We consider the problem of simulating diffusion bridges, i.e. diffusion processes that are conditioned to initialize and terminate at two given states. Diffusion bridge simulation has applications in diverse scientific fields and plays a crucial role for statistical inference of discretely-observed diffusions. This is known to be a challenging problem that has received much attention in the last two decades. In this work, we first show that the time-reversed diffusion bridge process can be simulated if one can time-reverse the unconditioned diffusion process. We introduce a variational formulation to learn this time-reversal that relies on a score matching method to circumvent intractability. We then consider another iteration of our proposed methodology to approximate the Doob's $h$-transform defining the diffusion bridge process. As our approach is generally applicable under mild assumptions on the underlying diffusion process, it can easily be used to improve the proposal bridge process within existing methods and frameworks. We discuss algorithmic considerations and extensions, and present some numerical results.

In reinforcement learning (RL), an agent must explore an initially unknown environment in order to learn a desired behaviour. When RL agents are deployed in real world environments, safety is of primary concern. Constrained Markov decision processes (CMDPs) can provide long-term safety constraints; however, the agent may violate the constraints in an effort to explore its environment. This paper proposes a model-based RL algorithm called Explicit Explore, Exploit, or Escape ($E^{4}$), which extends the Explicit Explore or Exploit ($E^{3}$) algorithm to a robust CMDP setting. $E^4$ explicitly separates exploitation, exploration, and escape CMDPs, allowing targeted policies for policy improvement across known states, discovery of unknown states, as well as safe return to known states. $E^4$ robustly optimises these policies on the worst-case CMDP from a set of CMDP models consistent with the empirical observations of the deployment environment. Theoretical results show that $E^4$ finds a near-optimal constraint-satisfying policy in polynomial time whilst satisfying safety constraints throughout the learning process. We discuss robust-constrained offline optimisation algorithms as well as how to incorporate uncertainty in transition dynamics of unknown states based on empirical inference and prior knowledge.

This formula was devised and penned by respected Thomas Bayes, renowned statistician.It is an arithmetical formula for determining conditional probability. Conditional probability is the likelihood of an outcome occurring, based on a previous outcome occurring. This might be a bit brain-teasing as you are working backwards. Bayes' theorem may be derived from the definition of conditional probability, P(Do not launch Stock price increases) 0.4 0.30 0.12 Thus, there is a 57% probability that the company's share price will increase. Bayes' Theorem has several forms.

This course is about the fundamental concepts of machine learning, focusing on regression, SVM, decision trees and neural networks. These topics are getting very hot nowadays because these learning algorithms can be used in several fields from software engineering to investment banking. Learning algorithms can recognize patterns which can help detect cancer for example or we may construct algorithms that can have a very good guess about stock prices movement in the market. In each section we will talk about the theoretical background for all of these algorithms then we are going to implement these problems together. We will use Python with SkLearn, Keras and TensorFlow.

This course is about the fundamental concepts of machine learning, focusing on regression, SVM, decision trees and neural networks. These topics are getting very hot nowadays because these learning algorithms can be used in several fields from software engineering to investment banking. Learning algorithms can recognize patterns which can help detect cancer for example or we may construct algorithms that can have a very good guess about stock prices movement in the market. In each section we will talk about the theoretical background for all of these algorithms then we are going to implement these problems together. We will use Python with SkLearn, Keras and TensorFlow.

This paper focuses on learning rate analysis of Nystr\"{o}m regularization with sequential sub-sampling for $\tau$-mixing time series. Using a recently developed Banach-valued Bernstein inequality for $\tau$-mixing sequences and an integral operator approach based on second-order decomposition, we succeed in deriving almost optimal learning rates of Nystr\"{o}m regularization with sequential sub-sampling for $\tau$-mixing time series. A series of numerical experiments are carried out to verify our theoretical results, showing the excellent learning performance of Nystr\"{o}m regularization with sequential sub-sampling in learning massive time series data. All these results extend the applicable range of Nystr\"{o}m regularization from i.i.d. samples to non-i.i.d. sequences.

Hamiltonian Monte Carlo (HMC) methods are widely used to draw samples from unnormalized target densities due to high efficiency and favorable scalability with respect to increasing space dimensions. However, HMC struggles when the target distribution is multimodal, because the maximum increase in the potential energy function (i.e., the negative log density function) along the simulated path is bounded by the initial kinetic energy, which follows a half of the $\chi_d^2$ distribution, where d is the space dimension. In this paper, we develop a Hamiltonian Monte Carlo method where the constructed paths can travel across high potential energy barriers. This method does not require the modes of the target distribution to be known in advance. Our approach enables frequent jumps between the isolated modes of the target density by continuously varying the mass of the simulated particle while the Hamiltonian path is constructed. Thus, this method can be considered as a combination of HMC and the tempered transitions method. Compared to other tempering methods, our method has a distinctive advantage in the Gibbs sampler settings, where the target distribution changes at each step. We develop a practical tuning strategy for our method and demonstrate that it can construct globally mixing Markov chains targeting high-dimensional, multimodal distributions, using mixtures of normals and a sensor network localization problem.

Experimental advances enabling high-resolution external control create new opportunities to produce materials with exotic properties. In this work, we investigate how a multi-agent reinforcement learning approach can be used to design external control protocols for self-assembly. We find that a fully decentralized approach performs remarkably well even with a "coarse" level of external control. More importantly, we see that a partially decentralized approach, where we include information about the local environment allows us to better control our system towards some target distribution. We explain this by analyzing our approach as a partially-observed Markov decision process. With a partially decentralized approach, the agent is able to act more presciently, both by preventing the formation of undesirable structures and by better stabilizing target structures as compared to a fully decentralized approach.

We consider off-policy evaluation (OPE) in Partially Observable Markov Decision Processes (POMDPs), where the evaluation policy depends only on observable variables and the behavior policy depends on unobservable latent variables. Existing works either assume no unmeasured confounders, or focus on settings where both the observation and the state spaces are tabular. As such, these methods suffer from either a large bias in the presence of unmeasured confounders, or a large variance in settings with continuous or large observation/state spaces. In this work, we first propose novel identification methods for OPE in POMDPs with latent confounders, by introducing bridge functions that link the target policy's value and the observed data distribution. In fully-observable MDPs, these bridge functions reduce to the familiar value functions and marginal density ratios between the evaluation and the behavior policies. We next propose minimax estimation methods for learning these bridge functions. Our proposal permits general function approximation and is thus applicable to settings with continuous or large observation/state spaces. Finally, we construct three estimators based on these estimated bridge functions, corresponding to a value function-based estimator, a marginalized importance sampling estimator, and a doubly-robust estimator. Their nonasymptotic and asymptotic properties are investigated in detail.

Obstacle avoidance for small unmanned aircraft is vital for the safety of future urban air mobility (UAM) and Unmanned Aircraft System (UAS) Traffic Management (UTM). There are many techniques for real-time robust drone guidance, but many of them solve in discretized airspace and control, which would require an additional path smoothing step to provide flexible commands for UAS. To provide a safe and efficient computational guidance of operations for unmanned aircraft, we explore the use of a deep reinforcement learning algorithm based on Proximal Policy Optimization (PPO) to guide autonomous UAS to their destinations while avoiding obstacles through continuous control. The proposed scenario state representation and reward function can map the continuous state space to continuous control for both heading angle and speed. To verify the performance of the proposed learning framework, we conducted numerical experiments with static and moving obstacles. Uncertainties associated with the environments and safety operation bounds are investigated in detail. Results show that the proposed model can provide accurate and robust guidance and resolve conflict with a success rate of over 99%.