` 包裹，长文用 `
`/`
` 划分小节。 3. 列表用 `
- `，引用用 `
`，代码用 `
`，重要句子可用 ``。
4. 保留原文所有信息，不得增删改内容。
5. 不要输出 ``/``/`` 等文档级标签。
* 正文内容：
"We study hybrid online learning with stochastic features and adversarial labels when the feature distribution is unknown. Prior work gave stochastic sequential covers of log-size $O(d\log^2 T)$ for a general VC class and logarithmic bounds only under stronger star-type structure. We show that finite VC dimension alone is enough: every binary class of VC dimension $d$ admits a distribution-blind exact cover with log-size $O(d\log T+\log(1/δ))$, matching the worst-case $Ω(d\log(T/d))$ dependence on the horizon. The key ingredient is a new local cube weight that counts Boolean subcubes incident to each realizable trace. Unlike the equal splits used in earlier constructions, we split mass unevenly according to these cube weights; a coordinate-deletion inequality shows that VC dimension controls the total cost of these splits. This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret for general VC partitions with unknown i.i.d. features, and an exact polynomial-time implementation for planar affine halfspaces. We also show that the binary phenomenon does not extend from VC dimension to pseudo-dimension: a fixed pseudo-dimension-one sigmoid class requires $Ω(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$."
2. 解构正文内容：
* 这是一段学术摘要，没有显式的“列表”或“引用”，但包含多个逻辑层次和关键结论。
* 主题：混合在线学习，随机特征，对抗标签，未知特征分布。
* 段落1：介绍研究背景和先前工作。
"We study hybrid online learning with stochastic features and adversarial labels when the feature distribution is unknown. Prior work gave stochastic sequential covers of log-size $O(d\log^2 T)$ for a general VC class and logarithmic bounds only under stronger star-type structure."
* 段落2：本文的核心贡献（主要结果）。
"We show that finite VC dimension alone is enough: every binary class of VC dimension $d$ admits a distribution-blind exact cover with log-size $O(d\log T+\log(1/δ))$, matching the worst-case $Ω(d\log(T/d))$ dependence on the horizon." -> 这句话很重要，可以用 `` 包裹，或者将 "We show that finite VC dimension alone is enough" 部分加粗。
* 段落3：关键方法/技术。
"The key ingredient is a new local cube weight that counts Boolean subcubes incident to each realizable trace. Unlike the equal splits used in earlier constructions, we split mass unevenly according to these cube weights; a coordinate-deletion inequality shows that VC dimension controls the total cost of these splits."
* 段落4：具体应用/推论。
"This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret for general VC partitions with unknown i.i.d. features, and an exact polynomial-time implementation for planar affine halfspaces."
* 段落5：负结果/进一步讨论。
"We also show that the binary phenomenon does not extend from VC dimension to pseudo-dimension: a fixed pseudo-dimension-one sigmoid class requires $Ω(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$."
3. 设计HTML结构：
* 由于是单段摘要，可以加一个 `` 作为标题，比如 `
研究背景与核心贡献
`，但规则说“不得增删改内容”，所以不能随便加标题，除非为了划分小节。
* 规则2：“每个自然段落用包裹，长文用
/
划分小节”。
* 既然原文没有明确的小节标题，我可以将原文的自然段落用 `
` 包裹。
* 如果要把长文划分小节，我可以提取第一句作为小节标题吗？不行，规则4“不得增删改内容”。
* 那就直接用 `
` 包裹每个自然段。原文看起来是一整段，但我可以按照语义将其拆分成几个 `
` 标签，只要不改变字词。
* 或者，既然它是一整段，我就用一个 `
` 包裹整段？不，通常摘要会分段。为了排版美观，按句群或逻辑块拆分成多个 `
` 是合理的，只要文字完全一致。
* 让我们把原文拆成5个逻辑段落（对应上面的5个句子群）。
* 加粗关键结论：
* We show that finite VC dimension alone is enough: every binary class...
* The key ingredient is a new local cube weight...
* This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret...
* 数学公式：规则提到“代码用
”，这里公式是用 `$...$` 包裹的 LaTeX 格式，不是代码块。规则3说“代码用”，但通常数学公式不需要变成代码块，除非特别要求。规则3“代码用”，这里没有代码，只有数学公式。为了安全起见，保持公式原样，或者把公式当作行内代码？不，原文是 `$O(d\log^2 T)$`，直接保留文本即可，不需要转义。如果严格说“代码”，这里并没有编程语言代码。所以直接输出文本。
* 等等，规则说“重要句子可用”。
* 让我们仔细拆分段落：
原文：
We study hybrid online learning with stochastic features and adversarial labels when the feature distribution is unknown. Prior work gave stochastic sequential covers of log-size $O(d\log^2 T)$ for a general VC class and logarithmic bounds only under stronger star-type structure.
We show that finite VC dimension alone is enough: every binary class of VC dimension $d$ admits a distribution-blind exact cover with log-size $O(d\log T+\log(1/δ))$, matching the worst-case $Ω(d\log(T/d))$ dependence on the horizon.
The key ingredient is a new local cube weight that counts Boolean subcubes incident to each realizable trace. Unlike the equal splits used in earlier constructions, we split mass unevenly according to these cube weights; a coordinate-deletion inequality shows that VC dimension controls the total cost of these splits.
This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret for general VC partitions with unknown i.i.d. features, and an exact polynomial-time implementation for planar affine halfspaces.
We also show that the binary phenomenon does not extend from VC dimension to pseudo-dimension: a fixed pseudo-dimension-one sigmoid class requires $Ω(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$.
4. 应用规则检查：
- 只输出HTML片段本身？是。
- 不要Markdown围栏、不要解释？是。
- 每个自然段落用包裹？是，按逻辑拆分成5个
。
- 长文用
/
划分小节？由于原文没有小标题，且不能增删改内容，强行加标题违反规则4。所以只用
和。
- 列表内容用
`，代码用 ``，重要句子可用 ``。 4. 保留原文所有信息，不得增删改内容。 5. 不要输出 ``/``/`` 等文档级标签。 * 正文内容： "We study hybrid online learning with stochastic features and adversarial labels when the feature distribution is unknown. Prior work gave stochastic sequential covers of log-size $O(d\log^2 T)$ for a general VC class and logarithmic bounds only under stronger star-type structure. We show that finite VC dimension alone is enough: every binary class of VC dimension $d$ admits a distribution-blind exact cover with log-size $O(d\log T+\log(1/δ))$, matching the worst-case $Ω(d\log(T/d))$ dependence on the horizon. The key ingredient is a new local cube weight that counts Boolean subcubes incident to each realizable trace. Unlike the equal splits used in earlier constructions, we split mass unevenly according to these cube weights; a coordinate-deletion inequality shows that VC dimension controls the total cost of these splits. This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret for general VC partitions with unknown i.i.d. features, and an exact polynomial-time implementation for planar affine halfspaces. We also show that the binary phenomenon does not extend from VC dimension to pseudo-dimension: a fixed pseudo-dimension-one sigmoid class requires $Ω(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$." 2. 解构正文内容： * 这是一段学术摘要，没有显式的“列表”或“引用”，但包含多个逻辑层次和关键结论。 * 主题：混合在线学习，随机特征，对抗标签，未知特征分布。 * 段落1：介绍研究背景和先前工作。 "We study hybrid online learning with stochastic features and adversarial labels when the feature distribution is unknown. Prior work gave stochastic sequential covers of log-size $O(d\log^2 T)$ for a general VC class and logarithmic bounds only under stronger star-type structure." * 段落2：本文的核心贡献（主要结果）。 "We show that finite VC dimension alone is enough: every binary class of VC dimension $d$ admits a distribution-blind exact cover with log-size $O(d\log T+\log(1/δ))$, matching the worst-case $Ω(d\log(T/d))$ dependence on the horizon." -> 这句话很重要，可以用 `` 包裹，或者将 "We show that finite VC dimension alone is enough" 部分加粗。 * 段落3：关键方法/技术。 "The key ingredient is a new local cube weight that counts Boolean subcubes incident to each realizable trace. Unlike the equal splits used in earlier constructions, we split mass unevenly according to these cube weights; a coordinate-deletion inequality shows that VC dimension controls the total cost of these splits." * 段落4：具体应用/推论。 "This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret for general VC partitions with unknown i.i.d. features, and an exact polynomial-time implementation for planar affine halfspaces." * 段落5：负结果/进一步讨论。 "We also show that the binary phenomenon does not extend from VC dimension to pseudo-dimension: a fixed pseudo-dimension-one sigmoid class requires $Ω(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$." 3. 设计HTML结构： * 由于是单段摘要，可以加一个 `` 作为标题，比如 `
研究背景与核心贡献
`，但规则说“不得增删改内容”，所以不能随便加标题，除非为了划分小节。 * 规则2：“每个自然段落用包裹，长文用
/
划分小节”。 * 既然原文没有明确的小节标题，我可以将原文的自然段落用 `
` 包裹。 * 如果要把长文划分小节，我可以提取第一句作为小节标题吗？不行，规则4“不得增删改内容”。 * 那就直接用 `
` 包裹每个自然段。原文看起来是一整段，但我可以按照语义将其拆分成几个 `
` 标签，只要不改变字词。 * 或者，既然它是一整段，我就用一个 `
` 包裹整段？不，通常摘要会分段。为了排版美观，按句群或逻辑块拆分成多个 `
` 是合理的，只要文字完全一致。 * 让我们把原文拆成5个逻辑段落（对应上面的5个句子群）。 * 加粗关键结论： * We show that finite VC dimension alone is enough: every binary class... * The key ingredient is a new local cube weight... * This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret... * 数学公式：规则提到“代码用
”，这里公式是用 `$...$` 包裹的 LaTeX 格式，不是代码块。规则3说“代码用”，但通常数学公式不需要变成代码块，除非特别要求。规则3“代码用”，这里没有代码，只有数学公式。为了安全起见，保持公式原样，或者把公式当作行内代码？不，原文是 `$O(d\log^2 T)$`，直接保留文本即可，不需要转义。如果严格说“代码”，这里并没有编程语言代码。所以直接输出文本。 * 等等，规则说“重要句子可用”。 * 让我们仔细拆分段落： 原文： We study hybrid online learning with stochastic features and adversarial labels when the feature distribution is unknown. Prior work gave stochastic sequential covers of log-size $O(d\log^2 T)$ for a general VC class and logarithmic bounds only under stronger star-type structure. We show that finite VC dimension alone is enough: every binary class of VC dimension $d$ admits a distribution-blind exact cover with log-size $O(d\log T+\log(1/δ))$, matching the worst-case $Ω(d\log(T/d))$ dependence on the horizon. The key ingredient is a new local cube weight that counts Boolean subcubes incident to each realizable trace. Unlike the equal splits used in earlier constructions, we split mass unevenly according to these cube weights; a coordinate-deletion inequality shows that VC dimension controls the total cost of these splits. This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret for general VC partitions with unknown i.i.d. features, and an exact polynomial-time implementation for planar affine halfspaces. We also show that the binary phenomenon does not extend from VC dimension to pseudo-dimension: a fixed pseudo-dimension-one sigmoid class requires $Ω(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$. 4. 应用规则检查： - 只输出HTML片段本身？是。 - 不要Markdown围栏、不要解释？是。 - 每个自然段落用包裹？是，按逻辑拆分成5个
。 - 长文用
/
划分小节？由于原文没有小标题，且不能增删改内容，强行加标题违反规则4。所以只用
和。 - 列表内容用