Hot-Uphill Cold-Downhill Gradient Noise

Integrate this into SGD, Adam, or AdamW after computing the usual deterministic parameter update. Maintain an exponential moving average of minibatch loss and gradient norm. Define the estimated uphill signal as DeltaL_hat_t=L_t-EMA(L)_t, optionally adding a term for directional disagreement, such as 1-cos(g_t,g_(t-1)). Map it through s_t=sigmoid(DeltaL_hat_t/tau), clip s_t to [0,1], and set T_t=T_c+(T_h-T_c)s_t. Pseudocode is: compute minibatch gradient g_t; update loss and gradient EMAs; calculate s_t; sample xi_t from a standard normal distribution; apply theta <- theta - eta times the AdamW update plus sqrt(2 eta T_t) times xi_t. Estimate the temperature bounds empirically rather than assuming physical units: choose T_c so downhill loss variance is 10 to 30 percent of baseline SGD variance, then sweep T_h/T_c over {1,2,4,8}. The first cheap experiment is a two-layer MLP on MNIST or CIFAR-10, comparing SGD, constant-temperature Langevinized SGD, and this controller at equal compute. The quantitative prediction is that escape from deliberately initialized bad basins improves as T_h/T_c increases, but final-basin fluctuations remain lower than with globally hot noise. Measure escape probability versus T_h/T_c and identify the transition where injected uphill variance exceeds local gradient-noise variance; the best ratio should be finite rather than maximal.