When to Sense and Control? A Time-adaptive Approach for Continuous-Time RL

Authors: lenart  treven
bhavya  sukhija
yarden  as
florian  dörfler
andreas  krause
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 09:44:07

Abstract: Reinforcement learning (RL) excels in optimizing policies for discrete-time Markov decision processes (MDP). However, various systems are inherently continuous in time, making discrete-time MDPs an inexact modeling choice. 
In many applications, such as greenhouse control or medical treatments, each interaction (measurement or switching of action) involves manual intervention and thus is inherently costly. Therefore, 
we generally prefer a time-adaptive approach with fewer interactions with the system.
In this work, we formalize an RL framework, 
**T**ime-**a**daptive **Co**ntrol \& **S**ensing (**TaCoS**), that tackles this challenge by optimizing over policies that besides control predict the duration of its application. Our formulation results in an extended MDP that any standard RL algorithm can solve.
We demonstrate that state-of-the-art RL algorithms trained on TaCoS drastically reduce the interaction amount over their discrete-time counterpart while retaining the same or improved performance, and exhibiting robustness over discretization frequency.
Finally, we propose OTaCoS, an efficient model-based algorithm for our setting. We show that OTaCoS enjoys sublinear regret for systems with sufficiently smooth dynamics and empirically results in further sample-efficiency gains.

Main paper: 
 introduction: 
 Nearly all state-of-the-art RL algorithms (Schulman et al., 2017; Haarnoja et al., 2018; Lillicrap et al., 2015; Schulman et al., 2015) were developed for discrete-time MDPs. Nevertheless, continuoustime systems are ubiquitous in nature, ranging from robotics, biology, medicine, environment and sustainability etc. (cf. Spong et al., 2006; Jones et al., 2009; Lenhart and Workman, 2007; Panetta and Fister, 2003; Turchetta et al., 2022) . Such systems can be naturally modeled with stochastic differential equations (SDEs), but computational approaches necessitate discretization. Furthermore, in many applications, obtaining measurements and switching actions is expensive. For instance, consider a greenhouse of fruits or medical treatment recommendations. In both cases, each measurement (crop inspection, medical exam) or switching of actions (climate control, treatment adjustment) typically involves costly human intervention. Hence, minimizing such interactions with the underlying system is desirable. This underlying challenge is rarely addressed in the RL literature.In practice, a time-equidistant discretization frequency is set, often manually, adjusted to the underlying system's characteristic time scale. This is challenging, however, especially for unknown/uncertain systems, and systems with multiple dominant time scales (Engquist et al., 2007) . Therefore, for many real-world applications having a global frequency of control is inadequate and wasteful. For example, in medicine, patient monitoring often requires higher frequency interaction during the onset of illness and lower frequency interactions as the patient recovers (Kaandorp and Koole, 2007) .In this work, we address this limitation of standard RL methods and propose a novel RL framework, Time-adaptive Control & Sensing (TACOS). TACOS reduces a general continuous-time RL problem with underlying SDE dynamics to an equivalent discrete-time MDP, that can be solved with any RL algorithm, including standard policy gradient methods like PPO and SAC (Schulman et al., 2017; Haarnoja et al., 2018) . We summarize our contributions below. Contributions 1. We reformulate the problem of time-adaptive continuous time RL to an equaivalent discrete-time MDP that can be solved with standard RL algorithms. 2. Using our formulation, we extend standard policy gradient techniques (Haarnoja et al. (2018) and Schulman et al. (2017) ) to the time-adaptive setting. Our empirical results on standard RL benchmarks (Freeman et al., 2021) show that TACOS outperforms its discrete-time counterpart in terms of policy performance, computational cost, and sample efficiency. 3. To further improve sample efficiency, we propose a model-based RL algorithm, OTACOS.OTACOS uses well-calibrated probabilistic models to capture epistemic uncertainty and, similar to Curi et al. (2020) and Treven et al. (2023) , leverages the principle of optimism in the face of uncertainty to guide exploration during learning. We theoretically prove that OTACOS suffers no regret and empirically demonstrate its sample efficiency. 
problem statement: 
 We consider a general nonlinear continuous time dynamical system with continuous state X ⊂ R dx and action U ⊂ R du space. The underlying dynamics are governed by a (controllable) SDE:EQUATIONHere x t ∈ X is the state at time t, u t ∈ U the control input, f * , g * are unknown drift and diffusion functions and B t is a standard Brownian motion in R d B . Our goal is to find a control policy π U : X × T → U which maximizes an unknown reward b * (x t , u t ) over a fixed horizonT def = [0, T ], i.e., max π∈Π E t∈T b * (x t , π U (x t , t))dt ,where the expectation is taken w.r.t. the policy and stochastic dynamics and Π is the class of policies2 over which we search.In practice, we can only measure the system state and execute control policies in discrete points in time. In this work, we focus on problems where state measurement and control are synchronized in time. We refer to these synchronized time points as interactions in the following parts of this paper. Synchronizing state measurement and control contrasts standard time-adaptive approaches such as event-triggered control (Heemels et al., 2021) , where the state is measured arbitrarily high frequency and control inputs are changed only so often to ensure stability. It is also in contrast to the complementary setting, where control inputs are changing at an arbitrarily high frequency but measurements are collected adaptively in time (Treven et al., 2023) . An adaptive control approach as Heemels et al. (2021) is very important for many real-world applications but similarly, an adaptive measurement strategy is crucial for efficient learning in RL (Treven et al., 2023) . Our approach treats both of these requirements jointly.We consider two different scenarios for continuous-time control: (i) Penalizing interactions with some cost, (ii) bounded number of interactions, i.e., hard constraint on control/measurement steps.Interaction cost We consider the setting where every interaction we take has an inherent cost c(x t , u t ) > 0. Note that we consider this cost structure for its simplicity and TACOS works for more general cost functions that depend on the duration of application for the action u t or the previous action u t-1 and thus captures many practical real-world settings. We define this task more formally belowEQUATIONt i = π T (x ti-1 , t i-1 ) + t i-1 , t 0 = 0, t K = T, ∀(x, t) ∈ X × T ; π T (x, t) ∈ [t min , t max ].Here t min > 0 is the minimal duration for which we have to apply the control, t max ∈ [t min , T ] the maximum duration, and π T is a policy that predicts the duration of applying the action.Bounded number of interactions In this setting, the number of interactions with the system is limited by a known amount K. Intuitively, this represents a scenario where we have a finite budget for the inputs that we can apply and have to decide on the best strategy to space these K inputs over the full horizon. A formal definition of this task is given belowEQUATIONt i = π T (x ti-1 , t i-1 , i -1) + t i-1 , t 0 = 0, t K = T, ∀(x, t, i) : π T (x, t, i) ∈ [t min , t max ].In the absence of the transition costs or the bound on the number of interactions, intuitively the policy would propose to interact with the system as frequently as possible, i.e., every t min seconds. The additional costs/constraints ensure that we do not converge to this trivial (but unrealistic) solution.Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: In Section 5 we highlight the assumptions of our work, which also correspond to the limitations of our theoretical analysis and also the setting for which our algorithm yields theoretical guarantees. Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. 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tacos: time adaptive control or sensing: 
 In the following, we reformulate the continuous-time problem as an equivalent discrete-time MDP.We first denote the state and running reward flows of Equation ( 1). The state flow by applying action u k for t k time reads:x k+1 = Ξ(x k , u k , t k ), Ξ(x, u, t) def = x + t 0 f * (x s , u)ds + t 0 g * (x s , u)dB s .We assume that every time we interact with the system, we also obtain the integrated reward and define the reward flow asEQUATIONDue to the stochasticity of (B t ) t∈T , the state flow Ξ(x, u, t) and the reward flow Ξ b * (x, u, t) are stochastic. For ease of notation, we denoteΦ f * (x k , u k , t k ) def = E [Ξ(x k , u k , t k )] , Φ b * (x k , u k , t k ) def = E [Ξ b * (x k , u k , t k )] w x k def = Ξ(x k , u k , t k ) -Φ(x k , u k , t k ), w b * k def = Ξ b * (x k , u k , t k ) -Φ b * (x k , u k , t k ), and the concatenated state and reward flow function, and noise as:EQUATIONIn this work, we search for policies that return the next control we apply and also the time for how long to apply the control.We convert the problem with interaction costs to a standard MDP which any RL algorithm for continuous state-action spaces can solve. To this end, we restrict ourselves to a policy class:Π IC = {π : X × T → U × T | π T (•, t) ∈ [t min , t max ], π is L π -Lipschitz} .For simplicity, we denote by π T the component of the policy that predicts the duration of applying the action and with π U the component that predicts the action value. The policies we consider map state x and time-to-go t to control u and the time τ for how long we apply the action u. We define the augmented state s = (x, b, t), where x is the state, b integrated reward and t time-to-go. With the introduced notation we arrive at a discrete-time MDP problem formulationmax π∈Π IC V π,Φ * (x 0 , T ) = max π∈Π IC E K-1 k=0 r(s k , π(s k )) (6) s.t. s k+1 = Ψ Φ * (s k , π(s k ), w k ), s 0 = (x 0 , 0, T ), K-1 k=0 π T (x k , t k ) = T,where we have (a) We add a constant switch cost of 0.1 and significantly reduce the number of interactions from 200 to 24. Initially, the policy applies maximal bang-bang torque for longer times, until the pendulum reaches the top. On the top, we measure and change the controller at a higher frequency in order to keep the pendulum stable, at the position with the highest reward. Ψ Φ * (s k , π(s k ), w k ) = (Φ * (x k , π(x k , t k )) + w k , t k -π T (x k , t k )) r(s k , π(s k )) = Ξ b * (x k , π(x k , t k )) -c(x k , π U (x k , t k )).The second setting is similar to the one studied by Ni and Jang (2022) . In this case, we consider the following class of policies:Π BI = {π : X × T × N → U × T | ∀k ∈ [K] : π(•, •, k) is L π -Lipschitz} .For an augmented state s = (x, b, t, k), our policies map states x, time-to-go t, number of past interactions k to a controller u and the time duration τ for applying the action. Here the optimal control problem readsmax π∈Π BI V π,Φ * (x 0 , T ) = max π∈Π BI E K-1 k=0 r(s k , π(s k )) (7) s.t. s k+1 = Ψ Φ * (s k , π(s k ), w k ), s 0 = (x 0 , 0, T, 0),where,Ψ Φ * (s k , π(s k ), w k ) = (Φ * (x k , π(x k , t k , k)) + w k , t k -π T (x k , t k , k), k + 1) r(s k , π(s k )) = Ξ b * (x k , π(x k , t k , k)).In the following, we provide a simple proposition which shows that our reformulated problem is equivalent to its continuous-time counterpart from Section 2. Proposition 1. The problem in Equation (2) and 3 are equivalent to Equation (6) and 7, respectively.Figure 1 depicts the influence of interaction cost and K on the controller's performance for the pendulum environment.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Justification: This paper proposes a method to improve exploration in reinforcement learning in the nonepisodic setting, and is not tied to specific applications. As such, it shares the many potential societal consequences that are associated with reinforcement learning and automation as a whole, spanning from environmental impact to concerns on ethics and alignment.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
tacos with model-free rl algorithms: 
 We now illustrate the performance of TACOS on several well-studied robotic RL tasks. We consider the RC car (Kabzan et al., 2020) , Greenhouse (Tap, 2000) , Pendulum, Reacher, Halfcheetah and Humanoid environments from Brax (Freeman et al., 2021) . Thus our experiments range from environments necessitating time-adaptive control like the Greenhouse, a realistic and highly dynamic race car simulation, and a very high dimensional RL task like the Humanoid. 3We investigate both the bounded number of interactions and interaction cost settings in our experiments. In particular, we study how the bound K affects the performance of TACOS and compare it to the standard equidistant baseline. We further study the interplay between the stochasticity of the environments (magnitude of g * ) and interaction costs and the influence of t min on TACOS. For all experiments in this section, we combine SAC with TACOS (SAC-TACOS). TACOS performs significantly better than equidistant discretization, especially for small values of K.How does the bound on the number of interactions K affect TACOS? We analyze the bounded number of interactions setting (cf. Section 3.2) of TACOS, studying the relationship between the number of interactions and the achieved episode reward. We compare our algorithm with the standard equidistant time discretization approach which splits the whole horizon T into T /K discrete time steps at which an interaction takes place. We evaluate the two methods in the greenhouse and pendulum environments. For the pendulum, we consider the swing-up and swing-down tasks. The results are reported in Figure 2 . The time-adaptive approach performs significantly better than the standard equidistant time discretization. This is particularly the case for the greenhouse and pendulum swing-down tasks. Both tasks involve driving the system to a stable equilibrium and thus, while high-frequency interaction might be necessary at the initial stages, a fairly low interaction frequency can be maintained when the system has reached the equilibrium state. This demonstrates the practical benefits of time-adaptive control. How does the interaction cost magnitude influence TACOS? We investigate the setting from Section 3.1 with interaction costs. In our experiments, we always pick a constant cost, i.e., c(x, u) = C. We study the influence of C on the episode reward and on the number of interactions that the policy has with the system within an episode. We again evaluate this on the greenhouse and pendulum environment. For the pendulum, we consider the swing-up task. The results are presented in the first row of Figure 3 . Noticeably, increasing C reduces the number of interactions. The decrease is drastic for the greenhouse environment since it can be controlled with considerably fewer interactions without having any effect on the performance. Generally, we observe that decreasing the number of interactions, that is, increasing C, also results in a slight decline in episode reward.How does environment stochasticity influence the number of interactions? We analyze the influence of the environment's stochasticity, i.e., the magnitude of the diffusion term g * , on the episode reward and number of interactions on TACOS. Intuitively, the more stochastic the environment, the more interactions we would require to stabilize the system. We again evaluate our method on the greenhouse and pendulum swing-up tasks. The results are reported in the second row of Figure 3 . The results verify our intuition that more stochasticity in the environment generally leads to more interactions. However, we observe that the policy is still able to achieve high rewards for a wide range of magnitude of g * . This showcases the robustness and adaptability of TACOS to stochastic environments. In the second row, we plot the episode reward versus the physical time in seconds spent in the environment for SAC-TACOS, SAC, and SAC-MC for a specific evaluation frequency 1 /teval. We exclude PPO-TACOS in this plot as it, being on-policy, requires significantly more samples than the off-policy methods. While all methods perform equally well for standard discretization (denoted with 1/t * ), our method is robust to interaction frequency and does not suffer a performance drop when we decrease t min .How does t min influence TACOS? As highlighted in Section 1, picking the right discretization for interactions is a challenging task. We show that TACOS can naturally alleviate this issue and adaptively pick the frequency of interaction while also being more computationally and data-efficient. Moreover, we show that TACOS is robust to the choice of t min , which represents the minimal duration an action has to be applied, i.e., its inverse is the highest frequency at which we can control the system. In this experiment, we consider SAC-TACOS and compare it to the standard SAC algorithm. TACOS adaptively picks the number of interactions and therefore during an episode of time T , it effectively collects less data than the standard discrete-time RL algorithm. 4 This makes comparison to the discrete-time setting challenging since environment interactions and physical time on the environment are not linearly related for TACOS as opposed to the standard discrete-time setting. Nevertheless, to be fair to the discrete-time method, we give SAC more physical time on the system for all environments, effectively resulting in the collection of more data for learning. Since the standard SAC algorithm updates the policy relative to the data amount, we consider a version of SAC, SAC-MC (SAC more compute), which leverages the additional data it collects to perform more gradient updates. This version essentially performs more policy updates than SAC-TACOS and thus is computationally more expensive. Furthermore, to demonstrate the generality of our framework, we also combine TACOS with PPO (PPO-TACOS).We report the performance after convergence across different t min in the first row of Figure 4 . From our experiment, we conclude that SAC-TACOS and PPO-TACOS are robust to the choice of t min and perform equally well when t min is decreased, i.e., frequency is increased. This is in contrast to the standard RL methods, which have a significant drop in performance at high frequencies. This observation is also made in prior work (Hafner et al., 2019) . Crucially, this highlights the sensitivity of the standard RL methods to the frequency of interaction. In the second row of Figure 4 we show the learning curve of the methods for a specific frequency 1 /teval. From the curve, we conclude that SAC-TACOS achieves higher rewards with significantly less physical time on the environment. We believe this is because our method explores more efficiently (akin to Dabney et al., 2020; Eberhard et al., 2022) , and also learns a much stronger/continuous-time representation of the underlying MDP.Interestingly, at the default frequency used in the benchmarks 1 /t * , all methods perform similarly. However, slightly decreasing the frequency already leads to a drastic drop in performance for all methods. Intuitively, decreasing the frequency prevents us from performing the necessary fine-grained control and obtaining the highest performance.While we have access to the optimal frequency 1 /t * for these benchmarks, for a general and unknown system it is very difficult to estimate this frequency. Furthermore, as we observe in our experiments, picking a very high frequency is also not an option when using standard RL algorithms. We believe this is where TACOS excels as it adaptively picks the frequency of interaction, thereby relieving the problem designer of this decision. 
efficient exploration for tacos via model-based rl: 
 In this section, we propose a novel model-based RL algorithm for TACOS called Optimistic TACOS (OTACOS). We analyze the episodic setting, where we interact with the system in episodes n = 1, . . . , N . In episode n, we execute the policy π n , collect measurements and integrated rewards (x n,0 , b n,0 ), . . . , (x n,kn , b n,kn ), and prepare the data We assume Φ * ∈ n≥0 M n with probability at least 1δ, where statistical model M n is defined asD n = {(z n,1 , y n,1 ), . . . , (z n,kn , y n,kn )}, where z n,i = (x n,i-1 , u n,i-1 , t n,i-1 ) and y n,i = (x n,i , b n,i ).M n def = f : Z → R dx+1 | ∀z ∈ Z, ∀j ∈ {1, . . . , d x + 1} : |µ n,j (z) -f j (z)| ≤ β n (δ)σ n,j (z) ,Here, µ n,j and σ n,j denote the j-th element in the vector-valued mean and standard deviation functions µ n and σ n respectively, and β n (δ) ∈ R ≥0 is a scalar function that depends on the confidence level δ ∈ (0, 1] and which is monotonically increasing in n.Similar to model-based RL algorithms for the discrete-time setting (Kakade et al., 2020; Curi et al., 2020; Sukhija et al., 2024) , we follow the principle of optimism in the face of uncertainty and select the policy π n for both settings of TACOS (cf. Sections 3.1 and 3.2) by solving:EQUATIONwhere □ ∈ {IC, BI} is the appropriate policy class from Section 3. Running OTACOS for N episodes, we measure the performance via the regret:R N = N n=1 V π * ,Φ * (x 0 , T ) -V πn,Φ * (x 0 , T ) .Here π * is the optimal policy from the class of policies we optimize over. Any kind of regret bound requires certain assumptions on the regularity of the underlying dynamics (1).Assumption 1 (Dynamics model). Given any norm ∥•∥, we assume that the drift f * , and diffusion g * are L f * and L g * -Lipschitz continuous, respectively, with respect to the induced metric. We further assume sup z∈Z ∥g * (z)∥ F ≤ A.Assumption 1 ensures the existence of the SDE (1) solution under policy π n . To provide bounds on the performance of OTACOS for settings Sections 3.1 and 3.2 we also need some assumptions on the noise and reward model.Assumption 2 (Reward and noise model for Section 3.1 Setting). Given any norm ∥•∥, we assume that running reward b is L b -Lipschitz continuous, with respect to the induced metric. We further assume boundedness of the reward 0 ≤ b * (x, u) ≤ B, and interaction cost 0 ≤ c(x, u) ≤ C. The dynamics noise is independent and follows: w x k ∼ N 0, σ 2 (x k , u k , t k )I dx . Assumption 3 (Reward and noise model for Section 3.2 Setting). Given any norm ∥•∥, we assume that the running reward b is L b -Lipschitz continuous, w.r.t. to the induced metric.Finally, we assume that we learn a well-calibrated model of the unknown flow Φ * . Assumption 4 (Well calibration assumption). Our learned model is an all-time-calibrated statistical model of Φ * , i.e., there exists an increasing sequence of (β n (δ)) n≥0 such that our model satisfies the well-calibration condition, cf., Definition 1.Analogous assumptions are made for model-based RL algorithms in the discrete-time setting (Curi et al., 2020; Sukhija et al., 2024) . This calibration assumption is satisfied if Φ * can be represented with Gaussian Process (GP) (Williams and Rasmussen, 2006; Kirschner and Krause, 2018) models. Theorem 2. Consider the setting from Section 3.1 and let Assumption 1, 2, and Assumption 4 hold. Then we have with probability at least 1δ:R N ≤ O β N -1 T 3/2 N I NNow consider, the setting with a bounded number of switches K, and let Assumption 1, 3, and Assumption 4 hold. Then, we get with probability at least 1δR N ≤ O β K N -1 Ke D(L f * +L 2 g * )(1+Lπ)T K N I N ,where D is a constant. Here, with I N we denote the model-complexity after observing N points (Curi et al., 2020) , which quantifies the difficulty of learning Φ * . For GPs, it behaves similar to the maximum information gain γ N (Srinivas et al., 2009) , i.e., implying sublinear regret for several common kernels (Vakili et al., 2021) .As a proof of concept, we evaluate OTACOS on the pendulum and RC car environment for the interaction cost setting. 5 As baselines, we adapt common model-based RL methods such as PETS (Chua et al., 2018) and planning with the mean to TACOS. We call them PETS-TACOS and MEAN-TACOS, respectively. The result is reported in Figure 5 . From the figure, we conclude that OTACOS is more sample efficient than other model-based baselines and SAC-TACOS (SAC-TACOS requires circa 6000 episodes for the pendulum and 2000 for the RC car). 2023) proposes a similar problem as ours from Section 3.1, where they specify a cost on the number of interactions. However, their solution is based on a heuristic, where a measurement is taken when the variance of the potential reward surpasses a prespecified threshold. On the contrary, we directly tackle this problem at hand and propose a general framework for time-adaptive control that does not rely on any heuristics. Karimi (2023) adapt SAC (Haarnoja et al., 2018) to include a regularization term, which effectively adds a cost for every discrete interaction. Ni and Jang (2022) induce a soft-constraint on the duration τ of each action in the environment. However, all the aforementioned works propose heuristic techniques to minimize interactions, whereas we formalize the problem systematically for the more general case of SDEs and show that it has an underlying MDP structure that any RL algorithm can leverage. In addition, we propose a no-regret model-based RL algorithm for this setting and analyze its sample complexity.Temporal abstractions are considered also in the framework of options (Sutton et al., 1999; Mankowitz et al., 2014; Mann and Mannor, 2014; Harb et al., 2018) . However, a key difference to TACOS is that in the options framework, the agent measures the state even between the controller switches.Learning to repeat actions Several works observe that repeating actions in the discrete-time MDPs problems such as Atari (Mnih et al., 2013; Braylan et al., 2015) or Cartpole (Hafner et al., 2019) significantly increase the speed of learning. However, the action repeat is fixed through the entire rollout and treated as a hyperparameter. 2020) empirically show that repeating the actions helps with the exploration, effectively having a similar effect that colored noise exploration has over the standard white noise exploration (Eberhard et al., 2022) .Continuous-time RL Following the seminal work of Doya (2000) and the advances in Neural ODEs of Chen et al. (2018) , continuous-time RL has regained interest (Cranmer et al., 2020; Greydanus et al., 2019; Yildiz et al., 2021; Lutter et al., 2021) . Moreover, modeling in continuous-time is found to be particularly useful when learning from different data sources where each source is collected at a different frequency (Burns et al., 2023; Zheng et al., 2023 ). An important line of work exists for modeling continuous dynamics for the case when states and actions are discrete, called Markov Jump Processes (Kallianpur and Sundar, 2014; Berger, 1993; Huang et al., 2019; Seifner and Sanchez, 2023) . Another line of work that is close to ours is event and self-Triggered Control (Astrom and Bernhardsson, 2002; Anta and Tabuada, 2010; Heemels et al., 2012 Heemels et al., , 2021)) , where they model continuous-time control systems by implementing changes to the input only when stability is at risk, ensuring efficient and timely interventions.  
conclusion and discussion: 
 We study the problem of time-adaptive RL for continuous-time systems with continuous state and action spaces. We investigate two practical settings where each interaction has an inherent cost and where we have a hard constraint on the number of interactions. We propose a novel RL framework, TACOS, and show that both of these settings result in extended MDPs which can be solved with standard RL algorithms. In our experiments, we show that combining standard RL algorithms with TACOS results in a significant reduction in the number of interactions without having any effect on the performance for the interaction cost setting. Furthermore, for the second setting, TACOS achieves considerably better control performance despite having a small budget for the number of interactions. Moreover, we show that TACOS improves robustness to a large range of interaction frequencies, and generally improves sample complexity of learning. Finally, we propose, OTACOS, a no-regret model-based RL algorithm for TACOS and show that it has further sample efficiency gains. 
safeguards: 
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 Appendix: In this section, we prove Theorem 2 for OTACOS. We separate the section into two parts; proof for the transaction cost setting (Appendix A.1) and the proof for the bounded number of switches setting (Appendix A.2).We start with the definitions of model complexity and sub-Gaussian random vector that we will use extensively in this section.Definition 2 (Model Complexity). We define the model complexity as is defined by Curi et al. (2020) .EQUATION)Definition 3. A random variable x ∈ R is said to be sub-Gaussian with variance proxy σ 2 if E[x] = 0 and we have:E[e tx ] ≤ e σ 2 t 2 2 , ∀t ∈ RA random vector x ∈ R d is said to be sub Gaussian with variance proxy σ 2 if for any e ∈ R d , ∥e∥ 2 = 1 the random variable x ⊤ e is σ 2 sub Gaussian. We write x ∼ subG σ 2 .In the following, we will be distinguishing between the state of the augmented MDP s and the true state of the dynamical system x. The augmented state at time step k includes the true state of the system, x k , the integrated reward b k between k -1 and k, and the time to left to go t k , i.e.,s k = [x ⊤ k , b k , t k ] ⊤ .We prove our regret bound for the transition cost case in the following. We start with the difference lemma which adapts Sukhija et al. (2024, Lemma 2) to our setting. Lemma 3 (Difference lemma). Define V πn,Φ (x, τ ) asE π,Φ   K(τ )-1 k≥0 r(s k , π(s k )) x 0 = x   ;whereK(τ )-1 k=0 π T (x k , t k ) = τthat is the total reward starting with time to go τ and state x for the policy π and dynamics Φ. Here the expectation w.r.t. π, Φ represents the expectation of the underlying trajectory induced by the policy π on the dynamics Φ. Then we have for all π, Φ ′ , Φ * , x 0 , T < 0;EQUATIONwhere x k+1 is the state ofs k+1 = Ψ Φ ′ (s k , π(s k ), w k ) and x k+1 is the state of s k+1 = Ψ Φ * (s k , π(s k ), w k ).Proof.V π,Φ * (x 0 , T ) = E π,Φ *   k≥0 r(s k , π(s k ))   = E π,Φ *   r(s 0 , π(s 0 )) + k≥1 r(s k , π(s k ))   = E π,Φ * [r(s k , π(s 0 )) + V π,Φ * (x 1 , t 1 )] = E π,Φ * [r(s k , π(s 0 )) + V π,Φ ′ ( x 1 , t 1 ) -V π,Φ ′ (x 0 , T )] + + E π,Φ * [V π,Φ (x 0 , T ) -V π,Φ ′ ( x 1 , t 1 ) + V π,Φ * (x 1 , t 1 )] = V π,Φ ′ (x 0 , T ) + E π,Φ * [V π,Φ (x 1 , t 1 ) -V π,Φ ′ ( x 1 , t 1 )]Next, we relate the regret at each episode to the model epistemic uncertainty using Lemma 3 and Lemma 7. Corollary 8. Let Assumption 1 -2 and Assumption 4 hold, then we have for all n ≥ 0 with probability at least 1δ.EQUATIONProof. From Lemma 4 we have:V πn,Φ * (x 0 , T ) -V π * ,Φ * (x 0 , T ) ≤ E   k≥0 V πn,Φn (x n,k+1 , t n,k+1 ) -V πn,Φn ( x n,k+1 , t n,k+1 )   .Lemma 6 can be applied to positive function g. We hence make a transformation and apply it to g(•) = V πn,Φn (•, t n,k+1 ) + C tmin T , which is positive due to Lemma 5. Moreover, ∀x ∈ X ;g(•) = V πn,Φn (•, t n,k+1 ) + C t min T ≤ Bt n,k+1 + C t min T ≤ T (B + C t min).Applying Lemma 6 we obtain:V πn,Φn (x n,k+1 , t n,k+1 ) -V πn,Φn ( x n,k+1 , t n,k+1 ) ≤ T σ B + C t min E [∥x n,k+1 -x n,k+1 ∥]Finally, applying Lemma 7 we arrive at:V πn,Φ * (x 0 , T ) -V π * ,Φ * (x 0 , T ) ≤ 2 √ d x β n-1 T σ B + C t min E   k≥0 ∥σ n-1 (x n,k , π n (x n,k , t n,k ))∥  Now we can prove our regret bound for the transition cost case. Theorem 9. Let Assumption 1 -2 and Assumption 4 hold, then we have for all n ≥ 0 with probability at least 1δ.R N = N n=1 V πn,Φ * (x 0 , T ) -V π * ,Φ * (x 0 , T ) ≤ 2 √ d x β N -1 T 3/2 σ 2 t min B + C t min N I NProof. We compute:R N = N n=1 V πn,Φ * (x 0 , T ) -V π * ,Φ * (x 0 , T ) ≤ 2 √ d x T σ B + C t min N n=1 β n-1 E   k≥0 ∥σ n-1 (x n,k , π n (x n,k , t n,k ))∥   ≤ 2 √ d x β N -1 T σ B + C t min E   N n=1 k≥0 ∥σ n-1 (x n,k , π n (x n,k , t n,k ))∥   ≤ 2 √ d x β N -1 T σ B + C t min T N t min E   N n=1 k≥0 ∥σ n-1 (x n,k , π n (x n,k , t n,k ))∥ 2   ≤ 2 √ d x β N -1 T 3/2 σ √ t min B + C t min N I NHere the first inequality follows because of Corollary 8, the second inequality follows due to the monotonicity of sequence (β n ) n≥0 , the third inequality follows by Cauchy-Schwarz and the last one by maximizing the term in expectation.Our regret R N is sublinear if β N -1 √ N I N is sublinear. For general well-calibrated models this is tough to verify. However, for Gaussian process dynamics, I N is equal to (up to constant factors) the maximum information gain γ N (Srinivas et al., 2009) (c.f., Curi et al. (2020, Lemma 17) ). The maximum information gain is sublinear for a rich class of kernels (Vakili et al., 2021) , i.e., yielding sublinear regret for OTACOS (see Sukhija et al. (2024, Theorem 2) for more detail).We overload the notation in this section and add number of switches to the value function, such that we have V πn,Φ * (x 0 , T, 0) = V πn,Φ * (x 0 , T ) Lemma 10 (Per episode regret bound). We have:V πn,Φ * (x 0 , T, 0) -V π * ,Φ * (x 0 , T, 0) ≤ ≤ E K-1 k=0 V πn,Φn (x n,k+1 , t n,k+1 , k + 1) -V πn,Φn ( x n,k+1 , t n,k+1 , k + 1) , where x n,k+1 is the state of one step hallucinated component s n,k+1 = Ψ Φn (s n,k , π n (s n,k ), w n,k ) and x n,k+1 is the state of s n,k+1 = Ψ Φ * (s n,k , π n (s n,k ), w n,k ). Proof. V πn,Φ * (x 0 , T, 0) = E   k≥0 r(s n,k , π n (s n,k ))   = E   r(s n,0 , π n (s n,0 )) + k≥1 r(s n,k , π n (s n,k ))   = E [r(s n,k , π n (s n,0 )) + V πn,Φ * (x n,1 , t n,1 , 1)] = E [r(s n,k , π n (s n,0 )) + V πn,Φn (x n,1 , t n,1 , 1) -V πn,Φn (x 0 , T, 0)] + + E [V πn,Φn (x 0 , T, 0) -V πn,Φn (x n,1 , t n,1 , 1) + V πn,Φ * (x n,1 , t n,1 , 1)] = V πn,Φn (x 0 , T, 0) + E [V πn,Φn ( x n,1 , t n,1 , 1) -V πn,Φn (x n,1 , t n,1 , 1)] + E [V πn,Φ * (x n,1 , t n,1 , 1) -V πn,Φn (x n,1 , t n,1 , 1)]Hence we have:V πn,Φ * (x 0 , T, 0) -V πn,Φn (x 0 , T, 0) = = E [V πn,Φn ( x n,1 , t n,1 , 1) -V πn,Φn (x n,1 , t n,1 , 1)] + E [V πn,Φ * (x n,1 , t n,1 , 1) -V πn,Φn (x n,1 , t n,1 , 1)]Repeating the step inductively the result follows and using V πn,Φ * (x n,K , t n,K , K) = 0 we prove the lemma.In principle, we could assume that the noise w k is Gaussian and then with the same analysis obtain the regret bound. However, stochastic flows are in many cases not exactly Gaussian but only sub-Gaussian. For such noise we need can not apply Lemma 6 and need to escort to different analysis. First we show that under mild assumptions on the SDE dynamics functions f * and g * the resulting noise w k is sub-Gaussian.To derive this result we will follow the work of Djellout et al. (2004) . We present the results in quite informal way, for more rigorous statements we refer the reader to Djellout et al. (2004) . Definition 4 (Wasserstein distance). Let (E, d E ) be a metric space and let µ, ν be two probability measures on E. We define:W p (µ, ν) = inf γ∈Γ(µ,ν) E (x,y)∼γ [d(x, y) p ] 1 pDefinition 5 (Kullback-Leibler divergence). Let (E, d E ) be a metric space and let µ, ν be two probability measures on E. We define:H(ν||µ) = E x∼ν log dν(x) dµ(x), if ν ≪ µDefinition 6 (L p -transportation cost information inequality). Let (E, d E ) be a metric space and let µ be a probability measure on E. We say that µ satisfy the L p -transportation cost information inequality, and for short write µ ∈ T p (C), if there exists a constant C such that for any measure ν on E we have:W p (µ, ν) ≤ 2CH(ν||µ).We now state an important theroem of Bobkov and Götze (1999) that we will use later.Theorem 11 (From Bobkov and Götze (1999) ). Let (E, d E ) be a metric space and let µ be a probability measure on E. We have that µ ∈ T 1 (C) if and only if for any µ-integrable and L F -Lipschitz function F : (E, d E ) → R and for any λ ∈ R we have:E x∼µ e λ(F (x)-Ex∼µ[F (x)]) ≤ e λ 2 2 CL 2 FNext, we provide a condition under which Ξ(x, u, t) is sub-Gaussian random variable for any t ∈ T .Corollary 12 (Adjusted Corollary 4.1 of Djellout et al. ( 2004)). Assumesup x∈R dx u∈R du ∥g * (x, u)∥ F ≤ A, ∥f * (x, u) -f * ( x, u)∥ ≤ L f * ∥(x, u) -( x, u)∥ ,and denote the law of (Ξ(x, u, t)) t∈T on the space C(T , R dx ) (space of continuous functions from T to R dx ) by P x . Then, there exist a constant C = C(A, L f * , T ) such that P x ∈ T 1 (C) on the space C(T , R dx ) equipped with the metric:d(γ 1 , γ 2 ) = sup t∈[0,T ] ∥γ 1 (t) -γ 2 (t)∥Lets e be a(ny) unit vector in R dx and define:F e,t : C(T , R dx ) → R F e,t : γ → γ(t) ⊤ eWe have:|F e,t (γ 1 ) -F e,t (γ 2 )| = (γ 1 (t) -γ 2 (t)) ⊤ e ≤ ∥γ 1 (t) -γ 2 (t)∥ ∥e∥ = ∥γ 1 (t) -γ 2 (t)∥ ≤ sup t∈T ∥γ 1 (t) -γ 2 (t)∥ = d(γ 1 , γ 2 )Therefore for any e, t the function F e,t is 1-Lipschitz. Since we haveE[|F e,t (γ)|] = C(T ,R dx ) |γ(t)| dP x (γ) = E[ Ξ(x, u, t) ⊤ e ] < ∞the function F e,t is also P x -integrable. Combining the latter observation with the Theorem 11 we obtain that for any e ∈ R dx and any t ∈ T we have:E Ξ(x,u,t) e λ(Ξ(x,u,t) ⊤ e-E[Ξ(x,u,t) ⊤ e]) = E γ∼Px e λ(Fe,t(γ)-E γ∼Px [Fe,t(γ)]) ≤ e λ 2 2 CHence under the assumption of Theorem 2 for Bounded number of switches setting we have that for any t ∈ T the random variable Ξ(x, u, t) -E [Ξ(x, u, t)] ∼ subG (C). The variance proxy C depends on A, L f * , T .To apply analysis for the case when noise w k is any sub-Gaussian we also need to show that the dynamics function Φ * is Lipschitz. We first start with some general results. Lemma 13. Let f : R n → R m , A ⊂ [n] and denote B = A C . If we have:• ∥f (x A , x B ) -f ( x A , x B )∥ 2 ≤ L A ∥x A -x A ∥ 2 , • ∥f (x A , x B ) -f (x A , x B )∥ 2 ≤ L B ∥x B -x B ∥ 2 , then f is 2(L A + L B ) Lipschitz.Proof. We have:∥f (x) -f ( x)∥ 2 = ∥f (x A , x B ) -f ( x A , x B )∥ 2 = ∥f (x A , x B ) -f ( x A , x B ) + f ( x A , x B ) -f ( x A , x B )∥ 2 ≤ L A ∥x A -x A ∥ 2 + L B ∥x B -x B ∥ 2 ≤ (L A + L B ) (∥x A -x A ∥ 2 + ∥x B -x B ∥ 2 ) ≤ 2(L A + L B ) x A -x A x B -x B 2 = 2(L A + L B ) ∥x -x∥ 2 Lemma 14 (Lipschitzness of Φ f * ). There exists a positive constant L Φ f such that the flow Φ f * is L Φ f -Lipschitz.Proof. We will first prove coordinate-wise Lipschitzness. We observe:1. Lipschitness in time:Φ f * (x, u, t) -Φ f * (x, u, t) = t 0 E[f * (x s , u)]ds - t 0 E[f * (x s , u)]ds ≤ t t E [∥f * (x s , u)∥] ds ≤ F t -t 2.Lipschitness in state x: To prove this, consider the δx t = Ξ(x, u, t) -Ξ( x, u, t), then we have dδx t = (f * (x t , u) -f * ( x t , u))dt + (g * (x t , u) -f * ( x t , u))dB t = δf * t dt + δg * t dB t . Note that ∥δf * t ∥ ≤ L f * ∥δx t ∥E[y t ] = t 0 2E[δx ⊤ s δf * s ] + E[tr(δg * s (δg * s ) ⊤ )]ds ≤ t 0 2E [∥δx s ∥ ∥δf * s ∥] + E[∥δg * s ∥ 2 ]ds ≤ t 0 (2L f * + L 2 g * )E[∥δx s ∥ 2 ]dsNote that y t = ∥δx t ∥ 2 , so we can apply Grönwall's inequality to getE ∥δx t ∥ 2 ≤ ∥δx 0 ∥ 2 e (2L f * +L 2 g * )t .Moreover,∥E[δx t ]∥ ≤ E ∥δx t ∥ 2 ≤ ∥δx 0 ∥ e 2L f * +L 2 g * 2 t ≤ ∥δx 0 ∥ e 2L f * +L 2 g * 2 T .Hence we have:∥Φ f * (x, u, t) -Φ f * ( x, u, t)∥ ≤ ∥x -x∥ e 2L f * +L 2 g * 2 T .3. Lipschitness in action u: We denote δx t = Ξ(x, u, t) -Ξ(x, u, t) and δu = uu Following the same steps as in the proof of Lipschitzness in state we arrive at:dy t = 2δx ⊤ t (δf * t dt + δg * t dB t ) + tr(δg * t (δg * t ) ⊤ )dt Integration yields: E[y t ] = t 0 2E[δx ⊤ s δf * s ] + E[tr(δg * s (δg * s ) ⊤ )]ds ≤ t 0 2E [∥δx s ∥ ∥δf * s ∥] + E[∥δg * s ∥ 2 ]ds ≤ t 0 2E [L f * ∥δx s ∥ (∥δx s ∥ + ∥δu∥)] + E 2L 2 g * ∥δx s ∥ 2 + ∥δu∥ 2 ds ≤ t 0 (3L f * + 2L 2 g * )E [y s ] + (L f * + 2L 2 g * ) ∥δu∥ ds,where we used (a + b) 2 ≤ 2a 2 + 2b 2 and ab ≤ 1 2 (a 2 + b 2 ). Applying Grönwall's inequality results in:E ∥δx t ∥ 2 ≤ ∥δu∥ 2 (L f * + 2L 2 g * )e (3L f * +2L 2 g * )t ≤ ∥δu∥ 2 (L f * + 2L 2 g * )e (3L f * +2L 2 g * )TApplying Lemma 13 on 2. and 3. we have thatΦ f * (•, •, t) is 2 e 2L f * +L 2 g * 2 T + L f * + 2L 2 g * e 3L f * +2L 2 g * 2 T -Lipschitz.Applying Lemma 13 on 1. and Φ f * (•, •, t) and bounding 2 ≤ 4 we finally obtain thatΦ f * is 4 e 2L f * +L 2 g * 2 T + L f * + 2L 2 g * e 3L f * +2L 2 g * 2 T + F -Lipschitz. Corollary 15 (Lipschitzness of the Φ b * ). The cost flow Φ b * is O e C1(L f * +L 2 g * )T -Lipschitz, where C 1 is a constant.Proof. Same as in the proof of Lemma 14 we first show coordinate-wise Lipschitzness.1. We first show Lipschitness in time:Φ b * (x, u, t) -Φ b * (x, u, t) = E t t b * (x s , u)ds ≤ E t t |b * (x s , u)| ds ≤ E B(t -t) = B(t -t).2. To obtain Lipschitzness in state observe:|Φ b * (x, u, t) -Φ b * ( x, u, t)| = E t 0 b * (Ξ(x, u, s), u) -b * (Ξ( x, u, s), u)ds ≤ E t 0 |b * (Ξ(x, u, s), u) -b * (Ξ( x, u, s), u)| ds ≤ L b * E t 0 ∥Ξ(x, u, s) -Ξ( x, u, s)∥ ds ≤ L b * t 0 E ∥Ξ(x, u, s) -Ξ( x, u, s)∥ 2 ds ≤ L b * ∥x -x∥ t 0 e 2L f * +L 2 g * 2 s ds = 2L b * 2L f * + L 2 g * e 2L f * +L 2 g * 2 t -1 ∥x -x∥ 3.Finally, for Lipschitzness in action observe:|Φ b * (x, u, t) -Φ b * (x, u, t)| = E t 0 b * (Ξ(x, u, s), u) -b * (Ξ(x, u, s), u)ds ≤ E t 0 |b * (Ξ(x, u, s), u) -b * (Ξ(x, u, s), u)| ds ≤ L b * E t 0 ∥Ξ(x, u, s) -Ξ(x, u, s)∥ + ∥u -u∥ ds ≤ L b * t ∥u -u∥ + L b * t 0 E ∥Ξ(x, u, s) -Ξ(x, u, s)∥ 2 ds ≤ L b * t ∥u -u∥ + L b * ∥u -u∥ L f * + 2L 2 g * t 0 e 3L f * +2L 2 g * 2 s ds = L b *   t + 2 L f * + 2L 2 g * 3L f * + 2L 2 g * e 3L f * +2L 2 g * 2 t -1   ∥u -u∥ ≤ L b *   T + 2 L f * + 2L 2 g * 3L f * + 2L 2 g * e 3L f * +2L 2 g * 2 T -1   ∥u -u∥Applying Lemma 13 result follows.Corollary 16 (Lipschitzness of Φ * ). The unknown functionΦ * is L Φ = L Φ f + L Φ b = O e D(L f * +L 2 g * )T -Lipschitz, where D is constant.Lemma 17 (Per episode regret bound (general sub-Gaussian noise)). Consider the setting with a bounded number of switches K, and let Assumption 1, 3, and Assumption 4 hold. Then, we get with probability at least 1δ:V πn,Φ * (x 0 , T, K) -V π * ,Φ * (x 0 , T, K) ≤ ≤ O L K-1 σ β K n-1 e C(L f * +L 2 g * )(1+Lπ)T K E K k=0 ∥σ n-1 (x n,k , π n (x n,k , t n,k , k))∥ 2Proof. Applying Lemma 5 of Curi et al. (2020) the result follows.Theorem 18. Consider the setting with a bounded number of switches K, and let Assumption 1, 3, and Assumption 4 hold. Then, we get with probability at least 1δ:R N = N n=1 V πn,Φ * (x 0 , T, K) -V π * ,Φ * (x 0 , T, K) ≤ O L K-1 σ β K N -1 √ Ke C(L f * +L 2 g * )(1+Lπ)T K N I NProof. We apply Lemma 17 and Cauchy-Schwarz:R N = N n=1 V πn,Φ * (x 0 , T, K) -V π * ,Φ * (x 0 , T, K) ≤ N n=1 O L K-1 σ β K n-1 e C(L f * +L 2 g * )(1+Lπ)T K E K k=0 ∥σ n-1 (x n,k , π n (x n,k , t n,k , k))∥ 2 ≤ O L K-1 σ β K N -1 e C(L f * +L 2 g * )(1+Lπ)T K E N n=1 K k=0 ∥σ n-1 (x n,k , π n (x n,k , t n,k , k))∥ 2 ≤ O L K-1 σ β K N -1 e C(L f * +L 2 g * )(1+Lπ)T K √ K N I NHere we first applied Lemma 17. Then we used the monotonicity of (β n ) n≥0 sequence. In the last step we first applied maximum over the collected data, then Cauchy-Schwarz inequality and finally the definition of model complexity.Figure 6 : Pendulum swing-down task. Row 1: 4 interactions, optimized interaction times, Row 2: 5 interactions, optimized interaction times, Row 3: 4 equidistant interactions, Row 4: 4 equidistant interactions.Figure 7 : When stochasticity of the environments increases, we need more interactions at the unstable equilibrium (Pendulum on top). The stochasticity scale goes from 0.1 to 0.5 to 1.0 from top to bottom row respectively.1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? paper's contributions and scope? Answer: [Yes] Justification: We highlight the problem setting, algorithm, and theoretical and empirical results in the abstract and introduction. Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.We assume that Π is the set of Lπ-Lipschitz policiesX ⊂ R 244 , U ⊂ R 17 . We provide our implementation at https://github.com/lasgroup/TaCoS.A standard RL algorithm would collect T /t min data points per episode.The code is available at https://github.com/lasgroup/model-based-rl. 


Decision: The reviewers found the work novel, well-written, and well-supported by experiments. They all had questions that were properly answered by the authors, thus, they all raised their scores during the rebuttals. It would be great if the authors revise the paper to include these answers that properly addressed the reviewers' questions.

Reviews: review: paper_summary: paper_summary: Reinforcement learning (RL) is effective for discrete-time Markov decision processes (MDPs), but many systems operate continuously in time, making discrete-time MDPs less accurate. In applications like greenhouse control or medical treatments, each interaction is costly due to manual intervention. To address this, we propose a time-adaptive approach, Time-adaptive Control & Sensing (TACOS), which optimizes policies that also predict the duration of their application. This results in an extended MDP solvable by standard RL algorithms. TACOS significantly reduces interactions while maintaining or improving performance and robustness. We also introduce OTACOS, a model-based algorithm with sublinear regret for smooth dynamic systems, offering further sample-efficiency gains.
strength_weakness: strength_weakness: My questions are listed as below
questions: questions: Thank you for the opportunity to review your paper. I have a few questions and observations outlined below.

1. Real-world applications and global frequencies.

- The discussion around the requirement for different global frequencies in real-world applications is intriguing (line 32), as it highlights the limitations of discrete-time sampling (lines 30-32). However, it seems this limitation is not fully addressed by continuous-time control (or the TACOS algorithm). The fundamental difference between discrete-time control and continuous-time control lies in the amount of environmental information included to compute the policy. For instance, within the total time interval $[0,T]$, discrete-time control computes the policy based on $K$ sampled data points {$t_1, t_2, \cdots, t_K$}, resulting in an optimal policy only for those specific sampled times. In contrast, continuous-time control computes an optimal policy over the entire continuous duration $[0,T]$ and then applies it at discrete times (constrained by Equations (2) and (3)). Utilizing the policy derived from a continuous-time formulation in a discrete-time setting would provide better generalization ability to obtain higher rewards for $t \notin$ {$ t_1,t_2,\cdots,t_K$} but ***does not necessarily address the limitation mentioned in lines 30-32.***

2. Definition of Notations:

- The paper lacks precise definitions of certain notations. For example, the policy $\pi: \mathcal{X} \to \mathcal{U}$ maps states to control inputs, but the exact meaning of $\pi_{\mathcal{T}}$ is unclear. In line 80, it is stated that "$\pi_{\mathcal{T}}$ is a policy that predicts the duration of applying the action." This suggests $\pi_{\mathcal{T}}$ is a prediction variable rather than a policy, as the term "policy" typically refers to decisions or actions taken. Additionally, while $t_{i} = \pi_{\mathcal{T}}(x_{t_{i-1}}) + t_{i-1}$ is mentioned, the role of $\pi_{\mathcal{T}}$ in the objective function needs further clarification.

Also have some minor issues.. 

- Line 65-66 and 68-69 discuss "real-time inference" or "adaptive control approach," which is a known challenge in real-world reinforcement learning. Including references to related literature would help readers better understand these contributions as follows. 

[1] Dulac-Arnold, G., Mankowitz, D., and Hester, T.Challenges of real-world reinforcement learning.arXiv preprint arXiv:1904.12901, 2019.

[2] Hyunin Lee, Ming Jin, Javad Lavaei, Somayeh Sojoudi, Pausing Policy Learning in Non-stationary Reinforcement Learning. ICML 2024 

[3] Al-Shedivat, M., Bansal, T., Burda, Y., Sutskever, I., Mordatch, I., and Abbeel, P.Continuous adaptation via meta-learning in nonstationary and competitive environments.In ICLR, 2018.

- Also please discuss what SDE stands for.
limitations: limitations: No potential negative societal impact.
paper_summary: paper_summary: This paper proposes a novel time-adaptive RL method framework (TaCoS) for continuous-time systems with continuous state and action spaces. The framework shows that the settings of interactions having costs or a budget of interactions, can be formulated as extended MDPs, that can be solved with standard RL algorithms. The paper theoretically demonstrates this, and empirically verifies this. The paper also empirically demonstrates that TaCoS works across a range of interaction frequencies, and proposes OTaCoS a model-based RL approach.
strength_weakness: strength_weakness: * L154: “Intuitively, the more stochastic the environment, the more interactions we would require to stabilize the system.” the argument should either have a reference or be empirically verified.
* Minor: Missing related work reference of (Nam et al. 2021).

References:

* Nam, HyunJi Alex, Scott Fleming, and Emma Brunskill. "Reinforcement learning with state observation costs in action-contingent noiselessly observable Markov decision processes." Advances in Neural Information Processing Systems 34 (2021): 15650-15666.
questions: questions: * Is the action applied from the policy kept constant until the next interaction? I presume the action is kept constant. Have you considered parameterizing the action and enabling non-constant actions or even continuous trajectories of actions between interaction times as a form of open-loop planning?
limitations: limitations: Yes, they are discussed in Section 5.
paper_summary: paper_summary: The paper introduces a framework for reinforcement learning named Time-adaptive Control & Sensing (TACOS). The TACOS framework reformulates the problem of continuous-time RL into an equivalent discrete-time Markov decision process (MDP) that standard RL algorithms can solve. Additionally, a model-based version of TACOS, named OTACOS, is proposed to reduce the sample complexity.
strength_weakness: strength_weakness: 1. Although the problem posed by the paper is novel and interesting, the solution simply involves incorporating time $t$ into the action space for learning, which a little lacks novelty.
2. The empirical validation is somewhat limited in diversity, primarily focusing on controlled synthetic environments. This limitation might affect the generalizability of the results to more complex or noisy real-world systems.
questions: questions: The model learned here differs from the typical models in model-based reinforcement learning, as it incorporates time as a transition variable, making the learning process more challenging. Could the authors demonstrate the learning performance of the model itself?
limitations: limitations: The paper does not show a discussion of limitations. Authors could enhance their discussion by explicitly addressing the potential limitations in real-world scenarios that involve complex, noisy, or non-stationary environments.
paper_summary: paper_summary: The paper focuses on continuous-time reinforcement learning. Typically the control and the measurements of continuous-time systems occur at discrete time points. These interventions generally come with a cost. The paper proposes a time-adaptive approach to reduce the number of these interventions in an optimal manner. Therefore they formulate the problem as an MDP, for which standard RL algorithms can be used. Additionally the authors propose a model-based algorithm for their setting.
strength_weakness: strength_weakness: 1. The references in the paper lack consistency in formatting. Conference names are sometimes written in full, sometimes abbreviated, and occasionally omitted altogether.

2. One major issue is the lack of details on the learning process. It would be interesting to compare the learning curves of the equidistant and the time-adaptive approach.

3. The experimental section leaves some questions open -> See Questions


Minor Weaknesses:
- Figure 3 comes before Figure 2
- Page 5, Line 134 says time discretization splits the whole horizon in T/K discrete time points. I think it should be K time points with length T/K
- It is difficult to discern the decline in interactions for the greenhouse task in Figure 3 first row, which coincides with the drop in episode reward.
questions: questions: 1. Regarding the experiments for Figure 2: After what time fraction are the greenhouse and pendulum swing-down usually in the stable equilibrium? As described, if the systems are in the stable equilibrium, the discretized approach continues with interactions, while the time-adaptive approach needs less. It would be interesting to compare the interactions for the time until the stable equilibrium is reached.

2. Figure 2, Pendulum Swing-up: Why does the equidistant approach perform so poorly? In Figure 1, the task could be solved with K=5 number of interactions, without high-frequency changes. Intuitively the equidistant approach should perform well with more than 10 interactions.

3. How many iterations were used for learning the equidistant policy and the SAC-TaCos policy in Figure 2? It appears that learning the equidistant policy should be easier, as it has smaller input and output spaces.

4. In the first row of Figure 3, the interaction cost increases, yet the reward does not decline significantly. It would be interesting to identify the point at which the reward starts to drop.
limitations: limitations: Limitations have been adressed adequately.

Comments: title: Official Comment
Official Comment
Response to reviewer BvQD
Acknowledging reviewer’s active engagement
Response to Reviewer XgGZ
Follow up on rebuttal
Follow up on rebuttal
Response to reviewer's comments
comment: Thanks for raising our score and the active engagement in the review process!
Thanks for raising our score and the active engagement in the review process!
Thank you for addressing my concerns and questions. I have raised my score.
Thank you for adressing my concerns. I appreciate the novelty of the work and therefore increase my score accordingly.
Thanks for your active engagement during the rebuttals, for acknowledging our contribution, and for increasing our score. 

We evaluated our algorithm on well-established deep RL benchmarks with varying complexity/dimensionality (c.f., Figure 4). Particularly, we also tested our algorithm on the humanoid environment from the OpenAI gym (244 dimensional state, 17 dimensional input) to demonstrate the scalability of TACOS.
In all the environments, we see a clear benefit of the TACOS framework.  We hope this also convinces the reviewer regarding the scalability of our approach.
Thank you for your replay! 

The significant improvements achieved by the algorithm in these environments is impressive, and I also recognize the novelty of the paper. I raised my score accordingly. The reason I questioned the complexity of the environment is because I think that while it is easy to understand improvements in some environments after adding the time adaptive control framework. Once the environment becomes more complex, the increased decision complexity could offset the gains in performance, or even lead to worse performance. This concern makes me worried about the scalability of the algorithm. That's why I didn't give a higher score.
Thanks for your active engagement in the rebuttal period and for increasing our score. We will revise the paper based on your feedback as suggested.
Thanks for the update. 

My questions are fully addressed -- **I have raised my score to 7 and stands on the accept side** -- contributions are reasonable. Thanks for the active engagement. **Please revise the paper based on four questions** if the paper gets accepted. Good luck!
Thanks again for your questions and active participation in the rebuttal period. 

The time $t_0$ is typically $0$ (initial time). Next, based on the state $s_0$ (which includes the time to-go), the policy outputs an action $a_0$ and how long it should be applied $\Delta t_0$. Therefore, $t_1 = t_0 + \Delta t_0$. Then at $t_1$ the state is measured again, the action $a_1$ and $\Delta t_1$ are returned by the policy and we get $t_2 = t_1 + \Delta t_1$. This process is then repeated at $t_2$ and so on.

How does the policy decide on $\Delta t$? In the setting with interaction costs (section 3.1), there is a cost every time the system changes its control/measures the system. This incentivizes the policy to have large values for $\Delta t$ (keep the same action for as long as possible -> change the input as few times as possible/minimize interactions). However, the policy has to also maximize the reward and thus it cannot pick arbitrarily high values for $\Delta t$. In a typical RL fashion, the policy learns to trade off these two objectives. 

Our formulation, TACOS, shows that such problems can be solved by any off-the-shelf RL algorithm (section 3.1 and section 3.2). Moreover, our contributions are as follows:

1) TACOS reformulates the time-adaptive RL setting (where policy decides when to interact with the system) to a standard discrete-time MDP for which any RL algorithm can be used (section 3.1 and section 3.2).

2) In our experiments, we show that sota model-free RL methods such as SAC and PPO can be applied off-the-shelf with the TACOS reformulation and perform much better than the standard discrete-time control setting, i.e., we show the benefits of time-adaptive RL and also that typical RL methods can be simply used for this setting.

3) We also show that for the time-adaptive setting, the regret bounds from mode-based RL for the discrete-time setting are also retained (section 5, theorem 2). 

In summary, our reformulation (TACOS) allows us to study time-adaptive RL problems with our standard and well-developed discrete-time RL toolset.  Further, the TACOS framework performs much better than discrete-time RL, while enjoying the same guarantees. We will further clarify this in the paper. Hope this addresses the concern of the reviewer. We'd appreciate the reviewer revising our score.
Dear Author,

Thank you for the clarification. Question 1 is almost fully addressed -- The reviewer just has one final question regarding how to determine the interaction time sequence {$t_1, t_2, \dots, t_K$} and the number of interactions $K$.

**I am inclined to support the acceptance of this paper. However, before I raise my score, I would like to kindly ask the author to clearly highlight the contributions of this work.** It appears that my Question 1 is more closely related to the advantages of continuous control over discrete-time control. It would be beneficial to understand the specific contributions of this work, whether it be in the formulation, experimental results, or another aspect -- please feel free to be creative in your explanation -- particularly in comparison with other continuous-time reinforcement learning algorithms.

Thank you.
Dear Reviewer,

We hope this message finds you well. We have noticed that our detailed rebuttal, addressing each of the concerns raised in your review, has not yet received any feedback. We understand the demanding nature of the review process and appreciate the time and effort invested in evaluating our work.

We kindly urge you to consider our responses to your questions, as we believe they adeptly address your concerns. With some days left, we hope we can still have a fruitful rebuttal period.

Thank you, 

Authors
Dear Reviewer,

We hope this message finds you well. We have noticed that our detailed rebuttal, addressing each of the concerns raised in your review, has not yet received any feedback. We understand the demanding nature of the review process and appreciate the time and effort invested in evaluating our work.

We kindly urge you to consider our responses to your questions, as we believe they adeptly address your concerns. With some days left, we hope we can still have a fruitful rebuttal period.

Thank you, 
Authors
Thanks for actively participating in the review process. We are glad that we could address most of your concerns. Below is our response to your question 1.

**Answer to Q1**: We would like to clarify that our framework, unlike standard RL, allows the policy to choose the interaction times $t_{1}, \dots, t_{k}$. Moreover, the algorithm works in the following way; 

At $t_2$, the policy outputs based on the state $s_{t_2}$ the action $a_{t_2}$ and how long it should be applied, i.e., $t_3 - t_2$.  In our experiments, this action is kept constant for the whole duration (zero order hold). Hence, the action remains the same between $[t_{2}, t_{3})$. Our framework allows the policy to be adaptive, i.e., the policy adaptively selects the frequency of control based on the state, rather than having a constant frequency (standard discrete-time control). 
In a typical RL fashion, the policy is optimized by directly interacting with the system and it tries to maximize the expected reward.  Therefore, it knows what frequency of control the system requires at the state $s_{t_2}$ from interactions. If the system is very noisy, intuitively you would expect the policy to exhibit more control. This is also observed in our experiments (c.f., Figure 3 in the main paper and Figure 6 in the PDF with additional experiments). 

Our method is better than standard discrete time control since our policy is adaptively selecting the times at which control is exhibited ($t_2$, $t_3$, etc). In discrete time this is specified by the problem designer ($t_i^{discrete} \neq t_i^{TACOS}$) which might be suboptimal. This is also what we show in our experiments (c.f., Figure 4 in the paper and Figure 7, 8 in the additional experiments).

Furthermore, while in all our experiments we focus on zero-order control, one could consider a different control strategy, e.g., linear interpolation w.r.t. time as suggested by the reviewer. We stuck with zero-order control cause of its simplicity and because it performed well for all experiments.

We hope this addresses your concerns and kindly ask you to consider revising our score. For any remaining questions, we are happy to provide further clarification.
Dear Authors,

Thank you for the update.

***Question 1:*** I appreciate the direction you're aiming for with your contribution. However, I still have some concerns that I'd like to clarify. To ensure I understand correctly: In continuous-time control, we have interaction times  $t \in $ {$ t_1, t_2, \ldots, t_K$}. This means that the action executed at time $t_2$ fully considers the duration between $t_2 $and $t_3$. However, if the system requires a different interaction frequency and needs to take action at any time $t^\prime \in [t_2, t_3]$, how does the continuous-time control compute the action at time $t^\prime$?

- If the action is indeed a function of time $t$, it makes sense that the action at time $t^\prime$ is computable. However, in this case, how is this method superior to simply interpolating between the actions executed at $t_2$ and $t_3$ in "discrete time sampling"?

***Question 2:*** Thank you. This is fully addressed. Please use that notation in the revised version if accepted.

***Question 3:*** Thank you. This is fully addressed. Please add the reference in the revised version if accepted.

***Question 4:*** Thank you. This is also fully addressed.