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[EQUATION] The maximum of [MATH] is [MATH] , achieved at [MATH] . Hence, [MATH] which is less than [MATH] provided [MATH] and [MATH] are both sufficiently large. |
The remaining results in this section pertain to the case in which the principal’s and agent’s utilities are independent random variables. |
Proposition 2 For any [MATH] there exists a product distribution of the agent’s and principal’s utilities such that the marginal distribution of each party’s utility is atomless, and if the principal cannot observe the agent’s utility then the principal’s expected utility is always less than [MATH] |
regardless of the choice of mechanism. Proof. Let [MATH] under the uniform measure. For a sample point [MATH] we will be defining the principal’s utility to be [MATH] where the function [MATH] is defined as follows. For some [MATH] let [MATH] map the interval [MATH] to |
[MATH] and let [MATH] map interval [MATH] to [MATH] . The restriction of [MATH] to each of the two subintervals is linear. Thus, the distribution of [MATH] is a mixture of two uniform distributions, on the intervals [MATH] and |
[MATH] , with the mixture weights being [MATH] and [MATH] , respectively. By Lemma we can restrict our attention to single proposal mechanisms. Let [MATH] denote the eligible set of the mechanism, and let [MATH] denote its characteristic function. |
The agent’s utility [MATH] will be defined as a function of both [MATH] and [MATH] , but with the property that conditional on the values of |
[MATH] and [MATH] , the value of [MATH] is always uniformly distributed. Such a function [MATH] can be constructed as follows. For [MATH] |
let [MATH] denote the intersection of [MATH] with [MATH] ; this set is measurable for almost every [MATH] . If the measure of [MATH] |
[MATH] belongs to [MATH] then [MATH] . If [MATH] then [EQUATION] Let [MATH] denote the probability that a random [MATH] satisfies [MATH] Let [MATH] be the probability that a random sample [MATH] belongs to the eligible set. Conditional on belonging to the eligible set, a random sample point [MATH] has probability at mo... |
[MATH] and [MATH] , conditional on the set of eligible points sampled by the agent being non-empty, the conditional distribution of [MATH] and [MATH] |
for the agent’s proposal (assuming best response to the mechanism) is equal to the conditional distribution of a single random sample [MATH] |
conditioned on belonging to [MATH] . Hence, if [MATH] denotes the value of [MATH] when the agent’s proposal [MATH] is a best response to the mechanism, we have |
[EQUATION] Meanwhile, if [MATH] then [EQUATION] Let [MATH] Ignoring [MATH] terms that vanish as [MATH] and [MATH] terms that vanish as [MATH] we have |
[EQUATION] The right side of ( 27 ) is maximized at [MATH] , where it equals [MATH] Thus, for any mechanism that is obvlivious to the value of [MATH] |
[MATH] , which is less than [MATH] for [MATH] and [MATH] sufficiently large. Proposition 3 Let [MATH] denote the solution to [MATH] For any [MATH] there exists a product distribution of the agent’s and principal’s utilities such that the principal’s expected utility is always less than |
[MATH] , regardless of the choice of mechanism. Proof. The proof is by contradiction. It is known (Hill and Kertz,, 1982 ; Kertz,, 1986 that there is no prophet inequality with factor [MATH] |
for i.i.d. distributions. In other words, there exists a distribution [MATH] such that if [MATH] and i.i.d. with distribution [MATH] , then for every stopping rule [MATH] we have |
[EQUATION] To prove the proposition we will consider the sample space [MATH] with sample points denoted by [MATH] under the product distribution where [MATH] is distributed according to [MATH] and [MATH] is uniformly dstributed in [MATH] . We will show how to transform any mechanism that guarantees utility at least [MA... |
The transformation is simple to describe. If there is a mechanism guaranteeing utility at least [MATH] for the principal, we can assume it is a single proposal mechanism by Lemma |
. Let [MATH] denote the eligible set of the mechanism. The stopping rule [MATH] is specified as follows. First, it draws [MATH] i.i.d. uniform random samples from [MATH] |
and sorts them into decreasing order, numbering them as [MATH] Then the stopping time [MATH] is defined as the minimum [MATH] such that [MATH] |
Observe that the distribution of the sequence of pairs [MATH] obtained when running the stopping rule is identical to the distribution over sequences that one obtains by drawing [MATH] i.i.d. samples from |
[MATH] and arranging them in order of decreasing [MATH] . This is because the random variables [MATH] are i.i.d., so the distribution of the random sequence [MATH] is invariant under permutations. Thus, the process of first pairing |
[MATH] with [MATH] and then sorting by [MATH] generates the same distribution over sequences of pairs as if one first sorts [MATH] and then pairs |
[MATH] with [MATH] Next, observe that when the sequence of pairs [MATH] is drawn as in the preceding paragraph, both the stopping rule and the agent’s best response to the single proposal mechanism select the pair |
[MATH] with maximum [MATH] , or they select [MATH] if there is no pair [MATH] . Hence, the principal’s expected payoff when running the mechanism is equal to |
[MATH] . Our assumption that the mechanism guarantees at least [MATH] to the principal thus implies that [MATH] contradicting ( 28 ). |
# Source: arxiv 1806.06973 # Title: On the Bias of Reed-Muller Codes over Odd Prime Fields # Sections: all # Downloaded: 2026-03-03T01:45:31.113334+00:00 |
On the Bias of Reed-Muller Codes over Odd Prime Fields Abstract We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, such polynomials have exponentially small bias. This also yields an exponential tail bound on the weight distribution of ... |
Our exponential tail bounds on the bias can be used to derive exponential lower bounds on the time for space-bounded learning of bounded-degree polynomials from their evaluations over odd prime fields. |
Introduction Reed-Muller codes are among the oldest error correcting codes, first introduced by Muller and Reed in the 1950s. These codes were initially defined in terms of bounded-degree multivariate polynomials over [MATH] but the same definition can be applied over any finite field. To be more precise, the [MATH] Re... |
[MATH] , denoted [MATH] , takes the message as the coefficients of some [MATH] -variate polynomial of degree at most [MATH] over [MATH] and the encoding is simply the evaluation of that polynomial over all possible inputs chosen from [MATH] |
A function [MATH] is balanced if elements of [MATH] occurs an equal number of times as an output of [MATH] The bias of a function [MATH] with co-domain [MATH] is a measure of the fractional deviation of [MATH] from being balanced. Since each codeword in a Reed-Muller code is the evaluation of a (polynomial) function ov... |
Some elements of a Reed-Muller code are very far from balanced (for example the 0 polynomial yields the all-0 codeword, and the codeword for the polynomial [MATH] has value 1 much more frequently than average) but since, as we might expect, randomly-chosen polynomials behave somewhat like randomly-chosen functions, mos... |
so we will only need to prove the statement for odd prime fields. We now define bias formally and discuss its applications. In the case that [MATH] the bias of [MATH] |
[EQUATION] More generally, for [MATH] a prime, [MATH] and [MATH] , we define the [MATH] -th order bias of [MATH] as [EQUATION] Prior uses of bias over these larger co-domains often focus only on the case of a single [MATH] (e.g., |
) since they consider structural implications of bias. However, the use of different values of [MATH] is essential for the applications of bias to bounding the imbalance of functions and codewords since, for [MATH] , one can have functions with 1st-order bias 0 that are very far from balanced. It turns out that it is n... |
[MATH] for all [MATH] (or, equivalently, all integers [MATH] with [MATH] since [MATH] in order to bound the imbalance: A standard exponential summation argument (e.g., Proposition 2.1 in |
), shows that for every [MATH] [EQUATION] For Reed-Muller codes, the bias of a codeword exactly determines its fraction (number of non-zero entries, which is called the weight of the codeword. (In the case of [MATH] the bias is determined by the weight but that is not true for [MATH] for odd prime [MATH] .) The distrib... |
[MATH] plays a critical role in many applications in coding theory and in many other applications in theoretical computer science. As a consequence, the weight distribution of Reed-Muller codes over [MATH] has been the subject of considerable study. For degrees [MATH] and [MATH] , the exact weight distribution (and hen... |
[MATH] has been known for roughly 50 years For other degrees, precise bounds are only known for weights up to 2.5 times the minimum distance of such codes |
but this is very far from the balanced regime. For general constant degrees, Kaufman, Lovett and Porat give a somewhat tight bound on the weight distribution for Reed-Muller codes over [MATH] , and Abbe, Shpilka, and Wigderson |
generalize the result to linear degrees. These results yield tail bounds for the number of codewords with bias approaching 0 and, using the cases for arbitrarily small constant bias, imply good bounds for list-decoding algorithms |
Ben-Eliezer, Hod, and Lovett proved sharper bounds showing that the fraction of codewords with more than exponentially small bias (of the form [MATH] for constant [MATH] ) is at most [MATH] for constant [MATH] where |
[MATH] is the dimension of the code. (For [MATH] they also showed that this fraction of codewords is tight by exhibiting a set of codewords in |
[MATH] of size [MATH] for [MATH] that has such a bias.) This bound was used by to show that learning bounded degree polynomials over |
[MATH] from their evaluations with success probability [MATH] requires space [MATH] or time [MATH] Our Results We generalize the results of Ben-Eliezer, Hod, and Lovett |
to show that only an exponentially small fraction of polynomials over prime fields can have non-negligible bias. Formally speaking, let [MATH] denote the set of polynomials of degree at most [MATH] |
in [MATH] variables over [MATH] and let [MATH] denote the set of monic monomials of degree at most [MATH] in [MATH] variables. (The Reed-Muller code [MATH] |
has dimension [MATH] and satisfies [MATH] .) Our main result is the following theorem: Theorem 1.1 For any [MATH] there are constants [MATH] depending on |
[MATH] such that for any odd prime [MATH] for all integers [MATH] and all [MATH] , we have [EQUATION] Using this theorem together with the methods of our companion paper |
or of , we obtain that any algorithm that learns polynomials over [MATH] of degree at most [MATH] with probability at least [MATH] from their evaluations on random inputs either requires time [MATH] or space |
[MATH] For the details, see The following corollary of Theorem 1.1 is also immediate: Corollary 1.2 For any [MATH] there are constants [MATH] |
such that for any odd prime [MATH] and integers [MATH] [MATH] with [MATH] , the number of codewords of [MATH] of weight at most [MATH] is at most |
[MATH] There is a limit to the amount that Theorem 1.1 can be improved, as shown by the following proposition: Proposition 1.3 For any [MATH] there are constants [MATH] and [MATH] |
depending on [MATH] such that for all integers [MATH] and all [MATH] , we have [EQUATION] As part of our proof of Theorem 1.1 we must prove the following tight bound on the rank of the evaluations of monomials of degree at most [MATH] on sets of points. Alternatively this can be seen as the extremal dimension of the sp... |
Lemma 1.4 Let [MATH] be a subset of [MATH] such that [MATH] Then the dimension of the subspace spanned by [MATH] is at least [MATH] |
Though this is all that we require to prove Theorem 1.1 , we prove it as a special case of a more general theorem that gives an exact extremal characterization of the dimension of the span of truncated Reed-Muller codes of all sizes. This generalizes a characterization for the case of [MATH] proved by Keevash and Sudak... |
Theorem 1.5 Let [MATH] and let [MATH] For [MATH] with [MATH] [EQUATION] where [MATH] consists of the [MATH] lexicographically minimal vectors in [MATH] (This is equality when [MATH] is also lexicographically minimal.) |
Thus, the extremal value of the dimension is a function [MATH] that is independent of [MATH] . As part of the proof of Theorem 1.5 , we characterize a variety of properties of [MATH] |
Proof Overview Our basic approach is a generalization of the high level outline of to odd prime fields, though parts of the argument are substantially more complex: |
We begin by using a moment method, showing that that [MATH] is bounded for suitable [MATH] Because we are dealing with odd prime fields rather than [MATH] we restrict ourselves to the case that [MATH] is even. For bounding these high moments, we reduce the problem to lower bounding the rank of certain random matrices (... |
For the case of [MATH] handled in a similar property to Lemma 2.4 (Lemma 4 in ), which follows from an extremal characterization of [MATH] |
polynomial evaluations by Keevash and Sudakov , was independently shown to follow more simply via an algorithmic construction that avoids consideration of any subset size that is not a power of 2. Unfortunately, this simpler algorithmic construction seems to break down completely for the case of odd prime fields. |
We instead provide the full extremal characterization for all set sizes, analogous to the Keevash and Sudakov characterization for [MATH] . This is the major source of technical difficulty in our paper. Like Keevash and Sudakov, we show that the proof of our extremal characterization is equivalent to proving the sub-ad... |
terms for arbitrary [MATH] rather than just over the two terms required for the case of [MATH] Discussion and Related Work Prior to our work, the main approach to analyzing the bias of polynomials over arbitrary prime fields has been to take a structural point of view. The general idea is to show that polynomials of la... |
). Green and Tao initiated the modern study of the relationship between the bias and the structure of polynomials over finite fields. Kaufman, Lovett, and Porat |
used this approach to obtain their bounds on bias over [MATH] Over general prime fields, Haramaty and Shpilka gave sharper structural properties for polynomials of degrees [MATH] In papers |
for constant degree and for large degree, Bhowmick and Lovett generalized the result of to show that if a degree [MATH] polynomial [MATH] has large bias, then [MATH] can be expressed as a function of a constant number of polynomials of degree at most [MATH] These bounds are sufficient to analyze the list-decoding prope... |
An open problem that remains from our work, as well as that of Ben-Eliezer, Hod, and Lovett is whether the amount of the bias can be improved still further by removing the [MATH] factor from the exponent in the bias in the statement of Theorem 1.1 for some range of values of [MATH] growing with [MATH] Though Propositio... |
show that a large number of polynomials have bias [MATH] , we would need to extend them to say that for all [MATH] there is a [MATH] such that the conclusion of the proposition holds in order to rule out improving the bias in Theorem 1.1 |
Organization The proof of Theorem 1.1 except for the proof of Lemma 1.4 is in Section Section also contains the proof of Proposition 1.3 In Section we reduce the proof of Lemma 1.4 , and that of the general extremal rank property of Theorem 1.5 , to proving the sub-additivity of the arithmetic function [MATH] In Sectio... |
The bias of random polynomials over odd prime fields In this section we prove Theorem 1.1 To provide tail bounds on the bias, we first characterize its high moments, focusing on even moments to ensure that they are real-valued. |
Lemma 2.1 Let [MATH] be an odd prime and [MATH] For [MATH] let [MATH] and [MATH] be chosen uniformly at random from [MATH] Then [EQUATION] |
Proof. Note that [MATH] therefore [MATH] So we have [EQUATION] For each [MATH] let [MATH] denote the coefficient of [MATH] in [MATH] We identify [MATH] with its vector of coefficients [MATH] and choose [MATH] uniformly by choosing the [MATH] uniformly. Therefore |
[EQUATION] where the second equality follows since [MATH] for all [MATH] Now let us look at the probability [EQUATION] We view [MATH] as arbitrary fixed values and we will upper bound this probability following the analysis of a similar probability in |
That is, we will upper bound the probability that this holds by considering a special subset [MATH] that allows us to derive a linear system whose rank will bound the probability that the constraints indexed by [MATH] all hold. |
We divide [MATH] arbitrarily into two disjoint parts [MATH] and [MATH] with [MATH] [MATH] consists of all monomials of degree are most [MATH] that have degree 1 on [MATH] and degree at most [MATH] on [MATH] |
We use the following properties of the [MATH] , whose proof we defer to later, to show that [MATH] contains a significant fraction of all monomials in [MATH] |
Proposition 2.2 If [MATH] for some [MATH] then (a) there exists a constant [MATH] such that for sufficiently large [MATH] if [MATH] then |
[EQUATION] (b) If [MATH] there exist constants [MATH] such that for sufficiently large [MATH] [EQUATION] Corollary 2.3 Let [MATH] . If [MATH] for some [MATH] then there exists a constant [MATH] such that for sufficiently large [MATH] |
[EQUATION] Proof. The equality follows immediately from the definition of [MATH] Let [MATH] . Then [EQUATION] and setting [MATH] yields the claim. |
Let [MATH] denote the event that [MATH] for all [MATH] To simply notation, since we think of [MATH] as fixed, for each [MATH] define [MATH] by |
[MATH] Since any [MATH] is of the form [MATH] for some [MATH] and [MATH] a monomial of degree at most [MATH] on [MATH] [MATH] requires that |
[EQUATION] where for [MATH] , we write [MATH] for [MATH] restricted to the coordinates in [MATH] We view these constraints as a system of linear equations over the set of variables [MATH] for [MATH] and [MATH] whose coefficients are given by the values of [MATH] for [MATH] |
for all [MATH] Observe that for different values of [MATH] we get separate and independent subsystems of equations with precisely the same coefficients but potentially different constant terms [MATH] since [MATH] depends on both [MATH] and [MATH] Therefore the probability that [MATH] is a solution is the product of the... |
For each [MATH] , there is a [MATH] matrix [MATH] for a system of linear equations on [MATH] for each [MATH] , having one constraint for each polynomial [MATH] of degree at most |
[MATH] on [MATH] Observe that [MATH] In particular, it follows that [EQUATION] We now see that for almost all choices of [MATH] , if [MATH] is at least a constant factor larger than [MATH] then the rank of [MATH] is large. This follows by replacing [MATH] by [MATH] [MATH] by [MATH] [MATH] by [MATH] and |
[MATH] by [MATH] in the following lemma. Lemma 2.4 For any [MATH] there is a constant [MATH] such that there exist constants [MATH] and [MATH] such that for [MATH] and [MATH] if [MATH] is chosen uniformly at random from |
[MATH] , then the matrix [MATH] given by [MATH] then [EQUATION] We first show how to use Lemma 2.4 to prove Theorem 1.1 Proof of Theorem 1.1 |
Let [MATH] , and set [MATH] and [MATH] and [MATH] as in Lemma 2.4 Let [MATH] We first bound the expected value of [MATH] By Lemma 2.1 |
and the definition of event [MATH] we have [EQUATION] Let [MATH] and [MATH] Now by definition, [EQUATION] by ( ). Observe that [MATH] so we can apply Lemma 2.4 with [MATH] [MATH] , and |
[MATH] to derive that [EQUATION] Therefore, [EQUATION] Now, for sufficiently large [MATH] , by Proposition 2.2 (a), [MATH] and by Corollary 2.3 |
[MATH] Therefore, [EQUATION] for some constant [MATH] Now we can apply Markov’s inequality to obtain that for any [MATH] [EQUATION] |
By definition, [MATH] for a fixed [MATH] Therefore, by Proposition 2.2 (b), [MATH] By choosing [MATH] , we obtain that [MATH] and setting [MATH] we derive that |
[EQUATION] as required. It remains to prove Lemma 2.4 and Proposition 2.2 We first prove Lemma 2.4 using Lemma 1.4 The proof of Lemma 1.4 is quite involved and forms the bulk of the paper. Its proof is the subsequent sections. |
Proof of Lemma 2.4 using Lemma 1.4 Let [MATH] for [MATH] and let [MATH] be the minimum of [MATH] from Proposition 2.2 and [MATH] |
from Corollary 2.3 Fix [MATH] We will first check the probability that an arbitrary fixed set of [MATH] columns spans the whole matrix, and then apply a union bound to obtain the final result. |
Let [MATH] denote the linear space spanned by those [MATH] columns. Recall that each column of [MATH] is the evaluation of all monomials of degree at most [MATH] at some point [MATH] (Since [MATH] , distinct elements of [MATH] have distinct evaluations.) |
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