Third level navigation class

You don’t need to write your custom Walker for that…

Let’s take a look on built-in Walker_Nav_Menu. You’ll find this function:

public function start_lvl( &$output, $depth = 0, $args = array() ) {
    if ( isset( $args->item_spacing ) && 'discard' === $args->item_spacing ) {
        $t="";
        $n = '';
    } else {
        $t = "\t";
        $n = "\n";
    }
    $indent = str_repeat( $t, $depth );

    // Default class.
    $classes = array( 'sub-menu' );

    /**
     * Filters the CSS class(es) applied to a menu list element.
     *
     * @since 4.8.0
     *
     * @param array    $classes The CSS classes that are applied to the menu `<ul>` element.
     * @param stdClass $args    An object of `wp_nav_menu()` arguments.
     * @param int      $depth   Depth of menu item. Used for padding.
     */
    $class_names = join( ' ', apply_filters( 'nav_menu_submenu_css_class', $classes, $args, $depth ) );
    $class_names = $class_names ? ' class="' . esc_attr( $class_names ) . '"' : '';

    $output .= "{$n}{$indent}<ul$class_names>{$n}";
}

As you can see, there is a filter (nav_menu_submenu_css_class) in there that will allow you to modify classes of the list item.

So all you have to do is to add your filter to that hook:

function change_ul_item_classes_in_nav( $classes, $args, $depth ) {
    if ( 0 == $depth ) {
        $classes[] = 'sub';
    }
    if ( 1 == $depth ) {
        $classes[] = 'subsub';
    }
    // ...
    return $classes;
}
add_filter( 'nav_menu_submenu_css_class', 'change_ul_item_classes_in_nav', 10, 3 );