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Injecting weight noise during training is a simple technique that has been proposed for almost two decades. However, little is known about its convergence behavior. This paper studies the convergence of two weight noise injection-based training algorithms, multiplicative weight noise injection with weight decay and additive weight noise injection with weight decay. We consider that they are applied to multilayer perceptrons either with linear or sigmoid output nodes. Let w(t) be the weight vector, let V(w) be the corresponding objective function of the training algorithm, let α >; 0 be the weight decay constant, and let μ(t) be the step size. We show that if μ(t)→ 0, then with probability one E[||w(t)||22] is bound and limt→∞||w(t)||2 exists. Based on these two properties, we show that if μ(t)→ 0, Σtμ(t)=∞, and Σtμ(t)2 <; ∞, then with probability one these algorithms converge. Moreover, w(t) converges with probability one to a point where ∇wV(w)=0.