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# 数学代写|凸优化代写Convex Optimization代考|ESE605 ESTIMATING LINEAR FORMS ON UNIONS OF CONVEX SETS

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## 数学代写|凸优化代写Convex Optimization代考|ESTIMATING LINEAR FORMS ON UNIONS OF CONVEX SETS

The key to the subsequent developments in this section and in Sections $3.3$ and $3.4$ is the following simple observation. Let $\mathcal{P}=\left{P_x: x \in \mathcal{X}\right}$ be a parametric family of distributions on $\mathbf{R}^d, \mathcal{X}$ being a convex subset of some $\mathbf{R}^m$. Suppose that given a linear form $g^T x$ on $\mathbf{R}^m$ and an observation $\omega \sim P_x$ stemming from unknown signal $x \in \mathcal{X}$, we want to recover $g^T x$, and intend to use for this purpose an affine function $h^T \omega+\kappa$ of the observation. How do we ensure that the recovery, with a given probability $1-\epsilon$, deviates from $g^T x$ by at most a given margin $\rho$, for all $x \in \mathcal{X} ?$

Let us focus on one “half” of the answer: how to ensure that the probability of the event $h^T \omega+\kappa>g^T x+\rho$ does not exceed $\epsilon / 2$, for every $x \in \mathcal{X}$. The answer becomes easy when assuming that we have at our disposal an upper bound on the exponential moments of the distributions from the family – a function $\Phi(h ; x)$ such that
$$\ln \left(\int \mathrm{e}^{h^T \omega} P_x(d \omega)\right) \leq \Phi(h ; x) \forall\left(h \in \mathbf{R}^n, x \in \mathcal{X}\right) .$$
Indeed, for obvious reasons, in this case the $P_x$-probability of the event $h^T \omega+\kappa-$ $g^T x>\rho$ is at most
$$\exp \left{\Phi(h ; x)-\left[g^T x+\rho-\kappa\right]\right}$$

## 数学代写|凸优化代写Convex Optimization代考|The problem

Let $\mathcal{O}=\left(\Omega, \Pi,\left{p_\mu(\cdot): \mu \in \mathcal{M}\right}, \mathcal{F}\right)$ be a simple observation scheme (see Section 2.4.2). The problem we consider in this section is as follows:
We are given a positive integer $K$ and $I$ nonempty convex compact sets $X_j \subset \mathbf{R}^n$, along with affine mappings $A_j(\cdot): \mathbf{R}^n \rightarrow \mathbf{R}^M$ such that $A_j(x) \in$ $\mathcal{M}$ whenever $x \in X_j, 1 \leq j \leq I$. In addition, we are given a linear function

$g^T x$ on $\mathbf{R}^n$. Given random observation
$$\omega^K=\left(\omega_1, \ldots, \omega_K\right)$$
with $\omega_k$ drawn, independently across $k$, from $p_{A_j(x)}$ with $j \leq I$ and $x \in X_j$, we want to recover $g^T x$.

It should be stressed that we do not know $j$ and $x$ underlying our observation.
Given reliability tolerance $\epsilon \in(0,1)$, we quantify the performance of a candidate estimate – a Borel function $\widehat{g}(\cdot): \Omega \rightarrow \mathbf{R}$-by the worst-case, over $j$ and $x$, width of a $(1-\epsilon)$-confidence interval. Precisely, we say that $\widehat{g}(\cdot)$ is $(\rho, \epsilon)$-reliable if
$$\forall\left(j \leq I, x \in X_j\right): \operatorname{Prob}{\omega \sim p{A_j(x)}}\left{\left|\widehat{g}(\omega)-g^T x\right|>\rho\right} \leq \epsilon .$$
We define the $\epsilon$-risk of the estimate as
$$\operatorname{Risk}\epsilon[\widehat{g}]=\inf {\rho: \widehat{g} \text { is }(\rho, \epsilon) \text {-reliable }} \text {, }$$ i.e., $\operatorname{Risk}\epsilon[\widehat{g}]$ is the smallest $\rho$ such that $\widehat{g}$ is $(\rho, \epsilon)$-reliable.

## 数学代写|凸优化代写凸优化代考| estimation LINEAR FORMS ON union OF凸集

.凸集的线性形式

$$\ln \left(\int \mathrm{e}^{h^T \omega} P_x(d \omega)\right) \leq \Phi(h ; x) \forall\left(h \in \mathbf{R}^n, x \in \mathcal{X}\right) .$$

$g^T x$ on $\mathbf{R}^n$。假设随机观察
$$\omega^K=\left(\omega_1, \ldots, \omega_K\right)$$

$$\forall\left(j \leq I, x \in X_j\right): \operatorname{Prob}{\omega \sim p{A_j(x)}}\left{\left|\widehat{g}(\omega)-g^T x\right|>\rho\right} \leq \epsilon .$$
$\epsilon$-估算的风险为
$$\operatorname{Risk}\epsilon[\widehat{g}]=\inf {\rho: \widehat{g} \text { is }(\rho, \epsilon) \text {-reliable }} \text {, }$$ 例如， $\operatorname{Risk}\epsilon[\widehat{g}]$ 是最小的 $\rho$ 如此这般 $\widehat{g}$ 是 $(\rho, \epsilon)$-可靠的。

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