Overview

We explore the Skip-Gram with Negative Sampling algorithm and implement it in JAX from scratch

Introduction

In this short post we are going to do a deep dive on the Skip-Gram with Negative Sampling (SGNS) algorithm. SGNS is one of the two algorithms used in Word2Vec that generates dense static embeddings. These embeddings have a fixed dimension, canonically 300 dimensional, and each token in our vocabular is statically assigned one and only one embedding irrepsective of multiple word senses.

The key aspects to understading how embeddings are learned under this paradigm are:

Understanding the concept of a sliding window
Understanding how training examples are generated
Understanding how the loss function works

Therefore, we are going to focus on the above aspects throughout this post. First, though, we need to find some data to use for when we get are hands dirty in the hands-on portion of this post.

Download and Prepare Data

To train our SGNS embeddings we are going to need some data. For this we will use an open source library called (Project Gutenberg)[https://www.gutenberg.org], which has a collection of books that are freely available for use. We are going to include five political books as our corpus. You can download and lightly pre-process the text before saving the data to a text file with the following code snippet.

Download

# Imports
import jax
import jax.numpy as jnp
import numpy as np
from jax import random, jit, grad
from flax import linen as nn
from typing import Tuple, List
from nltk.tokenize import word_tokenize, sent_tokenize
from collections import Counter
import tensorflow as tf
import optax as optim
from tqdm import tqdm
import requests

def clean_gutenberg_text(text: str) -> str:
    """
    Helper that cleans the downloaded Project Gutenberg text

    Parameters
    ----------
    text: 
        The downloaded text
    """
    # Find start marker
    start_idx = text.find("*** START OF")
    if start_idx != -1:
        text = text[start_idx:]
        # Skip the marker line
        text = text.split("\n", 1)[1]

    # Find end marker
    end_idx = text.find("*** END OF")
    if end_idx != -1:
        text = text[:end_idx]

    # Remove repeated blank lines
    lines = [line.strip() for line in text.splitlines()]
    lines = [line for line in lines if line]  # drop empty
    return "\n".join(lines)

urls = [
    "https://www.gutenberg.org/cache/epub/18/pg18.txt",      # Federalist Papers
    "https://www.gutenberg.org/files/815/815-0.txt",         # Democracy in America
    "https://www.gutenberg.org/files/1232/1232-0.txt",       # The Prince
    "https://www.gutenberg.org/files/132/132-0.txt",         # Art of War
    "https://www.gutenberg.org/files/34901/34901-0.txt"      # On Liberty
]

merged_text = ""
for url in urls:
    raw_text = requests.get(url).text
    clean_text = clean_gutenberg_text(raw_text)
    merged_text += clean_text + "\n\n"

with open("political_corpus.txt", "w", encoding="utf-8") as f:
    f.write(merged_text)

Preprocessing Utilities

We are going to need a few helper functions:

A way to tokenize our text and remove low occurence tokens
A way to map a token to an index
A way to map an index to a token

Below you can see these functions defined.

def tokenize_and_filter(text: str, min_occurrence: int) -> list[str]:
    """
    Tokenizes input text and filters out tokens that occur less than or equal to `min_occurrence` times.

    Parameters
    ----------
    text: str
        Input text to be tokenized.
    min_occurrence: int
        Minimum number of times a token must appear in the text to be included.

    Returns
    -------
    list[str]: 
        A list of tokens that occur more than `min_occurrence` times.
    """
    # Convert text to lowercase and tokenize
    tokens = word_tokenize(text.lower().strip())

    # Count occurrences of each token
    token_counts = Counter(tokens)

    # Filter tokens based on minimum occurrence threshold
    filtered_tokens = [token for token in tokens if token_counts[token] > min_occurrence]

    return filtered_tokens

def build_word_index(text: str, min_occurence: int = 0) -> tuple[dict[str, int], list[str]]:
    """
    Builds a word-to-index mapping (vocabulary) from input text, including a special <unk> token.

    Parameters
    ----------
    text: str
        Input text to extract vocabulary from.
    min_occurrence: int
        Minimum number of times a token must appear in the text to be included.

    Returns
    -------
    Tuple[Dict[str, int], List[str]]: 
        - word_to_index: A dictionary mapping each word to a unique integer ID.
        - vocabulary: A sorted list of unique vocabulary words including "<unk>".
    """
    # Tokenize and get unique words
    unique_tokens = set(tokenize_and_filter(text, min_occurrence=min_occurence))

    # Add special token for unknown words
    vocabulary = sorted(unique_tokens) + ["<unk>"]

    # Create word-to-index mapping
    word_to_index = {word: idx for idx, word in enumerate(vocabulary)}

    return word_to_index, vocabulary

def tokens_to_ids(text: str, word_to_index: dict[str, int], min_occurence: int = 0) -> list[int]:
    """
    Converts a text string into a list of integer token IDs using a provided word-to-index mapping.
    Unknown tokens are mapped to the ID of the "<unk>" token.

    Parameters
    ----------
    text: str
        Input text to convert.
    word_to_index: dict[str, int]
        A dictionary mapping words to their corresponding integer IDs.
    min_occurrence: int
        Minimum number of times a token must appear in the text to be included.

    Returns
    -------
    list[int]:
        A list of integer IDs representing the tokenized input text.
    """
    tokens = tokenize_and_filter(text, min_occurrence=min_occurence)
    token_ids = [word_to_index.get(token, word_to_index["<unk>"]) for token in tokens]
    return token_ids

Skip-Gram With Negative Sampling (SGNS)

There are really two parts to understanding how SGNS embeddings come together. First, is understanding how the training examples are generated. Second, is understanding how the loss function is computed.

Skip-Gram Pairs

We call our training example Skip-Gram Pairs, and what that means is that we are pairing our tokens into (inputs and targets). The way that we accomplish this is by selecting a context (targets) window size that determines how many targets each center word (input) will have and selecting the maximum length of any given sequence of tokens.

For example, our context window size is 2 and our sequence of tokens lets say is:

“The quick brown fox jumps over the lazy dog.”

We start by sliding our window over each token to contruct our Skip-Gram Pairs. Our window size is equal to context_window_size \(\times\) 2 + 1 = 5. As we slide our window over the sequence, we need to ensure that we our center position is on a token that allows 2 tokens to the left and 2 tokens to the right to be included as our context tokens. For our example sequence, this means the first center word that we can accomodate with our window size is the token “brown”. So we will have the following:

“The quick brown fox jumps”
“quick brown fox jumps over”
“brown fox jumps over the”
“fox jumps over the lazy”
“jumps over the lazy dog”

We can represent the above Skip-Gram pairs in a tabular format. Below is an example of the first window of skip-gram pairs.

Table 1: Tabular representation of Skip-Gram pair

Center	Context
brown	the
brown	quick
brown	fox
brown	jumps

def generate_skipgram_pairs(sentences: list[str], word_to_index: dict[str, int], max_sequence_length: int, context_window_size: int) -> tuple[jnp.ndarray, jnp.ndarray]:
    """
    Generate training pairs for the Skip-Gram model from a batch of input texts.

    Parameters
    ----------
    sentences: list[str]
        A list of input text strings.
    word_to_index: dict[str, int]
        A mapping from tokens (words) to their corresponding integer IDs.
    max_sequence_length: int
        The maximum number of tokens to use per input text.
    context_window_size: int
        The number of words to use on each side of the center word as context.

    Returns
    -------
    tuple[jnp.ndarray, jnp.ndarray]: 
        - center_words: 1D array of center word IDs (inputs).
        - context_words: 1D array of corresponding context word IDs (targets).
    """
    center_words, context_words = [], []

    for text in sentences:
        text = text.strip()
        token_ids = tokens_to_ids(text, word_to_index)

        # Skip sequences too short for the context window
        if len(token_ids) < context_window_size * 2 + 1:
            continue

        # Truncate long sequences
        if len(token_ids) > max_sequence_length:
            token_ids = token_ids[:max_sequence_length]

        # Slide a window over the token sequence
        for i in range(len(token_ids) - context_window_size * 2):
            window = token_ids[i: i + context_window_size * 2 + 1]
            center = window[context_window_size]
            context = window[:context_window_size] + window[context_window_size + 1:]

            for context_word in context:
                center_words.append(center)
                context_words.append(context_word)

    center_tensor = jnp.array(center_words, dtype=jnp.int32)
    context_tensor = jnp.array(context_words, dtype=jnp.int32)

    return center_tensor, context_tensor

Loss Function

Now that we understand how we are generating our examples, let’s go over the loss function. We start by setting up our unigram probabilities that are used when we need to sample negative examples. We adjust the raw frequency counts to diversify the words we are sampling as our negative examples otherwise we almost always sample stop words. During the forward pass we simply take the dot product between our center words and context words as a measure of similarity. For our positive samples, we then compute the logistic loss

\[ l(logits, labels) = max(logits, 0) - logits \times labels + log(1 + e^{-|logits|}) \]

where all the labels will be set to one, since this is our positive example. On the otherhand, we sample negative examples (15 in our case) and again compute the dot product between our center words and now our negative example. However, this time when we compute the loss we set the labels to zero since these are negative examples. We proceed by summing our loss over all the negative examples and we take the average similar to the positive example. Finally, we sum both positive and negative losses together to produce our final loss.

In the original paper the loss is written in terms of a sum of expectations:

\[ \mathbb{E}_{(w,c) \sim D} \left[ \log \sigma \big( v_w \cdot v_c \big) \right] \;+\; k \, \mathbb{E}_{c' \sim P_n} \left[ \log \sigma \big( - v_w \cdot v_{c'} \big) \right] \]

Where \(k\) is the number of negative examples and the left portion accounts for the positive example and the right portion account for the negative examples.

class SGNSLoss:
    BETA = 0.75
    NUM_SAMPLES = 15

    def __init__(self, vocabulary, id_counts, key):
        self.vocabulary = vocabulary
        self.vocab_len = len(vocabulary)
        self.key = key

        # Precompute unigram distribution
        freqs = jnp.array([id_counts[i] for i in range(self.vocab_len - 1)], dtype=jnp.float32)
        transformed = freqs ** self.BETA
        self.unigram_probs = transformed / transformed.sum()

    def forward(self, params, batch):
        center_ids, context_ids = batch
        center = params[center_ids]  # (batch_size, embed_dim)
        context = params[context_ids]  # (batch_size, embed_dim)

        # Positive logits
        true_logits = jnp.sum(center * context, axis=-1)
        loss = self._bce_loss_with_logits(true_logits, jnp.ones_like(true_logits))

        # Negative sampling
        key, subkey = random.split(self.key)
        neg_ids = random.categorical(subkey, jnp.log(self.unigram_probs), shape=(self.NUM_SAMPLES, center.shape[0]))
        neg_embeds = params[neg_ids]  # (num_samples, batch_size, embed_dim)

        # Compute loss for each negative sample
        def neg_loss_fn(neg):
            logits = jnp.sum(center * neg, axis=-1)
            return self._bce_loss_with_logits(logits, jnp.zeros_like(logits))

        neg_losses = jax.vmap(neg_loss_fn)(neg_embeds)
        total_neg_loss = jnp.sum(neg_losses, axis=0)  # sum over negative samples

        return loss + total_neg_loss.mean()

    @staticmethod
    def _bce_loss_with_logits(logits, labels):
        return jnp.mean(jnp.maximum(logits, 0) - logits * labels + jnp.log1p(jnp.exp(-jnp.abs(logits))))

Learning Embeddings

Now that we know how Skip-Gram with Negative sampling works, how the training examples are generated, and how the loss function works, we can get to the hands-on portion and learn the weights of our embeddings matrix.

Define Embedding Layer

Below we define our Embeddings matrix using Flax. This is a very simple look-up table essentially that gets initialized using the LeCun Normal initialization method which is scaled standard normal distribution \(N(0, \frac{1}{d})\) where d is the dimension of our embeddings matrix.

class SkipGramEmbeddings(nn.Module):
    vocab_size: int
    embed_len: int

    @nn.compact
    def __call__(self, center: jnp.ndarray, context: jnp.ndarray) -> tuple[jnp.ndarray, jnp.ndarray]:
        embedding = nn.Embed(num_embeddings=self.vocab_size, features=self.embed_len)
        word_embeds = embedding.embedding  # Retrieve embedding matrix
        return word_embeds[center], word_embeds[context]

Load and Process Dataset

We are ready to load and process the data that we scraped earlier. We are going to apply the helper functions we defined above to

Get a mapping from a token to the numerical index and vice versa
Get a set representing our total vocabulary
Tokenize our corpus into list of sentences
Generate our Skip-Gram pairs
Compute the frequencies of each token

with open('./political_corpus.txt', 'r', encoding='utf-8') as file:
    text = file.read()

word_to_index, vocabulary = build_word_index(text=text, min_occurence=10)
index_to_word = {v:k for k,v in word_to_index.items()}
sentences = sent_tokenize(text.lower().strip())
dataset = generate_skipgram_pairs(sentences=sentences, word_to_index=word_to_index, max_sequence_length=200, context_window_size=2)
id_counts = dict(
    Counter(tokens_to_ids(text, word_to_index))
)

Define Hyperparameters & Training Loop

Next we define our hyperparameters. We are going to set the embedding dimension to 300 to match what was used in the original implementation and we will go for 30 epochs with a batch size of 64.

We are going to use tensorflow-datasets’s data loader to iterate through our examples during training. We need to write up our update function to update the weights in our embeddings matrix and the training loop itself.

For the update, we simply apply the forward pass of the SGNS then compute the logistic loss, and finally calculate the gradient with respect to the parameters in our matrix. JAX will handle the computation of the gradients for us and we are using Optax to handle the updates.

One thing to mention is that we are using the Adaptive Moment Estimation (adam) optimizer with a learning rate of 1e-4. Finally, we need a helper function that will allow us to grab the nearest neighbors of a given token to inspect if our embeddings are actually learning anything useful.

# Hyperparameters
vocab_size = len(vocabulary)
embed_len = 300
data_len = len(dataset[0])
batch_size = 64
epochs = 30
learning_rate = 1e-4

# set up the data loader and our PRNG keys
dataloader = tf.data.Dataset.from_tensor_slices((dataset[0], dataset[1])).shuffle(100).batch(batch_size).prefetch(1)
model_rng = random.PRNGKey(0)
sgns_rng = random.PRNGKey(1)

# Dummy data to init our variabels with
dummy_center = jnp.zeros((batch_size,), dtype=jnp.int32)
dummy_context = jnp.zeros((batch_size,), dtype=jnp.int32)

# Init our embeddings, variables, and parameters
model = SkipGramEmbeddings(vocab_size, embed_len)
variables = model.init(model_rng, dummy_center, dummy_context)
params = variables['params']['Embed_0']['embedding']
sgns = SGNSLoss(vocabulary=vocabulary, id_counts=id_counts, key=sgns_rng)

# Define our parameters
optimizer = optim.adam(learning_rate)
opt_init = optimizer.init
opt_update = optimizer.update
apply_updates = optim.apply_updates

@jit
def update(params, opt_state, batch):
    """
    Simple function to update embeddings weights
    """
    loss, g = jax.value_and_grad(sgns.forward)(params, batch)  # compute loss + gradient
    updates, opt_state = opt_update(g, opt_state)
    params = apply_updates(params, updates)
    return opt_state, params, g, loss

def train(params):
    """
    simple training loop
    """
    opt_state = opt_init(params)
    train_it = dataloader.as_numpy_iterator()
    for epoch in range(epochs):
        print(f'Beginning epoch: {epoch + 1}/{epochs}')
        epoch_loss = 0
        for i, batch in enumerate(tqdm(train_it)):
            opt_state, params, g, loss = update(params, opt_state, batch)
            epoch_loss += loss
        avg_loss = epoch_loss / (i + 1)
        log_step(epoch, params, g, avg_loss)
        train_it = dataloader.as_numpy_iterator()
    return params

def log_step(epoch, params, g, loss):
    """
    Simple function to print out summaries after each epoch
    """
    print(f'EPOCH: {epoch + 1} | LOSS: {loss:.4f}')
    # Log embeddings!
    print('\nLearned embeddings:')
    print(f'word: "tyranny" neighbors: {nearest_neighbors("tyranny", params)}\n')

def nearest_neighbors(word: str, vectors: jnp.ndarray, k:int = 3) -> List[str]:
    """
    Finds k nearest neighbors to the given word using cosine similarity.

    Parameters
    ----------
    word: str
        Target word as a string.
    vectors:  
        Trained Embeddings
    k: int
        Number of nearest neighbors to return

    Returns
    -------
    List[str]
        Nearest neighbor tokens to word token
    """
    index = tokens_to_ids(word, word_to_index)
    query = vectors[index[0]]

    # Normalize vectors and query
    vectors_norm = vectors / jnp.linalg.norm(vectors, axis=1, keepdims=True)
    query_norm = query / jnp.linalg.norm(query)

    # Compute cosine similarities
    similarities = jnp.dot(vectors_norm, query_norm)

    # Get top 10 indices (excluding the word itself)
    top_indices = jnp.argsort(similarities)[::-1][1:k+1]

    # Map back to words
    return [index_to_word[idx] for idx in np.array(top_indices)]

Train

Alright, let’s get this show on the road. This can take a bit of time to complete. On my laptop it takes about an hour to finish training. While training after each epoch we will print out the learned embeddings for the word “tyranny” to see how our learning is evolving.

trained_params = train(params)

Beginning epoch: 1/30


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30508it [03:36, 141.01it/s]

EPOCH: 1 | LOSS: 9.3147

Learned embeddings:
word: "tyranny" neighbors: ['questions', 'economy', 'offenses']

Beginning epoch: 2/30


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EPOCH: 2 | LOSS: 8.6413

Learned embeddings:
word: "tyranny" neighbors: ['questions', 'regulations', 'introduce']

Beginning epoch: 3/30


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EPOCH: 3 | LOSS: 8.4056

Learned embeddings:
word: "tyranny" neighbors: ['questions', 'tendency', 'introduce']

Beginning epoch: 4/30


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EPOCH: 4 | LOSS: 8.2679

Learned embeddings:
word: "tyranny" neighbors: ['questions', 'tendency', 'monarchy']

Beginning epoch: 5/30


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EPOCH: 5 | LOSS: 8.1764

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'questions', 'monarchy']

Beginning epoch: 6/30


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EPOCH: 6 | LOSS: 8.1126

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'questions', 'monarchy']

Beginning epoch: 7/30


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EPOCH: 7 | LOSS: 8.0694

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'questions', 'temporary']

Beginning epoch: 8/30


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EPOCH: 8 | LOSS: 8.0293

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'questions', 'temporary']

Beginning epoch: 9/30


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23713it [02:46, 142.41it/s]
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23788it [02:47, 141.82it/s]
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23818it [02:47, 141.96it/s]
23833it [02:47, 141.98it/s]
23848it [02:47, 141.96it/s]
23863it [02:47, 141.70it/s]
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23908it [02:48, 141.96it/s]
23923it [02:48, 141.92it/s]
23938it [02:48, 141.86it/s]
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24838it [02:54, 142.03it/s]
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24868it [02:54, 141.91it/s]
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24898it [02:55, 142.20it/s]
24913it [02:55, 141.95it/s]
24928it [02:55, 142.18it/s]
24943it [02:55, 141.95it/s]
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24988it [02:55, 142.12it/s]
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25033it [02:56, 142.00it/s]
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25213it [02:57, 141.95it/s]
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25273it [02:57, 141.97it/s]
25288it [02:57, 142.11it/s]
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25318it [02:58, 142.09it/s]
25333it [02:58, 142.25it/s]
25348it [02:58, 142.20it/s]
25363it [02:58, 142.10it/s]
25378it [02:58, 142.18it/s]
25393it [02:58, 142.09it/s]
25408it [02:58, 142.15it/s]
25423it [02:58, 142.17it/s]
25438it [02:58, 142.24it/s]
25453it [02:59, 142.05it/s]
25468it [02:59, 142.11it/s]
25483it [02:59, 141.79it/s]
25498it [02:59, 141.91it/s]
25513it [02:59, 142.01it/s]
25528it [02:59, 142.08it/s]
25543it [02:59, 142.02it/s]
25558it [02:59, 142.05it/s]
25573it [02:59, 141.38it/s]
25588it [02:59, 141.52it/s]
25603it [03:00, 142.19it/s]
25618it [03:00, 142.35it/s]
25633it [03:00, 142.07it/s]
25648it [03:00, 141.89it/s]
25663it [03:00, 142.02it/s]
25678it [03:00, 142.08it/s]
25693it [03:00, 142.27it/s]
25708it [03:00, 142.41it/s]
25723it [03:00, 142.37it/s]
25738it [03:01, 142.13it/s]
25753it [03:01, 142.21it/s]
25768it [03:01, 142.20it/s]
25783it [03:01, 142.21it/s]
25798it [03:01, 142.25it/s]
25813it [03:01, 141.55it/s]
25828it [03:01, 142.36it/s]
25843it [03:01, 142.10it/s]
25858it [03:01, 142.08it/s]
25873it [03:01, 142.24it/s]
25888it [03:02, 142.27it/s]
25903it [03:02, 142.18it/s]
25918it [03:02, 141.85it/s]
25933it [03:02, 141.98it/s]
25948it [03:02, 142.05it/s]
25963it [03:02, 141.93it/s]
25978it [03:02, 142.12it/s]
25993it [03:02, 142.02it/s]
26008it [03:02, 141.95it/s]
26023it [03:03, 142.09it/s]
26038it [03:03, 142.00it/s]
26053it [03:03, 142.14it/s]
26068it [03:03, 142.15it/s]
26083it [03:03, 141.93it/s]
26098it [03:03, 141.77it/s]
26113it [03:03, 141.92it/s]
26128it [03:03, 142.21it/s]
26143it [03:03, 142.20it/s]
26158it [03:03, 141.96it/s]
26173it [03:04, 142.21it/s]
26188it [03:04, 142.00it/s]
26203it [03:04, 141.62it/s]
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26233it [03:04, 142.22it/s]
26248it [03:04, 142.19it/s]
26263it [03:04, 141.98it/s]
26278it [03:04, 141.48it/s]
26293it [03:04, 141.88it/s]
26308it [03:05, 142.11it/s]
26323it [03:05, 142.06it/s]
26338it [03:05, 142.24it/s]
26353it [03:05, 142.08it/s]
26368it [03:05, 142.23it/s]
26383it [03:05, 142.11it/s]
26398it [03:05, 142.04it/s]
26413it [03:05, 142.13it/s]
26428it [03:05, 142.15it/s]
26443it [03:05, 142.12it/s]
26458it [03:06, 141.93it/s]
26473it [03:06, 141.90it/s]
26488it [03:06, 142.15it/s]
26503it [03:06, 142.18it/s]
26518it [03:06, 142.39it/s]
26533it [03:06, 142.09it/s]
26548it [03:06, 142.19it/s]
26563it [03:06, 142.29it/s]
26578it [03:06, 142.22it/s]
26593it [03:07, 142.21it/s]
26608it [03:07, 142.32it/s]
26623it [03:07, 142.32it/s]
26638it [03:07, 142.03it/s]
26653it [03:07, 142.16it/s]
26668it [03:07, 142.10it/s]
26683it [03:07, 141.99it/s]
26698it [03:07, 141.88it/s]
26713it [03:07, 141.89it/s]
26728it [03:07, 141.80it/s]
26743it [03:08, 141.91it/s]
26758it [03:08, 141.93it/s]
26773it [03:08, 141.95it/s]
26788it [03:08, 141.84it/s]
26803it [03:08, 142.13it/s]
26818it [03:08, 141.82it/s]
26833it [03:08, 141.66it/s]
26848it [03:08, 141.76it/s]
26863it [03:08, 141.20it/s]
26878it [03:09, 141.49it/s]
26893it [03:09, 141.44it/s]
26908it [03:09, 141.74it/s]
26923it [03:09, 141.92it/s]
26938it [03:09, 141.89it/s]
26953it [03:09, 142.04it/s]
26968it [03:09, 141.80it/s]
26983it [03:09, 141.64it/s]
26998it [03:09, 141.90it/s]
27013it [03:10, 141.30it/s]
27028it [03:10, 142.41it/s]
27043it [03:10, 142.09it/s]
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27088it [03:10, 141.96it/s]
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27313it [03:12, 141.94it/s]
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27418it [03:12, 142.12it/s]
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27478it [03:13, 142.29it/s]
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27523it [03:13, 142.31it/s]
27538it [03:13, 142.26it/s]
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27568it [03:13, 142.21it/s]
27583it [03:14, 142.24it/s]
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27673it [03:14, 142.33it/s]
27688it [03:14, 142.12it/s]
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27718it [03:14, 142.28it/s]
27733it [03:15, 142.11it/s]
27748it [03:15, 142.08it/s]
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27778it [03:15, 142.04it/s]
27793it [03:15, 141.83it/s]
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27823it [03:15, 142.16it/s]
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27868it [03:16, 141.40it/s]
27883it [03:16, 142.17it/s]
27898it [03:16, 142.30it/s]
27913it [03:16, 142.16it/s]
27928it [03:16, 142.25it/s]
27943it [03:16, 142.33it/s]
27958it [03:16, 142.21it/s]
27973it [03:16, 141.44it/s]
27988it [03:16, 142.06it/s]
28003it [03:16, 141.91it/s]
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28033it [03:17, 142.18it/s]
28048it [03:17, 141.61it/s]
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28393it [03:19, 141.94it/s]
28408it [03:19, 142.11it/s]
28423it [03:19, 142.01it/s]
28438it [03:20, 142.29it/s]
28453it [03:20, 142.22it/s]
28468it [03:20, 142.34it/s]
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28513it [03:20, 141.87it/s]
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28618it [03:21, 142.10it/s]
28633it [03:21, 142.13it/s]
28648it [03:21, 142.11it/s]
28663it [03:21, 141.99it/s]
28678it [03:21, 142.22it/s]
28693it [03:21, 141.66it/s]
28708it [03:21, 142.08it/s]
28723it [03:22, 142.31it/s]
28738it [03:22, 142.34it/s]
28753it [03:22, 142.23it/s]
28768it [03:22, 142.15it/s]
28783it [03:22, 141.88it/s]
28798it [03:22, 142.02it/s]
28813it [03:22, 141.96it/s]
28828it [03:22, 142.02it/s]
28843it [03:22, 142.00it/s]
28858it [03:22, 142.13it/s]
28873it [03:23, 141.96it/s]
28888it [03:23, 142.13it/s]
28903it [03:23, 142.04it/s]
28918it [03:23, 142.02it/s]
28933it [03:23, 141.89it/s]
28948it [03:23, 141.98it/s]
28963it [03:23, 141.23it/s]
28978it [03:23, 142.08it/s]
28993it [03:23, 142.13it/s]
29008it [03:24, 142.16it/s]
29023it [03:24, 142.09it/s]
29038it [03:24, 142.13it/s]
29053it [03:24, 141.82it/s]
29068it [03:24, 142.00it/s]
29083it [03:24, 142.11it/s]
29098it [03:24, 142.22it/s]
29113it [03:24, 142.05it/s]
29128it [03:24, 142.11it/s]
29143it [03:25, 142.07it/s]
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29188it [03:25, 142.37it/s]
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29233it [03:25, 141.89it/s]
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29293it [03:26, 142.19it/s]
29308it [03:26, 142.12it/s]
29323it [03:26, 142.18it/s]
29338it [03:26, 142.38it/s]
29353it [03:26, 142.41it/s]
29368it [03:26, 142.41it/s]
29383it [03:26, 142.23it/s]
29398it [03:26, 142.30it/s]
29413it [03:26, 141.94it/s]
29428it [03:27, 142.09it/s]
29443it [03:27, 142.24it/s]
29458it [03:27, 142.12it/s]
29473it [03:27, 142.14it/s]
29488it [03:27, 142.13it/s]
29503it [03:27, 141.82it/s]
29518it [03:27, 142.22it/s]
29533it [03:27, 142.16it/s]
29548it [03:27, 142.31it/s]
29563it [03:27, 142.00it/s]
29578it [03:28, 142.02it/s]
29593it [03:28, 141.82it/s]
29608it [03:28, 141.84it/s]
29623it [03:28, 142.33it/s]
29638it [03:28, 142.32it/s]
29653it [03:28, 142.13it/s]
29668it [03:28, 142.13it/s]
29683it [03:28, 141.84it/s]
29698it [03:28, 142.17it/s]
29713it [03:29, 142.04it/s]
29728it [03:29, 142.20it/s]
29743it [03:29, 142.11it/s]
29758it [03:29, 142.23it/s]
29773it [03:29, 141.91it/s]
29788it [03:29, 142.03it/s]
29803it [03:29, 142.13it/s]
29818it [03:29, 142.21it/s]
29833it [03:29, 142.10it/s]
29848it [03:29, 142.14it/s]
29863it [03:30, 141.96it/s]
29878it [03:30, 142.14it/s]
29893it [03:30, 142.06it/s]
29908it [03:30, 142.18it/s]
29923it [03:30, 141.96it/s]
29938it [03:30, 142.25it/s]
29953it [03:30, 141.99it/s]
29968it [03:30, 142.07it/s]
29983it [03:30, 142.09it/s]
29998it [03:31, 142.00it/s]
30013it [03:31, 142.08it/s]
30028it [03:31, 142.12it/s]
30043it [03:31, 141.89it/s]
30058it [03:31, 142.09it/s]
30073it [03:31, 141.97it/s]
30088it [03:31, 142.13it/s]
30103it [03:31, 141.86it/s]
30118it [03:31, 141.68it/s]
30133it [03:31, 141.68it/s]
30148it [03:32, 141.87it/s]
30163it [03:32, 142.01it/s]
30178it [03:32, 142.11it/s]
30193it [03:32, 142.16it/s]
30208it [03:32, 141.74it/s]
30223it [03:32, 141.86it/s]
30238it [03:32, 142.04it/s]
30253it [03:32, 141.87it/s]
30268it [03:32, 142.12it/s]
30283it [03:33, 141.87it/s]
30298it [03:33, 141.85it/s]
30313it [03:33, 142.32it/s]
30328it [03:33, 142.11it/s]
30343it [03:33, 142.24it/s]
30358it [03:33, 142.24it/s]
30373it [03:33, 141.98it/s]
30388it [03:33, 141.47it/s]
30403it [03:33, 142.16it/s]
30418it [03:33, 142.11it/s]
30433it [03:34, 141.95it/s]
30448it [03:34, 142.11it/s]
30463it [03:34, 141.93it/s]
30478it [03:34, 141.86it/s]
30493it [03:34, 141.92it/s]
30508it [03:34, 141.91it/s]
30508it [03:34, 142.15it/s]

EPOCH: 9 | LOSS: 7.9989

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'temporary', 'questions']

Beginning epoch: 10/30


0it [00:00, ?it/s]
17it [00:00, 166.73it/s]
34it [00:00, 151.58it/s]
50it [00:00, 146.98it/s]
65it [00:00, 145.36it/s]
80it [00:00, 144.54it/s]
95it [00:00, 143.66it/s]
110it [00:00, 143.23it/s]
125it [00:00, 142.68it/s]
140it [00:00, 142.66it/s]
155it [00:01, 142.62it/s]
170it [00:01, 142.60it/s]
185it [00:01, 142.56it/s]
200it [00:01, 142.30it/s]
215it [00:01, 142.02it/s]
230it [00:01, 142.19it/s]
245it [00:01, 142.05it/s]
260it [00:01, 142.32it/s]
275it [00:01, 142.42it/s]
290it [00:02, 142.32it/s]
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30485it [03:33, 142.98it/s]
30500it [03:33, 143.01it/s]
30508it [03:33, 142.82it/s]

EPOCH: 10 | LOSS: 7.9710

Learned embeddings:
word: "tyranny" neighbors: ['temporary', 'tendency', 'perceived']

Beginning epoch: 11/30


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29825it [03:29, 142.63it/s]
29840it [03:29, 142.87it/s]
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29870it [03:29, 142.61it/s]
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29900it [03:29, 142.55it/s]
29915it [03:29, 142.57it/s]
29930it [03:29, 142.71it/s]
29945it [03:29, 142.46it/s]
29960it [03:30, 142.55it/s]
29975it [03:30, 142.52it/s]
29990it [03:30, 142.69it/s]
30005it [03:30, 142.73it/s]
30020it [03:30, 142.70it/s]
30035it [03:30, 142.85it/s]
30050it [03:30, 142.70it/s]
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30080it [03:30, 143.02it/s]
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30350it [03:32, 142.60it/s]
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30395it [03:33, 142.76it/s]
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30440it [03:33, 142.65it/s]
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30485it [03:33, 142.89it/s]
30500it [03:33, 142.92it/s]
30508it [03:33, 142.66it/s]

EPOCH: 11 | LOSS: 7.9501

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'perceived', 'temporary']

Beginning epoch: 12/30


0it [00:00, ?it/s]
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34it [00:00, 153.29it/s]
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1955it [00:13, 142.69it/s]
1970it [00:13, 142.79it/s]
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2015it [00:14, 142.90it/s]
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30508it [03:33, 142.90it/s]

EPOCH: 12 | LOSS: 7.9301

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'temporary', 'perceived']

Beginning epoch: 13/30


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25526it [02:55, 146.14it/s]
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25826it [02:57, 146.22it/s]
25841it [02:57, 146.34it/s]
25856it [02:57, 146.18it/s]
25871it [02:57, 146.25it/s]
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25901it [02:57, 146.12it/s]
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29711it [03:23, 146.02it/s]
29726it [03:23, 145.80it/s]
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29786it [03:24, 146.05it/s]
29801it [03:24, 146.13it/s]
29816it [03:24, 146.10it/s]
29831it [03:24, 146.00it/s]
29846it [03:24, 146.07it/s]
29861it [03:24, 146.05it/s]
29876it [03:24, 146.26it/s]
29891it [03:25, 146.19it/s]
29906it [03:25, 146.24it/s]
29921it [03:25, 146.06it/s]
29936it [03:25, 146.30it/s]
29951it [03:25, 146.15it/s]
29966it [03:25, 146.15it/s]
29981it [03:25, 146.18it/s]
29996it [03:25, 146.01it/s]
30011it [03:25, 146.09it/s]
30026it [03:26, 146.10it/s]
30041it [03:26, 146.01it/s]
30056it [03:26, 146.08it/s]
30071it [03:26, 146.15it/s]
30086it [03:26, 146.10it/s]
30101it [03:26, 146.18it/s]
30116it [03:26, 146.17it/s]
30131it [03:26, 146.10it/s]
30146it [03:26, 146.07it/s]
30161it [03:26, 146.18it/s]
30176it [03:27, 146.17it/s]
30191it [03:27, 146.08it/s]
30206it [03:27, 146.08it/s]
30221it [03:27, 146.12it/s]
30236it [03:27, 145.97it/s]
30251it [03:27, 146.00it/s]
30266it [03:27, 146.16it/s]
30281it [03:27, 146.22it/s]
30296it [03:27, 146.22it/s]
30311it [03:27, 146.17it/s]
30326it [03:28, 146.19it/s]
30341it [03:28, 146.12it/s]
30356it [03:28, 146.01it/s]
30371it [03:28, 146.03it/s]
30386it [03:28, 146.16it/s]
30401it [03:28, 146.13it/s]
30416it [03:28, 146.11it/s]
30431it [03:28, 146.19it/s]
30446it [03:28, 146.12it/s]
30461it [03:28, 146.01it/s]
30476it [03:29, 146.19it/s]
30491it [03:29, 146.04it/s]
30506it [03:29, 146.17it/s]
30508it [03:29, 145.75it/s]

EPOCH: 13 | LOSS: 7.9133

Learned embeddings:
word: "tyranny" neighbors: ['tendency', 'assembly', 'monarchy']

Beginning epoch: 14/30


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EPOCH: 14 | LOSS: 7.9004

Learned embeddings:
word: "tyranny" neighbors: ['monarchy', 'perceived', 'temporary']

Beginning epoch: 15/30


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EPOCH: 15 | LOSS: 7.8864

Learned embeddings:
word: "tyranny" neighbors: ['questions', 'monarchy', 'temporary']

Beginning epoch: 16/30


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EPOCH: 16 | LOSS: 7.8762

Learned embeddings:
word: "tyranny" neighbors: ['monarchy', 'temporary', 'perceived']

Beginning epoch: 17/30


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30443it [03:28, 146.20it/s]
30458it [03:28, 146.26it/s]
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30488it [03:28, 146.38it/s]
30503it [03:28, 146.30it/s]
30508it [03:28, 146.30it/s]

EPOCH: 17 | LOSS: 7.8665

Learned embeddings:
word: "tyranny" neighbors: ['monarchy', 'perceived', 'assembly']

Beginning epoch: 18/30


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EPOCH: 18 | LOSS: 7.8526

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'monarchy', 'temporary']

Beginning epoch: 19/30


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29866it [03:28, 142.87it/s]
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29911it [03:28, 142.93it/s]
29926it [03:28, 142.95it/s]
29941it [03:28, 143.09it/s]
29956it [03:28, 142.97it/s]
29971it [03:28, 143.09it/s]
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30001it [03:28, 142.67it/s]
30016it [03:29, 142.86it/s]
30031it [03:29, 142.87it/s]
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30481it [03:32, 142.82it/s]
30496it [03:32, 142.71it/s]
30508it [03:32, 143.55it/s]

EPOCH: 19 | LOSS: 7.8435

Learned embeddings:
word: "tyranny" neighbors: ['monarchy', 'perceived', 'temporary']

Beginning epoch: 20/30


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30485it [03:33, 142.91it/s]
30500it [03:33, 142.68it/s]
30508it [03:33, 142.67it/s]

EPOCH: 20 | LOSS: 7.8391

Learned embeddings:
word: "tyranny" neighbors: ['monarchy', 'perceived', 'temporary']

Beginning epoch: 21/30


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1955it [00:13, 142.71it/s]
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29900it [03:29, 142.77it/s]
29915it [03:29, 142.72it/s]
29930it [03:29, 142.82it/s]
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29960it [03:29, 142.88it/s]
29975it [03:29, 142.71it/s]
29990it [03:29, 142.75it/s]
30005it [03:30, 142.77it/s]
30020it [03:30, 142.82it/s]
30035it [03:30, 142.81it/s]
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30350it [03:32, 142.87it/s]
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30395it [03:32, 142.77it/s]
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30440it [03:33, 142.65it/s]
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30485it [03:33, 142.72it/s]
30500it [03:33, 142.88it/s]
30508it [03:33, 142.81it/s]

EPOCH: 21 | LOSS: 7.8320

Learned embeddings:
word: "tyranny" neighbors: ['monarchy', 'perceived', 'temporary']

Beginning epoch: 22/30


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30508it [03:34, 141.96it/s]

EPOCH: 22 | LOSS: 7.8214

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'temporary', 'monarchy']

Beginning epoch: 23/30


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30508it [03:32, 143.88it/s]

EPOCH: 23 | LOSS: 7.8163

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'represented', 'monarchy']

Beginning epoch: 24/30


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EPOCH: 24 | LOSS: 7.8129

Learned embeddings:
word: "tyranny" neighbors: ['represented', 'temporary', 'monarchy']

Beginning epoch: 25/30


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30508it [03:32, 143.73it/s]

EPOCH: 25 | LOSS: 7.8041

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'tendency', 'monarchy']

Beginning epoch: 26/30


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EPOCH: 26 | LOSS: 7.8054

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'monarchy', 'represented']

Beginning epoch: 27/30


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30508it [03:33, 142.89it/s]

EPOCH: 27 | LOSS: 7.7946

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'tendency', 'monarchy']

Beginning epoch: 28/30


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30508it [03:28, 146.20it/s]

EPOCH: 28 | LOSS: 7.7901

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'monarchy', 'tendency']

Beginning epoch: 29/30


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EPOCH: 29 | LOSS: 7.7878

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'tendency', 'monarchy']

Beginning epoch: 30/30


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EPOCH: 30 | LOSS: 7.7858

Learned embeddings:
word: "tyranny" neighbors: ['perceived', 'monarchy', 'tendency']

Inspect learned embeddings

Let’s print out a few neighbor tokens to a few selected tokens and check if these make any sense.

Looking at the results of the nearest neighbors we can see that there is some sense to the learned vector embeddings. The nearest neighbors to the state of Pennsylvania are other states, the nearest neigbor to nobles are the church, and that of war is criminal.

nearest_neighbors("pennsylvania", trained_params, 5)

['connecticut', 'georgia', 'rhode', 'civilization', '1832']

nearest_neighbors("nobles", trained_params, 5)

['church', 'orsini', 'traditions', 'decrees', 'whites']

nearest_neighbors("war", trained_params, 5)

['criminal', 'alexander', 'civilization', 'partial', 'blood']

Conclusion

In this short post we did a deep dive on the Skip-Gram with Negative sampling algorithm. Specifically, we focused on how training examples are generated, how the loss function is defined, and putting it all together manually using open-source scraped data with JAX.