Contrastive Learning - Learning Module

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Positive and Negative Pairs: The Foundation of Contrastive Learning

The Fundamental Unit of Contrastive Learning

At its core, contrastive learning is about comparison. We learn representations by contrasting similar things (positive pairs) against dissimilar things (negative pairs). This simple principle—learning what goes together and what doesn't—underlies all contrastive methods.

But this simplicity hides deep questions: What makes a good positive pair? How do we sample effective negatives? What happens when our assumptions about positives and negatives are violated? This page explores these foundational concepts that determine contrastive learning success or failure.

What You Will Learn

By the end of this page, you will understand: (1) How positive pairs encode invariances we want to learn, (2) The spectrum of negative sampling strategies, (3) The false negative problem and its solutions, (4) Hard negative mining and its tradeoffs, and (5) How pair selection affects learned representations.

Positive Pairs: Defining What Should Be Invariant

A positive pair consists of two inputs that should have similar representations. The choice of positive pairs implicitly defines what the representation should be invariant to.

The Invariance Principle

When we train a model to give similar representations to $(x, x^+)$, we're saying: "Whatever differs between $x$ and $x^+$ is not important for representation."

For augmentation-based positives:

$x$ = original image
$x^+$ = augmented version (cropped, color-shifted, etc.)

The representation becomes invariant to the augmentation transformations.

Sources of Positive Pairs

Positive Pair Construction Strategies
Strategy	How Positives Are Created	Invariance Learned	Example Methods
Data Augmentation	Apply random transforms to same image	Color, scale, crop, orientation	SimCLR, MoCo, BYOL
Temporal Adjacency	Nearby frames in video	Short-term motion, viewpoint	CPC, VideoMoCo
Multi-view	Different camera angles of same scene	Viewpoint, lighting	Multi-view learning
Cross-modal	Different modalities of same content	Modality-specific noise	CLIP (image-text)
Supervised	Same-class samples as positives	Intra-class variation	Supervised Contrastive

The Semantic Hypothesis

Augmentation-based positives rely on a critical assumption: semantic content is preserved under augmentations. When we crop an image of a dog, it's still a dog. When we change the colors, it's still the same dog.

This assumption can break in subtle ways:

Problematic cases:

Random crop captures only background → semantic content lost
Grayscale on color-diagnostic objects (e.g., traffic lights) → removes key information
Aggressive rotation on orientation-sensitive objects → changes meaning

The art of contrastive learning is designing augmentations that:

Create sufficient variation for learning
Preserve semantics relevant for downstream tasks

Augmentation-Task Alignment

The invariances you encode through augmentation should match your downstream task. If you make representations invariant to color (via color jitter), they won't work well for tasks where color matters (e.g., distinguishing ripe vs unripe fruit). Design augmentations with your end task in mind.

Negative Pairs: Preventing Representation Collapse

Negative pairs are samples that should have dissimilar representations. Without negatives, the trivial solution is to map everything to the same point—perfect positive alignment but useless representations.

The Role of Negatives

Negatives serve as a repulsive force in representation space:

$$\mathcal{L} = -\log \frac{e^{sim(q, k^+)/\tau}}{e^{sim(q, k^+)/\tau} + \sum_{i} e^{sim(q, k^-_i)/\tau}}$$

The denominator pushes the query away from negatives. More negatives = more repulsion from more directions = more uniform representation distribution.

Negative Sampling Strategies

Negative Sampling Approaches

•Random Negatives (Uniform) — Sample uniformly from the dataset. Simple and provides diverse negatives but may be too easy for the model.
•In-Batch Negatives — Use other samples in the mini-batch. Efficient (no extra computation) but diversity limited by batch size.
•Memory Bank — Store embeddings of all samples, sample negatives from this bank. Large negative pool but embeddings become stale.
•Queue-Based (MoCo) — FIFO queue of recent embeddings. Fresh negatives with momentum encoder consistency.
•Hard Negative Mining — Explicitly select negatives that are similar to the query. Stronger signal but risks instability.

How Many Negatives?

The number of negatives has a profound impact:

Negatives	Effect	Practical Consideration
Too few (<100)	Weak repulsion; easy task; less learning	May lead to collapse
Moderate (256-4K)	Balanced learning; good for smaller setups	Works with careful tuning
Many (4K-64K)	Strong repulsion; diverse negatives	Requires compute or memory bank
Very many (>64K)	Diminishing returns; redundancy	May not help beyond this point

The InfoNCE bound on mutual information ($\log(K+1)$) explains why: with $K=65535$ negatives, we can estimate up to ~11 nats of MI. Beyond this, more negatives provide marginal gains.

The False Negative Problem

A critical but often overlooked issue: some negatives aren't truly negative. When sampling negatives randomly, some may actually share semantic content with the query.

Understanding False Negatives

Consider ImageNet with 1000 classes and balanced classes:

In a batch of 4096 images, expect ~4 samples per class
For any query, ~4 "negatives" are actually same-class samples

These false negatives create conflicting gradients:

Positive loss: pull these together
Negative loss: push these apart

The model receives contradictory supervision.

Impact on Learning

False Negative Effects
Aspect	Effect of False Negatives	Severity
Gradient direction	Conflicting signals slow learning	Moderate
Representation structure	Same-class items pushed apart	High for fine-grained tasks
Loss optimization	Lower bound on achievable loss	Creates floor
Cluster formation	Clusters may split along false boundaries	Depends on FN rate

Mitigation Strategies

1. Temperature Scaling

Higher temperature softens the softmax distribution, reducing the weight on any single negative: $$p_i = \frac{e^{s_i/\tau}}{\sum_j e^{s_j/\tau}}$$

With high $\tau$, even false negatives with high similarity don't dominate gradients.

2. Debiased Contrastive Loss

Chuang et al. (2020) proposed explicitly correcting for false negatives: $$\mathcal{L}_{\text{debiased}} = -\log \frac{e^{f(x, x^+)/\tau}}{e^{f(x, x^+)/\tau} + N \cdot (g^- - \tau^+ / N \cdot g^+)}$$

where $g^+, g^-$ estimate positive and negative density.

3. Supervised Contrastive Loss

When labels are available, explicitly exclude same-class samples from negatives: $$\mathcal{L}{\text{sup}} = \sum{i} \frac{-1}{|P(i)|} \sum_{p \in P(i)} \log \frac{e^{z_i \cdot z_p / \tau}}{\sum_{a \neq i} e^{z_i \cdot z_a / \tau}}$$

where $P(i)$ is the set of all positives for sample $i$.

Practical Guidance

For most applications, moderately higher temperature (0.1-0.2 instead of 0.07) provides robustness to false negatives with minimal performance loss. True debiasing requires density estimation which adds complexity. If you have labels, supervised contrastive loss is the cleanest solution.

Hard Negative Mining: The Double-Edged Sword

Not all negatives contribute equally to learning. Hard negatives—samples that are similar to the query but semantically different—provide the strongest learning signal.

The Gradient Perspective

Recall the gradient of InfoNCE with respect to negative similarity: $$\frac{\partial \mathcal{L}}{\partial f(q, k^-_i)} = p^-_i$$

where $p^-_i$ is the softmax probability assigned to negative $i$. Hard negatives (high similarity) receive higher softmax probability and thus contribute more to the gradient.

Hard Negative Mining Strategies

Hard Negative Approaches

•Online Mining — Within each batch, weight negatives by similarity. Computationally free but limited by batch diversity.
•Semi-Hard Mining — Select negatives that are hard but not too hard (within a margin). Avoids focusing on outliers or label noise.
•Cross-Batch Mining — Maintain buffer of hard negatives across batches. Decouples hard negative count from batch size.
•Curriculum Mining — Gradually increase negative difficulty during training. Start easy, progressively challenge the model.
•Synthesis — Generate hard negatives via adversarial perturbation or mixup. Creates targeted challenge examples.

The Risks of Hard Negatives

Hard negative mining can backfire:

1. False Negative Amplification Hard negatives may actually be false negatives—same-class samples that look similar. Mining them amplifies the false negative problem.

2. Sampling Bias Overemphasizing hard negatives can bias representations toward distinguishing within difficult pairs while ignoring broader structure.

3. Training Instability Very hard negatives create large gradients that can destabilize training, especially early when the encoder is poor.

hard_negative_mining.py
Python
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import torch
import torch.nn.functional as F
 
def hard_negative_contrastive_loss(
    query,
    positive,
    negatives,
    temperature=0.07,
    hard_negative_weight=0.5,
    num_hard=None
):
    """
    Contrastive loss with hard negative emphasis.
    
    Args:
        query: Query embeddings (batch, dim)
        positive: Positive embeddings (batch, dim)
        negatives: Negative embeddings (batch, num_neg, dim)
        temperature: Temperature scaling
        hard_negative_weight: How much to weight hard negatives
        num_hard: Number of hardest negatives to emphasize
    
    Returns:
        Weighted contrastive loss
    """
    batch_size, num_neg, dim = negatives.shape
    
    # Normalize
    query = F.normalize(query, dim=-1)
    positive = F.normalize(positive, dim=-1)
    negatives = F.normalize(negatives, dim=-1)
    
    # Positive similarity
    pos_sim = (query * positive).sum(dim=-1, keepdim=True) / temperature
    
    # Negative similarities
    neg_sim = torch.bmm(negatives, query.unsqueeze(-1)).squeeze(-1) / temperature
    
    if num_hard is None:
        num_hard = num_neg // 4  # Top 25% as hard negatives
    
    # Find hardest negatives (highest similarity)
    hard_neg_sim, hard_indices = neg_sim.topk(num_hard, dim=-1)
    easy_neg_sim = neg_sim  # Keep all for baseline
    
    # Standard loss (all negatives)
    logits_all = torch.cat([pos_sim, neg_sim], dim=-1)
    loss_all = F.cross_entropy(
        logits_all,
        torch.zeros(batch_size, dtype=torch.long, device=query.device)
    )
    
    # Hard negative loss
    logits_hard = torch.cat([pos_sim, hard_neg_sim], dim=-1)
    loss_hard = F.cross_entropy(
        logits_hard,
        torch.zeros(batch_size, dtype=torch.long, device=query.device)
    )
    
    # Weighted combination
    loss = (1 - hard_negative_weight) * loss_all + hard_negative_weight * loss_hard
    
    return loss

Use Hard Negatives Carefully

In most scenarios, the implicit hard negative weighting from low temperature is sufficient. Explicit hard negative mining is most valuable when you have verified labels (to avoid false negatives) or are working in domains with very clear semantic boundaries.

How Pairs Shape Representations

The choice of positive and negative pairs fundamentally determines what the learned representation captures.

Positive Pair Strength and Representation Structure

Easy positives (minor augmentations):

Model learns minimal invariances
Representations may retain augmentation-sensitive features
Training converges quickly but representations may not transfer well

Hard positives (aggressive augmentations):

Model must learn strong invariances
Representations focus on high-level semantics
Risk: may discard useful information if augmentations destroy semantics

The Alignment-Uniformity Tradeoff

Wang & Isola (2020) characterized representations by two properties:

Alignment: Positive pairs should be close $$\mathcal{L}{\text{align}} = \mathbb{E}{(x, x^+)} |f(x) - f(x^+)|^2$$

Uniformity: Representations should cover the hypersphere uniformly $$\mathcal{L}{\text{uniform}} = \log \mathbb{E}{x, y} e^{-2|f(x) - f(y)|^2}$$

Strong positives improve alignment but may hurt uniformity (everything pulled to same region). Many negatives improve uniformity but may hurt alignment (too much repulsion).

Pair Selection Impact on Representation
Configuration	Alignment	Uniformity	Downstream Effect
Easy positives, few negatives	Moderate	Poor	Clustered but collapsed representation
Easy positives, many negatives	Good	Good	Well-spread but shallow features
Hard positives, few negatives	Poor	Poor	Potential collapse
Hard positives, many negatives	Good	Excellent	Semantic, transferable representations

Practical Implications

The interplay of positive difficulty and negative quantity explains several empirical observations:

SimCLR needs large batches because its strong augmentations create hard positives that require many negatives to prevent collapse.
Weaker augmentations need fewer negatives because the contrastive task is easier—the model can distinguish with less repulsive force.
Transfer learning benefits from harder positives because the model must learn semantic features to solve the task.
In-domain performance may prefer easier positives because task-relevant augmentation-sensitive features are preserved.

Summary: Pairs as the Language of Contrastive Learning

Positive and negative pairs are more than implementation details—they are the vocabulary through which we communicate to the model what we want it to learn.

Key Takeaways

•Positive pairs define invariances — Whatever varies between positives becomes an invariance of the representation.
•Negatives prevent collapse — They provide the repulsive force that spreads representations across the embedding space.
•False negatives corrupt learning — Same-class negatives create conflicting gradients; temperature and debiasing help.
•Hard negatives accelerate learning — But they must be used carefully to avoid amplifying false negatives.
•Alignment and uniformity must balance — Too much of either degrades representation quality.
•Pair selection should match downstream tasks — Design pairs to encode task-relevant invariances.

Foundation Complete

You now understand the fundamental unit of contrastive learning—the pair. This understanding is essential for diagnosing problems, designing new methods, and adapting contrastive learning to new domains. Next, we'll examine data augmentation in depth—the practical mechanism for creating effective positive pairs.