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# 数学代写|凸优化代写Convex Optimization代考|ESE605 A modification

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## 数学代写|凸优化代写Convex Optimization代考|A modification

From the computational viewpoint, an obvious shortcoming of the construction presented in the previous section is the necessity to operate with $M(N+1)$ hypotheses, which might require computing as many as $O\left(M^2 N^2\right)$ detectors. We are about to present a modified construction, where we deal at most $N+1$ times with just $M$ hypotheses at a time (i.e., with the total of at most $O\left(M^2 N\right)$ detectors). The idea is to replace simultaneously processing all hypotheses $H_{i j}, i j \in \mathcal{J}$, with processing them in stages $j=0,1, \ldots$, with stage $j$ operating only with the hypotheses $H_{i j}$, $i=1, \ldots, M$

The implementation of this idea is as follows. In the situation of Section 2.5.3, given the same entities $\Gamma,(\alpha, \beta), H_{i j}, X_{i j}$, ij $\in \mathcal{J}$, as at the beginning of Section 2.5.3.1 and specifying closeness $\mathcal{C}_{\alpha, \beta}$ according to (2.87), we now act as follows.
Preprocessing. For $j=0,1, \ldots, N$

we identify the set $\mathcal{I}j=\left{i \leq M: X{i j} \neq \emptyset\right}$ and stop if this set is empty. If this set is nonempty,

we specify the closeness $\mathcal{C}{\alpha \beta}^j$ on the set of hypotheses $H{i j}, i \in \mathcal{I}j$, as a “slice” of the closeness $\mathcal{C}{\alpha, \beta}$ :
$H_{i j}$ and $H_{i^{\prime} j}$ (equivalently, $i$ and $\left.i^{\prime}\right)$ are $\mathcal{C}{\alpha, \beta^j}^j$-close to each other if $\left(i j, i^{\prime} j\right)$ are $\mathcal{C}{\alpha, \beta}$-close, that is,
$$\left|x_i-x_{i^{\prime}}\right| \leq 2 \bar{\alpha} r_j+\beta, \bar{\alpha}=\frac{\alpha-1}{2} .$$

We build the optimal detectors $\phi_{i j, i^{\prime} j}$, along with their risks $\epsilon_{i j, i^{\prime} j}$, for all $i, i^{\prime} \in \mathcal{I}j$ such that $\left(i, i^{\prime}\right) \notin \mathcal{C}{\alpha, \beta}^j$. If $\epsilon_{i j, i^{\prime} j}=1$ for a pair $i, i^{\prime}$ of the latter type, that is, $A\left(X_{i j}\right) \cap A\left(X_{i^{\prime} j}\right) \neq \emptyset$, we claim that $(\alpha, \beta)$ is inadmissible and stop. Otherwise we find the smallest $K=K_j$ such that the spectral norm of the symmetric $M \times M$ matrix $E^{j K}$ with the entries
$$E_{i i^{\prime}}^{j K}= \begin{cases}\epsilon_{i j, i^{\prime} j}^K, & i \in \mathcal{I}j, i^{\prime} \in \mathcal{I}_j,\left(i, i^{\prime}\right) \notin \mathcal{C}{\alpha, \beta}^j \ 0, & \text { otherwise }\end{cases}$$
does not exceed $\bar{\epsilon}=\epsilon /(N+1)$. We then use the machinery of Section 2.5.2.3 to build detector-based test $\mathcal{T}{\mathcal{C}{\alpha, \beta}^j}^{K_j}$, which decides on the hypotheses $H_{i j}, i \in \mathcal{I}j$, with $\mathcal{C}{\alpha, \beta^j}^j$-risk not exceeding $\bar{\epsilon}$

## 数学代写|凸优化代写Convex Optimization代考|Near-optimality

We augment the above constructions with the following
Proposition 2.35. Let for some positive integer $\bar{K}, \epsilon \in(0,1 / 2)$, and a pair $(a, b) \geq$ 0 there exist an inference $\omega^{\bar{K}} \mapsto i\left(\omega^{\bar{K}}\right) \in{1, \ldots, M}$ such that whenever $x_* \in X$, we have
$$\operatorname{Prob}{\omega^R \sim P{x+}^R}\left{\left|x_-x_{i\left(\omega^R\right)}\right| \leq a \rho\left(x_\right)+b\right} \geq 1-\epsilon \text {. }$$
Then the pair $(\alpha=2 a+3, \beta=2 b)$ is admissible in the sense of Section 2.5.3.1 (and thus – in the sense of Section 2.5.3.2), and for the constructions in Sections 2.5.3.1 and 2.5.3.2 one has
$$K(\alpha, \beta) \leq \text { Ceil }\left(2 \frac{1+\ln (M(N+1)) / \ln (1 / \epsilon)}{1-\frac{\ln (4(1-\epsilon))}{\ln (1 / \epsilon)}} \bar{K}\right)$$
Proof. Consider the situation of Section 2.5.3.1 (the situation of Section 2.5.3.2 can be processed in a completely similar way). Observe that with $\alpha, \beta$ as above, there exists a simple test deciding on a pair of hypotheses $H_{i j}, H_{i^{\prime} j^{\prime}}$ which are not Indeed, the desired test $\mathcal{T}$ is as follows: given $i j \in \mathcal{J}, i^{\prime} j^{\prime} \in \mathcal{J}$, and observation $\omega^K$, we compute $i\left(\omega^K\right)$ and accept $H_{i j}$ if and only if $\left|x_{i\left(\omega^R\right)}-x_i\right| \leq(a+1) r_j+b$, and accept $H_{i^{\prime} j^{\prime}}$ otherwise. Let us check that the risk of this test indeed is at most є. Assume, first, that $H_{i j}$ takes place. The $P_{x_}^{\bar{K}}$-probability of the event $$\mathcal{E}:\left|x_{i\left(\omega^R\right)}-x_\right| \leq a \rho\left(x_\right)+b$$ is at least $1-\epsilon$ due to the origin of $i(\cdot)$, and $\left|x_i-x_\right| \leq r_j$ since $H_{i j}$ takes place, implying that $\rho\left(x_\right) \leq r_j$ by the definition of $\rho(\cdot)$. Thus, in the case of $\mathcal{E}$ it holds $$\left|x_{i\left(\omega^R\right)}-x_i\right| \leq\left|x_{i\left(\omega^R\right)}-x_\right|+\left|x_i-x_\right| \leq a \rho\left(x_\right)+b+r_j \leq(a+1) r_j+b .$$

## 数学代写|凸优化代写凸优化代考|修改

$H_{i j}$和$H_{i^{\prime} j}$(相当于，如果$\left(i j, i^{\prime} j\right)$是$\mathcal{C}{\alpha, \beta}$ -close，则$i$和$\left.i^{\prime}\right)$彼此是$\mathcal{C}{\alpha, \beta^j}^j$，即
$$\left|x_i-x_{i^{\prime}}\right| \leq 2 \bar{\alpha} r_j+\beta, \bar{\alpha}=\frac{\alpha-1}{2} .$$

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