of the distribution of cumulative regrets at a finite short time, the central quantity of interest in an MAB algorithm, as well as the intricate dynamical behaviors of the model. Our analytical results, in good agreement with simulations, point to the emergence of an interesting multimodal regret distribution, with large regrets resulting from excess exploitation of sub-optimal arms due to an initial unlucky output from the optimal one.

The digitalization of the reproductive body has engaged myriads of cutting-edge technologies in supporting people to know and tackle their intimate health. Generally understood as female technologies (aka female-oriented technologies or 'FemTech'), these products and systems collect a wide range of intimate data which are processed, transferred, saved and shared with other parties. In this paper, we explore how the "data-hungry" nature of this industry and the lack of proper safeguarding mechanisms, standards, and regulations for vulnerable data can lead to complex harms or faint agentic potential. We adopted mixed methods in exploring users' understanding of the security and privacy (SP) of these technologies. Our findings show that while users can speculate the range of harms and risks associated with these technologies, they are not equipped and provided with the technological skills to protect themselves against such risks. We discuss a number of approaches, including participatory threat modelling and SP by design, in the context of this work and conclude that such approaches are critical to protect users in these sensitive systems.

Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization'' (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem, or some multiflow problems. We clarify the interconnections of some problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in an undirected graph without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we provide a polynomial-time algorithm to the closely related and well-studied Minimum-weight Odd $\{s,t\}$-Join problem for non-negative weights, whose complexity, however, was not known; more generally, we solve the Minimum-weight Odd $T$-Join problem in FPT time when parameterized by $|T|$. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is only conjectured to be polynomially solvable. The analogous problems for digraphs are also considered.

We study the Pareto frontier of two archetypal objectives in multi-armed bandits, namely, regret minimization (RM) and best arm identification (BAI) with a fixed horizon. It is folklore that the balance between exploitation and exploration is crucial for both RM and BAI, but exploration is more critical in achieving the optimal performance for the latter objective. To this end, we design and analyze the BoBW-lil'UCB$(\gamma)$ algorithm. Complementarily, by establishing lower bounds on the regret achievable by any algorithm with a given BAI failure probability, we show that (i) no algorithm can simultaneously perform optimally for both the RM and BAI objectives, and (ii) BoBW-lil'UCB$(\gamma)$ achieves order-wise optimal performance for RM or BAI under different values of $\gamma$. Our work elucidates the trade-off more precisely by showing how the constants in previous works depend on certain hardness parameters. Finally, we show that BoBW-lil'UCB outperforms a close competitor UCB$_\alpha$ (Degenne et al., 2019) in terms of the time complexity and the regret on diverse datasets such as MovieLens and Published Kinase Inhibitor Set.