{"pdfData":[[{"bbox":[54,52,233,70],"type":"title","angle":0,"index":0,"level":1,"text":"# 测验：牛顿法与优化进阶","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"aecd6d19-fd1d-46a5-b9d5-4ed1ad9ab28d","page_size":[595,841],"block_position":"0-0"},{"bbox":[54,80,98,93],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"共 30 题 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"d71142ba-b142-4243-b58e-ae8a6b12dc32","page_size":[595,841],"block_position":"0-1"},{"bbox":[54,106,289,211],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"1. 优化问题里，牛顿法本质上是在求什么的根？\nA. $f(x)=0$ B. $f'(x)=0$ C. $f''(x)=0$ D. 积分=0 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"a5967cea-6d30-4e2b-994c-fcccbf5abf4d","page_size":[595,841],"block_position":"0-2"},{"bbox":[53,230,226,334],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"2. 一维优化牛顿法的迭代公式是?\nA. $x_{n+1}=x_{n}-f(x_{n})/f'(x_{n})$ B. $x_{n+1}=x_{n}-f'(x_{n})/f''(x_{n})$ C. $x_{n+1}=x_{n}-\\eta f'(x_{n})$ D. $x_{n+1}=x_{n}+f'(x_{n})/f''(x_{n})$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"d8530d8e-ebfa-4f22-862b-157a0e44f34b","page_size":[595,841],"block_position":"0-3"},{"bbox":[50,352,337,456],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"3. 多维优化牛顿法写作 $x \\leftarrow x - H^{-1} \\cdot$ 梯度， $H$ 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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"e3b7f31a-03de-4deb-9601-dce315f3ced1","page_size":[595,841],"block_position":"0-4"},{"bbox":[53,476,398,581],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"4. 对二次函数 $f(x) = (x - 3)^2 + 1$ ，从 $x_0 = 0$ 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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"f581a17d-756b-4a82-90d2-2e6269ca6f95","page_size":[595,841],"block_position":"0-5"},{"bbox":[54,598,235,703],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"6ef78981-bdf8-499a-a470-5637b31b2b14","page_size":[595,841],"block_position":"0-6"},{"bbox":[54,720,333,781],"type":"text","angle":0,"index":7,"merge_prev":false,"text":"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） ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":0,"id":"54d081e6-646b-4752-a52e-f4f574690648","page_size":[595,841],"block_position":"0-7"}],[{"bbox":[70,51,145,87],"type":"text","angle":0,"index":0,"merge_prev":false,"text":"C. 永不收敛  \nD. 与 $\\eta$ 相同 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"e8147de5-0d54-4599-ad3d-57c25ebeabe6","page_size":[595,841],"block_position":"1-0"},{"bbox":[54,106,246,211],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"c8501640-d890-475c-b7e2-a51f2c3f1bd2","page_size":[595,841],"block_position":"1-1"},{"bbox":[54,230,257,334],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"8f7ee416-b1fc-45b8-90f1-fb633e1105f0","page_size":[595,841],"block_position":"1-2"},{"bbox":[50,352,304,457],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"9. $f(x, y) = x^{2} - y^{2}$ 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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"7ac305fc-85f6-4ce2-a795-22a9101a2f59","page_size":[595,841],"block_position":"1-3"},{"bbox":[54,476,327,580],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"fcfc7877-771e-48d7-96cf-526a7e589c13","page_size":[595,841],"block_position":"1-4"},{"bbox":[54,598,327,703],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"b84218c9-dbc8-43db-8b3d-6ef2d71a6565","page_size":[595,841],"block_position":"1-5"},{"bbox":[55,720,350,780],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":1,"id":"d58bbb16-c162-4282-932f-a184e96cd4c2","page_size":[595,841],"block_position":"1-6"}],[{"bbox":[69,51,167,87],"type":"text","angle":0,"index":0,"merge_prev":false,"text":"C. 随机性抵消缩放  \nD. 只用到符号 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"2eedb1e5-d1cc-4ae4-90ea-87f658206b3d","page_size":[595,841],"block_position":"2-0"},{"bbox":[55,106,233,211],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"13. 阻尼牛顿与纯牛顿的差别是?\nA. 不用 Hessian\nB. 步长 $\\alpha \\leqslant 1$ ，可能缩小牛顿步\nC. 改成随机梯度\nD. 只用一阶 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"1b6550da-e64a-4817-8571-5678cd3a0414","page_size":[595,841],"block_position":"2-1"},{"bbox":[54,230,317,334],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"14. Levenberg-Marquardt 在 H 上加 $\\lambda I$ 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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"bb327c03-1cc1-4d08-97f3-412ce809cddc","page_size":[595,841],"block_position":"2-2"},{"bbox":[52,352,255,457],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"c08a7335-c774-4f16-81ca-1ea8ec587f54","page_size":[595,841],"block_position":"2-3"},{"bbox":[54,476,229,581],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"d7edfe80-3408-4549-97f8-97b77644a671","page_size":[595,841],"block_position":"2-4"},{"bbox":[55,599,271,703],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"17. L-BFGS 相对 BFGS 主要省了什么?\nA. 省梯度\nB. 不存 $n \\times n$ 矩阵，只记最近 m 对向量\nC. 省掉线搜索\nD. 变成一阶 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"6966c4dd-c9d9-4bd3-b0d7-dd1c926f4f5b","page_size":[595,841],"block_position":"2-5"},{"bbox":[55,721,295,781],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":2,"id":"24495752-9524-4929-8fdb-b6b7299d10bb","page_size":[595,841],"block_position":"2-6"}],[{"bbox":[69,51,179,86],"type":"text","angle":0,"index":0,"merge_prev":false,"text":"C. 牛顿法不能并行\nD. GPU 不支持二阶导 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"5ad6c51e-0585-40aa-a42f-0255931d180a","page_size":[595,841],"block_position":"3-0"},{"bbox":[54,107,279,211],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"b0bef2b4-61f7-4c18-8e96-138700a5a9b8","page_size":[595,841],"block_position":"3-1"},{"bbox":[54,230,211,333],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"6f32e079-bd3f-48a8-87a5-47da17bb5806","page_size":[595,841],"block_position":"3-2"},{"bbox":[51,352,283,457],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"21. 牛顿步里若某步 $f''(x_n) = 0$ ，正确处理是？A.除以0继续B.改用梯度步或 $H + \\lambda I$ C.立刻判定已收敛D.把学习率乘10 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"8e75a97d-f30e-4275-9b7d-87ed4ddcc01f","page_size":[595,841],"block_position":"3-3"},{"bbox":[54,476,257,580],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"22. $f(x) = x^4 / 4 - x^2 / 2$ 的驻点中， $x = 0$ 是？\nA. 全局最小\nB. 局部极大\nC. 不是驻点\nD. 鞍点且 $f' > 0$ ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"5ca0d3fd-c232-48fc-a81f-0013935b9bf6","page_size":[595,841],"block_position":"3-4"},{"bbox":[54,598,283,703],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"23. 从 $x_0 = 0.1$ 对上题做牛顿法，更可能冲向？A. $x = 1$ B. $x = 0$ （极大）C. $x = 100$ D. 随机 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"db1e7af3-0088-41a8-881f-3dee5773ba2d","page_size":[595,841],"block_position":"3-5"},{"bbox":[54,721,284,780],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"24. sklearn 逻辑回归 solver='lbfgs' 属于?\nA. 纯 SGD\nB. 有限内存拟牛顿 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":3,"id":"300a0fd4-3394-422b-a6b2-817c61174156","page_size":[595,841],"block_position":"3-6"}],[{"bbox":[69,51,193,86],"type":"text","angle":0,"index":0,"merge_prev":false,"text":"C. 牛顿-共轭梯度的别名\nD. 遗传算法 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"670d4a33-029b-4c79-989d-861cbeed47cd","page_size":[595,841],"block_position":"4-0"},{"bbox":[54,107,371,211],"type":"text","angle":0,"index":1,"merge_prev":false,"text":"25. 牛顿每一步都要解 $H \\mathrm{d} = -$ 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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"e6745da5-beb8-442b-8fc7-7a72ca513fb8","page_size":[595,841],"block_position":"4-1"},{"bbox":[54,230,262,333],"type":"text","angle":0,"index":2,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"403d59e4-cd4a-47ea-bba8-8bde70bd46d8","page_size":[595,841],"block_position":"4-2"},{"bbox":[51,352,295,456],"type":"text","angle":0,"index":3,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"e94bdbcb-bcd5-48bc-ba15-85ed195ad6ec","page_size":[595,841],"block_position":"4-3"},{"bbox":[54,476,311,580],"type":"text","angle":0,"index":4,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"8201c3e5-308c-4c18-994d-d0421e2fbdb8","page_size":[595,841],"block_position":"4-4"},{"bbox":[54,598,402,703],"type":"text","angle":0,"index":5,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"6bcdf3ee-a590-4829-8005-9049ed92525c","page_size":[595,841],"block_position":"4-5"},{"bbox":[54,721,321,780],"type":"text","angle":0,"index":6,"merge_prev":false,"text":"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 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":4,"id":"303f6662-5304-4216-9bda-8dc5dd893f29","page_size":[595,841],"block_position":"4-6"}],[{"bbox":[70,51,223,87],"type":"text","angle":0,"index":0,"merge_prev":false,"text":"C.所有机器学习都必须用牛顿D.Hessian正定等于全局最优 ","color":{"line":"rgba(13, 83, 222, 1)","fill":"rgba(13, 83, 222, 1)"},"page_idx":5,"id":"d7e7f90e-9760-4b18-951a-6d8cd670f9e9","page_size":[595,841],"block_position":"5-0"}]],"mergeConnections":[]}