exploration_exploitation - Obsidian Publish

# Exploration-Exploitation Trade-off ## Overview The exploration-exploitation trade-off is a fundamental dilemma in decision-making systems: whether to exploit current knowledge for immediate rewards or explore to gather more information for potentially better future rewards. Links to: - [[decision_theory]] - Theoretical framework - [[active_inference]] - Free energy perspective - [[reinforcement_learning]] - Learning framework ## Active Inference Perspective In active inference, this trade-off emerges naturally through the Expected Free Energy components: 1. **Exploitation (Pragmatic Value)** - Maximizes expected reward - Uses current beliefs - Links to [[pragmatic_value]] 2. **Exploration (Epistemic Value)** - Reduces uncertainty - Gathers information - Links to [[epistemic_value]] ## Mathematical Formulation The trade-off is controlled by balancing epistemic and pragmatic terms: $G(\pi) = \underbrace{\text{epistemic}}_{\text{exploration}} + \underbrace{\text{pragmatic}}_{\text{exploitation}}$ Links to: - [[expected_free_energy]] - Full formulation - [[efe_components]] - Component details - [[information_gain]] - Exploration measure ## Implementation ```python def compute_explore_exploit_ratio( epistemic: float, pragmatic: float, temperature: float = 1.0 ) -> float: """Compute exploration-exploitation ratio. Args: epistemic: Epistemic value (exploration) pragmatic: Pragmatic value (exploitation) temperature: Temperature parameter Returns: Ratio of exploration vs exploitation """ return temperature * (epistemic / (abs(pragmatic) + 1e-10)) ``` Links to: - [[temperature_parameter]] - Control parameter - [[numerical_methods]] - Implementation details - [[policy_selection]] - Action selection ## Control Mechanisms ### Temperature Parameter - Controls exploration tendency - Higher values favor exploration - Lower values favor exploitation - Links to: - [[softmax_function]] - Policy selection - [[annealing_schedule]] - Temperature dynamics - [[optimization_parameters]] - Parameter tuning ### Adaptive Strategies - Dynamic temperature adjustment - Uncertainty-based exploration - Information-seeking policies - Links to: - [[adaptive_control]] - Control theory - [[uncertainty_estimation]] - Uncertainty measures - [[information_seeking]] - Active strategies ## Applications 1. **Active Inference** - Natural emergence from free energy - Balanced through precision - Links to [[active_inference_applications]] 2. **Reinforcement Learning** - ε-greedy strategies - Thompson sampling - Upper confidence bounds - Links to: - [[epsilon_greedy]] - Basic strategy - [[thompson_sampling]] - Bayesian approach - [[ucb_algorithms]] - Confidence bounds 3. **Multi-armed Bandits** - Classic exploration problem - Online learning setting - Links to: - [[bandit_problems]] - Problem formulation - [[regret_minimization]] - Optimization goal - [[optimal_stopping]] - When to stop exploring ## Analysis Methods 1. **Performance Metrics** - Cumulative reward - Information gain - Regret bounds - Links to: - [[reward_analysis]] - Reward metrics - [[information_metrics]] - Information measures - [[regret_analysis]] - Regret computation 2. **Visualization** - Exploration trajectories - Value landscapes - Decision boundaries - Links to: - [[trajectory_plots]] - Path visualization - [[value_visualization]] - Value plotting - [[decision_visualization]] - Choice analysis ## Related Concepts - [[multi_agent_learning]] - Multiple learners - [[hierarchical_exploration]] - Structured exploration - [[optimal_control]] - Control theory - [[information_theory]] - Information measures - [[decision_making]] - Decision processes ## Advanced Theoretical Framework ### Information-Theoretic Foundation **Definition** (Information-Theoretic Exploration): The exploration benefit can be quantified using mutual information: $I(\pi) = H(O) - \mathbb{E}_\pi[H(O|S)]$ where $H(O)$ is the marginal entropy of observations and $H(O|S)$ is the conditional entropy given states. **Theorem** (Optimal Exploration Strategy): For a finite horizon $T$, the optimal exploration strategy maximizes: $\pi^* = \arg\max_\pi \sum_{t=0}^T \gamma^t \left[\mathbb{E}[R_t|\pi] + \beta I_t(\pi)\right]$ where $\gamma$ is a discount factor, $R_t$ is immediate reward, and $\beta$ controls exploration intensity. ```python class AdvancedExplorationExploitation: """Advanced framework for exploration-exploitation with rigorous mathematical foundation.""" def __init__(self, state_dim: int, action_dim: int, exploration_horizon: int = 10, discount_factor: float = 0.99): """Initialize advanced exploration-exploitation system. Args: state_dim: Dimension of state space action_dim: Dimension of action space exploration_horizon: Planning horizon for exploration discount_factor: Temporal discount factor """ self.state_dim = state_dim self.action_dim = action_dim self.horizon = exploration_horizon self.gamma = discount_factor # Initialize belief state and uncertainty estimates self.belief_state = np.zeros(state_dim) self.uncertainty_matrix = np.eye(state_dim) # Information gain tracking self.information_history = [] self.reward_history = [] def compute_expected_information_gain(self, policy: np.ndarray, belief_state: np.ndarray, uncertainty: np.ndarray) -> float: """Compute expected information gain for a policy. Information gain quantifies reduction in uncertainty: IG = H(θ) - E[H(θ|o)] Args: policy: Policy to evaluate belief_state: Current belief about state uncertainty: Uncertainty covariance matrix Returns: Expected information gain """ # Prior entropy prior_entropy = 0.5 * np.log((2 * np.pi * np.e)**self.state_dim * np.linalg.det(uncertainty)) # Expected posterior entropy (simplified approximation) # Full computation would require integrating over observation space observation_noise = 0.1 * np.eye(self.state_dim) posterior_uncertainty = np.linalg.inv( np.linalg.inv(uncertainty) + np.linalg.inv(observation_noise) ) posterior_entropy = 0.5 * np.log( (2 * np.pi * np.e)**self.state_dim * np.linalg.det(posterior_uncertainty) ) return prior_entropy - posterior_entropy def compute_epistemic_value(self, actions: np.ndarray, belief_states: np.ndarray, uncertainties: np.ndarray) -> np.ndarray: """Compute epistemic value for different actions. Epistemic value measures information-seeking utility: V_epistemic(a) = E[IG(θ; o) | a] Args: actions: Array of candidate actions belief_states: Corresponding belief states uncertainties: Uncertainty estimates Returns: Epistemic values for each action """ epistemic_values = np.zeros(len(actions)) for i, action in enumerate(actions): # Simulate observation under this action predicted_observation = self._predict_observation(action, belief_states[i]) # Compute information gain ig = self._compute_mutual_information( belief_states[i], uncertainties[i], predicted_observation ) epistemic_values[i] = ig return epistemic_values def compute_pragmatic_value(self, actions: np.ndarray, belief_states: np.ndarray, reward_function: Callable) -> np.ndarray: """Compute pragmatic value for different actions. Pragmatic value measures immediate utility: V_pragmatic(a) = E[R(s,a) | belief] Args: actions: Array of candidate actions belief_states: Corresponding belief states reward_function: Reward function R(s,a) Returns: Pragmatic values for each action """ pragmatic_values = np.zeros(len(actions)) for i, action in enumerate(actions): # Expected reward under current belief expected_reward = reward_function(belief_states[i], action) pragmatic_values[i] = expected_reward return pragmatic_values def thompson_sampling_exploration(self, num_samples: int = 100, temperature: float = 1.0) -> Dict[str, Any]: """Implement Thompson sampling for exploration. Thompson sampling draws samples from posterior and acts greedily. Args: num_samples: Number of posterior samples temperature: Exploration temperature Returns: Thompson sampling results """ # Sample from posterior belief posterior_samples = np.random.multivariate_normal( self.belief_state, self.uncertainty_matrix, num_samples ) # Compute action values for each sample action_values = np.zeros((num_samples, self.action_dim)) for i, sample in enumerate(posterior_samples): for j in range(self.action_dim): action = np.zeros(self.action_dim) action[j] = 1.0 action_values[i, j] = self._evaluate_action(sample, action) # Softmax action selection with temperature probabilities = scipy.special.softmax(action_values / temperature, axis=1) selected_actions = np.array([ np.random.choice(self.action_dim, p=prob) for prob in probabilities ]) return { 'posterior_samples': posterior_samples, 'action_values': action_values, 'action_probabilities': probabilities, 'selected_actions': selected_actions, 'exploration_diversity': np.std(selected_actions) } def upper_confidence_bound(self, confidence_level: float = 0.95, exploration_bonus: float = 1.0) -> Dict[str, Any]: """Implement Upper Confidence Bound exploration. UCB balances exploitation and exploration via confidence intervals. Args: confidence_level: Confidence level for bounds exploration_bonus: Multiplier for exploration term Returns: UCB analysis results """ # Compute confidence intervals confidence_radius = scipy.stats.norm.ppf((1 + confidence_level) / 2) action_means = np.zeros(self.action_dim) action_uncertainties = np.zeros(self.action_dim) for j in range(self.action_dim): action = np.zeros(self.action_dim) action[j] = 1.0 # Mean value estimate action_means[j] = self._evaluate_action(self.belief_state, action) # Uncertainty estimate (simplified) action_uncertainties[j] = np.sqrt( action.T @ self.uncertainty_matrix @ action ) # Upper confidence bounds ucb_values = (action_means + exploration_bonus * confidence_radius * action_uncertainties) # Select action with highest UCB best_action = np.argmax(ucb_values) return { 'action_means': action_means, 'action_uncertainties': action_uncertainties, 'ucb_values': ucb_values, 'selected_action': best_action, 'confidence_intervals': np.column_stack([ action_means - confidence_radius * action_uncertainties, action_means + confidence_radius * action_uncertainties ]) } def compute_regret_bounds(self, time_horizon: int, suboptimality_gaps: np.ndarray) -> Dict[str, float]: """Compute theoretical regret bounds for exploration strategies. Args: time_horizon: Total time horizon suboptimality_gaps: Gap between optimal and suboptimal actions Returns: Regret bound analysis """ # UCB regret bound: O(√(K T log T)) K = self.action_dim ucb_regret_bound = np.sqrt(K * time_horizon * np.log(time_horizon)) # Thompson sampling regret bound (Bayesian setting) thompson_regret_bound = np.sqrt(K * time_horizon) # ε-greedy regret bound epsilon = 0.1 # Typical exploration rate epsilon_greedy_bound = epsilon * time_horizon + (1 - epsilon) * np.sum(suboptimality_gaps) return { 'ucb_regret_bound': ucb_regret_bound, 'thompson_regret_bound': thompson_regret_bound, 'epsilon_greedy_bound': epsilon_greedy_bound, 'best_theoretical_bound': min(ucb_regret_bound, thompson_regret_bound), 'suboptimality_gaps': suboptimality_gaps } def adaptive_exploration_rate(self, time_step: int, performance_history: List[float], uncertainty_history: List[float]) -> float: """Compute adaptive exploration rate based on performance and uncertainty. Args: time_step: Current time step performance_history: History of performance metrics uncertainty_history: History of uncertainty measures Returns: Adaptive exploration rate """ # Base exploration rate (decreasing with time) base_rate = 1.0 / np.sqrt(time_step + 1) # Performance-based adjustment if len(performance_history) > 1: recent_improvement = performance_history[-1] - performance_history[-2] performance_factor = 1.0 + np.tanh(recent_improvement) else: performance_factor = 1.0 # Uncertainty-based adjustment current_uncertainty = uncertainty_history[-1] if uncertainty_history else 1.0 uncertainty_factor = np.sqrt(current_uncertainty) # Combined adaptive rate adaptive_rate = base_rate * performance_factor * uncertainty_factor return np.clip(adaptive_rate, 0.01, 0.99) # Keep in reasonable bounds def information_directed_sampling(self, action_space: np.ndarray, information_ratio: float = 1.0) -> Dict[str, Any]: """Implement information-directed sampling strategy. IDS optimizes the ratio of information gain to squared regret. Args: action_space: Available actions information_ratio: Weight for information vs regret Returns: IDS results """ num_actions = len(action_space) # Compute information gains and regrets information_gains = np.zeros(num_actions) squared_regrets = np.zeros(num_actions) for i, action in enumerate(action_space): # Information gain ig = self.compute_expected_information_gain( action, self.belief_state, self.uncertainty_matrix ) information_gains[i] = ig # Squared regret (simplified) optimal_value = np.max([self._evaluate_action(self.belief_state, a) for a in action_space]) action_value = self._evaluate_action(self.belief_state, action) squared_regrets[i] = (optimal_value - action_value)**2 # Information-directed ratio ids_ratios = information_gains / (squared_regrets + 1e-10) # Select action maximizing information ratio best_action_idx = np.argmax(ids_ratios) return { 'information_gains': information_gains, 'squared_regrets': squared_regrets, 'ids_ratios': ids_ratios, 'selected_action': action_space[best_action_idx], 'selected_action_index': best_action_idx } def _predict_observation(self, action: np.ndarray, state: np.ndarray) -> np.ndarray: """Predict observation given action and state.""" # Simplified observation model return state + 0.1 * action + 0.05 * np.random.randn(self.state_dim) def _compute_mutual_information(self, state: np.ndarray, uncertainty: np.ndarray, observation: np.ndarray) -> float: """Compute mutual information between state and observation.""" # Simplified mutual information computation # Full computation requires proper integration return 0.5 * np.log(np.linalg.det(uncertainty) + 1e-10) def _evaluate_action(self, state: np.ndarray, action: np.ndarray) -> float: """Evaluate action value for given state.""" # Simplified value function return -np.linalg.norm(state - action)**2 ### Multi-Armed Bandit Extensions class ContextualBandit: """Contextual bandit with sophisticated exploration strategies.""" def __init__(self, context_dim: int, num_arms: int, regularization: float = 1.0): """Initialize contextual bandit. Args: context_dim: Dimension of context vectors num_arms: Number of bandit arms regularization: Regularization parameter """ self.context_dim = context_dim self.num_arms = num_arms self.reg = regularization # Initialize linear models for each arm self.arm_weights = [np.zeros(context_dim) for _ in range(num_arms)] self.arm_covariances = [regularization * np.eye(context_dim) for _ in range(num_arms)] def linear_ucb(self, context: np.ndarray, confidence_width: float = 1.0) -> Dict[str, Any]: """Linear Upper Confidence Bound for contextual bandits. Args: context: Current context vector confidence_width: Width of confidence intervals Returns: LinUCB results """ ucb_values = np.zeros(self.num_arms) confidence_widths = np.zeros(self.num_arms) for arm in range(self.num_arms): # Predicted reward predicted_reward = self.arm_weights[arm] @ context # Confidence width confidence = confidence_width * np.sqrt( context @ np.linalg.inv(self.arm_covariances[arm]) @ context ) confidence_widths[arm] = confidence # Upper confidence bound ucb_values[arm] = predicted_reward + confidence # Select arm with highest UCB selected_arm = np.argmax(ucb_values) return { 'selected_arm': selected_arm, 'ucb_values': ucb_values, 'confidence_widths': confidence_widths, 'predicted_rewards': [w @ context for w in self.arm_weights] } def update_arm_model(self, arm: int, context: np.ndarray, reward: float) -> None: """Update linear model for selected arm. Args: arm: Selected arm index context: Observed context reward: Observed reward """ # Update covariance matrix self.arm_covariances[arm] += np.outer(context, context) # Update weight vector (normal equations) # In practice, would use incremental updates for efficiency inv_cov = np.linalg.inv(self.arm_covariances[arm]) self.arm_weights[arm] = inv_cov @ ( self.arm_covariances[arm] @ self.arm_weights[arm] + reward * context ) ### Hierarchical Exploration class HierarchicalExploration: """Hierarchical exploration for structured environments.""" def __init__(self, hierarchy_levels: int, state_abstraction: Callable, action_abstraction: Callable): """Initialize hierarchical exploration. Args: hierarchy_levels: Number of abstraction levels state_abstraction: Function mapping states to abstract states action_abstraction: Function mapping actions to abstract actions """ self.levels = hierarchy_levels self.state_abs = state_abstraction self.action_abs = action_abstraction # Exploration at each level self.level_explorers = [ AdvancedExplorationExploitation(state_dim=10, action_dim=5) for _ in range(hierarchy_levels) ] def hierarchical_policy_selection(self, state: np.ndarray, exploration_weights: np.ndarray) -> Dict[str, Any]: """Select policy using hierarchical exploration. Args: state: Current state exploration_weights: Weights for each hierarchy level Returns: Hierarchical policy selection results """ level_policies = [] level_values = [] current_state = state for level in range(self.levels): # Abstract state at this level abstract_state = self.state_abs(current_state, level) # Get policy at this level explorer = self.level_explorers[level] actions = np.eye(explorer.action_dim) # Simple action space epistemic_values = explorer.compute_epistemic_value( actions, [abstract_state] * len(actions), [explorer.uncertainty_matrix] * len(actions) ) pragmatic_values = explorer.compute_pragmatic_value( actions, [abstract_state] * len(actions), lambda s, a: -np.linalg.norm(s - a)**2 ) # Combined values combined_values = (exploration_weights[level] * epistemic_values + (1 - exploration_weights[level]) * pragmatic_values) # Select action at this level selected_action_idx = np.argmax(combined_values) selected_action = actions[selected_action_idx] level_policies.append(selected_action) level_values.append(combined_values) # Update state for next level current_state = self.action_abs(current_state, selected_action, level) return { 'level_policies': level_policies, 'level_values': level_values, 'final_action': level_policies[-1], 'exploration_trace': exploration_weights } # Example usage and validation def validate_advanced_exploration_exploitation(): """Validate advanced exploration-exploitation framework.""" # Test advanced exploration-exploitation explorer = AdvancedExplorationExploitation(state_dim=5, action_dim=3) # Test Thompson sampling thompson_result = explorer.thompson_sampling_exploration(num_samples=50) print("Thompson sampling diversity:", thompson_result['exploration_diversity']) # Test UCB ucb_result = explorer.upper_confidence_bound() print("UCB selected action:", ucb_result['selected_action']) # Test contextual bandit bandit = ContextualBandit(context_dim=4, num_arms=5) context = np.random.randn(4) linucb_result = bandit.linear_ucb(context) print("LinUCB selected arm:", linucb_result['selected_arm']) # Test regret bounds subopt_gaps = np.array([0.1, 0.2, 0.05]) regret_bounds = explorer.compute_regret_bounds(1000, subopt_gaps) print("UCB regret bound:", regret_bounds['ucb_regret_bound']) if __name__ == "__main__": validate_advanced_exploration_exploitation() ``` ### Game-Theoretic Exploration **Definition** (Nash Exploration Equilibrium): In multi-agent settings, the exploration equilibrium satisfies: $\pi_i^* \in \arg\max_{\pi_i} \mathbb{E}\left[U_i(\pi_i, \pi_{-i}^*) + \beta I_i(\pi_i, \pi_{-i}^*)\right]$ where $U_i$ is utility and $I_i$ is information gain for agent $i$. ```python class GameTheoreticExploration: """Game-theoretic approach to multi-agent exploration.""" def __init__(self, num_agents: int, state_dim: int, action_dim: int): """Initialize game-theoretic exploration. Args: num_agents: Number of agents state_dim: State space dimension action_dim: Action space dimension """ self.num_agents = num_agents self.state_dim = state_dim self.action_dim = action_dim # Agent beliefs and uncertainties self.agent_beliefs = [np.zeros(state_dim) for _ in range(num_agents)] self.agent_uncertainties = [np.eye(state_dim) for _ in range(num_agents)] def compute_nash_exploration_equilibrium(self, utility_functions: List[Callable], information_weights: np.ndarray, max_iterations: int = 100) -> Dict[str, Any]: """Compute Nash equilibrium for exploration game. Args: utility_functions: Utility function for each agent information_weights: Information seeking weights max_iterations: Maximum iterations for convergence Returns: Nash exploration equilibrium """ # Initialize policies policies = [np.ones(self.action_dim) / self.action_dim for _ in range(self.num_agents)] for iteration in range(max_iterations): new_policies = [] for agent in range(self.num_agents): # Compute best response best_response = self._compute_best_response( agent, policies, utility_functions[agent], information_weights[agent] ) new_policies.append(best_response) # Check convergence policy_change = sum(np.linalg.norm(new_policies[i] - policies[i]) for i in range(self.num_agents)) policies = new_policies if policy_change < 1e-6: break return { 'equilibrium_policies': policies, 'iterations_to_convergence': iteration + 1, 'final_policy_change': policy_change } def _compute_best_response(self, agent: int, other_policies: List[np.ndarray], utility_function: Callable, info_weight: float) -> np.ndarray: """Compute best response for an agent.""" # Simplified best response computation # Full implementation would solve optimization problem action_utilities = np.zeros(self.action_dim) for action in range(self.action_dim): # Utility component utility = utility_function( self.agent_beliefs[agent], action, other_policies ) # Information component info_gain = self._compute_info_gain(agent, action, other_policies) action_utilities[action] = utility + info_weight * info_gain # Softmax policy return scipy.special.softmax(action_utilities) def _compute_info_gain(self, agent: int, action: int, other_policies: List[np.ndarray]) -> float: """Compute information gain for agent's action.""" # Simplified information gain computation return np.random.exponential(1.0) # Placeholder ``` ### Optimal Exploration Theory **Theorem** (Fundamental Exploration Bound): For any exploration policy $\pi$ and finite horizon $T$, the cumulative regret is bounded by: $R_T(\pi) \leq \sum_{k: \Delta_k > 0} \frac{8\log T}{\Delta_k} + \left(1 + \frac{\pi^2}{3}\right) \sum_{k=1}^K \Delta_k$ where $\Delta_k$ is the suboptimality gap for action $k$. **Proof Sketch**: Uses concentration inequalities and the principle of optimism in the face of uncertainty. ```python class OptimalExplorationTheory: """Theoretical framework for optimal exploration with provable guarantees.""" def __init__(self, horizon: int, confidence_level: float = 0.95, suboptimality_tolerance: float = 0.01): """Initialize optimal exploration theory framework. Args: horizon: Planning horizon T confidence_level: Confidence level for bounds suboptimality_tolerance: Tolerance for suboptimal actions """ self.T = horizon self.delta = 1 - confidence_level self.epsilon = suboptimality_tolerance # Theoretical constants self.exploration_constants = { 'ucb': 2.0, 'thompson': 1.0, 'epsilon_greedy': 1.0 } def compute_information_ratio(self, algorithm: str, problem_instance: Dict[str, Any]) -> Dict[str, float]: """Compute information ratio for exploration algorithm. Information ratio quantifies efficiency of information acquisition: Γ_t = (Regret_t)² / I_t where I_t is information gain at time t. Args: algorithm: Exploration algorithm name problem_instance: Problem specification Returns: Information ratio analysis """ # Extract problem parameters K = problem_instance.get('num_arms', 10) gaps = np.array(problem_instance.get('suboptimality_gaps', [0.1] * K)) # Algorithm-specific analysis if algorithm == 'thompson_sampling': info_ratio_bound = self._thompson_sampling_info_ratio(gaps) elif algorithm == 'ucb': info_ratio_bound = self._ucb_info_ratio(gaps) elif algorithm == 'information_directed_sampling': info_ratio_bound = self._ids_info_ratio(gaps) else: raise ValueError(f"Unknown algorithm: {algorithm}") # Compute regret bound using information ratio regret_bound = self._compute_regret_from_info_ratio( info_ratio_bound, self.T, gaps ) return { 'information_ratio_bound': info_ratio_bound, 'regret_bound': regret_bound, 'algorithm': algorithm, 'problem_complexity': np.sum(1 / gaps**2) } def analyze_sample_complexity(self, accuracy: float, confidence: float, problem_structure: Dict[str, Any]) -> Dict[str, int]: """Analyze sample complexity for different exploration objectives. Computes number of samples needed to achieve (ε,δ)-PAC guarantees. Args: accuracy: Accuracy parameter ε confidence: Confidence parameter δ problem_structure: Structure of exploration problem Returns: Sample complexity bounds for different objectives """ # Pure exploration (best arm identification) gaps = np.array(problem_structure.get('gaps', [0.1])) complexity_factor = problem_structure.get('complexity', 1.0) pure_exploration_bound = int( (32 * np.log(2 / confidence) * complexity_factor) / (accuracy**2) ) # Regret minimization regret_minimization_bound = int( (8 * np.log(self.T) * np.sum(1 / gaps)) / accuracy ) # Fixed confidence fixed_confidence_bound = int( (2 * np.log(2 / confidence) * complexity_factor) / (accuracy**2) ) # Fixed budget fixed_budget_bound = self.T # Given fixed budget return { 'pure_exploration': pure_exploration_bound, 'regret_minimization': regret_minimization_bound, 'fixed_confidence': fixed_confidence_bound, 'fixed_budget': fixed_budget_bound, 'theoretical_lower_bound': self._compute_lower_bound(accuracy, confidence, gaps) } def multi_objective_exploration(self, objectives: List[Callable], objective_weights: np.ndarray, constraint_functions: List[Callable] = None) -> Dict[str, Any]: """Analyze multi-objective exploration with theoretical guarantees. Considers trade-offs between multiple exploration objectives subject to constraints. Args: objectives: List of objective functions objective_weights: Weights for objectives constraint_functions: Optional constraint functions Returns: Multi-objective exploration analysis """ # Pareto frontier analysis pareto_analysis = self._compute_pareto_frontier( objectives, objective_weights ) # Scalarization approaches scalarization_analysis = self._analyze_scalarization_methods( objectives, objective_weights ) # Constraint handling if constraint_functions: constrained_analysis = self._analyze_constrained_exploration( objectives, constraint_functions ) else: constrained_analysis = {'type': 'unconstrained'} # Theoretical guarantees theoretical_guarantees = self._derive_multi_objective_bounds( objectives, objective_weights, constraint_functions ) return { 'pareto_analysis': pareto_analysis, 'scalarization_analysis': scalarization_analysis, 'constrained_analysis': constrained_analysis, 'theoretical_guarantees': theoretical_guarantees } def contextual_exploration_theory(self, context_dimension: int, feature_map: Callable, regularity_assumptions: Dict[str, Any]) -> Dict[str, Any]: """Theoretical analysis of contextual exploration. Analyzes exploration in contextual settings with function approximation. Args: context_dimension: Dimension of context space feature_map: Feature mapping function φ(x,a) regularity_assumptions: Assumptions about problem regularity Returns: Contextual exploration theory """ # Feature dimension feature_dim = regularity_assumptions.get('feature_dimension', context_dimension) # Smoothness parameters smoothness = regularity_assumptions.get('smoothness', 1.0) # Covering number analysis covering_analysis = self._analyze_covering_numbers( context_dimension, feature_dim, smoothness ) # Regret bounds for contextual bandits contextual_regret_bounds = self._compute_contextual_regret_bounds( feature_dim, smoothness, covering_analysis ) # Information gain bounds info_gain_bounds = self._compute_contextual_info_gain_bounds( feature_dim, smoothness ) # Eluder dimension analysis eluder_analysis = self._analyze_eluder_dimension( feature_map, feature_dim ) return { 'covering_analysis': covering_analysis, 'regret_bounds': contextual_regret_bounds, 'information_gain_bounds': info_gain_bounds, 'eluder_analysis': eluder_analysis, 'sample_complexity': self._compute_contextual_sample_complexity( feature_dim, smoothness, self.T ) } def _thompson_sampling_info_ratio(self, gaps: np.ndarray) -> float: """Compute information ratio bound for Thompson sampling.""" return 1.0 # Known theoretical result def _ucb_info_ratio(self, gaps: np.ndarray) -> float: """Compute information ratio bound for UCB.""" return 2.0 # Known theoretical result def _ids_info_ratio(self, gaps: np.ndarray) -> float: """Compute information ratio bound for Information Directed Sampling.""" return 1.0 # Optimal information ratio def _compute_regret_from_info_ratio(self, info_ratio: float, horizon: int, gaps: np.ndarray) -> float: """Compute regret bound from information ratio.""" return np.sqrt(info_ratio * horizon * np.log(horizon)) def _compute_lower_bound(self, accuracy: float, confidence: float, gaps: np.ndarray) -> int: """Compute information-theoretic lower bound.""" return int(np.sum(1 / gaps**2) * np.log(1 / confidence)) def _compute_pareto_frontier(self, objectives: List[Callable], weights: np.ndarray) -> Dict[str, Any]: """Compute Pareto frontier for multi-objective exploration.""" # Simplified Pareto analysis num_points = 100 pareto_points = [] for i in range(num_points): # Vary weights w = np.random.dirichlet(weights) # Optimize weighted sum (simplified) objective_values = [w[j] * np.random.rand() for j in range(len(objectives))] pareto_points.append(objective_values) return { 'pareto_points': pareto_points, 'dominated_points': [], # Would compute dominated points 'efficient_frontier': pareto_points # Simplified } def _analyze_scalarization_methods(self, objectives: List[Callable], weights: np.ndarray) -> Dict[str, Any]: """Analyze different scalarization approaches.""" return { 'weighted_sum': 'Linear scalarization', 'chebyshev': 'Min-max scalarization', 'epsilon_constraint': 'Constraint method', 'achievement_scalarization': 'Goal programming' } def _analyze_constrained_exploration(self, objectives: List[Callable], constraints: List[Callable]) -> Dict[str, Any]: """Analyze exploration under constraints.""" return { 'constraint_type': 'inequality', 'lagrangian_approach': 'Available', 'penalty_methods': 'Available', 'barrier_methods': 'Available' } def _derive_multi_objective_bounds(self, objectives: List[Callable], weights: np.ndarray, constraints: List[Callable] = None) -> Dict[str, Any]: """Derive theoretical bounds for multi-objective exploration.""" return { 'regret_bounds': 'O(√T log T)', 'constraint_violation_bounds': 'O(√T)' if constraints else 'N/A', 'convergence_rate': 'O(1/√T)' } # Example usage and validation def validate_optimal_exploration_theory(): """Validate optimal exploration theory framework.""" print("Optimal Exploration Theory Validation") print("=" * 50) # Initialize theory framework theory = OptimalExplorationTheory(horizon=1000) # Test information ratio analysis problem = { 'num_arms': 5, 'suboptimality_gaps': [0.1, 0.2, 0.05, 0.15, 0.3] } for algorithm in ['thompson_sampling', 'ucb']: info_ratio_result = theory.compute_information_ratio(algorithm, problem) print(f"{algorithm} information ratio: {info_ratio_result['information_ratio_bound']:.4f}") print(f"{algorithm} regret bound: {info_ratio_result['regret_bound']:.4f}") # Test sample complexity analysis sample_complexity = theory.analyze_sample_complexity( accuracy=0.1, confidence=0.05, problem_structure={'gaps': [0.1, 0.2], 'complexity': 10} ) print(f"Pure exploration sample complexity: {sample_complexity['pure_exploration']}") print(f"Regret minimization sample complexity: {sample_complexity['regret_minimization']}") # Test contextual exploration contextual_analysis = theory.contextual_exploration_theory( context_dimension=5, feature_map=lambda x, a: np.concatenate([x, [a]]), regularity_assumptions={'feature_dimension': 6, 'smoothness': 1.0} ) print("Contextual exploration analysis completed") if __name__ == "__main__": validate_optimal_exploration_theory() ``` ### Learning-Theoretic Perspective **Definition** (Exploration Complexity): The exploration complexity $\mathcal{C}(\epsilon, \delta)$ is the minimum number of samples needed to identify an $\epsilon$-optimal policy with probability at least $1-\delta$. **Theorem** (Instance-Dependent Lower Bound): For any algorithm and problem instance with gaps $\{\Delta_k\}$: $\mathbb{E}[T_{\epsilon,\delta}] \geq \frac{\log(1/\delta)}{2} \sum_{k: \Delta_k > \epsilon} \frac{1}{\Delta_k^2}$ This establishes fundamental limits on sample efficiency for exploration. ## References - [[sutton_barto_2018]] - Reinforcement Learning - [[friston_2017]] - Active Inference - [[gittins_1979]] - Bandit Processes - [[thrun_1992]] - Exploration Strategies - [[bubeck_cesa_bianchi_2012]] - Regret Analysis - [[russo_van_roy_2018]] - Information Ratio - [[lattimore_szepesvari_2020]] - Bandit Algorithms - [[agrawal_goyal_2012]] - Thompson Sampling Theory