From cab3a98d5e27193fa13e846ca50401e776fef042 Mon Sep 17 00:00:00 2001 From: Andrey Vasnetsov Date: Fri, 25 Mar 2022 09:50:03 +0100 Subject: [PATCH] triplet loss article (#27) * triplet loss article * add main file --- ARTICLE_CHECLLIST.md | 4 + .../content/articles/filtrable-hnsw.md | 2 +- .../articles/neural-search-tutorial.md | 2 +- .../content/articles/triplet-loss.md | 299 +++++++++++++ .../triplet-loss/contrastive.gif | Bin 0 -> 1971489 bytes .../articles_data/triplet-loss/icon.svg | 37 ++ .../triplet-loss/init_embeddings.png | Bin 0 -> 16882 bytes .../triplet-loss/loss_objective.png | Bin 0 -> 39564 bytes .../articles_data/triplet-loss/preview.png | Bin 0 -> 24497 bytes .../triplet-loss/triplet-loss.png | Bin 0 -> 39564 bytes .../triplet-loss/triplet-loss.svg | 416 ++++++++++++++++++ .../articles_data/triplet-loss/triplet.gif | Bin 0 -> 1775054 bytes 12 files changed, 758 insertions(+), 2 deletions(-) create mode 100644 qdrant-landing/content/articles/triplet-loss.md create mode 100644 qdrant-landing/static/articles_data/triplet-loss/contrastive.gif create mode 100644 qdrant-landing/static/articles_data/triplet-loss/icon.svg create mode 100644 qdrant-landing/static/articles_data/triplet-loss/init_embeddings.png create mode 100644 qdrant-landing/static/articles_data/triplet-loss/loss_objective.png create mode 100644 qdrant-landing/static/articles_data/triplet-loss/preview.png create mode 100644 qdrant-landing/static/articles_data/triplet-loss/triplet-loss.png create mode 100644 qdrant-landing/static/articles_data/triplet-loss/triplet-loss.svg create mode 100644 qdrant-landing/static/articles_data/triplet-loss/triplet.gif diff --git a/ARTICLE_CHECLLIST.md b/ARTICLE_CHECLLIST.md index 30ba7642b..70bfec5c0 100644 --- a/ARTICLE_CHECLLIST.md +++ b/ARTICLE_CHECLLIST.md @@ -7,6 +7,10 @@ What to prepare before publishing a new article? * [ ] - Small preview icon - similar to [this](./qdrant-landing/static/articles_data/neural-search-tutorial/tutorial.svg) * transparent BG * White color only + * Find icon - https://www.flaticon.com/ + * Download png (for free) + * Convert with - https://www.freeconvert.com/png-to-svg/ + * Change color to white, background to transparent ## Publish to diff --git a/qdrant-landing/content/articles/filtrable-hnsw.md b/qdrant-landing/content/articles/filtrable-hnsw.md index f4e0efa1c..835399f7f 100644 --- a/qdrant-landing/content/articles/filtrable-hnsw.md +++ b/qdrant-landing/content/articles/filtrable-hnsw.md @@ -5,7 +5,7 @@ description: How to make ANN search with custom filtering? Search in selected su external_link: https://blog.vasnetsov.com/posts/categorical-hnsw/ preview_image: /articles_data/filtrable-hnsw/preview.png small_preview_image: /articles_data/filtrable-hnsw/global-network.svg -weight: 30 +weight: 60 author: Andrei Vasnetsov author_link: https://blog.vasnetsov.com/ --- diff --git a/qdrant-landing/content/articles/neural-search-tutorial.md b/qdrant-landing/content/articles/neural-search-tutorial.md index ce51a5f94..3e6e4e703 100644 --- a/qdrant-landing/content/articles/neural-search-tutorial.md +++ b/qdrant-landing/content/articles/neural-search-tutorial.md @@ -5,7 +5,7 @@ description: Our step-by-step guide on how to build a neural search service with external_link: https://blog.qdrant.tech/neural-search-tutorial-3f034ab13adc preview_image: /articles_data/neural-search-tutorial/preview.png small_preview_image: /articles_data/neural-search-tutorial/tutorial.svg -weight: 10 +weight: 50 author: Andrei Vasnetsov author_link: https://blog.vasnetsov.com/ --- diff --git a/qdrant-landing/content/articles/triplet-loss.md b/qdrant-landing/content/articles/triplet-loss.md new file mode 100644 index 000000000..f5ef0a1fa --- /dev/null +++ b/qdrant-landing/content/articles/triplet-loss.md @@ -0,0 +1,299 @@ +--- +title: Triplet Loss - Advanced Intro +short_description: "What are the advantages of Triplet Loss and how to efficiently implement it?" +description: "What are the advantages of Triplet Loss over Contrastive loss and how to efficiently implement it?" +preview_image: /articles_data/triplet-loss/preview.png +small_preview_image: /articles_data/triplet-loss/icon.svg +weight: 10 +author: Yusuf Sarıgöz +author_link: https://medium.com/@yusufsarigoz +date: 2022-03-24T15:12:00+03:00 +--- + +## What is Triplet Loss? + +Triplet Loss was first introduced in [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://arxiv.org/abs/1503.03832) in 2015, +and it has been one of the most popular loss functions for supervised similarity or metric learning ever since. +In its simplest explanation, Triplet Loss encourages that dissimilar pairs be distant from any similar pairs by at least a certain margin value. +Mathematically, the loss value can be calculated as +$L=max(d(a,p) - d(a,n) + m, 0)$, where: + +- $p$, i.e., positive, is a sample that has the same label as $a$, i.e., anchor, +- $n$, i.e., negative, is another sample that has a label different from $a$, +- $d$ is a function to measure the distance between these three samples, +- and $m$ is a margin value to keep negative samples far apart. + +The paper uses Euclidean distance, but it is equally valid to use any other distance metric, e.g., cosine distance. + +The function has a learning objective that can be visualized as in the following: + +{{< figure src=/articles_data/triplet-loss/loss_objective.png caption="Triplet Loss learning objective" >}} + +Notice that Triplet Loss does not have a side effect of urging to encode anchor and positive samples into the same point +in the vector space as in Contrastive Loss. +This lets Triplet Loss tolerate some intra-class variance, unlike Contrastive Loss, +as the latter forces the distance between an anchor and any positive essentially to $0$. +In other terms, Triplet Loss allows to stretch clusters in such a way as to include outliers +while still ensuring a margin between samples from different clusters, e.g., negative pairs. + +Additionally, Triplet Loss is less greedy. Unlike Contrastive Loss, +it is already satisfied when different samples are easily distinguishable from similar ones. It does not change the distances in a positive cluster if +there is no interference from negative examples. +This is due to the fact that Triplet Loss tries to ensure a margin between distances of negative pairs and distances of positive pairs. +However, Contrastive Loss takes into account the margin value only when comparing dissimilar pairs, +and it does not care at all where similar pairs are at that moment. +This means that Contrastive Loss may reach a local minimum earlier, +while Triplet Loss may continue to organize the vector space in a better state. + +Let's demonstrate how two loss functions organize the vector space by animations. +For simpler visualization, the vectors are represented by points in a 2-dimensional space, +and they are selected randomly from a normal distribution. + +{{< figure src=/articles_data/triplet-loss/contrastive.gif caption="Animation that shows how Contrastive Loss moves points in the course of training." >}} + +{{< figure src=/articles_data/triplet-loss/triplet.gif caption="Animation that shows how Triplet Loss moves points in the course of training." >}} + + +From mathematical interpretations of the two-loss functions, it is clear that Triplet Loss is theoretically stronger, +but Triplet Loss has additional tricks that help it work better. +Most importantly, Triplet Loss introduce online triplet mining strategies, e.g., automatically forming the most useful triplets. + +## Why triplet mining matters? + +The formulation of Triplet Loss demonstrates that it works on three objects at a time: + +- `anchor`, +- `positive` - a sample that has the same label as the anchor, +- and `negative` - a sample with a different label from the anchor and the positive. + +In a naive implementation, we could form such triplets of samples at the beginning of each epoch +and then feed batches of such triplets to the model throughout that epoch. This is called "offline strategy." +However, this would not be so efficient for several reasons: +- It needs to pass $3n$ samples to get a loss value of $n$ triplets. +- Not all these triplets will be useful for the model to learn anything, e.g., yielding a positive loss value. +- Even if we form "useful" triplets at the beginning of each epoch with one of the methods that I will be implementing in this series, +they may become "useless" at some point in the epoch as the model weights will be constantly updated. + +Instead, we can get a batch of $n$ samples and their associated labels, +and form triplets on the fly. That is called "online strategy." Normally, this gives +$n^3$ possible triplets, but only a subset of such possible triplets will be actually valid. Even in this case, +we will have a loss value calculated from much more triplets than the offline strategy. + +Given a triplet of `(a, p, n)`, it is valid only if: + +- `a` and `p` has the same label, +- `a` and `p` are distinct samples, +- and `n` has a different label from `a` and `p`. + +These constraints may seem to be requiring expensive computation with nested loops, +but it can be efficiently implemented with tricks such as distance matrix, masking, and broadcasting. +The rest of this series will focus on the implementation of these tricks. + + +## Distance matrix + +A distance matrix is a matrix of shape $(n, n)$ to hold distance values between all possible +pairs made from items in two $n$-sized collections. +This matrix can be used to vectorize calculations that would need inefficient loops otherwise. +Its calculation can be optimized as well, and we will implement [Euclidean Distance Matrix Trick (PDF)](https://www.robots.ox.ac.uk/~albanie/notes/Euclidean_distance_trick.pdf) +explained by Samuel Albanie. You may want to read this three-page document for +the full intuition of the trick, but a brief explanation is as follows: + +- Calculate the dot product of two collections of vectors, e.g., embeddings in our case. +- Extract the diagonal from this matrix that holds the squared Euclidean norm of each embedding. +- Calculate the squared Euclidean distance matrix based on the following equation: $||a - b||^2 = ||a||^2 - 2 ⟨a, b⟩ + ||b||^2$ +- Get the square root of this matrix for non-squared distances. + +We will implement it in PyTorch, so let's start with imports. + + +```python +import torch +import torch.nn as nn +import torch.nn.functional as F + +eps = 1e-8 # an arbitrary small value to be used for numerical stability tricks +``` + +--- + +```python +def euclidean_distance_matrix(x): + """Efficient computation of Euclidean distance matrix + + Args: + x: Input tensor of shape (batch_size, embedding_dim) + + Returns: + Distance matrix of shape (batch_size, batch_size) + """ + # step 1 - compute the dot product + + # shape: (batch_size, batch_size) + dot_product = torch.mm(x, x.t()) + + # step 2 - extract the squared Euclidean norm from the diagonal + + # shape: (batch_size,) + squared_norm = torch.diag(dot_product) + + # step 3 - compute squared Euclidean distances + + # shape: (batch_size, batch_size) + distance_matrix = squared_norm.unsqueeze(0) - 2 * dot_product + squared_norm.unsqueeze(1) + + # get rid of negative distances due to numerical instabilities + distance_matrix = F.relu(distance_matrix) + + # step 4 - compute the non-squared distances + + # handle numerical stability + # derivative of the square root operation applied to 0 is infinite + # we need to handle by setting any 0 to eps + mask = (distance_matrix == 0.0).float() + + # use this mask to set indices with a value of 0 to eps + distance_matrix += mask * eps + + # now it is safe to get the square root + distance_matrix = torch.sqrt(distance_matrix) + + # undo the trick for numerical stability + distance_matrix *= (1.0 - mask) + + return distance_matrix +``` + +## Invalid triplet masking + +Now that we can compute a distance matrix for all possible pairs of embeddings in a batch, +we can apply broadcasting to enumerate distance differences for all possible triplets and represent them in a tensor of shape `(batch_size, batch_size, batch_size)`. +However, only a subset of these $n^3$ triplets are actually valid as I mentioned earlier, +and we need a corresponding mask to compute the loss value correctly. +We will implement such a helper function in three steps: + +- Compute a mask for distinct indices, e.g., `(i != j and j != k)`. +- Compute a mask for valid anchor-positive-negative triplets, e.g., `labels[i] == labels[j] and labels[j] != labels[k]`. +- Combine two masks. + + +```python +def get_triplet_mask(labels): + """compute a mask for valid triplets + + Args: + labels: Batch of integer labels. shape: (batch_size,) + + Returns: + Mask tensor to indicate which triplets are actually valid. Shape: (batch_size, batch_size, batch_size) + A triplet is valid if: + `labels[i] == labels[j] and labels[i] != labels[k]` + and `i`, `j`, `k` are different. + """ + # step 1 - get a mask for distinct indices + + # shape: (batch_size, batch_size) + indices_equal = torch.eye(labels.size()[0], dtype=torch.bool, device=labels.device) + indices_not_equal = torch.logical_not(indices_equal) + # shape: (batch_size, batch_size, 1) + i_not_equal_j = indices_not_equal.unsqueeze(2) + # shape: (batch_size, 1, batch_size) + i_not_equal_k = indices_not_equal.unsqueeze(1) + # shape: (1, batch_size, batch_size) + j_not_equal_k = indices_not_equal.unsqueeze(0) + # Shape: (batch_size, batch_size, batch_size) + distinct_indices = torch.logical_and(torch.logical_and(i_not_equal_j, i_not_equal_k), j_not_equal_k) + + # step 2 - get a mask for valid anchor-positive-negative triplets + + # shape: (batch_size, batch_size) + labels_equal = labels.unsqueeze(0) == labels.unsqueeze(1) + # shape: (batch_size, batch_size, 1) + i_equal_j = labels_equal.unsqueeze(2) + # shape: (batch_size, 1, batch_size) + i_equal_k = labels_equal.unsqueeze(1) + # shape: (batch_size, batch_size, batch_size) + valid_indices = torch.logical_and(i_equal_j, torch.logical_not(i_equal_k)) + + # step 3 - combine two masks + mask = torch.logical_and(distinct_indices, valid_indices) + + return mask +``` + +## Batch-all strategy for online triplet mining + +Now we are ready for actually implementing Triplet Loss itself. +Triplet Loss involves several strategies to form or select triplets, and the simplest one is +to use all valid triplets that can be formed from samples in a batch. +This can be achieved in four easy steps thanks to utility functions we've already implemented: + +- Get a distance matrix of all possible pairs that can be formed from embeddings in a batch. +- Apply broadcasting to this matrix to compute loss values for all possible triplets. +- Set loss values of invalid or easy triplets to $0$. +- Average the remaining positive values to return a scalar loss. + +I will start by implementing this strategy, and more complex ones will follow as separate posts. + + +```python +class BatchAllTtripletLoss(nn.Module): + """Uses all valid triplets to compute Triplet loss + + Args: + margin: Margin value in the Triplet Loss equation + """ + def __init__(self, margin=1.): + super().__init__() + self.margin = margin + + def forward(self, embeddings, labels): + """computes loss value. + + Args: + embeddings: Batch of embeddings, e.g., output of the encoder. shape: (batch_size, embedding_dim) + labels: Batch of integer labels associated with embeddings. shape: (batch_size,) + + Returns: + Scalar loss value. + """ + # step 1 - get distance matrix + # shape: (batch_size, batch_size) + distance_matrix = euclidean_distance_matrix(embeddings) + + # step 2 - compute loss values for all triplets by applying broadcasting to distance matrix + + # shape: (batch_size, batch_size, 1) + anchor_positive_dists = distance_matrix.unsqueeze(2) + # shape: (batch_size, 1, batch_size) + anchor_negative_dists = distance_matrix.unsqueeze(1) + # get loss values for all possible n^3 triplets + # shape: (batch_size, batch_size, batch_size) + triplet_loss = anchor_positive_dists - anchor_negative_dists + self.margin + + # step 3 - filter out invalid or easy triplets by setting their loss values to 0 + + # shape: (batch_size, batch_size, batch_size) + mask = get_triplet_mask(labels) + triplet_loss *= mask + # easy triplets have negative loss values + triplet_loss = F.relu(triplet_loss) + + # step 4 - compute scalar loss value by averaging positive losses + num_positive_losses = (triplet_loss > eps).float().sum() + triplet_loss = triplet_loss.sum() / (num_positive_losses + eps) + + return triplet_loss +``` + +## Conclusion + +I mentioned that Triplet Loss is different from Contrastive Loss not only mathematically but also in its sample selection strategies, and I implemented the batch-all strategy for 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