Continuous Coin Betting Optimizer

A learning-rate-free, dual-averaging stochastic subgradient optimization algorithm that reduces d-dimensional minimization of a function F(w) from initial point w_1 to betting on d continuous coins whose outcomes are the negative stochastic subgradients g_{t,i} in [-L_i, L_i] (E[g_t] in partial[-F(w_t)]): for each coordinate i, it accumulates absolute subgradient sum G_{t,i} = G_{t-1,i} + |g_{t,i}| (G_{0,i} = L_i), cumulative reward Reward_{t,i} = Reward_{t-1,i} + (w_{t,i} - w_{1,i}) g_{t,i}, and signed subgradient sum theta_{t,i} = theta_{t-1,i} + g_{t,i}, bets signed wealth fraction beta_{t,i} = (2 sigma(2 theta_{t,i} / (G_{t,i} + L_i)) - 1) / L_i where sigma(x) = 1 / (1 + exp(-x)), and updates parameters via w_{t+1,i} = w_{1,i} + beta_{t,i} (L_i + Reward_{t,i}), with deep-network variant COCOB-Backprop adapting L_{t,i} = max(L_{t-1,i}, |g_{t,i}|) and clamping Reward_{t,i} >= 0 online.