In recent years, deep neural networks (DNNs) have achieved remarkable success in various tasks such as image recognition, natural language processing, and speech recognition. However, despite their excellent performance, these models have certain limitations, such as their lack of uncertainty quantification (UQ). Much of UQ for DNNs is based on a Bayesian approach that models network weights as random variables [22] or has involved ensembles of networks [18]. Gaussian processes (GPs) provide natural UQ, but they lack the representation learning that makes DNNs successful. Standard GPs are known to scale poorly with large datasets, but GP approximations are plentiful and one such method is the Vecchia approximation [31, 16], which uses nearest-neighbor conditioning sets to exploit conditional independence among the data.

We present MULTIGAIN 2.0, a major extension to the controller synthesis tool MultiGain, built on top of the probabilistic model checker PRISM. This new version extends MultiGain's multi-objective capabilities, by allowing for the formal verification and synthesis of controllers for probabilistic systems with multi-dimensional long-run average reward structures, steady-state constraints, and linear temporal logic properties. Additionally, MULTIGAIN 2.0 provides an approach for finding finite memory solutions and the capability for two- and three-dimensional visualization of Pareto curves to facilitate trade-off analysis in multi-objective scenarios

A topological neural network (TNN), which takes data from a Tychonoff topological space instead of the usual finite dimensional space, is introduced. As a consequence, a distributional neural network (DNN) that takes Borel measures as data is also introduced. Combined these new neural networks facilitate things like recognizing long range dependence, heavy tails and other properties in stochastic process paths or like acting on belief states produced by particle filtering or hidden Markov model algorithms. The veracity of the TNN and DNN are then established herein by a strong universal approximation theorem for Tychonoff spaces and its corollary for spaces of measures. These theorems show that neural networks can arbitrarily approximate uniformly continuous functions (with respect to the sup metric) associated with a unique uniformity. We also provide some discussion showing that neural networks on positive-finite measures are a generalization of the recent deep learning notion of deep sets.

We consider the finite sum minimization of $n$ strongly convex and smooth functions with Lipschitz continuous Hessians in $d$ dimensions. In many applications where such problems arise, including maximum likelihood estimation, empirical risk minimization, and unsupervised learning, the number of observations $n$ is large, and it becomes necessary to use incremental or stochastic algorithms whose per-iteration complexity is independent of $n$. Of these, the incremental/stochastic variants of the Newton method exhibit superlinear convergence, but incur a per-iteration complexity of $O(d^3)$, which may be prohibitive in large-scale settings. On the other hand, the incremental Quasi-Newton method incurs a per-iteration complexity of $O(d^2)$ but its superlinear convergence rate has only been characterized asymptotically. This work puts forth the Sharpened Lazy Incremental Quasi-Newton (SLIQN) method that achieves the best of both worlds: an explicit superlinear convergence rate with a per-iteration complexity of $O(d^2)$. Building upon the recently proposed Sharpened Quasi-Newton method, the proposed incremental variant incorporates a hybrid update strategy incorporating both classic and greedy BFGS updates. The proposed lazy update rule distributes the computational complexity between the iterations, so as to enable a per-iteration complexity of $O(d^2)$. Numerical tests demonstrate the superiority of SLIQN over all other incremental and stochastic Quasi-Newton variants.

This study presents our team PathToMyHeart's contribution to the George B. Moody PhysioNet Challenge 2022. Two models are implemented. The first model is a Dual Bayesian ResNet (DBRes), where each patient's recording is segmented into overlapping log mel spectrograms. These undergo two binary classifications: present versus unknown or absent, and unknown versus present or absent. The classifications are aggregated to give a patient's final classification. The second model is the output of DBRes integrated with demographic data and signal features using XGBoost.DBRes achieved our best weighted accuracy of $0.771$ on the hidden test set for murmur classification, which placed us fourth for the murmur task. (On the clinical outcome task, which we neglected, we scored 17th with costs of $12637$.) On our held-out subset of the training set, integrating the demographic data and signal features improved DBRes's accuracy from $0.762$ to $0.820$. However, this decreased DBRes's weighted accuracy from $0.780$ to $0.749$. Our results demonstrate that log mel spectrograms are an effective representation of heart sound recordings, Bayesian networks provide strong supervised classification performance, and treating the ternary classification as two binary classifications increases performance on the weighted accuracy.

Centralized Training with Decentralized Execution (CTDE) has recently emerged as a popular framework for cooperative Multi-Agent Reinforcement Learning (MARL), where agents can use additional global state information to guide training in a centralized way and make their own decisions only based on decentralized local policies. Despite the encouraging results achieved, CTDE makes an independence assumption on agent policies, which limits agents from adopting global cooperative information from each other during centralized training. Therefore, we argue that the existing CTDE framework cannot fully utilize global information for training, leading to an inefficient joint-policy exploration and even suboptimal results. In this paper, we introduce a novel Centralized Advising and Decentralized Pruning (CADP) framework for multi-agent reinforcement learning, that not only enables an efficacious message exchange among agents during training but also guarantees the independent policies for execution. Firstly, CADP endows agents the explicit communication channel to seek and take advice from different agents for more centralized training. To further ensure the decentralized execution, we propose a smooth model pruning mechanism to progressively constrain the agent communication into a closed one without degradation in agent cooperation capability. Empirical evaluations on StarCraft II micromanagement challenge and Google Research Football benchmarks and and across different MARL backbones demonstrate that the proposed framework achieves superior performance compared with the state-of-the-art counterparts. Our code is available at https://github.com/zyh1999/CADP.

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimization challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.

Machine Learning and Deep Learning have achieved an impressive standard today, enabling us to answer questions that were inconceivable a few years ago. Besides these successes, it becomes clear, that beyond pure prediction, which is the primary strength of most supervised machine learning algorithms, the quantification of uncertainty is relevant and necessary as well. While first concepts and ideas in this direction have emerged in recent years, this paper adopts a conceptual perspective and examines possible sources of uncertainty. By adopting the viewpoint of a statistician, we discuss the concepts of aleatoric and epistemic uncertainty, which are more commonly associated with machine learning. The paper aims to formalize the two types of uncertainty and demonstrates that sources of uncertainty are miscellaneous and can not always be decomposed into aleatoric and epistemic. Drawing parallels between statistical concepts and uncertainty in machine learning, we also demonstrate the role of data and their influence on uncertainty.

We exploit a formal correspondence between thermodynamics and inference, where the number of samples can be thought of as the inverse temperature, to define a "learning capacity'' which is a measure of the effective dimensionality of a model. We show that the learning capacity is a tiny fraction of the number of parameters for many deep networks trained on typical datasets, depends upon the number of samples used for training, and is numerically consistent with notions of capacity obtained from the PAC-Bayesian framework. The test error as a function of the learning capacity does not exhibit double descent. We show that the learning capacity of a model saturates at very small and very large sample sizes; this provides guidelines, as to whether one should procure more data or whether one should search for new architectures, to improve performance. We show how the learning capacity can be used to understand the effective dimensionality, even for non-parametric models such as random forests and $k$-nearest neighbor classifiers.

Autonomous systems often have logical constraints arising, for example, from safety, operational, or regulatory requirements. Such constraints can be expressed using temporal logic specifications. The system state is often partially observable. Moreover, it could encompass a team of multiple agents with a common objective but disparate information structures and constraints. In this paper, we first introduce an optimal control theory for partially observable Markov decision processes (POMDPs) with finite linear temporal logic constraints. We provide a structured methodology for synthesizing policies that maximize a cumulative reward while ensuring that the probability of satisfying a temporal logic constraint is sufficiently high. Our approach comes with guarantees on approximate reward optimality and constraint satisfaction. We then build on this approach to design an optimal control framework for logically constrained multi-agent settings with information asymmetry. We illustrate the effectiveness of our approach by implementing it on several case studies.