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The ongoing deep learning revolution has allowed computers to outclass humans in various games and perceive features imperceptible to humans during classification tasks. Current machine learning techniques have clearly distinguished themselves in specialized tasks. However, we have yet to see robots capable of performing multiple tasks at an expert level. Most work in this field is focused on the development of more sophisticated learning algorithms for a robot's controller given a largely static and presupposed robotic design. By focusing on the development of robotic bodies, rather than neural controllers, I have discovered that robots can be designed such that they overcome many of the current pitfalls encountered by neural controllers in multitask settings. Through this discovery, I also present novel metrics to explicitly measure the learning ability of a robotic design and its resistance to common problems such as catastrophic interference. Traditionally, the physical robot design requires human engineers to plan every aspect of the system, which is expensive and often relies on human intuition. In contrast, within the field of evolutionary robotics, evolutionary algorithms are used to automatically create optimized designs, however, such designs are often still limited in their ability to perform in a multitask setting. The metrics created and presented here give a novel path to automated design that allow evolved robots to synergize with their controller to improve the computational efficiency of their learning while overcoming catastrophic interference. Overall, this dissertation intimates the ability to automatically design robots that are more general purpose than current robots and that can perform various tasks while requiring less computation.

A well-established approach for inferring full displacement and stress fields from possibly sparse data is to calibrate the parameter of a given constitutive model using a Bayesian update. After calibration, a (stochastic) forward simulation is conducted with the identified model parameters to resolve physical fields in regions that were not accessible to the measurement device. A shortcoming of model calibration is that the model is deemed to best represent reality, which is only sometimes the case, especially in the context of the aging of structures and materials. While this issue is often addressed with repeated model calibration, a different approach is followed in the recently proposed statistical Finite Element Method (statFEM). Instead of using Bayes' theorem to update model parameters, the displacement is chosen as the stochastic prior and updated to fit the measurement data more closely. For this purpose, the statFEM framework introduces a so-called model-reality mismatch, parametrized by only three hyperparameters. This makes the inference of full-field data computationally efficient in an online stage: If the stochastic prior can be computed offline, solving the underlying partial differential equation (PDE) online is unnecessary. Compared to solving a PDE, identifying only three hyperparameters and conditioning the state on the sensor data requires much fewer computational resources. This paper presents two contributions to the existing statFEM approach: First, we use a non-intrusive polynomial chaos method to compute the prior, enabling the use of complex mechanical models in deterministic formulations. Second, we examine the influence of prior material models (linear elastic and St.Venant Kirchhoff material with uncertain Young's modulus) on the updated solution. We present statFEM results for 1D and 2D examples, while an extension to 3D is straightforward.

We provide non-asymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input arbitrary estimation algorithms for the target parameter and nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from machine learning to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can provide rates under weaker assumptions than in previous works and accommodate settings in which the target parameter belongs to a complex nonparametric class. We provide conditions on the metric entropy of the nuisance and target classes such that oracle rates of the same order as if we knew the nuisance parameter are achieved.

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with structured quadrature points (QMC), which provide stable $L_2$ reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a RKHS of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the $d$-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.

In this paper paired comparison models with stochastic background are investigated. We focus on the models which allow three options for choice and the parameters are estimated by maximum likelihood method. The existence and uniqueness of the estimator is a key issue of the evaluation. In the case of two options, a necessary and sufficient condition is given by Ford in the Bradley-Terry model. We generalize this statement for the set of strictly log-concave distribution. Although in the case of three options necessary and sufficient condition is not known, there are two different sufficient conditions which are formulated in the literature. In this paper we generalize them, moreover we compare these conditions. Their capacities to indicate the existence of the maximum are analyzed by a large number of computer simulations. These simulations support that the new condition indicates the existence of the maximum much more frequently then the previously known ones,

Estimating weights in the synthetic control method involves an optimization procedure that simultaneously selects and aligns control units in order to closely match the treated unit. However, this simultaneous selection and alignment of control units may lead to a loss of efficiency in the synthetic control method. Another concern arising from the aforementioned procedure is its susceptibility to under-fitting due to imperfect pretreatment fit. It is not uncommon for the linear combination, using nonnegative weights, of pre-treatment period outcomes for the control units to inadequately approximate the pre-treatment outcomes for the treated unit. To address both of these issues, this paper proposes a simple and effective method called Synthetic Matching Control (SMC). The SMC method begins by performing the univariate linear regression to establish a proper match between the pre-treatment periods of the control units and the treated unit. Subsequently, a SMC estimator is obtained by synthesizing (taking a weighted average) the matched controls. To determine the weights in the synthesis procedure, we propose an approach that utilizes a criterion of unbiased risk estimator. Theoretically, we show that the synthesis way is asymptotically optimal in the sense of achieving the lowest possible squared error. Extensive numerical experiments highlight the advantages of the SMC method.

In clustering problems, a central decision-maker is given a complete metric graph over vertices and must provide a clustering of vertices that minimizes some objective function. In fair clustering problems, vertices are endowed with a color (e.g., membership in a group), and the features of a valid clustering might also include the representation of colors in that clustering. Prior work in fair clustering assumes complete knowledge of group membership. In this paper, we generalize prior work by assuming imperfect knowledge of group membership through probabilistic assignments. We present clustering algorithms in this more general setting with approximation ratio guarantees. We also address the problem of "metric membership", where different groups have a notion of order and distance. Experiments are conducted using our proposed algorithms as well as baselines to validate our approach and also surface nuanced concerns when group membership is not known deterministically.

Injuries to the knee joint are very common for long-distance and frequent runners, an issue which is often attributed to fatigue. We address the problem of fatigue detection from biomechanical data from different sources, consisting of lower extremity joint angles and ground reaction forces from running athletes with the goal of better understanding the impact of fatigue on the biomechanics of runners in general and on an individual level. This is done by sequentially testing for change in a datastream using a simple martingale test statistic. Time-uniform probabilistic martingale bounds are provided which are used as thresholds for the test statistic. Sharp bounds can be developed by a hybrid of a piece-wise linear- and a law of iterated logarithm- bound over all time regimes, where the probability of an early detection is controlled in a uniform way. If the underlying distribution of the data gradually changes over the course of a run, then a timely upcrossing of the martingale over these bounds is expected. The methods are developed for a setting when change sets in gradually in an incoming stream of data. Parameter selection for the bounds are based on simulations and methodological comparison is done with respect to existing advances. The algorithms presented here can be easily adapted to an online change-detection setting. Finally, we provide a detailed data analysis based on extensive measurements of several athletes and benchmark the fatigue detection results with the runners' individual feedback over the course of the data collection. Qualitative conclusions on the biomechanical profiles of the athletes can be made based on the shape of the martingale trajectories even in the absence of an upcrossing of the threshold.

As a computational alternative to Markov chain Monte Carlo approaches, variational inference (VI) is becoming more and more popular for approximating intractable posterior distributions in large-scale Bayesian models due to its comparable efficacy and superior efficiency. Several recent works provide theoretical justifications of VI by proving its statistical optimality for parameter estimation under various settings; meanwhile, formal analysis on the algorithmic convergence aspects of VI is still largely lacking. In this paper, we consider the common coordinate ascent variational inference (CAVI) algorithm for implementing the mean-field (MF) VI towards optimizing a Kullback--Leibler divergence objective functional over the space of all factorized distributions. Focusing on the two-block case, we analyze the convergence of CAVI by leveraging the extensive toolbox from functional analysis and optimization. We provide general conditions for certifying global or local exponential convergence of CAVI. Specifically, a new notion of generalized correlation for characterizing the interaction between the constituting blocks in influencing the VI objective functional is introduced, which according to the theory, quantifies the algorithmic contraction rate of two-block CAVI. As illustrations, we apply the developed theory to a number of examples, and derive explicit problem-dependent upper bounds on the algorithmic contraction rate.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.