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In this study, we address the issue of enabling an artificial intelligence agent to execute complex language instructions within virtual environments. In our framework, we assume that these instructions involve intricate linguistic structures and multiple interdependent tasks that must be navigated successfully to achieve the desired outcomes. To effectively manage these complexities, we propose a hierarchical framework that combines the deep language comprehension of large language models with the adaptive action-execution capabilities of reinforcement learning agents. The language module (based on LLM) translates the language instruction into a high-level action plan, which is then executed by a pre-trained reinforcement learning agent. We have demonstrated the effectiveness of our approach in two different environments: in IGLU, where agents are instructed to build structures, and in Crafter, where agents perform tasks and interact with objects in the surrounding environment according to language commands.

We develop a new method for creating prediction sets that combines the flexibility of conformal methods with an estimate of the conditional distribution $P_{Y \mid X}$. Most existing methods, such as conformalized quantile regression and probabilistic conformal prediction, only offer marginal coverage guarantees. Our approach extends these methods to achieve conditional coverage, which is essential for many practical applications. While exact conditional guarantees are impossible without assumptions on the data distribution, we provide non-asymptotic bounds that explicitly depend on the quality of the available estimate of the conditional distribution. Our confidence sets are highly adaptive to the local structure of the data, making them particularly useful in high heteroskedasticity situations. We demonstrate the effectiveness of our approach through extensive simulations, showing that it outperforms existing methods in terms of conditional coverage and improves the reliability of statistical inference in a wide range of applications., Comment: 23 pages

Uncertainty quantification (UQ) is becoming increasingly recognized as a critical component of applications that rely on machine learning (ML). The rapid proliferation of large language models (LLMs) has stimulated researchers to seek efficient and effective approaches to UQ in text generation tasks, as in addition to their emerging capabilities, these models have introduced new challenges for building safe applications. As with other ML models, LLMs are prone to make incorrect predictions, ``hallucinate'' by fabricating claims, or simply generate low-quality output for a given input. UQ is a key element in dealing with these challenges. However research to date on UQ methods for LLMs has been fragmented, with disparate evaluation methods. In this work, we tackle this issue by introducing a novel benchmark that implements a collection of state-of-the-art UQ baselines, and provides an environment for controllable and consistent evaluation of novel techniques by researchers in various text generation tasks. Our benchmark also supports the assessment of confidence normalization methods in terms of their ability to provide interpretable scores. Using our benchmark, we conduct a large-scale empirical investigation of UQ and normalization techniques across nine tasks and shed light on the most promising approaches., Comment: Roman Vashurin, Ekaterina Fadeeva, Artem Vazhentsev contributed equally

For an arbitrary equidimensional quiver representation, we proposed the method of construction of a system of free generators of the field of $U$-invariants. The construction of the section and system of generators depends on the choice of a map that assign to each vertex one of the arrows incident to it., Comment: 14 pages

This study focuses on statistical inference for compound models of the form $X=\xi_1+\ldots+\xi_N$, where $N$ is a random variable denoting the count of summands, which are independent and identically distributed (i.i.d.) random variables $\xi_1, \xi_2, \ldots$. The paper addresses the problem of reconstructing the distribution of $\xi$ from observed samples of $X$'s distribution, a process referred to as decompounding, with the assumption that $N$'s distribution is known. This work diverges from the conventional scope by not limiting $N$'s distribution to the Poisson type, thus embracing a broader context. We propose a nonparametric estimate for the density of $\xi$, derive its rates of convergence and prove that these rates are minimax optimal for suitable classes of distributions for $\xi$ and $N$. Finally, we illustrate the numerical performance of the algorithm on simulated examples., Comment: 21 page, 2 figures

The general linear group GL(n) acts on the direct sum of $m$ copies of Mat(n) by the adjoint action. The action of GL(n) induces the action of the unitriangular subgroup U. We present the system of free generators of the field of U-invariants., Comment: 8 pages

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of $\textit{spectral algorithms}$ specified by profile $h(\lambda)$, and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile $h(\lambda)$ for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the half-line $x>0$ with a constant initial data and with Dirichlet or Neumann boundary conditions. In the case of Dirichlet boundary condition we prove that a nonlinear algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written strictly convex and coercive function. Therefore, there exists a unique minimum point of the potential, coordinates of this point determine free boundaries and provide the desired solution. This result is also extended to the case of infinitely many phase transitions. In the case of Neumann boundary condition we demonstrate that the problem may have solutions with different numbers (called types) of phase transitions. For each fixed type $n$ the system for determination of the free boundaries is again gradient and the corresponding potential is proved to be strictly convex and coercive, but in some wider non-physical domain. On the base of these properties it is proved that there exists a unique solution of Stefan-Neumann problem, and we also provide precise conditions to specify the type of the solution. Bibliography$:$ $5$ titles., Comment: arXiv admin note: text overlap with arXiv:2401.15338

Large language models (LLMs) are notorious for hallucinating, i.e., producing erroneous claims in their output. Such hallucinations can be dangerous, as occasional factual inaccuracies in the generated text might be obscured by the rest of the output being generally factually correct, making it extremely hard for the users to spot them. Current services that leverage LLMs usually do not provide any means for detecting unreliable generations. Here, we aim to bridge this gap. In particular, we propose a novel fact-checking and hallucination detection pipeline based on token-level uncertainty quantification. Uncertainty scores leverage information encapsulated in the output of a neural network or its layers to detect unreliable predictions, and we show that they can be used to fact-check the atomic claims in the LLM output. Moreover, we present a novel token-level uncertainty quantification method that removes the impact of uncertainty about what claim to generate on the current step and what surface form to use. Our method Claim Conditioned Probability (CCP) measures only the uncertainty of a particular claim value expressed by the model. Experiments on the task of biography generation demonstrate strong improvements for CCP compared to the baselines for seven LLMs and four languages. Human evaluation reveals that the fact-checking pipeline based on uncertainty quantification is competitive with a fact-checking tool that leverages external knowledge., Comment: Accepted to ACL-2024 (Findings). Ekaterina Fadeeva, Aleksandr Rubashevskii, and Artem Shelmanov contributed equally

We relate polyhedral products of spaces to graph products in algebraic categories via the loop homology and Lie algebras associated with central series. By way of application, we describe the restricted Lie algebra associated with the lower 2-central series of a right-angled Coxeter group and identify its universal enveloping algebra with the loop homology of the Davis-Januszkiewicz space., Comment: 14 pages

Uncertainty quantification in predictive modeling often relies on ad hoc methods as there is no universally accepted formal framework for that. This paper introduces a theoretical approach to understanding uncertainty through statistical risks, distinguishing between aleatoric (data-related) and epistemic (model-related) uncertainties. We explain how to split pointwise risk into Bayes risk and excess risk. In particular, we show that excess risk, related to epistemic uncertainty, aligns with Bregman divergences. To turn considered risk measures into actual uncertainty estimates, we suggest using the Bayesian approach by approximating the risks with the help of posterior distributions. We tested our method on image datasets, evaluating its performance in detecting out-of-distribution and misclassified data using the AUROC metric. Our results confirm the effectiveness of the considered approach and offer practical guidance for estimating uncertainty in real-world applications.

We study self-similar solutions of a multi-phase Stefan problem, first in the case of one space variable, and then in the radial multidimensional case. In both these cases we prove that a nonlinear algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written coercive function. Therefore, there exists a minimum point of the potential, coordinates of this point determine free boundaries and provide the desired solution. Moreover, in one-dimensional case the potential is proved to be strictly convex and this implies the uniqueness of the solution. In contrary, in the multidimensional case the potential is not convex but the uniqueness of our solution remains true, it follows from the general theory. Bibliography: 3 titles.

We consider a system of Ising spins s=1/2 with nonmagnetic impurities with charge associated with pseudospin S=1. The charge density is fixed pursuant to the concentration n. Analysis of the thermodynamic properties in the one-dimensional case showed the presence of so-called pseudotransitions at the boundaries between the staggered charge ordering and (anti)ferromagnetic ordering. In the case of n=0, a "1st order" pseudotransition was discovered. This type of pseudotransition is inherent for a series of other one-dimensional frustrated models. However, for n != 0 we discovered a new type of "2nd order" pseudotransition, which had not previously been observed in other systems., Comment: 6 pages, 4 figures

The properties of the ground state of one of the simplest models of frustrated magnetic systems, a dilute Ising chain in a magnetic field, are considered for all values of the concentration of charged non-magnetic impurities. An analytical method is proposed for calculating the residual entropy of frustrated states, including states at the boundaries between the phases of the ground state, which is based on the Markov property of the system under consideration and allows direct generalization to other one-dimensional spin models with Ising-type interactions. The properties of local distributions and concentration dependences of the composition, correlation functions, magnetization and entropy of the phases of the ground state of the model are investigated. It is shown that the field-induced transition from the antiferromagnetic ground state to the frustrated one is accompanied by charge ordering and the absence of pseudo-transitions in the dilute Ising chain is proved., Comment: 9 pages, 3 figures, 2 tables; XXVII International Symposium "Nanophysics and Nanoelectronics"

We argue that the unusual properties of a wide class of materials based on Jahn-Teller 3d and 4d ions with different crystal and electronic structures, from quasi-two-dimensional unconventional superconductors (cuprates, nickelates, ferropnictides/chalcogenides, ruthenate SrRuO4), manganites with local superconductivity to 3D ferrates (CaSr)FeO3, nickelates RNiO3 and silver oxide AgO with unusual charge and magnetic order can be explained within a single scenario. The properties of these materials are related to the instability of their highly symmetric Jahn-Teller "progenitors" with the ground orbital E-state to charge transfer with anti-Jahn-Teller disproportionation and the formation of a system of effective local composite spin-singlet or spin-triplet, electronic or hole bosons moving in a non-magnetic or magnetic lattice. These unusual systems are characterized by an extremely rich variety of phase states from non-magnetic and magnetic insulators to unusual metallic and superconducting states., Comment: 6 pages, 1 table. arXiv admin note: text overlap with arXiv:2306.06612

The Multi-Agent Pathfinding (MAPF) problem involves finding a set of conflict-free paths for a group of agents confined to a graph. In typical MAPF scenarios, the graph and the agents' starting and ending vertices are known beforehand, allowing the use of centralized planning algorithms. However, in this study, we focus on the decentralized MAPF setting, where the agents may observe the other agents only locally and are restricted in communications with each other. Specifically, we investigate the lifelong variant of MAPF, where new goals are continually assigned to the agents upon completion of previous ones. Drawing inspiration from the successful AlphaZero approach, we propose a decentralized multi-agent Monte Carlo Tree Search (MCTS) method for MAPF tasks. Our approach utilizes the agent's observations to recreate the intrinsic Markov decision process, which is then used for planning with a tailored for multi-agent tasks version of neural MCTS. The experimental results show that our approach outperforms state-of-the-art learnable MAPF solvers. The source code is available at https://github.com/AIRI-Institute/mats-lp., Comment: The paper is accepted to AAAI-2024 conference

Conformal Prediction (CP) stands out as a robust framework for uncertainty quantification, which is crucial for ensuring the reliability of predictions. However, common CP methods heavily rely on data exchangeability, a condition often violated in practice. Existing approaches for tackling non-exchangeability lead to methods that are not computable beyond the simplest examples. This work introduces a new efficient approach to CP that produces provably valid confidence sets for fairly general non-exchangeable data distributions. We illustrate the general theory with applications to the challenging setting of federated learning under data heterogeneity between agents. Our method allows constructing provably valid personalized prediction sets for agents in a fully federated way. The effectiveness of the proposed method is demonstrated in a series of experiments on real-world datasets., Comment: 28 pages

In modern federated learning, one of the main challenges is to account for inherent heterogeneity and the diverse nature of data distributions for different clients. This problem is often addressed by introducing personalization of the models towards the data distribution of the particular client. However, a personalized model might be unreliable when applied to the data that is not typical for this client. Eventually, it may perform worse for these data than the non-personalized global model trained in a federated way on the data from all the clients. This paper presents a new approach to federated learning that allows selecting a model from global and personalized ones that would perform better for a particular input point. It is achieved through a careful modeling of predictive uncertainties that helps to detect local and global in- and out-of-distribution data and use this information to select the model that is confident in a prediction. The comprehensive experimental evaluation on the popular real-world image datasets shows the superior performance of the model in the presence of out-of-distribution data while performing on par with state-of-the-art personalized federated learning algorithms in the standard scenarios.

Magnetic reconnection is one of the most universal processes in space plasma that is responsible for charged particle acceleration, mixing and heating of plasma populations. In this paper we consider a triggering process of reconnection that is driven by interaction of two discontinuities: solar wind rotational discontinuity and tangential discontinuity at the Earth's magnetospheric boundary, magnetopause. Combining the multispacecraft measurements and global hybrid simulations, we show that solar wind discontinuities may drive the magnetopause reconnection and cause the mixing of the solar wind and magnetosphere plasmas around the magnetopause, well downstream of the solar wind flow. Since large-amplitude discontinuities are frequently observed in the solar wind and predicted for various stellar winds, our results of reconnection driven by the discontinuity-discontinuity interaction may have a broad application beyond the magnetosphere.

Recent advancements in the capabilities of large language models (LLMs) have paved the way for a myriad of groundbreaking applications in various fields. However, a significant challenge arises as these models often "hallucinate", i.e., fabricate facts without providing users an apparent means to discern the veracity of their statements. Uncertainty estimation (UE) methods are one path to safer, more responsible, and more effective use of LLMs. However, to date, research on UE methods for LLMs has been focused primarily on theoretical rather than engineering contributions. In this work, we tackle this issue by introducing LM-Polygraph, a framework with implementations of a battery of state-of-the-art UE methods for LLMs in text generation tasks, with unified program interfaces in Python. Additionally, it introduces an extendable benchmark for consistent evaluation of UE techniques by researchers, and a demo web application that enriches the standard chat dialog with confidence scores, empowering end-users to discern unreliable responses. LM-Polygraph is compatible with the most recent LLMs, including BLOOMz, LLaMA-2, ChatGPT, and GPT-4, and is designed to support future releases of similarly-styled LMs., Comment: Accepted at EMNLP-2023

Object-centric architectures usually apply a differentiable module to the entire feature map to decompose it into sets of entity representations called slots. Some of these methods structurally resemble clustering algorithms, where the cluster's center in latent space serves as a slot representation. Slot Attention is an example of such a method, acting as a learnable analog of the soft k-means algorithm. Our work employs a learnable clustering method based on the Gaussian Mixture Model. Unlike other approaches, we represent slots not only as centers of clusters but also incorporate information about the distance between clusters and assigned vectors, leading to more expressive slot representations. Our experiments demonstrate that using this approach instead of Slot Attention improves performance in object-centric scenarios, achieving state-of-the-art results in the set property prediction task., Comment: 17 pages, 6 figures

Visual object navigation using learning methods is one of the key tasks in mobile robotics. This paper introduces a new representation of a scene semantic map formed during the embodied agent interaction with the indoor environment. It is based on a neural network method that adjusts the weights of the segmentation model with backpropagation of the predicted fusion loss values during inference on a regular (backward) or delayed (forward) image sequence. We have implemented this representation into a full-fledged navigation approach called SkillTron, which can select robot skills from end-to-end policies based on reinforcement learning and classic map-based planning methods. The proposed approach makes it possible to form both intermediate goals for robot exploration and the final goal for object navigation. We conducted intensive experiments with the proposed approach in the Habitat environment, which showed a significant superiority in navigation quality metrics compared to state-of-the-art approaches. The developed code and used custom datasets are publicly available at github.com/AIRI-Institute/skill-fusion.

Molecular conformation optimization is crucial to computer-aided drug discovery and materials design. Traditional energy minimization techniques rely on iterative optimization methods that use molecular forces calculated by a physical simulator (oracle) as anti-gradients. However, this is a computationally expensive approach that requires many interactions with a physical simulator. One way to accelerate this procedure is to replace the physical simulator with a neural network. Despite recent progress in neural networks for molecular conformation energy prediction, such models are prone to distribution shift, leading to inaccurate energy minimization. We find that the quality of energy minimization with neural networks can be improved by providing optimization trajectories as additional training data. Still, it takes around $5 \times 10^5$ additional conformations to match the physical simulator's optimization quality. In this work, we present the Gradual Optimization Learning Framework (GOLF) for energy minimization with neural networks that significantly reduces the required additional data. The framework consists of an efficient data-collecting scheme and an external optimizer. The external optimizer utilizes gradients from the energy prediction model to generate optimization trajectories, and the data-collecting scheme selects additional training data to be processed by the physical simulator. Our results demonstrate that the neural network trained with GOLF performs on par with the oracle on a benchmark of diverse drug-like molecules using $50$x less additional data., Comment: Published as a conference paper at ICLR2024 (Poster)

There have recently been significant advances in the problem of unsupervised object-centric representation learning and its application to downstream tasks. The latest works support the argument that employing disentangled object representations in image-based object-centric reinforcement learning tasks facilitates policy learning. We propose a novel object-centric reinforcement learning algorithm combining actor-critic and model-based approaches to utilize these representations effectively. In our approach, we use a transformer encoder to extract object representations and graph neural networks to approximate the dynamics of an environment. The proposed method fills a research gap in developing efficient object-centric world models for reinforcement learning settings that can be used for environments with discrete or continuous action spaces. Our algorithm performs better in a visually complex 3D robotic environment and a 2D environment with compositional structure than the state-of-the-art model-free actor-critic algorithm built upon transformer architecture and the state-of-the-art monolithic model-based algorithm.

Model predictive control (MPC) may provide local motion planning for mobile robotic platforms. The challenging aspect is the analytic representation of collision cost for the case when both the obstacle map and robot footprint are arbitrary. We propose a Neural Potential Field: a neural network model that returns a differentiable collision cost based on robot pose, obstacle map, and robot footprint. The differentiability of our model allows its usage within the MPC solver. It is computationally hard to solve problems with a very high number of parameters. Therefore, our architecture includes neural image encoders, which transform obstacle maps and robot footprints into embeddings, which reduce problem dimensionality by two orders of magnitude. The reference data for network training are generated based on algorithmic calculation of a signed distance function. Comparative experiments showed that the proposed approach is comparable with existing local planners: it provides trajectories with outperforming smoothness, comparable path length, and safe distance from obstacles. Experiment on Husky UGV mobile robot showed that our approach allows real-time and safe local planning. The code for our approach is presented at https://github.com/cog-isa/NPField together with demo video.

This paper presents a novel approach to address the challenge of online temporal memory learning for decision-making under uncertainty in non-stationary, partially observable environments. The proposed algorithm, Distributed Hebbian Temporal Memory (DHTM), is based on factor graph formalism and a multicomponent neuron model. DHTM aims to capture sequential data relationships and make cumulative predictions about future observations, forming Successor Features (SF). Inspired by neurophysiological models of the neocortex, the algorithm utilizes distributed representations, sparse transition matrices, and local Hebbian-like learning rules to overcome the instability and slow learning process of traditional temporal memory algorithms like RNN and HMM. Experimental results demonstrate that DHTM outperforms LSTM and a biologically inspired HMM-like algorithm, CSCG, in the case of non-stationary datasets. Our findings suggest that DHTM is a promising approach for addressing the challenges of online sequence learning and planning in dynamic environments., Comment: 20 pages, 9 figures

The intensive development of technology for fabrication semiconducting CVD diamond layers poses an important task of developing a precise and non-destructive method for estimation the boron content in thin epitaxial layers. For bulk and uniformly doped diamond samples, the infrared optical spectroscopy successfully performs such a role. Here we propose a correct method to determine the boron concentration in CVD homoepitaxial diamond layers from FTIR spectra. The method is the natural advancement of the existing technique for bulk samples. The feature of the novel technique is the accurate accounting of passing radiation through a multilayered structure with different thicknesses of absorbing media for special absorbing mechanisms. For this situation, an expression for the effective optical density is obtained. We have demonstrated the benefit of the method for a set of samples with CVD homoepitaxial layers grown on various HPHT substrates with and without nitrogen impurity. The measured FTIR spectra were subdivided into relevant sections responsible for the specific absorption mechanisms, and the correct amplitudes of the boron absorption peaks were derived. The data obtained from FTIR spectra is thoroughly compared to the charge carrier concentration derived from electrical capacitance-voltage measurements., Comment: 18 pages, 7 figures, 1 table

This paper presents an adaptive transformer model named SegmATRon for embodied image semantic segmentation. Its distinctive feature is the adaptation of model weights during inference on several images using a hybrid multicomponent loss function. We studied this model on datasets collected in the photorealistic Habitat and the synthetic AI2-THOR Simulators. We showed that obtaining additional images using the agent's actions in an indoor environment can improve the quality of semantic segmentation. The code of the proposed approach and datasets are publicly available at https://github.com/wingrune/SegmATRon., Comment: 14 pages, 6 figures

We consider the occupation measure of the cut points of a simple random walk on a $d$-dimensional cubic lattice for $d = 2, 3$, and we show that the scaling limit of the occupation measure in weak topology is the natural fractal measure on the Brownian cut points defined via its Minkowski content., Comment: 57 pages, 12 figures

Multi-agent Pathfinding (MAPF) problem generally asks to find a set of conflict-free paths for a set of agents confined to a graph and is typically solved in a centralized fashion. Conversely, in this work, we investigate the decentralized MAPF setting, when the central controller that posses all the information on the agents' locations and goals is absent and the agents have to sequientially decide the actions on their own without having access to a full state of the environment. We focus on the practically important lifelong variant of MAPF, which involves continuously assigning new goals to the agents upon arrival to the previous ones. To address this complex problem, we propose a method that integrates two complementary approaches: planning with heuristic search and reinforcement learning through policy optimization. Planning is utilized to construct and re-plan individual paths. We enhance our planning algorithm with a dedicated technique tailored to avoid congestion and increase the throughput of the system. We employ reinforcement learning to discover the collision avoidance policies that effectively guide the agents along the paths. The policy is implemented as a neural network and is effectively trained without any reward-shaping or external guidance. We evaluate our method on a wide range of setups comparing it to the state-of-the-art solvers. The results show that our method consistently outperforms the learnable competitors, showing higher throughput and better ability to generalize to the maps that were unseen at the training stage. Moreover our solver outperforms a rule-based one in terms of throughput and is an order of magnitude faster than a state-of-the-art search-based solver., Comment: 12 pages, 11 figures

The q-state Potts model on a diamond chain has mathematical significance in analyzing phase transitions and critical behaviors in diverse fields, including statistical physics, condensed matter physics, and materials science. By focusing on the 3-state Potts model on a diamond chain, we reveal rich and analytically solvable behaviors without phase transitions at finite temperatures. Upon investigating thermodynamic properties such as internal energy, entropy, specific heat, and correlation length, we observe sharp changes near zero temperature. Magnetic properties, including magnetization and magnetic susceptibility, display distinct behaviors that provide insights into spin configurations in different phases. However, the Potts model lacks genuine phase transitions at finite temperatures, in line with the Peierls argument for one-dimensional systems. Nonetheless, in the general case of an arbitrary $q$-state, magnetic properties such as correlation length, magnetization, and magnetic susceptibility exhibit intriguing remnants of a zero-temperature phase transition at finite temperatures. Furthermore, residual entropy uncovers unusual frustrated regions at zero-temperature phase transitions. This feature leads to the peculiar thermodynamic properties of phase boundaries, including a sharp entropy change resembling a first-order discontinuity without an entropy jump, and pronounced peaks in second-order derivatives of free energy, suggestive of a second-order phase transition divergence, but without singularities. This unusual behavior is also observed in the correlation length at the pseudo-critical temperature, which could potentially be misleading as a divergence., Comment: 17 pages, 12 figures

Prediction with the possibility of abstention (or selective prediction) is an important problem for error-critical machine learning applications. While well-studied in the classification setup, selective approaches to regression are much less developed. In this work, we consider the nonparametric heteroskedastic regression problem and develop an abstention procedure via testing the hypothesis on the value of the conditional variance at a given point. Unlike existing methods, the proposed one allows to account not only for the value of the variance itself but also for the uncertainty of the corresponding variance predictor. We prove non-asymptotic bounds on the risk of the resulting estimator and show the existence of several different convergence regimes. Theoretical analysis is illustrated with a series of experiments on simulated and real-world data.

Fundamental studies of the interaction of chiral light with chiral matter are important for the development of techniques that allow handedness-selective optical detection of chiral organic molecules. One approach to achieve this goal is the creation of a Fabry-P\'erot cavity that supports eigenmodes with a desired electromagnetic handedness, which interacts differently with left and right molecular enantiomers. In this paper, we theoretically study chiral Fabry-P\'erot cavities with mirrors comprising one-dimensional photonic crystal slabs made of van der Waals As$_2$S$_3$, a material with one of the highest known in-plane anisotropy. By utilizing the anisotropy degree of freedom provided by As$_2$S$_3$, we design Fabry-P\'erot cavities with constitutional and configurational geometrical chiralities. We demonstrate that in cavities with constitutional chirality, electromagnetic modes of left or right handedness exist due to the chirality of both mirrors, often referred to as handedness preserving mirrors in the literature. At the same time, cavities with configurational chirality support modes of both handednesses due to chiral morphology of the entire structure, set by the twist angle between the optical axes of the upper and lower non-chiral anisotropic mirrors. The developed chiral Fabry-P\'erot cavities can be tuned to the technologically available distance between the mirrors by properly twisting them, making such systems a prospective platform for the coupling of chiral light with chiral matter., Comment: 33 pages, 9 figures

We study multi-phase Stefan problem with increasing Riemann initial data and with generally negative latent specific heats for the phase transitions. We propose the variational formulation of self-similar solutions, which allows to find precise conditions for existence and uniqueness of the solution., Comment: 9 pages

While reinforcement learning algorithms have had great success in the field of autonomous navigation, they cannot be straightforwardly applied to the real autonomous systems without considering the safety constraints. The later are crucial to avoid unsafe behaviors of the autonomous vehicle on the road. To highlight the importance of these constraints, in this study, we compare two learnable navigation policies: safe and unsafe. The safe policy takes the constraints into account, while the other does not. We show that the safe policy is able to generate trajectories with more clearance (distance to the obstacles) and makes less collisions while training without sacrificing the overall performance., Comment: 4 pages, 5 figures

Community detection is one of the most critical problems in modern network science. Its applications can be found in various fields, from protein modeling to social network analysis. Recently, many papers appeared studying the problem of overlapping community detection, where each node of a network may belong to several communities. In this work, we consider Mixed-Membership Stochastic Block Model (MMSB) first proposed by Airoldi et al. (2008). MMSB provides quite a general setting for modeling overlapping community structure in graphs. The central question of this paper is to reconstruct relations between communities given an observed network. We compare different approaches and establish the minimax lower bound on the estimation error. Then, we propose a new estimator that matches this lower bound. Theoretical results are proved under fairly general conditions on the considered model. Finally, we illustrate the theory in a series of experiments.