Current time-synchronous sequence-to-sequence automatic speech recognition (ASR) models are trained by using sequence level cross-entropy that sums over all alignments. Due to the discriminative formulation, incorporating the right label context into the training criterion's gradient causes normalization problems and is not mathematically well-defined. The classic hybrid neural network hidden Markov model (NN-HMM) with its inherent generative formulation enables conditioning on the right label context. However, due to the HMM state-tying the identity of the right label context is never modeled explicitly. In this work, we propose a factored loss with auxiliary left and right label contexts that sums over all alignments. We show that the inclusion of the right label context is particularly beneficial when training data resources are limited. Moreover, we also show that it is possible to build a factored hybrid HMM system by relying exclusively on the full-sum criterion. Experiments were conducted on Switchboard 300h and LibriSpeech 960h.

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