[Paper Review] A Note on Unconditional Subexponential-time Pseudo-deterministic Algorithms for BPP Search Problems

Dhiraj Holden|arXiv (Cornell University)|Jul 18, 2017

Complexity and Algorithms in Graphs7 references3 citations

TL;DR

This paper presents the first unconditional pseudo-deterministic algorithm for all search-BPP problems that runs in subexponential time for infinitely many input lengths. By leveraging pseudorandom generators under PSPACE hardness assumptions and canonical sampling techniques, it ensures that for any polynomial-time samplable distribution, the algorithm outputs a unique correct solution with high probability over both inputs and randomness.

ABSTRACT

We show the first unconditional pseudo-determinism result for all of search-BPP. Specifically, we show that every BPP search problem can be computed pseudo-deterministically on average for infinitely many input lengths. In other words, for infinitely many input lengths and for any polynomial-time samplable distribution our algorithm succeeds in producing a unique answer (if one exists) with high probability over the distribution and the coins tossed.

Motivation & Objective

• To establish the existence of pseudo-deterministic algorithms for all search-BPP problems in subexponential time.
• To achieve average-case uniqueness of output across all polynomial-time samplable distributions.
• To resolve the open question of whether pseudo-determinism can be achieved unconditionally for general BPP search problems.
• To extend prior results on pseudo-deterministic algorithms for specific problems to the full class search-BPP.
• To explore the connection between pseudorandomness, circuit lower bounds, and average-case computation.

Proposed method

• The proof proceeds by case analysis on whether PSPACE is contained in subexponential-time BPP.
• In the case where PSPACE ⊆ BPTIME(e_{1/2+ε}(n)), subexponential circuit lower bounds for BPE are derived, enabling pseudo-deterministic algorithms via padding and known results.
• In the case where PSPACE ⊈ BPTIME(e_{1/2+ε}(n)), a pseudorandom generator (PRG) with stretch e_{1/2+ε}(n) is constructed that is indistinguishable from uniform by any uniform algorithm in time e_{1/2+ε}(n).
• The algorithm uses the PRG to sample randomness and iterates through its outputs, running the original BPP search algorithm and verifying solutions with a BPP verifier.
• The algorithm outputs the first valid solution found, and uniqueness is guaranteed by the indistinguishability of the PRG from uniform, ensuring high probability of consistent output.
• The construction relies on fractional exponential functions e_α(n) to express subexponential time bounds and uses canonical sampling techniques from prior work.

Experimental results

Research questions

• RQ1 Can every search-BPP problem be solved with a pseudo-deterministic algorithm running in subexponential time for infinitely many input lengths?
• RQ2 Is it possible to achieve average-case uniqueness of output across all polynomial-time samplable distributions without relying on computational assumptions?
• RQ3 Does the existence of pseudo-deterministic algorithms for general search-BPP imply new circuit lower bounds?
• RQ4 Can the result be extended to work for all input lengths, not just infinitely many?
• RQ5 Can a worst-case pseudo-deterministic algorithm be constructed for search-BPP using current pseudorandomness techniques?

Key findings

• Every relation in search-BPP is contained in HeurPsdTIME(e_{1/2+ε}(n), 1/poly(n)) for infinitely many input lengths and every polynomial-time samplable distribution.
• The algorithm achieves pseudo-determinism by using a pseudorandom generator that is indistinguishable from uniform under the assumption that PSPACE is not contained in subexponential-time BPP.
• When PSPACE ⊆ BPTIME(e_{1/2+ε}(n)), subexponential-size circuit lower bounds for BPE are derived, enabling pseudo-deterministic computation via known techniques.
• The algorithm outputs a unique correct solution with probability at least 1 - 1/poly(n) over the input distribution and randomness, ensuring high consistency.
• The proof relies on the existence of a pseudorandom generator with stretch e_{1/2+ε}(n) that cannot be distinguished from uniform in time e_{1/2+ε}(n) under the same assumption.
• The result is unconditional, as it does not require unproven complexity assumptions beyond the standard hierarchy collapse or hardness assumptions.