Outreach at Ravenswood Middle School


Over the course of the quarter students in EE276 have been preparing for an outreach event at Ravenswood Middle School. The class will be teaching about a wide range of topics related to information theory. Some teams are talking about mapping political landscapes while others delve into the theory of code breaking. There are groups demonstrating applications to the military when flying jets and others showing information theory through Fortnite. From all of us at Stanford, we are excited to share what we’ve learned this quarter and help inspire the next group of scientists and engineers!

We hope everyone is getting excited for a great afternoon!

Online Lecture: Information-theoretic Lower Bounds

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Lecture 9: Multiple Hypothesis Testing: More Examples

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(Warning: These materials may be subject to lots of typos and errors. We are grateful if you could spot errors and leave suggestions in the comments, or contact the author at yjhan@stanford.edu.) 

In the last lecture we have seen the general tools and concrete examples of reducing the statistical estimation problem to multiple hypothesis testing. In these examples, the loss function is typically straightforward and the construction of the hypotheses is natural. However, other than statistical estimation the loss function may become more complicated (e.g., the excess risk in learning theory), and the hypothesis constructions may be implicit. To better illustrate the power of the multiple hypothesis testing, this lecture will be exclusively devoted to more examples in potentially non-statistical problems. 

1. Example I: Density Estimation 

This section considers the most fundamental problem in nonparametric statistics, i.e., the density estimation. It is well-known in the nonparametric statistics literature that, if a density on the real line has Hölder smoothness parameter s>0, it then can be estimated within L_2 accuracy \Theta(n^{-s/(2s+1)}) based on n samples. However, in this section we will show that this is no longer the correct minimax rate when we generalize to other notions of smoothness and other natural loss functions. 

Fix an integer k\ge 1 and some norm parameter p\in [1,\infty], the Sobolev space \mathcal{W}^{k,p}(L) on [0,1] is defined as 

\mathcal{W}^{k,p}(L) := \{f\in C[0,1]: \|f\|_p + \|f^{(k)}\|_p \le L \},

where the derivative f^{(k)} is defined in terms of distributions (f not necessarily belongs to C^k[0,1]). Our target is to determine the minimax rate R_{n,k,p,q}^\star of estimating the density f\in \mathcal{W}^{k,p}(L) based on n i.i.d. observations from f, under the general L_q loss with q\in [1,\infty). For simplicity we suppress the dependence on L and assume that L is a large positive constant.

The main result of this section is as follows: 

R_{n,k,p,q}^\star = \begin{cases} \Omega(n^{-\frac{k}{2k+1}}) & \text{if } q<(1+2k)p, \\ \Omega((\frac{\log n}{n})^{\frac{k-1/p + 1/q}{2(k-1/p) + 1/q}} ) & \text{if } q \ge (1+2k)p. \end{cases}

This rate is also tight. We see that as k,p are fixed and q increases from 1 to \infty, the minimax rate for density estimation increases from n^{-k/(2k+1)} to (\log n/n)^{(k-1/p)/(2(k-1/p)+1) }. We will show that these rates correspond to the “dense” and the “sparse” cases, respectively.

1.1. Dense case: q<(1+2k)p 

We first look at the case with q<(1+2k)p, which always holds for Hölder smoothness where the norm parameter is p=\infty. The reason why it is called the “dense” case is that in the hypothesis construction, the density is supported everywhere on [0,1], and the difficulty is to classify the directions of the bumps around some common density. Specifically, let h\in (0,1) be a bandwidth parameter to be specified later, we consider 

f_v(x) := 1 + \sum_{i=1}^{h^{-1}} v_i\cdot h^sg\left(\frac{x - (i-1)h}{h}\right)

for v\in \{\pm 1\}^{1/h}, where g is a fixed smooth function supported on [0,1] with \int_0^1 g(x)dx=0, and s>0 is some tuning parameter to ensure that f_v\in \mathcal{W}^{k,p}(L). Since

f_v^{(k)}(x) = \sum_{i=1}^{h^{-1}} v_i\cdot h^{s-k}g^{(k)}\left(\frac{x - (i-1)h}{h}\right),

we conclude that the choice s=k is sufficient. To invoke the Assoaud’s lemma, first note that for q=1 the separation condition is fulfilled with \Delta = \Theta(h^{k+1}). Moreover, for any v,v'\in \{\pm 1\}^{1/h} with d_{\text{H}}(v,v') = 1, we have

\chi^2(f_v^{\otimes n}, f_{v'}^{\otimes n}) + 1 \le \left( 1 + \frac{ \|f_v - f_{v'}\|_2^2 }{1 - h^k\|g\|_\infty} \right)^n = \left(1 + \frac{ 4h^{2k+1}\|g\|_2^2 }{1 - h^k\|g\|_\infty} \right)^n.

Hence, the largest h>0 to ensure that \chi^2(f_v^{\otimes n}, f_{v'}^{\otimes n}) = O(1) is h = \Theta(n^{-1/(2k+1)}), which for q=1 gives the minimax lower bound

R_{n,k,p,q}^\star = \Omega(n^{-\frac{k}{2k+1}} ).

This lower bound is also valid for the general case q\ge 1 due to the monotonicity of L_q norms.

1.2. Sparse case: q\ge (1+2k)p 

Next we turn to the sparse case q\ge (1+2k)p, where the name of “sparse” comes from the fact that for the hypotheses we will construct densities with only one bump on a small interval, and the main difficulty is to locate that interval. Specifically, let h\in (0,1), s>0 be bandwidth and smoothness parameters to be specified later, and we consider 

f_i(x) := h^sg\left(\frac{x-(i-1)h}{h}\right)\cdot 1\left((i-1)h<x\le ih\right) + (1-h^{s+1}),

where i\in [h^{-1}], and g is a fixed smooth density on [0,1]. Clearly f_i(x) is a density on [0,1] for all i\in [M], and

f_i^{(k)}(x) = h^{s-k}g^{(k)} \left(\frac{x-(i-1)h}{h}\right)\cdot 1\left((i-1)h<x\le ih\right).

Consequently, \|f_i^{(k)}\|_p = h^{s-k+1/p}\|g^{(k)}\|_p, and the Sobolev ball requirement leads to the choice s = k-1/p.

Next we check the conditions of Fano’s inequality. Since 

\|f_i - f_j\|_q = h^{k-1/p+1/q} \cdot 2^{1/q}\|g\|_q, \qquad \forall i\neq j\in [h^{-1}],

the separation condition is fulfilled with \Delta = \Theta(h^{k-1/p+1/q}). Moreover, since

D_{\text{KL}}(f_i \| f_j) \le \frac{\|f_i - f_j\|_2^2}{1-h^s} = \frac{2\|g\|_2^2}{1-h^s}\cdot h^{2(k-1/p)+1},

we conclude that I(V;X)= O(nh^{2(k-1/p)+1}). Consequently, Fano’s inequality gives

R_{n,k,p,q}^\star = \Omega\left(h^{k-1/p+1/q}\left(1 - \frac{O(nh^{2(k-1/p)+1})+\log 2}{\log(1/h)} \right) \right),

and choosing h = \Theta((\log n/n)^{1/(2(k-1/p)+1)} ) gives the desired result.

2. Example II: Aggregation 

In estimation or learning problems, sometimes the learner is given a set of candidate estimators or predictors, and she aims to aggregate them into a new estimate based on the observed data. In scenarios where the candidates are not explicit, aggregation procedures can still be employed based on sample splitting, where the learner splits the data into independent parts, uses the first part to construct the candidates and the second part to aggregate them. 

In this section we restrict ourselves to the regression setting where there are n i.i.d. observations x_1,\cdots,x_n\sim P_X, and 

y_i = f(x_i )+ \xi_i, \qquad i\in [n],

where \xi_i\sim \mathcal{N}(0,1) is the independent noise, and f: \mathcal{X}\rightarrow {\mathbb R} is an unknown regression function with \|f\|_\infty \le 1. There is also a set of candidates \mathcal{F}=\{f_1,\cdots,f_M\}, and the target of aggregation is to find some \hat{f}_n (not necessarily in \mathcal{F}) to minimize

\mathop{\mathbb E}_f \left(\|\hat{f}_n - f\|_{L_2(P_X)}^2 - \inf_{\lambda\in \Theta} \|f_\lambda - f\|_{L_2(P_X)}^2\right),

where the expectation is taken over the random observations (x_1,y_1),\cdots,(x_n,y_n), f_\lambda(x) := \sum_{i=1}^M \lambda_i f_i(x) for any \lambda \in {\mathbb R}^M, and \Theta\subseteq {\mathbb R}^p is a suitable subset corresponding to different types of aggregation. For a fixed data distribution P_X, the minimax rate of aggregation is defined as the minimum worst-case excess risk over all bounded functions f and candidate functions \{f_1,\cdots,f_M\}:

R_{n,M}^{\Theta} := \sup_{f_1,\cdots,f_M} \inf_{\hat{f}_n} \sup_{\|f\|_\infty\le 1} \mathop{\mathbb E}_f \left(\|\hat{f}_n - f\|_{L_2(P_X)}^2 - \inf_{\lambda\in \Theta} \|f_\lambda - f\|_{L_2(P_X)}^2\right).

Some special cases are in order:

  1. When \Theta = {\mathbb R}^M, the estimate \hat{f}_n is compared with the best linear aggregates and this is called linear aggregation, with optimal rate denoted as R^{\text{L}}
  2. When \Theta = \Lambda^M = \{\lambda\in {\mathbb R}_+^M: \sum_{i=1}^M \lambda_i\le 1 \}, the estimate \hat{f}_n is compared with the best convex combination of candidates (and zero) and this is called convex aggregation, with optimal rate denoted as R^{\text{C}}
  3. When \Theta = \{e_1,\cdots,e_M\}, the set of canonical vectors, the estimate \hat{f}_n is compared with the best candidate in \mathcal{F} and this is called model selection aggregation, with optimal rate denoted as R^{\text{MS}}

The main result in this section is summarized in the following theorem. 

Theorem 1 If there is a cube \mathcal{X}_0\subseteq \mathcal{X} such that P_X admits a density lower bounded from below w.r.t. the Lebesgue measure on \mathcal{X}_0, then

R_{n,M}^\Theta = \begin{cases} \Omega(1\wedge \frac{M}{n}) &\text{if } \Theta = {\mathbb R}^M, \\ \Omega(1\wedge (\frac{M}{n} + \sqrt{\frac{1}{n}\log(\frac{M}{\sqrt{n}}+1)}) ) & \text{if } \Theta = \Lambda^M, \\ \Omega(1\wedge \frac{\log M}{n}) & \text{if } \Theta = \{e_1,\cdots,e_M\}. \end{cases}

We remark that the rates in Theorem 1 are all tight. In the upcoming subsections we will show that although the loss function of aggregation becomes more complicated, the idea of multiple hypothesis testing can still lead to tight lower bounds. 

2.1. Linear aggregation 

Since the oracle term \inf_{\lambda \in {\mathbb R}^d} \|f_\lambda - f\|_{L^2(P_X)}^2 is hard to deal with, a natural idea would be to consider a well-specified model such that this term is zero. Since P_X admits a density, we may find a partition \mathcal{X} = \cup_{i=1}^M \mathcal{X}_i such that P_X(\mathcal{X}_i) = M^{-1} for all i. Consider the candidate functions f_i(x) = 1(x\in \mathcal{X}_i), and for v\in \{0, 1\}^M, let 

f_v(x) := \gamma\sum_{i=1}^M v_if_i(x),

where \gamma\in (0,1) is to be specified.

To apply the Assoaud’s lemma, note that for the loss function 

L(f,\hat{f}) := \|\hat{f} - f\|_{L_2(P_X)}^2 - \inf_{\lambda\in {\mathbb R}^p} \|f_\lambda - f\|_{L_2(P_X)}^2,

the orthogonality of the candidates (f_i)_{i\in [M]} implies that the separability condition holds for \Delta = \gamma^2/2M. Moreover,

\max_{d_{\text{H}}(v,v')=1 } D_{\text{KL}}(P_{f_v}^{\otimes n} \| P_{f_{v'}}^{\otimes n}) = \max_{d_{\text{H}}(v,v')=1 } \frac{n\|f_v - f_{v'}\|_{L_2(P_X)}^2}{2} = O\left(\frac{n\gamma^2}{M}\right),

Therefore, the Assoud’s lemma (with Pinsker’s inequality) gives

R_{n,M}^{\text{L}} \ge \frac{\gamma^2}{4}\left(1 - O\left( \sqrt{\frac{n\gamma^2}{M}}\right)\right).

Choosing \gamma = 1\wedge \sqrt{M/n} completes the proof.

2.2. Convex aggregation 

The convex aggregation differs only with the linear aggregation in the sense that the linear coefficients must be non-negative and the sum is at most one. Note that the only requirement in the previous arguments is the orthogonality of (f_i)_{i\in [M]} under L_2(P_X), we may choose any orthonormal functions (f_i)_{i\in [M]} under L_2(dx) with \max_i\|f_i\|_\infty = O(1) (existence follows from the cube assumption) and use the density lower bound assumption to conclude that \|f_i\|_{L_2(P_X)} = \Omega(\|f_i\|_2) (to ensure the desired separation). Then the choice of \gamma above becomes O(n^{-1/2}), and we see that the previous arguments still hold for convex aggregation if M = O(\sqrt{n}). Hence, it remains to prove that when M = \Omega(\sqrt{n})

R_{n,M}^{\text{C}} = \Theta\left(1\wedge \sqrt{\frac{1}{n}\log\left(\frac{M}{\sqrt{n}}+1\right) }\right).

Again we consider the well-specified case where 

f_v(x) := \gamma\sum_{i=1}^M v_if_i(x),

with the above orthonormal functions (f_i), a constant scaling factor \gamma = O(1), and a uniform size-m subset of (v_i) being 1/m and zero otherwise (m\in \mathbb{N} is to be specified). Since the vector v is no longer the vertex of a hypercube, we apply the generalized Fano’s inequality instead of Assoaud. First, applying Lemma 4 in Lecture 8 gives

I(V; X) \le \mathop{\mathbb E}_V D_{\text{KL}}(P_{f_V}^{\otimes n} \| P_{f_0}^{\otimes n}) = O\left(\frac{n}{m}\right).

Second, as long as m\le M/3, using similar arguments as in the sparse linear regression example in Lecture 8, for \Delta = c/m with a small constant c>0 we have

p_\Delta := \sup_{f} \mathop{\mathbb P}\left(\|f_V - f\|_{L^2(P_X)}^2 \le \Delta \right) \le \left(\frac{M}{m}+1\right)^{c'm},

with c'>0. Therefore, the generalized Fano’s inequality gives

R_{n,M}^{\text{C}} \ge \frac{c}{m}\left(1 - \frac{O(n/m) + \log 2}{c'm\log\left(\frac{M}{m}+1\right)} \right),

and choosing m \asymp \sqrt{n/\log(M/\sqrt{n}+1)} = O(M) completes the proof.

2.3. Model selection aggregation 

As before, we consider the well-specified case where we select orthonormal functions (\phi_i)_{i\in [M]} on L_2(dx), and let f_i = \gamma \phi_i, f\in \{f_1,\cdots,f_M\}. The orthonormality of (\phi_i) gives \Delta = \Theta(\gamma^2) for the separation condition of the original Fano’s inequality, and 

\max_{i\neq j} D_{\text{KL}}(P_{f_i}^{\otimes n}\| P_{f_j}^{\otimes n}) = \max_{i\neq j} \frac{\|f_i - f_j\|_{L_2(P_X)}^2}{2} = O\left(n\gamma^2\right).

Hence, Fano’s inequality gives

R_{n,M}^{\text{MS}} = \Omega\left(\gamma^2 \left(1 -\frac{n\gamma^2 + \log 2}{\log M}\right)\right),

and choosing \gamma^2 \asymp 1\wedge n^{-1}\log M completes the proof.

We may show a further result that any model selector cannot attain the above optimal rate of model selection aggregation. Hence, even to compare with the best candidate function in \mathcal{F}, the optimal aggregate \hat{f} should not be restricted to \mathcal{F}. Specifically, we have the following result, whose proof is left as an exercise. 

Exercise 1 Under the same assumptions of Theorem 1, show that

\sup_{f_1,\cdots,f_M} \inf_{\hat{f}_n\in\{f_1,\cdots,f_M\} } \sup_{\|f\|_\infty\le 1} \mathop{\mathbb E}_f \left(\|\hat{f}_n - f\|_{L_2(P_X)}^2 - \min_{i\in [M]} \|f_i - f\|_{L_2(P_X)}^2\right) = \Omega\left(1\wedge \sqrt{\frac{\log M}{n}}\right).

3. Example III: Learning Theory 

Consider the classification problem where there are n training data (x_1,y_1), \cdots, (x_n, y_n) i.i.d. drawn from an unknown distribution P_{XY}, with y_i \in \{\pm 1\}. There is a given collection of classifiers \mathcal{F} consisting of functions f: \mathcal{X}\rightarrow \{\pm 1\}, and given the training data, the target is to find some classifier \hat{f}\in \mathcal{F} with a small excess risk 

R_{\mathcal{F}}(P_{XY}, \hat{f}) = \mathop{\mathbb E}\left[P_{XY}(Y \neq \hat{f}(X)) - \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X))\right],

which is the difference in the performance of the chosen classifer and the best classifier in the function class \mathcal{F}. In the definition of the excess risk, the expectation is taken with respect to the randomness in the training data. The main focus of this section is to characterize the minimax excess risk of a given function class \mathcal{F}, i.e.,

R_{\text{pes}}^\star(\mathcal{F}) := \inf_{\hat{f}}\sup_{P_{XY}} R_{\mathcal{F}}(P_{XY}, \hat{f}).

The subscript “pes” here stands for “pessimistic”, where P_{XY} can be any distribution over \mathcal{X} \times \{\pm 1\} and there may not be a good classifier in \mathcal{F}, i.e., \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X)) may be large. We also consider the optimistic scenario where there exists a perfect (error-free) classifer in \mathcal{F}. Mathematically, denoting by \mathcal{P}_{\text{opt}}(\mathcal{F}) the collection of all probability distributions P_{XY} on \mathcal{X}\times \{\pm 1\} such that \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X))=0, the minimax excess risk of a given function class \mathcal{F} in the optimistic case is defined as

R_{\text{opt}}^\star(\mathcal{F}) := \inf_{\hat{f}}\sup_{P_{XY} \in \mathcal{P}_{\text{opt}}(\mathcal{F})} R_{\mathcal{F}}(P_{XY}, \hat{f}).

The central claim of this section is the following: 

Theorem 2 Let the VC dimension of \mathcal{F} be d. Then

R_{\text{\rm opt}}^\star(\mathcal{F}) = \Omega\left(1\wedge \frac{d}{n}\right), \qquad R_{\text{\rm pes}}^\star(\mathcal{F}) = \Omega\left(1\wedge \sqrt{\frac{d}{n}}\right).

Recall that the definition of VC dimension is as follows: 

Definition 3 For a given function class \mathcal{F} consisting of mappings from \mathcal{X} to \{\pm 1\}, the VC dimension of \mathcal{F} is the largest integer d such that there exist d points from \mathcal{X} which can be shattered by \mathcal{F}. Mathematically, it is the largest d>0 such that there exist x_1,\cdots,x_d\in \mathcal{X}, and for all v\in \{\pm 1\}^d, there exists a function f_v\in \mathcal{F} such that f_v(x_i) = v_i for all i\in [d]

VC dimension plays a significant role in statistical learning theory. For example, it is well-known that for the empirical risk minimization (ERM) classifier 

\hat{f}^{\text{ERM}} = \arg\min_{f\in \mathcal{F}} \frac{1}{n}\sum_{i=1}^n 1(y_i \neq f(x_i)),

we have

\mathop{\mathbb E}\left[P_{XY}(Y \neq \hat{f}^{\text{ERM}}(X)) - \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X))\right] \le 1\wedge \begin{cases} O(d/n) & \text{in optimistic case}, \\ O(\sqrt{d/n}) & \text{in pessimistic case}. \end{cases}

Hence, Theorem 2 shows that the ERM classifier attains the minimax excess risk for all function classes, and the VC dimension exactly characterizes the difficulty of the learning problem.

3.1. Optimistic case 

We first apply the Assoaud’s lemma to the optimistic scenario. By the definition of VC dimension, there exist points x_1,\cdots,x_d\in \mathcal{X} and functions f_v\in \mathcal{F} such that for all v\in \{\pm 1\}^d and i\in [d], we have f_v(x_i) = v_i. Consider 2^{d-1} hypotheses indexed by u\in \{\pm 1\}^{d-1}: the distribution P_X is always 

P_X(\{x_i\}) = p > 0, \quad \forall i\in [d-1], \qquad P_X(\{x_d\}) = 1-p(d-1),

where p\in (0,(d-1)^{-1}) is to be specified later. For the conditional distribution P_{Y|X}, let Y = f_{(u,1)}(X) hold almost surely under the joint distribution P_u. Clearly this is the optimistic case, for there always exists a perfect classifier in \mathcal{F}.

We first examine the separation condition in Assoaud’s lemma, where the loss function here is 

L(P_{XY}, \hat{f}) := P_{XY}(Y \neq \hat{f}(X)) - \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X)).

For any u,u'\in \{\pm 1\}^{d-1} and any \hat{f}\in \mathcal{F}, we have

\begin{array}{rcl} L(P_u, \hat{f}) + L(P_{u'}, \hat{f}) &=& p\cdot \sum_{i=1}^{d-1} 1(\hat{f}(x_i) \neq f_u(x_i)) + 1(\hat{f}(x_i) \neq f_{u'}(x_i)) \\ &=& p\cdot \sum_{i=1}^{d-1} 1(\hat{f}(x_i) \neq u_i) + 1(\hat{f}(x_i) \neq u_i') \\ &\ge & p\cdot d_{\text{H}}(u,u'), \end{array}

and therefore the separation condition holds for \Delta = p. Moreover, if u and u' only differ in the i-th component, then P_u and P_{u'} are completely indistinguishable if x_i does not appear in the training data. Hence, by coupling,

\max_{d_{\text{H}}(u,u')=1 } \|P_u^{\otimes n} - P_{u'}^{\otimes n}\|_{\text{\rm TV}} \le \left(1 - p\right)^n.

Therefore, Assoaud’s lemma gives

R_{\text{opt}}^\star(\mathcal{F}) \ge \frac{(d-1)p}{2}\left(1 - (1-p)^n\right),

and choosing p = \min\{n^{-1},(d-1)^{-1}\} yields to the desired lower bound.

3.2. Pessimistic case 

The analysis of the pessimistic case only differs in the construction of the hypotheses. As before, fix x_1,\cdots,x_d\in \mathcal{X} and f_v\in \mathcal{F} such that f_v(x_i) = v_i for all v\in \{\pm 1\}^d and i\in [d]. Now let P_X be the uniform distribution on \{x_1,\cdots,x_d\}, and under P_v, the conditional probability P_{Y|X} is 

P_v(Y= v_i | X=x_i) = \frac{1}{2} + \delta, \qquad \forall i\in [d],

and \delta\in (0,1/2) is to be specified later. In other words, the classifier f_v only outperforms the random guess by a margin of \delta under P_v.

Again we apply the Assoaud’s lemma for the lower bound of R_{\text{pes}}^\star(\mathcal{F}). First note that for all v\in \{\pm 1\}^d

\min_{f^\star \in \mathcal{F}} P_v(Y \neq f^\star(X)) = \frac{1}{2} - \delta.

Hence, for any v\in \{\pm 1\}^d and any f\in \mathcal{F},

\begin{array}{rcl} L(P_v, f) &=& \frac{1}{d}\sum_{i=1}^d \left[\frac{1}{2} + \left(2\cdot 1(f(x_i) = v_i) - 1\right)\delta \right] - \left(\frac{1}{2}-\delta\right) \\ &=& \frac{2\delta}{d}\sum_{i=1}^d 1(f(x_i) = v_i). \end{array}

By triangle inequality, the separation condition is fulfilled with \Delta = 2\delta/d. Moreover, direct computation yields

\max_{d_{\text{H}}(v,v')=1 } H^2(P_v, P_{v'}) = \frac{1}{d}\left(\sqrt{\frac{1}{2}+\delta} - \sqrt{\frac{1}{2}-\delta}\right)^2 = O\left(\frac{\delta^2}{d}\right),

and therefore tensorization gives

\max_{d_{\text{H}}(v,v')=1 } H^2(P_v^{\otimes n}, P_{v'}^{\otimes n}) = 2 - 2\left(1 - O\left(\frac{\delta^2}{d}\right) \right)^n.

Finally, choosing \delta = (1\wedge \sqrt{d/n})/2 gives the desired result.

3.3. General case 

We may interpolate between the pessimistic and optimistic cases as follows: for any given \varepsilon\in (0,1/2], we restrict to the set \mathcal{P}(\mathcal{F},\varepsilon) of joint distributions with 

\sup_{P_{XY}\in \mathcal{P}(\mathcal{F},\varepsilon) }\inf_{f^\star\in \mathcal{F}} P_{XY}(Y\neq f^\star(X))\le \varepsilon.

Then we may define the minimax excess risk over \mathcal{P}(\mathcal{F},\varepsilon) as

R^\star(\mathcal{F},\varepsilon) := \inf_{\hat{f}}\sup_{P_{XY} \in \mathcal{P}(\mathcal{F},\varepsilon)} R_{\mathcal{F}}(P_{XY}, \hat{f}).

Clearly the optimistic case corresponds to \varepsilon =0, and the pessimistic case corresponds to \varepsilon = 1/2. Similar to the above arguments, we have the following lower bound on R^\star(\mathcal{F},\varepsilon) , whose proof is left as an exercise.

Exercise 2 Show that when the VC dimension of \mathcal{F} is d, then

R^\star(\mathcal{F},\varepsilon) = \Omega\left(1\wedge \left(\frac{d}{n} + \sqrt{\frac{d}{n}\cdot \varepsilon}\right) \right).

4. Example IV: Stochastic Optimization 

Consider the following oracle formulation of the convex optimizaton: let f: \Theta\rightarrow {\mathbb R} be the convex objective function, and we aim to find the minimum value \min_{\theta\in\Theta} f(\theta). To do so, the learner can query some first-order oracle adaptively, where given the query \theta_t\in \Theta the oracle outputs a pair (f(\theta_t), g(\theta_t)) consisting of the objective value at \theta_t (zeroth-order information) and a sub-gradient of f at \theta_t (first-order information). The queries can be made in an adaptive manner where \theta_t can depend on all previous outputs of the oracle. The target of convex optimization is to determine the minimax optimization gap after T queries defined as 

R_T^\star(\mathcal{F}, \Theta) = \inf_{\theta_T}\sup_{f\in \mathcal{F}} \left( f(\theta_{T+1}) - \min_{\theta\in\Theta} f(\theta) \right),

where \mathcal{F} is a given class of convex functions, and the final query \theta_{T+1} can only depend on the past outputs of the functions.

Since there is no randomness involved in the above problem, the idea of multiple hypothesis testing cannot be directly applied here. Therefore, in this section we consider a simpler problem of stochastic optimization, and we postpone the general case to later lectures. Specifically, suppose that the above first-order oracle \mathcal{O} is stochastic, i.e., it only outputs (\widehat{f}(\theta_t), \widehat{g}(\theta_t)) such that \mathop{\mathbb E}[\widehat{f}(\theta_t)] = f(\theta_t), and \mathop{\mathbb E}[\widehat{g}(\theta_t)] \in \partial f(\theta_t). Let \mathcal{F} be the set of all convex L-Lipschitz functions (in L_2 norm), \Theta = \{\theta\in {\mathbb R}^d: \|\theta\|_2\le R \}, and assume that the subgradient estimate returned by the oracle \mathcal{O} always satisfies \|\widehat{g}(\theta_t)\|_2\le L as well. Now the target is to determine the quantity 

R_{T,d,L,R}^\star = \inf_{\theta_T} \sup_{f \in \mathcal{F}} \sup_{\mathcal{O}} \left(\mathop{\mathbb E} f(\theta_{T+1}) - \min_{\theta\in\Theta} f(\theta)\right),

where the expectation is taken over the randomness in the oracle output. The main result in this section is summarized in the following theorem:

Theorem 4

R_{T,d,L,R}^\star = \Omega\left(\frac{LR}{\sqrt{T}}\right).

Since it is well-known that the stochastic gradient descent attains the optimization gap O(LR/\sqrt{T}) for convex Lipschitz functions, the lower bound in Theorem 4 is tight. 

Now we apply the multiple hypothesis testing idea to prove Theorem 4. Since the randomness only comes from the oracle, we should design the stochastic oracle carefully to reduce the optimization problem to testing. A natural way is to choose some function F(\theta;X) such that F(\cdot;X) is convex L-Lipschitz for all X, and f(\theta) = \mathop{\mathbb E}_P[F(\theta;X)]. In this way, the oracle can simply generate i.i.d. X_1,\cdots,X_T\sim P, and reveals the random vector (F(\theta_t;X_t), \nabla F(\theta_t;X_t)) to the learner at the t-th query. Hence, the proof boils down to find a collection of probability distributions (P_v)_{v\in \mathcal{V}} such that they are separated apart while hard to distinguish based on T observations. 

We first look at the separation condition. Since the loss function of this problem is L(f,\theta_{T+1}) = f(\theta_{T+1}) - \min_{\theta\in\Theta} f(\theta), we see that 

\inf_{\theta_{T+1}\in \Theta} L(f,\theta_{T+1}) + L(g,\theta_{T+1}) = \min_{\theta\in\Theta} (f(\theta) + g(\theta)) - \min_{\theta\in\Theta} f(\theta) - \min_{\theta\in\Theta} g(\theta),

which is known as the optimization distance d_{\text{opt}}(f,g). Now consider

F(\theta;X) = L\left|\theta_i - \frac{Rx_i}{\sqrt{d}}\right| \qquad \text{if }x = \pm e_i,

and for v\in \{\pm 1\}^d, let f_v(\theta) = \mathop{\mathbb E}_{P_v}[F(\theta;X)] with P_v defined as

P_v(X = v_ie_i) = \frac{1+\delta}{2d}, \quad P_v(X= -v_ie_i) = \frac{1-\delta}{2d}, \qquad \forall i\in [d].

Clearly F(\cdot;X) is convex and L-Lipschitz, and for \|\theta\|_\infty \le R/\sqrt{d} we have

f_v(\theta) = \frac{LR}{\sqrt{d}} - \frac{\delta L}{d}\sum_{i=1}^d v_i\theta_i.

Hence, it is straightforward to verify that \min_{\|\theta\|_2\le R} f_v(\theta) = LR(1-\delta)/\sqrt{d}, and 

d_{\text{opt}}(f_v,f_{v'}) = \frac{2\delta LR}{d}\cdot d_{\text{H}}(v,v').

In other words, the separation condition in the Assoaud’s lemma is fulfilled with \Delta = 2\delta LR/d. Moreover, simple algebra gives

\max_{d_{\text{H}}(v,v') =1} D_{\text{KL}}(P_v^{\otimes T}\| P_{v'}^{\otimes T}) = O(T\delta^2),

and therefore Assoaud’s lemma gives

R_{T,d,L,R}^\star \ge d\cdot \frac{\delta LR}{d}\left(1 - O(\delta\sqrt{T})\right),

and choosing \delta \asymp T^{-1/2} completes the proof of Theorem 4. In fact, the above arguments also show that if the L_2 norm is replaced by the L_\infty norm, then

R_{T,d,L,R}^\star = \Theta\left(LR\sqrt{\frac{d}{T}}\right).

5. Bibliographic Notes 

The example of nonparametric density estimation over Sobolev balls is taken from Nemirovski (2000), which also contains the tight minimax linear risk among all linear estimators. For the upper bound, linear procedures fail to achieve the minimax rate whenever q>p, and some non-linearities are necessary for the minimax estimators either in the function domain (Lepski, Mammen and Spokoiny (1997)) or the wavelet domain (Donoho et al. (1996)). 

The linear and convex aggregations are proposed in Nemirovski (2000) for the adaptive nonparametric thresholding estimators, and the concept of model selection aggregation is due to Yang (2000). For the optimal rates of different aggregations (together with upper bounds), we refer to Tsybakov (2003) and Leung and Barron (2006). 

The examples from statistical learning theory and stochastic optimization are similar in nature. The results of Theorem 2 and the corresponding upper bounds are taken from Vapnik (1998), albeit with a different proof language. For general results of the oracle complexity of convex optimization, we refer to the wonderful book Nemirovksi and Yudin (1983) and the lecture note Nemirovski (1995). The current proof of Theorem 4 is due to Agarwal et al. (2009). 

  1. Arkadi Nemirovski, Topics in non-parametric statistics. Ecole d’Eté de Probabilités de Saint-Flour 28 (2000): 85. 
  2. Oleg V. Lepski, Enno Mammen, and Vladimir G. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. The Annals of Statistics 25.3 (1997): 929–947. 
  3. David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, and Dominique Picard, Density estimation by wavelet thresholding. The Annals of Statistics (1996): 508–539. 
  4. Yuhong Yang, Combining different procedures for adaptive regression. Journal of multivariate analysis 74.1 (2000): 135–161. 
  5. Alexandre B. Tsybakov, Optimal rates of aggregation. Learning theory and kernel machines. Springer, Berlin, Heidelberg, 2003. 303–313. 
  6. Gilbert Leung, and Andrew R. Barron. Information theory and mixing least-squares regressions. IEEE Transactions on Information Theory 52.8 (2006): 3396–3410. 
  7. Vlamimir Vapnik, Statistical learning theory. Wiley, New York (1998): 156–160. 
  8. Arkadi Nemirovski, and David Borisovich Yudin. Problem complexity and method efficiency in optimization. 1983. 
  9. Arkadi Nemirovski, Information-based complexity of convex programming. Lecture Notes, 1995. 
  10. Alekh Agarwal, Martin J. Wainwright, Peter L. Bartlett, and Pradeep K. Ravikumar, Information-theoretic lower bounds on the oracle complexity of convex optimization. Advances in Neural Information Processing Systems, 2009. 

Lecture 8: Multiple Hypothesis Testing: Tree, Fano and Assoaud

Blog, Online Lectures

(Warning: These materials may be subject to lots of typos and errors. We are grateful if you could spot errors and leave suggestions in the comments, or contact the author at yjhan@stanford.edu.) 

In the last three lectures we systematically introduced the Le Cam’s two-point method with generalizations and various examples. The main idea of the two-point method is to reduce the problem in hand to a binary hypothesis testing problem, where the hypotheses may be either single or composite. This approach works when the target is to estimate a scalar (even when the underlying statistical model may involve high-dimensional parameters), while it typically fails when the target is to recover a high-dimensional vector of parameters. For example, even in the simplest Gaussian location model X\sim \mathcal{N}(\theta, I_p) where the target is to estimate \theta under the mean squared loss, the best two-point approach gives a lower bound R_p^\star = \Omega(1) while we have known from Lecture 4 that the minimax risk is R_p^\star = p. To overcome this difficulty, recall from the minimax theorem that a suitable prior should always work, which by discretization motivates the idea of multiple hypothesis testing. 

In this lecture, we develop the general theory of reducing problems into multiple hypothesis testing, and present several tools including tree-based methods, Assoaud’s lemma and Fano’s inequality. In the next two lectures we will enumerate more examples (possibly beyond the minimax estimation in statistics) and variations of the above tools. 

1. General Tools of Multiple Hypothesis Testing 

This section presents the general theory of applying multiple hypothesis testing to estimation problems, as well as some important tools such as tree, Fano and Assoaud. Recall from the two-point method that the separability and indistinguishability conditions are of utmost importance to apply testing arguments, the above tools require different separability conditions and represent the indistinguishability condition in terms of different divergence functions. 

1.1. Tree-based method and Fano’s inequality 

We start with the possibly simplest separability condition, i.e., the chosen parameters (hypothese) \theta_1, \cdots, \theta_m\in \Theta are pairwise seperated. The following lemma shows that the minimax estimation error is lower bounded in terms of the average test error. 

Lemma 1 (From estimation to testing) Let L: \Theta \times \mathcal{A} \rightarrow {\mathbb R}_+ be any loss function, and there exist \theta_1,\cdots,\theta_m\in \Theta such that

L(\theta_i, a) + L(\theta_j, a) \ge \Delta, \qquad \forall i\neq j\in [m], a \in \mathcal{A}.


\inf_{\hat{\theta}} \sup_{\theta\in\Theta} \mathop{\mathbb E}_\theta L(\theta,\hat{\theta})\ge \frac{\Delta}{2}\cdot \inf_{\Psi}\frac{1}{m}\sum_{i=1}^m \mathop{\mathbb P}_{\theta_i}(\Psi\neq i),

where the infimum is taken over all measurable tests \Psi: \mathcal{X} \rightarrow [m]

Proof: For any given estimator \hat{\theta}, we construct a test \Psi as follows: 

\Psi(X) \in \arg\min_{i\in [m]} L(\theta_i, \hat{\theta}(X)).

Then the separability condition implies that

L(\theta_i, \hat{\theta}(X)) \ge \frac{\Delta}{2}\cdot 1(\Psi(X) \neq i), \qquad \forall i\in [m].

Then the rest follows from that the maximum is lower bounded by the average. \Box 

Remark 1 The following weaker separability condition also suffices for Lemma 1 to hold: for any i\neq j\in [m] and a\in \mathcal{A}, the inequality L(\theta_i, a)\le \Delta/2 always implies that L(\theta_j, a)\ge \Delta/2

The remaining quantity in Lemma 1 is the optimal average test error \inf_{\Psi}m^{-1}\sum_{i=1}^m \mathop{\mathbb P}_{\theta_i}(\Psi\neq i), for which we are looking for a lower bound. Recall that when m=2, this quantity equals to (1- \|P_{\theta_1} - P_{\theta_2}\|_{\text{TV}})/2 by Le Cam’s first lemma. For general m\ge 2, we have the following two different lower bounds. 

Lemma 2 (Tree-based inequality) Let T=([m], E) be any undirected tree on vertex set [m] with edge set E. Then

\inf_{\Psi}\frac{1}{m}\sum_{i=1}^m \mathop{\mathbb P}_{\theta_i}(\Psi\neq i) \ge \frac{1}{m}\sum_{(i,j)\in E} (1 - \|P_{\theta_i} - P_{\theta_j}\|_{\text{\rm TV}}).

Proof: It is straightforward to see that 

\inf_{\Psi}\frac{1}{m}\sum_{i=1}^m \mathop{\mathbb P}_{\theta_i}(\Psi\neq i) = 1 - \frac{1}{m}\int \max\{dP_{\theta_1}(x), \cdots, dP_{\theta_m}(x)\}. \ \ \ \ \ (1)

It is also easy (left as an exercise) to establish the following elementary inequality: for any reals x_1, \cdots, x_m and any tree T=([m], E), we have 

\sum_{i=1}^m x_i - \max_{i\in [m]} x_i \ge \sum_{(i,j)\in E} \min\{x_i, x_j\}.

Now using \int \min\{dP, dQ \} = 1 - \|P-Q\|_{\text{TV}} gives the desired inequality. \Box 

Lemma 3 (Fano’s inequality) Let \bar{P} := m^{-1}\sum_{i=1}^m P_i. Then

\inf_{\Psi}\frac{1}{m}\sum_{i=1}^m \mathop{\mathbb P}_{\theta_i}(\Psi\neq i) \ge 1 - \frac{1}{\log m}\left(\frac{1}{m}\sum_{i=1}^m D_{\text{\rm KL}}(P_i \| \bar{P}) +\log 2\right).

Remark 2 If we introduce auxiliary random variables V\sim \mathsf{Uniform}([m]) and X with P_{X|V=i} := P_{\theta_i}, then

\frac{1}{m}\sum_{i=1}^m D_{\text{\rm KL}}(P_i \| \bar{P}) = I(V;X),

where I(V;X) denotes the mutual information between V and X

Proof: We present two proofs of Lemma 3. The first proof builds upon the representation (1) and is more analytical, while the second proof makes use of the data-processing inequality, which is essentially the classical information-theoretic proof of Fano’s inequality. 

(First Proof) By (1), it suffices to prove that 

\mathop{\mathbb E}_{\bar{P}} \left[ \max_{i\in [m]} \left\{\frac{dP_i}{d\bar{P}} \right\} \right] \le \frac{1}{\log m}\left(\sum_{i=1}^m \mathop{\mathbb E}_{\bar{P}} \left[\frac{dP_i}{d\bar{P}}\log \frac{dP_i}{d\bar{P}}\right] + m\log 2\right).

By the linearity of expectation, it further suffices to prove that for any non-negative reals x_1, \cdots, x_m with \sum_{i=1}^m x_i = m, we have 

\max_{i\in [m]} x_i \le \frac{1}{\log m}\left(\sum_{i=1}^m x_i\log x_i + m\log 2\right).

To establish this inequality, let t:=\max_{i\in [m]} x_i. Then by the convexity of x\mapsto x\log x, Jensen’s inequality gives 

\sum_{i=1}^m x_i\log x_i \ge t\log t + (m-t)\log\frac{m-t}{m-1} \ge m\log \frac{m}{2} - (m-t)\log m.

Plugging in the above inequality completes the proof of Lemma 3. 

(Second Proof) Introduce the auxiliary random variables V and X as in Remark 2. For any fixed X, consider the kernel \mathsf{K}_X: [m]\rightarrow \{0,1\} which sends i\in [m] to 1(\Psi(X) = i), then \mathsf{K}_X\circ P_V is the Bernoulli distribution with parameter m^{-1}, and \mathsf{K}_X\circ P_{V|X} is the Bernoulli distribution with parameter p_X := \mathop{\mathbb P}_{V|X}(V = \Psi(X)). By the data-processing inequality of KL divergence, 

\begin{array}{rcl} D_{\text{KL}}(P_{V|X} \| P_V) & \ge & p_X \log(mp_X) + (1-p_X)\log (1-p_X) \\ &\ge& p_X\log m - \log 2. \end{array}

Now taking expectation on X at both sides gives 

I(V;X) = \mathop{\mathbb E}_X D_{\text{KL}}(P_{V|X} \| P_V) \ge \mathop{\mathbb P}(V = \Psi(X))\cdot \log m - \log 2,

and rearranging completes the proof. \Box

Lemma 2 decomposes a multiple hypothesis testing problem into several binary testing problems, which cannot outperform the best two-point methods in typical scenarios but can be useful when there is some external randomness associated with each P_{\theta_i} (see the bandit example later). Lemma 3 is the well-known Fano’s inequality involving the mutual information, and the additional \log m term is the key difference in multiple hypothesis testing. Hence, the typical lower bound arguments are to apply Lemma 1 together with Lemma 3 (or sometimes Lemma 2), with the following lemma which helps to upper bound the mutual information. 

Lemma 4 (Variational representation of mutual informatino)

I(V;X) = \inf_{Q_X} \mathop{\mathbb E}_V [D_{\text{\rm KL}}(P_{X|V} \| Q_X )].

Proof: Simply verify that 

I(V;X) = \mathop{\mathbb E}_V [D_{\text{\rm KL}}(P_{X|V} \| Q_X )] - D_{\text{KL}}(P_X\|Q_X). \Box

1.2. Assoaud’s lemma 

Instead of the previous pairwise separation condition, Assoaud’s lemma builds upon a different one where the hypotheses are essentially the vertices of an m-dimensional hypercube. Specifically, we shall require that the distance between parameters v and v'\in \{\pm 1\}^p is proportional to their Hamming distance d_{\text{H}}(v,v') := \sum_{i=1}^p 1(v_i \neq v_i'), which becomes quite natural when the parameter of the statistical model lies in an m-dimensional space. 

Theorem 5 (Assoaud’s Lemma) Let L: \Theta\times \mathcal{A} \rightarrow {\mathbb R}_+ be any function. If there exist parameters (\theta_v)\subseteq \Theta indexed by v\in \{\pm1\}^p such that

L(\theta_v, a) + L(\theta_{v'}, a) \ge \Delta\cdot d_{\text{\rm H}}(v,v'), \qquad \forall v,v'\in \{\pm 1\}^p, a\in\mathcal{A},


\inf_{\hat{\theta}} \sup_{\theta\in \Theta} \mathop{\mathbb E}_\theta[L(\theta,\hat{\theta})] \ge \frac{\Delta}{2}\sum_{j=1}^p \left(1 - \|P_{j,+} - P_{j,-}\|_{\text{\rm TV}}\right),


P_{j,+} := \frac{1}{2^{p-1}}\sum_{v\in \{\pm 1\}^p: v_j = 1} P_{\theta_v}, \quad P_{j,-} := \frac{1}{2^{p-1}}\sum_{v\in \{\pm 1\}^p: v_j = -1} P_{\theta_v}.

Proof: For a given estimator \hat{\theta}, define 

\hat{v} := \arg\min_{v\in \{\pm 1\}^p} L(\theta_v, \hat{\theta}).

Then it is straightforward to see that

L(\theta_v, \hat{\theta}) \ge \frac{L(\theta_v,\hat{\theta}) + L(\theta_{\hat{v}}, \hat{\theta})}{2} \ge \frac{\Delta}{2}\sum_{j=1}^p 1(v_j \neq \hat{v}_j), \qquad \forall v\in \{\pm 1\}^p.

The rest of the proof follows from Le Cam’s first lemma and the fact that the maximum is no smaller than the average. \Box

In most scenarios, the total variation distance between the mixtures P_{j,+} and P_{j,-} is hard to compute directly, and the following corollaries (based on joint convexity of the total variation distance and Cauchy–Schwartz) are often the frequently presented versions of Assoaud’s lemma. 

Corollary 6 Under the conditions of Theorem 5, we have

\inf_{\hat{\theta}} \sup_{\theta\in \Theta} \mathop{\mathbb E}_\theta[L(\theta,\hat{\theta})] \ge \frac{p\Delta}{2}\cdot \left(1 - \max_{v,v'\in \{\pm 1\}^p: d_{\text{\rm H}}(v,v')=1} \|P_{\theta_v} - P_{\theta_{v'}}\|_{\text{\rm TV}}\right).

Corollary 7 Under the conditions of Theorem 5, we have

\inf_{\hat{\theta}} \sup_{\theta\in \Theta} \mathop{\mathbb E}_\theta[L(\theta,\hat{\theta})] \ge \frac{p\Delta}{2}\cdot \left(1 - \left(\frac{1}{p2^p}\sum_{j=1}^p \sum_{v\in \{\pm 1\}^p} \|P_{\theta_v} - P_{\theta_{\bar{v}^j}} \|_{\text{\rm TV}}^2 \right)^{1/2} \right),

where \bar{v}^j is the resulting binary vector after flipping the j-th coordinate of v

The idea behind the Assoaud’s lemma is to apply a two-point argument for each coordinate of the parameter and then sum up all coordinates. Hence, Assoaud’s lemma typically improves over the best two-point argument by a factor of the dimension p, given that the hypercube-type separation condition in Theorem 5 holds. 

1.3. Generalized Fano’s inequality 

In the Fano’s inequality and Assoaud’s lemma above, we introduce a random variable V which is uniformly distributed on the set [m] or the hypercube \{\pm 1\}^p. Moreover, the separation condition must hold for each pair of the realizations of V. In general, we may wish to consider some non-uniform random variable V, and only require that the separation condition holds for most pairs. This is the focus of the following generalized Fano’s inequality. 

Theorem 8 (Generalized Fano’s Inequality) Let L: \Theta\times \mathcal{A}\rightarrow {\mathbb R}_+ be any loss function, and \pi be any probability distribution on \Theta. For any \Delta>0, define

p_{\Delta} := \sup_{a\in \mathcal{A}} \pi(\{\theta\in \Theta: L(\theta,a)\le \Delta \}).

Then for V\sim \pi, we have

\inf_{\hat{\theta}}\sup_{\theta\in\Theta} \mathop{\mathbb E}_\theta[L(\theta,\hat{\theta})] \ge \Delta\left(1 - \frac{I(V;X) + \log 2}{\log(1/p_{\Delta})}\right).

Proof: By Markov’s inequality, it suffices to show that for any estimator \hat{\theta}

\mathop{\mathbb P}\left(L(V, \hat{\theta}(X))\le \Delta \right)\le \frac{I(V;X) + \log 2}{\log(1/p_\Delta)}.

We again use the second proof of Lemma 3. For any fixed X, consider the deterministic kernel \mathsf{K}_X: \Theta\rightarrow \{0,1\} which sends V to 1( L(V,\hat{\theta}(X))\le \Delta ). Then the composition K_X\circ P_V is a Bernoulli distribution with parameter p_X\le p_\Delta, and the composition K_X\circ P_{V|X} is a Bernoulli distribution with parameter q_X := P_{V|X}(L(V,\hat{\theta}(X))\le \Delta). Then by the data processing inequality of KL divergence,

\begin{array}{rcl} D_{\text{KL}}(P_{V|X}\|P_V) &\ge& q_X\log \frac{q_X}{p_X} + (1-q_X)\log \frac{1-q_X}{1-p_X} \\ &\ge & q_X\log \frac{q_X}{p_\Delta} + (1-q_X)\log (1-q_X) \\ &\ge & q_X\log(1/p_\Delta) - \log 2. \end{array}

Since \mathop{\mathbb E}[q_X] = \mathop{\mathbb P}(L(V,\hat{\theta}(X))\le \Delta), taking expectation on X at both sides gives the desired inequality. \Box

To see that Theorem 8 is indeed a generalization to the original Fano’s inequality, simply note that when V\sim \mathsf{Uniform}([m]) and the separation condition in Lemma 1 holds, we have p_\Delta = m^{-1} and therefore the denominator \log(1/p_\Delta) = \log m becomes the log-cardinality. Hence, in the generalized Fano’s inequality, the verification of the seperation condition becomes upper bounding the probability p_\Delta

2. Applications 

In this section we present some applications of the above tools to statistical examples. Here the applications are mostly simple and straightforward, and we defer some more sophisticated examples in other domains to the next lecture. 

2.1. Example I: Gaussian mean estimation 

We start from the possibly simplest example of Gaussian mean estimation. Consider X\sim \mathcal{N}(\theta,\sigma^2 I_p) with known \sigma, and the target is the estimate the vector \theta\in {\mathbb R}^p under the squared \ell_2 loss. Let R_{p,\sigma}^\star be the corresponding minimax risk. 

We first show that any two-point argument fails. In fact, if the two points are chosen to be \theta_1, \theta_2\in {\mathbb R}^p, the two-point argument gives 

R_{p,\sigma}^\star \ge \frac{\|\theta_1 - \theta_2\|_2^2}{2}\left(1-\Phi\left(\frac{\|\theta_1 - \theta_2\|_2}{2\sigma}\right)\right),

where \Phi(\cdot) is the normal CDF. Optimization only gives R_{p,\sigma}^\star = \Omega(\sigma^2).

Now we show that Assoaud’s lemma can give us the rate-optimal lower bound R_{p,\sigma}^\star= \Omega(p\sigma^2). Let P_v = \mathcal{N}(\delta v, \sigma^2 I_p) for v\in \{\pm 1\}^p, where \delta>0 is a parameter to be chosen later. Since 

\|\delta v - \delta v'\|_2^2 = 4\delta^2\cdot d_{\text{H}}(v,v'),

the separation condition in Theorem 5 holds with \Delta = 4\delta^2. Consequently, Corollary 6 gives

R_{p,\sigma}^\star \ge 2p\delta^2\cdot \left(1 - \Phi\left(\frac{\delta}{\sigma}\right)\right),

and choosing \delta =\Theta(\sigma) gives R_{p,\sigma}^\star= \Omega(p\sigma^2).

We can also establish the same lower bound using the generalized Fano’s inequality. Let V\sim \mathsf{Uniform}(\{\pm\delta\}^p), then Lemma 4 gives 

I(V;X) \le \mathop{\mathbb E} D_{\text{KL}}(\mathcal{N}(V,\sigma^2 I_p) \| \mathcal{N}(0, \sigma^2I_p)) = \frac{\mathop{\mathbb E} \|V\|_2^2}{2\sigma^2} = \frac{p\delta^2}{2\sigma^2}.

Since for any \theta_0\in {\mathbb R}^p with v_0 = \text{sign}(\theta_0)\in \{\pm 1\}^p (break ties arbitrarily), we have

\|V - \theta_0\|_2^2 \ge \delta^2 \cdot d_{\text{H}}(\text{sign}(V), v_0),

choosing \Delta = p\delta^2/3 gives

\mathop{\mathbb P}(\|V-\theta_0\|_2^2 \le \Delta) \le \mathop{\mathbb P}(d_{\text{H}}(\text{sign}(V), v_0)\le p/3 ) \le e^{-cp}

for some numerical constant c>0, where the last inequality is given by sub-Gaussian concentration. Consequently, p_\Delta \le e^{-cp}, and Theorem 8 gives

R_{p,\sigma}^\star \ge \frac{p\delta^2}{3}\left(1 - \frac{p\delta^2/(2\sigma^2) + \log 2}{cp}\right).

Again, choosing \delta = c'\sigma with small c'>0 gives R_{p,\sigma}^\star = \Omega(p\sigma^2).

Remark 3 The original version of Fano’s inequality can also be applied here, where \{\theta_1,\cdots,\theta_m\} is a maximal packing of \{\pm \delta\}^p with \ell_2 packing distance O(\delta\sqrt{p})

2.2. Example II: Sparse linear regression 

Consider the following sparse linear regression Y\sim \mathcal{N}(X\theta, \sigma^2 I_n), where X\in {\mathbb R}^{n\times p} is a fixed design matrix, and \theta\in {\mathbb R}^p is a sparse vector with \|\theta\|_0\le s. Here s\in [1,p] is a known sparsity parameter, and we are interested in the minimax risk R_{n,p,s,\sigma}^\star(X) of estimating \theta under the squared \ell_2 loss. Note that when X = I_p, this problem is reduced to the sparse Gaussian mean estimation. 

We apply the generalized Fano’s inequality to this problem. Since a natural difficulty of this problem is to recover the support of \theta, we introduce the random vector V=\delta v_S, where \delta>0 is a parameter to be specified later, v\sim \mathsf{Uniform}\{\pm 1\}^p, v_S denotes the restriction of v onto S, and S\subseteq [p] is uniformly chosen from all size-s subsets of [p]. Clearly, 

\mathop{\mathbb E}[VV^\top] = \delta^2 \mathop{\mathbb E}[v_Sv_S^\top] = \frac{s\delta^2}{p}I_p.

By Lemma 4,

\begin{array}{rcl} I(V;Y) &\le & \mathop{\mathbb E} D_{\text{KL}}(\mathcal{N}(XV, \sigma^2 I_n) \| \mathcal{N}(0, \sigma^2 I_n)) \\ &=& \frac{\mathop{\mathbb E}\|XV\|_2^2}{2\sigma^2} \\ &=& \frac{\text{Trace}(X^\top X\cdot \mathop{\mathbb E}[VV^\top]) }{2\sigma^2}\\ &=& \frac{s\delta^2}{2p\sigma^2}\|X\|_{\text{F}}^2, \end{array}

where \|X\|_{\text{F}} := \sqrt{\text{Trace}(X^\top X)} is the Frobenius norm of X. Now it remains to upper bound p_\Delta for \Delta = s\delta^2/12. Fix any \theta_0\in {\mathbb R}^p such that \|V - \theta_0\|_2^2\le \Delta holds with non-zero probability (otherwise the upper bound is trivial), and by symmetry we may assume that \|\theta_0 - \delta1_{[s]}\|_2^2 \le \Delta. Now by triangle inequality, \|\theta_0 - \delta v_S\|_2^2 \le \Delta implies that \|v_S - 1_{[s]}\|_2^2 \le s/3. Hence,

\begin{array}{rcl} p_\Delta &\le& \mathop{\mathbb P}(\|v_S - 1_{[s]}\|_2^2\le s/3) \\ &\le& \frac{1}{2^s\binom{p}{s}}\sum_{k=\lceil 2s/3 \rceil}^s \binom{s}{k} \binom{p-k}{s-k}2^{s-k} \\ &\le& \left(s/(ep)\right)^{cs} \end{array}

for a numerical constant c>0 after some algebra. Consequently, Theorem 8 gives

R_{n,p,s,\sigma}^\star(X) \ge \frac{s\delta^2}{12}\left(1 - \frac{ \frac{s\delta^2}{2p\sigma^2}\|X\|_{\text{F}}^2 + \log 2}{cs\log(ep/s)} \right).

Finally, choosing \delta^2 = c'p\sigma^2\log(ep/s)/\|X\|_{\text{F}}^2 for some small constant c'>0 gives

R_{n,p,s,\sigma}^\star (X)= \Omega\left(\frac{sp\sigma^2\log(ep/s)}{\|X\|_{\text{F}}^2} \right).

In the special cases where X= I_p or X consists of i.i.d. \mathcal{N}(0,n^{-1}) entries, we arrive at the tight lower bound \Omega(s\sigma^2\log(ep/s)) for both sparse Gaussian mean estimation and compressed sensing.

2.3. Example III: Multi-armed bandit 

Next we revisit the example of the multi-armed bandit in Lecture 5. Let T be the time horizon, K be the total number of arms. For each i\in [K], the reward of pulling arm i at time t is an independent random variable r_{t,i}\sim \mathcal{N}(\mu_i, 1). The target of the learner is to devise a policy (\pi_t)_{t\in [T]} where \pi_t\in [K] is the arm to pull at time t, and \pi_t may depend on the entire observed history (\pi_\tau)_{\tau<t} and (r_{\tau,\pi_\tau})_{\tau<t}. The learner would like minimize the following worst-case regret: 

R_{T,K}^\star = \inf_{\pi} \sup_{\max_{i\in [K]}|\mu_i| \le \sqrt{K} } \mathop{\mathbb E}\left[T \max_{1\le i\le K} \mu_i - \sum_{t=1}^T \mu_{t,\pi_t}\right],

where \max_{i\in [K]}|\mu_i|\le \sqrt{K} is a technical condition. As in Lecture 5, our target is to show that

R_{T,K}^\star = \Omega(\sqrt{KT})

via multiple hypothesis testing.

To prove the lower bound, a natural idea is to construct K hypotheses where the i-th arm is the optimal arm under the i-th hypothesis. Specifically, we set 

\begin{array}{rcl} \mu^{(1)} &=& (\delta, 0, 0, 0, \cdots, 0), \\ \mu^{(2)} &=& (\delta, 2\delta, 0, 0, \cdots, 0), \\ \mu^{(3)} &=& (\delta, 0, 2\delta, 0, \cdots, 0), \\ \vdots & & \vdots \\ \mu^{(K)} &=& (\delta, 0, 0, 0, \cdots, 2\delta), \end{array}

where \delta>0 is some parameter to be specified later. The construction is not entirely symmetric, and the reward distribution under the first arm is always \mathcal{N}(\delta,1).

For a mean vector \mu and policy \pi, let L(\mu,\pi) = T \max_{1\le i\le K} \mu_i - \sum_{t=1}^T \mu_{t,\pi_t} be the non-negative loss function for this online learning problem. Then clearly the separation condition in Lemma 1 is fulfilled with \Delta = \delta T. Next, we apply Lemma 2 to a star graph ([K], \{(1,2),(1,3),\cdots,(1,K) \}) with center 1 gives 

\begin{array}{rcl} R_{T,K}^\star &\overset{(a)}{\ge} & \frac{\delta T}{2}\cdot \frac{1}{K}\sum_{i=2}^K \left(1 - \|P_{\mu^{(1)}} - P_{\mu^{(i)}} \|_{\text{TV}}\right) \\ &\overset{(b)}{\ge} & \frac{\delta T}{4K} \sum_{i=2}^K \exp\left( -D_{\text{KL}}(P_{\mu^{(1)}} \| P_{\mu^{(i)}} ) \right) \\ &\overset{(c)}{=}& \frac{\delta T}{4K} \sum_{i=2}^K \exp\left( - 2\delta^2 \mathop{\mathbb E}_1(T_i)\right) \\ &\overset{(d)}{\ge}& \frac{\delta T}{4}\cdot \frac{K-1}{K} \exp\left(-2\delta^2\cdot \frac{\mathop{\mathbb E}_1(T_2 + \cdots + T_K)}{K-1} \right) \\ &\overset{(e)}{\ge} & \frac{\delta T}{4}\cdot \frac{K-1}{K}\exp\left(-\frac{2T\delta^2}{K-1}\right), \end{array}

where in step (a) we denote by P_{\mu^{(i)}} the distribution of the observed rewards under mean vector \mu^{(i)}, step (b) follows from the inequality 1 - \|P-Q\|_{\text{TV}}\ge \frac{1}{2}\exp(-D_{\text{KL}}(P\|Q)) presented in Lecture 5, step (c) is the evaluation of the KL divergence where \mathop{\mathbb E}_1(T_i) denotes the expected number of pullings of arm i under the mean vector \mu^{(1)}, step (d) follows from Jensen’s inequality, and step (e) is due to the deterministic inequality T_2 + \cdots + T_K\le T. Now choosing \delta = \Theta(\sqrt{K/T}) gives the desired lower bound.

The main reason why we apply Lemma 2 instead of the Fano’s inequality and choose the same reward distribution for arm 1 at all times is to deal with the random number of pullings of different arms under different reward distributions. In the above way, we may stick to the expectation \mathop{\mathbb E}_1 and apply the deterministic inequality T_2 + \cdots + T_K\le T to handle the randomness. 

2.4. Example IV: Gaussian mixture estimation 

Finally we look at a more involved example in nonparametric statistics. Let f be a density on {\mathbb R} which is a Gaussian mixture, i.e., f = g * \mathcal{N}(0,1) for some density g, where * denotes the convolution. We consider the estimation of the density f given n i.i.d. observations X_1,\cdots,X_n from f, and we denote by R_n^\star the minimax risk under the squared L_2 loss between real-valued functions. The central claim is that 

R_n^\star = \Omega\left(\frac{(\log n)^{1/2}}{n}\right),

which is slightly larger than the (squared) parametric rate n^{-1}.

Before proving this lower bound, we first gain some insights from the upper bound. In the problem formulation, the only restriction on the density f is that f must be a Gaussian mixture. Since convolution makes function smoother, f should be fairly smooth, and harmonic analysis suggests that the extent of smoothness is reflected in the speed of decay of the Fourier transform of f. Let \widehat{f} be the Fourier transform of f, we have 

\widehat{f}(\omega) = \widehat{g}(\omega)\cdot \frac{1}{\sqrt{2\pi}}e^{-\omega^2/2},

and \|\widehat{g}\|_\infty \le \|g\|_1=1. Hence, if we truncate \widehat{f}(\omega) to zero for all |\omega| > 2\sqrt{\log n}, the L_2 approximation error would be smaller than O(n^{-1}). This suggests us to consider the kernel K on {\mathbb R} with

\widehat{K}(\omega) = 1(|\omega|\le 2\sqrt{\log n}).

Then the density estimator \mathop{\mathbb P}_n * K has mean f * K, which by Parseval’s identity has squared bias

\|f - f*K\|_2^2 = \|\widehat{f}(1-\widehat{K}) \|_2^2 = O(n^{-1}).

Moreover, by Parseval again,

\mathop{\mathbb E}\|\mathop{\mathbb P}_n * K - f * K\|_2^2 \le \frac{\|K\|_2^2}{n} = \frac{\|\widehat{K}\|_2^2}{n} = O\left(\frac{\sqrt{\log n}}{n}\right),

and a triangle inequality leads to the desired result.

The upper bound analysis motivates us to apply Fourier-type arguments in the lower bound. For v\in \{\pm 1\}^p (with integer p to be specified later), consider the density 

f_v = f_0 + \left(\delta\cdot \sum_{i=1}^p v_ig_i \right) * \phi,

where f_0 is some centered probability density, \delta>0 is some parameter to be specified later, g_1,\cdots,g_p are perturbation functions with \int g_i = 0, and \phi is the PDF of \mathcal{N}(0,1). To apply the Assouad’s lemma on this hypercube structure, we require the following orthogonality condition

\int_{{\mathbb R}} (g_i*\phi)(x) (g_j*\phi)(x)dx = 1(i=j)

to ensure the separation condition in Theorem 5 with \Delta = \Theta(\delta^2). By Plancherel’s identity, the above condition is equivalent to

\int_{{\mathbb R}} \widehat{g}_i(\omega)\widehat{g}_j(\omega)\phi^2(\omega)d\omega = 1(i=j).

Recall from Lecture 7 that the Hermite polynomials are orthogonal under the normal distribution, a natural candidate of \widehat{g}_i(\omega) is \widehat{g}_i(\omega)\propto \phi(\omega)H_{2i-1}(2\omega), where we have used that \phi(\omega)^4 \propto \phi(2\omega), and we restrict to odd degrees to ensure that \int g_i = \widehat{g}_i(0) = 0. This choice enjoys another nice property where the inverse Fourier transform of \widehat{g}_i has the following closed-form expression for g_i:

g_i(x) = \sqrt{2}(2\pi)^{3/4}\sqrt{\frac{3^{2i-1}}{(2i-1)!}}\cdot\phi(x) H_{2i-1}\left(\frac{2x}{\sqrt{3}}\right).

Moreover, since H_k(x)\sim \sqrt{k!}e^{x^2/4} for large x, we have \|g_i\|_\infty = \Theta(3^i). Therefore, we must have p=O(\log n) to ensure the non-negativity of the density f_v.

Now the only remaining condition is to check that the \chi^2-divergence between neighboring vertices is at most O(1), which is essentially equivalent to 

n\delta^2\cdot \int_{{\mathbb R}} \frac{(g_i * \phi(x))^2}{f_0(x)}dx = O(1).

A natural choice of f_0 is the PDF of \mathcal{N}(0,\sigma^2), with \sigma>0 to be specified. Splitting the above integral over {\mathbb R} into |x|\le C\sigma and |x|> C\sigma for some large constant C>0, the orthogonality relation of g_i gives

\int_{|x|\le C\sigma} \frac{(g_i * \phi(x))^2}{f_0(x)}dx \le \sqrt{2\pi}e^{C^2/2}\sigma \cdot \int_{{\mathbb R}} (g_i * \phi(x))^2dx = O(\sigma).

To evaluate the integral over |x|>C\sigma, recall that g_i(x) behaves as 3^ie^{-cx^2} for large |x|. Hence, the natural requirement \sigma^2 = \Omega(p) leads to the same upper bound O(\sigma) for the second integral, and therefore the largest choice of \delta is \delta^2 = (n\sigma)^{-1}.

To sum up, we have arrived at the lower bound 

R_{n}^\star = \Omega\left(\frac{p}{n\sigma}\right)

subject to the constraints \sigma^2 = \Omega(\sqrt{p}) and p = O(\log n). Hence, the optimal choices for the auxiliary parameters are p = \Theta(\log n), \sigma = \Theta(\sqrt{\log n}), leading to the desired lower bound.

3. Bibliographic Notes 

The lower bound technique based on hypothesis testing was pioneered by Ibragimov and Khas’minskii (1977), who also applied Fano’s lemma (Fano (1952)) to a statistical setting. The Assouad’s lemma is due to Assoaud (1983), and we also refer to a survey paper Yu (1997). The tree-based technique (Lemma 2 and the analysis of the multi-armed bandit) is due to Gao et al. (2019), and the generalized Fano’s inequality is motivated by the distance-based Fano’s inequality (Duchi and Wainwright (2013)) and the current form is presented in the lecture note Duchi (2019). 

For the examples, the lower bound of sparse linear regression is taken from Candes and Davenport (2013), and we also refer to Donoho and Johnstone (1994), Raskutti, Wainwright and Yu (2011), and Zhang, Wainwright and Jordan (2017) for related results. The full proof of the Gaussian mixture example is referred to Kim (2014). 

  1. Il’dar Abdullovich Ibragimov and Rafail Zalmanovich Khas’minskii. On the estimation of an infinite-dimensional parameter in Gaussian white noise. Doklady Akademii Nauk. Vol. 236. No. 5. Russian Academy of Sciences, 1977. 
  2. Robert M. Fano, Class notes for transmission of information. Massachusetts Institute of Technology, Tech. Rep (1952). 
  3. Patrick Assouad, Densité et dimension. Annales de l’Institut Fourier. Vol. 33. No. 3. 1983. 
  4. Bin Yu, Assouad, Fano, and Le Cam. Festschrift for Lucien Le Cam. Springer, New York, NY, 1997. 423–435. 
  5. Zijun Gao, Yanjun Han, Zhimei Ren, and Zhengqing Zhou, Batched multi-armed bandit problem. Advances in Neural Information Processing Systems, 2019. 
  6. John C. Duchi, and Martin J. Wainwright. Distance-based and continuum Fano inequalities with applications to statistical estimation. arXiv preprint arXiv:1311.2669 (2013). 
  7. John C. Duchi, Lecture notes for statistics 311/electrical engineering 377. 2019. 
  8. Emmanuel J. Candes, and Mark A. Davenport. How well can we estimate a sparse vector? Applied and Computational Harmonic Analysis 34.2 (2013): 317–323. 
  9. David L. Donoho, and Iain M. Johnstone. Minimax risk over \ell_p-balls for \ell_q-error. Probability Theory and Related Fields 99.2 (1994): 277–303. 
  10. Garvesh Raskutti, Martin J. Wainwright, and Bin Yu. Minimax rates of estimation for high-dimensional linear regression over \ell_q-balls. IEEE transactions on information theory 57.10 (2011): 6976–6994. 
  11. Yuchen Zhang, Martin J. Wainwright, and Michael I. Jordan. Optimal prediction for sparse linear models? Lower bounds for coordinate-separable M-estimators. Electronic Journal of Statistics 11.1 (2017): 752-799. 
  12. Arlene K. H. Kim. Minimax bounds for estimation of normal mixtures. Bernoulli 20.4 (2014): 1802–1818. 

Journal for High Schoolers in 2019

Journal for High Schoolers

In 2019, from June to August, 40 high school students attended the STEM to SHTEM (Science, Humanities, Technology, Engineering and Mathematics) summer program hosted by Prof. Tsachy Weissman and the Stanford Compression Forum. During this summer program, the high schoolers pursued fun research projects in various domains under the supervision of 18 mentors, where the entire collection of the high schoolers’ reports can be found below.