The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Explicitly ORDER BY After GROUP BY (modified query below): By default, the database sorts all 'GROUP BY col1, col2, ...' queries as if you specified 'ORDER BY col1, col2, ...' in the query as well. If a query includes a GROUP BY clause but you want to avoid the overhead of sorting the result, you can suppress sorting by specifying 'ORDER BY NULL'.

Optimal indexes for this query:

ALTER TABLE `UserLoginHistory` ADD INDEX `userloginhistory_idx_logdate_userid` (`LogDate`,`UserId`);
ALTER TABLE `UserLoginHistory` ADD INDEX `userloginhistory_idx_userid_logdate` (`UserId`,`LogDate`);

The optimized query:

SELECT
        ul.LogDate,
        ul.UserId 
    FROM
        UserLoginHistory ul 
    WHERE
        NOT EXISTS (
            SELECT
                * 
            FROM
                UserLoginHistory ulPrevious 
            WHERE
                ulPrevious.LogDate < ul.LogDate 
                AND ul.UserId = ulPrevious.UserId
        ) 
    GROUP BY
        ul.LogDate,
        ul.UserId 
    ORDER BY
        NULL