Consider designing a linear classifier \[ y = sign(f(x; w, b)), \quad f(x; w, b) = w^T x + b \] on a dataset \( D = \{(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)\}, x_i \in \mathbb{R}^d, y_i \in \{+1, -1\}, i = 1, 2, \dots, N \). Recall that the sign function outputs \( +1 \) if the argument is positive, and \( -1 \) if the argument is non-positive. The parameters \( w \) and \( b \) are updated as per the following training algorithm: \[ w_{new} = w_{old} + y_n x_n, \quad b_{new} = b_{old} + y_n \] whenever sign\( (f(x_n; w_{old}, b_{old})) \neq y_n \). In other words, whenever the classifier wrongly predicts a sample \( (x_n, y_n) \) from the dataset, \( w_{old} \) gets updated to \( w_{new} \), and likewise \( b_{old} \) gets updated to \( b_{new} \). Consider the case \( (x_n, +1), f(x_n; w_{old}, b_{old}) < 0 \). Then: