Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity

We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires n^{epsilon k} time, for some constant epsilon>1/2, to count (note that these conjectures are significantly weaker than the usual ones made on these problems) on randomized machines for all but finitely many input lengths, then we have the following derandomizations: - BPP can be decided in polynomial time using only n^alpha random bits on average over any efficient input distribution, for any constant alpha>0 - BPP can be decided in polynomial time with no randomness on average over the uniform distribution This answers an open question of Ball et al. (STOC \u2717) in the positive of whether derandomization can be achieved from conjectures from fine-grained complexity theory. More strongly, these derandomizations improve over all previous ones achieved from worst-case uniform assumptions by succeeding on all but finitely many input lengths. Previously, derandomizations from worst-case uniform assumptions were only know to succeed on infinitely many input lengths. It is specifically the structure and moderate hardness of the k-Orthogonal Vectors and k-CLIQUE problems that makes removing this restriction possible. Via this uniform derandomization, we connect the problem-centric and resource-centric views of complexity theory by showing that exact hardness assumptions about specific problems like k-CLIQUE imply quantitative and qualitative relationships between randomized and deterministic time. This can be either viewed as a barrier to proving some of the main conjectures of fine-grained complexity theory lest we achieve a major breakthrough in unconditional derandomization or, optimistically, as route to attain such derandomizations by working on very concrete and weak conjectures about specific problems

Tighter Connections between Derandomization and Circuit Lower Bounds

We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization: - general derandomization of promiseBPP (connected to Boolean circuits), - derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (connected to arithmetic circuit lower bounds over the same field), and - derandomization of PIT over the integers (connected to arithmetic circuit lower bounds over the integers). We show how to make these connections uniform equivalences, although at the expense of using somewhat less common versions of complexity classes and for a less studied notion of inclusion. Our main results are as follows: 1. We give the first proof that a non-trivial (nondeterministic subexponential-time) algorithm for PIT over a fixed finite field yields arithmetic circuit lower bounds. 2. We get a similar result for the case of PIT over the integers, strengthening a result of Jansen and Santhanam [JS12] (by removing the need for advice). 3. We derive a Boolean circuit lower bound for NEXP intersect coNEXP from the assumption of sufficiently strong non-deterministic derandomization of promiseBPP (without advice), as well as from the assumed existence of an NP-computable non-empty property of Boolean functions useful for proving superpolynomial circuit lower bounds (in the sense of natural proofs of [RR97]); this strengthens the related results of [IKW02]. 4. Finally, we turn all of these implications into equivalences for appropriately defined promise classes and for a notion of robust inclusion/separation (inspired by [FS11]) that lies between the classical "almost everywhere" and "infinitely often" notions

Agnostic Learning from Tolerant Natural Proofs

We generalize the "learning algorithms from natural properties" framework of [CIKK16] to get agnostic learning algorithms from natural properties with extra features. We show that if a natural property (in the sense of Razborov and Rudich [RR97]) is useful also against functions that are close to the class of "easy" functions, rather than just against "easy" functions, then it can be used to get an agnostic learning algorithm over the uniform distribution with membership queries. * For AC0[q], any prime q (constant-depth circuits of polynomial size, with AND, OR, NOT, and MODq gates of unbounded fanin), which happens to have a natural property with the requisite extra feature by [Raz87, Smo87, RR97], we obtain the first agnostic learning algorithm for AC0[q], for every prime q. Our algorithm runs in randomized quasi-polynomial time, uses membership queries, and outputs a circuit for a given Boolean function f that agrees with f on all but at most polylog(n)*opt fraction of inputs, where opt is the relative distance between f and the closest function h in the class AC0[q]. * For the ideal case, a natural proof of strongly exponential correlation circuit lower bounds against a circuit class C containing AC0[2] (i.e., circuits of size exp(Omega(n)) cannot compute some n-variate function even with exp(-Omega(n)) advantage over random guessing) would yield a polynomial-time query agnostic learning algorithm for C with the approximation error O(opt)

Hardness Amplification for Non-Commutative Arithmetic Circuits

We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire. This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism

Lifting for Constant-Depth Circuits and Applications to MCSP

Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas. We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC?-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest. We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC? circuit lower bounds

Stability is Stable: Connections between Replicability, Privacy, and Adaptive Generalization

The notion of replicable algorithms was introduced in Impagliazzo et al. [STOC '22] to describe randomized algorithms that are stable under the resampling of their inputs. More precisely, a replicable algorithm gives the same output with high probability when its randomness is fixed and it is run on a new i.i.d. sample drawn from the same distribution. Using replicable algorithms for data analysis can facilitate the verification of published results by ensuring that the results of an analysis will be the same with high probability, even when that analysis is performed on a new data set. In this work, we establish new connections and separations between replicability and standard notions of algorithmic stability. In particular, we give sample-efficient algorithmic reductions between perfect generalization, approximate differential privacy, and replicability for a broad class of statistical problems. Conversely, we show any such equivalence must break down computationally: there exist statistical problems that are easy under differential privacy, but that cannot be solved replicably without breaking public-key cryptography. Furthermore, these results are tight: our reductions are statistically optimal, and we show that any computational separation between DP and replicability must imply the existence of one-way functions. Our statistical reductions give a new algorithmic framework for translating between notions of stability, which we instantiate to answer several open questions in replicability and privacy. This includes giving sample-efficient replicable algorithms for various PAC learning, distribution estimation, and distribution testing problems, algorithmic amplification of $\delta$ in approximate DP, conversions from item-level to user-level privacy, and the existence of private agnostic-to-realizable learning reductions under structured distributions.Comment: STOC 2023, minor typos fixe

Teaching how to program using automated assessment and functional glossy games (Experience Report)

Our department has long been an advocate of the functional-first school of programming and has been teaching Haskell as a first language in introductory programming course units for 20 years. Although the functional style is largely beneficial, it needs to be taught in an enthusiastic and captivating way to fight the unusually high computer science drop-out rates and appeal to a heterogeneous population of students.This paper reports our experience of restructuring, over the last 5 years, an introductory laboratory course unit that trains hands-on functional programming concepts and good software development practices. We have been using game programming to keep students motivated, and following a methodology that hinges on test-driven development and continuous bidirectional feedback. We summarise successes and missteps, and how we have learned from our experience to arrive at a model for comprehensive and interactive functional game programming assignments and a general functionally-powered automated assessment platform, that together provide a more engaging learning experience for students. In our experience, we have been able to teach increasingly more advanced functional programming concepts while improving student engagement.The authors would like to thank the precursors of the 20-year functional programming culture and FPro unit at our university, and all the instructors and TAs that have been involved in the PLab unit throughout the years. This work is financed by the ERDFs European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme within project POCI-01-0145-FEDER-006961, and by National Funds through the Portuguese funding agency, FCT s Fundacao para a Ciencia e a Tecnologia as part of project UID/EEA/50014/2013

'Divided they stand, divided they fail': opposition politics in Morocco

The literature on democratization emphasises how authoritarian constraints usually lead genuine opposition parties and movements to form alliances in order to make demands for reform to the authoritarian regime. There is significant empirical evidence to support this theoretical point. While this trend is partly visible in the Middle East and North Africa, such coalitions are usually short-lived and limited to a single issue, never reaching the stage of formal and organic alliances. This article, using the case of Morocco, seeks to explain this puzzle by focusing on ideological and strategic differences that exist between the Islamist and the secular/liberal sectors of civil society, where significant opposition politics occurs. In addition, this article also aims to explain how pro-democracy strategies of the European Union further widen this divide, functioning as a key obstacle to democratic reforms

The incidence of hip, forearm, humeral, ankle, and vertebral fragility fractures in Italy: results from a 3-year multicenter study

INTRODUCTION: We aimed to assess the incidence and hospitalization rate of hip and "minor" fragility fractures in the Italian population. METHODS: We carried out a 3-year survey at 10 major Italian emergency departments to evaluate the hospitalization rate of hip, forearm, humeral, ankle, and vertebral fragility fractures in people 45 years or older between 2004 and 2006, both men and women. These data were compared with those recorded in the national hospitalizations database (SDO) to assess the overall incidence of fragility fractures occurring at hip and other sites, including also those events not resulting in hospital admissions. RESULTS: We observed 29,017 fractures across 3 years, with hospitalization rates of 93.0% for hip fractures, 36.3% for humeral fractures, 31.3% for ankle fractures, 22.6% for forearm/wrist fractures, and 27.6% for clinical vertebral fractures. According to the analyses performed with the Italian hospitalization database in year 2006, we estimated an annual incidence of 87,000 hip, 48,000 humeral, 36,000 ankle, 85,000 wrist, and 155,000 vertebral fragility fractures in people aged 45 years or older (thus resulting in almost 410,000 new fractures per year). Clinical vertebral fractures were recorded in 47,000 events per year. CONCLUSIONS: The burden of fragility fractures in the Italian population is very high and calls for effective preventive strategies