We leverage multilevel Monte Carlo (MLMC) to improve the performance of multi-step look-ahead Bayesian optimization (BO) methods that involve nested expectations and maximizations. The complexity rate of naive Monte Carlo degrades for nested operations, whereas MLMC is capable of achieving the canonical Monte Carlo convergence rate for this type of problem, independently of dimension and without any smoothness assumptions. Our theoretical study focuses on the approximation improvements for one- and two-step look-ahead acquisition functions, but, as we discuss, the approach is generalizable in various ways, including beyond the context of BO. Findings are verified numerically and the benefits of MLMC for BO are illustrated on several benchmark examples. Code is available here https://github.com/Shangda-Yang/MLMCBO.

This paper describes an architecture for predicting the price of cryptocurrencies for the next seven days using the Adaptive Network Based Fuzzy Inference System (ANFIS). Historical data of cryptocurrencies and indexes that are considered are Bitcoin (BTC), Ethereum (ETH), Bitcoin Dominance (BTC.D), and Ethereum Dominance (ETH.D) in The architectural performance designed in this paper has been compared with different inputs and neural network models in terms of statistical evaluation criteria. Finally, the proposed method can predict the price of digital currencies in a short time. NTRODUCTION Digital currency is a form of electronic money that operates on the internet and possesses most of the attributes of conventional money, except for its physical absence. A subset of digital currency is cryptocurrency, which is encrypted by specific algorithms. These cryptocurrencies often utilize blockchain technology to record transactions [1]. The main distinction between cryptocurrencies and other digital currencies is the level of security of the former.

Privacy concerns have led to a surge in the creation of synthetic datasets, with diffusion models emerging as a promising avenue. Although prior studies have performed empirical evaluations on these models, there has been a gap in providing a mathematical characterization of their privacy-preserving capabilities. To address this, we present the pioneering theoretical exploration of the privacy preservation inherent in discrete diffusion models (DDMs) for discrete dataset generation. Focusing on per-instance differential privacy (pDP), our framework elucidates the potential privacy leakage for each data point in a given training dataset, offering insights into how the privacy loss of each point correlates with the dataset's distribution. Our bounds also show that training with $s$-sized data points leads to a surge in privacy leakage from $(\epsilon, O(\frac{1}{s^2\epsilon}))$-pDP to $(\epsilon, O(\frac{1}{s\epsilon}))$-pDP of the DDM during the transition from the pure noise to the synthetic clean data phase, and a faster decay in diffusion coefficients amplifies the privacy guarantee. Finally, we empirically verify our theoretical findings on both synthetic and real-world datasets.

This paper introduces Bayesian Flow Networks (BFNs), a new class of generative model in which the parameters of a set of independent distributions are modified with Bayesian inference in the light of noisy data samples, then passed as input to a neural network that outputs a second, interdependent distribution. Starting from a simple prior and iteratively updating the two distributions yields a generative procedure similar to the reverse process of diffusion models; however it is conceptually simpler in that no forward process is required. Discrete and continuous-time loss functions are derived for continuous, discretised and discrete data, along with sample generation procedures. Notably, the network inputs for discrete data lie on the probability simplex, and are therefore natively differentiable, paving the way for gradient-based sample guidance and few-step generation in discrete domains such as language modelling. The loss function directly optimises data compression and places no restrictions on the network architecture. In our experiments BFNs achieve competitive log-likelihoods for image modelling on dynamically binarized MNIST and CIFAR-10, and outperform all known discrete diffusion models on the text8 character-level language modelling task.

Pervasive across diverse domains, stochastic systems exhibit fluctuations in processes ranging from molecular dynamics to climate phenomena. The Langevin equation has served as a common mathematical model for studying such systems, enabling predictions of their temporal evolution and analyses of thermodynamic quantities, including absorbed heat, work done on the system, and entropy production. However, inferring the Langevin equation from observed trajectories remains challenging, particularly for nonlinear and high-dimensional systems. In this study, we present a comprehensive framework that employs Bayesian neural networks for inferring Langevin equations in both overdamped and underdamped regimes. Our framework first provides the drift force and diffusion matrix separately and then combines them to construct the Langevin equation. By providing a distribution of predictions instead of a single value, our approach allows us to assess prediction uncertainties, which can prevent potential misunderstandings and erroneous decisions about the system. We demonstrate the effectiveness of our framework in inferring Langevin equations for various scenarios including a neuron model and microscopic engine, highlighting its versatility and potential impact.

In advancing the understanding of decision-making processes, Inverse Reinforcement Learning (IRL) have proven instrumental in reconstructing animal's multiple intentions amidst complex behaviors. Given the recent development of a continuous-time multi-intention IRL framework, there has been persistent inquiry into inferring discrete time-varying rewards with IRL. To tackle the challenge, we introduce Latent (Markov) Variable Inverse Q-learning (L(M)V-IQL), a novel class of IRL algorthms tailored for accommodating discrete intrinsic reward functions. Leveraging an Expectation-Maximization approach, we cluster observed expert trajectories into distinct intentions and independently solve the IRL problem for each. Demonstrating the efficacy of L(M)V-IQL through simulated experiments and its application to different real mouse behavior datasets, our approach surpasses current benchmarks in animal behavior prediction, producing interpretable reward functions. This advancement holds promise for neuroscience and cognitive science, contributing to a deeper understanding of decision-making and uncovering underlying brain mechanisms.

Weakly supervised learning generally faces challenges in applicability to various scenarios with diverse weak supervision and in scalability due to the complexity of existing algorithms, thereby hindering the practical deployment. This paper introduces a general framework for learning from weak supervision (GLWS) with a novel algorithm. Central to GLWS is an Expectation-Maximization (EM) formulation, adeptly accommodating various weak supervision sources, including instance partial labels, aggregate statistics, pairwise observations, and unlabeled data. We further present an advanced algorithm that significantly simplifies the EM computational demands using a Non-deterministic Finite Automaton (NFA) along with a forward-backward algorithm, which effectively reduces time complexity from quadratic or factorial often required in existing solutions to linear scale. The problem of learning from arbitrary weak supervision is therefore converted to the NFA modeling of them. GLWS not only enhances the scalability of machine learning models but also demonstrates superior performance and versatility across 11 weak supervision scenarios. We hope our work paves the way for further advancements and practical deployment in this field.

Ecological rationality refers to the notion that humans are rational agents adapted to their environment. However, testing this theory remains challenging due to two reasons: the difficulty in defining what tasks are ecologically valid and building rational models for these tasks. In this work, we demonstrate that large language models can generate cognitive tasks, specifically category learning tasks, that match the statistics of real-world tasks, thereby addressing the first challenge. We tackle the second challenge by deriving rational agents adapted to these tasks using the framework of meta-learning, leading to a class of models called ecologically rational meta-learned inference (ERMI). ERMI quantitatively explains human data better than seven other cognitive models in two different experiments. It additionally matches human behavior on a qualitative level: (1) it finds the same tasks difficult that humans find difficult, (2) it becomes more reliant on an exemplar-based strategy for assigning categories with learning, and (3) it generalizes to unseen stimuli in a human-like way. Furthermore, we show that ERMI's ecologically valid priors allow it to achieve state-of-the-art performance on the OpenML-CC18 classification benchmark.

The expected loss is an upper bound to the model generalization error which admits robust PAC-Bayes bounds for learning. However, loss minimization is known to ignore misspecification, where models cannot exactly reproduce observations. This leads to significant underestimates of parameter uncertainties in the large data, or underparameterized, limit. We analyze the generalization error of near-deterministic, misspecified and underparametrized surrogate models, a regime of broad relevance in science and engineering. We show posterior distributions must cover every training point to avoid a divergent generalization error and derive an ensemble {ansatz} that respects this constraint, which for linear models incurs minimal overhead. The efficient approach is demonstrated on model problems before application to high dimensional datasets in atomistic machine learning. Parameter uncertainties from misspecification survive in the underparametrized limit, giving accurate prediction and bounding of test errors.

Bayesian optimisation requires fitting a Gaussian process model, which in turn requires specifying hyperparameters - most of the theoretical literature assumes those hyperparameters are known. The commonly used maximum likelihood estimator for hyperparameters of the Gaussian process is consistent only if the data fills the space uniformly, which does not have to be the case in Bayesian optimisation. Since no guarantees exist regarding the correctness of hyperparameter estimation, and those hyperparameters can significantly affect the Gaussian process fit, theoretical analysis of Bayesian optimisation with unknown hyperparameters is very challenging. Previously proposed algorithms with the no-regret property were only able to handle the special case of unknown lengthscales, reproducing kernel Hilbert space norm and applied only to the frequentist case. We propose a novel algorithm, HE-GP-UCB, which is the first algorithm enjoying the no-regret property in the case of unknown hyperparameters of arbitrary form, and which supports both Bayesian and frequentist settings. Our proof idea is novel and can easily be extended to other variants of Bayesian optimisation. We show this by extending our algorithm to the adversarially robust optimisation setting under unknown hyperparameters. Finally, we empirically evaluate our algorithm on a set of toy problems and show that it can outperform the maximum likelihood estimator.