The problem of online linear regression is considered from an individual sequence perspective, where the aim is to control the square loss predictive regret with respect to the best linear predictor \theta \top x_t simultaneously for every sequence of covariate vectors x_t \in R d and outcomes y_t \in R in some constraint set. This is naturally formulated as a sequential game between the forecaster and an adversarial environment. In previous work [1], this problem was addressed in the "fixed-design" case, where the horizon T and the sequence of covariate vectors x_1 T is known in advance. The exact minimax strategy (MMS) was introduced and shown to be minimax optimal under natural constraint sets on the label sequence (such as ellipse-constrained labels). The MMS strategy consists in some form of least squares, but where the inverse cumulative covariance matrix \Pi_t {-1} is replaced by a shrunk version P_t that takes future instance into account.

Using Bayes's theorem, we derive a unit-wise recurrence as well as a backward recursion similar to the forward-backward algorithm. The resulting Bayesian recurrent units can be integrated as recurrent neural networks within deep learning frameworks, while retaining a probabilistic interpretation from the direct correspondence with hidden Markov models. Whilst the contribution is mainly theoretical, experiments on speech recognition indicate that adding the derived units at the end of state-of-the-art recurrent architectures can improve the performance at a very low cost in terms of trainable parameters.

This paper considers safe robot mission planning in uncertain dynamical environments. This problem arises in applications such as surveillance, emergency rescue, and autonomous driving. It is a challenging problem due to modeling and integrating dynamical uncertainties into a safe planning framework, and finding a solution in a computationally tractable way. In this work, we first develop a probabilistic model for dynamical uncertainties. Then, we provide a framework to generate a path that maximizes safety for complex missions by incorporating the uncertainty model. We also devise a Monte Carlo method to obtain a safe path efficiently. Finally, we evaluate the performance of our approach and compare it to potential alternatives in several case studies.

We begin by reiterating that common neural network activation functions have simple Bayesian origins. In this spirit, we go on to show that Bayes's theorem also implies a simple recurrence relation; this leads to a Bayesian recurrent unit with a prescribed feedback formulation. We show that introduction of a context indicator leads to a variable feedback that is similar to the forget mechanism in conventional recurrent units. A similar approach leads to a probabilistic input gate. The Bayesian formulation leads naturally to the two pass algorithm of the Kalman smoother or forward-backward algorithm, meaning that inference naturally depends upon future inputs as well as past ones. Experiments on speech recognition confirm that the resulting architecture can perform as well as a bidirectional recurrent network with the same number of parameters as a unidirectional one. Further, when configured explicitly bidirectionally, the architecture can exceed the performance of a conventional bidirectional recurrence.

The digital telecommunications receiver is an important context for inference methodology, the key objective being to minimize the expected loss function in recovering the transmitted information. For that criterion, the optimal decision is the Bayesian minimum-risk estimator. However, the computational load of the Bayesian estimator is often prohibitive and, hence, efficient computational schemes are required. The design of novel schemes, striking new balances between accuracy and computational load, is the primary concern of this thesis. Two popular techniques, one exact and one approximate, will be studied. The exact scheme is a recursive one, namely the generalized distributive law (GDL), whose purpose is to distribute all operators across the conditionally independent (CI) factors of the joint model, so as to reduce the total number of operators required. In a novel theorem derived in this thesis, GDL, if applicable, will be shown to guarantee such a reduction in all cases. An associated lemma also quantifies this reduction. For practical use, two novel algorithms, namely the no-longer-needed (NLN) algorithm and the generalized form of the Markovian Forward-Backward (FB) algorithm, recursively factorizes and computes the CI factors of an arbitrary model, respectively. The approximate scheme is an iterative one, namely the Variational Bayes (VB) approximation, whose purpose is to find the independent (i.e. zero-order Markov) model closest to the true joint model in the minimum Kullback-Leibler divergence (KLD) sense. Despite being computationally efficient, this naive mean field approximation confers only modest performance for highly correlated models. A novel approximation, namely Transformed Variational Bayes (TVB), will be designed in the thesis in order to relax the zero-order constraint in the VB approximation, further reducing the KLD of the optimal approximation.