Efficient Large Deviation Estimation Based on Importance Sampling

Abstract

We present a complete framework for determining the asymptotic (or logarithmic) efficiency of estimators of large deviation probabilities and rate functions based on importance sampling. The framework relies on the idea that importance sampling in that context is fully characterized by the joint large deviations of two random variables: the observable defining the large deviation probability of interest and the likelihood factor (or Radon–Nikodym derivative) connecting the original process and the modified process used in importance sampling. We recover with this framework known results about the asymptotic efficiency of the exponential tilting and obtain new necessary and sufficient conditions for a general change of process to be asymptotically efficient. This allows us to construct new examples of efficient estimators for sample means of random variables that do not have the exponential tilting form. Other examples involving Markov chains and diffusions are presented to illustrate our results.

Appendices

Appendix A: Convex Analysis

We collect in this section basic results of convex analysis used in the paper in relation to the rate function $I_Q^B(w)$, defined in (50), and its Legendre–Fenchel transform $\lambda _Q^B(k)$, defined in (57). Both are functions of a single real variable, so we state the necessary results only for this simple case. We assume further that all convex functions are proper closed convex functions. For more general results and proofs, we refer to [76,77,78].

1.1 Subdifferentials

Let $f:{\mathbb {R}}\rightarrow {\bar{{\mathbb {R}}}}$ be a real function taking values in the set of extended reals ${\bar{{\mathbb {R}}}}$. The subdifferential $\partial f(x)$ of f at the point x is the set of all values $k\in {\mathbb {R}}$ such that

$$\begin{aligned} f(y)\ge f(x) +k(y-x) \end{aligned}$$

(A.1)

for all $y\in {\mathbb {R}}$ [76, Sect. 23]. Put differently, and as illustrated in Fig. 7a, $\partial f(x)$ is the set of slopes of all possible supporting lines of f at x. If f has not supporting line at x, then $\partial f(x)=\emptyset$. We will see next that this may happen when f is nonconvex.

For convex functions, subdifferentials exist everywhere in the domain of f(x), except possibly at boundary points [76, Theorem 23.4]. For this class of functions, we have in fact $\partial f(x) = [f'(x^-),f'(x^+)]$, where $f'(x^-)$ is the left-derivative and $f'(x^+)$ the right-derivative [76, Theorem 24.3]. If these are equal, f is differentiable at x so that $\partial f(x) = \{f'(x)\}$ [76, Theorem 25.1]. In all cases, $\partial f(x)$ is a closed convex interval [76, p. 215].

Fig. 7

a Function f(x) with a unique supporting line at the point a, no supporting line at the point b, and many supporting lines at the point c, leading to $\partial f(a)=\{f'(a)\}$, $\partial f(b)=\emptyset$, and $\partial f(c)=[f'(c^-),f'(c^+)]$. b Function f(x) and its convex envelope $f^{**}(x)$

1.2 Legendre–Fenchel Transforms

The Legendre–Fenchel transform of f is the real function defined by

$$\begin{aligned} f^*(k) = \sup _{x\in {\mathbb {R}}} \{kx-f(x)\},\qquad k\in {\mathbb {R}}. \end{aligned}$$

(A.2)

This function is also called the dual or conjugate of f and has the property of being convex [76, Theorem 12.2]. The double dual or biconjugate of f is the Legendre–Fenchel of $f^*$:

$$\begin{aligned} f^{**}(x) = \sup _{k\in {\mathbb {R}}} \{kx-f^*(k)\}. \end{aligned}$$

(A.3)

This is also a convex function, corresponding to the convex envelope or convex hull of f [77, Theorem 11.1], as illustrated in Fig. 7b.

With this geometric interpretation of $f^{**}$, it is natural to say that x is a convex point of f if $f(x)=f^{**}(x)$ and a nonconvex point of f if $f(x)\ne f^{**}(x)$. An important result proved in [68, Lem. 4.1] is that the set of convex points coincides with the set of points admitting supporting lines, except possibly at boundary points. With this proviso, we then have $f(x)=f^{**}(x)$ if and only if $\partial f(x)\ne \emptyset$. This is illustrated in Fig. 7a. The same result also implies that, if $f(x)=f^{**}(x)$, then $\partial f(x)=\partial f^{**}(x)$.

In this paper, we deal with rate functions, which always have at least one global minimum. Denoting one such minimizer by $x^*$, we then have $0\in \partial f(x^*)$. Hence, $x^*$ is a convex point such that $f(x^*)=f^{**}(x^*)$ and $\partial f(x^*)=\partial f^{**}(x^*)$.

1.3 Duality

The proof of our main result, Theorem 4, is based on another important result about convex functions stating (see [76, Cor. 23.5.1] or [77, Prop. 11.3]) that

$$\begin{aligned} k\in \partial f(x)\iff x\in \partial f^*(k). \end{aligned}$$

(A.4)

This property expresses a form of duality or conjugacy between the slopes of f and the slopes of $f^*$, illustrated in Fig. 8a. From this result, it is easy to see that convex, affine parts of f correspond to cusps of $f^*$, and vice versa, as shown in Fig. 8b.

The duality in (A.4) also holds for $f^{**}$, since this function is convex and is the Legendre–Fenchel transform of $f^*$. Therefore,

$$\begin{aligned} k\in \partial f^{**}(x)\iff x\in \partial f^*(k). \end{aligned}$$

(A.5)

This result implies that $f^*$ has a cusp also when f is nonconvex, as shown in Fig. 8, since $f^{**}$ is affine where f is nonconvex. Thus, $f^*$ has a cusp either if f is affine or f is nonconvex.

Since subdifferentials of f and $f^{**}$ match at convex points, it is also clear from (A.5) that the first duality (A.4) holds locally at these points even if f is not globally convex. We use this result in this paper when dealing with the subdifferential of $I_Q^B$ at its global minimum $w^*$, which is a convex point, as mentioned. In this case, the first duality result can be applied at that point even though $I_Q^B$ might be nonconvex at other points, as in Figs. 2c or 6.

Fig. 8

a Illustration of the duality between the slopes of f(x) and the slopes of its Legendre–Fenchel transform $f^*(k)$. b Functions with affine or nonconvex parts give rise to a Legendre–Fenchel transform having a cusp

Appendix B: Contraction Principle

The contraction principle is an important result in large deviation theory relating the rate functions of random variables that can be mapped to one another. Let $(A_n)_{n>0}$ be a sequence of random variables satisfying the LDP with good rate function $I_A$ and let $(B_n)_{n>0}$ be another sequence such that $B_n=f(A_n)$ with f continuous. Then $(B_n)_{n>0}$ also satisfies the LDP with good rate function

$$\begin{aligned} I_B(b)=\inf _{a:f(a)=b} I_A(a). \end{aligned}$$

(A.6)

See [6, Theorem 4.2.1] for details.

Instead of considering a single continuous function f as the contraction, one can also consider a sequence $(f_n)_{n>0}$ of continuous functions. In this case, the contraction principle also applies provided that $f_n$ is “close enough” to f with respect to $P_n$. To be more precise, let ${\mathcal {A}}$ denote the space of $A_n$ and define

$$\begin{aligned} \Gamma _{n,\delta }=\{a\in {\mathcal {A}}: \Vert f_n(a)-f(a)\Vert >\delta \} \end{aligned}$$

(A.7)

as the set of points for which $f_n$ differs from f by at least $\delta >0$ with respect to any metric $\Vert \cdot \Vert$ on ${\mathcal {B}}$, the space of $B_n$. Then, according to [6, Cor. 4.2.21], $B_n=f_n(A_n)$ satisfies the LDP with good rate function $I_B$ given by (A.6) with f as the contraction if, for all $\delta >0$,

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{n}\log P_n(\Gamma _{n,\delta })=-\infty . \end{aligned}$$

(A.8)

This condition only means that the probability that $f_n$ differs from f decreases faster than exponentially with n in the large deviation limit. This is met in most cases when $f_n$ is smooth and $I_A$ is a good rate function.

Two particular applications of this result are considered in the paper.

Example 4

Consider two real random variables $A_n$ and $B_n$ related by the simple rescaling $B_n=c_n A_n$ with $c_n\rightarrow 1$ as $n\rightarrow \infty$. Here, the limit function is the identity $f(a)=a$, so one expects $A_n$ and $B_n$ to have the same rate function. This is verified by noting that, for every $M>0$, there exists $n_0=n_0(M,\delta )$ such that for all $n\ge n_0$, one has $\Gamma _{n,\delta }\subseteq (-\infty ,-M]\cup [M,\infty )$. Therefore, from the definition of the LDP, we obtain

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{1}{n}\log P_n(\Gamma _{n,\delta })\le -\inf _{|a|\ge M} I_A(a). \end{aligned}$$

(A.9)

But, since the rate function $I_A$ of $A_n$ is good, it is coercive, so that

$$\begin{aligned} \lim _{|a|\rightarrow \infty } I_A(a)=\infty . \end{aligned}$$

(A.10)

Therefore, the limit on the left-hand side of (A.9) must give $-\infty$, implying $I_B(b) = I_A(b)$ from the condition (A.8).

Example 5

Let $B_n =f(A_n)+c_n$ with $c_n\rightarrow c$. Then the rate function of $B_n$ is obtained from (A.6) with the contraction $B_n=f(A_n)+c$. This follows trivially because the distance between $f(a)+c_n$ and $f(a)+c$ is constant in a. Since $c_n\rightarrow c$, there must be an n beyond which $|c_n-c|<\delta$, leading to $P_n(\Gamma _{n,\delta })=0$, so the condition (A.8) is also satisfied.

These results also hold if $\Gamma _{n,\delta }$ is defined on a subset of ${\mathcal {A}}$, since any restriction or constraint on $A_n$ can be included in the definition of $f_n$. This arises, for example, when considering the contraction of $J_Q(m,w)$ to $I_Q^B(w)$, which involves the restriction $m\in B$.