In this work, we study the low-rank MDPs with adversarially changed losses in the full-information feedback setting. In particular, the unknown transition probability kernel admits a low-rank matrix decomposition \citep{REPUCB22}, and the loss functions may change adversarially but are revealed to the learner at the end of each episode. We propose a policy optimization-based algorithm POLO, and we prove that it attains the $\widetilde{O}(K^{\frac{5}{6}}A^{\frac{1}{2}}d\ln(1+M)/(1-\gamma)^2)$ regret guarantee, where $d$ is rank of the transition kernel (and hence the dimension of the unknown representations), $A$ is the cardinality of the action space, $M$ is the cardinality of the model class that contains all the plausible representations, and $\gamma$ is the discounted factor. Notably, our algorithm is oracle-efficient and has a regret guarantee with no dependence on the size of potentially arbitrarily large state space. Furthermore, we also prove an $\Omega(\frac{\gamma^2}{1-\gamma} \sqrt{d A K})$ regret lower bound for this problem, showing that low-rank MDPs are statistically more difficult to learn than linear MDPs in the regret minimization setting. To the best of our knowledge, we present the first algorithm that interleaves representation learning, exploration, and exploitation to achieve the sublinear regret guarantee for RL with nonlinear function approximation and adversarial losses.

In this paper, we provide the first deterministic algorithm that achieves $1/2$-approximation for monotone submodular maximization subject to a knapsack constraint, while making a number of queries that scales only linearly with the size of the ground set $n$. Moreover, our result automatically paves the way for developing a linear-time deterministic algorithm that achieves the tight $1-1/e$ approximation guarantee for monotone submodular maximization under a cardinality (size) constraint. To complement our positive results, we also show strong information-theoretic lower bounds. More specifically, we show that when the maximum cardinality allowed for a solution is constant, no deterministic or randomized algorithm making a sub-linear number of function evaluations can guarantee any constant approximation ratio. Furthermore, when the constraint allows the selection of a constant fraction of the ground set, we show that any algorithm making fewer than $\Omega(n/\log(n))$ function evaluations cannot perform better than an algorithm that simply outputs a uniformly random subset of the ground set of the right size. We extend our results to the general case of maximizing a monotone submodular function subject to the intersection of a $p$-set system and multiple knapsack constraints. Finally, we evaluate the performance of our algorithms on multiple real-life applications, including movie recommendation, location summarization, Twitter text summarization, and video summarization.

-- In this work we introduce a novel adaptive anomaly detection framework specifically designed for monitoring sequential random finite set (RFS) observations. Our approach effectively distinguishes between In-Control data (normal) and Out-Of-Control data (anomalies) by detecting deviations from the expected statistical behavior of the process. The primary contributions of this study include the development of an innovative RFS-based framework that not only learns the normal behavior of the data-generating process online but also dynamically adapts to behavioral shifts to accurately identify abnormal point patterns. T o achieve this, we introduce a new class of RFS-based posterior distributions, named Power Discounting Posteriors (PD), which facilitate adaptation to systematic changes in data while enabling anomaly detection of point pattern data through a novel predictive posterior density function. The effectiveness of the proposed approach is demonstrated by extensive qualitative and quantitative simulation experiments.

Discrete fuzzy numbers, and in particular those defined over a finite chain $L_n = \{0, \ldots, n\}$, have been effectively employed to represent linguistic information within the framework of fuzzy systems. Research on total (admissible) orderings of such types of fuzzy subsets, and specifically those belonging to the set $\mathcal{D}_1^{L_n\rightarrow Y_m}$ consisting of discrete fuzzy numbers $A$ whose support is a closed subinterval of the finite chain $L_n = \{0, 1, \ldots, n\}$ and whose membership values $A(x)$, for $x \in L_n$, belong to the set $Y_m = \{ 0 = y_1 < y_2 < \cdots < y_{m-1} < y_m = 1 \}$, has facilitated the development of new methods for constructing logical connectives, based on a bijective function, called $\textit{pos function}$, that determines the position of each $A \in \mathcal{D}_1^{L_n\rightarrow Y_m}$. For this reason, in this work we revisit the problem by introducing algorithms that exploit the combinatorial structure of total (admissible) orders to compute the $\textit{pos}$ function and its inverse with exactness. The proposed approach achieves a complexity of $\mathcal{O}(n^{2} m \log n)$, which is quadratic in the size of the underlying chain ($n$) and linear in the number of membership levels ($m$). The key point is that the dominant factor is $m$, ensuring scalability with respect to the granularity of membership values. The results demonstrate that this formulation substantially reduces computational cost and enables the efficient implementation of algebraic operations -- such as aggregation and implication -- on the set of discrete fuzzy numbers.

Estimating optimal phylogenetic trees or hierarchical clustering trees from metric data is an important problem in evolutionary biology and data analysis. Intuitively, the goodness-of-fit of a metric space to a tree depends on its inherent treeness, as well as other metric properties such as intrinsic dimension. Existing algorithms for embedding metric spaces into tree metrics provide distortion bounds depending on cardinality. Because cardinality is a simple property of any set, we argue that such bounds do not fully capture the rich structure endowed by the metric. We consider an embedding of a metric space into a tree proposed by Gromov. By proving a stability result, we obtain an improved additive distortion bound depending only on the hyperbolicity and doubling dimension of the metric. We observe that Gromov's method is dual to the well-known single linkage hierarchical clustering (SLHC) method. By means of this duality, we are able to transport our results to the setting of SLHC, where such additive distortion bounds were previously unknown.

Because cardinality is a simple property of any set, we argue that such bounds do not fully capture the rich structure endowed by the metric.

We propose and analyze a regularization approach for structured prediction problems.

In fact, the additive noise model assumes that all categories of the variables are placed in the "right" order; furthermore, if