Video

Transcript

um hi my name is lake um

i’ll be talking about breaking contrast

models with the sets card game

this project is motivated by a

particular failure mode of clip

consider this text prompt which asks for

a red cube on top of a green cube

on top of a blue cube the image forms a

natural pairing with the text

if i shuffle the colors in both the

image and the text i get another pairing

if i continue this procedure to generate

all six pairings of texts and

images and feed them into clip i’d

expect

clip to predict a six by six logics

matrix with a strong diagonal

but the logits predicted are actually

quite even it seems that

when there are multiple entities

relations and attributes

clip struggles my intuition for why this

happens

has to do with the dot product retrieval

layer between query and key

representation vectors in standard

contrasted models

we can view this layer as performing

linear classification

in which a linear boundary q is encoded

to separate the key data points

under this perspective the vc dimension

limitation applies

for quotes we have six keys one query

may match with three of them as positive

ps

but there are actually two to the power

six number of ways to assign some keys

as positive

and others as negative as long as our

hypothesis class can throw all 2 to the

power 6 number of queries at the keys

this is the idea of shattering in

general

if our queries are limited by the vector

dimension d

the vc theory states that they cannot

match with all possible subsets of d

plus one keys in data sets like imagenet

there are only a thousand labels in the

contrasting model setup this corresponds

to a thousand static queries

for example match all the docs so

model performance is far from hitting

the limits imposed by the vector’s vc

dimension

but on data sets like clever many

dynamic queries can be formed

such as identifying all the scenes with

one or more blue objects

here we need to subset the scenes in

many dynamic ways

so we may hit the limit imposed by the

vector’s vc dimension

i also have a secondary intuition that’s

related to the poor approximation of

full rank

query key matrices using vectors that

are shorter than the rank

feel free to ask me about it to validate

my intuitions

i set up two architectures one is a

contrastive model which i encode

queries and keys separately and score

the compatibility using the dot product

retrieval layer

the other is a non-contrastive model

which scores each query key pair as a

continuous sequence

the contrasted model uses an eight layer

transformer to encode the query symbols

and an embedding lookup to encode the

key symbols

while the non-contrasting model uses a

four layer transformer to encode the

concatenated query key symbols

the contrastive model has 17 million

parameters while the non-contrastive

model has only half of that

my experiment goal is to show that

contrasting models with limited vector

representation dimensions

are worse than non-contrasted models

that uses half the parameters

task wise i borrowed the well-known sex

card game

it suits my purpose because of some nice

extendable properties

each card has multiple attributes and

possible values for attributes

these dimensions are scalable any pairs

of cards can form a query that evaluates

to a key card

in a complete set of three cards on any

attribute the cards either have to

be the same or all different

this query key matrix has a regular

pattern but it is still full rank

to extend this game i introduced the

star

regular expressions such that queries

can evaluate to subsets of key cards

in addition i introduced a set union and

symmetric difference operators

to make the queries and return subsets

even more dynamic i’m going to show an

example game

and how the models would play it here’s

a deck of 27 cards

each with 3 attributes and three

possible values per attribute

each query has eight pairs of cards and

set union operators between them

depending on the query the matching

subset of keys and its size

vary this particular query returns 13

matching positive keys

a different query may return a different

smaller or larger subset of keys

such as 3 or 21. a training example is a

tuple of query symbols

and a key symbol sampled from the

matching positive key

we can sample different queries and

their keys this way to make a batch

the contrasted model uses the info nce

training objective which normalizes the

dot product

over all the query key pairings in this

batch size by batch size query

key matrix the scores are penalized

across

both columns and rows the

non-contrasting model uses a

conventional cross-entropy objective

the scores are penalized across keys in

the support

at test time the models are given

queries such as this one

and they score the compatibility of this

query with each of the 27 keys

there are 13 ground truth keys for this

query so one crude metric is to measure

how much your top 13 predictions overlap

with the

13 round trip keys i call this metric

the top k prediction ground truth

overlap for now

a more sensitive metric is to compare

the kl diversions

between the normalized predicted scores

and a ground truth distribution

constructed from dividing one among all

the ground truth keys evenly

putting all of this together i have

seven contrasted models with vector

representation dimension from 512 down

to four

they all have 17 million parameters and

a non-contrastive model with half the

size

i have five different games that are

based on a 27 card deck

from the sets game in its original form

to the afford mentioned extensions and a

final game of shattering 27 numbers

they have increasing levels of

difficulties characterized by the

respective total number of unique key

subsets the queries

should separate we tried each game on

all the models

here’s a results plot on the y-axis we

have kl

divergence loss on the x-axis we have

those five games ordered from left to

right with increasing levels of

difficulty

measured by the total number of key

subsets the craze should separate

in powers of two the first game is the

original set

it requires the queries to separate 2 to

the power 4.57 subsets of keys

all the models did very well the second

game

includes a star regular expression and

requires the queries to separate

2 to the power 6 number subsets of keys

so the contrastive model with vector

dimension 4 starts to do poorly

as we introduce a set union and

symmetric

difference operators the models need to

separate between

2 to the power 21 and 23 subsets of keys

so the contrastive models with vector

dimension 8 to 20

also do worse and worse finally

on shattering 27 numbers the model needs

to satisfy a vc dimension of at least 27

so all the models with vector dimensions

less than 27

fails to varying extents

notice that throughout these schemes the

contrasting model

models with vector dimension 5 12 27

and the non-contrasted model with half

the parameters performed

consistently well the top k overlap

metric

shows a similar agreeing trend here

higher is better so the plot looks like

the kl

loss plot flipped upside down to

understand why the models with shorter

vector representations are worse i

zoomed into one game for some analysis

this is a representative query from

erroneous predictions

it matches with 13 ground truth positive

keys

and 14 round trip negative keys

the perfect model normalizes the dot

products and distribute the probability

mass

evenly among the positive keys and

nothing among the negative keys

the contrasted model with vector

representation dimension

5 12 and 27 are not far from perfect

but if we drop the dimension to 20 or

below we start to see

mass moving to the negative keys until

the dimension 4

which performs about the same as a

completely even distribution

it seems that as we drop the vector

representation dimension

the model becomes lessly able to develop

a preference

among the keys using entropy of

predictions as a measure of this

phenomenon

what we saw from a previous example can

generalize to all the queries that

matches with

10 to positive ground through positive

key cards

the entropy of predictions increases as

vector dimension decreases

notice that queries in this game return

to

return up to 2 to the power 21 subsets

of keys

and starting with dimension 20 entropy

grows more dramatically

this trend is also reflected in the

aggregate of all queries in the test

sets

these entropies are increasingly or

monotonically as vector dimension

decreases

in addition to this 27 card version i

also trained an 81 card

version of this particular game the

setup is similar

the contrastive models range from vector

representation dimension

512 down to 16 or around

17 million parameters the

non-contrastive model is half the size

here are the plots the trends are

similar

kl loss decreases as the vector

dimension increases and top k overlap

metric increases with the vector

dimension

notice that the game has 81 cards so in

theory with a vector dimension of 81

or above we should be able to solve this

game perfectly

and we do see some agreeing trends here

here are a few things i learned from

training these models

more difficult games require more gentle

learning rate schedules

cosign schedules help to unstructured

models from local

minima in some cases efficiency of

conversions

depends on initialization schemes

using dot products instead of cosign

similarity

and temperature works better for this

data

overall i think these observations are

more specific to the nature of this

synthetic game

than contrastive models

besides what i just told you i also

worked on some adjacent topics

during the past six months they include

a variable binding

in which i did some literature reviews

on tensor products

and a study of dolly and eclipse failure

modes

and then i looked at a point mutual

information for classification

specifically comparing the info mce

versus the cross-entropy objective for

zero shot classification on toy problems

and i looked at a various uh versions of

game rules and operators

and observed that their ranks

the rank of their query key distribution

measures these changes

finally i want to uh thank you for

listening

and also uh thank my mentor gabe for

giving me a lot of valuable guidances

and

and feedback in these six months

um and and i’ll be update i’ll be

updating my blog

over the weekend um with the uh with the

final with the final blog post so feel

free to

check it out next week also feel free to

contact me for

uh further questions so now i look over

to the q a for for questions

um

so there is a question on uh on the

intuition

behind entropy going up

as the embedding dimension goes down

and so

let’s see let me go

to the

so uh from the error analysis that i saw

i saw that um

as the vector dimension goes down the

the vector size the vectors themselves

become less able to

encode enough signals that allow them to

uh kind of separate the the different

keys

um but i would be curious to do more

investigation on that front but in

general the observation is that as the

vector dimension goes down

the models become less and less able to

develop a preference among the keys

and then so there’s a question on the

intuition on rank

let me also go to the corresponding

slide

so the intuition is that if our query

key matrix it has

rank uh has a is full rank

so in this case if i have uh if i have

less keys than my queries this full rank

matrix would have the rank equal to

the support of the keys i’ll let’s say

27

and if my um if my

vector representation sizes are less

than 27

when i take dot products between my

queries and keys

it’s mathematically equivalent to doing

a matrix multiplication

of the keys matrix that has size

support keys by by a number that’s less

than support size of keys

with the query matrix that

has a dimension that on the on one

dimension that’s less than

support of keys and other being the

support of queries

so these vectors are shorter than the

than the full rank

27 and so um so when we do a matrix

multiplication between these two

matrices

the the m prime matrix is not going to

be

full rank it would have at at most the

rank of

the whatever the the vector

representation

dimension is um and then there is also a

question of

if the non-contrastive uh oh

so i i see that my time is up so i’m

gonna uh

uh hand the time over to sam and uh look

at the

look at the the third q a question of

line and make a reply