Approximate Degree, Weight, and Indistinguishability

Abstract

We prove that the OR function on {-1,1\} n can be pointwise approximated with error ε by a polynomial of degree O ( k ) and weight 2 O ( n log (1/ε)/k) , for any k ≥ √ n log (1/ε). This result is tight for any k ≤ (1-Ω (1)) n . Previous results were either not tight or had ε = Ω (1). In general, we obtain a tight approximate degree-weight result for any symmetric function. Building on this, we also obtain an approximate degree-weight result for bounded-width CNF. For these two classes no such result was known. We prove that the \( \mathsf {OR} \) function on \( \lbrace -1,1\rbrace ^n \) can be pointwise approximated with error \( \epsilon \) by a polynomial of degree \( O(k) \) and weight \( 2^{O(n \log (1/\epsilon) /k)} \) , for any \( k \ge \sqrt {n \log (1/\epsilon)} \) . This result is tight for any \( k \le (1-\Omega (1))n \) . Previous results were either not tight or had \( \epsilon = \Omega (1) \) . In general, we obtain a tight approximate degree-weight result for any symmetric function. Building on this, we also obtain an approximate degree-weight result for bounded-width \( \mathsf {CNF} \) . For these two classes no such result was known. One motivation for such results comes from the study of indistinguishability. Two distributions \( P \) , \( Q \) over \( n \) -bit strings are \( (k,\delta) \) -indistinguishable if their projections on any \( k \) bits have statistical distance at most \( \delta \) . The above approximations give values of \( (k,\delta) \) that suffice to fool \( \mathsf {OR} \) , symmetric functions, and bounded-width \( \mathsf {CNF} \) , and the first result is tight for all \( k \) while the second result is tight for \( k \le (1-\Omega (1))n \) . We also show that any two \( (k, \delta) \) -indistinguishable distributions are \( O(n^{k/2}\delta) \) -close to two distributions that are \( (k,0) \) -indistinguishable, improving the previous bound of \( O(n)^k \delta \) . Finally, we present proofs of some known approximate degree lower bounds in the language of indistinguishability, which we find more intuitive.