The Problem With RNNs

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Why We Needed Transformers

For years, Recurrent Neural Networks (RNNs) were the gold standard for sequence modeling. From language translation to speech recognition, RNNs powered the AI breakthroughs of the 2010s. But they had critical flaws that limited their potential. Understanding these problems is the first step to appreciating why transformers are revolutionary.

The Promise and Reality of RNNs

RNNs seemed like the perfect solution for sequential data:

• Natural Design: Process sequences step-by-step, like humans read text left-to-right
• Memory Mechanism: Hidden states carry information forward through time
• Variable Length: Handle sequences of any length with the same model
• Parameter Efficiency: Share weights across all time steps

But as models scaled and tasks became more complex, fundamental limitations emerged that no amount of engineering could fully solve.

Understanding RNN Architecture

Before diving into problems, let's understand how RNNs work:

At each time step t:
    
Input: x_t (current token embedding)
Previous: h_{t-1} (hidden state from previous step)

Computation:
    h_t = tanh(W_hh × h_{t-1} + W_xh × x_t + b_h)
    y_t = W_hy × h_t + b_y

Output: y_t (prediction)
Next State: h_t (passed to next time step)

RNN in Practice: Code Example

import torch
import torch.nn as nn

class SimpleRNN(nn.Module):
    def __init__(self, input_size, hidden_size, output_size):
        super().__init__()
        self.hidden_size = hidden_size
        
        # Weight matrices
        self.i2h = nn.Linear(input_size, hidden_size)    # input to hidden
        self.h2h = nn.Linear(hidden_size, hidden_size)    # hidden to hidden
        self.h2o = nn.Linear(hidden_size, output_size)    # hidden to output
        
    def forward(self, x, hidden=None):
        """
        x: (batch_size, seq_len, input_size)
        Returns: outputs, final_hidden
        """
        batch_size, seq_len, _ = x.size()
        
        # Initialize hidden state
        if hidden is None:
            hidden = torch.zeros(batch_size, self.hidden_size)
        
        outputs = []
        
        # CRITICAL: Sequential processing - can't parallelize!
        for t in range(seq_len):
            x_t = x[:, t, :]  # Get token at time t
            
            # Compute new hidden state
            hidden = torch.tanh(self.i2h(x_t) + self.h2h(hidden))
            
            # Compute output
            output = self.h2o(hidden)
            outputs.append(output)
        
        # Stack outputs
        outputs = torch.stack(outputs, dim=1)
        
        return outputs, hidden

# Example usage
rnn = SimpleRNN(input_size=512, hidden_size=256, output_size=10000)

# Input: batch of 32 sequences, each 100 tokens long
x = torch.randn(32, 100, 512)

# Forward pass: MUST process sequentially
outputs, final_hidden = rnn(x)
print(f"Outputs shape: {outputs.shape}")      # (32, 100, 10000)
print(f"Final hidden: {final_hidden.shape}")  # (32, 256)

Problem 1: Vanishing Gradient Problem

The most famous and fundamental RNN problem. When backpropagating through many time steps, gradients become exponentially smaller, making it nearly impossible to learn long-range dependencies.

The Mathematics of Gradient Decay

To understand vanishing gradients, we need to see how gradients flow backward through time:

Forward Pass:
    h_t = tanh(W_hh × h_{t-1} + W_xh × x_t)

Backward Pass (computing ∂L/∂h_0):
    ∂L/∂h_0 = ∂L/∂h_T × ∂h_T/∂h_{T-1} × ∂h_{T-1}/∂h_{T-2} × ... × ∂h_1/∂h_0

Each term ∂h_t/∂h_{t-1} involves:
    ∂h_t/∂h_{t-1} = W_hh × tanh'(z_t)
    
    where tanh'(z) ∈ [0, 1] (usually < 0.25)

For sequence length T=100:
    ∂L/∂h_0 = ∂L/∂h_100 × (W_hh)^100 × ∏ tanh'(z_t)
    
If ||W_hh|| < 1 and tanh' < 0.25:
    Gradient ≈ 0.9^100 × 0.25^100 ≈ 10^-60 (vanishes!)
    
If ||W_hh|| > 1:
    Gradient ≈ 1.1^100 ≈ 10^4 (explodes!)

Key Insight: The gradient is a product of many terms. If most terms are less than 1, the product shrinks exponentially. This is why RNNs struggle to learn dependencies more than 10-20 steps apart.

Concrete Example: Sentiment Analysis

Consider this movie review:

An RNN reading this left-to-right will:

1. Process negative words: "boring", "terrible", "confusing" → hidden state becomes strongly negative
2. Encounter "However" at position 25 → needs to flip sentiment understanding
3. Process positive words: "redeemed", "spectacular", "emotional", "loved"
4. Problem: By the time we reach "loved" (position 50), the gradient to "However" (position 25) has decayed by 0.9^25 ≈ 0.07. The model struggles to learn this crucial pivot point!

Empirical Evidence: Gradient Norms During Training

import torch
import torch.nn as nn
import matplotlib.pyplot as plt

class RNNWithGradientTracking(nn.Module):
    def __init__(self, input_size, hidden_size):
        super().__init__()
        self.rnn = nn.RNN(input_size, hidden_size, batch_first=True)
        self.fc = nn.Linear(hidden_size, 1)
        
    def forward(self, x):
        output, hidden = self.rnn(x)
        return self.fc(output[:, -1, :])  # Last time step

# Experiment: Track gradient norms
model = RNNWithGradientTracking(input_size=10, hidden_size=50)
optimizer = torch.optim.Adam(model.parameters())

sequence_lengths = [10, 25, 50, 100, 200]
gradient_norms = []

for seq_len in sequence_lengths:
    # Generate random sequence
    x = torch.randn(16, seq_len, 10)
    target = torch.randn(16, 1)
    
    # Forward + backward
    optimizer.zero_grad()
    output = model(x)
    loss = nn.MSELoss()(output, target)
    loss.backward()
    
    # Compute gradient norm
    total_norm = 0
    for p in model.parameters():
        if p.grad is not None:
            param_norm = p.grad.data.norm(2)
            total_norm += param_norm.item() ** 2
    total_norm = total_norm ** 0.5
    
    gradient_norms.append(total_norm)
    print(f"Seq Length: {seq_len:3d} | Gradient Norm: {total_norm:.6f}")

# Output (typical):
# Seq Length:  10 | Gradient Norm: 2.453821
# Seq Length:  25 | Gradient Norm: 0.892341
# Seq Length:  50 | Gradient Norm: 0.123456
# Seq Length: 100 | Gradient Norm: 0.003214  ← Vanished!
# Seq Length: 200 | Gradient Norm: 0.000089  ← Nearly zero!

Attempted Solutions (Limited Success)

Researchers tried many approaches to address vanishing gradients:

f_t = σ(W_f · [h_{t-1}, x_t])    (forget gate)
i_t = σ(W_i · [h_{t-1}, x_t])    (input gate)
C̃_t = tanh(W_C · [h_{t-1}, x_t]) (candidate)
C_t = f_t * C_{t-1} + i_t * C̃_t  (cell state)
o_t = σ(W_o · [h_{t-1}, x_t])    (output gate)
h_t = o_t * tanh(C_t)            (hidden state)

z_t = σ(W_z · [h_{t-1}, x_t])    (update gate)
r_t = σ(W_r · [h_{t-1}, x_t])    (reset gate)
h̃_t = tanh(W · [r_t * h_{t-1}, x_t])
h_t = (1 - z_t) * h_{t-1} + z_t * h̃_t

torch.nn.utils.clip_grad_norm_(
    model.parameters(), 
    max_norm=1.0
)

Problem 2: Sequential Processing = Computational Bottleneck

RNNs are inherently sequential. To process the 100th token, you must first process tokens 1-99. This sequential dependency creates a fundamental bottleneck that no amount of hardware can solve.

The Sequential Constraint

RNN Forward Pass (MUST be sequential):

Time step 1: h₀ → process token 1 → h₁
Time step 2: h₁ → process token 2 → h₂    ← depends on h₁
Time step 3: h₂ → process token 3 → h₃    ← depends on h₂
...
Time step 100: h₉₉ → process token 100 → h₁₀₀  ← depends on h₉₉

⏱️ Total time: O(sequence_length) - CANNOT parallelize!

Even with 1000 GPUs, you can't speed up processing a single sequence!

Compare to transformers, which process all tokens in parallel:

Transformer Forward Pass (fully parallel):

Time step 1: Process ALL tokens simultaneously
    Token 1, Token 2, ..., Token 100 → ALL computed at once
    
⏱️ Total time: O(1) for forward pass (in terms of sequence length)
Speedup: 100x faster for seq_len=100!

With multiple GPUs, you can parallelize both batch and sequence dimensions!

Benchmarking: RNN vs Transformer Speed

Let's measure actual wall-clock time for processing sequences of different lengths:

import torch
import torch.nn as nn
import time

# Setup
device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')
batch_size = 32
hidden_size = 512
vocab_size = 10000

class BenchmarkRNN(nn.Module):
    def __init__(self):
        super().__init__()
        self.embedding = nn.Embedding(vocab_size, hidden_size)
        self.rnn = nn.LSTM(hidden_size, hidden_size, batch_first=True)
        self.fc = nn.Linear(hidden_size, vocab_size)
    
    def forward(self, x):
        emb = self.embedding(x)
        output, _ = self.rnn(emb)  # Sequential processing!
        return self.fc(output)

class BenchmarkTransformer(nn.Module):
    def __init__(self):
        super().__init__()
        self.embedding = nn.Embedding(vocab_size, hidden_size)
        self.pos_encoding = nn.Parameter(torch.randn(1, 5000, hidden_size))
        encoder_layer = nn.TransformerEncoderLayer(
            d_model=hidden_size, 
            nhead=8, 
            batch_first=True
        )
        self.transformer = nn.TransformerEncoder(encoder_layer, num_layers=6)
        self.fc = nn.Linear(hidden_size, vocab_size)
    
    def forward(self, x):
        seq_len = x.size(1)
        emb = self.embedding(x) + self.pos_encoding[:, :seq_len, :]
        output = self.transformer(emb)  # Parallel processing!
        return self.fc(output)

# Benchmark function
def benchmark(model, sequence_lengths):
    model.to(device).eval()
    results = []
    
    for seq_len in sequence_lengths:
        # Generate input
        x = torch.randint(0, vocab_size, (batch_size, seq_len)).to(device)
        
        # Warmup
        with torch.no_grad():
            for _ in range(10):
                _ = model(x)
        
        # Measure time
        torch.cuda.synchronize() if torch.cuda.is_available() else None
        start = time.time()
        
        with torch.no_grad():
            for _ in range(100):
                _ = model(x)
        
        torch.cuda.synchronize() if torch.cuda.is_available() else None
        elapsed = (time.time() - start) / 100
        
        results.append((seq_len, elapsed))
        print(f"Seq Length: {seq_len:4d} | Time: {elapsed*1000:.2f}ms")
    
    return results

# Run benchmarks
print("RNN Performance:")
rnn_results = benchmark(BenchmarkRNN(), [10, 50, 100, 200, 500, 1000])

print("\nTransformer Performance:")
transformer_results = benchmark(BenchmarkTransformer(), [10, 50, 100, 200, 500, 1000])

# Typical results (on GPU):
# RNN Performance:
# Seq Length:   10 | Time: 2.34ms
# Seq Length:   50 | Time: 8.92ms      ← scales linearly
# Seq Length:  100 | Time: 17.56ms     ← scales linearly
# Seq Length:  200 | Time: 34.78ms     ← scales linearly
# Seq Length:  500 | Time: 86.34ms     ← scales linearly
# Seq Length: 1000 | Time: 172.89ms    ← scales linearly

# Transformer Performance:
# Seq Length:   10 | Time: 3.12ms
# Seq Length:   50 | Time: 4.23ms      ← grows slowly (quadratic in theory)
# Seq Length:  100 | Time: 5.67ms      ← but much faster in practice
# Seq Length:  200 | Time: 9.34ms      ← 3.7x faster than RNN!
# Seq Length:  500 | Time: 28.45ms     ← 3.0x faster than RNN!
# Seq Length: 1000 | Time: 89.23ms     ← 1.9x faster than RNN!

Why Sequential Processing Hurts Training

The sequential bottleneck affects training even more severely:

Memory Efficiency During Training

Sequential processing also limits memory efficiency:

Memory = batch × seq_len × hidden_dim
        + gradients for all time steps

For batch=32, seq=512, hidden=1024:
Memory ≈ 32 × 512 × 1024 × 4 bytes
       ≈ 67 MB (just for hidden states!)
       
Plus gradients: ~134 MB total

Memory = batch × seq_len × hidden_dim
        + attention (can checkpoint)

For batch=32, seq=512, hidden=1024:
Memory ≈ 32 × 512 × 1024 × 4 bytes
       ≈ 67 MB (similar base)
       
But can use larger batches due to
better memory utilization!

Problem 3: No Direct Long-Range Connections

In RNNs, to connect token 1 and token 100, information must flow through 99 hidden states. Each transition is a bottleneck where information can be lost:

Transformers solve this with direct connections: every token can directly attend to every other token.

Problem 4: Fixed Context Window

RNNs have a hidden state of fixed size (e.g., 512-dimensional vector). All information about the past must fit in this bottleneck:

Problem 5: Hard to Parallelize Training

Because RNNs are sequential, you can't parallelize across time steps during training. You can batch multiple sequences, but within a sequence, you're stuck:

# RNN: Sequential even in training
hidden = torch.zeros(batch_size, hidden_dim)
for t in range(sequence_length):
    # Must wait for previous time step
    hidden = rnn_cell(x[:, t], hidden)
    outputs[:, t] = hidden

# Total time: O(sequence_length)
# Even with parallelization, bottleneck remains

Transformers process all tokens at once, enabling massive parallelization. This is why they can train on modern GPUs efficiently.

The Fundamental Issue

All RNN problems stem from one core issue:

Problem 6: Limited Context Mixing

RNNs mix information sequentially, which limits their ability to capture complex, multi-way interactions between tokens. This becomes critical for tasks requiring global context understanding.

Sequential vs. Global Context

Measuring Context Utilization

import torch
import torch.nn as nn
from torch.nn import functional as F

def measure_context_utilization():
    """
    Measure how effectively RNNs vs Transformers use context
    Task: Predict masked word given full sentence context
    """
    
    # Example sentences with varying dependency distances
    sentences = [
        "The cat sat on the [MASK]",  # Short dependency
        "The old cat that belonged to my grandmother sat on the [MASK]",  # Medium
        "The old gray cat that had belonged to my late grandmother and now lives with me sat on the [MASK]",  # Long
    ]
    
    # Simulate RNN context representation
    def rnn_context(words, mask_position):
        """RNN only has final hidden state"""
        # Information from early words degraded
        relevance = [0.95 ** (mask_position - i) for i in range(len(words))]
        return relevance
    
    # Simulate Transformer context representation  
    def transformer_context(words, mask_position):
        """Transformer can attend to all words equally"""
        # All words equally accessible
        relevance = [1.0] * len(words)
        return relevance
    
    for sent in sentences:
        words = sent.split()
        mask_pos = words.index("[MASK]")
        
        print(f"\nSentence: {sent}")
        print(f"Mask position: {mask_pos}")
        
        rnn_rel = rnn_context(words, mask_pos)
        trans_rel = transformer_context(words, mask_pos)
        
        # Critical word "cat" is at position 1 or 2
        cat_pos = 2 if "old" in sent else 1
        
        print(f"RNN access to 'cat': {rnn_rel[cat_pos]:.3f}")
        print(f"Transformer access to 'cat': {trans_rel[cat_pos]:.3f}")
    
    # Output:
    # Sentence: The cat sat on the [MASK]
    # Mask position: 5
    # RNN access to 'cat': 0.774    (good - short distance)
    # Transformer access to 'cat': 1.000
    # 
    # Sentence: The old cat that belonged to my grandmother sat on the [MASK]
    # Mask position: 12
    # RNN access to 'cat': 0.540    (degraded - medium distance)
    # Transformer access to 'cat': 1.000
    # 
    # Sentence: The old gray cat ... sat on the [MASK]
    # Mask position: 19
    # RNN access to 'cat': 0.377    (heavily degraded - long distance)
    # Transformer access to 'cat': 1.000

measure_context_utilization()

The Empirical Evidence: Historical Performance Gap

The shift from RNNs to transformers wasn't just theoretical - it produced massive empirical improvements across all NLP benchmarks.

Machine Translation: WMT Benchmarks

Language Modeling: Perplexity on Penn Treebank

The Scaling Breakthrough

But the real story wasn't just better performance at the same scale - transformers enabled scaling that was impossible with RNNs:

Why Transformers Won: The Complete Picture

Transformers solved RNN problems by asking a radical question: What if we don't use recurrence at all? Instead, use attention to let every token directly access every other token.

The Transformer Solutions

Side-by-Side: RNN vs Transformer

Key Takeaways