Losses

Standard Implicit Loss Functions

A Collie model can’t train without a loss function, and Collie comes out-of-the-box with two different standard loss function calculations: Bayesian Personalized Ranking (BPR) loss and Hinge loss.

In its simplest form, each loss function accepts as input a prediction score for a positive item (an item a user has interacted with), and a prediction score for a negative item (an item a user has not interacted with). While the mathematical details differ between each loss, generally, all implicit losses will punish a model for ranking a negative item higher than a positive item. The severity of this punishment differs across loss functions, as shown in the example below.

import torch

from collie.loss import bpr_loss, hinge_loss


# an ideal loss case
positive_score = torch.tensor([3.0])
negative_score = torch.tensor([1.5])

print('BPR Loss:  ', bpr_loss(positive_score, negative_score))
print('Hinge Loss:', hinge_loss(positive_score, negative_score))

print('\n-----\n')

# a less-than-ideal loss case
positive_score = torch.tensor([1.5])
negative_score = torch.tensor([3.0])

print('BPR Loss:  ', bpr_loss(positive_score, negative_score))
print('Hinge Loss:', hinge_loss(positive_score, negative_score))

BPR Loss:   tensor(0.2157)
Hinge Loss: tensor(0.)

-----

BPR Loss:   tensor(1.4860)
Hinge Loss: tensor(8.7500)

Adaptive Implicit Loss Functions

Some losses extend this idea by being “adaptive,” or accepting multiple negative item prediction scores for each positive score. These losses are typically much more punishing than “non-adaptive” losses, since they allow more opportunities for the model to incorrectly rank a negative item higher than a positive one. These losses include Adaptive Bayesian Personalized Ranking loss, Adaptive Hinge loss, and Weighted Approximately Ranked Pairwise (WARP) loss.

import torch

from collie.loss import adaptive_bpr_loss, adaptive_hinge_loss, warp_loss


# an ideal loss case
positive_score = torch.tensor([3.0])
many_negative_scores = torch.tensor([[1.5], [0.5], [1.0]])

print('Adaptive BPR Loss:  ', adaptive_bpr_loss(positive_score, many_negative_scores))
print('Adaptive Hinge Loss:', adaptive_hinge_loss(positive_score, many_negative_scores))
print('WARP Loss:          ', warp_loss(positive_score, many_negative_scores, num_items=3))

print('\n-----\n')

# a less-than-ideal loss case
positive_score = torch.tensor([1.5])
many_negative_scores = torch.tensor([[2.0], [3.0], [2.5]])

print('Adaptive BPR Loss:  ', adaptive_bpr_loss(positive_score, many_negative_scores))
print('Adaptive Hinge Loss:', adaptive_hinge_loss(positive_score, many_negative_scores))
print('WARP Loss:          ', warp_loss(positive_score, many_negative_scores, num_items=3))
print('WARP Loss:          ', warp_loss(positive_score, many_negative_scores, num_items=30))

print('\n-----\n')

# a case where multiple negative items gives us greater opportunity to correct the model
positive_score = torch.tensor([1.5])
many_negative_scores = torch.tensor([[1.0], [4.0], [1.49]])

print('Adaptive BPR Loss:  ', adaptive_bpr_loss(positive_score, many_negative_scores))
print('Adaptive Hinge Loss:', adaptive_hinge_loss(positive_score, many_negative_scores))
print('WARP Loss:          ', warp_loss(positive_score, many_negative_scores, num_items=3))
print('WARP Loss:          ', warp_loss(positive_score, many_negative_scores, num_items=30))

Adaptive BPR Loss:   tensor(0.2157)
Adaptive Hinge Loss: tensor(0.)
WARP Loss:           tensor(0.)

-----

Adaptive BPR Loss:   tensor(1.4860)
Adaptive Hinge Loss: tensor(8.7500)
WARP Loss:           tensor(4.3636)
WARP Loss:           tensor(31.1301)

-----

Adaptive BPR Loss:   tensor(1.7782)
Adaptive Hinge Loss: tensor(15.7500)
WARP Loss:           tensor(0.8510)
WARP Loss:           tensor(4.5926)

Partial Credit Loss Functions

If you have item metadata available, you might reason that not all losses should be equal. For example, say you are training a recommendation system on MovieLens data, where users interact with different films, and you are comparing a positive item, Star Wars, with two negative items: Star Trek and Legally Blonde.

Normally, the loss for Star Wars compared with Star Trek, and Star Wars compared with Legally Blonde would be equal. But, as humans, we know that Star Trek is closer to Star Wars (both being space western films) than Legally Blonde is (a romantic comedy that does not have space elements), and would want our loss function to account for that 5.

For these scenarios, all loss functions in Collie support partial credit calculations, meaning we can provide metadata to reduce the potential loss for certain items with matching metadata. This is best seen through an example below:

import torch

# we'll just look at ``bpr_loss`` for this, but note that this works with
# all loss functions in Collie
from collie.loss import bpr_loss


# positive item is Star Wars
star_wars_score = torch.tensor([1.0])

# negative items are Star Trek and Legally Blonde
star_trek_score = torch.tensor([3.0])
legally_blonde_score = torch.tensor([3.0])

print('Star Wars vs Star Trek Loss:      ', end='')
print(bpr_loss(positive_scores=star_wars_score, negative_scores=star_trek_score))

print('Star Wars vs Legally Blonde Loss: ', end='')
print(bpr_loss(positive_scores=star_wars_score, negative_scores=legally_blonde_score))

print('\n-----\n')

# now let's apply a partial credit calculation to the loss
metadata_weights = {'genre': 0.25}

# categorically encode Sci-Fi as ``0`` and Comedy as ``1`` and
# order values by Star Wars, Star Trek, Legally Blonde
metadata = {'genre': torch.tensor([0, 0, 1])}

print('Star Wars vs Star Trek Partial Credit Loss:      ', end='')
print(bpr_loss(positive_scores=star_wars_score,
               negative_scores=star_trek_score,
               positive_items=torch.tensor([0]),
               negative_items=torch.tensor([1]),
               metadata=metadata,
               metadata_weights=metadata_weights))

print('Star Wars vs Legally Blonde Partial Credit Loss: ', end='')
print(bpr_loss(positive_scores=star_wars_score,
               negative_scores=legally_blonde_score,
               positive_items=torch.tensor([0]),
               negative_items=torch.tensor([2]),
               metadata=metadata,
               metadata_weights=metadata_weights))

Star Wars vs Star Trek Loss:      tensor(1.6566)
Star Wars vs Legally Blonde Loss: tensor(1.6566)

-----

Star Wars vs Star Trek Partial Credit Loss:      tensor(1.0287)
Star Wars vs Legally Blonde Partial Credit Loss: tensor(1.6566)

See Tutorials for a more in-depth example using partial credit loss functions.

Standard Losses

BPR Loss

collie.loss.bpr_loss(positive_scores: tensor, negative_scores: tensor, num_items: Optional[Any] = None, positive_items: Optional[tensor] = None, negative_items: Optional[tensor] = None, metadata: Optional[Dict[str, tensor]] = {}, metadata_weights: Optional[Dict[str, float]] = {}) → tensor[source]

Modified Bayesian Personalised Ranking 1.

See ideal_difference_from_metadata docstring for more info on how metadata is used.

Modified from torchmf and Spotlight:

Parameters

  • positive_scores (torch.tensor, 1-d) – Tensor containing predictions for known positive items of shape 1 x batch_size

  • negative_scores (torch.tensor, 1-d) – Tensor containing scores for a single sampled negative item of shape 1 x batch_size

  • num_items (Any) – Ignored, included only for compatability with WARP loss

  • positive_items (torch.tensor, 1-d) – Tensor containing ids for known positive items of shape 1 x batch_size. This is only needed if metadata is provided

  • negative_items (torch.tensor, 1-d) – Tensor containing ids for randomly-sampled negative items of shape 1 x batch_size. This is only needed if metadata is provided

  • metadata (dict) – Keys should be strings identifying each metadata type that match keys in metadata_weights. Values should be a torch.tensor of shape (num_items x 1). Each tensor should contain categorical metadata information about items (e.g. a number representing the genre of the item)

  • metadata_weights (dict) –

    Keys should be strings identifying each metadata type that match keys in metadata. Values should be the amount of weight to place on a match of that type of metadata, with the sum of all values <= 1. e.g. If metadata_weights = {'genre': .3, 'director': .2}, then an item is:

    • a 100% match if it’s the same item,

    • a 50% match if it’s a different item with the same genre and same director,

    • a 30% match if it’s a different item with the same genre and different director,

    • a 20% match if it’s a different item with a different genre and same director,

    • a 0% match if it’s a different item with a different genre and different director, which is equivalent to the loss without any partial credit

Returns

loss

Return type

torch.tensor

References

1

Hildesheim et al. “BPR: Bayesian Personalized Ranking from Implicit Feedback.” BPR | Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, 1 June 2009, dl.acm.org/doi/10.5555/1795114.1795167.

Hinge Loss

collie.loss.hinge_loss(positive_scores: tensor, negative_scores: tensor, num_items: Optional[Any] = None, positive_items: Optional[tensor] = None, negative_items: Optional[tensor] = None, metadata: Optional[Dict[str, tensor]] = {}, metadata_weights: Optional[Dict[str, float]] = {}) → tensor[source]

Modified hinge pairwise loss function 2.

See ideal_difference_from_metadata docstring for more info on how metadata is used.

Modified from Spotlight: https://github.com/maciejkula/spotlight/blob/master/spotlight/losses.py

Parameters

  • positive_scores (torch.tensor, 1-d) – Tensor containing scores for known positive items

  • negative_scores (torch.tensor, 1-d) – Tensor containing scores for a single sampled negative item

  • num_items (Any) – Ignored, included only for compatability with WARP loss

  • positive_items (torch.tensor, 1-d) – Tensor containing ids for known positive items of shape 1 x batch_size. This is only needed if metadata is provided

  • negative_items (torch.tensor, 1-d) – Tensor containing ids for randomly-sampled negative items of shape 1 x batch_size. This is only needed if metadata is provided

  • metadata (dict) – Keys should be strings identifying each metadata type that match keys in metadata_weights. Values should be a torch.tensor of shape (num_items x 1). Each tensor should contain categorical metadata information about items (e.g. a number representing the genre of the item)

  • metadata_weights (dict) –

    Keys should be strings identifying each metadata type that match keys in metadata. Values should be the amount of weight to place on a match of that type of metadata, with the sum of all values <= 1. e.g. If metadata_weights = {'genre': .3, 'director': .2}, then an item is:

    • a 100% match if it’s the same item,

    • a 50% match if it’s a different item with the same genre and same director,

    • a 30% match if it’s a different item with the same genre and different director,

    • a 20% match if it’s a different item with a different genre and same director,

    • a 0% match if it’s a different item with a different genre and different director, which is equivalent to the loss without any partial credit

Returns

loss

Return type

torch.tensor

References

2

“Hinge Loss.” Wikipedia, Wikimedia Foundation, 5 Mar. 2021, en.wikipedia.org/wiki/ Hinge_loss.

Adaptive Losses

Adaptive BPR Loss

collie.loss.adaptive_bpr_loss(positive_scores: tensor, many_negative_scores: tensor, num_items: Optional[Any] = None, positive_items: Optional[tensor] = None, negative_items: Optional[tensor] = None, metadata: Optional[Dict[str, tensor]] = {}, metadata_weights: Optional[Dict[str, float]] = {}) → tensor[source]

Modified adaptive BPR loss function.

Approximates WARP loss by taking the maximum of negative predictions for each user and sending this to BPR loss.

See ideal_difference_from_metadata docstring for more info on how metadata is used.

Parameters

  • positive_scores (torch.tensor, 1-d) – Tensor containing scores for known positive items of shape num_negative_samples x batch_size

  • many_negative_scores (torch.tensor, 2-d) – Iterable of tensors containing scores for many (n > 1) sampled negative items of shape num_negative_samples x batch_size. More tensors increase the likelihood of finding ranking-violating pairs, but risk overfitting

  • num_items (Any) – Ignored, included only for compatability with WARP loss

  • positive_items (torch.tensor, 1-d) – Tensor containing ids for known positive items of shape num_negative_samples x batch_size. This is only needed if metadata is provided

  • negative_items (torch.tensor, 2-d) – Tensor containing ids for sampled negative items of shape num_negative_samples x batch_size. This is only needed if metadata is provided

  • metadata (dict) – Keys should be strings identifying each metadata type that match keys in metadata_weights. Values should be a torch.tensor of shape (num_items x 1). Each tensor should contain categorical metadata information about items (e.g. a number representing the genre of the item)

  • metadata_weights (dict) –

    Keys should be strings identifying each metadata type that match keys in metadata. Values should be the amount of weight to place on a match of that type of metadata, with the sum of all values <= 1. e.g. If metadata_weights = {'genre': .3, 'director': .2}, then an item is:

    • a 100% match if it’s the same item,

    • a 50% match if it’s a different item with the same genre and same director,

    • a 30% match if it’s a different item with the same genre and different director,

    • a 20% match if it’s a different item with a different genre and same director,

    • a 0% match if it’s a different item with a different genre and different director, which is equivalent to the loss without any partial credit

Returns

loss

Return type

torch.tensor

Adaptive Hinge Loss

collie.loss.adaptive_hinge_loss(positive_scores: tensor, many_negative_scores: tensor, num_items: Optional[Any] = None, positive_items: Optional[tensor] = None, negative_items: Optional[tensor] = None, metadata: Optional[Dict[str, tensor]] = {}, metadata_weights: Optional[Dict[str, float]] = {}) → tensor[source]

Modified adaptive hinge pairwise loss function 3.

Approximates WARP loss by taking the maximum of negative predictions for each user and sending this to hinge loss.

See ideal_difference_from_metadata docstring for more info on how metadata is used.

Modified from Spotlight: https://github.com/maciejkula/spotlight/blob/master/spotlight/losses.py

Parameters

  • positive_scores (torch.tensor, 1-d) – Tensor containing scores for known positive items of shape num_negative_samples x batch_size

  • many_negative_scores (torch.tensor, 2-d) – Iterable of tensors containing scores for many (n > 1) sampled negative items of shape num_negative_samples x batch_size. More tensors increase the likelihood of finding ranking-violating pairs, but risk overfitting

  • num_items (Any) – Ignored, included only for compatability with WARP loss

  • positive_items (torch.tensor, 1-d) – Tensor containing ids for known positive items of shape num_negative_samples x batch_size. This is only needed if metadata is provided

  • negative_items (torch.tensor, 2-d) – Tensor containing ids for sampled negative items of shape num_negative_samples x batch_size. This is only needed if metadata is provided

  • metadata (dict) – Keys should be strings identifying each metadata type that match keys in metadata_weights. Values should be a torch.tensor of shape (num_items x 1). Each tensor should contain categorical metadata information about items (e.g. a number representing the genre of the item)

  • metadata_weights (dict) –

    Keys should be strings identifying each metadata type that match keys in metadata. Values should be the amount of weight to place on a match of that type of metadata, with the sum of all values <= 1. e.g. If metadata_weights = {'genre': .3, 'director': .2}, then an item is:

    • a 100% match if it’s the same item,

    • a 50% match if it’s a different item with the same genre and same director,

    • a 30% match if it’s a different item with the same genre and different director,

    • a 20% match if it’s a different item with a different genre and same director,

    • a 0% match if it’s a different item with a different genre and different director, which is equivalent to the loss without any partial credit

Returns

loss

Return type

torch.tensor

References

3

Kula, Maciej. “Loss Functions.” Loss Functions - Spotlight Documentation, maciejkula.github.io/spotlight/losses.html.

WARP Loss

collie.loss.warp_loss(positive_scores: tensor, many_negative_scores: tensor, num_items: int, positive_items: Optional[tensor] = None, negative_items: Optional[tensor] = None, metadata: Optional[Dict[str, tensor]] = {}, metadata_weights: Optional[Dict[str, float]] = {}) → tensor[source]

Modified WARP loss function 4.

See http://www.thespermwhale.com/jaseweston/papers/wsabie-ijcai.pdf for loss equation.

See ideal_difference_from_metadata docstring for more info on how metadata is used.

Parameters

  • positive_scores (torch.tensor, 1-d) – Tensor containing scores for known positive items of shape num_negative_samples x batch_size

  • many_negative_scores (torch.tensor, 2-d) – Iterable of tensors containing scores for many (n > 1) sampled negative items of shape num_negative_samples x batch_size. More tensors increase the likelihood of finding ranking-violating pairs, but risk overfitting

  • num_items (int) – Total number of items in the dataset

  • positive_items (torch.tensor, 1-d) – Tensor containing ids for known positive items of shape num_negative_samples x batch_size. This is only needed if metadata is provided

  • negative_items (torch.tensor, 2-d) – Tensor containing ids for sampled negative items of shape num_negative_samples x batch_size. This is only needed if metadata is provided

  • metadata (dict) – Keys should be strings identifying each metadata type that match keys in metadata_weights. Values should be a torch.tensor of shape (num_items x 1). Each tensor should contain categorical metadata information about items (e.g. a number representing the genre of the item)

  • metadata_weights (dict) –

    Keys should be strings identifying each metadata type that match keys in metadata. Values should be the amount of weight to place on a match of that type of metadata, with the sum of all values <= 1. e.g. If metadata_weights = {'genre': .3, 'director': .2}, then an item is:

    • a 100% match if it’s the same item,

    • a 50% match if it’s a different item with the same genre and same director,

    • a 30% match if it’s a different item with the same genre and different director,

    • a 20% match if it’s a different item with a different genre and same director,

    • a 0% match if it’s a different item with a different genre and different director, which is equivalent to the loss without any partial credit

Returns

loss

Return type

torch.tensor

References

4

Weston et al. WSABIE: Scaling Up To Large Vocabulary Image Annotation. www.thespermwhale.com/jaseweston/papers/wsabie-ijcai.pdf.

Footnotes

5

If it were up to the author of this library, everyone would be recommended Legally Blonde. It is a fantastic film.