We present a novel approach for motion planning in mobile robotics under sensing and motion uncertainty based on state lattices with graduated fidelity. The probability of collision is reliably estimated considering the robot shape, and the fidelity adapts to the complexity of the environment, improving the planning efficiency while maintaining the performance. Safe and optimal paths are found with an informed search algorithm, for which a novel multi-resolution heuristic is presented. Results for different scenarios and robot shapes are given, showing the validity of the proposed methods.

The ability of overparameterized deep networks to generalize well has been linked to the fact that stochastic gradient descent (SGD) finds solutions that lie in flat, wide minima in the training loss -- minima where the output of the network is resilient to small random noise added to its parameters. So far this observation has been used to provide generalization guarantees only for neural networks whose parameters are either \textit{stochastic} or \textit{compressed}. In this work, we present a general PAC-Bayesian framework that leverages this observation to provide a bound on the original network learned -- a network that is deterministic and uncompressed. What enables us to do this is a key novelty in our approach: our framework allows us to show that if on training data, the interactions between the weight matrices satisfy certain conditions that imply a wide training loss minimum, these conditions themselves {\em generalize} to the interactions between the matrices on test data, thereby implying a wide test loss minimum. We then apply our general framework in a setup where we assume that the pre-activation values of the network are not too small (although we assume this only on the training data). In this setup, we provide a generalization guarantee for the original (deterministic, uncompressed) network, that does not scale with product of the spectral norms of the weight matrices -- a guarantee that would not have been possible with prior approaches.

Given unpaired data from multiple domains, a key challenge is to efficiently exploit these data sources for modeling a target domain. Variants of this problem have been studied in many contexts, such as cross-domain translation and domain adaptation. We propose AlignFlow, a generative modeling framework for learning from multiple domains via normalizing flows. The use of normalizing flows in AlignFlow allows for a) flexibility in specifying learning objectives via adversarial training, maximum likelihood estimation, or a hybrid of the two methods; and b) exact inference of the shared latent factors across domains at test time. We derive theoretical results for the conditions under which AlignFlow guarantees marginal consistency for the different learning objectives. Furthermore, we show that AlignFlow guarantees exact cycle consistency in mapping datapoints from one domain to another. Empirically, AlignFlow can be used for data-efficient density estimation given multiple data sources and shows significant improvements over relevant baselines on unsupervised domain adaptation.

Active learning (AL) concerns itself with learning a model from as few labelled data as possible through actively and iteratively querying an oracle with selected unlabelled samples. In this paper, we focus on a popular type of AL in which the utility of a sample is measured by a specified goal achieved by the retrained model after accounting for the sample's marginal influence. Such AL strategies attract a lot of attention thanks to their intuitive motivations, yet they typically suffer from impractically high computational costs due to their need for many iterations of model retraining. With the help of influence functions, we present an effective approximation that bypasses model retraining altogether, and propose a general efficient implementation that makes such AL strategies applicable in practice, both in the serial and the more challenging batch-mode setting. Additionally, we present theoretical analyses which call into question a common practice widely adopted in the field. Finally, we carry out empirical studies with both synthetic and real-world datasets to validate our discoveries as well as showcase the potentials and issues with such goal-oriented AL strategies.

We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thus, the blessing of compositionality mitigates the curse of dimensionality. On the other hand, a theorem called good propagation of errors allows to `lift' theorems about shallow networks to those about deep networks with an appropriate choice of norms, smoothness, etc. We illustrate this in three contexts where each channel in the deep network calculates a spherical polynomial, a non-smooth ReLU network, or another zonal function network related closely with the ReLU network.

Recurrent neural networks (RNNs) have been extraordinarily successful for prediction with sequential data. To tackle highly variable and noisy real-world data, we introduce Particle Filter Recurrent Neural Networks (PF-RNNs), a new RNN family that explicitly models uncertainty in its internal structure: while an RNN relies on a long, deterministic latent state vector, a PF-RNN maintains a latent state distribution, approximated as a set of particles. For effective learning, we provide a fully differentiable particle filter algorithm that updates the PF-RNN latent state distribution according to the Bayes rule. Experiments demonstrate that the proposed PF-RNNs outperform the corresponding standard gated RNNs on a synthetic robot localization dataset and 10 real-world sequence prediction datasets for text classification, stock price prediction, etc.

While tasks could come with varying number of instances in realistic settings, the existing meta-learning approaches for few-shot classfication assume even task distributions where the number of instances for each task and class are fixed. Due to such restriction, they learn to equally utilize the meta-knowledge across all the tasks, even when the number of instances per task and class largely varies. Moreover, they do not consider distributional difference in unseen tasks at the meta-test time, on which the meta-knowledge may have varying degree of usefulness depending on the task relatedness. To overcome these limitations, we propose a novel meta-learning model that adaptively balances the effect of the meta-learning and task-specific learning, and also class-specific learning within each task. Through the learning of the balancing variables, we can decide whether to obtain a solution close to the initial parameter or far from it. We formulate this objective into a Bayesian inference framework and solve it using variational inference. Our Bayesian Task-Adaptive Meta-Learning (Bayesian-TAML) significantly outperforms existing meta-learning approaches on benchmark datasets for both few-shot and realistic class- and task-imbalanced datasets, with especially higher gains on the latter.

Mixtures of experts probabilistically divide the input space into regions, where the assumptions of each expert, or conditional model, need only hold locally. Combined with Gaussian process (GP) experts, this results in a powerful and highly flexible model. We focus on alternative mixtures of GP experts, which model the joint distribution of the inputs and targets explicitly. We highlight issues of this approach in multi-dimensional input spaces, namely, poor scalability and the need for an unnecessarily large number of experts, degrading the predictive performance and increasing uncertainty. We construct a novel model to address these issues through a nested partitioning scheme that automatically infers the number of components at both levels. Multiple response types are accommodated through a generalised GP framework, while multiple input types are included through a factorised exponential family structure. We show the effectiveness of our approach in estimating a parsimonious probabilistic description of both synthetic data of increasing dimension and an Alzheimer's challenge dataset.

Determinantal point processes (DPPs) have attracted substantial attention as an elegant probabilistic model that captures the balance between quality and diversity within sets. DPPs are conventionally parameterized by a positive semi-definite kernel matrix, and this symmetric kernel encodes only repulsive interactions between items. These so-called symmetric DPPs have significant expressive power, and have been successfully applied to a variety of machine learning tasks, including recommendation systems, information retrieval, and automatic summarization, among many others. Efficient algorithms for learning symmetric DPPs and sampling from these models have been reasonably well studied. However, relatively little attention has been given to nonsymmetric DPPs, which relax the symmetric constraint on the kernel. Nonsymmetric DPPs allow for both repulsive and attractive item interactions, which can significantly improve modeling power, resulting in a model that may better fit for some applications. We present a method that enables a tractable algorithm, based on maximum likelihood estimation, for learning nonsymmetric DPPs from data composed of observed subsets. Our method imposes a particular decomposition of the nonsymmetric kernel that enables such tractable learning algorithms, which we analyze both theoretically and experimentally. We evaluate our model on synthetic and real-world datasets, demonstrating improved predictive performance compared to symmetric DPPs, which have previously shown strong performance on modeling tasks associated with these datasets.

We propose a new framework of imposing monotonicity constraints in a Bayesian non-parametric setting. Our approach is based on numerical solutions of stochastic differential equations [Hedge, 2019]. We derive a non-parametric model of monotonic functions that allows for interpretable priors and principled quantification of hierarchical uncertainty. We demonstrate the efficacy of the proposed model by providing competitive results to other probabilistic models of monotonic functions on a number of benchmark functions. In addition, we consider the utility of a monotonic constraint in hierarchical probabilistic models, such as deep Gaussian processes. These typically suffer difficulties in modelling and propagating uncertainties throughout the hierarchy that can lead to hidden layers collapsing to point estimates. We address this by constraining hidden layers to be monotonic and present novel procedures for learning and inference that maintain uncertainty. We illustrate the capacity and versatility of the proposed framework on the task of temporal alignment of time-series data where it is beneficial to preserve the uncertainty in the temporal warpings.