[
    [
        {
            "type": "title",
            "content": {
                "title_content": [
                    {
                        "type": "text",
                        "content": "测验：牛顿法与优化进阶"
                    }
                ],
                "level": 1
            },
            "bbox": [
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        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "共 30 题"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "1. 优化问题里，牛顿法本质上是在求什么的根？\nA.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x)=0"
                    },
                    {
                        "type": "text",
                        "content": " \nB.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f'(x)=0"
                    },
                    {
                        "type": "text",
                        "content": " \nC.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f''(x)=0"
                    },
                    {
                        "type": "text",
                        "content": " \nD. 积分=0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "2. 一维优化牛顿法的迭代公式是?\nA.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{n+1}=x_{n}-f(x_{n})/f'(x_{n})"
                    },
                    {
                        "type": "text",
                        "content": " \nB.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{n+1}=x_{n}-f'(x_{n})/f''(x_{n})"
                    },
                    {
                        "type": "text",
                        "content": " \nC.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{n+1}=x_{n}-\\eta f'(x_{n})"
                    },
                    {
                        "type": "text",
                        "content": " \nD.   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_{n+1}=x_{n}+f'(x_{n})/f''(x_{n})"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "3. 多维优化牛顿法写作  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x \\leftarrow x - H^{-1} \\cdot"
                    },
                    {
                        "type": "text",
                        "content": " 梯度， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "H"
                    },
                    {
                        "type": "text",
                        "content": " 是什么？A. 雅可比矩阵B. Hessian（二阶偏导矩阵）C. 协方差矩阵D. 单位矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "4. 对二次函数  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = (x - 3)^2 + 1"
                    },
                    {
                        "type": "text",
                        "content": "，从  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_0 = 0"
                    },
                    {
                        "type": "text",
                        "content": " 出发，牛顿法几步到最小值？A. 1步B. 约20步C. 不收敛D. 取决于学习率"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "5. 梯度下降是一阶方法，牛顿法是？\nA. 零阶（只用 f）\nB. 一阶\nC. 二阶（用到 Hessian）\nD. 三阶"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "6. 牛顿法在根（或非退化极小）附近的局部收敛速度是？A. 线性B. 二阶（误差近似平方）"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "C. 永不收敛  \nD. 与  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\eta"
                    },
                    {
                        "type": "text",
                        "content": " 相同"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "7. Hessian 正定，对应驻点的几何是？\nA. 局部极小（碗）\nB. 局部极大（倒碗）\nC. 鞍点（马鞍）\nD. 无法判断"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "8. Hessian 有正有负特征值时，驻点是？\nA. 全局最小\nB. 鞍点\nC. 一定发散\nD. 一定是极大"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "9.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x, y) = x^{2} - y^{2}"
                    },
                    {
                        "type": "text",
                        "content": " 在原点处，牛顿法一步会到哪里？A. 某个极小值B. 原点（鞍点）C. 无穷远D. 随机"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "10. 同一函数上，梯度下降从鞍点附近出发通常会怎样？\nA. 同样停在鞍点\nB. 沿负曲率方向滑走\nC. 一定到全局最小\nD. 立刻发散到 NaN"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "11. 病态山谷（狭长等高线）里，梯度下降常见现象是？\nA. 一步到位\nB. 之字形振荡\nC. 自动选对学习率\nD. 不需要迭代"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "12. 精确牛顿法（α=1）对线性坐标缩放为什么相对不敏感？\nA. 它不用导数\nB. Hessian 把各方向曲率编进步长"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "C. 随机性抵消缩放  \nD. 只用到符号"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "13. 阻尼牛顿与纯牛顿的差别是?\nA. 不用 Hessian\nB. 步长   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\alpha \\leqslant 1"
                    },
                    {
                        "type": "text",
                        "content": " ，可能缩小牛顿步\nC. 改成随机梯度\nD. 只用一阶"
                    }
                ]
            },
            "bbox": [
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                126,
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                250
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "14. Levenberg-Marquardt 在 H 上加  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "\\lambda I"
                    },
                    {
                        "type": "text",
                        "content": " 的作用是？A. 把牛顿和梯度下降连起来B. 计算特征值C. 做正则化分类D. 替代损失"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "15. 拟牛顿（如 BFGS）的核心想法是？\nA. 随机丢弃梯度\nB. 不真算 Hessian，用梯度差估曲率\nC. 只做坐标下降\nD. 改用积分"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "16. 一维拟牛顿对应的古典方法是?\nA. 二分法\nB. 割线法（用差商代 f'）\nC. 蒙特卡洛\nD. FFT"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "17. L-BFGS 相对 BFGS 主要省了什么?\nA. 省梯度\nB. 不存   "
                    },
                    {
                        "type": "equation_inline",
                        "content": "n \\times n"
                    },
                    {
                        "type": "text",
                        "content": "  矩阵，只记最近 m 对向量\nC. 省掉线搜索\nD. 变成一阶"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "18. 深度学习训练很少用纯牛顿法，首要原因是？A. 牛顿法数学错误B. Hessian 太大，存/求逆代价过高"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "C. 牛顿法不能并行\nD. GPU 不支持二阶导"
                    }
                ]
            },
            "bbox": [
                115,
                60,
                300,
                102
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "19. Adam 算二阶优化方法吗？\nA. 算，因为它用了二阶矩\nB. 不算，没有 Hessian，仍是自适应一阶\nC. 算，等价于 BFGS\nD. 不算，它是零阶"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "20. BFGS 四个字母来自？\nA. 四个超参数\nB. 四位提出者的姓氏首字母\nC. 矩阵分解步骤\nD. 贝叶斯公式"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "21. 牛顿步里若某步  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f''(x_n) = 0"
                    },
                    {
                        "type": "text",
                        "content": "，正确处理是？A.除以0继续B.改用梯度步或  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "H + \\lambda I"
                    },
                    {
                        "type": "text",
                        "content": " C.立刻判定已收敛D.把学习率乘10"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "22.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f(x) = x^4 / 4 - x^2 / 2"
                    },
                    {
                        "type": "text",
                        "content": " 的驻点中， "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = 0"
                    },
                    {
                        "type": "text",
                        "content": " 是？\nA. 全局最小\nB. 局部极大\nC. 不是驻点\nD. 鞍点且  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "f' > 0"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "23. 从  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x_0 = 0.1"
                    },
                    {
                        "type": "text",
                        "content": " 对上题做牛顿法，更可能冲向？A.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = 1"
                    },
                    {
                        "type": "text",
                        "content": " B.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = 0"
                    },
                    {
                        "type": "text",
                        "content": " （极大）C.  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "x = 100"
                    },
                    {
                        "type": "text",
                        "content": " D. 随机"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "24. sklearn 逻辑回归 solver='lbfgs' 属于?\nA. 纯 SGD\nB. 有限内存拟牛顿"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "C. 牛顿-共轭梯度的别名\nD. 遗传算法"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "25. 牛顿每一步都要解  "
                    },
                    {
                        "type": "equation_inline",
                        "content": "H \\mathrm{d} = -"
                    },
                    {
                        "type": "text",
                        "content": " 梯度。这和后面哪一课直接衔接？A. 排列组合B. 线性方程组与高斯消元C. 贝叶斯D. 图论"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "26. 「二次收敛」的直观理解，哪句正确？\nA. 每步误差减半\nB. 误差大约变成上次误差的平方\nC. 步数必须是 2 的幂\nD. 学习率必须是 0.2"
                    }
                ]
            },
            "bbox": [
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                395
            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "27. 为什么 mini-batch 训练更不适合精确牛顿?\nA. batch 没有梯度\nB. 噪声梯度让 Hessian 估计更不稳定\nC. batch 一定非凸\nD. GPU 算不了矩阵"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "28. 拟牛顿要求近似矩阵 B 尽量保持正定，是为了？\nA. 让步长必须为 1\nB. 保证搜索方向是下降方向\nC. 让特征值全为负\nD. 减少内存"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "29. 相机标定、Bundle Adjustment 里常用的高斯牛顿 / LM，本质是？\nA. 随机搜索\nB. 最小二乘上的牛顿类方法\nC. 纯 Adam\nD. 离散枚举"
                    }
                ]
            },
            "bbox": [
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            ]
        },
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "30. 下列哪句概括今天最准确？\nA. 牛顿法已过时，不必学\nB. 牛顿用法是曲率；问题太大就估曲率或退回一阶"
                    }
                ]
            },
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            ]
        }
    ],
    [
        {
            "type": "paragraph",
            "content": {
                "paragraph_content": [
                    {
                        "type": "text",
                        "content": "C.所有机器学习都必须用牛顿D.Hessian正定等于全局最优"
                    }
                ]
            },
            "bbox": [
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            ]
        }
    ]
]