Santiago Paternain

Santiago Paternain, Juan Andrés Bazerque, Austin Small et al.

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.

Weiqin Chen, Dharmashankar Subramanian, Santiago Paternain

In this paper, we consider the problem of learning safe policies for probabilistic-constrained reinforcement learning (RL). Specifically, a safe policy or controller is one that, with high probability, maintains the trajectory of the agent in a given safe set. We establish a connection between this probabilistic-constrained setting and the cumulative-constrained formulation that is frequently explored in the existing literature. We provide theoretical bounds elucidating that the probabilistic-constrained setting offers a better trade-off in terms of optimality and safety (constraint satisfaction). The challenge encountered when dealing with the probabilistic constraints, as explored in this work, arises from the absence of explicit expressions for their gradients. Our prior work provides such an explicit gradient expression for probabilistic constraints which we term Safe Policy Gradient-REINFORCE (SPG-REINFORCE). In this work, we provide an improved gradient SPG-Actor-Critic that leads to a lower variance than SPG-REINFORCE, which is substantiated by our theoretical results. A noteworthy aspect of both SPGs is their inherent algorithm independence, rendering them versatile for application across a range of policy-based algorithms. Furthermore, we propose a Safe Primal-Dual algorithm that can leverage both SPGs to learn safe policies. It is subsequently followed by theoretical analyses that encompass the convergence of the algorithm, as well as the near-optimality and feasibility on average. In addition, we test the proposed approaches by a series of empirical experiments. These experiments aim to examine and analyze the inherent trade-offs between the optimality and safety, and serve to substantiate the efficacy of two SPGs, as well as our theoretical contributions.

Weiqin Chen, Dharmashankar Subramanian, Santiago Paternain

This paper considers the problem of learning safe policies in the context of reinforcement learning (RL). In particular, we consider the notion of probabilistic safety. This is, we aim to design policies that maintain the state of the system in a safe set with high probability. This notion differs from cumulative constraints often considered in the literature. The challenge of working with probabilistic safety is the lack of expressions for their gradients. Indeed, policy optimization algorithms rely on gradients of the objective function and the constraints. To the best of our knowledge, this work is the first one providing such explicit gradient expressions for probabilistic constraints. It is worth noting that the gradient of this family of constraints can be applied to various policy-based algorithms. We demonstrate empirically that it is possible to handle probabilistic constraints in a continuous navigation problem.

Weiqin Chen, Xinjie Zhang, Sandipan Mishra et al.

Offline reinforcement learning (RL) learns effective policies from a static target dataset. The performance of state-of-the-art offline RL algorithms notwithstanding, it relies on the size of the target dataset, and it degrades if limited samples in the target dataset are available, which is often the case in real-world applications. To address this issue, domain adaptation that leverages auxiliary samples from related source datasets (such as simulators) can be beneficial. However, establishing the optimal way to trade off the limited target dataset and the large-but-biased source dataset while ensuring provably theoretical guarantees remains an open challenge. To the best of our knowledge, this paper proposes the first framework that theoretically explores the impact of the weights assigned to each dataset on the performance of offline RL. In particular, we establish performance bounds and the existence of the optimal weight, which can be computed in closed form under simplifying assumptions. We also provide algorithmic guarantees in terms of convergence to a neighborhood of the optimum. Notably, these results depend on the quality of the source dataset and the number of samples in the target dataset. Our empirical results on the well-known Procgen and MuJoCo benchmarks substantiate the theoretical contributions in this work.

Weiqin Chen, Nhan Huu Pham, Michael Robert Glass et al.

Reinforcement learning (RL) has demonstrated significant promise in enhancing the reasoning capabilities of Text2SQL LLMs, especially with advanced algorithms such as GRPO and DAPO. However, the performance of these methods is highly sensitive to the design of reward functions. Inappropriate rewards can lead to reward hacking, where models exploit loopholes in the reward structure to achieve high scores without genuinely solving the task. This work considers a constrained RL framework for Text2SQL that incorporates natural and interpretable reward and constraint signals, while dynamically balancing trade-offs among them during the training. We establish the theoretical guarantees of our constrained RL framework and our numerical experiments on the well-known Text2SQL datasets substantiate the improvement of our approach over the state-of-the-art RL-trained LLMs.

Zhengyao Gu, Jonathan Light, Raul Astudillo et al.

Post-training large foundation models with reinforcement learning typically relies on massive and heterogeneous datasets, making effective curriculum learning both critical and challenging. In this work, we propose ACTOR-CURATOR, a scalable and fully automated curriculum learning framework for reinforcement learning post-training of large language models (LLMs). ACTOR-CURATOR learns a neural curator that dynamically selects training problems from large problem banks by directly optimizing for expected policy performance improvement. We formulate problem selection as a non-stationary stochastic bandit problem, derive a principled loss function based on online stochastic mirror descent, and establish regret guarantees under partial feedback. Empirically, ACTOR-CURATOR consistently outperforms uniform sampling and strong curriculum baselines across a wide range of challenging reasoning benchmarks, demonstrating improved training stability and efficiency. Notably, it achieves relative gains of 28.6% on AIME2024 and 30.5% on ARC-1D over the strongest baseline and up to 80% speedup. These results suggest that ACTOR-CURATOR is a powerful and practical approach for scalable LLM post-training.

Anmol Dwivedi, Santiago Paternain, Ali Tajer

This paper considers the sequential design of remedial control actions in response to system anomalies for the ultimate objective of preventing blackouts. A physics-guided reinforcement learning (RL) framework is designed to identify effective sequences of real-time remedial look-ahead decisions accounting for the long-term impact on the system's stability. The paper considers a space of control actions that involve both discrete-valued transmission line-switching decisions (line reconnections and removals) and continuous-valued generator adjustments. To identify an effective blackout mitigation policy, a physics-guided approach is designed that uses power-flow sensitivity factors associated with the power transmission network to guide the RL exploration during agent training. Comprehensive empirical evaluations using the open-source Grid2Op platform demonstrate the notable advantages of incorporating physical signals into RL decisions, establishing the gains of the proposed physics-guided approach compared to its black box counterparts. One important observation is that strategically~\emph{removing} transmission lines, in conjunction with multiple real-time generator adjustments, often renders effective long-term decisions that are likely to prevent or delay blackouts.

Jushan Chen, Jonathan Fried, Santiago Paternain

We present a framework leveraging a novel variant of the model-based diffusion algorithm to minimize the time required for a redundant dual-arm robot configuration to follow a desired relative Cartesian path. Our prior work proposed a bi-level optimization approach for the dual-arm problem, where we derived the analytical solution to the lower-level convex sub-problem and solved the high-level nonconvex problem using a primal-dual approach. However, the gradient-based nature leads to a large computation overhead, and it prohibits directly imposing an $L_{\infty}$ Cartesian error constraint along the joint trajectory due to the sparsity of the gradient. In this work, we propose a diffusion-based framework that relies on probabilistic sampling to tackle the aforementioned challenges in the nonconvex high-level problem, leading to a 35x reduction in the runtime and 34\% less Cartesian error compared to our prior work.

Shima Rabiei, Sandipan Mishra, Santiago Paternain

This paper considers the problem of zero-shot safety guarantees for cascade dynamical systems. These are systems where a subset of the states (the inner states) affects the dynamics of the remaining states (the outer states) but not vice-versa. We define safety as remaining on a set deemed safe for all times with high probability. We propose to train a safe RL policy on a reduced-order model, which ignores the dynamics of the inner states, but it treats it as an action that influences the outer state. Thus, reducing the complexity of the training. When deployed in the full system the trained policy is combined with a low-level controller whose task is to track the reference provided by the RL policy. Our main theoretical contribution is a bound on the safe probability in the full-order system. In particular, we establish the interplay between the probability of remaining safe after the zero-shot deployment and the quality of the tracking of the inner states. We validate our theoretical findings on a quadrotor navigation task, demonstrating that the preservation of the safety guarantees is tied to the bandwidth and tracking capabilities of the low-level controller.

Jonathan Light, Wei Cheng, Benjamin Riviere et al.

Inference scaling methods for LLMs often rely on decomposing problems into steps (or groups of tokens), followed by sampling and selecting the best next steps. However, these steps and their sizes are often predetermined or manually designed based on domain knowledge. We propose dynamic decomposition, a method that adaptively and automatically partitions solution and reasoning traces into manageable steps during inference. By more effectively allocating compute -- particularly through subdividing challenging steps and prioritizing their sampling -- dynamic decomposition significantly improves inference efficiency. Experiments on benchmarks such as APPS, MATH, and LiveCodeBench demonstrate that dynamic decomposition outperforms static approaches, including token-level, sentence-level, and single-step decompositions, reducing the pass@10 error rate by 5.0%, 6.7%, and 10.5% respectively. These findings highlight the potential of dynamic decomposition to improve a wide range of inference scaling techniques.

Weiqin Chen, James Onyejizu, Long Vu et al.

Primal-dual methods have a natural application in Safe Reinforcement Learning (SRL), posed as a constrained policy optimization problem. In practice however, applying primal-dual methods to SRL is challenging, due to the inter-dependency of the learning rate (LR) and Lagrangian multipliers (dual variables) each time an embedded unconstrained RL problem is solved. In this paper, we propose, analyze and evaluate adaptive primal-dual (APD) methods for SRL, where two adaptive LRs are adjusted to the Lagrangian multipliers so as to optimize the policy in each iteration. We theoretically establish the convergence, optimality and feasibility of the APD algorithm. Finally, we conduct numerical evaluation of the practical APD algorithm with four well-known environments in Bullet-Safey-Gym employing two state-of-the-art SRL algorithms: PPO-Lagrangian and DDPG-Lagrangian. All experiments show that the practical APD algorithm outperforms (or achieves comparable performance) and attains more stable training than the constant LR cases. Additionally, we substantiate the robustness of selecting the two adaptive LRs by empirical evidence.

Weiqin Chen, Xinjie Zhang, Dharmashankar Subramanian et al.

Transformer models (TMs) have exhibited remarkable in-context reinforcement learning (ICRL) capabilities, allowing them to generalize to and improve in previously unseen environments without re-training or fine-tuning. This is typically accomplished by imitating the complete learning histories of a source RL algorithm over a substantial amount of pretraining environments, which, however, may transfer suboptimal behaviors inherited from the source algorithm/dataset. Therefore, in this work, we address the issue of inheriting suboptimality from the perspective of dataset preprocessing. Motivated by the success of the weighted empirical risk minimization, we propose a simple yet effective approach, learning history filtering (LHF), to enhance ICRL by reweighting and filtering the learning histories based on their improvement and stability characteristics. To the best of our knowledge, LHF is the first approach to avoid source suboptimality by dataset preprocessing, and can be combined with the current state-of-the-art (SOTA) ICRL algorithms. We substantiate the effectiveness of LHF through a series of experiments conducted on the well-known ICRL benchmarks, encompassing both discrete environments and continuous robotic manipulation tasks, with three SOTA ICRL algorithms (AD, DPT, DICP) as the backbones. LHF exhibits robust performance across a variety of suboptimal scenarios, as well as under varying hyperparameters and sampling strategies. Notably, the superior performance of LHF becomes more pronounced in the presence of noisy data, indicating the significance of filtering learning histories.

Anmol Dwivedi, Ali Tajer, Santiago Paternain et al.

Electricity grid's resiliency and climate change strongly impact one another due to an array of technical and policy-related decisions that impact both. This paper introduces a physics-informed machine learning-based framework to enhance grid's resiliency. Specifically, when encountering disruptive events, this paper designs remedial control actions to prevent blackouts. The proposed Physics-Guided Reinforcement Learning (PG-RL) framework determines effective real-time remedial line-switching actions, considering their impact on power balance, system security, and grid reliability. To identify an effective blackout mitigation policy, PG-RL leverages power-flow sensitivity factors to guide the RL exploration during agent training. Comprehensive evaluations using the Grid2Op platform demonstrate that incorporating physical signals into RL significantly improves resource utilization within electric grids and achieves better blackout mitigation policies - both of which are critical in addressing climate change.

Weiqin Chen, Santiago Paternain

Pretrained foundation models have exhibited extraordinary in-context learning performance, allowing zero-shot generalization to new tasks not encountered during pretraining. In the case of reinforcement learning (RL), in-context RL (ICRL) emerges when pretraining FMs on decision-making problems in an autoregressive-supervised manner. Nevertheless, current state-of-the-art ICRL algorithms, like Algorithm Distillation, Decision Pretrained Transformer and Decision Importance Transformer, impose stringent requirements on the pretraining dataset concerning the source policies, context information, and action labels. Notably, these algorithms either demand optimal policies or require varying degrees of well-trained behavior policies for all pretraining environments. This significantly hinders the application of ICRL to real-world scenarios, where acquiring optimal or well-trained policies for a substantial volume of real-world training environments can be intractable. To overcome this challenge, we introduce a novel approach, termed State-Action Distillation (SAD), that allows to generate an effective pretraining dataset guided solely by random policies. In particular, SAD selects query states and corresponding action labels by distilling outstanding state-action pairs from the entire state and action spaces by using random policies within a trust horizon, and then inherits the classical autoregressive-supervised mechanism during pretraining. To the best of our knowledge, this is the first work that enables effective ICRL under random policies and random contexts. We also establish quantitative analysis of the trustworthiness as well as the performance guarantees of SAD. Moreover, our empirical results across multiple popular ICRL benchmark environments demonstrate that, on average, SAD outperforms the best baseline by 236.3% in the offline evaluation and by 135.2% in the online evaluation.

Sergio Rozada, Santiago Paternain, Juan Andres Bazerque et al.

In pursuit of reinforcement learning systems that could train in physical environments, we investigate multi-task approaches as a means to alleviate the need for massive data acquisition. In a tabular scenario where the Q-functions are collected across tasks, we model our learning problem as optimizing a higher order tensor structure. Recognizing that close-related tasks may require similar actions, our proposed method imposes a low-rank condition on this aggregated Q-tensor. The rationale behind this approach to multi-task learning is that the low-rank structure enforces the notion of similarity, without the need to explicitly prescribe which tasks are similar, but inferring this information from a reduced amount of data simultaneously with the stochastic optimization of the Q-tensor. The efficiency of our low-rank tensor approach to multi-task learning is demonstrated in two numerical experiments, first in a benchmark environment formed by a collection of inverted pendulums, and then into a practical scenario involving multiple wireless communication devices.

Weiqin Chen, Mark S. Squillante, Chai Wah Wu et al.

We devise a control-theoretic reinforcement learning approach to support direct learning of the optimal policy. We establish various theoretical properties of our approach, such as convergence and optimality of our analog of the Bellman operator and Q-learning, a new control-policy-variable gradient theorem, and a specific gradient ascent algorithm based on this theorem within the context of a specific control-theoretic framework. We empirically evaluate the performance of our control theoretic approach on several classical reinforcement learning tasks, demonstrating significant improvements in solution quality, sample complexity, and running time of our approach over state-of-the-art methods.

Leopoldo Agorio, Sean Van Alen, Miguel Calvo-Fullana et al.

We address the conflicting requirements of a multi-agent assignment problem through constrained reinforcement learning, emphasizing the inadequacy of standard regularization techniques for this purpose. Instead, we recur to a state augmentation approach in which the oscillation of dual variables is exploited by agents to alternate between tasks. In addition, we coordinate the actions of the multiple agents acting on their local states through these multipliers, which are gossiped through a communication network, eliminating the need to access other agent states. By these means, we propose a distributed multi-agent assignment protocol with theoretical feasibility guarantees that we corroborate in a monitoring numerical experiment.

Arindam Chowdhury, Santiago Paternain, Gunjan Verma et al.

We propose a learning-based framework for efficient power allocation in ad hoc interference networks under episodic constraints. The problem of optimal power allocation -- for maximizing a given network utility metric -- under instantaneous constraints has recently gained significant popularity. Several learnable algorithms have been proposed to obtain fast, effective, and near-optimal performance. However, a more realistic scenario arises when the utility metric has to be optimized for an entire episode under time-coupled constraints. In this case, the instantaneous power needs to be regulated so that the given utility can be optimized over an entire sequence of wireless network realizations while satisfying the constraint at all times. Solving each instance independently will be myopic as the long-term constraint cannot modulate such a solution. Instead, we frame this as a constrained and sequential decision-making problem, and employ an actor-critic algorithm to obtain the constraint-aware power allocation at each step. We present experimental analyses to illustrate the effectiveness of our method in terms of superior episodic network-utility performance and its efficiency in terms of time and computational complexity.

Sergio Rozada, Santiago Paternain, Antonio G. Marques

Value-function (VF) approximation is a central problem in Reinforcement Learning (RL). Classical non-parametric VF estimation suffers from the curse of dimensionality. As a result, parsimonious parametric models have been adopted to approximate VFs in high-dimensional spaces, with most efforts being focused on linear and neural-network-based approaches. Differently, this paper puts forth a a parsimonious non-parametric approach, where we use stochastic low-rank algorithms to estimate the VF matrix in an online and model-free fashion. Furthermore, as VFs tend to be multi-dimensional, we propose replacing the classical VF matrix representation with a tensor (multi-way array) representation and, then, use the PARAFAC decomposition to design an online model-free tensor low-rank algorithm. Different versions of the algorithms are proposed, their complexity is analyzed, and their performance is assessed numerically using standardized RL environments.

Luiz F. O. Chamon, Santiago Paternain, Miguel Calvo-Fullana et al.

Though learning has become a core component of modern information processing, there is now ample evidence that it can lead to biased, unsafe, and prejudiced systems. The need to impose requirements on learning is therefore paramount, especially as it reaches critical applications in social, industrial, and medical domains. However, the non-convexity of most modern statistical problems is only exacerbated by the introduction of constraints. Whereas good unconstrained solutions can often be learned using empirical risk minimization, even obtaining a model that satisfies statistical constraints can be challenging. All the more so, a good one. In this paper, we overcome this issue by learning in the empirical dual domain, where constrained statistical learning problems become unconstrained and deterministic. We analyze the generalization properties of this approach by bounding the empirical duality gap -- i.e., the difference between our approximate, tractable solution and the solution of the original (non-convex) statistical problem -- and provide a practical constrained learning algorithm. These results establish a constrained counterpart to classical learning theory, enabling the explicit use of constraints in learning. We illustrate this theory and algorithm in rate-constrained learning applications arising in fairness and adversarial robustness.

Miguel Calvo-Fullana, Luiz F. O. Chamon, Santiago Paternain

Safety is a critical feature of controller design for physical systems. When designing control policies, several approaches to guarantee this aspect of autonomy have been proposed, such as robust controllers or control barrier functions. However, these solutions strongly rely on the model of the system being available to the designer. As a parallel development, reinforcement learning provides model-agnostic control solutions but in general, it lacks the theoretical guarantees required for safety. Recent advances show that under mild conditions, control policies can be learned via reinforcement learning, which can be guaranteed to be safe by imposing these requirements as constraints of an optimization problem. However, to transfer from learning safety to learning safely, there are two hurdles that need to be overcome: (i) it has to be possible to learn the policy without having to re-initialize the system; and (ii) the rollouts of the system need to be in themselves safe. In this paper, we tackle the first issue, proposing an algorithm capable of operating in the continuing task setting without the need of restarts. We evaluate our approach in a numerical example, which shows the capabilities of the proposed approach in learning safe policies via safe exploration.

Miguel Calvo-Fullana, Santiago Paternain, Luiz F. O. Chamon et al.

A common formulation of constrained reinforcement learning involves multiple rewards that must individually accumulate to given thresholds. In this class of problems, we show a simple example in which the desired optimal policy cannot be induced by any weighted linear combination of rewards. Hence, there exist constrained reinforcement learning problems for which neither regularized nor classical primal-dual methods yield optimal policies. This work addresses this shortcoming by augmenting the state with Lagrange multipliers and reinterpreting primal-dual methods as the portion of the dynamics that drives the multipliers evolution. This approach provides a systematic state augmentation procedure that is guaranteed to solve reinforcement learning problems with constraints. Thus, as we illustrate by an example, while previous methods can fail at finding optimal policies, running the dual dynamics while executing the augmented policy yields an algorithm that provably samples actions from the optimal policy.

Clark Zhang, Santiago Paternain, Alejandro Ribeiro

Data driven models of dynamical systems help planners and controllers to provide more precise and accurate motions. Most model learning algorithms will try to minimize a loss function between the observed data and the model's predictions. This can be improved using prior knowledge about the task at hand, which can be encoded in the form of constraints. This turns the unconstrained model learning problem into a constrained one. These constraints allow models with finite capacity to focus their expressive power on important aspects of the system. This can lead to models that are better suited for certain tasks. This paper introduces the constrained Sufficiently Accurate model learning approach, provides examples of such problems, and presents a theorem on how close some approximate solutions can be. The approximate solution quality will depend on the function parameterization, loss and constraint function smoothness, and the number of samples in model learning.

Luiz F. O. Chamon, Santiago Paternain, Alejandro Ribeiro

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.

Santiago Paternain, Juan Andres Bazerque, Alejandro Ribeiro

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.

Luiz F. O. Chamon, Santiago Paternain, Miguel Calvo-Fullana et al.

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.

Santiago Paternain, Miguel Calvo-Fullana, Luiz F. O. Chamon et al.

In this paper, we study the learning of safe policies in the setting of reinforcement learning problems. This is, we aim to control a Markov Decision Process (MDP) of which we do not know the transition probabilities, but we have access to sample trajectories through experience. We define safety as the agent remaining in a desired safe set with high probability during the operation time. We therefore consider a constrained MDP where the constraints are probabilistic. Since there is no straightforward way to optimize the policy with respect to the probabilistic constraint in a reinforcement learning framework, we propose an ergodic relaxation of the problem. The advantages of the proposed relaxation are threefold. (i) The safety guarantees are maintained in the case of episodic tasks and they are kept up to a given time horizon for continuing tasks. (ii) The constrained optimization problem despite its non-convexity has arbitrarily small duality gap if the parametrization of the policy is rich enough. (iii) The gradients of the Lagrangian associated with the safe-learning problem can be easily computed using standard policy gradient results and stochastic approximation tools. Leveraging these advantages, we establish that primal-dual algorithms are able to find policies that are safe and optimal. We test the proposed approach in a navigation task in a continuous domain. The numerical results show that our algorithm is capable of dynamically adapting the policy to the environment and the required safety levels.

Santiago Paternain, Luiz F. O. Chamon, Miguel Calvo-Fullana et al.

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.

Bruno A. Angélico, Luiz F. O. Chamon, Santiago Paternain et al.

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.

Maria Peifer, Luiz. F. O. Chamon, Santiago Paternain et al.

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.

Clark Zhang, Arbaaz Khan, Santiago Paternain et al.

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.

Alec Koppel, Santiago Paternain, Cedric Richard et al.

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.