text stringlengths 448 13k | label int64 0 1 | doc_idx stringlengths 8 14 |
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abstract: 'We study strong-coupling lattice QCD with staggered-Wilson fermions, with emphasis on discrete symmetries and possibility of their spontaneous breaking. We perform hopping parameter expansion and effective potential analyses in the strong-coupling limit. From gap equations we find nonzero pion condensate... | 1 | member_4 |
been recently shown that the flavored-mass terms can also be constructed for staggered fermions [@KS; @Suss; @Sha] in Ref. [@Adams1; @Adams2; @Hoel]. The original purpose of introducing these terms was establishment of the index theorem with staggered fermions [@Adams1]. A bonus here is that staggered fermions with the... | 1 | member_4 |
application of the overlap [@GW; @Neu] and domain-wall [@Kap; @FuSh] versions, both built on the Wilson-type kernel. Thus, in order to judge applicability of these new lattice fermions, it is essential to investigate the Aoki phase in the staggered-Wilson fermions. The phase structure for the staggered-Wilson fermion w... | 1 | member_4 |
reproduced. Lastly, we discuss parity-flavor symmetry breaking for 2-flavor cases. These results suggest that we can take a chiral limit by tuning a mass parameter in lattice QCD with staggered-Wilson fermions as with the Wilson fermion.
This paper is organized as follows. In Sec. \[sec:SWF\], we review staggered flav... | 1 | member_4 |
to a second derivative term as $\sim a\int dx^{4} \bar{\psi}D^{2}_{\mu}\psi$ up to $\mathcal{O}(a^2)$ errors. Thus we can regard them as cousins of Wilson fermion.
There are also non-trivial flavored-mass terms for staggered fermions, which split 4 tastes into branches and satisfy $\gamma_{5}$ hermiticity. Since $\gam... | 1 | member_4 |
positive mass and the other two with negative mass. These two branches correspond to $+1$ and $-1$ eigenvalues of $\gamma_{5}$ in the taste space. The latter splits them into one with positive mass, two with zero mass and the other one with negative mass. We note that $M_{A}$ and $M_{H}$ are also derived from the flavo... | 1 | member_4 |
we discuss discrete symmetry of staggered-Wilson fermions. A potential problem with staggered-Wilson fermions in lattice QCD is discrete symmetry breaking. As discussed in [@Adams2; @Hoel], the discrete symmetry possessed by the original staggered fermion is broken to its subgroup both in the Adams-type and Hoelbling-t... | 1 | member_4 |
\bar{\chi_{x}} \to S_{R}(R^{-1}x)\bar{\chi}_{R^{-1}x},\,\,\,\,
U_{\nu, x} \to U_{\nu, Rx}
\ ,
\label{rot1}\end{aligned}$$ where $R_{\rho\sigma}$ is the rotation $x_{\rho}\to x_{\sigma}$, $x_{\sigma}\to -x_{\rho}$, $x_{\tau}\to x_{\tau}$, $\tau \not= \rho, \sigma$ and $S_{R}(x)={1\over{2}}[1\pm\eta_{\rho}(x)\eta_{\sigma... | 1 | member_4 |
follows. We first consider parity. Both staggered-Wilson fermions are invariant under $$\mathcal{I}_{s}\mathcal{S}_{4}\sim
\exp(ip_{4})\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{5}\,\phi(-{\bf p},p_{4})\sim
\exp(ip_{4})\Gamma_{4}\,\phi(-{\bf p},p_{4})
\ ,
\label{parity}$$ with $\mathcal{I}_{s}\equiv \mathcal{I}_{1}\mathca... | 1 | member_4 |
fermions have proper charge conjugation symmetry.
$N_f$ $\mathcal{S}$$\&$$\mathcal{I}$-subgroup $\mathcal{R}$-subgroup $P$ $C$ $SW_{4}$
----------- ------- ---------------------... | 1 | member_4 |
it could not lead to a correct continuum theory, and we would need to tune parameters to restore Lorentz symmetry. Indeed the recent study on symmetries of staggered-Wilson fermions by Sharpe [@Steve] reports that recovery of Lorentz symmetry requires fine-tuning of coefficients in the gluonic sector in lattice QCD wit... | 1 | member_4 |
intensively discussed in [@Rev; @PdF].
Hopping Parameter Expansion {#sec:HPE}
===========================
In this section we investigate parity-phase structure in lattice QCD with staggered-Wilson fermions in the framework of hopping parameter expansion (HPE) in the strong-coupling regime [@AokiP]. In the hopping par... | 1 | member_4 |
the Hoelbling fermion. Black circles stand for the leading one-point function $\langle \chi_x \bar{\chi}_x \rangle_0$ while white circles stand for $\langle \chi_x \bar{\chi}_x \rangle$ which include next-leading and higher hopping terms. By summing up higher contributions, we obtain the second equality.[]{data-label="... | 1 | member_4 |
it successfully includes certain kinds of diagrams to all orders of $K$ thanks to a self-consistent approach. We thus expect that it works to figure out existence of Aoki phase. We note that this approximation especially works well for a small hopping parameter $K\ll1$. In Fig. \[FR-H\], we depict Feynman rules in the ... | 1 | member_4 |
\chi_x^a \bar{\chi}_x^b \rangle
& \equiv - \delta^{ab} \Sigma_x
{\nonumber}\\
&= \langle \chi_x^a \bar{\chi}_x^b \rangle_0
{\nonumber}\\
&
+ \sum_{\pm \mu}
(-1) (K \eta_{\mu,x})^2
\langle (\chi^a \bar{\chi})_x \rangle_0
U_{\mu,x}
\langle (\chi \bar{\chi})_{x+\hat{\mu}} \rangle_0
U_{\mu,x}^\... | 1 | member_4 |
{1}{2^3 \sqrt{3}} \right)^2
(-1) \left( K \eta_{\rho,x} \right)^2
\langle (\chi^a \bar{\chi})_x \rangle_0
\mathcal{W}_{\mu\nu,x}^{(2)}
{\nonumber}\\
& \times
\langle (\chi \bar{\chi})_{x+\hat{\mu}+\hat{\nu}} \rangle_0
U_{\rho,x+\hat{\mu}+\hat{\nu}}
\langle (\chi \bar{\chi})_{x+\hat{\mu}+\hat{\... | 1 | member_4 |
\displaystyle \frac {1}{24} (Kr)^2 \cdot 4 \cdot 3
\left( \sigma +i \epsilon_x \pi \right)^2
\ ,
\label{HPE-HoelSelf1}$$ which yields $- \sigma = -1 + 16 K^2 \pi^2$ and $- i \pi = - 8K^2 \cdot 2 i \sigma \pi$. For simplicity, we have set $r=2\sqrt{2}$ to make the equation Eq. (\[HPE-HoelSelf1\]) simpler. Of course we ... | 1 | member_4 |
derive the following $\mathcal{O}(K^{3})$ equation for a two-point function.
![Feynman diagram for mesonic two-point functions for $\mathcal{O}(K^{3})$ self-consistent equation with the Hoelbling fermion. []{data-label="Two-H"}](HPEwHoelbling-Mass.eps){height="4cm"}
$$\begin{aligned}
\mathcal{S}(0,x)
= &\langle {{\... | 1 | member_4 |
+ e^{ip_\mu} \right)
{\nonumber}\\
&+ \left( 2 K r \displaystyle \frac{1}{2^3 \sqrt{3}} \right)^2
\sum_{\mu \neq \nu} \left( e^{-i(p_\mu+p_\nu)} + e^{i(p_\mu+p_\nu)}
+ e^{-i(p_\mu-p_\nu)} + e^{i(p_\mu-p_\nu)} \right)\biggr] \mathcal{S}(p)
\ .
\label{HPE-HoelSelf2}\end{aligned}$$ We finally obtain the meson pro... | 1 | member_4 |
parity-symmetric and parity-broken phases at $|K|=1/4$, which is consistent with the result from the one-point function in Eq. (\[cond\]). We note that the massless pion at the phase boundary is consistent with the scenario of second-order transition. We can also derive the sigma meson mass by substituting $p=(i m_\pi ... | 1 | member_4 |
First, we derive meson condensates from the one-point function $\mathcal{M}_{x}=\bar{\chi}_{x}\chi_{x}$. The equation for the one-point function is obtained as shown in Fig. \[One-A\], $$\begin{aligned}
- \Sigma_x
& \equiv
- \langle \mathcal{M}_x \rangle
{\nonumber}\\
& =
- \langle \mathcal{M}_x \rangle_0 + 2K^2 \s... | 1 | member_4 |
a non-trivial solution as $$\sigma = \displaystyle \frac{1}{16K^2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\pi = \pm \sqrt{ \displaystyle \frac{1}{16K^2} \left( 1- \displaystyle
\frac{1}{16K^2} \right) }
\ .
\label{condA}$$ It indicates that parity-broken phase appears in the range of the hopping parameter as $\mid{K}\mid > 1/4$... | 1 | member_4 |
&- \delta_{0x} N_c + K^2
\sum_{\pm \mu} \langle \chi_{\hat{\mu}}^a {{\bar{\chi}}}_{\hat{\mu}}^a
{{\bar{\chi}}}_x^b \chi_x^b \rangle
{\nonumber}\\
& - \left( 2 K r \displaystyle \frac{1}{4! \cdot 2^4} \right)^2
\sum_{\substack{\pm \mu, \pm \nu, \pm \rho,
\pm \sigma \\ (\mu \neq \nu \neq \rho \neq \sigma)}}
... | 1 | member_4 |
Here the pion mass becomes zero at $\mid{K}\mid = 1/4$ and becomes tachyonic in the range $\mid{K}\mid > 1/4$. It suggests that there occurs a second-order phase transition between parity-symmetric and broken phases at $|K|=1/4$, which is consistent with Eq. (\[condA\]). We can also derive the sigma meson mass by subst... | 1 | member_4 |
section we again begin with the Hoelbling case as exercise, and go on to Adams fermion with better discrete symmetry.
Hoelbling type {#hoelbling-type}
--------------
In the strong-coupling limit we can drop plaquette action. Then the partition function for meson fields $\mathcal{M}_x=({{\bar{\chi}}}_x \chi_x)/N$ with... | 1 | member_4 |
---
abstract: |
A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turán densit... | 1 | member_5 |
hypergraph $H$ is a pair $(V,E)$; $V$ is the vertex set, and $E\subseteq 2^V$ is the edge set. If all edges have the same cardinality $k$, then $H$ is a $k$-uniform hypergraph. Turán problems on $k$-uniform hypergraphs have been actively studied for many decades. However, Turán problems on non-uniform hypergraphs are r... | 1 | member_5 |
a subgraph; such a graph is called $H$-*free*. Katona et al. [@KNS] showed that $f(n,H)={{\rm ex}}(n,H)/{n\choose k}$ is a decreasing function of $n$. The limit $\displaystyle \pi(H)=\lim_{n\to \infty} f(n,H)$, which always exists, is called the [*Turán density*]{} of $H$.
For $k=2$, the graph case, Erdős-Stone-Simono... | 1 | member_5 |
In particular, Baber [@baber] recently found the Turán density of many $3$-uniform hypergraphs using flag algebra methods. For a more complete survey of methods and results on uniform hypergraphs see Peter Keevash’s survey paper [@KeevashSurvey].
A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an e... | 1 | member_5 |
we prove that blowing-up will not affect the Turán density. Using various techniques, we determine the Turán density of every $\{1,2\}$-hypergraph in section 4. Remarkably, the Turán densities of $\{1,2\}$-hypergraphs are in the set $$\bigg\{1,\frac{9}{8}, \frac{5}{4}, \frac{3}{2}, \frac{5}{3},
\ldots, 2-\frac{1}{k},\l... | 1 | member_5 |
one additional vertex, $\ast$, added to every edge of $H$. In a hypergraph Turán problem workshop hosted by the AIM Research Conference Center in 2011, the following conjecture was posed: $\displaystyle \lim_{t\to\infty}\pi(S^t(K^{r}_n))=0$. We conjecture $\displaystyle\lim_{t\to\infty} \pi(S^t(H))=|R(H)|-1$ holds for ... | 1 | member_5 |
$n$ vertices with $R(G)\subseteq R$. We simplify it to $G$ if $n$ and $R$ are clear under context.
Let $R$ be a fixed set of edge types. Let $H$ be an $R$-graph. The number of vertices in $H$ is denoted by $v(H):=|V(H)|$. Our goal is to measure the edge density of $H$ and be able to compare it (in a meaningful way) to... | 1 | member_5 |
non-uniform hypergraph $G$ on $n$ vertices, we define the Lubell function of $G$ as $$\label{eq:lubell}
h_{n}(G):=\sum_{F\in E(G)}\frac{1}{\binom{n}{|F|}}=\sum_{k\in R(G)}\frac{|E(H^{k})|}{\binom{n}{k}}.$$
The Lubell function is the expected number of edges hit by a random full chain. Namely, pick a uniformly rand... | 1 | member_5 |
set $\bigcup_{i\in R} \binom{[n]}{i}$. For example, $K^{\{k\}}_n$ is the complete $k$-uniform hypergraph. $K^{[k]}_n$ is the non-uniform hypergraph with all possible edges of cardinality at most $k$.
(0,0)–(2,1)–(2,-1)–cycle; at (0,0) \[vertex\_open\] (v1) \[label=above:[1]{}\] ; at (2,1) \[vertex\_open\] (v2) \[label... | 1 | member_5 |
always exists. When ${\mathcal{H}}$ contains one hypergraph $H$, then we write $\pi(H)$ instead of $\pi(\{H\})$.
Throughout, we will consider $n$ growing to infinity, and $R$ to be a fixed set (not growing with $n$). Note that $\pi(\mathcal{H})$ agrees with the *usual* definition of $$\displaystyle \pi(\mathcal{H})=\l... | 1 | member_5 |
}{\binom{n}{k_i}} \\
&= h_{n}(G_{n})\\
&=\pi_n({\mathcal{H}}).\end{aligned}$$ The sequence $\pi_n({\mathcal{H}})$ is non-negative and decreasing; therefore it converges.
For a fixed set $R:=\{k_1,k_2,\ldots, k_r\}$ (with $k_1<k_2<\cdots < k_r$), an [*$R$-flag*]{} is an $R$-graph containing exactly one edge of each... | 1 | member_5 |
a random $n$-permutation $\sigma$ uniformly. Let $X$ be the number of edges of $G^R_n$ hit by a random flag $\sigma(L)$. Note that each edge $F$ has probability $\frac{1}{{n\choose |F|}}$ of being hit by $\sigma(L)$. We have $$\label{eq:ef}
{{\rm E}}(X)=\sum_{F\in E(G)}\frac{1}{{n\choose |F|}}=h_n(G).$$ Since $G^R_... | 1 | member_5 |
$A$ be the set of all singleton edges. For any $x,y\in A$, $xy$ is not a 2-edge of $G$. We have $$\begin{aligned}
h_n(G)&\leq \frac{|A|}{n} + 1-\frac{{|A|\choose 2}}{{n\choose 2}} \\
&= 1+ \frac{|A|}{n} -\frac{|A|^2}{n^2} + O\left(\frac{1}{n}\right) \\
&\leq 1+ \frac{1}{4} + O\left(\frac{1}{n}\right).\end{a... | 1 | member_5 |
easy to check $h_n(G)=\frac{5}{4}+O\left(\frac{1}{n}\right)$ and that $G$ is $H$-free.
Now we prove the upper bound. Consider any $H$-free hypergraph $G$ of edge-type $\{2,3\}$ on $n$ vertices. Recall that $d_{2}(v)$ denotes the number of 2-edges that contain $v$. For each pair of 2-edges that intersect $v$ there is a... | 1 | member_5 |
can generalize this construction, giving the following lower bound for $S^k(K_2^{\{1,2\}})$ (the $k$-th suspension of $K_2^{\{1,2\}}$). The details of the computation are omitted. $$\label{eq:sk122}
\pi(S^k(K_2^{\{1,2\}}))\geq 1+ \frac{1}{2^{k+1}}.$$
\[conj:1\] For any $k\geq 2$, $\pi(S^k(K_2^{\{1,2\}}))= 1+ \frac... | 1 | member_5 |
a copy of $H$, so the number of copies of $H$ in $G$ is at least $\frac{a}{2r}{n\choose m}/{{n-v(H)\choose m-v(H)}}=b {n\choose v(H)}$ where $b:=\frac{a}{2r}{m\choose v(H)}^{-1}$. $\square$
Supersaturation can be used to show that “blowing up” does not change the Turán density $\pi(H)$ just like in the uniform cases.
... | 1 | member_5 |
; (v1)–(v2); at (2,-2) [$H(1,1,2)$]{};
In the blow-up $H(2,1,1)$ vertex 1 splits into vertices $v_{1,1}$ and $v_{1,2}$; vertex 2 becomes $v_2$ and vertex 3 becomes $v_3$. In the blow-up $H(1,1,2)$ vertex 3 splits into vertices $v_{3,1}$ and $v_{3,2}$; vertex 1 becomes $v_1$ and vertex 2 becomes $v_2$.
[**(Blowing up)... | 1 | member_5 |
monochromatic copy of $K^{v(H)}_{v(H)}(s)$, which gives a copy of $H(s)$ in $G$. $\square$
[**(Squeeze Theorem)**]{} Let $H$ be any hypergraph. If there exists a hypergraph $H^{\prime}$ and integer $s\geq 2$ such that $H^{\prime}\subseteq H\subseteq H^{\prime}(s)$ then $\pi(H)=\pi(H^{\prime})$.
[**Proof:**]{} One nee... | 1 | member_5 |
we will not assume this knowledge.
Since there is no copy of $H$ in $G_{n}$, it follows that $G_{n}[X_{n}]$ contains no copy of $H^{k}$. We have that $$\begin{aligned}
\pi(H) &=\lim_{n\to\infty} h_{n}(G_{n}) \\
&=\lim_{n\to\infty} \sum_{F\in H^{1}}\frac{1}{\binom{n}{1}} + \sum_{F\in H^{k}} \frac{1}{\binom... | 1 | member_5 |
and $g^{k}$ is $H^{k}$-free. Then $$E(G^{k})=\{F\in \binom{[n]}{k}:\text{either } F\in E(g^{k}) \text{ or } F\cap \bar{X}\neq \emptyset\}.$$ Then $G=G^{1}\cup G^{k}$ is $H$-free and (by choice of $x$) we have that $\displaystyle \lim_{n\to\infty}h_{n}(G)$ attains the upper bound of $\pi(H)$. $\square$
Let us now retur... | 1 | member_5 |
be contained in some blow-up of $K_{2}^{\{1,2\}}$ since $H^{2}$ is bipartite, i.e. there exists some $s>2$ such that $H\subseteq K_{2}^{\{1,2\}}(s)$. So, by the squeeze theorem we have $$\frac{5}{4}=\pi(K_{2}^{\{1,2\}}) \leq \pi(H) \leq \pi(K_{2}^{\{1,2\}}(s))=\frac{5}{4}.$$ Hence $\pi(H)=\frac{5}{4}$ as claimed. $\squ... | 1 | member_5 |
where $|X|=\frac{3n}{4}$ and $|\bar{X}|=\frac{n}{4}$. Let $$E(G)=\{\{x\}:x\in X\} \cup \{\{x,\bar{x}\}: x\in X \text{ and } \bar{x}\in \bar{X}\}.$$ It is clear that $G_{n}$ contains no closed paths of length $2k$ when $k\geq 1$. Also, $$\begin{aligned}
\lim_{n\to\infty} h_{n}(G_{n}) &= \lim_{n\to\infty} \frac{|X|}{\bin... | 1 | member_5 |
(0,-3) circle (1 cm); (3, 0) circle (1 cm); (3,-3) circle (1 cm); at (-1.5, 0) [$A_{1}$]{}; at (-1.5,-3) [$A_{2}$]{}; at (4.5, 0) [$B_{2}$]{}; at (4.5, -3) [$B_{1}$]{};
at (0, .5) \[vertex\_closed\] (v1) ; at (-.5, -.5) \[vertex\_closed\] (v2) ; at (.35,-.35) \[vertex\_closed\] (v3) ;
at (3, .5) \[vertex\_open\] (v4)... | 1 | member_5 |
in $Z_{n}$ has at most 1 neighbor in $X_{n}$. It follows that $$\begin{aligned}
\pi(\bar{P}_{4}) &=\lim_{n\to\infty} \pi_{n}(\bar{P}_{4}) \\
&= \lim_{n\to \infty} h_{n}(G_{n}) \\
&\leq \lim_{n\to\infty} \frac{|X_{n}|}{\binom{n}{1}} + \frac{|X_{n}|\cdot |Y_{n}|}{\binom{n}{2}} + \frac{|Y_{n}|\cdot |... | 1 | member_5 |
previous illustration. The difference is, in this case, $B_{1}$ (or $A_{1}$) is empty.
(0, 0) circle (1 cm); (0,-3) circle (1 cm); (3, 0) circle (1 cm); at (-1.5, 0) [$A_{1}$]{}; at (-1.5,-3) [$A_{2}$]{}; at (4.5, 0) [$B_{2}$]{};
at (0, .5) \[vertex\_closed\] (v1) ; at (-.5, -.5) \[vertex\_closed\] (v2) ; at (.35,-.3... | 1 | member_5 |
---
author:
- 'Mark Bun [^1]'
- 'Roi Livni [^2]'
- 'Shay Moran [^3]'
bibliography:
- 'biblio.bib'
title: |
An Equivalence Between Private Classification\
and Online Prediction
---
Introduction
============
This paper continues the study of the close relationship between differentially-private learning and onl... | 1 | member_6 |
social networks). Consequently, a large body of practical and theoretical work has been dedicated to understand which learning tasks can be performed by DP learning algorithms. The simplest and most extensively studied model of learning is the private PAC model [@Valiant84; @KasiviswanathanLNRS11], which captures binar... | 1 | member_6 |
label $\hat{y}_t\in\{\pm 1\}$, and finally learns whether its prediction was correct. The goal is to minimize the [*regret*]{}, namely the number of mistakes compared to the best expert in $\mathcal{H}$: $$\sum_{t=1}^T 1[y_t\neq \hat{y}_t] - \min_{h^*\in\mathcal{H}} \sum_{t=1}^T 1[y_t\neq h^*(x_t)].$$ In this context, ... | 1 | member_6 |
boil down to the notion of stability, which plays a key role in both topics. On one hand, the definition of differential privacy is itself a form of stability; it requires robustness of the output distribution of an algorithm when its input undergoes small changes. On the other hand, stability also arises as a central ... | 1 | member_6 |
following form of stability: for $\eta > 0$ and $n\in\mathbb{N}$, a learning algorithm ${\mathcal{A}}$ is [*$(n,\eta)$-globally stable*]{}[^4] with respect to a distribution ${\mathcal{D}}$ over examples if there exists an hypothesis $h$ whose frequency as an output is at least $\eta$. Namely, $$\Pr_{S\sim {\mathcal{D}... | 1 | member_6 |
The Littlestone dimension of ${\mathcal{H}}$ is $d$. More generally, any class of hypotheses that can be described by polynomial *equalities* of constant degree has finite Littlestone dimension.[^5] This can be generalized even further to classes that are definable in [*stable theories*]{}. This (different, still) noti... | 1 | member_6 |
\[sec:preliminaries\].
\[thm:main\] Let ${\mathcal{H}}\subseteq\{\pm 1\}^X$ be a class with Littlestone dimension $d$, let ${\varepsilon},\delta \in (0, 1)$ be privacy parameters, and let $\alpha,\beta \in (0, 1/2)$ be accuracy parameters. For $$n = O\left(\frac{16^d \cdot d^2 \cdot (d + \log(1/\beta\delta))}{\alpha{\... | 1 | member_6 |
to transform a learner in the realizable setting to a learner in the agnostic setting[^6]. We note that formally the transformation in [@alon2020closure] is stated for a constant ${\varepsilon}=O(1)$. Taking ${\varepsilon}=O(1)$ is without loss of generality as a standard “secrecy-of-the-sample” argument can be used to... | 1 | member_6 |
perspective, and should not be regarded as an efficient transformation between online and private learning. Indeed, the Littlestone dimension dependencies concealed by the $\tilde \Theta_d(\cdot)$ in the above bounds on the regret and sample complexities may be very different from one another. For example, there are cl... | 1 | member_6 |
generalizes “Occam’s Razor" for finite hypothesis classes to show that global stability is enough to imply similar generalization bounds in the realizable setting.
\[prop:gs\] Let ${\mathcal{H}}\subseteq\{\pm 1\}^X$ be a class, and assume that ${\mathcal{A}}$ is a learner for ${\mathcal{H}}$ (i.e. ${\operatorname{loss... | 1 | member_6 |
learning algorithm has oracle access, a globally-stable learner is one which is “weakly” pseudo-deterministic in that it produces some fixed output with probability bounded away from zero. A different model of pseudo-deterministic learning, in the context of learning from membership queries, was defined and studied by ... | 1 | member_6 |
n_0$.
Then there exists a constant $C_{\mathcal{H}}$ such that for every $\alpha, \beta, {\varepsilon}, \delta \in (0, 1)$ there exists an $({\varepsilon}, \delta)$-differentially private learner for ${\mathcal{H}}$ with $$\Pr_{S\sim {\mathcal{D}}^{n}}[{\operatorname{loss}}_{{\mathcal{D}}}({{\mathcal{A}}}(S)) > \alpha... | 1 | member_6 |
then produces a strong learner by making roughly $T \approx \log(1/\beta)/\alpha^2$ calls to the weak learner. By composition of differential privacy, this gives an $({\varepsilon}, 0)$-differentially private strong learner with sample complexity roughly $n_0 \cdot \log(1/\beta)/\alpha^2{\varepsilon}$.
What goes wrong... | 1 | member_6 |
a globally-stable algorithm, and (ii) we then show how to generically obtain a differentially-private learner from any globally-stable learner.
Step 1: Finite Littlestone Dimension $\implies$ Globally-Stable Learning
------------------------------------------------------------------------
Let ${\mathcal{H}}$ be a con... | 1 | member_6 |
guarantees that $M\leq d$ always. Consequently, there is $0\leq i \leq d$ such that $$\Pr[M=i] \geq \frac{1}{d+1}.$$ Note that we can identify, with high probability, an $i$ such that $\Pr[M=i] \geq 1/2d$ by running ${\mathcal{A}}$ on $O(d)$ samples from ${\mathcal{D}}^n$. We next describe how to handle each of the $d+... | 1 | member_6 |
$d-1$ (by assumption, this event occurs with probability at least $(1/2d)^2$) and consider the following two possibilities:
- $\Pr[f_1=f_2]\geq\frac{1}{4}$,
- $\Pr[f_1=f_2] < \frac{1}{4}$.
If (i) holds then using a simple calculation one can show that there is $h$ such that $\Pr[A(S) = h] \geq \frac{1}{(2d)^2}\c... | 1 | member_6 |
(the next one would be that $\Pr[M=d-2] \geq \frac{1}{2d}$, and so on). The proof we present in Section \[sec:LSstable\] is based on a similar idea of performing “random contests,” although the construction becomes more complex to handle other issues, such as generalization, which were not addressed here. For more deta... | 1 | member_6 |
(based on the exponential mechanism) on a fresh set of examples to identify such an accurate hypothesis from the short list.
Preliminaries {#sec:preliminaries}
=============
PAC Learning
------------
We use standard notation from statistical learning; see, e.g., [@Shalev14book]. Let $X$ be any “domain” set and consi... | 1 | member_6 |
also use the following notation: for samples $S,T$, let $S\circ T$ denote the combined sample obtained by appending $T$ to the end of $S$.
Online Learning {#sec:online}
---------------
#### Littlestone Dimension.
The Littlestone dimension is a combinatorial parameter that captures mistake and regret bounds in online... | 1 | member_6 |
Algorithm (${\mathsf{SOA}}$).
The simplest setting in which learnability is captured by the Littlestone dimension is called the [*mistake-bound model*]{} [@Littlestone87online]. Let ${\mathcal{H}}\subseteq \{\pm 1\}^X$ be a fixed hypothesis class known to the learner. The learning process takes place in a sequence of ... | 1 | member_6 |
= {\mathcal{H}}$.
2. For trials $t = 1, 2, \dots$:
- For each $b \in \{\pm 1\}$ and $x \in X$, let ${\mathcal{H}}_t^b(x) = \{h \in {\mathcal{H}}_t : h(x) = b\}$. Define $h : X \to \{\pm 1\}$ by $h_t(x) = {\operatorname{argmax}}_b {\operatorname{Ldim}}({\mathcal{H}}_t^{b}(x))$.
- Receive instance $x_t$.
... | 1 | member_6 |
rule of the ${\mathsf{SOA}}$ to obtain $h_{t+1}$.
- Else, set $h_{t+1}$ as follows: $h_{t+1}(x_{t+1}) = y_{t+1}$, and $h_{t+1}(x)=h_t(x)$ for every $x\neq x_{t+1}$.
Thus, upon observing a non-realizable sequence, this update rule locally updates the maintained predictor $h_t$ to agree with the last example.
Differ... | 1 | member_6 |
$\delta=0$ is also referred to as [*pure differential privacy*]{}. Thus, a class ${\mathcal{H}}$ is privately learnable if it is PAC learnable by an algorithm $A$ that is $({\varepsilon}(n),\delta(n))$-differentially private with ${\varepsilon}(n) \leq 0.1$, and $\delta(n) \leq n^{-\omega(1)} $.
Globally-Stable Learni... | 1 | member_6 |
sample obtained by appending $T$ to the end of $S$. Define a sequence of distributions ${\mathcal{D}}_k$ for $k\geq 0$ as follows:
[**Distributions ${\mathcal{D}}_k$**]{}\
Let $n$ denote an “auxiliary sample” size (to be fixed later) and let ${\mathcal{D}}$ denote the target realizable distribution over examples. The... | 1 | member_6 |
that ${\mathcal{D}}_k$ is well-defined and consider a sample $S$ drawn from ${\mathcal{D}}_k$. The size of $S$ is $\lvert S\rvert = k\cdot(n + 1)$. Among these $k\cdot(n+1)$ examples there are $k\cdot n$ examples drawn from ${\mathcal{D}}$ and $k$ examples which are generated in Item 3(iv). We will refer to these $k$ e... | 1 | member_6 |
---
abstract: 'We present high performance implementations of the QR and the singular value decomposition of a batch of small matrices hosted on the GPU with applications in the compression of hierarchical matrices. The one-sided Jacobi algorithm is used for its simplicity and inherent parallelism as a building block f... | 1 | member_7 |
is an $m \times m$ orthonormal matrix whose columns $U_i$ are called the left singular vectors. $\Sigma$ is an $m \times n$ diagonal matrix whose diagonal entries $\sigma_i$ are called the singular values and are sorted in decreasing order. $V$ is an $n \times n$ orthonormal matrix whose columns $V_i$ are called the ri... | 1 | member_7 |
context of hierarchical matrix operations, effective compression relies on the ability to perform the computation of large batches of independent SVDs of small matrices of low numerical rank. Randomized methods [@halko2011finding] are well suited for computing a truncated SVD of these types of matrices and are built on... | 1 | member_7 |
larger matrix sizes that must reside in global memory. Section \[sec:randomized\] details the implementation of the batched randomized SVD routine. We then discuss some details of the application to hierarchical matrix compression in Section \[sec:application\]. We conclude and discuss future work in Section \[sec:conc... | 1 | member_7 |
the orthogonality of the resulting $Q$ factor suffers with the condition number of the matrix. Another method is based on Givens rotations, where entries in the subdiagonal part of the matrix are zeroed out to form the triangular factor and the rotations are accumulated to form the orthogonal factor. This method is ver... | 1 | member_7 |
combination of both to produce a decomposition $B = U_B \Sigma V_B^T$. The complete SVD is then determined as $A = (Q_U U_B)
\Sigma (Q_V V_B)^T$ during the backward transformation. These methods require significant algorithmic and programming effort to become robust and efficient while still suffering from a loss of r... | 1 | member_7 |
= rot(G)$ $A_{ij} = A_{ij} R$ \[alg:jacobi:rot\]
GPU Optimization Considerations
-------------------------------
GPU kernels are launched by specifying a grid configuration which lets us organize threads into blocks and blocks into a grid. Launching a GPU kernel causes a short stall (as much as 10 microseconds) as th... | 1 | member_7 |
as the compute capability of the card ([@wilt2013cuda] for more details). While not a requirement for good performance [@volkov2010better], it is generally a good idea to aim for high occupancy.
Memory on the GPU is organized into a hierarchy of memory spaces as shown in Figure \[fig:memory\_hierarchy\]. At the bottom... | 1 | member_7 |
As such, it is common to use blocking techniques in many algorithms, where a block of data is brought in from global memory and processed in one of the faster memories.
![The memory hierarchy of a modern GPU.[]{data-label="fig:memory_hierarchy"}](memory.pdf){width="45.00000%"}
Related Work
------------
Batched GPU r... | 1 | member_7 |
In this section, we discuss implementation details of our batched QR kernel and compare it with other implementations from the MAGMA 2.2 [@tnld10] and CUBLAS 8 [@nvidia-cublas] libraries.
Implementation
--------------
One benefit of the Householder algorithm is that the application of reflectors to the trailing matri... | 1 | member_7 |
each other’s registers. Figure \[fig:register\_storage\_reduction\] shows the data layout for a theoretical warp of size 8 with 4 columns in registers and a warp reduction using shuffles. Once we factor the panel, we can apply the reflectors to the trailing sub-matrix in a separate kernel that is optimized for performi... | 1 | member_7 |
maximize the use of the reflectors which have been stored in registers. Figure \[fig:qr\_fig\] shows one step of a panel factorization and the application of its reflectors to the trailing submatrix. Since threads are limited to 1024 per block on current architectures, we use the approach developed in [@journals/concur... | 1 | member_7 |
triangular factor R and reflectors V which are used to update the trailing sub-matrix M.[]{data-label="fig:qr_fig"}](qr_fig.pdf){width="65.00000%"}
Performance
-----------
Figures \[fig:batch\_qr\] and \[fig:batch\_qr\_rect\] show the performance of our batched QR for 1000 square and rectangular matrices with a panel... | 1 | member_7 |
is hosted in registers and analyze the performance of the resulting kernel.
Implementation
--------------
In this implementation, to avoid repeated global memory accesses, we attempt to fit the matrix in register memory using the same layout as the panel in the QR factorization, i.e. one row per thread; however, the ... | 1 | member_7 |
as the convergence test is performed redundantly in each thread. Finally, the column update is done in parallel by each thread on its own register data. As with the QR kernel, we keep occupancy up for the smaller matrix sizes by assigning multiple SVD operations to a single block of threads with each operation assigned... | 1 | member_7 |
30, it is obvious that the register approach will not suffice for larger matrix sizes. This leads us to our next implementation based on the slower but more parallel-friendly shared memory.
[.45]{} ![Performance of the batched register memory SVD on a P100 GPU for 1000 matrices of varying size in single and double pre... | 1 | member_7 |
memory consumption, where potentially only a few warps will be active in a multiprocessor. Instead, we exploit the inherent parallelism of the one-sided Jacobi to assign a warp to a pair of columns, i.e., there are $n/2$ warps processing an $m \times n$ matrix stored in shared memory. There are a total of $n(n-1)/2$ pa... | 1 | member_7 |
warp reductions can be used. This follows our observation in Section \[subsec:reg\_perf\] to assign as few threads as possible to process column pairs, frees up valuable resources and increases the overall performance of the reduction. Row padding is used to keep the rows at multiples of the warp size, and column paddi... | 1 | member_7 |
number of resident blocks become limited by the registers/block limits of the device, dropping to 2 and then 1 resident blocks. Performance increases steadily from there as we increase the number of threads assigned to the operation until we reach a matrix size of $64 \times 64$ where we reach the block limit of 1024 t... | 1 | member_7 |
parallelism; however, since we implement a batched routine for independent operations, we will use the serial block Jacobi algorithm for individual matrices and rely on the parallelism of the batch processing. The parallel version, where multiple blocks are processed simultaneously, can still be used when the batch siz... | 1 | member_7 |
symmetric positive definite). Updating $A^{p+1}_{ij} = A^p_{ij} U^{(p)}_{ij}$ orthogonalizes the block columns, since we have $${A^{p+1}_{ij}}^T A^{p+1}_{ij} = {U^{(p)}_{ij}}^T {A^p_{ij}}^T A^p_{ij} U^{(p)}_{ij} = {U^{(p)}_{ij}}^T G^{(p)}_{ij} U^{(p)}_{ij} = \Lambda^{p}_{ij},$$ where $\Lambda^{p}_{ij}$ is a diagonal ma... | 1 | member_7 |
since the final sweep will not actually perform any rotations within the SVD of $G$. The entire batched operation will then converge when $e = \max e_l < \epsilon$, where $\epsilon$ is our convergence tolerance. This gives us the Gram matrix path of the batched block Jacobi Algorithm \[alg:block\_jacobi\] to compute th... | 1 | member_7 |
$A^{p+1}_{ij}$: $$A^p_{ij} = Q^p_{ij} R^p_{ij} = \left( Q^p_{ij} U^p_{ij} \Sigma^p_{ij} \right) {V^p_{ij}}^T = A^{p+1}_{ij} {V^p_{ij}}^T.$$ If the right singular vectors are needed, we can accumulate the action of $V^p_{ij}$ on the identity matrix. For our batched implementation, we use the batch QR routine developed i... | 1 | member_7 |
QR decompositions and the SVD of the Gram matrix. For the Gram matrix approach, the SVD is the most costly phase, even for the larger operations, while the QR and SVD decompositions take almost the same time for the larger matrices in the direct approach. Figure \[fig:block\_jacobi\_perf\] shows the performance of the ... | 1 | member_7 |
of sweeps required for convergence [@Oksa_2006] as well as by adaptively selecting pairs of block columns based on the computed offdiagonal norms of their Gram matrices. These changes are beyond the scope of this paper and will be the focus of future work.
[0.45]{} ![Profile of the different phases of the block Jacobi... | 1 | member_7 |
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