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Electronic Colloquium on Computational Complexity

Under the auspices of the Computational Complexity Foundation (CCF)

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Reports tagged with Learning theory:
TR96-030 | 31st March 1996
Meera Sitharam

Approximation from linear spaces and applications to complexity

We develop an analytic framework based on
linear approximation and point out how a number of complexity
related questions --
on circuit and communication
complexity lower bounds, as well as
pseudorandomness, learnability, and general combinatorics
of Boolean functions --
fit neatly into this framework. ... more >>>

TR98-075 | 9th December 1998
Adam Klivans, Dieter van Melkebeek

Graph Nonisomorphism has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses.

We establish hardness versus randomness trade-offs for a
broad class of randomized procedures. In particular, we create efficient
nondeterministic simulations of bounded round Arthur-Merlin games using
a language in exponential time that cannot be decided by polynomial
size oracle circuits with access to satisfiability. We show that every
language with ... more >>>

TR01-043 | 26th April 2001
Mikhail V. Vyugin, Vladimir Vyugin

Predictive complexity and information

A new notion of predictive complexity and corresponding amount of
information are considered.
Predictive complexity is a generalization of Kolmogorov complexity
which bounds the ability of any algorithm to predict elements of
a sequence of outcomes. We consider predictive complexity for a wide class
of bounded loss functions which ... more >>>

TR05-126 | 5th November 2005
Eric Allender, Lisa Hellerstein, Paul McCabe, Michael Saks

Minimizing DNF Formulas and AC0 Circuits Given a Truth Table

Comments: 2

For circuit classes R, the fundamental computational problem, Min-R,
asks for the minimum R-size of a boolean function presented as a truth
table. Prominent examples of this problem include Min-DNF, and
Min-Circuit (also called MCSP). We begin by presenting a new reduction
proving that Min-DNF is NP-complete. It is significantly ... more >>>

TR08-016 | 26th February 2008
Alexander Razborov, Alexander A. Sherstov

The Sign-Rank of AC^0

The sign-rank of a matrix A=[A_{ij}] with +/-1 entries
is the least rank of a real matrix B=[B_{ij}] with A_{ij}B_{ij}>0
for all i,j. We obtain the first exponential lower bound on the
sign-rank of a function in AC^0. Namely, let
f(x,y)=\bigwedge_{i=1}^m\bigvee_{j=1}^{m^2} (x_{ij}\wedge y_{ij}).
We show that the matrix [f(x,y)]_{x,y} has ... more >>>

TR13-151 | 7th November 2013
Mark Bun, Justin Thaler

Hardness Amplification and the Approximate Degree of Constant-Depth Circuits

Revisions: 3

We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to within constant error in a certain one-sided sense, then an OR of disjoint copies of that circuit ... more >>>

TR14-135 | 23rd October 2014
Noga Alon, Shay Moran, Amir Yehudayoff

Sign rank, VC dimension and spectral gaps

Revisions: 2

We study the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is $3$. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. Similar (slightly less accurate) statements hold for $d>2$ as well. We discuss the tightness of our methods, and describe ... more >>>

TR15-009 | 16th January 2015
Aloni Cohen, Shafi Goldwasser, Vinod Vaikuntanathan

Aggregate Pseudorandom Functions and Connections to Learning

Revisions: 1

In the first part of this work, we introduce a new type of pseudo-random function for which ``aggregate queries'' over exponential-sized sets can be efficiently answered. An example of an aggregate query may be the product of all function values belonging to an exponential-sized interval, or the sum of all ... more >>>

TR16-075 | 9th May 2016
Mark Bun, Justin Thaler

Improved Bounds on the Sign-Rank of AC$^0$

Revisions: 1

The sign-rank of a matrix $A$ with entries in $\{-1, +1\}$ is the least rank of a real matrix $B$ with $A_{ij} \cdot B_{ij} > 0$ for all $i, j$. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC$^0$, answering an ... more >>>

TR16-121 | 4th August 2016
Mark Bun, Justin Thaler

Approximate Degree and the Complexity of Depth Three Circuits

Revisions: 1

Threshold weight, margin complexity, and Majority-of-Threshold circuit size are basic complexity measures of Boolean functions that arise in learning theory, communication complexity, and circuit complexity. Each of these measures might exhibit a chasm at depth three: namely, all polynomial size Boolean circuits of depth two have polynomial complexity under the ... more >>>

TR19-027 | 1st March 2019
Mark Bun, Nikhil Mande, Justin Thaler

Sign-Rank Can Increase Under Intersection

The communication class $UPP^{cc}$ is a communication analog of the Turing Machine complexity class $PP$. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.

For a communication problem ... more >>>

TR19-041 | 7th March 2019
Srinivasan Arunachalam, Alex Bredariol Grilo, Aarthi Sundaram

Quantum hardness of learning shallow classical circuits

In this paper we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of PAC learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results.

1) Hardness of learning ... more >>>

TR20-185 | 1st December 2020
Srinivasan Arunachalam, Alex Grilo, Tom Gur, Igor Oliveira, Aarthi Sundaram

Quantum learning algorithms imply circuit lower bounds

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathrm{C}$ be a class of polynomial-size concepts, and suppose that $\mathrm{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - \gamma$ by a time $T$ quantum algorithm. ... more >>>

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