The computational complexity of finding stationary points in non-convex optimization

Abstract Finding approximate stationary points, i.e., points where the gradient is approximately zero, of non-convex but smooth objective functions f over unrestricted d -dimensional domains is one of the most fundamental problems in classical non-convex optimization. Nevertheless, the computational and query complexity of this problem are still not well understood when the dimension d of the problem is independent of the approximation error. In this paper, we show the following computational and query complexity results: The problem of finding approximate stationary points over unrestricted domains is $${\textsf{PLS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>PLS</mml:mi> </mml:math> -complete. For $$d = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we provide a zero-order algorithm for finding $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -approximate stationary points that requires at most $$O(1/\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> value queries to the objective function. We show that any algorithm needs at least $$\Omega (1/\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> queries to the objective function and/or its gradient to find $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -approximate stationary points when $$d=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Combined with the above, this characterizes the query complexity of this problem to be $$\Theta (1/\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . For $$d = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we provide a zero-order algorithm for finding $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -KKT points in constrained optimization problems that requires at most $$O(1/\sqrt{\varepsilon })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>ε</mml:mi> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> value queries to the objective function. This closes the gap between works of Bubeck and Mikulincer and Vavasis and characterizes the query complexity of this problem to be $$\Theta (1/\sqrt{\varepsilon })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>ε</mml:mi> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Combining our results with a recent result of Fearnley et al., we show that finding approximate KKT points in constrained optimization is reducible to finding approximate stationary points in unconstrained optimization but the converse is impossible.