scope_rl.ope.continuous.marginal_estimators#

State(-Action) Marginal Off-Policy Estimators for continuous action cases (designed for deterministic evaluation policies).

Classes

`DoubleReinforcementLearning`	Double Reinforcement Learning (DRL) estimator for continuous action space.
`StateActionMarginalDR`	State-Action Marginal Doubly Robust (SAM-DR) for continuous action spaces.
`StateActionMarginalIS`	State-Action Marginal Importance Sampling (SAM-IS) for continuous action spaces.
`StateActionMarginalSNDR`	State-Action Marginal Self-Normalized Doubly Robust (SAM-SNDR) for continuous action spaces.
`StateActionMarginalSNIS`	State-Action Marginal Self-Normalized Importance Sampling (SAM-SNIS) for continuous action spaces.
`StateMarginalDM`	Direct Method (DM) for continuous-action and stationary OPE (designed for deterministic evaluation policies).
`StateMarginalDR`	State Marginal Doubly Robust (SM-DR) for continuous action spaces.
`StateMarginalIS`	State Marginal Importance Sampling (SM-IS) for continuous action spaces.
`StateMarginalSNDR`	State Marginal Self-Normalized Doubly Robust (SM-SNDR) for continuous action spaces.
`StateMarginalSNIS`	State Marginal Self-Normalized Importance Sampling (SM-SNIS) for continuous action spaces.

class scope_rl.ope.continuous.marginal_estimators.DoubleReinforcementLearning(estimator_name='drl')[source]#

Double Reinforcement Learning (DRL) estimator for continuous action space.

Bases: scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.DoubleReinforcementLearning

Note

DRL estimates the policy value using state-action marginal importance weight and Q-function estimated by cross-fitting.

\[\hat{J}_{\mathrm{DRL}} (\pi; \mathcal{D}) := \frac{1}{n} \sum_{k=1}^K \sum_{i=1}^{n_k} \sum_{t=0}^{T-1} ( \rho^j(s_{t}^{(i)}, a_{t}^{(i)}) (r_{t}^{(i)} - Q^j(s_{t}^{(i)}, a_{t}^{(i)})) + \rho^j(s_{t-1}^{(i)}, a_{t-1}^{(i)}) Q^j(s_{t}^{(i)}, \pi(s_{t}^{(i)})) )\]

where \(\rho(s, a) \approx d^{\pi}(s, a) / d^{\pi_b}(s, a)\) is the state-action marginal importance weight, where \(d^{\pi}(s)\) is the marginal visitation probability of the policy \(\pi\) on \((s, a)\). \(Q(s, a)\) is the Q-function. \(K\) is the number of folds and \(\mathcal{D}_j\) is the \(j\)-th split of logged data consisting of \(n_k\) samples. \(\rho^j\) and \(Q^j\) are estimated on the subset of data used for OPE, i.e., \(\mathcal{D} \setminus \mathcal{D}_j\).

DRL achieves the semiparametric efficiency bound with a consistent value predictor.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="drl") – Name of the estimator.

References

Nathan Kallus and Masatoshi Uehara. “Double Reinforcement Learning for Efficient Off-Policy Evaluation in Markov Decision Processes.” 2020.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Methods

`estimate_interval`(step_per_trajectory, ...)	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(step_per_trajectory, ...)	Estimate the policy value of the evaluation policy.

estimate_policy_value(step_per_trajectory, reward, state_action_marginal_importance_weight, state_action_value_prediction, **kwargs)[source]#

Estimate the policy value of the evaluation policy.

Parameters:

step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_action_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state-action marginal distribution, i.e., \(d^{\pi}(s, a) / d^{\pi_b}(s, a)\)
state_action_value_prediction (array-like of shape (n_trajectories * step_per_trajectory, 2)) – \(\hat{Q}\) for the observed action and that chosen by the evaluation policy, i.e., (row 0) \(\hat{Q}(s_t, a_t)\) and (row 2) \(\hat{Q}(s_t, \pi(a | s_t))\).

Returns:

V_hat – Estimated policy value.

Return type:

float

estimate_interval(step_per_trajectory, reward, state_action_marginal_importance_weight, state_action_value_prediction, alpha=0.05, ci='bootstrap', n_bootstrap_samples=10000, random_state=None, **kwargs)[source]#

Estimate the confidence interval of the policy value by nonparametric bootstrap.

Parameters:

step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_action_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state-action marginal distribution, i.e., \(d^{\pi}(s, a) / d^{\pi_b}(s, a)\)
state_action_value_prediction (array-like of shape (n_trajectories * step_per_trajectory, 2)) – \(\hat{Q}\) for the observed action and that chosen by the evaluation policy, i.e., (row 0) \(\hat{Q}(s_t, a_t)\) and (row 2) \(\hat{Q}(s_t, \pi(a | s_t))\).
alpha (float, default=0.05) – Significance level. The value should be within [0, 1).
ci ({"bootstrap", "hoeffding", "bernstein", "ttest"}, default="bootstrap") – Method to estimate the confidence interval.
n_bootstrap_samples (int, default=10000 (> 0)) – Number of resampling performed in the bootstrap procedure.
random_state (int, default=None (>= 0)) – Random state.

Returns:

estimated_confidence_interval – Dictionary storing the estimated mean and upper-lower confidence bounds.

key: [
    mean,
    {100 * (1. - alpha)}% CI (lower),
    {100 * (1. - alpha)}% CI (upper),
]

Return type:

dict

class scope_rl.ope.continuous.marginal_estimators.StateMarginalDM[source]#

Direct Method (DM) for continuous-action and stationary OPE (designed for deterministic evaluation policies).

Bases: scope_rl.ope.BaseStateMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateMarginalDM

Note

DM estimates the policy value using an estimated initial state value as follows.

\[\hat{J}_{\mathrm{DM}} (\pi; \mathcal{D}) := \frac{1}{n} \sum_{i=1}^n \hat{Q}(s_0^{(i)}, \pi(s_0^{(i)})) = \frac{1}{n} \sum_{i=1}^n \hat{V}(s_0^{(i)}),\]

where \(\mathcal{D}=\{\{(s_t, a_t, r_t)\}_{t=0}^{T-1}\}_{i=1}^n\) is the logged dataset with \(n\) trajectories. \(T\) indicates step per episode. \(\hat{Q}(s_t, a_t)\) is the estimated Q value given a state-action pair. \(\hat{V}(s_t)\) is the estimated value function given a state.

DM has low variance compared to other estimators, but can produce larger bias due to approximation errors.

There are several methods to estimate \(\hat{Q}(s, a)\) such as Fitted Q Evaluation (FQE) (Le et al., 2019), Minimax Q-Function Learning (MQL) (Uehara et al., 2020), and Augmented Lagrangian Method (ALM) (Yang et al., 2020).

See also

The implementation of FQE is provided by d3rlpy. The implementations of Minimax Weight and Value Learning (including ALM) is available at scope_rl.ope.weight_value_learning.

Note

This function is different from DirectMethod in that the initial state is sampled from the stationary distribution \(d^{\pi}(s_0)\).

Parameters:: estimator_name (str, default="sm_dm") – Name of the estimator.

References

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Hoang Le, Cameron Voloshin, and Yisong Yue. “Batch Policy Learning under Constraints.” 2019.

Methods

`estimate_interval`(initial_state_value_prediction)	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(...)	Estimate the policy value of the evaluation policy.

estimate_policy_value(initial_state_value_prediction, **kwargs)[source]#

Estimate the policy value of the evaluation policy.

Parameters:: initial_state_value_prediction (array-like of shape (n_trajectories, )) – Estimated initial state value.
Returns:: V_hat – Estimated policy value.
Return type:: float

estimate_interval(initial_state_value_prediction, alpha=0.05, ci='bootstrap', n_bootstrap_samples=10000, random_state=None, **kwargs)[source]#

Estimate the confidence interval of the policy value by nonparametric bootstrap.

Parameters:

initial_state_value_prediction (array-like of shape (n_trajectories, )) – Estimated initial state value.
alpha (float, default=0.05) – Significance level. The value should be within [0, 1)
n_bootstrap_samples (int, default=10000 (> 0)) – Number of resampling performed in the bootstrap procedure.
random_state (int, default=None (>= 0)) – Random state.

Returns:

estimated_confidence_interval – Dictionary storing the estimated mean and upper-lower confidence bounds.

key: [
    mean,
    {100 * (1. - alpha)}% CI (lower),
    {100 * (1. - alpha)}% CI (upper),
]

Return type:

dict

class scope_rl.ope.continuous.marginal_estimators.StateMarginalIS(estimator_name='sm_is')[source]#

State Marginal Importance Sampling (SM-IS) for continuous action spaces.

Bases: scope_rl.ope.BaseStateMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateMarginalIS

Note

SM-IS estimates the policy value using state marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine State Marginal Importance Sampling and \(k\)-step PDIS as follows.

\[\hat{J}_{\mathrm{SM-IS}} (\pi; \mathcal{D}) := \frac{1}{n} \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t w_{0:t}^{(i)} \delta(\pi, a_{0:t}^{(i)}) r_t^{(i)} + \frac{1}{n} \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \rho(s_{t-k}^{(i)}) w_{t-k:t}^{(i)} \delta(\pi, a_{t-k:t}^{(i)}) r_t^{(i)},\]

where \(w_{t_1:t_2} := \prod_{t=t_1}^{t_2} (\pi(a_t | s_t) / \pi_0(a_t | s_t))\) and \(\rho(s) \approx d^{\pi}(s) / d^{\pi_b}(s)\) is the state-marginal importance weight, where \(d^{\pi}(s)\) is the marginal visitation probability of the policy \(\pi\) on \(s\). \(\delta(\pi, a_{t_1:t_2}) = \prod_{t=t_1}^{t_2} K(\pi(s_t), a_t)\) quantifies the similarity between the action logged in the dataset and that taken by the evaluation policy (\(K(\cdot, \cdot)\) is a kernel function). Note that the bandwidth of the kernel is an important hyperparameter; the variance of the above estimator often becomes small when the bandwidth of the kernel is large, while the bias often becomes large in those cases. Additionally, when \(k=0\), this estimator is reduced to the vanilla state marginal IS.

SM-IS corrects distribution shift between the behavior and evaluation policies. Moreover, SM-IS reduces the variance caused by trajectory-wise or per-decision importance weighting by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sm_is") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. “Breaking the Curse of Horizon: Infinite-Horizon Off-Policy Estimation.” 2018

Doina Precup, Richard S. Sutton, and Satinder P. Singh. “Eligibility Traces for Off-Policy Policy Evaluation.” 2000.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.

estimate_policy_value(n_step_pdis, step_per_trajectory, action, reward, state_marginal_importance_weight, pscore, evaluation_policy_action, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, **kwargs)[source]#

Estimate the policy value of the evaluation policy.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state marginal distribution, i.e., \(d^{\pi}(s) / d^{\pi_b}(s)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.

Returns:

V_hat – Estimated policy value.

Return type:

float

estimate_interval(n_step_pdis, step_per_trajectory, action, reward, state_marginal_importance_weight, pscore, evaluation_policy_action, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, alpha=0.05, ci='bootstrap', n_bootstrap_samples=10000, random_state=None, **kwargs)[source]#

Estimate the confidence interval of the policy value by nonparametric bootstrap.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state marginal distribution, i.e., \(d^{\pi}(s) / d^{\pi_b}(s)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.
alpha (float, default=0.05) – Significance level. The value should be within [0, 1).
ci ({"bootstrap", "hoeffding", "bernstein", "ttest"}, default="bootstrap") – Method to estimate the confidence interval.
n_bootstrap_samples (int, default=10000 (> 0)) – Number of resampling performed in the bootstrap procedure.
random_state (int, default=None (>= 0)) – Random state.

Returns:

estimated_confidence_interval – Dictionary storing the estimated mean and upper-lower confidence bounds.

key: [
    mean,
    {100 * (1. - alpha)}% CI (lower),
    {100 * (1. - alpha)}% CI (upper),
]

Return type:

dict

class scope_rl.ope.continuous.marginal_estimators.StateMarginalDR(estimator_name='sm_dr')[source]#

State Marginal Doubly Robust (SM-DR) for continuous action spaces.

Bases: scope_rl.ope.BaseStateMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateMarginalDR

Note

SM-DR estimates the policy value using state marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine state-marginal importance weighting and \(k\)-step DR as follows.

\[\begin{split}\hat{J}_{\mathrm{SM-DR}} (\pi; \mathcal{D}) &:= \frac{1}{n} \sum_{i=1}^n \hat{Q}(s_0^{(i)}, \pi(s_0^{(i)})) \\ & \quad \quad + \frac{1}{n} \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t w_{0:t}^{(i)} \delta(\pi, a_{0:t}^{(i)}) \left(r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)})) - \hat{Q}(s_t^{(i)}, a_t^{(i)}) \right) \\ & \quad \quad + \frac{1}{n} \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \rho(s_{t-k}^{(i)}) w_{t-k:t}^{(i)} \delta(\pi, a_{t-k:t}^{(i)}) \left(r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)})) - \hat{Q}(s_t^{(i)}, a_t^{(i)}) \right),\end{split}\]

where \(w_{t_1:t_2} := \prod_{t=t_1}^{t_2} (\pi(a_t | s_t) / \pi_0(a_t | s_t))\) and \(\rho(s) \approx d^{\pi}(s) / d^{\pi_b}(s)\) is the state-marginal importance weight, where \(d^{\pi}(s)\) is the marginal visitation probability of the policy \(\pi\) on \(s\). \(Q(s, a)\) is the state-action value. \(\delta(\pi, a_{t_1:t_2}) = \prod_{t=t_1}^{t_2} K(\pi(s_t), a_t)\) quantifies the similarity between the action logged in the dataset and that taken by the evaluation policy (\(K(\cdot, \cdot)\) is a kernel function). Note that the bandwidth of the kernel is an important hyperparameter; the variance of the above estimator often becomes small when the bandwidth of the kernel is large, while the bias often becomes large in those cases. Additionally, when \(k=0\), this estimator is reduced to the vanilla state marginal DR.

SM-DR corrects the distribution shift between the behavior and evaluation policies. Moreover, SM-DR reduces the variance caused by the trajectory-wise or per-decision importance weight by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sm_dr") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. “Breaking the Curse of Horizon: Infinite-Horizon Off-Policy Estimation.” 2018

Nan Jiang and Lihong Li. “Doubly Robust Off-policy Value Evaluation for Reinforcement Learning.” 2016.

Philip S. Thomas and Emma Brunskill. “Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning.” 2016.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.

estimate_policy_value(n_step_pdis, step_per_trajectory, action, reward, state_marginal_importance_weight, pscore, evaluation_policy_action, state_action_value_prediction, initial_state_value_prediction, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, **kwargs)[source]#

Estimate the policy value of the evaluation policy.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state marginal distribution, i.e., \(d^{\pi}(s) / d^{\pi_b}(s)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
state_action_value_prediction (array-like of shape (n_trajectories * step_per_trajectory, 2)) – \(\hat{Q}\) for the observed action and that chosen by the evaluation policy, i.e., (row 0) \(\hat{Q}(s_t, a_t)\) and (row 2) \(\hat{Q}(s_t, \pi(a | s_t))\).
initial_state_value_prediction (array-like of shape (n_trajectories, )) – Estimated initial state value.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.

Returns:

V_hat – Estimated policy value.

Return type:

float

estimate_interval(n_step_pdis, step_per_trajectory, action, reward, state_marginal_importance_weight, pscore, evaluation_policy_action, state_action_value_prediction, initial_state_value_prediction, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, alpha=0.05, ci='bootstrap', n_bootstrap_samples=10000, random_state=None, **kwargs)[source]#

Estimate the confidence interval of the policy value by nonparametric bootstrap.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state marginal distribution, i.e., \(d^{\pi}(s) / d^{\pi_b}(s)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
state_action_value_prediction (array-like of shape (n_trajectories * step_per_trajectory, 2)) – \(\hat{Q}\) for the observed action and that chosen by the evaluation policy, i.e., (row 0) \(\hat{Q}(s_t, a_t)\) and (row 2) \(\hat{Q}(s_t, \pi(a | s_t))\).
initial_state_value_prediction (array-like of shape (n_trajectories, )) – Estimated initial state value.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.
alpha (float, default=0.05) – Significance level. The value should be within [0, 1).
ci ({"bootstrap", "hoeffding", "bernstein", "ttest"}, default="bootstrap") – Method to estimate the confidence interval.
n_bootstrap_samples (int, default=10000 (> 0)) – Number of resampling performed in the bootstrap procedure.
random_state (int, default=None (>= 0)) – Random state.

Returns:

estimated_confidence_interval – Dictionary storing the estimated mean and upper-lower confidence bounds.

key: [
    mean,
    {100 * (1. - alpha)}% CI (lower),
    {100 * (1. - alpha)}% CI (upper),
]

Return type:

dict

class scope_rl.ope.continuous.marginal_estimators.StateMarginalSNIS(estimator_name='sm_snis')[source]#

State Marginal Self-Normalized Importance Sampling (SM-SNIS) for continuous action spaces.

Bases: scope_rl.ope.continuous.StateMarginalIS scope_rl.ope.BaseStateMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateMarginalSNIS

Note

SM-SNIS estimates the policy value using state marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine state-marginal importance weighting and \(k\)-step PDIS as follows.

\[\begin{split}\hat{J}_{\mathrm{SM-SNIS}} (\pi; \mathcal{D}) &:= \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t \frac{w_{0:t}^{(i)} \delta(\pi, a_{0:t}^{(i)})}{\sum_{i'=1}^n w_{0:t}^{(i')} \delta(\pi, a_{0:t}^{(i')})} r_t^{(i)} \\ & \quad \quad + \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \frac{\rho(s_{t-k}^{(i)}) w_{t-k:t}^{(i)} \delta(\pi, a_{t-k:t}^{(i+1)})}{\sum_{i'=1}^n \rho(s_{t-k}^{(i')}) w_{t-k:t}^{(i')} \delta(\pi, a_{t-k:t}^{(i')})} r_t^{(i)},\end{split}\]

SM-SNIS corrects the distribution shift between the behavior and evaluation policies. Moreover, SM-SNIS reduces the variance caused by trajectory-wise or per-decision importance weighting by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sm_snis") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. “Breaking the Curse of Horizon: Infinite-Horizon Off-Policy Estimation.” 2018

Doina Precup, Richard S. Sutton, and Satinder P. Singh. “Eligibility Traces for Off-Policy Policy Evaluation.” 2000.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.

class scope_rl.ope.continuous.marginal_estimators.StateMarginalSNDR(estimator_name='sm_sndr')[source]#

State Marginal Self-Normalized Doubly Robust (SM-SNDR) for continuous action spaces.

Bases: scope_rl.continuous.StateMarginalDR scope_rl.BaseStateMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateMarginalSNDR

Note

SM-SNDR estimates the policy value using state marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine state-marginal importance weighting and \(k\)-step PDIS as follows.

\[\begin{split}\hat{J}_{\mathrm{SM-SNDR}} (\pi; \mathcal{D}) &:= \frac{1}{n} \sum_{i=1}^n \hat{Q}(s_0^{(i)}, \pi(s_0^{(i)}) \\ & \quad \quad + \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t \frac{w_{0:t}^{(i)} \delta(\pi, a_{0:t}^{(i)})}{\sum_{i'=1}^n w_{0:t}^{(i')} \delta(\pi, a_{0:t}^{(i')})} \left(r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)}))) - \hat{Q}(s_t^{(i)}, a_t^{(i)}) \right) \\ & \quad \quad + \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \frac{\rho(s_{t-k}^{(i)}) w_{t-k:t}^{(i)} \delta(\pi, a_{t-k:t}^{(i)})}{\sum_{i'=1}^n \rho(s_{t-k}^{(i')}) w_{t-k:t}^{(i')} \delta(\pi, a_{t-k:t}^{(i')})} \left(r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)})) - \hat{Q}(s_t^{(i)}, a_t^{(i)}) \right),\end{split}\]

SM-SNDR corrects the distribution shift between the behavior and evaluation policies. Moreover, SM-SNDR reduces the variance caused by trajectory-wise or per-decision importance weighting by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sm_sndr") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. “Breaking the Curse of Horizon: Infinite-Horizon Off-Policy Estimation.” 2018

Nan Jiang and Lihong Li. “Doubly Robust Off-policy Value Evaluation for Reinforcement Learning.” 2016.

Philip S. Thomas and Emma Brunskill. “Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning.” 2016.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.

class scope_rl.ope.continuous.marginal_estimators.StateActionMarginalIS(estimator_name='sam_is')[source]#

State-Action Marginal Importance Sampling (SAM-IS) for continuous action spaces.

Bases: scope_rl.ope.BaseStateActionMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateActionMarginalIS

Note

SAM-IS estimates the policy value using state-action marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine state-marginal importance weighting and \(k\)-step PDIS as follows.

\[\begin{split}\hat{J}_{\mathrm{SAM-IS}} (\pi; \mathcal{D}) &:= \frac{1}{n} \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t w_{0:t}^{(i)} \delta(\pi, a_{t_1:t_2}^{(i)}) r_t^{(i)} \\ & \quad \quad + \frac{1}{n} \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \rho(s_{t-k}^{(i)}, a_{t-k}^{(i)}) w_{t-k+1:t}^{(i)} \delta(\pi, a_{t_1:t_2}^{(i)}) r_t^{(i)},\end{split}\]

where \(w_{t_1:t_2} := \prod_{t=t_1}^{t_2} (\pi(a_t | s_t) / \pi_0(a_t | s_t))\) and \(\rho(s, a) \approx d^{\pi}(s, a) / d^{\pi_b}(s, a)\) is the state-marginal importance weight, where \(d^{\pi}(s, a)\) is the marginal visitation probability of the policy \(\pi\) on \((s, a)\). \(\delta(\pi, a_{t_1:t_2}) = \prod_{t=t_1}^{t_2} K(\pi(s_t), a_t)\) quantifies the similarity between the action logged in the dataset and that taken by the evaluation policy (\(K(\cdot, \cdot)\) is a kernel function). Note that the bandwidth of the kernel is an important hyperparameter; the variance of the above estimator often becomes small when the bandwidth of the kernel is large, while the bias often becomes large in those cases. Additionally, when \(k=0\), this estimator is reduced to the vanilla state-action marginal IS.

SAM-IS corrects the distribution shift between the behavior and evaluation policies. Moreover, SAM-IS reduces the variance caused by trajectory-wise or per-decision importance weighting by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sam_is") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Doina Precup, Richard S. Sutton, and Satinder P. Singh. “Eligibility Traces for Off-Policy Policy Evaluation.” 2000.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.

estimate_policy_value(n_step_pdis, step_per_trajectory, action, reward, state_action_marginal_importance_weight, pscore, evaluation_policy_action, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, **kwargs)[source]#

Estimate the policy value of the evaluation policy.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_action_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state-action marginal distribution, i.e., \(d^{\pi}(s, a) / d^{\pi_b}(s, a)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.

Returns:

V_hat – Estimated policy value.

Return type:

float

estimate_interval(n_step_pdis, step_per_trajectory, action, reward, state_action_marginal_importance_weight, pscore, evaluation_policy_action, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, alpha=0.05, ci='bootstrap', n_bootstrap_samples=10000, random_state=None, **kwargs)[source]#

Estimate the confidence interval of the policy value by nonparametric bootstrap.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_action_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state-action marginal distribution, i.e., \(d^{\pi}(s, a) / d^{\pi_b}(s, a)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.
alpha (float, default=0.05) – Significance level. The value should be within [0, 1).
ci ({"bootstrap", "hoeffding", "bernstein", "ttest"}, default="bootstrap") – Method to estimate the confidence interval.
n_bootstrap_samples (int, default=10000 (> 0)) – Number of resampling performed in the bootstrap procedure.
random_state (int, default=None (>= 0)) – Random state.

Returns:

estimated_confidence_interval – Dictionary storing the estimated mean and upper-lower confidence bounds.

key: [
    mean,
    {100 * (1. - alpha)}% CI (lower),
    {100 * (1. - alpha)}% CI (upper),
]

Return type:

dict

class scope_rl.ope.continuous.marginal_estimators.StateActionMarginalDR(estimator_name='sam_dr')[source]#

State-Action Marginal Doubly Robust (SAM-DR) for continuous action spaces.

Bases: scope_rl.ope.BaseStateActionMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateActionMarginalDR

Note

SAM-DR estimates the policy value using state-action marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine state-marginal importance weighting and \(k\)-step PDIS as follows.

\[\begin{split}\hat{J}_{\mathrm{SAM-DR}} (\pi; \mathcal{D}) &:= \frac{1}{n} \sum_{i=1}^n \hat{Q}(s_0^{(i)}, \pi(s_0^{(i)})) \\ & \quad \quad + \frac{1}{n} \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t w_{0:t}^{(i)} \delta(\pi, a_{0:t}^{(i)}) \left( r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)})) - \hat{Q}(s_t^{(i)}, a_t^{(i)}) \right) \\ & \quad \quad + \frac{1}{n} \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \rho(s_{t-k}^{(i)}, a_{t-k}^{(i)}) w_{t-k+1:t}^{(i)} \delta(\pi, a_{t-k+1:t}^{(i)}) \left( r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)})) - \hat{Q}(s_t^{(i)}, a_t^{(i)}) \right),\end{split}\]

where \(w_{t_1:t_2} := \prod_{t=t_1}^{t_2} (\pi(a_t | s_t) / \pi_0(a_t | s_t))\) and \(\rho(s, a) \approx d^{\pi}(s, a) / d^{\pi_b}(s, a)\) is the state-marginal importance weight, where \(d^{\pi}(s, a)\) is the marginal visitation probability of the policy \(\pi\) on \((s, a)\). \(Q(s, a)\) is the state-action value. \(\delta(\pi, a_{t_1:t_2}) = \prod_{t=t_1}^{t_2} K(\pi(s_t), a_t)\) quantifies the similarity between the action logged in the dataset and that taken by the evaluation policy (\(K(\cdot, \cdot)\) is a kernel function). Note that the bandwidth of the kernel is an important hyperparameter; the variance of the above estimator often becomes small when the bandwidth of the kernel is large, while the bias often becomes large in those cases. Additionally, when \(k=0\), this estimator is reduced to the vanilla state-action marginal DR.

SAM-DR corrects the distribution shift between the behavior and evaluation policies. Moreover, SAM-DR reduces the variance caused by trajectory-wise or per-decision importance weighting by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sam_dr") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Nan Jiang and Lihong Li. “Doubly Robust Off-policy Value Evaluation for Reinforcement Learning.” 2016.

Philip S. Thomas and Emma Brunskill. “Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning.” 2016.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.

estimate_policy_value(n_step_pdis, step_per_trajectory, action, reward, state_action_marginal_importance_weight, pscore, evaluation_policy_action, state_action_value_prediction, initial_state_value_prediction, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, **kwargs)[source]#

Estimate the policy value of the evaluation policy.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_action_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state-action marginal distribution, i.e., \(d^{\pi}(s, a) / d^{\pi_b}(s, a)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
state_action_value_prediction (array-like of shape (n_trajectories * step_per_trajectory, 2)) – \(\hat{Q}\) for the observed action and that chosen by the evaluation policy, i.e., (row 0) \(\hat{Q}(s_t, a_t)\) and (row 2) \(\hat{Q}(s_t, \pi(a | s_t))\).
initial_state_value_prediction (array-like of shape (n_trajectories, )) – Estimated initial state value.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.

Returns:

V_hat – Estimated policy value.

Return type:

float

estimate_interval(n_step_pdis, step_per_trajectory, action, reward, state_action_marginal_importance_weight, pscore, evaluation_policy_action, state_action_value_prediction, initial_state_value_prediction, gamma=1.0, kernel='gaussian', bandwidth=1.0, action_scaler=None, alpha=0.05, ci='bootstrap', n_bootstrap_samples=10000, random_state=None, **kwargs)[source]#

Estimate the confidence interval of the policy value by nonparametric bootstrap.

Parameters:

n_step_pdis (int (>= 0)) – Number of initial steps whose rewards are estimated by step-wise importance weighting. When set to zero, the estimator is reduced to the vanilla state marginal IS.
step_per_trajectory (int (> 0)) – Number of timesteps in an episode.
action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the behavior policy.
reward (array-like of shape (n_trajectories * step_per_trajectory, )) – Observed immediate rewards.
state_action_marginal_importance_weight (array-like of shape (n_trajectories * step_per_trajectory, )) – Importance weight wrt the state-action marginal distribution, i.e., \(d^{\pi}(s, a) / d^{\pi_b}(s, a)\)
pscore (array-like of shape (n_trajectories * step_per_trajectory, )) – Conditional action choice probability of the behavior policy, i.e., \(\pi_b(a | s)\)
evaluation_policy_action (array-like of shape (n_trajectories * step_per_trajectory, action_dim)) – Action chosen by the evaluation policy.
state_action_value_prediction (array-like of shape (n_trajectories * step_per_trajectory, 2)) – \(\hat{Q}\) for the observed action and that chosen by the evaluation policy, i.e., (row 0) \(\hat{Q}(s_t, a_t)\) and (row 2) \(\hat{Q}(s_t, \pi(a | s_t))\).
initial_state_value_prediction (array-like of shape (n_trajectories, )) – Estimated initial state value.
gamma (float, default=1.0) – Discount factor. The value should be within (0, 1].
kernel ({"gaussian", "epanechnikov", "triangular", "cosine", "uniform"}) – Name of the kernel function to smooth importance weights.
bandwidth (float, default=1.0 (> 0)) – Bandwidth hyperparameter of the kernel function.
action_scaler (d3rlpy.preprocessing.ActionScaler, default=None) – Scaling factor of action.
alpha (float, default=0.05) – Significance level. The value should be within [0, 1).
ci ({"bootstrap", "hoeffding", "bernstein", "ttest"}, default="bootstrap") – Method to estimate the confidence interval.
n_bootstrap_samples (int, default=10000 (> 0)) – Number of resampling performed in the bootstrap procedure.
random_state (int, default=None (>= 0)) – Random state.

Returns:

estimated_confidence_interval – Dictionary storing the estimated mean and upper-lower confidence bounds.

key: [
    mean,
    {100 * (1. - alpha)}% CI (lower),
    {100 * (1. - alpha)}% CI (upper),
]

Return type:

dict

class scope_rl.ope.continuous.marginal_estimators.StateActionMarginalSNIS(estimator_name='sam_snis')[source]#

State-Action Marginal Self-Normalized Importance Sampling (SAM-SNIS) for continuous action spaces.

Bases: scope_rl.ope.continuous.StateActionMarginalIS scope_rl.ope.BaseStateActionMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateActionMarginalSNIS

Note

SAM-SNIS estimates the policy value using state-action marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine state-marginal importance weighting and \(k\)-step PDIS as follows.

\[\begin{split}\hat{J}_{\mathrm{SAM-SNIS}} (\pi; \mathcal{D}) &:= \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t \frac{w_{0:t}^{(i)} \delta(\pi, a_{0:t}^{(i)})}{\sum_{i'=1} w_{0:t}^{(i')} \delta(\pi, a_{0:t}^{(i')})} r_t^{(i)} \\ & \quad \quad + \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \frac{\rho(s_{t-k}^{(i)}, a_{t-k}^{(i)}) w_{t-k+1:t}^{(i)} \delta(\pi, a_{t-l+1:t}^{(i)})}{\sum_{i'=1}^n \rho(s_{t-k}^{(i')}, a_{t-k}^{(i')}) w_{t-k+1:t}^{(i')} \delta(\pi, a_{t-l+1:t}^{(i')})} r_t^{(i)},\end{split}\]

SAM-SNIS corrects the distribution shift between the behavior and evaluation policies. Moreover, SAM-SNIS reduces the variance caused by trajectory-wise or per-decision importance weighting by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sam_snis") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Doina Precup, Richard S. Sutton, and Satinder P. Singh. “Eligibility Traces for Off-Policy Policy Evaluation.” 2000.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.

class scope_rl.ope.continuous.marginal_estimators.StateActionMarginalSNDR(estimator_name='sam_sndr')[source]#

State-Action Marginal Self-Normalized Doubly Robust (SAM-SNDR) for continuous action spaces.

Bases: scope_rl.ope.continuous.StateActionMarginalDR scope_rl.ope.BaseStateActionMarginalOPEEstimator -> scope_rl.ope.BaseOffPolicyEstimator

Imported as: scope_rl.ope.continuous.StateActionMarginalSNDR

Note

SAM-SNDR estimates the policy value using state-action marginal importance weighting. Following SOPE (Yuan et al., 2021), we combine state-marginal importance weighting and \(k\)-step PDIS as follows.

\[\begin{split}\hat{J}_{\mathrm{SAM-SNDR}} (\pi; \mathcal{D}) &:= \frac{1}{n} \sum_{i=1}^n \hat{Q}(s_0^{(i)}, \pi(s_0^{(i)})) \\ & \quad \quad + \sum_{i=1}^n \sum_{t=0}^{k-1} \gamma^t \frac{w_{0:t}^{(i)} \delta(\pi, a_{0:t}^{(i)})}{\sum_{i'=1}^n w_{0:t}^{(i')} \delta(\pi, a_{0:t}^{(i')})} (r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)})) - \hat{Q}(s_t^{(i)}, a_t^{(i)})) \\ & \quad \quad + \sum_{i=1}^n \sum_{t=k}^{T-1} \gamma^t \frac{w(s_{t-k}^{(i)}, a_{t-k}^{(i)}) w_{t-k+1:t}^{(i)} \delta(\pi, a_{t-k+1:t}^{(i)})}{\sum_{i'=1}^n w(s_{t-k}^{(i')}, a_{t-k}^{(i')}) w_{t-k+1:t}^{(i')} \delta(\pi, a_{t-k+1:t}^{(i')})} (r_t^{(i)} + \gamma \hat{Q}(s_{t+1}^{(i)}, \pi(s_{t+1}^{(i)})) - \hat{Q}(s_t^{(i)}, a_t^{(i)})),\end{split}\]

where \(w_{t_1:t_2} := \prod_{t=t_1}^{t_2} (\pi(a_t | s_t) / \pi_0(a_t | s_t))\) and \(\rho(s, a) \approx d^{\pi}(s, a) / d^{\pi_b}(s, a)\) is the state-marginal importance weight, where \(d^{\pi}(s, a)\) is the marginal visitation probability of the policy \(\pi\) on \((s, a)\). \(Q(s, a)\) is the state-action value. \(\delta(\pi, a_{t_1:t_2}) = \prod_{t=t_1}^{t_2} K(\pi(s_t), a_t)\) quantifies the similarity between the action logged in the dataset and that taken by the evaluation policy (\(K(\cdot, \cdot)\) is a kernel function). Note that the bandwidth of the kernel is an important hyperparameter; the variance of the above estimator often becomes small when the bandwidth of the kernel is large, while the bias often becomes large in those cases. Additionally, when \(k=0\), this estimator is reduced to the vanilla state-action marginal SNDR.

SAM-SNDR corrects the distribution shift between the behavior and evaluation policies. Moreover, SAM-SNDR reduces the variance caused by trajectory-wise or per-decision importance weighting by considering the marginal distribution across various timesteps.

There are several ways to estimate the state(-action) marginal importance weight such as Augmented Lagrangian Method (ALM) (Yang et al., 2020) and Minimax Weight Learning (MWL) (Uehara et al., 2020).

See also

The implementations of such weight learning methods are available at scope_rl.ope.weight_value_learning.

Parameters:: estimator_name (str, default="sam_sndr") – Name of the estimator.

References

Christina J. Yuan, Yash Chandak, Stephen Giguere, Philip S. Thomas, and Scott Niekum. “SOPE: Spectrum of Off-Policy Estimators.” 2021.

Masatoshi Uehara, Jiawei Huang, and Nan Jiang. “Minimax Weight and Q-Function Learning for Off-Policy Evaluation.” 2020.

Mengjiao Yang, Ofir Nachum, Bo Dai, Lihong Li, and Dale Schuurmans. “Off-Policy Evaluation via the Regularized Lagrangian.” 2020.

Nathan Kallus and Angela Zhou. “Policy Evaluation and Optimization with Continuous Treatments.” 2019.

Nan Jiang and Lihong Li. “Doubly Robust Off-policy Value Evaluation for Reinforcement Learning.” 2016.

Philip S. Thomas and Emma Brunskill. “Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning.” 2016.

Methods

`estimate_interval`(n_step_pdis, ...[, gamma, ...])	Estimate the confidence interval of the policy value by nonparametric bootstrap.
`estimate_policy_value`(n_step_pdis, ...[, ...])	Estimate the policy value of the evaluation policy.