Entropy-Feedback Zeroth-Order Cooling

1. Integration point: implement this as a zeroth-order optimizer for a small neural network, derivative-free fine-tuning of a pretrained model, or optimization of prompts or low-rank adapter parameters. At each optimizer step, draw N perturbations of the current trainable parameter vector, evaluate every candidate on the same minibatch, and replace the ordinary fixed-temperature softmax weighting and hand-designed cooling schedule with the entropy controller above. Use antithetic perturbations when memory permits and keep a separate validation minibatch for monitoring collapse.

2. Pseudocode:

initialize theta, tau=tau0
for t in range(T):
    eps = Normal(0,I,N)
    candidates = theta + sigma * eps
    losses = [loss(model(candidate), minibatch) for candidate in candidates]
    z = -(losses - min(losses)) / tau
    w = softmax(z)
    H = -sum(w * log(w + 1e-12))
    h = H / log(N)
    b = sigmoid((h - hcrit) / delta)
    theta_mean = sum_i(w[i] * candidates[i])
    theta = (1-gamma)*theta + gamma*theta_mean
    tau = tau * exp(-alpha*(epsilon_floor + (1-epsilon_floor)*b))

3. Computed directly from the mechanism: normalized importance-weight entropy, effective sample size, and the thresholded temperature update. Estimated empirically: threshold h_c, cooling gain alpha, perturbation scale sigma, and damping gamma. Start with h_c=0.5, delta=0.05, alpha=0.05, epsilon=0.02, and gamma=1; tune only h_c in the first ablation. Clip the logits before softmax for numerical stability.

4. First cheap experiment: optimize a two-layer ReLU MLP on MNIST using N=64 or 128 Gaussian candidates per step and compare against the same zeroth-order optimizer with exponential cooling, constant temperature, and a manually tuned ESS floor. Run five seeds on a 10,000-example subset. The quantitative prediction is that normalized entropy will repeatedly approach h_c from above rather than falling immediately toward zero; the slope of log(tau_t) should change from approximately -alpha when h_t>h_c+2delta to approximately -alpha epsilon when h_t<h_c-2delta. The measured ESS at transition should be N^{h_c}; for N=128 and h_c=0.5, the predicted transition is ESS approximately 11.3. Reject the mechanism if the cooling-rate change is absent or transition ESS differs by more than 20%.