1 Introduction
Reinforcement learning algorithms generally fall into one of two categories: modelbased approaches, which build a predictive model of an environment and derive a controller from it, and modelfree techniques, which learn a direct mapping from states to actions. Modelfree methods have shown promise as a generalpurpose tool for learning complex policies from raw state inputs (Mnih et al., 2015; Lillicrap et al., 2016; Haarnoja et al., 2018), but their generality comes at the cost of efficiency. When dealing with realworld physical systems, for which data collection can be an arduous process, modelbased approaches are appealing due to their comparatively fast learning. However, model accuracy acts as a bottleneck to policy quality, often causing modelbased approaches to perform worse asymptotically than their modelfree counterparts.
In this paper, we study how to most effectively use a predictive model for policy optimization. We first formulate and analyze a class of modelbased reinforcement learning algorithms with improvement guarantees. Although there has been recent interest in monotonic improvement of modelbased reinforcement learning algorithms (Sun et al., 2018; Luo et al., 2019), most commonly used modelbased approaches lack the improvement guarantees that underpin many modelfree methods (Schulman et al., 2015). While it is possible to apply analogous techniques to the study of modelbased methods to achieve similar guarantees, it is more difficult to use such analysis to justify model usage in the first place due to pessimistic bounds on model error. However, we show that more realistic model error rates derived empirically allow us to modify this analysis to provide a more reasonable tradeoff on model usage.
Our main contribution is a practical algorithm built on these insights, which we call modelbased policy optimization (MBPO), that makes limited use of a predictive model to achieve pronounced improvements in performance compared to other modelbased approaches. More specifically, we disentangle the task horizon and model horizon by querying the model only for short rollouts. We empirically demonstrate that a large amount of these short modelgenerated rollouts can allow a policy optimization algorithm to learn substantially faster than recent modelbased alternatives while retaining the asymptotic performance of the most competitive modelfree algorithms. We also show that MBPO does not suffer from the same pitfalls as prior modelbased approaches, avoiding model exploitation and failure on longhorizon tasks. Finally, we empirically investigate different strategies for model usage, supporting the conclusion that careful use of short modelbased rollouts provides the most benefit to a reinforcement learning algorithm.
2 Related Work
Modelbased reinforcement learning methods are promising candidates for realworld sequential decisionmaking problems due to their data efficiency (Kaelbling et al., 1996). For example, Gaussian processes and timevarying linear dynamical systems provide excellent performance in the lowdata regime (Deisenroth & Rasmussen, 2011; Levine & Koltun, 2013; Kumar et al., 2016)
. Neural network predictive models, both deterministic
(Draeger et al., 1995; Nagabandi et al., 2018) and probabilistic (Gal et al., 2016; Depeweg et al., 2016), are appealing because they may allow for algorithms that combine the sample efficiency of a modelbased approach with the asymptotic performance of highcapacity function approximators, even in domains with complex observations such as images (Oh et al., 2015; Ebert et al., 2018; Kaiser et al., 2019). Our work uses an ensemble of probabilistic networks, as in Chua et al. (2018), although our model is employed to learn a policy rather than in the context of a recedinghorizon planning routine.Learned models may be incorporated into otherwise modelfree methods for improvements in data efficiency. For example, a modelfree policy can be used as an action proposal distribution within a modelbased planner (Piché et al., 2019). Conversely, model rollouts may be used to provide extra training examples for a Qfunction (Sutton, 1990), to improve the target value estimates of existing data points (Feinberg et al., 2018), or to provide additional context to a policy (Du & Narasimhan, 2019). However, the performance of such approaches rapidly degrades with increasing model error (Gu et al., 2016)
, motivating work that interpolates between different rollout lengths
(Buckman et al., 2018), tunes the ratio of real to modelgenerated data (Kalweit & Boedecker, 2017), or does not rely on model predictions (Heess et al., 2015). Our approach similarly tunes model usage during policy optimization, but we show that justifying nonnegligible model usage during most points in training requires consideration of the model’s ability to generalize outside of its training distribution.Prior methods have also explored incorporating computation that resembles modelbased planning but without constraining the intermediate predictions of the planner to match plausible environment observations (Tamar et al., 2016; Racanière et al., 2017; Oh et al., 2017; Silver et al., 2017). While such methods can reach asymptotic performance on par with modelfree approaches, they may not benefit from the sample efficiency of modelbased methods as they forgo the extra supervision used in standard modelbased methods.
The bottleneck in scaling modelbased approaches to complex tasks often lies in learning accurate predictive models of highdimensional dynamics (Atkeson & Schaal, 1997). Model ensembles have shown to be effective in preventing a policy or planning procedure from exploiting the inaccuracies of any single model (Rajeswaran et al., 2017; Kurutach et al., 2018; Clavera et al., 2018; Chua et al., 2018). Alternatively, a model may also be trained on its own outputs to avoid compounding error from multistep predictions (Talvitie, 2014, 2016) or predict many timesteps into the future (Whitney & Fergus, 2018). We empirically demonstrate that a combination of model ensembles with short model rollouts is sufficient to prevent exploitation of model inaccuracies.
Theoretical analysis of modelbased reinforcement learning algorithms has been considered by Sun et al. (2018) and Luo et al. (2019), who bound the discrepancy between returns under a model and those in the real environment of interest. Their approaches enforce a trust region around a reference policy, whereas we do not constrain the policy but instead consider rollout length based on estimated model generalization capacity. Alternate analyses have been carried out by incorporating the structure of the value function into the model learning (Farahmand et al., 2017) or by regularizing the model by controlling its Lipschitz constant (Asadi et al., 2018). Prior work has also constructed complexity bounds for modelbased approaches in the tabular setting (Szita & Szepesvari, 2010) and for the linear quadratic regulator (Dean et al., 2017), whereas we consider general nonlinear systems.
3 Background
We consider a Markov decision process (MDP), defined by the tuple
. and are the state and action spaces, respectively, and is the discount factor. The dynamics or transition distribution are denoted as , the initial state distribution as , and the reward function as . The goal of reinforcement learning is to find the optimal policy that maximizes the expected sum of discounted rewards, denoted by :The dynamics are assumed to be unknown. Modelbased reinforcement learning methods aim to construct a model of the transition distribution,
, using data collected from interaction with the MDP, typically using supervised learning. We additionally assume that the reward function has unknown form, and predict
as a learned function of and .4 Monotonic Improvement with Model Bias
In this section, we first lay out a general recipe for MBPO with monotonic improvement. This general recipe resembles or subsumes several prior algorithms and provides us with a concrete framework that is amenable to theoretical analysis. Described generically in Algorithm 1, MBPO optimizes a policy under a learned model, collects data under the updated policy, and uses that data to train a new model. While conceptually simple, the performance of MBPO can be difficult to understand; errors in the model can be exploited during policy optimization, resulting in large discrepancies between the predicted returns of the policy under the model and under the true dynamics.
4.1 Monotonic Modelbased Improvement
Our goal is to outline a principled framework in which we can provide performance guarantees for modelbased algorithms. To show monotonic improvement for a modelbased method, we wish to construct a bound of the following form:
denotes the returns of the policy in the true MDP, whereas denotes the returns of the policy under our model. Such a statement guarantees that, as long as we improve by at least under the model, we can guarantee improvement on the true MDP.
The gap between true returns and model returns,
, can be expressed in terms of two error quantities of the model: generalization error due to sampling, and distribution shift due to the updated policy encountering states not seen during model training. As the model is trained with supervised learning, sample error can be quantified by standard PAC generalization bounds, which bound the difference in expected loss and empirical loss by a constant with high probability
(ShalevShwartz & BenDavid, 2014). We denote this generalization error by , which can be estimated in practice by measuring the validation loss of the model. For our analysis, we denote distribution shift by the maximum totalvariation distance, , of the policy between iterations. In practice, we measure the KL divergence between policies, which we can relate to by Pinsker’s inequality. With these two sources of error controlled (generalization by , and distribution shift by ), we now present our bound:Theorem 4.1.
Let the expected TVdistance between two transition distributions be bounded at each timestep by and the policy divergence be bounded by . Then the true returns and model returns of the policy are bounded as:
Proof.
See Appendix A, Theorem A.1. ∎
This bound implies that as long as we improve the returns under the model by more than , we can guarantee improvement under the true returns.
4.2 Interpolating ModelBased and ModelFree Updates
Theorem 4.1 provides a useful relationship between model returns and true returns. However, it contains several issues regarding cases when the model error is high. First, there may not exist a policy such that , in which case improvement is not guaranteed. Second, the analysis relies on running full rollouts through the model, allowing model errors to compound. This is reflected in the bound by a factor scaling quadratically with the effective horizon, . In such cases, we can improve the algorithm by choosing to rely less on the model and instead more on real data collected from the true dynamics when the model is inaccurate.
In order to allow for dynamic adjustment between modelbased and modelfree rollouts, we introduce the notion of a branched rollout, in which we begin a rollout from a state under the previous policy’s state distribution and run steps according to under the learned model . This branched rollout structure resembles the scheme proposed in the original Dyna algorithm (Sutton, 1990), which can be viewed as a special case of a length 1 branched rollouts. Formally, we can view this as executing a nonstationary policy which begins a rollout by sampling actions from the previous policy . Then, at some specified time, we switch to unrolling the trajectory under the model and current policy for steps. Under such a scheme, the returns can be bounded as follows:
Theorem 4.2.
Given returns from the branched rollout method,
(1) 
Proof.
See Appendix A, Theorem A.3. ∎
4.3 Model Generalization in Practice
Theorem 4.2 would be most useful for guiding algorithm design if it could be used to determine an optimal model rollout length . While this bound does include two competing factors, one exponentially decreasing in and another scaling linearly with , the values of the associated constants prevent an actual tradeoff; taken literally, this lower bound is maximized when , corresponding to not using the model at all. One limitation of the analysis is pessimistic scaling of model error with respect to policy shift , as we do not make any assumptions about the generalization capacity or smoothness properties of the model (Asadi et al., 2018).
To better determine how well we can expect our model to generalize in practice, we empirically measure how the model error under new policies increases with policy change . We train a model on the state distribution of a datacollecting policy and then continue policy optimization while measuring the model’s loss on all intermediate policies during this optimization. Figure 1a shows that, as expected, the model error increases with the divergence between the current policy and the datacollecting policy . However, the rate of this increase depends on the amount of data collected by . We plot the local change in model error over policy change, , in Figure 1b. The decreasing dependence on policy shift shows that not only do models trained with more data perform better on their training distribution, but they also generalize better to nearby distributions.
The clear trend in model error growth rate suggests a way to modify the pessimistic bounds. In the previous analysis, we assumed access to only model error on the distribution of the most recent datacollecting policy and approximated the error on the current distribution as . If we can instead approximate the model error on the distribution of the current policy , which we denote as , we may use this directly. For example, approximating with a linear function of the policy divergence yields:
where is empirically estimated as in Figure 1. Equipped with an approximation of , the model’s error on the distribution of the current policy , we arrive at the following bound:
Theorem 4.3.
Under the branched rollout method, using model error under the updated policy , we have
(2) 
Proof.
See Appendix A, Theorem A.2. ∎
While this bound appears similar to Theorem 4.2, the important difference is that this version actually motivates model usage. More specifically, for sufficiently low . While this insight does not immediately suggest an algorithm design by itself, we can build on this idea to develop a method that makes limited use of truncated, but nonzerolength, model rollouts.
5 ModelBased Policy Optimization with Deep Reinforcement Learning
We now present a practical modelbased reinforcement learning algorithm based on the derivation in the previous section. Instantiating Algorithm 1 amounts to specifying three design decisions: (1) the parametrization of the model , (2) how the policy is optimized given model samples, and (3) how to query the model for samples for policy optimization.
Predictive model.
In our work, we use a bootstrap ensemble of dynamics models
. Each member of the ensemble is a probabilistic neural network whose outputs parametrize a Gaussian distribution with diagonal covariance:
. Individual probabilistic models capture aleatoric uncertainty, or the noise in the outputs with respect to the inputs. The bootstrapping procedure accounts for epistemic uncertainty, or uncertainty in the model parameters, which is crucial in regions when data is scarce and the model can by exploited by policy optimization. Chua et al. (2018) demonstrate that a proper handling of both of these uncertainties allows for asymptotically competitive modelbased learning. To generate a prediction from the ensemble, we simply select a model uniformly at random, allowing for different transitions along a single model rollout to be sampled from different dynamics models.Policy optimization.
We adopt softactor critic (SAC) (Haarnoja et al., 2018) as our policy optimization algorithm. SAC alternates between a policy evaluation step, which estimates using the Bellman backup operator, and a policy improvement step, which trains an actor by minimizing the expected KLdivergence .
Model usage.
Many recent modelbased algorithms have focused on the setting in which model rollouts begin from the initial state distribution (Kurutach et al., 2018; Clavera et al., 2018). While this may be a more faithful interpretation of Algorithm 1, as it is optimizing a policy purely under the state distribution of the model, this approach entangles the model rollout length with the task horizon. Because compounding model errors make extended rollouts difficult, these works evaluate on truncated versions of benchmarks. The branching strategy described in Section 4.2, in which model rollouts begin from the state distribution of a different policy under the true environment dynamics, effectively relieves this limitation. In practice, branching replaces few long rollouts from the initial state distribution with many short rollouts starting from replay buffer states.
A practical implementation of MBPO is described in Algorithm 2.^{1}^{1}1When SAC is used as the policy optimization algorithm, we must also perform gradient updates on the parameters of the functions, but we omit these updates for clarity. The primary differences from the general formulation in Algorithm 1 are length rollouts from replay buffer states in the place of optimization under the model’s state distribution and a fixed number of policy update steps in the place of an intractable argmax. Even when the horizon length
is short, we can perform many such short rollouts to yield a large set of model samples for policy optimization. This large set allows us to take many more policy gradient steps per environment sample (between 20 and 40) than is typically stable in modelfree algorithms. A full listing of the hyperparameters included in
Algorithm 2 for all evaluation environments is given in Appendix C.6 Experiments
Our experimental evaluation aims to study two primary questions: (1) How well does MBPO perform on benchmark reinforcement learning tasks, compared to stateoftheart modelbased and modelfree algorithms? (2) What conclusions can we draw about appropriate model usage?
6.1 Comparative Evaluation
In our comparisons, we aim to understand both how well our method compares to stateoftheart modelbased and modelfree methods and how our design choices affect performance. We compare to two stateoftheart modelfree methods, SAC (Haarnoja et al., 2018) and PPO (Schulman et al., 2017), both to establish a baseline and, in the case of SAC, measure the benefit of incorporating a model, as our modelbased method uses SAC for policy learning as well. For modelbased methods, we compare to PETS (Chua et al., 2018), which does not perform explicit policy learning, but directly uses the model for planning; STEVE (Buckman et al., 2018), which also uses shorthorizon modelbased rollouts, but incorporates data from these rollouts into value estimation rather than policy learning; and SLBO (Luo et al., 2019), a modelbased algorithm with performance guarantees that performs model rollouts from the initial state distribution. These comparisons represent the stateoftheart in both modelfree and modelbased reinforcement learning.
We evaluate MBPO and these baselines on a set of MuJoCo continuous control tasks (Todorov et al., 2012) commonly used to evaluate modelfree algorithms. Note that some recent works in modelbased reinforcement learning have used modified versions of these benchmarks, where the task horizon is chosen to be shorter so as to simplify the modeling problem (Kurutach et al., 2018; Clavera et al., 2018). We use the standard fulllength version of these tasks. MBPO also does not assume access to privileged information in the form of fully observable states or the reward function for offline evaluation.
Figure 2 shows the learning curves for all methods, along with asymptotic performance of algorithms which do not converge in the region shown. These results show that MBPO learns substantially faster, an order of magnitude faster on some tasks, than prior modelfree methods, while attaining comparable final performance. For example, MBPO’s performance on the Ant task at 300 thousand steps is the same as that of SAC at 3 million steps. On Hopper and Walker2d, MBPO requires the equivalent of 14 and 40 minutes, respectively, of simulation time if the simulator were running in real time. More crucially, MBPO learns on some of the higherdimensional tasks, such as Ant, which pose problems for purely modelbased approaches such as PETS (Chua et al., 2018).
6.2 Design Evaluation
We next make ablations and modifications to our method to better understand why MBPO outperforms prior approaches. Results for the following experiments are shown in Figure 3.
No model.
The ratio between the number of gradient updates and environment samples, , is much higher in MBPO than in comparable modelfree algorithms because the modelgenerated data reduces the risk of overfitting. We run standard SAC with similarly high ratios, but without model data, to ensure that the model is actually helpful. While increasing the number of gradient updates per sample taken in SAC does marginally speed up learning, we cannot match the sampleefficiency of our method without using the model. For hyperparameter settings of MBPO, see Appendix C.
Rollout horizon.
While the bestperforming rollout length schedule on the Hopper task linearly increases from to (Appendix C), we find that fixing the rollout length at 1 for the duration of training retains much of the benefit of our modelbased method. We also find that our model is accurate enough for 200step rollouts, although this performs worse than shorter values when used for policy optimization. 500step rollouts are too inaccurate for effective learning. While more precise finetuning is always possible, augmenting policy training data with singlestep model rollouts provides a baseline that is surprisingly difficult to beat and outperforms recent methods which perform longer rollouts from the initial state distribution. This result agrees with our theoretical analysis which prescribes short modelbased rollouts to mitigate compounding modeling errors.
Value expansion.
An alternative to using model rollouts for direct training of a policy is to improve the quality of target values of samples collected from the real environment. This technique is used in modelbased value expansion (MVE) (Feinberg et al., 2018) and STEVE (Buckman et al., 2018). While MBPO outperforms both of these approaches, there are other confounding factors making a headtohead comparison difficult, such as the choice of policy learning algorithm. To better determine the relationship between training on modelgenerated data and using model predictions to improve target values, we augment SAC with the step target objective:
in which and are model predictions and . We refer the reader to Feinberg et al. (2018) for further discussion of this approach. We verify that SAC also benefits from improved target values, and similar to our conclusions from MBPO, singlestep model rollouts () provide a surprisingly effective baseline. While modelgenerated data augmentation and value expansion are in principle complementary approaches, preliminary experiments did not show improvements to MBPO by using improved target value estimates.
Model exploitation.
We analyze the problem of “model exploitation,” which a number of recent works have raised as a primary challenge in modelbased reinforcement learning (Rajeswaran et al., 2017; Clavera et al., 2018; Kurutach et al., 2018). We plot empirical returns of a trained policy on the Hopper task under both the real environment and the model in Figure 4 (c) and find, surprisingly, that they are highly correlated, indicating that a policy trained entirely on modelpredicted transitions may not exploit the model at all if the rollouts are sufficiently short. This is likely because short rollouts are more likely to reflect the real dynamics (Figure 4 ab), reducing the opportunities for policies to rely on inaccuracies of model predictions. While the models for other environments are not necessarily as accurate as that for Hopper, we find across the board that model returns tend to underestimate real environment returns in MBPO.
7 Discussion
We have investigated the role of model usage in policy optimization procedures through both a theoretical and empirical lens. We have shown that, while it is possible to formulate modelbased reinforcement learning algorithms with monotonic improvement guarantees, such an analysis cannot necessarily be used to motivate using a model in the first place. However, an empirical study of model generalization shows that predictive models can indeed perform well outside of their training distribution. Incorporating a linear approximation of model generalization into the analysis gives rise to a more reasonable tradeoff that does in fact justify using the model for truncated rollouts. The algorithm stemming from this insight, MBPO, has asymptotic performance rivaling the best modelfree algorithms, learns substantially faster than prior modelfree or modelbased methods, and scales to long horizon tasks that often cause modelbased methods to fail. We experimentally investigate the tradeoffs associated with our design decisions, and find that model rollouts as short as a single step can provide pronounced benefits to policy optimization procedures.
Acknowledgements
We thank Anusha Nagabandi, Michael Chang, Chelsea Finn, Pulkit Agrawal, and Jacob Steinhardt for insightful discussions; Vitchyr Pong, Alex Lee, Kyle Hsu, and Aviral Kumar for feedback on an early draft of the paper; and Kristian Hartikainen for help with the SAC baseline. This research was partly supported by the NSF via IIS1651843, IIS1700697, and IIS1700696, the Office of Naval Research, ARL DCIST CRA W911NF1720181, and computational resource donations from Google. M.J. is supported by fellowships from the National Science Foundation and the Open Philanthropy Project. M.Z. is supported by an NDSEG fellowship.
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Appendix A Modelbased Policy Optimization with Performance Guarantees
In this appendix, we provide proofs for bounds presented in the main paper.
We begin with a standard bound on modelbased policy optimization, with bounded policy change and model error .
Theorem A.1 (MBPO performance bound).
Let the expected total variation between two transition distributions is bounded at each timestep by , and the policy divergences are bounded as . Then the returns are bounded as:
Proof.
Let denote the data collecting policy. Asis we can use Lemma B.3 to bound the returns, but it will require bounded model error under the new policy . Thus, we need to introduce by adding and subtracting , to get:
We can bound and both using Lemma B.3.
For , we apply Lemma B.3 using (no model error because both terms are under the true model), and obtain:
For , we apply Lemma B.3 using and obtain:
Adding these two bounds together yields the desired result. ∎
Next, we describe bounds for branched rollouts. We define a branched rollout as a rollout which begins under some policy and dynamics (either true or learned), and at some point in time switches to rolling out under a new policy and dynamics for steps. The point at which the branch is selected is weighted exponentially in time – that is, the probability of a branch point being selected is proportional to .
We first present the simpler bound where the model error is bounded under the new policy, which we label as . This bound is difficult to apply in practice as supervised learning will typically control model error under the dataset collected by the previous policy.
Theorem A.2.
Let the expected total variation between two the learned model is bounded at each timestep under the expectation of by , and the policy divergences are bounded as . Then under a branched rollouts scheme with a branch length of , the returns are bounded as:
Proof.
As in the proof for Theorem A.1, the proof for this theorem requires adding and subtracting the correct reference quantity and applying the corresponding returns bound (Lemma B.4).
The choice of reference quantity is a branched rollout which executes the old policy under the true dynamics until the branch point, then executes the new policy under the true dynamics for steps. We denote the returns under this scheme as . We can split the returns as follows:
We can bound both terms and using Lemma B.4.
accounts for the error from executing the old policy instead of the current policy. This term only suffers from error before the branch begins, and we can use Lemma B.4 and all other errors set to 0. This implies:
incorporates model error under the new policy incurred after the branch. Again we use Lemma B.4, setting and all other errors set to 0. This implies:
Adding and together completes the proof.
∎
The next bound is an analogue of Theorem A.2 except using model errors under the previous policy rather than the new policy .
Theorem A.3.
Let the expected total variation between two the learned model is bounded at each timestep under the expectation of by , and the policy divergences are bounded as . Then under a branched rollouts scheme with a branch length of , the returns are bounded as:
Proof.
This proof is a short extension of the proof for Theorem A.2. The only modification is that we need to bound in terms of the model error under the rather than .
Once again, we design a new reference rollout. We use a rollout that executes the old policy under the true dynamics until the branch point, then executes the old policy under the model for steps. We denote the returns under this scheme as . We can split as follows:
Once again, we bound both terms and using Lemma B.4.
The rollouts in differ in both model and policy after the branch. This can be bound using Lemma B.4 by setting and . This results in:
The rollouts in differ only in the policy after the branch (as they both rollout under the model). This can be bound using Lemma B.4 by setting and . This results in:
Adding from Theorem A.2 and , above completes the proof.
∎
Appendix B Useful Lemmas
In this section, we provide proofs for various lemmas used in our bounds.
Lemma B.1 (TVD of Joint Distributions).
Suppose we have two distributions and . We can bound the total variation distance of the joint as:
Alternatively, we have a tighter bound in terms of the expected TVD of the conditional:
Proof.
∎
Lemma B.2 (Markov chain TVD bound, timevarying).
Suppose the expected KLdivergence between two transition distributions is bounded as , and the initial state distributions are the same – . Then the distance in the state marginal is bounded as:
Proof.
We begin by bounding the TVD in statevisitation at time , which is denoted as .
Where we have defined , which we assume is upper bounded by . Assuming we are not modeling the initial state distribution, we can set . ∎
Lemma B.3 (Branched Returns bound).
Suppose the expected KLdivergence between two dynamics distributions is bounded as , and . Then the returns are bounded as:
Proof.
Lemma B.4 (Returns bound, branched rollout).
Assume we run a branched rollout of length . Before the branch (“pre” branch), we assume that the dynamics distributions are bounded as and after the branch as . Likewise, the policy divergence is bounded pre and post branch by and , repsectively. Then the Kstep returns are bounded as:
Proof.
We begin by bounding state marginals at each timestep, similar to Lemma B.3. Recall that Lemma B.2 implies that state marginal error at each timestep can be bounded by the state marginal error at the previous timestep, plus the divergence at the current timestep.
Thus, letting and denote the stateaction marginals, we can write:
For :
and for :