Geometric-Consensus Q-Network Training

Integrate this at the distributed deep-Q training step, immediately after each worker computes a minibatch TD statistic and before applying the shared update. Use N workers with identical online parameters theta and target parameters theta_minus. At epoch start, broadcast theta and theta_minus once. For each local minibatch on worker i, compute y_i=r+gamma(1-done) max_a Q_theta_minus(s_prime,a), the TD residual delta_i=y_i-Q_theta(s,a), and either a vector of per-action TD targets or a parameter-gradient/delta vector g_i. Do not all-reduce every minibatch. Instead, accumulate an epoch statistic z_i, such as the mean gradient over B local minibatches or a flattened parameter delta, then execute L neighbor exchanges: z_i <- sum_j W[i,j] z_j. Apply the resulting consensus statistic to the common model, for example theta <- theta + (1/N)sum_i z_i, or have every worker apply its locally mixed z_i. Pseudocode: initialize theta, theta_minus; repeat epochs: collect local z_i=0; for b=1..B: sample transitions, compute y_i and g_i, update z_i <- z_i + g_i/B; optionally perform local optimizer steps on a private copy; end; for l=1..L: z_i <- sum_j W[i,j]z_j; theta_i <- theta_i + eta z_i; synchronize theta_minus periodically. Estimate rho with power iteration on W-P: v <- random zero-mean vector; rho_hat <- ||(W-P)v||/||v||, repeatedly. Estimate Rbar from a high percentile of |y_i|, and choose L=ceil(log(epsilon_cons(1-gamma)/(C1*N*Rbar))/log(rho_hat)), clipped to [0,L_max], where epsilon_cons is the observed standard error of the local statistic. The paper's math supplies the geometric rho^L error budget; the neural-gradient variance, C1, and effective Rbar must be estimated empirically. First test on a 4- or 8-worker CartPole, Atari-5, or small DM Control benchmark using identical MLP DQN workers. Compare per-minibatch all-reduce, FedAvg-style local updates, and this periodic consensus at equal environment samples and equal communication bytes. Success means similar or better return and TD loss at substantially fewer communication rounds, with stable disagreement ||theta_i-theta_bar|| and ideally near-linear wall-clock sample throughput as N increases.