Local volatility is a versatile option pricing model due to its state dependent diffusion coefficient. Calibration is, however, non-trivial as it involves both proposing a hypothesis model of the latent function and a method for fitting it to data. In this paper we present novel Bayesian inference with Gaussian process priors. We obtain a rich representation of the local volatility function with a probabilistic notion of uncertainty attached to the calibrate. We propose an inference algorithm and apply our approach to S&P 500 market data.

Deep learning has achieved many breakthroughs in modern classification tasks. Numerous architectures have been proposed for different data structures but when it comes to the loss function, the cross-entropy loss is the predominant choice. Recently, several alternative losses have seen revived interests for deep classifiers. In particular, empirical evidence seems to promote square loss but a theoretical justification is still lacking. In this work, we contribute to the theoretical understanding of square loss in classification by systematically investigating how it performs for overparametrized neural networks in the neural tangent kernel (NTK) regime. Interesting properties regarding the generalization error, robustness, and calibration error are revealed. We consider two cases, according to whether classes are separable or not. In the general non-separable case, fast convergence rate is established for both misclassification rate and calibration error. When classes are separable, the misclassification rate improves to be exponentially fast. Further, the resulting margin is proven to be lower bounded away from zero, providing theoretical guarantees for robustness. We expect our findings to hold beyond the NTK regime and translate to practical settings. To this end, we conduct extensive empirical studies on practical neural networks, demonstrating the effectiveness of square loss in both synthetic low-dimensional data and real image data. Comparing to cross-entropy, square loss has comparable generalization error but noticeable advantages in robustness and model calibration.

Causal discovery aims to learn a causal graph from observational data. To date, most causal discovery methods require data to be stored in a central server. However, data owners gradually refuse to share their personalized data to avoid privacy leakage, making this task more troublesome by cutting off the first step. A puzzle arises: $\textit{how do we infer causal relations from decentralized data?}$ In this paper, with the additive noise model assumption of data, we take the first step in developing a gradient-based learning framework named DAG-Shared Federated Causal Discovery (DS-FCD), which can learn the causal graph without directly touching local data and naturally handle the data heterogeneity. DS-FCD benefits from a two-level structure of each local model. The first level learns the causal graph and communicates with the server to get model information from other clients, while the second level approximates causal mechanisms and personally updates from its own data to accommodate the data heterogeneity. Moreover, DS-FCD formulates the overall learning task as a continuous optimization problem by taking advantage of an equality acyclicity constraint, which can be naturally solved by gradient descent methods. Extensive experiments on both synthetic and real-world datasets verify the efficacy of the proposed method.

In this work we explore a new framework for approximate Bayesian inference in large datasets based on stochastic control. We advocate stochastic control as a finite time and low variance alternative to popular steady-state methods such as stochastic gradient Langevin dynamics (SGLD). Furthermore, we discuss and adapt the existing theoretical guarantees of this framework and establish connections to already existing VI routines in SDE-based models.

Neural Networks play a growing role in many science disciplines, including physics. Variational Autoencoders (VAEs) are neural networks that are able to represent the essential information of a high dimensional data set in a low dimensional latent space, which have a probabilistic interpretation. In particular the so-called encoder network, the first part of the VAE, which maps its input onto a position in latent space, additionally provides uncertainty information in terms of a variance around this position. In this work, an extension to the Autoencoder architecture is introduced, the FisherNet. In this architecture, the latent space uncertainty is not generated using an additional information channel in the encoder, but derived from the decoder, by means of the Fisher information metric. This architecture has advantages from a theoretical point of view as it provides a direct uncertainty quantification derived from the model, and also accounts for uncertainty cross-correlations. We can show experimentally that the FisherNet produces more accurate data reconstructions than a comparable VAE and its learning performance also apparently scales better with the number of latent space dimensions.

Humans and animals explore their environment and acquire useful skills even in the absence of clear goals, exhibiting intrinsic motivation. The study of intrinsic motivation in artificial agents is concerned with the following question: what is a good general-purpose objective for an agent? We study this question in dynamic partially-observed environments, and argue that a compact and general learning objective is to minimize the entropy of the agent's state visitation estimated using a latent state-space model. This objective induces an agent to both gather information about its environment, corresponding to reducing uncertainty, and to gain control over its environment, corresponding to reducing the unpredictability of future world states. We instantiate this approach as a deep reinforcement learning agent equipped with a deep variational Bayes filter. We find that our agent learns to discover, represent, and exercise control of dynamic objects in a variety of partially-observed environments sensed with visual observations without extrinsic reward.

Weak supervision (WS) frameworks are a popular way to bypass hand-labeling large datasets for training data-hungry models. These approaches synthesize multiple noisy but cheaply-acquired estimates of labels into a set of high-quality pseudolabels for downstream training. However, the synthesis technique is specific to a particular kind of label, such as binary labels or sequences, and each new label type requires manually designing a new synthesis algorithm. Instead, we propose a universal technique that enables weak supervision over any label type while still offering desirable properties, including practical flexibility, computational efficiency, and theoretical guarantees. We apply this technique to important problems previously not tackled by WS frameworks including learning to rank, regression, and learning in hyperbolic manifolds. Theoretically, our synthesis approach produces a consistent estimator for learning a challenging but important generalization of the exponential family model. Experimentally, we validate our framework and show improvement over baselines in diverse settings including real-world learning-to-rank and regression problems along with learning on hyperbolic manifolds.

In this research, we extend the universal reinforcement learning (URL) agent models of artificial general intelligence to quantum environments. The utility function of a classical exploratory stochastic Knowledge Seeking Agent, KL-KSA, is generalized to distance measures from quantum information theory on density matrices. Quantum process tomography (QPT) algorithms form the tractable subset of programs for modeling environmental dynamics. The optimal QPT policy is selected based on a mutable cost function based on algorithmic complexity as well as computational resource complexity. Instead of Turing machines, we estimate the cost metrics on a high-level language to allow realistic experimentation. The entire agent design is encapsulated in a self-replicating quine which mutates the cost function based on the predictive value of the optimal policy choosing scheme. Thus, multiple agents with pareto-optimal QPT policies evolve using genetic programming, mimicking the development of physical theories each with different resource trade-offs. This formal framework is termed Quantum Knowledge Seeking Agent (QKSA). Despite its importance, few quantum reinforcement learning models exist in contrast to the current thrust in quantum machine learning. QKSA is the first proposal for a framework that resembles the classical URL models. Similar to how AIXI-tl is a resource-bounded active version of Solomonoff universal induction, QKSA is a resource-bounded participatory observer framework to the recently proposed algorithmic information-based reconstruction of quantum mechanics. QKSA can be applied for simulating and studying aspects of quantum information theory. Specifically, we demonstrate that it can be used to accelerate quantum variational algorithms which include tomographic reconstruction as its integral subroutine.

Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. 2) For Gaussian RBNs, we additionally derive an analytic approximation, allowing for robust parameter optimisation and Bayesian inference. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations.

Probability is one of the most common terminologies, not only in mathematics but also in the real world. We use the word probability frequently. About seven years ago, I was in my secondary level of education and got introduced to the term probability as a topic of mathematics. At that time, I had solved so many mathematical problems regarding probability. Unfortunately, it did not seem interesting to me.