Joint-Particle Distributional Critic

(1) Integration point: implement this as a distributional RL critic replacing a scalar Q-network or independent quantile heads. Given a state embedding $h(s)$ and a latent particle $\xi_k\sim\mathcal N(0,I)$, an MLP outputs an $A$-dimensional vector $X_k=f_\theta(s,\xi_k)$, one return sample for every action. Use the same particle index across all action coordinates. The replay buffer must store coupled transitions $(R_a,S'_a)_{a=1}^{A}$ generated by one shared simulator random seed; ordinary single-action environments cannot test the joint advantage directly. (2) Pseudocode: sample a minibatch of coupled transitions and latent particles; compute $X_k=f_\theta(s,\xi_k)$; compute $\mu_b=K^{-1}\sum_k f_{\bar\theta}(s'_a,\xi_k)_b$ and choose $b^*(s'_a)=\arg\max_b\mu_b$; construct $Y_{a,k}=R_a+\gamma f_{\bar\theta}(S'_a,\xi'_k)_{b^*(S'_a)}$ using shared $\xi'_k$; draw $M$ random unit vectors $u_m$; project both $X_k$ and $Y_k$ onto every direction, sort the $K$ projections, and minimize the mean absolute sorted difference in the sliced-Wasserstein loss. Update $\theta$ with Adam and periodically copy it to $\bar\theta$. (3) Mathematical versus empirical quantities: the return recursion, shared-vector target, and Wasserstein objective are mathematical components; $K$, $M$, target-update period, and $\gamma$ are hyperparameters. Estimate action means by particle averages. Log empirical cross-action covariance $\widehat{\operatorname{Cov}}(X_{\cdot,a},X_{\cdot,b})$ and compare it with covariance from held-out coupled rollouts. (4) First cheap experiment: create a stochastic two-action bandit or gridworld in which both counterfactual rewards depend on a controllable shared Bernoulli or Gaussian exogenous variable. Compare scalar DQN, independent quantile regression, and the joint critic at equal network size and environment interactions. Then test a small vectorized control task with common-randomness simulator seeds. Measure held-out joint-return Wasserstein error, mean return, action-selection accuracy when means are close, and cross-action covariance calibration. Success means lower joint-law error or faster return improvement at equal interaction count; if shared-particle training gives no improvement over independently resampled transitions, reject the added architecture.