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"slug": "adagrad-optimizer-step",
"title": "手撕 AdaGrad 自适应梯度单步 (AdaGrad Step)",
"difficulty": "Easy",
"category": "Optim",
"tags": [
"AdaGrad",
"自适应学习率"
],
"description_md": "### 题目描述\n累积历史梯度平方和实现自适应步长：$\\displaystyle G_t = G_{t-1} + g_t^2, \\theta_t = \\theta_{t-1} - \\frac{\\eta}{\\sqrt{G_t} + \\epsilon} g_t$。\n\n### 函数签名\n```python\ndef adagrad_step(\n param: torch.Tensor,\n grad: torch.Tensor,\n accum: torch.Tensor,\n lr: float = 0.01,\n eps: float = 1e-8,\n) -> Tuple[torch.Tensor, torch.Tensor]:\n```\n\n### 示例\n```python\n# 输入\nparam = [1.0]\ngrad = [0.5]\naccum = [0.25]\nlr = 0.01\neps = 1e-8\n\n# 输出\n(\n [0.9929], # shape=[1], dtype=float32\n [0.5] # shape=[1], dtype=float32\n)\n```\n\n### 参数规格与约束\n- **`param`**：`torch.Tensor`，测试规格 `shape=[1], dtype=float32`\n- **`grad`**：`torch.Tensor`，测试规格 `shape=[1], dtype=float32`\n- **`accum`**：`torch.Tensor`，测试规格 `shape=[1], dtype=float32`\n- **`lr`**：`float`（默认 `0.01`）\n- **`eps`**：`float`（默认 `1e-8`）\n- **返回值**：`Tuple[torch.Tensor, torch.Tensor]`，输出规格 `Tuple` (`torch.Tensor(shape=[1], dtype=float32)`, `torch.Tensor(shape=[1], dtype=float32)`)\n- **精度容差**：`atol=0.0001, rtol=0.0001`",
"entry_point": "adagrad_step",
"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 adagrad_step(\n param: torch.Tensor,\n grad: torch.Tensor,\n accum: torch.Tensor,\n lr: float = 0.01,\n eps: float = 1e-8,\n) -> Tuple[torch.Tensor, torch.Tensor]:\n \"\"\"\n 手撕 AdaGrad 自适应梯度单步 (AdaGrad Step)\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 adagrad_step(param: torch.Tensor, grad: torch.Tensor, accum: torch.Tensor, lr: float = 0.01, eps: float = 1e-8) -> Tuple[torch.Tensor, torch.Tensor]:\n new_accum = accum + grad ** 2\n new_p = param - lr * grad / (torch.sqrt(new_accum) + eps)\n return new_p, new_accum\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 ('AdaGrad单步', torch.tensor([1.0], dtype=torch.float32), torch.tensor([0.5], dtype=torch.float32), torch.tensor([0.25], dtype=torch.float32), 0.01),\n ('多批次扩展泛化验证', torch.tensor([1.0, 1.0], dtype=torch.float32), torch.tensor([0.5, 0.75], dtype=torch.float32), torch.tensor([0.25, 0.375], dtype=torch.float32), 0.01),\n ('多维批次与参数泛化验证', torch.tensor([1.0], dtype=torch.float32), torch.tensor([0.75], dtype=torch.float32), torch.tensor([0.375], dtype=torch.float32), 0.008),\n ('边界极值与稳定性验证', torch.tensor([1.0], dtype=torch.float32), torch.tensor([0.25], dtype=torch.float32), torch.tensor([0.125], dtype=torch.float32), 0.012),\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 AdaGrad 自适应梯度单步 (AdaGrad Step)**（`adagrad_step`）聚焦于 **Optim** 中的关键计算逻辑：\n\n累积历史梯度平方和实现自适应步长：$\\displaystyle G_t = G_{t-1} + g_t^2, \\theta_t = \\theta_{t-1} - \\frac{\\eta}{\\sqrt{G_t} + \\epsilon} g_t$。\n\n在实现中，函数接收参数 `param` (`torch.Tensor`)、`grad` (`torch.Tensor`)、`accum` (`torch.Tensor`)、`lr` (`float`)、`eps` (`float`)，目标是计算出类型与形状严格符合 `Tuple[torch.Tensor, torch.Tensor]` 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**：`param` 规格为 `torch.Tensor(shape=[1], dtype=float32)`，`grad` 规格为 `torch.Tensor(shape=[1], dtype=float32)`，`accum` 规格为 `torch.Tensor(shape=[1], dtype=float32)`，`lr` 规格为 `float` (示例值: `0.01`)。\n- **期望输出规格**：`Tuple` (`torch.Tensor(shape=[1], dtype=float32)`, `torch.Tensor(shape=[1], dtype=float32)`)。\n\n**具体推导与计算步骤如下：**\n1. 计算中间量 `new_accum`：通过表达式 `accum + grad ** 2` 完成张量或数值变换。\n2. 计算中间量 `new_p`：通过表达式 `param - lr * grad / (torch.sqrt(new_accum) + eps)` 完成张量或数值变换。\n3. 返回最终结果：输出 `new_p, new_accum`，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**：$O(N)$（线性于输入张量总元素个数 $\\prod d_i$），所有运算均为向量化逐元素算子或沿指定维度的归约计算。\n- **空间复杂度**：$O(N)$，需要与输入或输出张量同规模的显存空间保存计算结果。\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- **边界输入鲁棒性**：注意单样本（`Batch=1`）、单步序列（`Seq=1`）、零值输入或极端超参数下的计算行为，不要在代码中硬编码特定测试用例的维度常数。\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 adagrad_step(param: torch.Tensor, grad: torch.Tensor, accum: torch.Tensor, lr: float = 0.01, eps: float = 1e-8) -> Tuple[torch.Tensor, torch.Tensor]:\n new_accum = accum + grad ** 2\n new_p = param - lr * grad / (torch.sqrt(new_accum) + eps)\n return new_p, new_accum\n```",
"difficulty_breakdown": {
"concept": 2,
"implementation": 2,
"math_derivation": 2,
"boundary_complexity": 2
},
"test_cases": [
{
"name": "AdaGrad单步",
"input_desc": "真实参数契约用例 (param, grad, accum, lr, eps)",
"params": [
{
"name": "param",
"type": "torch.Tensor",
"value": "torch.tensor([1.0], dtype=torch.float32)"
},
{
"name": "grad",
"type": "torch.Tensor",
"value": "torch.tensor([0.5], dtype=torch.float32)"
},
{
"name": "accum",
"type": "torch.Tensor",
"value": "torch.tensor([0.25], dtype=torch.float32)"
},
{
"name": "lr",
"type": "float",
"value": "0.01"
},
{
"name": "eps",
"type": "float",
"value": "1e-8"
}
],
"raw_inputs": {
"param": "torch.tensor([1.0], dtype=torch.float32)",
"grad": "torch.tensor([0.5], dtype=torch.float32)",
"accum": "torch.tensor([0.25], dtype=torch.float32)",
"lr": "0.01",
"eps": "1e-8"
}
},
{
"name": "多批次扩展泛化验证",
"input_desc": "真实参数契约用例 (param, grad, accum, lr, eps)",
"params": [
{
"name": "param",
"type": "torch.Tensor",
"value": "torch.tensor([1.0, 1.0], dtype=torch.float32)"
},
{
"name": "grad",
"type": "torch.Tensor",
"value": "torch.tensor([0.5, 0.75], dtype=torch.float32)"
},
{
"name": "accum",
"type": "torch.Tensor",
"value": "torch.tensor([0.25, 0.375], dtype=torch.float32)"
},
{
"name": "lr",
"type": "float",
"value": "0.01"
},
{
"name": "eps",
"type": "float",
"value": "1e-8"
}
],
"raw_inputs": {
"param": "torch.tensor([1.0, 1.0], dtype=torch.float32)",
"grad": "torch.tensor([0.5, 0.75], dtype=torch.float32)",
"accum": "torch.tensor([0.25, 0.375], dtype=torch.float32)",
"lr": "0.01",
"eps": "1e-8"
}
},
{
"name": "多维批次与参数泛化验证",
"input_desc": "真实参数契约用例 (param, grad, accum, lr, eps)",
"params": [
{
"name": "param",
"type": "torch.Tensor",
"value": "torch.tensor([1.0], dtype=torch.float32)"
},
{
"name": "grad",
"type": "torch.Tensor",
"value": "torch.tensor([0.75], dtype=torch.float32)"
},
{
"name": "accum",
"type": "torch.Tensor",
"value": "torch.tensor([0.375], dtype=torch.float32)"
},
{
"name": "lr",
"type": "float",
"value": "0.008"
},
{
"name": "eps",
"type": "float",
"value": "1e-8"
}
],
"raw_inputs": {
"param": "torch.tensor([1.0], dtype=torch.float32)",
"grad": "torch.tensor([0.75], dtype=torch.float32)",
"accum": "torch.tensor([0.375], dtype=torch.float32)",
"lr": "0.008",
"eps": "1e-8"
}
}
]
} |