Bregman divergence explained

In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions – notably as either values of the parameter of a parametric model or as a data set of observed values – the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance.

Bregman divergences are similar to metrics, but satisfy neither the triangle inequality (ever) nor symmetry (in general). However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as generalizations of least squares.

Bregman divergences are named after Russian mathematician Lev M. Bregman, who introduced the concept in 1967.

Definition

Let

F\colon\Omega\toR

be a continuously-differentiable, strictly convex function defined on a convex set

\Omega

The Bregman distance associated with F for points

p,q\in\Omega

is the difference between the value of F at point p and the value of the first-order Taylor expansion of F around point q evaluated at point p:

D_F(p,q)=F(p)-F(q)-\langle\nablaF(q),p-q\rangle.

Properties

Non-negativity:

D_F(p,q)\ge0

for all

. This is a consequence of the convexity of

Positivity: When

is strictly convex,

D_F(p,q)=0

iff

p=q

Uniqueness up to affine difference:

D_F=D_G

iff

F-G

is an affine function.

Convexity:

D_F(p,q)

is convex in its first argument, but not necessarily in the second argument. If F is strictly convex, then

D_F(p,q)

is strictly convex in its first argument.

- For example, Take f(x) = |x|, smooth it at 0, then take

y=1,x₁=0.1,x₂=-0.9,x₃=0.9x₁+0.1x₂

, then

D_f(y,x₃₎ ≈ 1>0.9D_f(y,x₁₎+0.1D_f(y,x₂₎ ≈ 0.2

Linearity: If we think of the Bregman distance as an operator on the function F, then it is linear with respect to non-negative coefficients. In other words, for

F_1,F₂

strictly convex and differentiable, and

λ\ge0

D
	F₁+λF₂

(p,q)=

D
	F₁

(p,q)+λ

D
	F₂

(p,q)

F^*

which is also strictly convex and continuously differentiable on some convex set

\Omega^*

. The Bregman distance defined with respect to

F^*

is dual to

D_F(p,q)

D
	F^*

(p^*,q^*)=D_F(q,p)

Here,

p^*=\nablaF(p)

and

q^*=\nablaF(q)

are the dual points corresponding to p and q.

Moreover, using the same notations :

D_F(p,q)=F(p)+F^*(q^*)-\langlep,q^*\rangle

Integral form: by the integral remainder form of Taylor's Theorem, a Bregman divergence can be written as the integral of the Hessian of

along the line segment between the Bregman divergence's arguments.

Mean as minimizer: A key result about Bregman divergences is that, given a random vector, the mean vector minimizes the expected Bregman divergence from the random vector. This result generalizes the textbook result that the mean of a set minimizes total squared error to elements in the set. This result was proved for the vector case by (Banerjee et al. 2005), and extended to the case of functions/distributions by (Frigyik et al. 2008). This result is important because it further justifies using a mean as a representative of a random set, particularly in Bayesian estimation.
Bregman balls are bounded, and compact if X is closed: Define Bregman ball centered at x with radius r by

B_f(x,r):=\left\{y\inX:D_f(y,x)\leqr\right\}

. When

X\subset\Rⁿ

is finite dimensional,

\forallx\inX

, if

is in the relative interior of

, or if

is locally closed at

(that is, there exists a closed ball

B(x,r)

centered at

, such that

B(x,r)\capX

is closed), then

B_f(x,r)

is bounded for all

. If

is closed, then

B_f(x,r)

is compact for all

Law of cosines:^[1]

For any

p,q,z

D_F(p,q)=D_F(p,z)+D_F(z,q)-(p-z)^T(\nablaF(q)-\nablaF(z))

Parallelogram law: for any

\theta,\theta_1,\theta₂

B_F\left(\theta₁:\theta\right)+B_F\left(\theta₂:\theta\right)=B_F\left(\theta₁:

	\theta₁+\theta₂
	2

\right)+B_F\left(\theta₂:

	\theta₁+\theta₂
	2

\right)+2B_F\left(

	\theta₁+\theta₂
	2

:\theta\right)

Bregman projection: For any

W\subset\Omega

, define the "Bregman projection" of

onto

P_W(q)=argmin_\omega\inD_F(\omega,q)

. Then

- if

is convex, then the projection is unique if it exists;

- if

is nonempty, closed, and convex and

\Omega\subset\Rⁿ

is finite dimensional, then the projection exists and is unique.^[2]

Generalized Pythagorean Theorem:^[1]

For any

v\in\Omega,a\inW

D_F(a,v)\geD_F(a,P_W(v))+D_F(P_W(v),v).

This is an equality if

P_W(v)

is in the relative interior of

In particular, this always happens when

is an affine set.

Lack of triangle inequality: Since the Bregman divergence is essentially a generalization of squared Euclidean distance, there is no triangle inequality. Indeed,

D_F(z,x)-D_F(z,y)-D_F(y,x)=\langle\nablaf(y)-\nablaf(x),z-y\rangle

, which may be positive or negative.

Proofs

Non-negativity and positivity: use Jensen's inequality.
Uniqueness up to affine difference: Fix some

x\in\Omega

, then for any other

y\in\Omega

, we have by definition

F(y)-G(y)=F(x)-G(x)+\langle\nablaF(x)-\nablaG(x),y-x\rangle

Convexity in the first argument: by definition, and use convexity of F. Same for strict convexity.
Linearity in F, law of cosines, parallelogram law: by definition.
Duality: See figure 1 of.^[3]
Bregman balls are bounded, and compact if X is closed:

Fix

x\inX

. Take affine transform on

, so that

\nablaf(x)=0

Take some

\epsilon>0

, such that

\partialB(x,\epsilon)\subsetX

. Then consider the "radial-directional" derivative of

on the Euclidean sphere

\partialB(x,\epsilon)

\langle\nablaf(y),(y-x)\rangle

for all

y\in\partialB(x,\epsilon)

Since

\partialB(x,\epsilon)\subset\Rⁿ

is compact, it achieves minimal value

\delta

at some

y_0\in\partialB(x,\epsilon)

Since

is strictly convex,

\delta>0

. Then

B_f(x,r)\subsetB(x,r/\delta)\capX

Since

D_f(y,x)

C¹

D_f

is continuous in

, thus

B_f(x,r)

is closed if

is.

Projection

P_W

is well-defined when

is closed and convex.

Fix

v\inX

. Take some

w\inW

, then let

r:=D_f(w,v)

. Then draw the Bregman ball

B_f(v,r)\capW

. It is closed and bounded, thus compact. Since

D_{f( ⋅ ,}v)

is continuous and strictly convex on it, and bounded below by

, it achieves a unique minimum on it.

Pythagorean inequality.

By cosine law,

D_f(w,v)-D_f(w,P_W(v))-D_f(P_W(v),v)=\langle\nabla_yD_f(y,

v)\|
	y=P_W(v)

,w-P_W(v)\rangle

, which must be

\geq0

, since

P_W(v)

minimizes

D_{f( ⋅ ,}v)

, and

is convex.

Pythagorean equality when

P_W(v)

is in the relative interior of

\langle\nabla_yD_f(y,

v)\|
	y=P_W(v)

,w-P_W(v)\rangle>0

, then since

is in the relative interior, we can move from

P_W(v)

in the direction opposite of

, to decrease

D_f(y,v)

, contradiction.

Thus

\langle\nabla_yD_f(y,

v)\|
	y=P_W(v)

,w-P_W(v)\rangle=0

Classification theorems

The only symmetric Bregman divergences on

X\subset\Rⁿ

are squared generalized Euclidean distances (Mahalanobis distance), that is,

D_f(y,x)=(y-x)^TA(y-x)

for some positive definite

.^[4]

The following two characterizations are for divergences on

\Gamma_n

, the set of all probability measures on

\{1,2,...,n\}

, with

n\geq2

Define a divergence on

\Gamma_n

as any function of type

D:\Gamma_n x \Gamma_n\to[0,infty]

, such that

D(x,x)=0

for all

x\in\Gamma_n

, then:

The only divergence on

\Gamma_n

that is both a Bregman divergence and an f-divergence is the Kullback–Leibler divergence.^[5]

n\geq3

, then any Bregman divergence on

\Gamma_n

that satisfies the data processing inequality must be the Kullback–Leibler divergence. (In fact, a weaker assumption of "sufficiency" is enough.) Counterexamples exist when

n=2

.Given a Bregman divergence

D_F

, its "opposite", defined by

	*(v,
D
	F

w)=D_F(w,v)

, is generally not a Bregman divergence. For example, the Kullback-Leiber divergence is both a Bregman divergence and an f-divergence. Its reverse is also an f-divergence, but by the above characterization, the reverse KL divergence cannot be a Bregman divergence.

Examples

	T
D
	F(x,y)=\tfrac{1}{2}(x-y)

Q(x-y)

is generated by the convex quadratic form

F(x)=\tfrac{1}{2}x^TQx

The canonical example of a Bregman distance is the squared Euclidean distance

D_F(x,y)=\|x-y\|²

. It results as the special case of the above, when

is the identity, i.e. for

F(x)=\|x\|²

. As noted, affine differences, i.e. the lower orders added in

, are irrelevant to

D_F

The generalized Kullback–Leibler divergence

D_F(p,q)=\sum_ip(i)log

	p(i)
	q(i)

-\sump(i)+\sumq(i)

is generated by the negative entropy function

F(p)=\sum_ip(i)logp(i)

When restricted to the simplex, the last two terms cancel, giving the usual Kullback–Leibler divergence for distributions.

The Itakura–Saito distance,

D_F(p,q)=\sum_i\left(

	p(i)
	q(i)

-log

	p(i)
	q(i)

-1\right)

is generated by the convex function

F(p)=-\sum_ilogp(i)

Generalizing projective duality

A key tool in computational geometry is the idea of projective duality, which maps points to hyperplanes and vice versa, while preserving incidence and above-below relationships. There are numerous analytical forms of the projective dual: one common form maps the point

p=(p_1,\ldotsp_d)

to the hyperplane

x_d+1=

	d
\sum
	1

2p_ix_i

. This mapping can be interpreted (identifying the hyperplane with its normal) as the convex conjugate mapping that takes the point p to its dual point

p^*=\nablaF(p)

, where F defines the d-dimensional paraboloid

x_d+1=\sum

	2
x
	i

If we now replace the paraboloid by an arbitrary convex function, we obtain a different dual mapping that retains the incidence and above-below properties of the standard projective dual. This implies that natural dual concepts in computational geometry like Voronoi diagrams and Delaunay triangulations retain their meaning in distance spaces defined by an arbitrary Bregman divergence. Thus, algorithms from "normal" geometry extend directly to these spaces (Boissonnat, Nielsen and Nock, 2010)

Generalization of Bregman divergences

Bregman divergences can be interpreted as limit cases of skewed Jensen divergences (see Nielsen and Boltz, 2011). Jensen divergences can be generalized using comparative convexity, and limit cases of these skewed Jensen divergences generalizations yields generalized Bregman divergence (see Nielsen and Nock, 2017).The Bregman chord divergence^[6] is obtained by taking a chord instead of a tangent line.

Bregman divergence on other objects

Bregman divergences can also be defined between matrices, between functions, and between measures (distributions). Bregman divergences between matrices include the Stein's loss and von Neumann entropy. Bregman divergences between functions include total squared error, relative entropy, and squared bias; see the references by Frigyik et al. below for definitions and properties. Similarly Bregman divergences have also been defined over sets, through a submodular set function which is known as the discrete analog of a convex function. The submodular Bregman divergences subsume a number of discrete distance measures, like the Hamming distance, precision and recall, mutual information and some other set based distance measures (see Iyer & Bilmes, 2012 for more details and properties of the submodular Bregman.)

For a list of common matrix Bregman divergences, see Table 15.1 in.^[7]

Applications

In machine learning, Bregman divergences are used to calculate the bi-tempered logistic loss, performing better than the softmax function with noisy datasets.^[8]

Bregman divergence is used in the formulation of mirror descent, which includes optimization algorithms used in machine learning such as gradient descent and the hedge algorithm.

References

Banerjee . Arindam . Merugu . Srujana . Dhillon . Inderjit S. . Ghosh . Joydeep . . 1705–1749 . Clustering with Bregman divergences . 6 . 2005.
Bregman . L. M. . 10.1016/0041-5553(67)90040-7 . USSR Computational Mathematics and Mathematical Physics . 200–217 . The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming . 7 . 1967 . 3.
Frigyik . Bela A. . Srivastava . Santosh . Gupta . Maya R. . 10.1109/TIT.2008.929943 . . 5130–5139 . Functional Bregman Divergences and Bayesian Estimation of Distributions . 54 . 2008 . 11 . dead . https://web.archive.org/web/20100812221422/http://www.ee.washington.edu/research/guptalab/publications/FrigyikSrivastavaGupta.pdf . 2010-08-12 . cs/0611123 . 1254 .
Iyer . Rishabh . Bilmes . Jeff . . Submodular-Bregman divergences and Lovász-Bregman divergences with Applications . 2012.
Book: Frigyik . Bela A. . Srivastava . Santosh . Gupta . Maya R. . University of Washington, Dept. of Electrical Engineering . UWEE Tech Report 2008-0001 . An Introduction to Functional Derivatives . 2008 . 2014-03-20 . https://web.archive.org/web/20170217025324/https://www2.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf . 2017-02-17 . dead .
Harremoës . Peter . Divergence and Sufficiency for Convex Optimization . Entropy . 10.3390/e19050206 . 19 . 5 . 206 . 2017. 1701.01010. 2017Entrp..19..206H. free .
Nielsen . Frank . Nock . Richard . The dual Voronoi diagrams with respect to representational Bregman divergences . 10.1109/ISVD.2009.15 . IEEE . Proc. 6th International Symposium on Voronoi Diagrams . 2009.
Nielsen . Frank . Nock . Richard . 0711.3242 . On the Centroids of Symmetrized Bregman Divergences . 2007 . cs.CG.
Nielsen . Frank . Boissonnat . Jean-Daniel . Nock . Richard . Visualizing Bregman Voronoi diagrams . Proc. 23rd ACM Symposium on Computational Geometry (video track) . 10.1145/1247069.1247089 . 2007 .
Boissonnat . Jean-Daniel . Jean-Daniel Boissonnat . Nielsen . Frank . Nock . Richard . Bregman Voronoi Diagrams . . 44 . 2 . 10.1007/s00454-010-9256-1 . 2010 . 281–307. 0709.2196. 1327029 .
Nielsen . Frank . Nock . Richard . On approximating the smallest enclosing Bregman Balls . 10.1145/1137856.1137931 . 485–486 . Proc. 22nd ACM Symposium on Computational Geometry . 2006.
Nielsen . Frank . Boltz . Sylvain . The Burbea-Rao and Bhattacharyya centroids . . 57 . 8 . 10.1109/TIT.2011.2159046 . 2011 . 5455–5466. 1004.5049. 14238708 .
Nielsen . Frank . Nock . Richard . Generalizing Skew Jensen Divergences and Bregman Divergences With Comparative Convexity . . 24 . 8 . 1702.04877 . 10.1109/LSP.2017.2712195 . 2017 . 1123–1127. 2017ISPL...24.1123N. 31899023 .

Notes and References

Web site: Learning with Bregman Divergences. utexas.edu. 19 August 2023.
Dhillon. Inderjit. Inderjit Dhillon. Tropp. Joel. Matrix Nearness Problems with Bregman Divergence. SIAM Journal on Matrix Analysis and Applications. 29. 4. 2008. Supposed D_\varphi is a Bregman divergence, supposed that is a finite collection of closed, convex sets whose intersection is nonempty. Given an input matrix Y our goal is to produce a matrix \mathbf in the intersection that diverges the least from \textbf, i.e. to solve \min_ D_\varphi(\mathbf;\mathbf) subject to \mathbf \in \big\cap_k C_k. Under mild conditions, the solution is unique and it has a variational characterization analogous with the characterization of an orthogonal projection onto a convex set" (see s2.4, page 1125 for more).
Nielsen . Frank . 2021-10-28 . Fast Approximations of the Jeffreys Divergence between Univariate Gaussian Mixtures via Mixture Conversions to Exponential-Polynomial Distributions . Entropy . 23 . 11 . 1417 . 10.3390/e23111417 . 34828115 . 8619509 . 2107.05901 . 2021Entrp..23.1417N . 1099-4300. free .
Nielsen . Frank. Boissonnat . Jean-Daniel . Jean-Daniel Boissonnat. Nock . Richard. September 2010. Bregman Voronoi Diagrams: Properties, Algorithms and Applications. Discrete & Computational Geometry. 44. 2. 281–307. 10.1007/s00454-010-9256-1. 0179-5376. 0709.2196. 1327029 .
Jiao . Jiantao . Courtade . Thomas . No . Albert . Venkat . Kartik . Weissman . Tsachy . December 2014 . Information Measures: the Curious Case of the Binary Alphabet . IEEE Transactions on Information Theory . 60 . 12 . 7616–7626 . 10.1109/TIT.2014.2360184 . 0018-9448. 1404.6810 . 13108908 .
Book: 1810.09113. Nielsen. Frank. Geometric Science of Information. Nock. Richard. The Bregman Chord Divergence. Lecture Notes in Computer Science. 2019. 11712. 299–308. 10.1007/978-3-030-26980-7_31. 978-3-030-26979-1. 53046425.
"Matrix Information Geometry", R. Nock, B. Magdalou, E. Briys and F. Nielsen, pdf, from this book
Ehsan Amid, Manfred K. Warmuth, Rohan Anil, Tomer Koren (2019). "Robust Bi-Tempered Logistic Loss Based on Bregman Divergences". Conference on Neural Information Processing Systems. pp. 14987-14996. pdf