热带半环上的多项式的VC维？

就像这个问题一样，我对热带和电路的 $\mathbf{BPP}$ 与 $\mathbf{P}$ / $\mathrm{poly}$ 问题感兴趣。这个问题简化为显示热带半环上多项式的VC维的上限（请参见下面的定理2）。 $(\max,+)$ $(\min,+)$

令为半环。甲零图案的序列的的多项式是子集为其中存在和使得对于所有 $R$ $(f_1,\ldots,f_m)$ $m$ $R[x_1,\ldots,x_n]$ $S\subseteq \{1,\ldots,m\}$ $x\in R^n$ $y\in R$ ，当且仅当。也就是说，究竟那些多项式的曲线图与必须击中点。（“零模式”，因为条件可以通过替换。）让 $i=1,\ldots,m$ $f_i(x)= y$ $i\in S$ $f_i$ $i\in S$ $(x,y)\in R^{n+1}$ $f_i(x)=y$ $f_i(x)-y=0$ =个度为多项式序列的最大零模式数。因此，。所述 Vapnik等-Chervonenkis维度的多项式是。 $Z(m)$ $m$ $d$ $0\leq Z(m)\leq 2^m$ $d$ $VC(n,d) := \max\{m\colon Z(m)=2^m\}$

注意：通常，VC维是针对集合的族定义为最大基数一组使得。装配到该帧中，我们可以与每对相关联设定的所有多项式度为其 ${\cal F}$ $|S|$ $S$ $\{F\cap S\colon F\in{\cal F}\}=2^S$ $(x,y)\in R^{n+1}$ $F_{x,y}$ $f$ $\leq d$ 成立。那么所有这样的集合的族的VC维度恰好是。 $f(x)=y$ ${\cal F}$ $F_{x,y}$ $VC(n,d)$

甲琐碎上界是（我们需要至少不同矢量具有所有可能的模式），但它在无限半环是无用的。为了在VC维度上具有良好的上限，我们需要在上具有良好的上限。在field上，这样的界限是已知的。 $m=VC(n,d)$ $m\leq n\log |R|$ $2^m$ $x\in R^n$ $2^m$ $Z(m)$

定理1： 在任何场，我们有 $R$ 。 $Z(m)\leq \binom{md+n}{n}$

Milnor，Heintz和 Warren 早些时候证明了类似的上限 ; 他们的证明使用了真实代数几何中的沉重技巧。相反，Ronyai，Babai和Ganapathy（我们在下面给出）的定理1的半页证明是线性代数的简单应用。

通过寻找小的满足 $m$ ，我们得到在任何领域都成立。考虑到与/，此处重要的是维数仅为度的对数。这很重要，因为多项式大小的电路可以计算指数级的多项式，并且因为Haussler在PAC学习中的结果（第114页的推论2） $\binom{md+n}{n} < 2^m$ $VC(n,d)=O(n\log d)$ $\mathbf{BPP}$ $\mathbf{P}$ $\mathrm{poly}$ $d$ 本文）得出以下结果（我们假设确定性电路被允许使用多数表决来输出其值）。

定理2： 适用于在任意半环电路，其中仅在多项式和。 $\mathbf{BPP}\subseteq \mathbf{P}/\mathrm{poly}$ $R$ $VC(n,d)$ $n$ $\log d$

有关Haussler的结果如何暗含定理2，请参见此处。

In particular, by Theorem 1, $\mathbf{BPP}\subseteq \mathbf{P}/\mathrm{poly}$ holds over any field. (Interesting is here only the case of infinite fields: for finite ones, much simpler arguments work: Chernoff bound then does the work.) But what about (infinite) semirings that are not fields, or even not rings? Motivated by dynamic programming, I am mainly interested in tropical $(\max,+)$ and $(\min,+)$ semirings, but other "non-field" (infinite) semirings are interesting as well. Note that, over the $(\max,+)$ semiring, a polynomial $f(x)=\sum_{a\in A} c_a\prod_{i=1}^n x_i^{a_i}$ with $A\subseteq\mathbb{N}$ and $c_a\in \mathbb{R}$ , turns into the maximization problem ; 度是（作为习惯）的最大的在所有。 $f(x)=\max_{a\in A}\ \{c_a+a_1x_1+a_2x_2+\cdots+a_nx_n\}$ $f$ $a_1+\cdots+a_n$ $a\in A$

问：是程度的VC维以上热带半环在多项式多项式？ $\leq d$ $n\log d$

我承认，要得到一个快速答案可能是一个相当困难的问题：热带代数相当“疯狂”。但是也许有人对为什么热带多项式（如果有的话）比实际多项式能产生更多的零模式有一些想法？还是为什么他们“不应该”？或一些相关参考。

Or, perhaps, the proof of Babai, Ronyai, and Ganapathy (below) can be somehow "twisted" to work over tropical semirings? Or over any other infinite semirings (which are not fields)?

Proof of Theorem 1: Assume that a sequence $(f_1,\ldots,f_m)$ has $p$ different zero-patterns, and let $v_1,\ldots,v_p\in R^n$ be witnesses to these zero-patterns. Let $S_i=\{k\colon f_k(v_i)\neq 0\}$ be a zero-pattern witnessed by the $i$ -th vector $v_i$ , and consider the polynomials $g_i:=\prod_{k\in S_i}f_k$ . We claim that these polynomials are linearly independent over our field. This claim completes the proof of the theorem since each $g_i$ has degree at most $D:=md$ , and the dimension of the space of polynomials of degree at most $D$ is $\binom{n+D}{D}$ . To prove the claim, it is enough to note that $g_i(v_j)\neq 0$ if and only if $S_i\subseteq S_j$ . Suppose contrariwise that a nontrivial linear relation $\lambda_1 g_i(x)+\cdots+\lambda_p g_p(x)=0$ exists. Let $j$ be a subscript such that $|S_j|$ is minimal among the $S_i$ with $\lambda_i\neq 0$ . Substitute $v_j$ in the relation. While $\lambda_jg_j(v_j)\neq 0$ , we have $\lambda_ig_i(v_j)=0$ for all $i\neq j$ , a contradiction. $\Box$

— Stasys
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I've realized that the answer to my question is - yes: the VC dimension of degree $\leq d$ polynomials on $n$ variables over any tropical semiring is at most a constant times $n^2\log(n+d)$ . This can be shown using Theorem 1 above. See here for details. So, BPP $\subseteq$ P/poly holds also for tropical circuits and, hence, also for "pure" dynamic programming algorithms.

N.B. (added 25.06.2019) In the mean time, I've resolved the problem completely in this paper. In such a generality, which I haven't even dreamed at the beginning. Tropical case is here just a very, very special case. And even more curiously: by just an appropriate combination of already know (deep in any respect) results of other authors.

What remains else to do in this (BPP vs. P/poly) direction? Besides the decrease of the size of resulting deterministic circuits (an interesting question in itself).

— Stasys
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