In this work, we present a sampling algorithm for single hidden layer neural networks. This algorithm is built upon a recursive series of Bayesian posteriors using a method we call Greedy Bayes. Sampling of the Bayesian posterior for neuron weight vectors $w$ of dimension $d$ is challenging because of its multimodality. Our algorithm to tackle this problem is based on a coupling of the posterior density for $w$ with an auxiliary random variable $\xi$. The resulting reverse conditional $w|\xi$ of neuron weights given auxiliary random variable is shown to be log concave. In the construction of the posterior distributions we provide some freedom in the choice of the prior. In particular, for Gaussian priors on $w$ with suitably small variance, the resulting marginal density of the auxiliary variable $\xi$ is proven to be strictly log concave for all dimensions $d$. For a uniform prior on the unit $\ell_1$ ball, evidence is given that the density of $\xi$ is again strictly log concave for sufficiently large $d$. The score of the marginal density of the auxiliary random variable $\xi$ is determined by an expectation over $w|\xi$ and thus can be computed by various rapidly mixing Markov Chain Monte Carlo methods. Moreover, the computation of the score of $\xi$ permits methods of sampling $\xi$ by a stochastic diffusion (Langevin dynamics) with drift function built from this score. With such dynamics, information-theoretic methods pioneered by Bakry and Emery show that accurate sampling of $\xi$ is obtained rapidly when its density is indeed strictly log-concave. After which, one more draw from $w|\xi$, produces neuron weights $w$ whose marginal distribution is from the desired posterior.

Causal discovery from observational data holds great promise, but existing methods rely on strong assumptions about the underlying causal structure, often requiring full observability of all relevant variables. We tackle these challenges by leveraging the score function $\nabla \log p(X)$ of observed variables for causal discovery and propose the following contributions. First, we generalize the existing results of identifiability with the score to additive noise models with minimal requirements on the causal mechanisms. Second, we establish conditions for inferring causal relations from the score even in the presence of hidden variables; this result is two-faced: we demonstrate the score's potential as an alternative to conditional independence tests to infer the equivalence class of causal graphs with hidden variables, and we provide the necessary conditions for identifying direct causes in latent variable models. Building on these insights, we propose a flexible algorithm for causal discovery across linear, nonlinear, and latent variable models, which we empirically validate.

Theory of Mind (ToM), the ability to attribute beliefs, intentions, or mental states to others, is a crucial feature of human social interaction. In complex environments, where the human sensory system reaches its limits, behaviour is strongly driven by ou r beliefs about the state of the world around us. Accessing others' mental states, e.g., beliefs and intentions, allows for more effective social interactions in natural contexts. Yet, these variables are not directly observable, making understanding ToM a challenging quest of interest for different fields, including psychology, machine learning and robotics. In this paper, we contribute to this topic by showing a developmental synergy between learning to predict low - level mental states (e.g., intentions, g oals) and attributing high - level ones (i.e., beliefs). Specifically, we assume that learning beliefs attribution can occur by observing one's own decision processes involving beliefs, e.g., in a partially observable environment. Using a simple feed - forward deep learning model, we show that, when learning to predict others' intentions and actions, more accurate predictions can be acquired earlier if beliefs attribution is learnt simultaneously. Furthermore, we show that the learning performance improves even when observed actors have a different embodiment than the observer and the gain is higher when observing beliefs - driven chunks of behaviour. We propose that our computational approach can inform the understanding of human social cognitive development and be relevant for the design of future adaptive social robots able to autonomously understand, assist, and learn from human interaction partners in novel natural environments and tasks.

The Expectation Maximization (EM) algorithm has been a cent ral part of the statistician's toolbox since being formalised by [ 22 ] as an effective general computational solution to the marginal maximum likelihood problem. At that time the algor ithm had been proposed previously in numerous special contexts, including that of empirical Bayes [ 27 ]. Empirical Bayes methods have received considerable attention in the m odern machine learning literature, where they are widely used to specify hyper-paramete rs in high-dimensional models. In recent years there has been a great deal of interest within the Bayesian statistics and machine learning communities in the construction of gradie nt flows, especially Wasserstein gradient flows, which underlie Langevin Monte Carlo algorit hms. Some recent work has focussed on the intersection of empirical Bayes type method s and gradient flow-based algorithms. Our aim is to demonstrate here that some of the tools, particularly those emerging from optimal transport and Wasserstein geometry, which hav e been developed in the context of these modern computational methods provide a natura l approach to the analysis of the EM algorithm itself--and many of its approximations. S uch analysis is quite direct, requires limited further technical work and yields state-o f-the-art conclusions under conditions which are, if anything, weaker than those ordinaril y employed in the quantitative analysis of EM algorithms. 1 In this paper we utilize the connection between EM and a coord inate-wise minimization algorithm applied to the free energy functional identified b y [ 43 ] to provide non-asymptotic error bounds for EM algorithms under an extended form of the l og-Sobolev inequality. To do this, we extend an argument commonly used to understand Eu clidean coordinate descent algorithms by comparison with gradient descent via the desc ent lemma [ 9, 8, 10 ], together with recently developed results for using and understandin g gradients on the product of Euclidean and Wasserstein spaces [ 13 ].

It is designed to help students and researchers to quickly familiarize themselves with the area and to provide a foundation for the development of university courses on the mathematics of deep learning. Our main goal in the composition of this book was to present various rigorous, but easy to grasp, results that help to build an understanding of fundamental mathematical concepts in deep learning. To achieve this, we prioritize simplicity over generality. As a mathematical introduction to deep learning, this book does not aim to give an exhaustive survey of the entire (and rapidly growing) field, and some important research directions are missing. In particular, we have favored mathematical results over empirical research, even though an accurate account of the theory of deep learning requires both.

Contracts are the economic framework which allows a principal to delegate a task to an agent -- despite misaligned interests, and even without directly observing the agent's actions. In many modern reinforcement learning settings, self-interested agents learn to perform a multi-stage task delegated to them by a principal. We explore the significant potential of utilizing contracts to incentivize the agents. We model the delegated task as an MDP, and study a stochastic game between the principal and agent where the principal learns what contracts to use, and the agent learns an MDP policy in response. We present a learning-based algorithm for optimizing the principal's contracts, which provably converges to the subgame-perfect equilibrium of the principal-agent game. A deep RL implementation allows us to apply our method to very large MDPs with unknown transition dynamics. We extend our approach to multiple agents, and demonstrate its relevance to resolving a canonical sequential social dilemma with minimal intervention to agent rewards.

Clustering algorithms frequently require the number of clusters to be chosen in advance, but it is usually not clear how to do this. To tackle this challenge when clustering within sequential data, we present a method for estimating the number of clusters when the data is a trajectory of a Block Markov Chain. Block Markov Chains are Markov Chains that exhibit a block structure in their transition matrix. The method considers a matrix that counts the number of transitions between different states within the trajectory, and transforms this into a spectral embedding whose dimension is set via singular value thresholding. The number of clusters is subsequently estimated via density-based clustering of this spectral embedding, an approach inspired by literature on the Stochastic Block Model. By leveraging and augmenting recent results on the spectral concentration of random matrices with Markovian dependence, we show that the method is asymptotically consistent - in spite of the dependencies between the count matrix's entries, and even when the count matrix is sparse. We also present a numerical evaluation of our method, and compare it to alternatives.

Off-policy evaluation (OPE) is widely applied in sectors such as pharmaceuticals and e-commerce to evaluate the efficacy of novel products or policies from offline datasets. This paper introduces a causal deepset framework that relaxes several key structural assumptions, primarily the mean-field assumption, prevalent in existing OPE methodologies that handle spatio-temporal interference. These traditional assumptions frequently prove inadequate in real-world settings, thereby restricting the capability of current OPE methods to effectively address complex interference effects. In response, we advocate for the implementation of the permutation invariance (PI) assumption. This innovative approach enables the data-driven, adaptive learning of the mean-field function, offering a more flexible estimation method beyond conventional averaging. Furthermore, we present novel algorithms that incorporate the PI assumption into OPE and thoroughly examine their theoretical foundations. Our numerical analyses demonstrate that this novel approach yields significantly more precise estimations than existing baseline algorithms, thereby substantially improving the practical applicability and effectiveness of OPE methodologies.

Resilience is defined as the ability of a network to resist, adapt, and quickly recover from disruptions, and to continue to maintain an acceptable level of services from users' perspective. With the advent of future radio networks, including advanced 5G and upcoming 6G, critical services become integral to future networks, requiring uninterrupted service delivery for end users. Unfortunately, with the growing network complexity, user mobility and diversity, it becomes challenging to scale current resilience management techniques that rely on local optimizations to large dense network deployments. This paper aims to address this problem by globally optimizing the resilience of a dense multi-cell network based on multi-agent deep reinforcement learning. Specifically, our proposed solution can dynamically tilt cell antennas and reconfigure transmit power to mitigate outages and increase both coverage and service availability. A multi-objective optimization problem is formulated to simultaneously satisfy resiliency constraints while maximizing the service quality in the network area in order to minimize the impact of outages on neighbouring cells. Extensive simulations then demonstrate that with our proposed solution, the average service availability in terms of user throughput can be increased by up to 50-60% on average, while reaching a coverage availability of 99% in best cases.

In this work we evaluate the utility of synthetic data for training automatic speech recognition (ASR). We use the ASR training data to train a text-to-speech (TTS) system similar to FastSpeech-2. With this TTS we reproduce the original training data, training ASR systems solely on synthetic data. For ASR, we use three different architectures, attention-based encoder-decoder, hybrid deep neural network hidden Markov model and a Gaussian mixture hidden Markov model, showing the different sensitivity of the models to synthetic data generation. In order to extend previous work, we present a number of ablation studies on the effectiveness of synthetic vs. real training data for ASR. In particular we focus on how the gap between training on synthetic and real data changes by varying the speaker embedding or by scaling the model size. For the latter we show that the TTS models generalize well, even when training scores indicate overfitting.