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| { | |
| "slug": "adaboost-alpha-update", | |
| "title": "手撕 AdaBoost 基分类器权重系数计算 (AdaBoost Alpha Calculation)", | |
| "difficulty": "Easy", | |
| "category": "ML", | |
| "tags": [ | |
| "集成学习", | |
| "AdaBoost", | |
| "基学习器权重" | |
| ], | |
| "description_md": "### 题目描述\n在二分类 AdaBoost 中，给定基分类器的加权错误率 $\\epsilon_m \\in (0, 0.5)$，计算该基分类器的表决权重 $\\displaystyle \\alpha_m = \\frac{1}{2} \\ln \\frac{1 - \\epsilon_m}{\\epsilon_m + \\epsilon_{eps}}$。\n\n### 函数签名\n```python\ndef adaboost_alpha(error_rate: float, eps: float = 1e-10) -> float:\n```\n\n### 示例\n```python\n# 输入\nerror_rate = 0.1\neps = 1e-10\n\n# 输出 (float)\n1.0986\n```\n\n### 参数规格与约束\n- **`error_rate`**：`float`\n- **`eps`**：`float`（默认 `1e-10`）\n- **返回值**：`float`，输出规格 `float` (示例值: `1.098612`)\n- **精度容差**：`atol=0.0001, rtol=0.0001`", | |
| "entry_point": "adaboost_alpha", | |
| "starter_code": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\nfrom typing import Optional, List, Tuple\n\ndef adaboost_alpha(error_rate: float, eps: float = 1e-10) -> float:\n \"\"\"\n 手撕 AdaBoost 基分类器权重系数计算 (AdaBoost Alpha Calculation)\n \"\"\"\n # 请在此编写代码\n pass\n", | |
| "reference_code": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\nfrom typing import Optional, List, Tuple\n\ndef adaboost_alpha(error_rate: float, eps: float = 1e-10) -> float:\n err = max(min(error_rate, 1.0 - eps), eps)\n return float(0.5 * math.log((1.0 - err) / err))\n", | |
| "forbidden_modules": [ | |
| "os", | |
| "sys", | |
| "subprocess", | |
| "shutil", | |
| "socket", | |
| "requests" | |
| ], | |
| "forbidden_calls": [], | |
| "time_limit": 3.0, | |
| "memory_limit_mb": 512, | |
| "atol": 0.0001, | |
| "rtol": 0.0001, | |
| "test_code": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\n\ncases = [\n ('错误率 0.1', 0.1),\n ('错误率 0.4', 0.4),\n ('多维批次与参数泛化验证', 0.08),\n ('边界极值与稳定性验证', 0.48),\n]\nfor idx, (name, *args) in enumerate(cases, 1):\n t0 = time.time()\n try:\n exp = ref_obj(*args)\n act = user_obj(*args)\n match, err, max_diff = compare_outputs(act, exp, atol=atol, rtol=rtol)\n passed, msg = match, err\n except Exception as e:\n passed, msg = False, f\"Runtime Exception: {str(e)}\"\n act, exp, max_diff = None, None, None\n cases_results.append({\n \"case_id\": idx, \"case_name\": name, \"passed\": passed,\n \"time_ms\": round((time.time() - t0) * 1000, 2), \"message\": msg,\n \"user_output_summary\": summarize_obj(act) if act is not None else None,\n \"expected_summary\": summarize_obj(exp) if exp is not None else None,\n \"max_diff\": max_diff,\n })\n", | |
| "is_featured": false, | |
| "editorial_md": "#### 💡 核心算法原理与数学推导\n**手撕 AdaBoost 基分类器权重系数计算 (AdaBoost Alpha Calculation)**（`adaboost_alpha`）聚焦于 **ML** 中的关键计算逻辑：\n\n在二分类 AdaBoost 中，给定基分类器的加权错误率 $\\epsilon_m \\in (0, 0.5)$，计算该基分类器的表决权重 $\\displaystyle \\alpha_m = \\frac{1}{2} \\ln \\frac{1 - \\epsilon_m}{\\epsilon_m + \\epsilon_{eps}}$。\n\n在实现中，函数接收参数 `error_rate` (`float`)、`eps` (`float`)，目标是计算出类型与形状严格符合 `float` 契约的结果。\n\n#### 📐 输入输出张量变换与步骤拆解\n- **输入基准规格**：`error_rate` 规格为 `float` (示例值: `0.1`)。\n- **期望输出规格**：`float` (示例值: `1.0986122886681098`)。\n\n**具体推导与计算步骤如下：**\n1. 计算中间量 `err`：通过表达式 `max(min(error_rate, 1.0 - eps), eps)` 完成张量或数值变换。\n2. 返回最终结果：输出 `float(0.5 * math.log((1.0 - err) / err))`，严格对齐函数签名要求的返回类型与维度。\n\n#### 📊 复杂度分析\n- **时间复杂度**：$O(1)$，仅涉及标量代数运算或闭式公式求值，执行耗时与模型规模参数数值大小无关。\n- **空间复杂度**：$O(1)$，仅使用常数级标量局部变量，无额外张量显存分配。\n\n#### ⚠️ 关键实现细节与易错点\n- **除零保护与数值平滑 (`eps`)**：在执行分母归一化、标准差缩放或对数运算时，必须按公式位置准确加上 `eps`（例如在平方根内 `sqrt(var + eps)` 还是分母外 `+ eps`），防止全零或常量输入触发 `NaN` / `Inf`。\n- **指数与对数域数值稳定性**：直接对大幅值 logits 计算 `exp` 易导致浮点数上溢（Overflow），对接近 0 的概率直接计算 `log` 易导致下溢；应优先使用 `F.log_softmax`、`F.logsigmoid` 或减去最大值的稳定化写法。\n- **纯向量化算子优先**：尽量使用 PyTorch 原生向量化张量算子替代低效的 Python 逐元素循环，以兼顾代码简洁性与执行性能。\n\n#### 💻 官方标准参考实现\n```python\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\nfrom typing import Optional, List, Tuple\n\ndef adaboost_alpha(error_rate: float, eps: float = 1e-10) -> float:\n err = max(min(error_rate, 1.0 - eps), eps)\n return float(0.5 * math.log((1.0 - err) / err))\n```", | |
| "difficulty_breakdown": { | |
| "concept": 2, | |
| "implementation": 1, | |
| "math_derivation": 2, | |
| "boundary_complexity": 2 | |
| }, | |
| "test_cases": [ | |
| { | |
| "name": "错误率 0.1", | |
| "input_desc": "真实参数契约用例 (error_rate, eps)", | |
| "params": [ | |
| { | |
| "name": "error_rate", | |
| "type": "float", | |
| "value": "0.1" | |
| }, | |
| { | |
| "name": "eps", | |
| "type": "float", | |
| "value": "1e-10" | |
| } | |
| ], | |
| "raw_inputs": { | |
| "error_rate": "0.1", | |
| "eps": "1e-10" | |
| } | |
| }, | |
| { | |
| "name": "错误率 0.4", | |
| "input_desc": "真实参数契约用例 (error_rate, eps)", | |
| "params": [ | |
| { | |
| "name": "error_rate", | |
| "type": "float", | |
| "value": "0.4" | |
| }, | |
| { | |
| "name": "eps", | |
| "type": "float", | |
| "value": "1e-10" | |
| } | |
| ], | |
| "raw_inputs": { | |
| "error_rate": "0.4", | |
| "eps": "1e-10" | |
| } | |
| }, | |
| { | |
| "name": "多维批次与参数泛化验证", | |
| "input_desc": "真实参数契约用例 (error_rate, eps)", | |
| "params": [ | |
| { | |
| "name": "error_rate", | |
| "type": "float", | |
| "value": "0.08" | |
| }, | |
| { | |
| "name": "eps", | |
| "type": "float", | |
| "value": "1e-10" | |
| } | |
| ], | |
| "raw_inputs": { | |
| "error_rate": "0.08", | |
| "eps": "1e-10" | |
| } | |
| } | |
| ] | |
| } |