In this paper, we consider the closed-loop control problem of nonlinear robotic systems in the presence of probabilistic uncertainties and disturbances. More precisely, we design a state feedback controller that minimizes deviations of the states of the system from the nominal state trajectories due to uncertainties and disturbances. Existing approaches to address the control problem of probabilistic systems are limited to particular classes of uncertainties and systems such as Gaussian uncertainties and processes and linearized systems. We present an approach that deals with nonlinear dynamics models and arbitrary known probabilistic uncertainties. We formulate the controller design problem as an optimization problem in terms of statistics of the probability distributions including moments and characteristic functions. In particular, in the provided optimization problem, we use moments and characteristic functions to propagate uncertainties throughout the nonlinear motion model of robotic systems. In order to reduce the tracking deviations, we minimize the uncertainty of the probabilistic states around the nominal trajectory by minimizing the trace and the determinant of the covariance matrix of the probabilistic states. To obtain the state feedback gains, we solve deterministic optimization problems in terms of moments, characteristic functions, and state feedback gains using off-the-shelf interior-point optimization solvers. To illustrate the performance of the proposed method, we compare our method with existing probabilistic control methods.
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Tracking body pose on-the-go could have powerful uses in fitness, mobile gaming, context-aware virtual assistants, and rehabilitation. However, users are unlikely to buy and wear special suits or sensor arrays to achieve this end. Instead, in this work, we explore the feasibility of estimating body pose using IMUs already in devices that many users own -- namely smartphones, smartwatches, and earbuds. This approach has several challenges, including noisy data from low-cost commodity IMUs, and the fact that the number of instrumentation points on a users body is both sparse and in flux. Our pipeline receives whatever subset of IMU data is available, potentially from just a single device, and produces a best-guess pose. To evaluate our model, we created the IMUPoser Dataset, collected from 10 participants wearing or holding off-the-shelf consumer devices and across a variety of activity contexts. We provide a comprehensive evaluation of our system, benchmarking it on both our own and existing IMU datasets.
We present a method for synthesizing dynamic, reduced-order output-feedback polynomial control policies for control-affine nonlinear systems which guarantees runtime stability to a goal state, when using visual observations and a learned perception module in the feedback control loop. We leverage Lyapunov analysis to formulate the problem of synthesizing such policies. This problem is nonconvex in the policy parameters and the Lyapunov function that is used to prove the stability of the policy. To solve this problem approximately, we propose two approaches: the first solves a sequence of sum-of-squares optimization problems to iteratively improve a policy which is provably-stable by construction, while the second directly performs gradient-based optimization on the parameters of the polynomial policy, and its closed-loop stability is verified a posteriori. We extend our approach to provide stability guarantees in the presence of observation noise, which realistically arises due to errors in the learned perception module. We evaluate our approach on several underactuated nonlinear systems, including pendula and quadrotors, showing that our guarantees translate to empirical stability when controlling these systems from images, while baseline approaches can fail to reliably stabilize the system.
Quantitative notions of bisimulation are well-known tools for the minimization of dynamical models such as Markov chains and ordinary differential equations (ODEs). In \emph{forward bisimulations}, each state in the quotient model represents an equivalence class and the dynamical evolution gives the overall sum of its members in the original model. Here we introduce generalized forward bisimulation (GFB) for dynamical systems over commutative monoids and develop a partition refinement algorithm to compute the coarsest one. When the monoid is $(\mathbb{R}, +)$, we recover %our framework recovers probabilistic bisimulation for Markov chains and more recent forward bisimulations for %systems of nonlinear ODEs. %ordinary differential equations. Using $(\mathbb{R}, \cdot)$ we get %When the monoid is $(\mathbb{R}, \cdot)$ we can obtain nonlinear reductions for discrete-time dynamical systems and ODEs %ordinary differential equations where each variable in the quotient model represents the product of original variables in the equivalence class. When the domain is a finite set such as the Booleans $\mathbb{B}$, we can apply GFB to Boolean networks (BN), a widely used dynamical model in computational biology. Using a prototype implementation of our minimization algorithm for GFB, we find disjunction- and conjunction-preserving reductions on 60 BN from two well-known repositories, and demonstrate the obtained analysis speed-ups. We also provide the biological interpretation of the reduction obtained for two selected BN, and we show how GFB enables the analysis of a large one that could not be analyzed otherwise. Using a randomized version of our algorithm we find product-preserving (therefore non-linear) reductions on 21 dynamical weighted networks from the literature that could not be handled by the exact algorithm.
Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper is to provide concentration inequalities on the distance between the sets of minimizers of the risks for a broad spectrum of estimation problems. In particular, the risks are defined on metric spaces through probability measures that are also supported on metric spaces. A particular attention will therefore be given to include unbounded spaces and non-convex cost functions that might also be unbounded. This work identifies a set of assumptions allowing to describe a regime that seem to govern the concentration in many estimation problems, where the empirical minimizers are stable. This stability can then be leveraged to prove parametric concentration rates in probability and in expectation. The assumptions are verified, and the bounds showcased, on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.
In financial engineering, portfolio optimization has been of consistent interest. Portfolio optimization is a process of modulating asset distributions to maximize expected returns and minimize risks. To obtain the expected returns, deep learning models have been explored in recent years. However, due to the deterministic nature of the models, it is difficult to consider the risk of portfolios because conventional deep learning models do not know how reliable their predictions can be. To address this limitation, this paper proposes a probabilistic model, namely predictive auxiliary classifier generative adversarial networks (PredACGAN). The proposed PredACGAN utilizes the characteristic of the ACGAN framework in which the output of the generator forms a distribution. While ACGAN has not been employed for predictive models and is generally utilized for image sample generation, this paper proposes a method to use the ACGAN structure for a probabilistic and predictive model. Additionally, an algorithm to use the risk measurement obtained by PredACGAN is proposed. In the algorithm, the assets that are predicted to be at high risk are eliminated from the investment universe at the rebalancing moment. Therefore, PredACGAN considers both return and risk to optimize portfolios. The proposed algorithm and PredACGAN have been evaluated with daily close price data of S&P 500 from 1990 to 2020. Experimental scenarios are assumed to rebalance the portfolios monthly according to predictions and risk measures with PredACGAN. As a result, a portfolio using PredACGAN exhibits 9.123% yearly returns and a Sharpe ratio of 1.054, while a portfolio without considering risk measures shows 1.024% yearly returns and a Sharpe ratio of 0.236 in the same scenario. Also, the maximum drawdown of the proposed portfolio is lower than the portfolio without PredACGAN.
This paper presents a deep learning based model predictive control (MPC) algorithm for systems with unmatched and bounded state-action dependent uncertainties of unknown structure. We utilize a deep neural network (DNN) as an oracle in the underlying optimization problem of learning based MPC (LBMPC) to estimate unmatched uncertainties. Generally, non-parametric oracles such as DNN are considered difficult to employ with LBMPC due to the technical difficulties associated with estimation of their coefficients in real time. We employ a dual-timescale adaptation mechanism, where the weights of the last layer of the neural network are updated in real time while the inner layers are trained on a slower timescale using the training data collected online and selectively stored in a buffer. Our results are validated through a numerical experiment on the compression system model of jet engine. These results indicate that the proposed approach is implementable in real time and carries the theoretical guarantees of LBMPC.
This paper focuses on the information freshness of finite-state Markov sources, using the uncertainty of information (UoI) as the performance metric. Measured by Shannon's entropy, UoI can capture not only the transition dynamics of the Markov source but also the different evolutions of information quality caused by the different values of the last observation. We consider an information update system with M finite-state Markov sources transmitting information to a remote monitor via m communication channels. Our goal is to explore the optimal scheduling policy to minimize the sum-UoI of the Markov sources. The problem is formulated as a restless multi-armed bandit (RMAB). We relax the RMAB and then decouple the relaxed problem into M single bandit problems. Analyzing the single bandit problem provides useful properties with which the relaxed problem reduces to maximizing a concave and piecewise linear function, allowing us to develop a gradient method to solve the relaxed problem and obtain its optimal policy. By rounding up the optimal policy for the relaxed problem, we obtain an index policy for the original RMAB problem. Notably, the proposed index policy is universal in the sense that it applies to general RMABs with bounded cost functions.
This paper attempts to characterize the kinds of physical scenarios in which an online learning-based cognitive radar is expected to reliably outperform a fixed rule-based waveform selection strategy, as well as the converse. We seek general insights through an examination of two decision-making scenarios, namely dynamic spectrum access and multiple-target tracking. The radar scene is characterized by inducing a state-space model and examining the structure of its underlying Markov state transition matrix, in terms of entropy rate and diagonality. It is found that entropy rate is a strong predictor of online learning-based waveform selection, while diagonality is a better predictor of fixed rule-based waveform selection. We show that these measures can be used to predict first and second-order stochastic dominance relationships, which can allow system designers to make use of simple decision rules instead of more cumbersome learning approaches under certain conditions. We validate our findings through numerical results for each application and provide guidelines for future implementations.
Uncertainty quantification in image restoration is a prominent challenge, mainly due to the high dimensionality of the encountered problems. Recently, a Bayesian uncertainty quantification by optimization (BUQO) has been proposed to formulate hypothesis testing as a minimization problem. The objective is to determine whether a structure appearing in a maximum a posteriori estimate is true or is a reconstruction artifact due to the ill-posedness or ill-conditioness of the problem. In this context, the mathematical definition of having a ``fake structure" is crucial, and highly depends on the type of structure of interest. This definition can be interpreted as an inpainting of a neighborhood of the structure, but only simple techniques have been proposed in the literature so far, due to the complexity of the problem. In this work, we propose a data-driven method using a simple convolutional neural network to perform the inpainting task, leading to a novel plug-and-play BUQO algorithm. Compared to previous works, the proposed approach has the advantage that it can be used for a wide class of structures, without needing to adapt the inpainting operator to the area of interest. In addition, we show through simulations on magnetic resonance imaging, that compared to the original BUQO's hand-crafted inpainting procedure, the proposed approach provides greater qualitative output images. Python code will be made available for reproducibility upon acceptance of the article.
Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.
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