**HPC-BBO**

Robot design is often a slow and difficult process requiring the iterative construction and testing of prototypes, with the goal of sequentially optimizing the design. For most robots, this process is further complicated by the need, when validating the capabilities of the hardware to solve the desired task, to already have an appropriate controller, which is in turn designed and tuned for the specific hardware. In this paper, we propose a novel approach, HPC-BBO, to efficiently and automatically design hardware configurations, and evaluate them by also automatically tuning the corresponding controller. HPC-BBO is based on a hierarchical Bayesian optimization process which iteratively optimizes morphology configurations (based on the performance of the previous designs during the controller learning process) and subsequently learns the corresponding controllers (exploiting the knowledge collected from optimizing for previous morphologies). Moreover, HPC-BBO can select a ‘batch’ of multiple morphology designs at once, thus parallelizing hardware validation and reducing the number of time-consuming production cycles. We validate HPC-BBO on the design of the morphology and controller for a simulated 6-legged microrobot. Experimental results show that HPC-BBO outperforms multiple competitive baselines, and yields a $360\%$ reduction in production cycles over standard Bayesian optimization, thus reducing the hypothetical manufacturing time of our microrobot from 21 to 4 months. … **Output-Constrained BNN**

Bayesian neural network (BNN) priors are defined in parameter space, making it hard to encode prior knowledge expressed in function space. We formulate a prior that incorporates functional constraints about what the output can or cannot be in regions of the input space. Output-Constrained BNNs (OC-BNN) represent an interpretable approach of enforcing a range of constraints, fully consistent with the Bayesian framework and amenable to black-box inference. We demonstrate how OC-BNNs improve model robustness and prevent the prediction of infeasible outputs in two real-world applications of healthcare and robotics. … **Deep Generative Model (DGM)**

We consider the semi-supervised clustering problem where crowdsourcing provides noisy information about the pairwise comparisons on a small subset of data, i.e., whether a sample pair is in the same cluster. We propose a new approach that includes a deep generative model (DGM) to characterize low-level features of the data, and a statistical relational model for noisy pairwise annotations on its subset. The two parts share the latent variables. To make the model automatically trade-off between its complexity and fitting data, we also develop its fully Bayesian variant. The challenge of inference is addressed by fast (natural-gradient) stochastic variational inference algorithms, where we effectively combine variational message passing for the relational part and amortized learning of the DGM under a unified framework. Empirical results on synthetic and real-world datasets show that our model outperforms previous crowdsourced clustering methods. … **EmbNum**

Among the fundamental questions in computer science, at least two have a deep impact on mathematics. What can computation compute How many steps does a computation require to solve an instance of the 3-SAT problem Our work addresses the first question, by introducing a new model called the x-machine. The x-machine executes Turing machine instructions and two special types of instructions. Quantum random instructions are physically realizable with a quantum random number generator. Meta instructions can add new states and add new instructions to the x-machine. A countable set of x-machines is constructed, each with a finite number of states and instructions; each x-machine can compute a Turing incomputable language, whenever the quantum randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan Turing posed the halting problem for Turing machines and proved that this problem is unsolvable for Turing machines. Consider an enumeration E_a(i) = (M_i, T_i) of all Turing machines M_i and initial tapes T_i. Does there exist an x-machine X that has at least one evolutionary path X –> X_1 –> X_2 –> . . . –> X_m, so at the mth stage x-machine X_m can correctly determine for 0 <= i <= m whether M_i’s execution on tape T_i eventually halts We demonstrate an x-machine Q(x) that has one such evolutionary path. The existence of this evolutionary path suggests that David Hilbert was not misguided to propose in 1900 that mathematicians search for finite processes to help construct mathematical proofs. Our refinement is that we cannot use a fixed computer program that behaves according to a fixed set of mechanical rules. We must pursue methods that exploit randomness and self-modification so that the complexity of the program can increase as it computes. …

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