Because our intuition about local minima is wrong in extremely high dimensional spaces. In two and three dimensions, local minima are common. In a million dimensions, local minima are rare. The intuitive explanation is that for a local minimum to exist, the function must be curving up (first derivative = 0, second derivative &gt;= 0) simultaneously in <i>every</i> dimension. It makes sense that as you add more dimensions this becomes less and less likely, and for a million dimensions it&#x27;s vanishingly unlikely.<p>What you get instead of local minima are saddle points, where some dimensions are curving up and some are curving down. Saddle points can also be problematic for optimization but they can be dealt with using fancy optimization techniques. For a more rigorous explanation, see <a href="http:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1406.2572" rel="nofollow">http:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1406.2572</a>