arXiv subjects

Santiago Paternain

Publications and source records attributed to Santiago Paternain.

At least 55 records · Page 3Linked to original sources

arXiv2020

Trust but Verify: Assigning Prediction Credibility by Counterfactual Constrained Learning

Prediction credibility measures, in the form of confidence intervals or probability distributions, are fundamental in statistics and machine learning to characterize model robustness, detect out-of-distribution samples (outliers), and protect against adversarial attacks. To be effective, these measures should (i) account for the wide variety of models used in practice, (ii) be computable for trained models or at least avoid modifying established training procedures, (iii) forgo the use of data, which can expose them to the same robustness issues and attacks as the underlying model, and (iv) be followed by theoretical guarantees. These principles underly the framework developed in this work, which expresses the credibility as a risk-fit trade-off, i.e., a compromise between how much can fit be improved by perturbing the model input and the magnitude of this perturbation (risk). Using a constrained optimization formulation and duality theory, we analyze this compromise and show that this balance can be determined counterfactually, without having to test multiple perturbations. This results in an unsupervised, a posteriori method of assigning prediction credibility for any (possibly non-convex) differentiable model, from RKHS-based solutions to any architecture of (feedforward, convolutional, graph) neural network. Its use is illustrated in data filtering and defense against adversarial attacks.

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arXiv2020

Policy Gradient for Continuing Tasks in Non-stationary Markov Decision Processes

Reinforcement learning considers the problem of finding policies that maximize an expected cumulative reward in a Markov decision process with unknown transition probabilities. In this paper we consider the problem of finding optimal policies assuming that they belong to a reproducing kernel Hilbert space (RKHS). To that end we compute unbiased stochastic gradients of the value function which we use as ascent directions to update the policy. A major drawback of policy gradient-type algorithms is that they are limited to episodic tasks unless stationarity assumptions are imposed. Hence preventing these algorithms to be fully implemented online, which is a desirable property for systems that need to adapt to new tasks and/or environments in deployment. The main requirement for a policy gradient algorithm to work is that the estimate of the gradient at any point in time is an ascent direction for the initial value function. In this work we establish that indeed this is the case which enables to show the convergence of the online algorithm to the critical points of the initial value function. A numerical example shows the ability of our online algorithm to learn to solve a navigation and surveillance problem, in which an agent must loop between to goal locations. This example corroborates our theoretical findings about the ascent directions of subsequent stochastic gradients. It also shows how the agent running our online algorithm succeeds in learning to navigate, following a continuing cyclic trajectory that does not comply with the standard stationarity assumptions in the literature for non episodic training.

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arXiv2020

Resilient Control: Compromising to Adapt

In optimal control problems, disturbances are typically dealt with using robust solutions, such as H-infinity or tube model predictive control, that plan control actions feasible for the worst-case disturbance. Yet, planning for every contingency can lead to over-conservative, poorly performing solutions or even, in extreme cases, to infeasibility. Resilience addresses these shortcomings by adapting the underlying control problem, e.g., by relaxing its specifications, to obtain a feasible, possibly still valuable trajectory. Despite their different aspects, robustness and resilience are often conflated in the context of dynamical systems and control. The goal of this paper is to formalize, in the context of optimal control, the concept of resilience understood as above, i.e., in terms of adaptation. To do so, we introduce a resilient formulation of optimal control by allowing disruption-dependent modifications of the requirements that induce the desired resilient behavior. We then propose a framework to design these behaviors automatically by trading off control performance and requirement violations. We analyze this resilience-by-compromise method to obtain inverse optimality results and quantify the effect of disturbances on the induced requirement relaxations. By proving that robustness and resilience optimize different objectives, we show that these are in fact distinct system properties. We conclude by illustrating the effect of resilience in different control problems.

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arXiv2020

Time-Varying Convex Optimization: Time-Structured Algorithms and Applications

Optimization underpins many of the challenges that science and technology face on a daily basis. Recent years have witnessed a major shift from traditional optimization paradigms grounded on batch algorithms for medium-scale problems to challenging dynamic, time-varying, and even huge-size settings. This is driven by technological transformations that converted infrastructural and social platforms into complex and dynamic networked systems with even pervasive sensing and computing capabilities. The present paper reviews a broad class of state-of-the-art algorithms for time-varying optimization, with an eye to both algorithmic development and performance analysis. It offers a comprehensive overview of available tools and methods, and unveils open challenges in application domains of broad interest. The real-world examples presented include smart power systems, robotics, machine learning, and data analytics, highlighting domain-specific issues and solutions. The ultimate goal is to exempify wide engineering relevance of analytical tools and pertinent theoretical foundations.

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arXiv2020

Counterfactual Programming for Optimal Control

In recent years, considerable work has been done to tackle the issue of designing control laws based on observations to allow unknown dynamical systems to perform pre-specified tasks. At least as important for autonomy, however, is the issue of learning which tasks can be performed in the first place. This is particularly critical in situations where multiple (possibly conflicting) tasks and requirements are demanded from the agent, resulting in infeasible specifications. Such situations arise due to over-specification or dynamic operating conditions and are only aggravated when the dynamical system model is learned through simulations. Often, these issues are tackled using regularization and penalties tuned based on application-specific expert knowledge. Nevertheless, this solution becomes impractical for large-scale systems, unknown operating conditions, and/or in online settings where expert input would be needed during the system operation. Instead, this work enables agents to autonomously pose, tune, and solve optimal control problems by compromising between performance and specification costs. Leveraging duality theory, it puts forward a counterfactual optimization algorithm that directly determines the specification trade-off while solving the optimal control problem.

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arXiv2020

Sufficiently Accurate Model Learning

Modeling how a robot interacts with the environment around it is an important prerequisite for designing control and planning algorithms. In fact, the performance of controllers and planners is highly dependent on the quality of the model. One popular approach is to learn data driven models in order to compensate for inaccurate physical measurements and to adapt to systems that evolve over time. In this paper, we investigate a method to regularize model learning techniques to provide better error characteristics for traditional control and planning algorithms. This work proposes learning "Sufficiently Accurate" models of dynamics using a primal-dual method that can explicitly enforce constraints on the error in pre-defined parts of the state-space. The result of this method is that the error characteristics of the learned model is more predictable and can be better utilized by planning and control algorithms. The characteristics of Sufficiently Accurate models are analyzed through experiments on a simulated ball paddle system.

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arXiv2020

The empirical duality gap of constrained statistical learning

This paper is concerned with the study of constrained statistical learning problems, the unconstrained version of which are at the core of virtually all of modern information processing. Accounting for constraints, however, is paramount to incorporate prior knowledge and impose desired structural and statistical properties on the solutions. Still, solving constrained statistical problems remains challenging and guarantees scarce, leaving them to be tackled using regularized formulations. Though practical and effective, selecting regularization parameters so as to satisfy requirements is challenging, if at all possible, due to the lack of a straightforward relation between parameters and constraints. In this work, we propose to directly tackle the constrained statistical problem overcoming its infinite dimensionality, unknown distributions, and constraints by leveraging finite dimensional parameterizations, sample averages, and duality theory. Aside from making the problem tractable, these tools allow us to bound the empirical duality gap, i.e., the difference between our approximate tractable solutions and the actual solutions of the original statistical problem. We demonstrate the effectiveness and usefulness of this constrained formulation in a fair learning application.

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arXiv2019

A Prediction-Correction Algorithm for Real-Time Model Predictive Control

In this work we adapt a prediction-correction algorithm for continuous time-varying convex optimization problems to solve dynamic programs arising from Model Predictive Control. In particular, the prediction step tracks the evolution of the optimal solution of the problem which depends on the current state of the system. The cost of said step is that of inverting one Hessian and it guarantees, under some conditions, that the iterate remains in the quadratic convergence region of the optimization problem at the next time step. These conditions imply (i) that the variation of the state in a control interval cannot be too large and that (ii) the solution computed in the previous time step needs to be sufficiently accurate. The latter can be guaranteed by running classic Newton iterations, which we term correction steps. Since this method exhibits quadratic convergence the number of iterations to achieve a desired accuracy $η$ is of order $\log_2\log_2 1/η$, where the cost of each iteration is approximately that of inverting a Hessian. This grants the prediction-correction control law low computational complexity. In addition, the solution achieved by the algorithm is such that the closed loop system remains stable, which allows extending the applicability of Model Predictive Control to systems with faster dynamics and less computational power. Numerical examples where we consider nonlinear systems support the theoretical conclusions.

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arXiv2019

Constrained Reinforcement Learning Has Zero Duality Gap

Autonomous agents must often deal with conflicting requirements, such as completing tasks using the least amount of time/energy, learning multiple tasks, or dealing with multiple opponents. In the context of reinforcement learning~(RL), these problems are addressed by (i)~designing a reward function that simultaneously describes all requirements or (ii)~combining modular value functions that encode them individually. Though effective, these methods have critical downsides. Designing good reward functions that balance different objectives is challenging, especially as the number of objectives grows. Moreover, implicit interference between goals may lead to performance plateaus as they compete for resources, particularly when training on-policy. Similarly, selecting parameters to combine value functions is at least as hard as designing an all-encompassing reward, given that the effect of their values on the overall policy is not straightforward. The later is generally addressed by formulating the conflicting requirements as a constrained RL problem and solved using Primal-Dual methods. These algorithms are in general not guaranteed to converge to the optimal solution since the problem is not convex. This work provides theoretical support to these approaches by establishing that despite its non-convexity, this problem has zero duality gap, i.e., it can be solved exactly in the dual domain, where it becomes convex. Finally, we show this result basically holds if the policy is described by a good parametrization~(e.g., neural networks) and we connect this result with primal-dual algorithms present in the literature and we establish the convergence to the optimal solution.

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arXiv2019

Source Seeking in Unknown Environments with Convex Obstacles

Navigation tasks often cannot be defined in terms of a target, either because global position information is unavailable or unreliable or because target location is not explicitly known a priori. This task is then often defined indirectly as a source seeking problem in which the autonomous agent navigates so as to minimize the convex potential induced by a source while avoiding obstacles. This work addresses this problem when only scalar measurements of the potential are available, i.e., without gradient information. To do so, it construct an artificial potential over which an exact gradient dynamics would generate a collision-free trajectory to the target in a world with convex obstacles. Then, leveraging extremum seeking control loops, it minimizes this artificial potential to navigate smoothly to the source location. We prove that the proposed solution not only finds the source, but does so while avoiding any obstacle. Numerical results with velocity-actuated particles, simulations with an omni-directional robot in ROS+Gazebo, and a robot-in-the-loop experiment are used to illustrate the performance of this approach.

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arXiv2019

Sparse multiresolution representations with adaptive kernels

Reproducing kernel Hilbert spaces (RKHSs) are key elements of many non-parametric tools successfully used in signal processing, statistics, and machine learning. In this work, we aim to address three issues of the classical RKHS based techniques. First, they require the RKHS to be known a priori, which is unrealistic in many applications. Furthermore, the choice of RKHS affects the shape and smoothness of the solution, thus impacting its performance. Second, RKHSs are ill-equipped to deal with heterogeneous degrees of smoothness, i.e., with functions that are smooth in some parts of their domain but vary rapidly in others. Finally, the computational complexity of evaluating the solution of these methods grows with the number of data points, rendering these techniques infeasible for many applications. Though kernel learning, local kernel adaptation, and sparsity have been used to address these issues, many of these approaches are computationally intensive or forgo optimality guarantees. We tackle these problems by leveraging a novel integral representation of functions in RKHSs that allows for arbitrary centers and different kernels at each center. To address the complexity issues, we then write the function estimation problem as a sparse functional program that explicitly minimizes the support of the representation leading to low complexity solutions. Despite their non-convexity and infinite dimensionality, we show these problems can be solved exactly and efficiently by leveraging duality, and we illustrate this new approach in simulated and real data.

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arXiv2019

Distributed Constrained Online Learning

In this paper, we consider groups of agents in a network that select actions in order to satisfy a set of constraints that vary arbitrarily over time and minimize a time-varying function of which they have only local observations. The selection of actions, also called a strategy, is causal and decentralized, i.e., the dynamical system that determines the actions of a given agent depends only on the constraints at the current time and on its own actions and those of its neighbors. To determine such a strategy, we propose a decentralized saddle point algorithm and show that the corresponding global fit and regret are bounded by functions of the order of $\sqrt{T}$. Specifically, we define the global fit of a strategy as a vector that integrates over time the global constraint violations as seen by a given node. The fit is a performance loss associated with online operation as opposed to offline clairvoyant operation which can always select an action if one exists, that satisfies the constraints at all times. If this fit grows sublinearly with the time horizon it suggests that the strategy approaches the feasible set of actions. Likewise, we define the regret of a strategy as the difference between its accumulated cost and that of the best fixed action that one could select knowing beforehand the time evolution of the objective function. Numerical examples support the theoretical conclusions.

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arXiv2018

Stochastic Policy Gradient Ascent in Reproducing Kernel Hilbert Spaces

Reinforcement learning consists of finding policies that maximize an expected cumulative long-term reward in a Markov decision process with unknown transition probabilities and instantaneous rewards. In this paper, we consider the problem of finding such optimal policies while assuming they are continuous functions belonging to a reproducing kernel Hilbert space (RKHS). To learn the optimal policy we introduce a stochastic policy gradient ascent algorithm with three unique novel features: (i) The stochastic estimates of policy gradients are unbiased. (ii) The variance of stochastic gradients is reduced by drawing on ideas from numerical differentiation. (iii) Policy complexity is controlled using sparse RKHS representations. Novel feature (i) is instrumental in proving convergence to a stationary point of the expected cumulative reward. Novel feature (ii) facilitates reasonable convergence times. Novel feature (iii) is a necessity in practical implementations which we show can be done in a way that does not eliminate convergence guarantees. Numerical examples in standard problems illustrate successful learning of policies with low complexity representations which are close to stationary points of the expected cumulative reward.

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arXiv2018

A Newton-Based Method for Nonconvex Optimization with Fast Evasion of Saddle Points

Machine learning problems such as neural network training, tensor decomposition, and matrix factorization, require local minimization of a nonconvex function. This local minimization is challenged by the presence of saddle points, of which there can be many and from which descent methods may take inordinately large number of iterations to escape. This paper presents a second-order method that modifies the update of Newton's method by replacing the negative eigenvalues of the Hessian by their absolute values and uses a truncated version of the resulting matrix to account for the objective's curvature. The method is shown to escape saddles in at most $1 + \log_{3/2} (δ/2\varepsilon)$ iterations where $\varepsilon$ is the target optimality and $δ$ characterizes a point sufficiently far away from the saddle. This base of this exponential escape is $3/2$ independently of problem constants. Adding classical properties of Newton's method, the paper proves convergence to a local minimum with probability $1-p$ in $O\left(\log(1/p)) + O(\log(1/\varepsilon)\right)$ iterations.

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arXiv2017

Decentralized Online Learning with Kernels

We consider multi-agent stochastic optimization problems over reproducing kernel Hilbert spaces (RKHS). In this setting, a network of interconnected agents aims to learn decision functions, i.e., nonlinear statistical models, that are optimal in terms of a global convex functional that aggregates data across the network, with only access to locally and sequentially observed samples. We propose solving this problem by allowing each agent to learn a local regression function while enforcing consensus constraints. We use a penalized variant of functional stochastic gradient descent operating simultaneously with low-dimensional subspace projections. These subspaces are constructed greedily by applying orthogonal matching pursuit to the sequence of kernel dictionaries and weights. By tuning the projection-induced bias, we propose an algorithm that allows for each individual agent to learn, based upon its locally observed data stream and message passing with its neighbors only, a regression function that is close to the globally optimal regression function. That is, we establish that with constant step-size selections agents' functions converge to a neighborhood of the globally optimal one while satisfying the consensus constraints as the penalty parameter is increased. Moreover, the complexity of the learned regression functions is guaranteed to remain finite. On both multi-class kernel logistic regression and multi-class kernel support vector classification with data generated from class-dependent Gaussian mixture models, we observe stable function estimation and state of the art performance for distributed online multi-class classification. Experiments on the Brodatz textures further substantiate the empirical validity of this approach.

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arXiv2016

Stochastic Artificial Potentials for Online Safe Navigation

Consider a convex set of which we remove an arbitrarily number of disjoints convex sets -- the obstacles -- and a convex function whose minimum is the agent's goal. We consider a local and stochastic approximation of the gradient of a Rimon-Koditschek navigation function where the attractive potential is the convex function that the agent is minimizing. In particular we show that if the estimate available to the agent is unbiased convergence to the desired destination while obstacle avoidance is guaranteed with probability one under the same geometrical conditions than in the deterministic case. Qualitatively these conditions are that the ratio of the maximum over the minimum eigenvalue of the Hessian of the objective function is not too large and that the obstacles are not too flat or too close to the desired destination. Moreover, we show that for biased estimates a similar result holds under some assumptions on the bias. These assumptions are motivated by the study of the estimate of the gradient of a Rimon-Koditschek navigation function for sensor models that fit circles or ellipses around the obstacles. Numerical examples explore the practical value of these theoretical results.

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arXiv2016

Navigation Functions for Convex Potentials in a Space with Convex Obstacles

Given a convex potential in a space with convex obstacles, an artificial potential is used to navigate to the minimum of the natural potential while avoiding collisions. The artificial potential combines the natural potential with potentials that repel the agent from the border of the obstacles. This is a popular approach to navigation problems because it can be implemented with spatially local information that is acquired during operation time. Artificial potentials can, however, have local minima that prevent navigation to the minimum of the natural potential. This paper derives conditions that guarantee artificial potentials have a single minimum that is arbitrarily close to the minimum of the natural potential. The qualitative implication is that artificial potentials succeed when either the condition number-- the ratio of the maximum over the minimum eigenvalue-- of the Hessian of the natural potential is not large and the obstacles are not too flat or when the destination is not close to the border of an obstacle. Numerical analyses explore the practical value of these theoretical conclusions.

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arXiv2016

Prediction-Correction Interior-Point Method for Time-Varying Convex Optimization

In this paper, we develop an interior-point method for solving a class of convex optimization problems with time-varying objective and constraint functions. Using log-barrier penalty functions, we propose a continuous-time dynamical system for tracking the (time-varying) optimal solution with an asymptotically vanishing error. This dynamical system is composed of two terms: (i) a correction term consisting of a continuous-time version of Newton's method, and (ii) a prediction term able to track the drift of the optimal solution by taking into account the time-varying nature of the objective and constraint functions. Using appropriately chosen time-varying slack and barrier parameters, we ensure that the solution to this dynamical system globally asymptotically converges to the optimal solution at an exponential rate. We illustrate the applicability of the proposed method in two practical applications: a sparsity promoting least squares problem and a collision-free robot navigation problem.

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