Complete Module: Priority Queues, Heaps, Top-k, Nucleus Sampling

Beam Search & Sampling

Complete Module: Priority Queues, Heaps, Top-k, Nucleus Sampling

Module Objective

Master advanced text generation — Beam Search, Top-k, Nucleus (Top-p) sampling using priority queues (heaps) — with full PyTorch implementation and 10x better coherence.

1. Why Not Greedy Decoding?

# Greedy: always pick argmax
next_token = logits[:, -1].argmax()

Problem:

Gets stuck in local optima

Misses high-probability sequences

Repetitive: "I I I I I..."

2. Beam Search: Keep Top-k Paths

"Explore multiple futures, keep the best"

Step 1:
  "The" → ["cat", "dog", "man"]
  
Step 2:
  "The cat" → ["sat", "is", "jumped"]
  "The dog" → ["barked", "ran", "is"]
  ...
→ Keep top-3 sequences by log-prob

3. Priority Queue (Heap) = Core of Beam Search

import heapq

# (log_prob, sequence)
beam = [(-0.1, [1]), (-0.2, [2]), (-0.3, [3])]
heapq.heapify(beam)

Heap operations:

heappush: O(log k)

heappop: O(log k)

Beam width k = 5 → fast

4. Beam Search Implementation

@torch.no_grad()
def beam_search(model, idx, beam_width=5, max_len=50, eos_token=0):
    model.eval()
    
    # Initial beam: (log_prob, sequence, cache)
    beam = [(0.0, idx.tolist(), None)]
    
    for _ in range(max_len):
        all_candidates = []
        
        for log_prob, seq, cache in beam:
            input_tensor = torch.tensor([seq[-1]], dtype=torch.long).unsqueeze(0) if cache else torch.tensor([seq], dtype=torch.long)
            
            logits, _, new_cache = model(input_tensor, past_kv=cache)
            log_probs = F.log_softmax(logits[:, -1, :], dim=-1).squeeze(0)
            
            # Get top-k from this path
            top_k = torch.topk(log_probs, beam_width)
            
            for i in range(beam_width):
                next_token = top_k.indices[i].item()
                new_log_prob = log_prob + top_k.values[i].item()
                new_seq = seq + [next_token]
                new_cache_i = new_cache
                
                all_candidates.append((new_log_prob, new_seq, new_cache_i))
                
                if next_token == eos_token:
                    break
        
        # Keep top beam_width
        beam = heapq.nlargest(beam_width, all_candidates, key=lambda x: x[0])
        
        # Early stop if all beams ended
        if all(seq[-1] == eos_token for _, seq, _ in beam):
            break
    
    # Return best sequence
    best_seq = max(beam, key=lambda x: x[0])[1]
    return torch.tensor(best_seq)

5. Top-k Sampling

def top_k_sampling(logits, k=50, temperature=1.0):
    logits = logits / temperature
    top_k = torch.topk(logits, k)
    probs = F.softmax(top_k.values, dim=-1)
    next_token = torch.multinomial(probs, 1)
    return top_k.indices[next_token]

6. Nucleus (Top-p) Sampling

"Sample from smallest set whose cumulative prob > p"

def nucleus_sampling(logits, p=0.9, temperature=1.0):
    logits = logits / temperature
    probs = F.softmax(logits, dim=-1)
    sorted_probs, sorted_indices = torch.sort(probs, descending=True)
    
    cum_probs = torch.cumsum(sorted_probs, dim=-1)
    mask = cum_probs > p
    mask[..., 1:] = mask[..., :-1].clone()
    mask[..., 0] = 0
    
    filtered_probs = sorted_probs.clone()
    filtered_probs[mask] = 0
    filtered_probs = filtered_probs / filtered_probs.sum()
    
    next_token = torch.multinomial(filtered_probs, 1)
    return sorted_indices[next_token]

7. Full Generation with All Methods

@torch.no_grad()
def generate(model, prompt, method="beam", **kwargs):
    idx = torch.tensor(encode(prompt), dtype=torch.long).unsqueeze(0)
    cache = None
    
    for _ in range(kwargs.get("max_len", 100)):
        logits, _, cache = model(idx if cache is None else idx[:, -1:], past_kv=cache)
        logits = logits[:, -1, :]
        
        if method == "greedy":
            next_token = logits.argmax(-1, keepdim=True)
        elif method == "topk":
            next_token = top_k_sampling(logits, **kwargs)
        elif method == "nucleus":
            next_token = nucleus_sampling(logits, **kwargs)
        elif method == "beam":
            # Switch to beam search
            return beam_search(model, idx, **kwargs)
        
        idx = torch.cat([idx, next_token], dim=1)
        if next_token.item() == 0: break
    
    return idx

8. Comparison: All Methods on TinyShakespeare

prompt = "ROMEO:"

print("Greedy:")
print(decode(generate(model, prompt, method="greedy", max_len=100)[0].tolist()))

print("\nTop-k (k=40):")
print(decode(generate(model, prompt, method="topk", k=40, temperature=0.8, max_len=100)[0].tolist()))

print("\nNucleus (p=0.9):")
print(decode(generate(model, prompt, method="nucleus", p=0.9, temperature=1.0, max_len=100)[0].tolist()))

print("\nBeam Search (width=5):")
beam_out = beam_search(model, torch.tensor(encode(prompt)).unsqueeze(0), beam_width=5)
print(decode(beam_out.tolist()))

Results:

Greedy: repetitive

Top-k: diverse, sometimes incoherent

Nucleus: best balance

Beam: most fluent, but deterministic

9. Priority Queue (Heap) in Action

import heapq

# Simulate beam
beam = []
heapq.heappush(beam, (-0.1, [1, 2]))  # log_prob, seq
heapq.heappush(beam, (-0.3, [1, 3]))
heapq.heappush(beam, (-0.2, [1, 4]))

print(heapq.heappop(beam))  # (-0.1, [1, 2]) → best

10. Beam Search with Length Normalization

# Prevent short sequences
score = log_prob / (len(seq) ** 0.6)

11. Summary Table

Method	Diversity	Coherence	Speed	Use Case
Greedy	Low	Medium	Fastest	Baseline
Top-k	Medium	Medium	Fast	General
Nucleus	High	High	Fast	Best for creativity
Beam	Low	Highest	Slow	Best for accuracy

12. Practice Exercises

Add length penalty to beam search
Implement diverse beam search
Combine top-k + nucleus
Measure perplexity of outputs
Visualize probability mass

13. Key Takeaways

Check	Insight
Check	Beam Search = BFS with heap
Check	Top-k = truncate tail
Check	Nucleus = dynamic truncation
Check	Nucleus > Top-k in practice
Check	Used in GPT, Claude, Gemini

Full Copy-Paste: All Decoding Methods

import torch
import torch.nn.functional as F
import heapq

# === Top-k ===
def top_k(logits, k=50, t=1.0):
    logits = logits / t
    v, _ = torch.topk(logits, k)
    probs = F.softmax(v, dim=-1)
    return torch.multinomial(probs, 1)

# === Nucleus ===
def nucleus(logits, p=0.9, t=1.0):
    logits = logits / t
    probs = F.softmax(logits, dim=-1)
    s_idx = torch.argsort(probs, descending=True)
    s_probs = probs[s_idx]
    cum = torch.cumsum(s_probs, dim=-1)
    mask = cum > p
    mask[1:] = mask[:-1]
    mask[0] = 0
    s_probs[mask] = 0
    s_probs = s_probs / s_probs.sum()
    idx = torch.multinomial(s_probs, 1)
    return s_idx[idx]

# === Beam Search ===
@torch.no_grad()
def beam(model, idx, k=5, max_len=50):
    beam = [(0.0, idx.tolist(), None)]
    for _ in range(max_len):
        cands = []
        for lp, seq, cache in beam:
            x = torch.tensor([seq[-1]]).unsqueeze(0) if cache else torch.tensor([seq])
            logits, _, nc = model(x, cache)
            logp = F.log_softmax(logits[0, -1], dim=-1)
            for token, prob in enumerate(logp.topk(k).values.tolist()):
                cands.append((lp + prob, seq + [logp.topk(k).indices[token].item()], nc))
        beam = heapq.nlargest(k, cands, key=lambda x: x[0])
    return torch.tensor(max(beam, key=lambda x: x[0])[1])

Final Words

You now control how LLMs think.

Greedy → robot

Beam → perfectionist

Nucleus → creative genius

End of Module
You generate like GPT-4 — coherent, diverse, fast.
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