Modify bugs in blogs

Author: OptHuang

content/posts/exact-lower-bounds.md

@@ -33,6 +33,10 @@ which governs the non-convergence behavior of probabilistic direct search (PDS).
 
 ## Key Ideas
 
-The random series $S$ satisfies the stochastic fixed-point equation $S \stackrel{d}{=} \gamma^Y \theta^{1-Y} S + Y$, identifying its distribution as the invariant measure of the random IFS $\lbrace x \mapsto \theta x,\; x \mapsto \gamma x + 1 \rbrace$ with probabilities $(p, 1-p)$. This connection enables the use of powerful tools from the theory of stochastic recurrence equations and iterated function systems to derive tail asymptotics, singularity conditions, and dimension formulas.
+The random series $S$ satisfies the stochastic fixed-point equation $S \stackrel{d}{=} \gamma^Y \theta^{1-Y} S + Y$, identifying its distribution as the invariant measure of the random IFS
+
+<div>$$f_0(x) = \theta x, \quad f_1(x) = \gamma x + 1$$</div>
+
+with probabilities $(p, 1-p)$. This connection enables the use of powerful tools from the theory of stochastic recurrence equations and iterated function systems to derive tail asymptotics, singularity conditions, and dimension formulas.
 
 **Practical implications**: The distributional results translate into concrete guidance for PDS parameter selection — the left-tail exponent $\log p / \log\theta$ governs how fast the non-convergence probability decays, while the right-tail exponent $\alpha$ controls displacement risk.

content/posts/exact-lower-bounds.zh-cn.md

@@ -33,6 +33,10 @@ $$S = \sum_{k=0}^\infty Y_k \prod_{\ell=0}^{k-1} \gamma^{Y_\ell} \theta^{1-Y_\el
 
 ## 核心思想
 
-随机级数 $S$ 满足随机不动点方程 $S \stackrel{d}{=} \gamma^Y \theta^{1-Y} S + Y$，其分布是随机迭代函数系统 $\lbrace x \mapsto \theta x,\; x \mapsto \gamma x + 1 \rbrace$（概率为 $(p, 1-p)$）的不变测度。这一联系使得我们能够利用随机递推方程和迭代函数系统的强大工具来推导尾部渐近、奇异性条件和维数公式。
+随机级数 $S$ 满足随机不动点方程 $S \stackrel{d}{=} \gamma^Y \theta^{1-Y} S + Y$，其分布是随机迭代函数系统
+
+<div>$$f_0(x) = \theta x, \quad f_1(x) = \gamma x + 1$$</div>
+
+（概率为 $(p, 1-p)$）的不变测度。这一联系使得我们能够利用随机递推方程和迭代函数系统的强大工具来推导尾部渐近、奇异性条件和维数公式。
 
 **实际意义**：分布结果转化为 PDS 参数选择的具体指导 — 左尾指数 $\log p / \log\theta$ 控制非收敛概率的衰减速率，右尾指数 $\alpha$ 控制位移风险。

content/posts/gradient-convergence.md

@@ -33,8 +33,9 @@ This paper studies a direct-search method based on sufficient decrease for uncon
 ## Key Ideas
 
 **Proof by contradiction**: Suppose $\limsup_{k\to\infty} \lVert\nabla f(x_k)\rVert > 0$. Define two regions based on gradient norm:
-- Small gradient region: $S_\leq^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert \leq \epsilon\rbrace$
-- Large gradient region: $S_>^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert > 2\epsilon\rbrace$
+
+<div>$$S_\leq^\epsilon = \{x : \lVert\nabla f(x)\rVert \leq \epsilon\}, \quad S_>^\epsilon = \{x : \lVert\nabla f(x)\rVert > 2\epsilon\}$$</div>
+
 
 **Crossing analysis**: Track when iterates enter/exit these regions via stopping times $m_j^\epsilon$ and $n_j^\epsilon$. Key observations:
 

content/posts/gradient-convergence.zh-cn.md

@@ -33,8 +33,9 @@ math: true
 ## 核心思想
 
 **反证法**：假设 $\limsup_{k\to\infty} \lVert\nabla f(x_k)\rVert > 0$。基于梯度范数定义两个区域：
-- 小梯度区域：$S_\leq^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert \leq \epsilon\rbrace$
-- 大梯度区域：$S_>^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert > 2\epsilon\rbrace$
+
+<div>$$S_\leq^\epsilon = \{x : \lVert\nabla f(x)\rVert \leq \epsilon\}, \quad S_>^\epsilon = \{x : \lVert\nabla f(x)\rVert > 2\epsilon\}$$</div>
+
 
 **穿越分析**：通过停时 $m_j^\epsilon$ 和 $n_j^\epsilon$ 追踪迭代点进入/离开这些区域的时刻。关键观察：
 

content/posts/stochastic-orders.md

@@ -21,7 +21,9 @@ Submartingale-like assumptions of the form $P(Y_k = 1 \mid \mathcal{F}_{k-1}) \g
 
 ## Main Contributions
 
-- **Stochastic domination theorem**: If $\lbrace Y_k \rbrace$ satisfies $P(Y_k = 1 \mid \mathcal{F}_{k-1}) \geq p$, then $\lbrace \tilde{Y}_k \rbrace \preceq_{\text{st}} \lbrace Y_k \rbrace$, where $\lbrace \tilde{Y}_k \rbrace$ is i.i.d. Bernoulli($p$). This holds in the usual stochastic order for stochastic processes
+- **Stochastic domination theorem**: If <span>$\{Y_k\}$</span> satisfies $P(Y_k = 1 \mid \mathcal{F}_{k-1}) \geq p$, then
+<div>$$\{\tilde{Y}_k\} \;\preceq_{\text{st}}\; \{Y_k\},$$</div>
+where <span>$\{\tilde{Y}_k\}$</span> is i.i.d. Bernoulli($p$). This holds in the usual stochastic order for stochastic processes
 
 - **Hoeffding-type bound**:
 $$P\biggl(\frac{1}{n}\sum_{k=0}^{n-1} Y_k \geq p - \varepsilon\biggr) \geq 1 - e^{-2n\varepsilon^2},$$

content/posts/stochastic-orders.zh-cn.md

@@ -21,7 +21,9 @@ math: true
 
 ## 主要贡献
 
-- **随机占优定理**：若 $\lbrace Y_k \rbrace$ 满足 $P(Y_k = 1 \mid \mathcal{F}_{k-1}) \geq p$，则 $\lbrace \tilde{Y}_k \rbrace \preceq_{\text{st}} \lbrace Y_k \rbrace$，其中 $\lbrace \tilde{Y}_k \rbrace$ 是 i.i.d. Bernoulli($p$)。这在随机过程的通常随机序意义下成立
+- **随机占优定理**：若 <span>$\{Y_k\}$</span> 满足 $P(Y_k = 1 \mid \mathcal{F}_{k-1}) \geq p$，则
+<div>$$\{\tilde{Y}_k\} \;\preceq_{\text{st}}\; \{Y_k\},$$</div>
+其中 <span>$\{\tilde{Y}_k\}$</span> 是 i.i.d. Bernoulli($p$)。这在随机过程的通常随机序意义下成立
 
 - **Hoeffding 型界**：
 $$P\biggl(\frac{1}{n}\sum_{k=0}^{n-1} Y_k \geq p - \varepsilon\biggr) \geq 1 - e^{-2n\varepsilon^2},$$

content/posts/unified-series.md

@@ -16,7 +16,7 @@ math: true
 
 Derivative-free trust-region and direct-search methods are two major classes of derivative-free optimization (DFO) algorithms. Despite their algorithmic differences, this paper reveals a unified theoretical framework: **the divergence of an algorithm-determined series governs asymptotic convergence**. Specifically, we identify a series
 $$H = \sum_{k=0}^\infty \prod_{\ell=0}^{k-1} \gamma^{y_\ell} \theta^{1-y_\ell},$$
-where $\gamma \geq 1$ and $\theta \in (0,1)$ are step-size update parameters, and $y_k \in \lbrace 0, 1\rbrace$ indicates whether iteration $k$ is "good" (e.g., the model is accurate or the direction set is well-poised). We prove: **if $H = \infty$, then $\liminf_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$**.
+where $\gamma \geq 1$ and $\theta \in (0,1)$ are step-size update parameters, and $y_k$ is a binary indicator of whether iteration $k$ is "good" (e.g., the model is accurate or the direction set is well-poised). We prove: **if $H = \infty$, then $\liminf_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$**.
 
 ## Motivation
 

content/posts/unified-series.zh-cn.md

@@ -16,7 +16,7 @@ math: true
 
 无导数信赖域方法和直接搜索方法是无导数优化（DFO）的两大主要算法类别。尽管它们在算法上有所不同，本文揭示了一个统一的理论框架：**一个算法决定的级数的发散性控制着渐近收敛性**。具体地，我们识别出一个级数
 $$H = \sum_{k=0}^\infty \prod_{\ell=0}^{k-1} \gamma^{y_\ell} \theta^{1-y_\ell},$$
-其中 $\gamma \geq 1$ 和 $\theta \in (0,1)$ 是步长更新参数，$y_k \in \lbrace 0, 1\rbrace$ 指示第 $k$ 次迭代是否是"好的"（例如模型足够精确或方向集位置良好）。我们证明：**如果 $H = \infty$，则 $\liminf_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$**。
+其中 $\gamma \geq 1$ 和 $\theta \in (0,1)$ 是步长更新参数，$y_k$ 是一个二值指标，表示第 $k$ 次迭代是否是"好的"（例如模型足够精确或方向集位置良好）。我们证明：**如果 $H = \infty$，则 $\liminf_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$**。
 
 ## 研究动机