Blame for data/problems/adaboost-alpha-update.json

{
  "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"
      }
    }
  ]
}