An accurate and simple closed-form formula, for backward-wave start-oscillation condition for a millimeter-wave helix traveling-wave tube amplifier was developed, using an artificial neural network (ANN) algorithm. The analysis considers the effects of circuit loss and also the variation of electron beam diameter corresponding to beam filling. The formula is simple and amenable to easy computation, even using a scientific calculator, and without resorting to exhaustive numerical iterative search followed in conventional analyses and, at the same time, without sacrificing the accuracy in results. The formula was validated against published results, and excellent accuracy was observed. The analysis has been further used for inferring some physical interpretations on the effects of beam-filling factor and circuit loss on the start-oscillation condition of a typical millimeter-wave helix traveling-wave tube.