深度强化学习与多目标优化的结合是一个前沿且富有挑战性的研究方向-育师

一、核心概念框架

1.1 多目标强化学习（MORL）定义

MORL=<S,A,P,R⃗,γ> MORL = <S, A, P, R⃗, γ>MORL=<S,A,P,R⃗,γ>
其中R⃗=[r1,r2,...,rm]是m维奖励向量其中 R⃗ = [r₁, r₂, ..., rₘ] 是m维奖励向量其中R⃗=[r1,r2,...,rm]是m维奖励向量
目标：找到帕累托最优策略集目标：找到帕累托最优策略集目标：找到帕累托最优策略集

二、主要技术路线

2.1 标量化方法（主流方法）

# 线性标量化示例classLinearScalarization:def__init__(self,weights):self.weights=weights# 权重向量defscalarize(self,reward_vector):returnnp.dot(self.weights,reward_vector)# 在DRL算法中的应用classMO_DQN:def__init__(self,scalarization_fn):self.scalarization=scalarization_fndefcompute_scalar_reward(self,vector_reward):returnself.scalarization.scalarize(vector_reward)

2.2 基于偏好的方法

# 条件网络架构classPreferenceConditionedNetwork(nn.Module):def__init__(self,state_dim,action_dim,pref_dim):super().__init__()# 将偏好向量与状态拼接self.net=nn.Sequential(nn.Linear(state_dim+pref_dim,256),nn.ReLU(),nn.Linear(256,action_dim))defforward(self,state,preference):x=torch.cat([state,preference],dim=-1)returnself.net(x)

2.3 帕累托前沿方法

classParetoDQN:def__init__(self,num_objectives):self.num_objectives=num_objectives# 多个Q网络，每个对应一个目标self.q_nets=[QNetwork()for_inrange(num_objectives)]defcompute_pareto_front(self,q_values_list):"""计算帕累托最优动作集"""# 实现非支配排序pass

三、完整实现方案

3.1 MORL算法架构

importtorchimportnumpyasnpfromtypingimportList,TupleclassMultiObjectivePPO:""" 多目标PPO算法实现 """def__init__(self,state_dim:int,action_dim:int,num_objectives:int,scalarization_method:str="linear"):self.num_objectives=num_objectives# Actor-Critic网络（多输出）self.actor=MultiOutputActor(state_dim,action_dim)self.critic=MultiOutputCritic(state_dim,num_objectives)# 标量化策略self.scalarization=self._init_scalarization(scalarization_method)# 经验回放缓冲区self.buffer=MultiObjectiveBuffer()def_init_scalarization(self,method:str):ifmethod=="linear":returnLinearScalarization()elifmethod=="chebyshev":returnChebyshevScalarization()elifmethod=="hypervolume":returnHypervolumeBasedScalarization()else:raiseValueError(f"Unknown scalarization method:{method}")defcompute_scalarized_advantages(self,vector_values):"""计算标量化优势函数"""scalar_values=self.scalarization(vector_values)advantages=scalar_values-scalar_values.mean()returnadvantagesdefupdate(self,batch):# 多目标策略梯度更新vector_values=self.critic(batch.states)advantages=self.compute_scalarized_advantages(vector_values)# PPO损失计算（多目标扩展）loss=self.compute_multi_objective_loss(batch,advantages,vector_values)# 优化步骤self.optimizer.zero_grad()loss.backward()self.optimizer.step()

3.2 帕累托优化层

classParetoOptimizationLayer(nn.Module):"""帕累托优化层，用于决策时选择非支配解"""def__init__(self,epsilon=0.1):super().__init__()self.epsilon=epsilon# 帕累托容忍度defforward(self,q_values:torch.Tensor)->torch.Tensor:""" 输入: [batch_size, num_actions, num_objectives] 输出: [batch_size, num_actions] 帕累托最优动作掩码 """batch_size,num_actions,num_obj=q_values.shape pareto_mask=torch.ones(batch_size,num_actions,dtype=torch.bool)foriinrange(batch_size):forjinrange(num_actions):forkinrange(num_actions):ifj!=k:# 检查支配关系ifself.dominates(q_values[i,k],q_values[i,j]):pareto_mask[i,j]=Falsebreakreturnpareto_maskdefdominates(self,a:torch.Tensor,b:torch.Tensor)->bool:"""判断a是否支配b"""# a支配b当且仅当在所有目标上都不差于b，且至少一个目标严格更好better_or_equal=(a>=b).all()strictly_better=(a>b).any()returnbetter_or_equalandstrictly_better

四、实用算法实现

4.1 多目标DDPG

classMODDPG:def__init__(self,num_objectives,preference_sampling='adaptive'):# 多Critic网络self.critics=[Critic()for_inrange(num_objectives)]self.actor=Actor()# 偏好采样策略self.preference_sampler=PreferenceSampler(method=preference_sampling,num_objectives=num_objectives)deftrain_step(self,batch):# 采样偏好权重weights=self.preference_sampler.sample()# 计算标量化Q值q_values=[]fori,criticinenumerate(self.critics):q_values.append(critic(batch.states,batch.actions))scalar_q=self.scalarize_q_values(q_values,weights)# 更新Actor（最大化标量化Q值）new_actions=self.actor(batch.states)actor_loss=-self.scalarize_q_values([critic(batch.states,new_actions)forcriticinself.critics],weights).mean()# 更新Criticsfori,criticinenumerate(self.critics):target_q=batch.rewards[:,i]+self.gamma*self.target_critics[i](batch.next_states,self.target_actor(batch.next_states))critic_loss=F.mse_loss(q_values[i],target_q.detach())critic_optimizers[i].zero_grad()critic_loss.backward()critic_optimizers[i].step()

4.2 基于进化策略的多目标优化

classMOES:"""多目标进化策略"""def__init__(self,policy,num_objectives):self.policy=policy self.num_objectives=num_objectives self.population=[]defevolve(self,env,generations=100):forgeninrange(generations):# 评估种群fitnesses=self.evaluate_population(env)# 非支配排序fronts=self.non_dominated_sort(fitnesses)# 拥挤度计算crowding_distances=self.calculate_crowding_distance(fronts)# 选择下一代new_population=self.selection(fronts,crowding_distances)# 变异和交叉self.population=self.variation(new_population)

五、评估指标系统

classMORLEvaluator:@staticmethoddefcompute_hypervolume(pareto_front,reference_point):"""计算超体积指标"""pass@staticmethoddefcompute_sparsity(pareto_front):"""计算帕累托前沿的稀疏性"""pass@staticmethoddefcompute_coverage(set1,set2):"""计算两个解集之间的覆盖率"""pass

六、应用实例：多目标机器人控制

classMultiObjectiveRobotEnv:def__init__(self):self.objectives=['energy_efficiency','task_completion','safety']defstep(self,action):# 计算多目标奖励rewards={'energy':-self.compute_energy_cost(action),'task':self.compute_task_progress(),'safety':self.compute_safety_score()}# 转换为向量reward_vector=np.array([rewards[obj]forobjinself.objectives])returnnext_state,reward_vector,done,info# 训练流程deftrain_morl_robot():env=MultiObjectiveRobotEnv()agent=MultiObjectivePPO(state_dim=env.observation_space.shape[0],action_dim=env.action_space.shape[0],num_objectives=3,scalarization_method='chebyshev')# 多偏好训练preferences=[[0.8,0.1,0.1],# 侧重能效[0.1,0.8,0.1],# 侧重任务完成[0.1,0.1,0.8],# 侧重安全]forprefinpreferences:agent.set_preference(pref)# 训练阶段forepisodeinrange(num_episodes):state=env.reset()whilenotdone:action=agent.select_action(state)next_state,reward_vec,done,_=env.step(action)agent.store_transition(state,action,reward_vec,next_state,done)agent.update()