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This paper introduces a novel training framework called Focused Discriminative Training (FDT) to further improve streaming word-piece end-to-end (E2E) automatic speech recognition (ASR) models trained using either CTC or an interpolation of CTC and attention-based encoder-decoder (AED) loss. The proposed approach presents a novel framework to identify and improve a model's recognition on challenging segments of an audio. Notably, this training framework is independent of hidden Markov models (HMMs) and lattices, eliminating the need for substantial decision-making regarding HMM topology, lexicon, and graph generation, as typically required in standard discriminative training approaches. Compared to additional fine-tuning with MMI or MWER loss on the encoder, FDT is shown to be more effective in achieving greater reductions in Word Error Rate (WER) on streaming models trained on LibriSpeech. Additionally, this method is shown to be effective in further improving a converged word-piece streaming E2E model trained on 600k hours of assistant and dictation dataset.