Bregman divergence

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 Lev M. Bregman, who introduced the concept in 1967.

Definition[]

Let ${\displaystyle F:\Omega \to \mathbb {R} }$ be a continuously-differentiable, strictly convex function defined on a closed convex set ${\displaystyle \Omega }$.

The Bregman distance associated with F for points ${\displaystyle p,q\in \Omega }$ 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:

${\displaystyle D_{F}(p,q)=F(p)-F(q)-\langle \nabla F(q),p-q\rangle .}$

Properties[]

• Non-negativity: ${\displaystyle D_{F}(p,q)\geq 0}$ for all p, q. This is a consequence of the convexity of F.
• Convexity:${\displaystyle D_{F}(p,q)}$ is convex in its first argument, but not necessarily in the second argument (see ^[1])
• 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 ${\displaystyle F_{1},F_{2}}$ strictly convex and differentiable, and ${\displaystyle \lambda \geq 0}$,

${\displaystyle D_{F_{1}+\lambda F_{2}}(p,q)=D_{F_{1}}(p,q)+\lambda D_{F_{2}}(p,q)}$

• Duality: The function F has a convex conjugate ${\displaystyle F^{*}}$. The Bregman distance defined with respect to ${\displaystyle F^{*}}$ has an interesting relationship to ${\displaystyle D_{F}(p,q)}$

${\displaystyle D_{F^{*}}(p^{*},q^{*})=D_{F}(q,p)}$

Here, ${\displaystyle p^{*}=\nabla F(p)}$ and ${\displaystyle q^{*}=\nabla F(q)}$ are the dual points corresponding to p and q.

• 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.
• Law of cosines:^[2]

For any ${\displaystyle p,q,z}$

${\displaystyle D_{F}(p,q)=D_{F}(p,z)+D_{F}(z,q)-(p-z)^{T}(\nabla F(q)-\nabla F(z))}$

• Generalized Pythagorean Theorem:^[3]

Consider the "Bregman projection" of ${\displaystyle q}$ onto a convex set ${\displaystyle \Omega }$: ${\displaystyle P_{\Omega }(q)={\text{argmin}}_{\omega \in \Omega }D_{F}(\omega ,q)}$. The Bregman divergence is an obtuse triangle in the sense

${\displaystyle D_{F}(p,q)\geq D_{F}(p,P_{\Omega }(q))+D_{F}(P_{\Omega }(q),q).}$

Examples[]

• Squared Euclidean distance ${\displaystyle D_{F}(x,y)=\|x-y\|^{2}}$ is the canonical example of a Bregman distance, generated by the convex function ${\displaystyle F(x)=\|x\|^{2}}$
• The squared Mahalanobis distance, ${\displaystyle D_{F}(x,y)={\tfrac {1}{2}}(x-y)^{T}Q(x-y)}$ which is generated by the convex function ${\displaystyle F(x)={\tfrac {1}{2}}x^{T}Qx}$. This can be thought of as a generalization of the above squared Euclidean distance.
• The generalized Kullback–Leibler divergence

${\displaystyle D_{F}(p,q)=\sum _{i}p(i)\log {\frac {p(i)}{q(i)}}-\sum p(i)+\sum q(i)}$

is generated by the negative entropy function

${\displaystyle F(p)=\sum _{i}p(i)\log p(i)}$

• The Itakura–Saito distance,

${\displaystyle D_{F}(p,q)=\sum _{i}\left({\frac {p(i)}{q(i)}}-\log {\frac {p(i)}{q(i)}}-1\right)}$

is generated by the convex function

${\displaystyle F(p)=-\sum _{i}\log p(i)}$

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 ${\displaystyle p=(p_{1},\ldots p_{d})}$ to the hyperplane ${\displaystyle x_{d+1}=\sum _{1}^{d}2p_{i}x_{i}}$. 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 ${\displaystyle p^{*}=\nabla F(p)}$, where F defines the d-dimensional paraboloid ${\displaystyle x_{d+1}=\sum x_{i}^{2}}$.

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^[4] is obtained by taking a chord instead of a tangent line. Bregman divergences have been generalized by considering a pair of generators.^[5] These so-called duo Bregman divergences are obtained as the Kullback-Leibler divergence between densities of nested exponential families.

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 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.^[6]

Applications[]

In machine learning, Bregman divergences are used to calculate the bi-tempered logistic loss, performing better than the softmax function with noisy datasets.^[7]

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[]