1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## Specify the parameters of Lipschitzness and smoothness.

1. Construct an example showing that the 0−1 loss function may suffer from local minima; namely, construct a training sample *S *∈ (*X *×{±1})*m *(say, for *X *= R2), for which there exist a vector w and some *_ >*0 such that

2. For any w_ such that *w*−*w *_ ≤*_ *we have *LS*(w) ≤ *LS *(w_) (where the loss here* *is the 0−1 loss). This means that w is a local minimum of *LS *.

3. There exists some w∗ such that *LS*(w∗) *<>**LS*(w). This means that w is not a global minimum of *LS *.

4. Consider the learning problemof logistic regression: Let*H*=*X *={x∈R*d *: x ≤ *B*}, for some scalar *B **> *0, let *Y *= {±1}, and let the loss function * *be defined a (w*, *(x*, **y*)) = log(1 + exp ( − *y*_w*,*x_)). Show that the resulting learning problem is both convex-Lipschitz-bounded and convex-smooth-bounded. Specify the parameters of Lipschitzness and smoothness.