21 research outputs found
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Bayesian Auction Design and Approximation
We study two classes of problems within Algorithmic Economics: revenue guarantees of simple mechanisms, and social welfare guarantees of auctions. We develop new structural and algorithmic tools for addressing these problems, and obtain the following results:
In the -unit model, four canonical mechanisms can be classified as: (i) the discriminating group, including Myerson Auction and Sequential Posted-Pricing, and (ii) the anonymous group, including Anonymous Reserve and Anonymous Pricing. We prove that any two mechanisms from the same group have an asymptotically tight revenue gap of 1 + θ(1 /√), while any two mechanisms from the different groups have an asymptotically tight revenue gap of θ(log ).
In the single-item model, we prove a nearly-tight sample complexity of Anonymous Reserve for every value distribution family investigated in the literature: [0, 1]-bounded, [1, ]-bounded, regular, and monotone hazard rate (MHR).
Remarkably, the setting-specific sample complexity poly(⁻¹) depends on the precision ∈ (0, 1), but not on the number of bidders ≥ 1. Further, in the two bounded-support settings, our algorithm allows correlated value distributions. These are in sharp contrast to the previous (nearly-tight) sample complexity results on Myerson Auction.
In the single-item model, we prove that the tight Price of Anarchy/Stability for First Price Auctions are both PoA = PoS = 1 - 1/² ≈ 0.8647
On Finer Separations Between Subclasses of Read-Once Oblivious ABPs
Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials
as products of univariate polynomials that have matrices as coefficients. In an
attempt to understand the landscape of algebraic complexity classes surrounding
ROABPs, we study classes of ROABPs based on the algebraic structure of these
coefficient matrices. We study connections between polynomials computed by
these structured variants of ROABPs and other well-known classes of polynomials
(such as depth-three powering circuits, tensor-rank and Waring rank of
polynomials).
Our main result concerns commutative ROABPs, where all coefficient matrices
commute with each other, and diagonal ROABPs, where all the coefficient
matrices are just diagonal matrices. In particular, we show a somewhat
surprising connection between these models and the model of depth-three
powering circuits that is related to the Waring rank of polynomials. We show
that if the dimension of partial derivatives captures Waring rank up to
polynomial factors, then the model of diagonal ROABPs efficiently simulates the
seemingly more expressive model of commutative ROABPs. Further, a commutative
ROABP that cannot be efficiently simulated by a diagonal ROABP will give an
explicit polynomial that gives a super-polynomial separation between dimension
of partial derivatives and Waring rank.
Our proof of the above result builds on the results of Marinari, M\"oller and
Mora (1993), and M\"oller and Stetter (1995), that characterise rings of
commuting matrices in terms of polynomials that have small dimension of partial
derivatives. The algebraic structure of the coefficient matrices of these
ROABPs plays a crucial role in our proofs.Comment: Accepted to STACS 202
Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits
Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS\u2708) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (?^[k] ? ? ?) and sum-product-of-constant-degree-polynomials (?^[k] ? ? ?^[?]), for constants k, ?, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC\u2705; Kayal & Saxena, CCC\u2706; Saxena & Seshadhri, FOCS\u2710, STOC\u2711); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP\u2711; Saha,Saxena & Saptharishi, Comput.Compl.\u2713; Forbes, FOCS\u2715; Kumar & Saraf, CCC\u2716); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS\u2709; Shpilka, STOC\u2719; Peleg & Shpilka, CCC\u2720, STOC\u2721). We solve two of the basic underlying open problems in this work.
We give the first polynomial-time PIT for ?^[k] ? ? ?. Further, we give the first quasipolynomial time blackbox PIT for both ?^[k] ? ? ? and ?^[k] ? ? ?^[?]. No subexponential time algorithm was known prior to this work (even if k = ? = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top ?-gate to ?
Hitting Sets for Orbits of Circuit Classes and Polynomial Families
The orbit of an n-variate polynomial f(?) over a field ? is the set {f(A?+?) : A ? GL(n,?) and ? ? ??}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of:
1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials.
2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ? ?}, which is complete for arithmetic formulas.
3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials.
4) Polynomials computable by occur-once formulas
Proof Complexity Lower Bounds from Algebraic Circuit Complexity
We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi (J. ACM, 2018), where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the Boolean setting for subsystems of Extended Frege proofs, where proof-lines are circuits from restricted Boolean circuit classes. Except one, all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). We give two general methods of converting certain algebraic circuit lower bounds into proof complexity ones. However, we need to strengthen existing lower bounds to hold for either the functional model or for multiplicities (see below). Our techniques are reminiscent of existing methods for converting Boolean circuit lower bounds into related proof complexity results, such as feasible interpolation. We obtain the relevant types of lower bounds for a variety of classes (sparse polynomials, depth-3 powering formulas, read-once oblivious algebraic branching programs, and multilinear formulas), and infer the relevant proof complexity results. We complement our lower bounds by giving short refutations of the previously studied subset-sum axiom using IPS subsystems, allowing us to conclude strict separations between some of these subsystems. Our first method is a functional lower bound, a notion due to Grigoriev and Razborov (Appl. Algebra Eng. Commun. Comput., 2000), which says that not only does a polynomial f require large algebraic circuits, but that any polynomial g agreeing with f on the Boolean cube also requires large algebraic circuits. For our classes of interest, we develop functional lower bounds where g(x¯¯¯) equals 1/p(x¯¯¯) where p is a constant-degree polynomial, which in turn yield corresponding IPS lower bounds for proving that p is nonzero over the Boolean cube. In particular, we show superpolynomial lower bounds for refuting variants of the subset-sum axiom in various IPS subsystems. Our second method is to give lower bounds for multiples, that is, to give explicit polynomials whose all (nonzero) multiples require large algebraic circuit complexity. By extending known techniques, we are able to obtain such lower bounds for our classes of interest, which we then use to derive corresponding IPS lower bounds. Such lower bounds for multiples are of independent interest, as they have tight connections with the algebraic hardness versus randomness paradigm
Cyclotomic Identity Testing and Applications
We consider the cyclotomic identity testing problem: given a polynomial
, decide whether is
zero, for a primitive complex -th root of unity and
integers . We assume that and are
represented in binary and consider several versions of the problem, according
to the representation of . For the case that is given by an algebraic
circuit we give a randomized polynomial-time algorithm with two-sided errors,
showing that the problem lies in BPP. In case is given by a circuit of
polynomially bounded syntactic degree, we give a randomized algorithm with
two-sided errors that runs in poly-logarithmic parallel time, showing that the
problem lies in BPNC. In case is given by a depth-2 circuit
(or, equivalently, as a list of monomials), we show that the cyclotomic
identity testing problem lies in NC. Under the generalised Riemann hypothesis,
we are able to extend this approach to obtain a polynomial-time algorithm also
for a very simple subclass of depth-3 circuits. We complement
this last result by showing that for a more general class of depth-3
circuits, a polynomial-time algorithm for the cyclotomic
identity testing problem would yield a sub-exponential-time algorithm for
polynomial identity testing. Finally, we use cyclotomic identity testing to
give a new proof that equality of compressed strings, i.e., strings presented
using context-free grammars, can be decided in coRNC: randomized NC with
one-sided errors
Methodik zur Integration von Vorwissen in die Modellbildung
This book describes how prior knowledge about dynamical systems and functions can be integrated in mathematical modelling. The first part comprises the transformation of the known properties into a mathematical model and the second part explains four approaches for solving the resulting constrained optimization problems. Numerous examples, tables and compilations complete the book
Lower Bounds by Birkhoff Interpolation
International audienceIn this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order Ω(√ d), and were obtained from arguments based on Wronskian determinants and "shifted derivatives." We obtain this improvement thanks to a new lower bound method based on Birkhoff interpolation (also known as "lacunary polynomial interpolation")