记录Ceres优化中遇到的一个奇怪问题

笔者在学习视觉SLAM十四讲的第6讲非线性优化时，运行使用Ceres拟合曲线的例程，发现当生成数据为如下代码时

  vector<double> x_data, y_data;      // 数据
  for (int i = 0; i < N; i++) {
    double x = i / 100.0;
    x_data.push_back(x);
    double real = exp(ar * x * x + br * x + cr);
    double gauss_noise = rng.gaussian(w_sigma * w_sigma);
    y_data.push_back(real + gauss_noise);
    // rng.gaussian(w_sigma * w_sigma);
  }

输出为

solve time cost = 0.000777462 seconds.
Ceres Solver Report: Iterations: 8, Initial cost: 1.597873e+06, Final cost: 5.096851e+01, Termination: CONVERGENCE
estimated a,b,c = 0.890908 2.1719 0.943628

当把注释去掉，即为如下代码时

vector<double> x_data, y_data; // 数据
  for (int i = 0; i < N; i++)
  {
    double x = i / 100.0;
    x_data.push_back(x);
    double real = exp(ar * x * x + br * x + cr);
    double gauss_noise = rng.gaussian(w_sigma * w_sigma);
    y_data.push_back(real + gauss_noise);
    rng.gaussian(w_sigma * w_sigma); //按理来说此行不应该影响结果。 此处替换为其他无关语句时，结果不受影响
  }

输出结果为

solve time cost = 0.000914116 seconds.
Ceres Solver Report: Iterations: 7, Initial cost: 1.599017e+06, Final cost: 5.808070e+01, Termination: CONVERGENCE
estimated a,b,c = 1.05448 1.92195 1.02832

即优化得到了一个更好的结果，重复运行了多次都是这样。

画出曲线，发现这两套参数拟合的结果与真实值都很接近。

将数据点增加到一万个时，有注释结果如下

solve time cost = 0.0271037 seconds.
Ceres Solver Report: Iterations: 8, Initial cost: 1.624262e+08, Final cost: 5.071043e+03, Termination: CONVERGENCE
estimated a,b,c = 1.02485 1.96084 1.01517

无注释即多运行一次rng.gaussian结果如下

solve time cost = 0.10454 seconds.
Ceres Solver Report: Iterations: 8, Initial cost: 1.623919e+08, Final cost: 4.873203e+03, Termination: CONVERGENCE
estimated a,b,c = 1.00244 1.99048 1.00779

可见数据量增大后，能够较好的拟合曲线。