Add question 238: SimCLR Contrastive Loss (NT-Xent)

build/238.json

@@ -0,0 +1,90 @@
+{
+  "id": "238",
+  "title": "SimCLR Contrastive Loss (NT-Xent)",
+  "difficulty": "medium",
+  "category": "Deep Learning",
+  "video": "",
+  "likes": "0",
+  "dislikes": "0",
+  "contributor": [
+    {
+      "profile_link": "https://github.com/BARALLL",
+      "name": "baralm"
+    }
+  ],
+  "description": "\\## NT-Xent Loss for Self-Supervised Contrastive Learning\n\n\n\nIn self-supervised contrastive learning frameworks like \\*\\*SimCLR\\*\\*, we learn meaningful representations without labels by:\n\n1\\. Creating two augmented \"views\" of each image\n\n2\\. Training the model to recognize that views of the \\*\\*same\\*\\* image should have similar embeddings\n\n3\\. While views of \\*\\*different\\*\\* images should have dissimilar embeddings\n\n\n\n\\### The Problem\n\nYou are given a batch of $N$ images. For each image, we generate 2 augmented views, resulting in a batch size of $2N$. The embeddings are organized in an \\*\\*interleaved\\*\\* fashion:\n\n\\- Rows $2k$ and $2k+1$ are two views of the same image $k$ (a positive pair).\n\n\\- Any other pair of rows constitutes a negative pair.\n\n\n\nFor a specific sample $i$, let $j$ be its positive pair. The \\*\\*NT-Xent (Normalized Temperature-scaled Cross-Entropy)\\*\\* loss for sample $i$ is defined as:\n\n\n\n$$\n\n\\\\ell\\_i = -\\\\log \\\\frac{\\\\exp(\\\\text{sim}(z\\_i, z\\_j) / \\\\tau)}{\\\\sum\\_{k=1}^{2N} \\\\mathbb{1}\\_{\\[k \\\\neq i]} \\\\exp(\\\\text{sim}(z\\_i, z\\_k) / \\\\tau)}\n\n$$\n\n\n\nWhere:\n\n\\- $z$ is the batch of L2-normalized embeddings.\n\n\\- $\\\\text{sim}(u, v) = u^\\\\top v$ (Cosine similarity, since $u, v$ are normalized).\n\n\\- $\\\\mathbb{1}\\_{\\[k \\\\neq i]}$ is an indicator function (returns 1 if $k \\\\neq i$, else 0). Effectively, we sum over all samples except the sample itself.\n\n\\- $\\\\tau$ is the temperature parameter.\n\n\n\nThe total loss is the arithmetic mean over all $2N$ samples: $L = \\\\frac{1}{2N} \\\\sum\\_{i=0}^{2N-1} \\\\ell\\_i$.\n\n\n\n\\### Your Task\n\nImplement the function `nt\\_xent\\_loss(z, temperature)` that computes the NT-Xent loss using vectorized NumPy operations.\n\n\n\n\\*\\*Input Format\\*\\*\n\n\\- `z`: A numpy array of shape `(2N, embedding\\_dim)` containing \\*\\*L2-normalized\\*\\* embeddings.\n\n  - \\*\\*Structure\\*\\*: The rows are interleaved such that `z\\[2k]` and `z\\[2k+1]` form a positive pair (two views of image $k$).\n\n  - Visually: `\\[View1\\_Img1, View2\\_Img1, View1\\_Img2, View2\\_Img2, ...]`.\n\n  - All other interactions `z\\[i]` and `z\\[j]` (where `j` is not the pair of `i`) are considered negatives.\n\n\\- `temperature`: A float scaling parameter ($\\\\tau > 0$).\n\n\n\n\\### Output Format\n\n\\- Returns `float`: The average NT-Xent loss over all $2N$ samples.\n\n\n\n\\### Note on Stability\n\n\\- You should implement the \\*\\*Log-Sum-Exp trick\\*\\* (subtracting the maximum value before exponentiation) to ensure numerical stability.\n\n\n\n\\### Constraints\n\n\\- $N \\\\geq 1$ (at least 1 image, so batch size $\\\\geq 2$)\n\n\\- `embedding\\_dim` $\\\\geq 1$\n\n\\- `temperature` $> 0$\n\n\\- Input embeddings are guaranteed to be L2-normalized\n\n\\- \\*\\*Performance:\\*\\* Avoid explicit `for` loops. Use matrix operations and broadcasting.",
+  "learn_section": "\n# Learn Section\n\n# Understanding NT-Xent Loss (Normalized Temperature-scaled Cross Entropy)\n\n### 1. The Intuition: \"Find Your Partner\"\nAt its core, Self-Supervised Learning (SSL) creates a \"pretext task\" from unlabeled data. \nImagine a crowded room of people (embeddings). Everyone has a generic twin. The goal of NT-Xent is to make you stand as close as possible to your twin (alignment), while pushing everyone else away (uniformity).\n\n*   **Positive pairs**: Different views (augmentations) of the SAME image → should be **SIMILAR**.\n*   **Negative pairs**: Views of DIFFERENT images → should be **DISSIMILAR**.\n\n### 2. Generating Views\nWe take a batch of $N$ images and generate 2 views for each, resulting in a batch size of $2N$.\n\n```text\nImage A ───────── Augment ──→ View A₁ ──┐\n          │                             ├── Should be near (Positive) ✓\n          └───── Augment ───→ View A₂ ──┘\n\nImage B ───────── Augment ──→ View B₁ ──┐\n                                        ├── Should be far (Negative) ✗\nImage A ───────── Augment ──→ View A₁ ──┘\n```\n\n### 3. The Math: A Classification Problem\nThe NT-Xent loss is essentially a **Softmax Cross-Entropy** loss. We treat the positive pair as the \"correct class\" and all other images in the batch as \"negative classes.\"\n\n$$\\ell_i = -\\log \\frac{\\exp(\\text{sim}(z_i, z_j) / \\tau)}{\\sum_{k \\neq i} \\exp(\\text{sim}(z_i, z_k) / \\tau)}$$\n\n**Breakdown:**\n*   **Numerator**: The score of the positive pair ($z_i, z_j$). We want this high.\n*   **Denominator**: The sum of scores of $z_i$ against *all* other samples (negatives + positive).\n*   **Goal**: By maximizing the numerator relative to the denominator, we force the model to learn unique features that distinguish sample $i$ from the crowd.\n\n### 4. Visualizing the Similarity Matrix\nThe implementation relies on a $(2N \\times 2N)$ similarity matrix. If we organize our batch as `[Cat1, Cat2, Dog1, Dog2]`, the ideal matrix looks like this:\n\n$$\n\\begin{bmatrix}\n\\text{Mask} & \\mathbf{\\text{High}} & \\text{Low} & \\text{Low} \\\\\n\\mathbf{\\text{High}} & \\text{Mask} & \\text{Low} & \\text{Low} \\\\\n\\text{Low} & \\text{Low} & \\text{Mask} & \\mathbf{\\text{High}} \\\\\n\\text{Low} & \\text{Low} & \\mathbf{\\text{High}} & \\text{Mask}\n\\end{bmatrix}\n$$\n\n1.  **The Diagonal (Masked)**: We explicitly ignore comparing an image to itself ($k \\neq i$).\n2.  **The Off-diagonals**: These are the **Positive Pairs**. We want to maximize these values.\n3.  **Everything else**: These are **Negatives**. We want to minimize these values.\n\n### 5. The Critical Role of Temperature ($\\tau$)\nThe temperature $\\tau$ scales the dot products before the softmax. It controls how much the model focuses on difficult examples.\n\n*   **High $\\tau$ (e.g., 1.0)**: The distribution is smoother. The model treats all negatives roughly equally.\n*   **Low $\\tau$ (e.g., 0.1)**: The distribution becomes sharp/peaky. The model ignores easy negatives and focuses heavily on **\"Hard Negatives\"** (images that look similar to the anchor but aren't).\n\n$$\\text{As } \\tau \\to 0: \\text{Loss approaches argmax (winner-take-all)}$$\n\n### 6. Why It Works: Alignment & Uniformity\nResearch shows this loss optimizes two specific geometric properties on the embedding hypersphere:\n1.  **Alignment**: Two views of the same image map to nearby points.\n2.  **Uniformity**: Feature vectors spread roughly uniformly across the sphere. This prevents **feature collapse**, where the model maps all images to the same constant vector to cheat the loss.\n\n### 7. Implementation Steps\n1.  **Forward Pass**: Get normalized embeddings $z$ (shape $2N \\times D$).\n2.  **Similarity**: Compute matrix $S = z \\cdot z^T$ (shape $2N \\times 2N$).\n3.  **Scale**: Divide $S$ by $\\tau$.\n4.  **Mask**: Set diagonal values to $-\\infty$ (so exp() becomes 0).\n5.  **Labels**: Create target labels. If batch is organized as `[View1_A, View1_B, ..., View2_A, View2_B...]`, then $i$ matches with $i + N$.\n6.  **Loss**: Apply Standard Cross Entropy.\n\n### 6. Numerical Stability (Log-Sum-Exp Trick)\nComputers struggle with large exponents. If $\\text{sim}=1.0$ and $\\tau=0.01$, then $e^{100}$ is huge. To prevent overflow, we use the identity:\n\n$$ \\log \\left( \\sum e^{x_i} \\right) = a + \\log \\left( \\sum e^{x_i - a} \\right) $$\n\nwhere $a = \\max(x)$.\nBy subtracting the maximum value from the logits before exponentiating, the largest term becomes $e^0 = 1$, preventing overflow while keeping the probabilities mathematically identical.\n\n### 8. Connection to InfoNCE\nNT-Xent is a specific form of InfoNCE (Noise Contrastive Estimation) loss, which has theoretical connections to maximizing mutual information between views.\n\n### References\n*   **SimCLR Paper**: [Chen et al., 2020](https://arxiv.org/abs/2002.05709)\n    ",
+  "starter_code": "def nt_xent_loss(z: np.ndarray, temperature: float) -> float:\n    \"\"\"\n    Compute the NT-Xent loss for contrastive learning.\n    \n    Args:\n        z: L2-normalized embeddings, shape (2N, embedding_dim)\n           Positive pairs: z[2k] and z[2k+1] are views of image k\n        temperature: Temperature scaling parameter (τ > 0)\n    \n    Returns:\n        The scalar NT-Xent loss value\n    \"\"\"\n    pass",
+  "solution": "def nt_xent_loss(z: np.ndarray, temperature: float) -> float:\n    N = z.shape[0]\n    sim = (z @ z.T) / temperature\n    \n    sim_exp = np.exp(sim - np.max(sim, axis=1, keepdims=True))\n    mask_diag = ~np.eye(N, dtype=bool)\n    denominator = np.sum(sim_exp * mask_diag, axis=1)\n    \n    indices = np.arange(N)\n    pos_indices = indices + 1 - 2 * (indices % 2)\n    numerator = sim_exp[indices, pos_indices]\n    \n    losses = -np.log(numerator / denominator)\n    \n    return float(np.mean(losses))",
+  "example": {
+    "input": "# N=2 (Total batch 4).\n# 0 and 1 are views of Cat. 2 and 3 are views of Dog.\n# 0 matches 1 (Positive). 0 mismatches 2 and 3 (Negatives).\n\nz = np.array([\n    [1.0, 0.0],  # 0: Cat View A\n    [1.0, 0.0],  # 1: Cat View B (Perfect match with 0)\n    [0.0, 1.0],  # 2: Dog View A (Orthogonal to 0)\n    [0.0, 1.0]   # 3: Dog View B (Orthogonal to 0)\n])\ntemperature = 0.5",
+    "output": "2.2395447662218846",
+    "reasoning": "Let's calculate loss for index 0 (Cat A):\n\n1. **Positive Pair**: Index 1. Sim = 1.0.\n\n2. **Negatives**: Index 2, Index 3. Sim = 0.0.\n\n3. **Terms**:\n\n   - Numerator (Positive): $\\exp(1.0 / 0.5) = \\exp(2) \\approx 7.389$\n\n   - Denominator (All $k \\neq 0$):\n\n     - Term 1 (Pos): $\\exp(1.0/0.5) \\approx 7.389$\n\n     - Term 2 (Neg): $\\exp(0.0/0.5) = 1$\n\n     - Term 3 (Neg): $\\exp(0.0/0.5) = 1$\n\n   - Denominator Sum: $7.389 + 1 + 1 = 9.389$\n\n4. **Loss for index 0**: $-\\log(7.389 / 9.389) \\approx 0.239$\n\n\n\nSince the setup is symmetric, all 4 indices have the same loss."
+  },
+  "test_cases": [
+    {
+      "test": "print(nt_xent_loss([[1, 0], [1, 0], [0, 1], [0, 1]], 1.0))",
+      "expected_output": "0.5514447139320511"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [1, 0]], 1.0))",
+      "expected_output": "0.0"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [1, 0], [1, 0], [1, 0]], 1.0))",
+      "expected_output": "1.0986122886681098"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [1, 0], [-1, 0], [-1, 0]], 0.5))",
+      "expected_output": "0.035976299748193295"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], 1.0))",
+      "expected_output": "1.0986122886681098"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [1, 0], [0, 1], [0, 1]], 100.0))",
+      "expected_output": "1.0919567454272663"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [1, 0], [0, 1], [0, 1]], 0.5))",
+      "expected_output": "0.23954476622188456"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [1, 0], [0, 1], [0, 1]], 2.0))",
+      "expected_output": "0.7943767694176431"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0, 0], [1, 0, 0], [0, 1, 0], [0, 1, 0], [0, 0, 1], [0, 0, 1]], 1.0))",
+      "expected_output": "0.904832441554448"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [0.8, 0.6], [0, 1], [-0.6, 0.8]], 1.0))",
+      "expected_output": "0.6735767888870939"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [-1, 0], [0, 1], [0, -1]], 1.0))",
+      "expected_output": "1.861994804058251"
+    },
+    {
+      "test": "print(nt_xent_loss([[1, 0], [0, 1], [0.99, 0.141], [0.141, 0.99]], 1.0))",
+      "expected_output": "1.4700659232878173"
+    },
+    {
+      "test": "print(nt_xent_loss([[1], [1]], 1.0))",
+      "expected_output": "0.0"
+    },
+    {
+      "test": "print(nt_xent_loss([[1.0, 0.0], [0.707, 0.707], [-1.0, 0.0], [-0.707, 0.707]], 1.0))",
+      "expected_output": "0.4528130954640332"
+    },
+    {
+      "test": "import numpy as np; np.random.seed(42); N = 1000; dim = 64; temperature = 0.1; embeddings = np.random.randn(2 * N, dim); z = embeddings / np.linalg.norm(embeddings, axis=1, keepdims=True); print(nt_xent_loss(z, temperature))",
+      "expected_output": "8.37825890387809"
+    },
+    {
+      "test": "import numpy as np; np.random.seed(42); N = 8; dim = 8192; temperature = 0.5; embeddings = np.random.randn(2 * N, dim); z = embeddings / np.linalg.norm(embeddings, axis=1, keepdims=True); print(nt_xent_loss(z, temperature))",
+      "expected_output": "2.7080214605287156"
+    }
+  ]
+}

questions/238_simclr_nt_xent/description.md

@@ -0,0 +1,96 @@
+\## NT-Xent Loss for Self-Supervised Contrastive Learning
+
+
+
+In self-supervised contrastive learning frameworks like \*\*SimCLR\*\*, we learn meaningful representations without labels by:
+
+1\. Creating two augmented "views" of each image
+
+2\. Training the model to recognize that views of the \*\*same\*\* image should have similar embeddings
+
+3\. While views of \*\*different\*\* images should have dissimilar embeddings
+
+
+
+\### The Problem
+
+You are given a batch of $N$ images. For each image, we generate 2 augmented views, resulting in a batch size of $2N$. The embeddings are organized in an \*\*interleaved\*\* fashion:
+
+\- Rows $2k$ and $2k+1$ are two views of the same image $k$ (a positive pair).
+
+\- Any other pair of rows constitutes a negative pair.
+
+
+
+For a specific sample $i$, let $j$ be its positive pair. The \*\*NT-Xent (Normalized Temperature-scaled Cross-Entropy)\*\* loss for sample $i$ is defined as:
+
+
+
+$$
+
+\\ell\_i = -\\log \\frac{\\exp(\\text{sim}(z\_i, z\_j) / \\tau)}{\\sum\_{k=1}^{2N} \\mathbb{1}\_{\[k \\neq i]} \\exp(\\text{sim}(z\_i, z\_k) / \\tau)}
+
+$$
+
+
+
+Where:
+
+\- $z$ is the batch of L2-normalized embeddings.
+
+\- $\\text{sim}(u, v) = u^\\top v$ (Cosine similarity, since $u, v$ are normalized).
+
+\- $\\mathbb{1}\_{\[k \\neq i]}$ is an indicator function (returns 1 if $k \\neq i$, else 0). Effectively, we sum over all samples except the sample itself.
+
+\- $\\tau$ is the temperature parameter.
+
+
+
+The total loss is the arithmetic mean over all $2N$ samples: $L = \\frac{1}{2N} \\sum\_{i=0}^{2N-1} \\ell\_i$.
+
+
+
+\### Your Task
+
+Implement the function `nt\_xent\_loss(z, temperature)` that computes the NT-Xent loss using vectorized NumPy operations.
+
+
+
+\*\*Input Format\*\*
+
+\- `z`: A numpy array of shape `(2N, embedding\_dim)` containing \*\*L2-normalized\*\* embeddings.
+
+  - \*\*Structure\*\*: The rows are interleaved such that `z\[2k]` and `z\[2k+1]` form a positive pair (two views of image $k$).
+
+  - Visually: `\[View1\_Img1, View2\_Img1, View1\_Img2, View2\_Img2, ...]`.
+
+  - All other interactions `z\[i]` and `z\[j]` (where `j` is not the pair of `i`) are considered negatives.
+
+\- `temperature`: A float scaling parameter ($\\tau > 0$).
+
+
+
+\### Output Format
+
+\- Returns `float`: The average NT-Xent loss over all $2N$ samples.
+
+
+
+\### Note on Stability
+
+\- You should implement the \*\*Log-Sum-Exp trick\*\* (subtracting the maximum value before exponentiation) to ensure numerical stability.
+
+
+
+\### Constraints
+
+\- $N \\geq 1$ (at least 1 image, so batch size $\\geq 2$)
+
+\- `embedding\_dim` $\\geq 1$
+
+\- `temperature` $> 0$
+
+\- Input embeddings are guaranteed to be L2-normalized
+
+\- \*\*Performance:\*\* Avoid explicit `for` loops. Use matrix operations and broadcasting.
+

questions/238_simclr_nt_xent/example.json

@@ -0,0 +1,5 @@
+{
+    "input": "# N=2 (Total batch 4).\n# 0 and 1 are views of Cat. 2 and 3 are views of Dog.\n# 0 matches 1 (Positive). 0 mismatches 2 and 3 (Negatives).\n\nz = np.array([\n    [1.0, 0.0],  # 0: Cat View A\n    [1.0, 0.0],  # 1: Cat View B (Perfect match with 0)\n    [0.0, 1.0],  # 2: Dog View A (Orthogonal to 0)\n    [0.0, 1.0]   # 3: Dog View B (Orthogonal to 0)\n])\ntemperature = 0.5",
+    "output": "2.2395447662218846",
+    "reasoning": "Let's calculate loss for index 0 (Cat A):\n\n1. **Positive Pair**: Index 1. Sim = 1.0.\n\n2. **Negatives**: Index 2, Index 3. Sim = 0.0.\n\n3. **Terms**:\n\n   - Numerator (Positive): $\\exp(1.0 / 0.5) = \\exp(2) \\approx 7.389$\n\n   - Denominator (All $k \\neq 0$):\n\n     - Term 1 (Pos): $\\exp(1.0/0.5) \\approx 7.389$\n\n     - Term 2 (Neg): $\\exp(0.0/0.5) = 1$\n\n     - Term 3 (Neg): $\\exp(0.0/0.5) = 1$\n\n   - Denominator Sum: $7.389 + 1 + 1 = 9.389$\n\n4. **Loss for index 0**: $-\\log(7.389 / 9.389) \\approx 0.239$\n\n\n\nSince the setup is symmetric, all 4 indices have the same loss."
+}