Beam-beam instability induced by random fluctuations in size of a strong beam has been studied both numerically and analytically. In contrast with stochastic particle motion due to overlapping of nonlinear beam-beam resonance islands, noise beam-beam instability exists at any value of beam-beam parameter. Two conditions are essential to initiate instability: beam-beam kick has to be a nonlinear function of coordinate and parameter of the kick has to be a subject of noise. Meanwhile, the value of beam emittance is conserved even in noise regime if beam-beam kick is a linear function of coordinate. It gives an idea to prevent beam-beam instability using linearization of beam-beam kick. Compensation scheme utilizing higher order component in field distribution of the focusing elements is suggested.