Beam-search Algorithm for Decoding RNN output

Question

I have been trying to understand the logic used by the beam-search algorithm in automatic speech recognition for the decoding part. The papers I've tried to follow are First-Pass Large Vocabulary Continuous Speech Recognition using Bi-Directional Recurrent DNNs, Lexicon-Free Conversational Speech Recognition with Neural Networks and Towards End-to-End Speech Recognition with Recurrent Neural Networks. The problem is that the idea behind the algorithm is not so easy to follow and there are a lot of typos in the pseudo-code provided in the papers. Also, this implementation from the second paper is incredible hard to follow and this one, from the last paper mentioned, doesn't includes a Language Model.

This is my implementation in Python, which fails because of some missing probabilities:

class BeamSearch(object):
"""
Decoder for audio to text.

From: https://arxiv.org/pdf/1408.2873.pdf (hardcoded)
"""
def __init__(self, alphabet='" abcdefghijklmnopqrstuvwxyz'):
    # blank symbol plus alphabet
    self.alphabet = '-' + alphabet
    # index of each char
    self.char_to_index = {c: i for i, c in enumerate(self.alphabet)}

def decode(self, probs, k=100):
    """
    Decoder.

    :param probs: matrix of size Windows X AlphaLength
    :param k: beam size
    :returns: most probable prefix in A_prev
    """
    # List of prefixs, initialized with empty char
    A_prev = ['']
    # Probability of a prefix at windows time t to ending in blank
    p_b = {('', 0): 1.0}
    # Probability of a prefix at windows time t to not ending in blank
    p_nb = {('', 0): 0.0}

    # for each time window t
    for t in range(1, probs.shape[0] + 1):
        A_new = []
        # for each prefix
        for s in Z:
            for c in self.alphabet:
                if c == '-':
                    p_b[(s, t)] = probs[t-1][self.char_to_index[self.blank]] *\
                                    (p_b[(s, t-1)] +\
                                        p_nb[(s, t-1)])
                    A_new.append(s)
                else:
                    s_new = s + c
                    # repeated chars
                    if len(s) > 0 and c == s[-1]:
                        p_nb[(s_new, t)] = probs[t-1][self.char_to_index[c]] *\
                                            p_b[(s, t-1)]
                        p_nb[(s, t)] = probs[t-1][self.char_to_index[c]] *\
                                            p_b[(s, t-1)]
                    # spaces
                    elif c == ' ':
                        p_nb[(s_new, t)] = probs[t-1][self.char_to_index[c]] *\
                                           (p_b[(s, t-1)] +\
                                            p_nb[(s, t-1)])
                    else:
                        p_nb[(s_new, t)] = probs[t-1][self.char_to_index[c]] *\
                                            (p_b[(s, t-1)] +\
                                                p_nb[(s, t-1)])
                        p_nb[(s, t)] = probs[t-1][self.char_to_index[c]] *\
                                            (p_b[(s, t-1)] +\
                                                p_nb[(s, t-1)])
                    if s_new not in A_prev:
                        p_b[(s_new, t)] = probs[t-1][self.char_to_index[self.blank]] *\
                                            (p_b[(s, t-1)] +\
                                                p_nb[(s, t-1)])
                        p_nb[(s_new, t)]  = probs[t-1][self.char_to_index[c]] *\
                                                p_nb[(s, t-1)]
                    A_new.append(s_new)
        A = A_new
        s_probs = map(lambda x: (x, (p_b[(x, t)] + p_nb[(x, t)])*len(x)), A_new)
        xs = sorted(s_probs, key=lambda x: x[1], reverse=True)[:k]
        Z, best_probs = zip(*xs)
    return Z[0], best_probs[0]

Any help will be really appreciated.

Answer 1

I implemented the beam search using the -inf initialization and also follow ctc_beam_search algorithm from the paper http://proceedings.mlr.press/v32/graves14.pdf... it is nearly similar to this except for the update of p_b for characters..the algorithm runs properly...even this algorithm would work if the initialization is present..

A_prev = ['']
p_b[('',0)] = 1
p_nb[('',0)] = 0
for alphabet in alphabets:
    p_b[(alphabet,0)] = -float("inf")
    p_nb[(alphabet,0)] = -float("inf")
for t in range(1,probs.shape[0] +1):
    A_new = []
    for s in A_prev:
        if s!='':
            try:                
                p_nb[(s,t)] = p_nb[(s,t-1)]*probs[t-1][char_map[s[-1:]]]
            except:
                p_nb[(s,t)] = p_nb[(s,t-1)]*probs[t-1][char_map['<SPACE>']]*pW(s)
            if s[:-1] in A_prev:
                p_nb[(s,t)] = p_nb[(s,t)]+pr(probs[t-1],s[-1:],s[:-1],t)
        p_b[(s,t)] = (p_nb[(s,t-1)]+p_b[(s,t-1)])*probs[t-1][0]
        if s=='':
            p_nb[(s,t)] = 0
        if s not in A_new:
            A_new.append(s)
        for c in alphabets:
            s_new = s+c
            p_b[(s_new,t)] = 0
            p_nb[(s_new,t)] = pr(probs[t-1],c,s,t)
            #print s_new,' ',p_nb[(s_new,t)]
            if s_new not in A_new:
                A_new.append(s_new)
    s_probs = map(lambda x: (x,(p_b[(x, t)]+ p_nb[(x, t)])), A_new)

Answer 2

The problem is that the block with s_new not in A_prev is referring to probabilities which wont exist for a new string generated. Initialize with -float("inf") for new strings' previous timestep i.e. s_new,t-1. You can put a try catch block where if the p[(s_new,t-1)] doesn't exist, it will use -infinity.