Effective-Exploration Bias Correction

Integrate this at the target-construction stage of a contextual-bandit reward model, not inside the neural encoder. Let a network q_theta(x,a) predict reward for context x and arm a, while the online policy selects arms with I_t=q_theta(x_t,a)+f_t/sqrt(N_a). Maintain per-arm or per-cluster count N_a, mean bar_r_a, and variance sigma_hat_a squared. At the end of each training window, compute c_a=sigma_hat_a/(sqrt(N_a) f_T). Estimate whether arm a is uniquely optimal by comparing its upper confidence estimate with the best competitor: u_a=bar_r_a+sigma_hat_a/sqrt(N_a), and u_minus_a=max_{b != a}(bar_r_b+sigma_hat_b/sqrt(N_b)). Set a soft gate g_a=sigmoid((u_minus_a-u_a)/tau), where tau is a temperature; g_a near one means likely non-unique-optimal and g_a near zero means likely uniquely optimal. Train with corrected regression targets y_i_corr=r_i+g_{A_i} c_{A_i}, or add this correction only to the arm-specific residual head while leaving the shared context representation unchanged. Pseudocode: collect (x_t,a_t,r_t); update N_a, bar_r_a, and sigma_hat_a; compute f_T, g_a, and c_a; replay minibatches with y_i_corr; periodically refresh statistics. The theorem supplies the sign and 1/f_T scaling, while sigma_hat_a, the optimality gates, and the effective f_T for nonstationary schedules are estimated empirically. First test on a synthetic contextual bandit with 10 arms, Gaussian rewards, a two-layer MLP, and UCB1 with f_t=sqrt(2 log t). Compare ordinary MSE with the correction using held-out uniformly randomized arm data. Success means lower held-out reward MSE and reduced signed calibration error for rarely selected arms at equal interaction budget, without degrading predictions for the uniquely best arm.