""" Day 113 - PyTorch 基础：Autograd 自动求导 ========================================== 本文件演示 PyTorch Autograd 自动微分引擎的工作原理和使用方法。 运行前请确保已安装 PyTorch: pip install torch """ import torch def main(): print("=" * 60) print("Day 113 - Autograd 自动求导基础") print("=" * 60) # ===================================================== # 1. 基础：requires_grad 与 backward # ===================================================== print("\n--- 1. 基础梯度计算 ---\n") # 创建需要梯度的 Tensor x = torch.tensor(3.0, requires_grad=True) print(f"x = {x}") print(f"requires_grad = {x.requires_grad}") # 前向传播 y = x ** 2 # y = x^2 print(f"y = x^2 = {y}") print(f"y.grad_fn = {y.grad_fn}") # 记录了创建 y 的操作 # 反向传播 y.backward() print(f"dy/dx = 2x = {x.grad}") # dy/dx = 2x = 6.0 # ===================================================== # 2. 多变量梯度 # ===================================================== print("\n--- 2. 多变量梯度 ---\n") x = torch.tensor([2.0, 3.0], requires_grad=True) print(f"x = {x}") # y = x1^2 + x2^3 y = x[0] ** 2 + x[1] ** 3 print(f"y = x1^2 + x2^3 = {y.item():.1f}") y.backward() print(f"dy/dx1 = 2*x1 = {x.grad[0].item():.1f} (应为 4.0)") print(f"dy/dx2 = 3*x2^2 = {x.grad[1].item():.1f} (应为 27.0)") # ===================================================== # 3. 计算图与 grad_fn # ===================================================== print("\n--- 3. 计算图与 grad_fn ---\n") x = torch.tensor(2.0, requires_grad=True) print(f"x: {x}, grad_fn: {x.grad_fn}") # None，因为 x 是叶子节点 y = x + 3 print(f"y = x + 3: {y.item()}, grad_fn: {y.grad_fn}") z = y * 2 print(f"z = y * 2: {z.item()}, grad_fn: {z.grad_fn}") w = z ** 2 print(f"w = z^2: {w.item()}, grad_fn: {w.grad_fn}") # 反向传播 w.backward() print(f"\n计算图: x → (+3) → y → (*2) → z → (^2) → w") print(f"dw/dx = 2*(z)*2 = {x.grad.item():.1f} (应为 32.0)") print(f"验证: 2 * (2*2 + 3) * 2 * 2 = {2 * (2*2 + 3) * 2 * 2}") # ===================================================== # 4. 梯度累积 # ===================================================== print("\n--- 4. 梯度累积问题与清零 ---\n") x = torch.tensor(2.0, requires_grad=True) # 第一次反向传播 y1 = x ** 2 y1.backward() print(f"第一次: x.grad = {x.grad.item()}") # 4.0 # 第二次反向传播（梯度会累积！） y2 = x ** 3 y2.backward() print(f"第二次（累积后）: x.grad = {x.grad.item()}") # 4.0 + 12.0 = 16.0 # 清零后重新计算 x.grad.zero_() y3 = x ** 2 y3.backward() print(f"清零后第三次: x.grad = {x.grad.item()}") # 4.0 # ===================================================== # 5. retain_graph # ===================================================== print("\n--- 5. retain_graph 参数 ---\n") x = torch.tensor(2.0, requires_grad=True) # 如果需要多次反向传播，需要 retain_graph=True y = x ** 2 y.backward(retain_graph=True) # 保留计算图 print(f"第一次反向传播: {x.grad.item()}") y.backward() # 再次反向传播 print(f"第二次反向传播（累积）: {x.grad.item()}") # ===================================================== # 6. torch.no_grad() 上下文管理器 # ===================================================== print("\n--- 6. torch.no_grad() ---\n") x = torch.tensor(2.0, requires_grad=True) print(f"x.requires_grad: {x.requires_grad}") # 默认情况下，所有操作都会被跟踪 y = x * 2 print(f"y.requires_grad: {y.requires_grad}") # 使用 no_grad 禁用跟踪 with torch.no_grad(): y = x * 2 print(f"no_grad 内: y.requires_grad: {y.requires_grad}") print(f"no_grad 内: y.grad_fn: {y.grad_fn}") # no_grad 的用途： # 1. 推理时不需要梯度，节省内存 # 2. 更新参数时不应该被记录到计算图中 # 3. 某些操作不需要梯度 print("\n用途：推理、参数更新、性能优化") # ===================================================== # 7. detach() 方法 # ===================================================== print("\n--- 7. detach() 方法 ---\n") x = torch.tensor(2.0, requires_grad=True) y = x ** 2 # detach 创建一个共享数据但不追踪梯度的新 Tensor z = y.detach() print(f"y: {y.item()}, requires_grad: {y.requires_grad}, grad_fn: {y.grad_fn}") print(f"z: {z.item()}, requires_grad: {z.requires_grad}, grad_fn: {z.grad_fn}") # z 和 y 共享底层数据 y.data.fill_(100) print(f"\ny 修改为 100 后 z = {z.item()}") # z 也变了，因为共享数据 # ===================================================== # 8. 梯度计算实例：复杂函数 # ===================================================== print("\n--- 8. 复杂函数梯度计算 ---\n") # 计算 f(x, y) = (x^2 + y^2) * sin(x*y) 的梯度 x = torch.tensor(1.0, requires_grad=True) y = torch.tensor(2.0, requires_grad=True) f = (x ** 2 + y ** 2) * torch.sin(x * y) print(f"f(x, y) = (x² + y²) * sin(x*y)") print(f"f(1, 2) = {f.item():.4f}") f.backward() print(f"\n∂f/∂x = {x.grad.item():.4f}") print(f"∂f/∂y = {y.grad.item():.4f}") # 手动验证（数值梯度） eps = 1e-5 x_val, y_val = 1.0, 2.0 fx = ((x_val + eps) ** 2 + y_val ** 2) * torch.tensor((x_val + eps) * y_val).sin() fy = (x_val ** 2 + y_val ** 2) * torch.tensor(x_val * (y_val + eps)).sin() f0 = (x_val ** 2 + y_val ** 2) * torch.tensor(x_val * y_val).sin() numerical_df_dx = (fx - f0) / eps numerical_df_dy = (fy - f0) / eps print(f"\n数值验证: ∂f/∂x ≈ {numerical_df_dx.item():.4f}") print(f"数值验证: ∂f/∂y ≈ {numerical_df_dy.item():.4f}") # ===================================================== # 9. 梯度裁剪 # ===================================================== print("\n--- 9. 梯度裁剪 ---\n") x = torch.tensor([10.0, 20.0], requires_grad=True) y = (x ** 2).sum() y.backward() print(f"原始梯度: {x.grad}") # 按范数裁剪 torch.nn.utils.clip_grad_norm_(x, max_norm=5.0) print(f"裁剪后梯度 (max_norm=5): {x.grad}") print(f"梯度范数: {x.grad.norm().item():.4f}") # ===================================================== # 10. 训练循环中的标准模式 # ===================================================== print("\n--- 10. 标准训练循环模式 ---\n") # 模拟一个简单的训练循环 w = torch.randn(1, requires_grad=True) for i in range(3): # 前向传播 x = torch.tensor(1.0) y_pred = w * x loss = (y_pred - 1.0) ** 2 # 反向传播前清零梯度 if w.grad is not None: w.grad.zero_() # 反向传播 loss.backward() # 更新参数 with torch.no_grad(): w -= 0.1 * w.grad print(f"Step {i+1}: loss={loss.item():.4f}, w={w.item():.4f}") print(f"\n最终 w ≈ 1.0: {w.item():.4f}") print("\n" + "=" * 60) print("Autograd 自动求导演示完成！") print("=" * 60) if __name__ == "__main__": main()