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e8703c66f3ce4ddf8c07388ae2706af9b510938d | subsection | 25 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | We have reduced our equation to \beta +\gamma =\frac{3}{2} ,
with \frac{1}{2}<\beta <\gamma <1, but now the same argument
used for \alpha shows that this is possible if and only if \beta =\frac{2}{3} ,
since otherwise the pole of order \beta k would not be a branched point,
which leads us to an absurd. Calling p_1,p_2,... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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369a165ea19e5fc09dbd87fb79c8358d60954f57 | subsection | 26 | 50 | Maps preserving a parabolic orbifold | In this chapter we will discuss the main consequences
of Lemma REF . We have a rational map
f:\mathbb {P}^1\rightarrow \mathbb {P}^1 that has an invariant orbifold \mathcal {O},
i.e. there exists a map \tilde{f}:\mathcal {O}\rightarrow \mathcal {O} such
that the following diagram commutes\begin{}[column sep=2.5pc,row s... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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36706b94873f1a8b3bce3f13b1d4407847f43b27 | subsection | 27 | 50 | Maps preserving a parabolic orbifold | We have a rational map
f:\mathbb {P}^1\rightarrow \mathbb {P}^1 that has an invariant orbifold \mathcal {O},
i.e. there exists a map \tilde{f}:\mathcal {O}\rightarrow \mathcal {O} such
that the following diagram commutes\begin{}[column sep=2.5pc,row sep=2pc]
\mathcal {O}{r}{\tilde{f}} {d}[swap]{p} & \mathcal {O}{d}{p} ... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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de5abf8267b9eb5f6ceaf6c5b80fe18beb8145fb | subsection | 28 | 50 | Construction of a torsor associated to a torsion line bundle | We begin recalling some general facts concerning line bundles
over an orbifold \mathcal {O}. We say that \pi :L \rightarrow \mathcal {O} is a torsion
line bundle of order n if L^{\otimes n} is trivial,
i.e. if there exists an isomorphism of line bundles on \mathcal {O}\begin{}[column sep=2.5pc,row sep=2pc]
L^{\otimes n... | {
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"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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dcc94f6563babf80d3ee72327dc03d56c22d9a7e | subsection | 29 | 50 | Construction of a torsor associated to a torsion line bundle | Note that L\mathop {|}_{U_i}^{\otimes n_i} is the trivial bundle on
U_i since such a representation has order dividing n_i.If we set Z=\coprod \limits _i p^{-1}(x_i) and U=\mathcal {O}\setminus Z,
then L is completely determined by the triple\left(L\mathop {\mid }\nolimits _{Z}, L \mathop {\mid }\nolimits _{U}, \phi \r... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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122211dfa80da0920a3190d82e4ea72ea253e249 | subsection | 30 | 50 | Construction of a torsor associated to a torsion line bundle | Better still the singular points of \mathcal {E}
lie over those of \mathcal {O}, and if the underlying
space of \mathcal {O} is \mathbb {P}^1 with y a singular point
of \mathcal {E} lying, in the above notation
(REF ),
over x_i then the local monodromy of \mathcal {E}
at y is the kernel of \rho _i.From the exact sequen... | {
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} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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63f91a6ce94b397042e2b5b6896098987600c41d | subsection | 31 | 50 | Construction of a torsor associated to a torsion line bundle | Therefore if L is a torsion line bundle of order
n we have the following commutative diagram:\begin{}[column sep=2.5pc,row sep=2pc]
L {r}{F_n} {d}[swap]{\pi } & \mathcal {O}\times {ld}{p_1} \\
\mathcal {O}&
\end{}and for any \lambda \in *
our \mu _n torsor is isomorphic to
\mathcal {E}=F_n^{-1}(\mathcal {O}\times \lamb... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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0415c914a529b38d23978927aa79962f3517bfb7 | subsection | 32 | 50 | Construction of a torsor associated to a torsion line bundle | We are going to prove that this property still holds if X
is an orbifold whose underlying space is \mathbb {P}^1.Let us consider an orbifold \mathcal {O} modelled on \mathbb {P}^1 whose
set of singular points is \lbrace x_1, \dots ,x_r\rbrace \subset \mathbb {P}^1,
each x_i having finite weight n_i. This means that the... | {
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}... | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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17036735975126995432a540e286bc7612178a27 | subsection | 33 | 50 | Construction of a torsor associated to a torsion line bundle | Indeed if each representation \rho _i is trivial, then
L\mid _{U_i} is trivial so the maps \phi are just
gluing with the trivial line bundle on the moduli of
U_i, i.e. the naive quotient of the \mu _{n_i}
action, identified with open subset of \mathbb {P}^1, and so
the kernel is a line bundle on \mathbb {P}^1.
We know,... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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d4a036c810187816cc7f25b4b897ff3b78f1edb9 | subsection | 34 | 50 | Construction of a torsor associated to a torsion line bundle | Therefore if L is a torsion line bundle of order
n we have the following commutative diagram:\begin{}[column sep=2.5pc,row sep=2pc]
L {r}{F_n} {d}[swap]{\pi } & \mathcal {O}\times {ld}{p_1} \\
\mathcal {O}&
\end{}and for any \lambda \in *
our \mu _n torsor is isomorphic to
\mathcal {E}=F_n^{-1}(\mathcal {O}\times \lamb... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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022eaea07b0f734c41509aa5b07b5ece0739c87a | subsection | 35 | 50 | Holomorphic differentials on a parabolic orbifold | In this section we show how the construction of Lemma REF
applies to the line bundle \Omega _{\mathcal {O}} of holomorphic differential
forms over a parabolic orbifold.Around a non-space like point x_i, in the above notation (REF ),
we have that \Omega _{\mathcal {O}}\mathop {|}_{U_i} is the \mu _{n_i}-module
\mathcal... | {
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"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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38457f095a3b92aefe34800f27e078ec7760624e | subsection | 36 | 50 | Holomorphic differentials on a parabolic orbifold | H^0(\Omega _{\mathcal {O}}(\log D)^{\otimes k})1) From the definition of n all the local representations (REF )
have order that divides n, so we need only to show that deg(\Omega _{\mathcal {O}})=0 (resp. deg(\Omega _{\mathcal {O}}(\log D))=0, but this follows from the fact that deg(\Omega _{\mathcal {O}})=-\chi (\math... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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71dd9e680ae2ff4eb4f3a4273069ad16813741fd | subsection | 37 | 50 | Holomorphic differentials on a parabolic orbifold | (X,\partial )=(\mathbb {P}^1, 0 + \infty )).We have seen in REF that if n= lcm\lbrace n_1,\dots ,n_r\rbrace ,
then \Omega _{\mathcal {O}} (resp. \Omega _{\mathcal {O}}(\log D)) is a torsion line bundle
of order n and we know from Lemma REF that it defines a unique \mu _n torsor.
On the other hand, the representation de... | {
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} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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6ce444aa7e4c45adacd217f563746a4d555a64f3 | subsection | 38 | 50 | Holomorphic differentials on a parabolic orbifold | Consequently
we obtain the following commutative diagram,[column sep=2.5pc,row sep=2pc]
E [r, ""] rd f*E d [r,"f"] E [d]O[r,"f"] Oand hence by composition we obtain a \mu _n-equivariant map F: E \rightarrow E.Corollary 4.4
Let f: \mathbb {P}^1\rightarrow \mathbb {P}^1 be a rational map of degree d>1 which satisfies As... | {
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"Jacopo Garofali"
] | [
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d94466e10174ac5281915d767d7627651fcf0e80 | subsection | 39 | 50 | Holomorphic differentials on a parabolic orbifold | It is a fact that the naive quotient E/G is isomorphic to \mathbb {P}^1, as the map f is necessarily ramified and the Riemann-Hurwitz formula gives \chi (E/G) >0.It follows that \# Ram_f=2\# G, so if the fiber of each p\in E/G consists of n_p distinct elements, each of order e_p=\# Stab_G(p), we can write Riemann-Hurwi... | {
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c6e09eabf4021c081f4eeb24772242ab093bef56 | subsection | 40 | 50 | Holomorphic differentials on a parabolic orbifold | \Omega _{\mathcal {O}}(\log D))
is a torsion line bundle of order n:=lcm\lbrace \nu _f(x): x \notin D\rbrace .
If we denote by p:\mathcal {O}\rightarrow \mathbb {P}^1 the natural projection, then \tilde{q}:=p^*q\in H^0(\Omega _{\mathcal {O}}^{\otimes k})
(resp. H^0(\Omega _{\mathcal {O}}(\log D)^{\otimes k})1) From th... | {
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} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
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e81cb80f2f84de7d07049c9808adf887a4eba5f9 | subsection | 41 | 50 | Holomorphic differentials on a parabolic orbifold | F:(\mathbb {P}^1, 0+\infty )\rightarrow (\mathbb {P}^1, 0+\infty )) such that the following diagram commutes:\begin{}[column sep=2.5pc,row sep=2pc]
(X, \partial ) {r}{F} {d}[swap]{\pi } & (X, \partial ) {d}{\pi } \\
(\mathcal {O}, D) {r}{f} & (\mathcal {O},D)
\end{}where (X,\partial )= (E, \emptyset ) (resp. (X,\partia... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
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3cd0ac323020badc308fa0c956600092727f342a | subsection | 42 | 50 | Holomorphic differentials on a parabolic orbifold | This follows from the fact that the isomorphism f^*\Omega _{\mathcal {O}} \mathop {\rightarrow }\limits ^{\sim } \Omega _{\mathcal {O}}
(resp. f^*\Omega _{\mathcal {O}}(\log D) \mathop {\rightarrow }\limits ^{\sim }
\Omega _{\mathcal {O}}(\log D))
affords an isomorphism of \mu _n-torsors f^*E \mathop {\rightarrow }\lim... | {
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} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
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cb5ee29d3414701e4a05bd2edcf64ab9902acd9a | subsection | 43 | 50 | Holomorphic differentials on a parabolic orbifold | In the case n=2 the condition above is empty, hence is generic.\textbf {Remark} 4.5
The orbifolds listed in Corollary REF are the only one which can
be realized as quotients of an elliptic curve E for the action of
a group of automorphisms of E.
In fact it is well known that the group of automorphisms of an elliptic c... | {
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58e8cfb810d6bb25f67aba44838bb3a631b3914f | subsection | 44 | 50 | Conclusions | We conclude this chapter formulating at first a structural theorem for the maps f with a parallel tensor q and finally discussing the invariant nature of q.Theorem 4.6 Let f: \mathbb {P}^1\rightarrow \mathbb {P}^1 be a rational map of degree d>1 which satisfies Assumption REF .Then, up to conjugation with an element of... | {
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6f0c4c35c2f8300c32445b33dabc4fcb29de0c13 | subsection | 45 | 50 | Conclusions | Consequently the only translations allowed are the solutions of\theta z \equiv z \;(mod\,\mathbb {Z}[\theta ])where \theta is a primitive n-th root of unity. [table] Table table contains the solutions of (REF ) lying in the fundamental domain of the elliptic curve. For n=2 they consist of the subgroup E[2] of 2-torsion... | {
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22e7eec5481b2c5428370c29b6f40bef957be620 | subsection | 46 | 50 | Conclusions | In case (2) they are (up to sign) the Tchebycheff polynomials P_n(z) defined by P_n(cosz)=cos(nz), since the cosine function is the universal covering map of * which commutes with z \mapsto -z.
Note that in each case we have shown that (modulo multiplication by an element of PGL_2() the action of f on \mathbb {P}^1, w... | {
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6a21c21da4b90bfaed45b664e827dd71a079d75f | subsection | 47 | 50 | Conclusions | \textit {f} is obtained as quotient of such automorphisms under the action of a discrete group of automorphisms of the complex plane.\\
\begin{}[htb]
\begin{}{|c|c|}
\hline Orbifold \mathcal {O} preserved by \textit {f} & Automorphism of inducing \textit {f} \\
\hline (\infty ,\infty ) & z \mapsto nz; n\in \mathbb {Z} ... | {
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2b15b84975450c924d2d8dd3d5cc7041fa5d96f0 | subsection | 48 | 50 | Conclusions | Recall that when \alpha \notin \mathbb {Z} we say that the elliptic curve E has complex multiplication and if multiplication by a complex \alpha , (which must be an integer in some imaginary quadratic field, as \alpha \Lambda \subset \Lambda , see ) is allowed in E, then the complex structure of E is completely determi... | {
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f1a944c86324382838d75e40b484459f906963c3 | subsection | 49 | 50 | Conclusions | We have seen in REF that \tilde{q}=p^*q is a holomorphic section of \Omega _{\mathcal {O}}^{\otimes k}, hence \pi ^*\tilde{q} is a constant multiple of dz^{\otimes k}.
It follows that the eigenvalue \lambda such that f^*q=\lambda q can be computed explicitly.
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e9f471e064e44d934a339209f85a5fd7a42e2dfb | abstract | 0 | 121 | Abstract | A long-standing problem in the theory of stochastic gradient descent (SGD) is
to prove that its without-replacement version RandomShuffle converges faster
than the usual with-replacement version. We present the first (to our
knowledge) non-asymptotic solution to this problem, which shows that after a
"reasonable" numbe... | {
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81a4b4118d3a978cd8019a99ba4d7f2292ea9c94 | subsection | 1 | 121 | Introduction | We consider stochastic optimization methods for the finite-sum problemF\left(x\right) := \frac{1}{n} \sum _{i=1}^n f_i\left(x\right),where each function f_i : \mathbb {R}^d\rightarrow \mathbb {R} is smooth and convex, and the sum F is strongly convex. A classical approach to solving (REF ) is stochastic gradient descen... | {
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36d4a2fd581fb72f386ed2ec3b2efca6917f0407 | subsection | 2 | 121 | Summary of results | We follow the common practice of reporting convergence rates depending on T, the number of calls to the (stochastic / incremental) gradient oracle. For instance, Sgd converges at the rate \mathcal {O}(\frac{1}{T}) for solving (REF ), ignoring logarithmic terms in the bound . The underlying argument is to view Sgd as st... | {
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"raw": "A. Rakhlin, O. Shamir, K. Sridharan, et al. Making gradient descent optimal for strongly convex stochastic optimization. In ICML. Citeseer, 2012.",
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a313559c50bc110f62395cd29a9112ad951d1a82 | subsection | 3 | 121 | Related work | conjecture a tantalizing matrix AM-GM inequality that underlies RandomShuffle's superiority over Sgd. While limited progress on this conjecture has been reported , , the correctness of the full conjecture is still wide open. With the technique of transductive Rademacher complexity, shows that Sgd is not worse than Rand... | {
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"raw": "B. Recht and C. Ré. Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences... | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
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94739a9119da92b1518a0544fb149c471e1f10fe | subsection | 4 | 121 | Background and problem setup | For problem (REF ), we assume the finite sum function F(x): \mathbb {R}^d\rightarrow \mathbb {R} is strongly convex, i.e.,F(x) \ge F(y) + \left\langle \nabla F(y), x-y \right\rangle + \tfrac{\mu }{2}\left\Vert x-y\right\Vert ^2,where x, y\in \mathbb {R}^d, and \mu >0 is the strong convexity parameter. Furthermore, we a... | {
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df8f7e435282df9bc2dfbe92309f76206fe5cd91 | subsection | 5 | 121 | The algorithms under study: | For both Sgd and RandomShuffle, we use \gamma as the step size, which is predetermined before the algorithms are run. The sequences generated by both methods are denoted as (x_k)_{k=0}^T; here x_0 is the initial point and T is the total number of iterations (i.e., number of stochastic gradients used).Sgd is defined as ... | {
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0107230f60b2265dcc984918ebc1d2e2fc391b41 | subsection | 6 | 121 | Convergence analysis of | The goal of this section is to build theoretical analysis for RandomShuffle. Specifically, we answer the following question: when can we show RandomShuffle to be better than Sgd? We begin by first analyzing quadratic functions in Section REF , where the analysis benefits from having a constant Hessian. Subsequently, in... | {
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c5f3b5476696260fce115c239dc9a769382c0dba | subsection | 7 | 121 | Body | We first consider the quadratic instance of (REF ), wheref_i(x) = \tfrac{1}{2}x^T A_i x + b_i^Tx,\qquad i=1,\ldots ,n,where A_i\in \mathbb {R}^{d\times d} is positive semi-definite, and b_i \in \mathbb {R}^d. We should notice often in analyzing strongly convex problems, the quadratic case presents a good example when t... | {
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273b1ef3b8640bed882c4ea530ec6e097c18325e | subsection | 8 | 121 | Body | Formally, we recover the main result of as the following:Corollary 2 As T\rightarrow \infty , RandomShuffle achieves asymptotic convergence rate \mathcal {O}\left(\frac{1}{T^2}\right) when run with the proper step size schedule.Next, we consider the more general case where each component function f_i is convex and the... | {
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840e95eeb9693e6c906fbd7cfe556e1535e37962 | subsection | 9 | 121 | Body | We have the following extension of our previous result:Theorem 5
Under the Polyak-Łojasiewicz condition, define condition number \kappa = L/\mu . So long as \frac{T}{\log T} > 16\kappa ^2 n, with step size \eta = \frac{2\log T}{ T\mu }, RandomShuffle achieves convergence rate:\operatorname{\mathrm {\mathbb {E}}}[\left... | {
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"raw": "O. Shamir. Without-replacement sampling for stochastic gradient methods. In Advances in Neural Information Processing Systems, pages 46–54, 2016.",
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fc69e206ee7ca8dbf4f7dbc062437ea4a429c926 | subsection | 10 | 121 | Body | Then there isF(\bar{x}) - F(x^*) \le \frac{2D\sqrt{nD\left(\left\Vert \Delta \right\Vert +L_H LD^2 + 2L_HDG\right)}}{\sqrt{T}} + \mathcal {O}\left(\left(\frac{n}{T}\right)^\frac{2}{3}\delta ^\frac{1}{3} + \left(\frac{n}{T}\right)^\frac{3}{4}\right).We have some discussion of this result:Firstly, it is interesting to se... | {
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831c27f63bffc33755f868f002bb391a70c09841 | subsection | 11 | 121 | Body | We leave this possibility (either improving upon the {1}{\sqrt{T}} dependence on T under existence of noise, or recovering {1}{T} when there is no noise) as an open question.For the completeness of the paper, we include the following analysis of Sgd under Polyak-Łojasiewicz condition.Theorem 8 For finite sum problem sa... | {
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29affe39a368fd0c8d3d1fed328cb0aa0621a2be | subsection | 12 | 121 | Understanding the dependence on | Since the motivation of building our convergence rate analysis is to show that RandomShuffle behaves better than Sgd, we would definitely hope that our convergence bounds have a better dependence on T compared to the \mathcal {O}(\frac{1}{T}) bound for Sgd. In an ideal situation, one may hope for a rate of the form \ma... | {
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d060657c769ee9ea17b022418efa3d8ed51f7217 | subsection | 13 | 121 | Understanding the dependence on | We directly have the following corollary:Corollary 3 Given the information of \mu , L, G, under the assumption T\ge n and constant step size, there is no step size choice that leads to a convergence rate \mathcal {O}\left(\frac{1}{T^{1+\delta }}\right) for \delta >0.This result indicates that in order to achieve a bett... | {
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"raw": "R. Johnson and T. Zhang. Accelerating stochastic gradient descent using predictive variance reduction. In Advances in neural information processing systems, pages 31... | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
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2f543b881f24ca47fbf7309db765b944d974cce1 | subsection | 14 | 121 | Sparse functions | In the literature on large-scale machine learning, sparsity is a common feature of data. When the data are sparse, each training data point has only a few non-zero features. Under such a setting, each iteration of Sgd only modifies a few dimensions of the decision variables. Some commonly occurring sparse problems incl... | {
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"raw": "B. Recht, C. Re, S. Wright, and F. Niu. Hogwild: A lock-free approach to parallelizing stochastic gradient descent. In Advances in neural information processing systems, pages 693–701, 2011.",
... | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
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de3f6e1fcf16f7e72db7c70796a94b0a0b275932 | subsection | 15 | 121 | Sparse functions | The key to proving Theorem REF lies in constructing a tighter bound for the error term in the main recursion (see §) by including a discount due to sparsity.We end this section by noting the following simple corollary:Corollary 4 When \rho = \mathcal {O}\left(\frac{1}{n}\right), there is some constant C only dependent ... | {
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f0edebac84cae55f6febf6ddf7dff8cd0b54a239 | subsection | 16 | 121 | Proof sketch of Theorem | In this section we provide a proof sketch for Theorem REF . The central idea is to establish an inequality\operatorname{\mathrm {\mathbb {E}}}\bigl [\left\Vert x^{t+1}_0 - x^*\right\Vert ^2\bigr ] \le (1-n\gamma \alpha _1)\left\Vert x^t_0 - x^*\right\Vert ^2 + n\gamma ^3 \alpha _2 + n^4\gamma ^4 \alpha _3,where x^t_0 a... | {
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39a897db87b952e3f78d73780635a4cca9e2d956 | subsection | 17 | 121 | Proof sketch of Theorem | Thus, the terms A^t_2 and A^t_4 are also random variables that depend on \sigma _t(\cdot ), and require taking expectations. The main body of our analysis involves bounding each of these terms separately.The term A_1^t can be easily bounded by exploiting the strong convexity of F, using a standard inequality (Theorem 2... | {
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282607955d6d37045d837f999ccefcb1197f6d2f | subsection | 18 | 121 | Proof sketch of Theorem | Specifically, by introducing second-order information and somewhat involved analysis, we obtain the following bound for A_2^t:Lemma 1
Over the randomness of the permutation, we have the inequality:-2\gamma \langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}[R^t] \rangle &\le \frac{1}{2}\gamma \mu (n-1)\left\Vert... | {
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792a9279eb55df24270e5ef9b91beea02e46a52a | subsection | 19 | 121 | Discussion of results | We discuss below our results in more detail, including their implications, strengths, and limitations. | {
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2fe0928d3bb4061ccab747e47f3729abd6c5911c | subsection | 20 | 121 | Comparison with | It is well-known that under strong convexity Sgd converges with a rate of \mathcal {O}\bigl (\frac{1}{T}\bigr ) . A direct comparison indicates the following fact: RandomShuffle is provably better than Sgd after \mathcal {O}(\sqrt{n}) epochs. This is an acceptable amount of epochs for even some of the largest data sets... | {
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"raw": "A. Rakhlin, O. Shamir, K. Sridharan, et al. Making gradient descent optimal for strongly convex stochastic optimization. In ICML. Citeseer, 2012.",
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e43533b0c1e2ca66339f995867fbf05a4352eabe | subsection | 21 | 121 | Deterministic variant. | When the algorithm is run in a deterministic fashion, i.e., the functions f_i are visited in a fixed order, better convergence rate than Sgd can also be achieved as T becomes large. For instance, a result in translates into a \mathcal {O}\bigl (\frac{n^2}{T^2}\bigr ) bound for the deterministic case. This directly imp... | {
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"raw": "M. Gürbüzbalaban, A. Ozdaglar, and P. Parrilo. Convergence rate of incremental gradient and newton methods. arXiv preprint arXiv:1510.08562, 2015a.",
"source_ref_id": "77a1eaebdd70c8b40c822000... | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
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bfbd89a532b2212bb863915e3b7cc9cd948ae454 | subsection | 22 | 121 | Epochs required. | It is also a limitation that our bound only holds after a certain number of epochs. Moreover, this number of epochs is dependent on \kappa (e.g., \mathcal {O}(\kappa ) epochs for the quadratic case). This limits the interest of our result to cases when the problem is not too ill-conditioned. Otherwise, such a number of... | {
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e5081ba69926cbe205fc042592736506b2cbe8aa | subsection | 23 | 121 | Dependence on | It should be noticed that \kappa can be large sometimes. Therefore, it may be informative to view our result in a \kappa -dependent form. In particular, we still assume D, L, L_H are constant, but no longer \mu . We use the bound G\le \max _i{\left\Vert \nabla f_i(x^*)\right\Vert + DL} and assume \max _i{\left\Vert \na... | {
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644b5a2bfa0612a7a1efbf3d4640946eacce5ed9 | subsection | 24 | 121 | Sparse data setting. | Notably, in the sparse setting (with sparsity factor \rho = \mathcal {O}\bigl (\frac{1}{n}\bigr )), the proven convergence rate is strictly better than the \mathcal {O}\bigl (\frac{1}{T}\bigr ) rate of Sgd. This result follows the following intuition: when each dimension is only touched by several functions, letting th... | {
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} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
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963838aa88fe1743829b9a2900bcde4bd19dbd1a | subsection | 25 | 121 | Extensions | In this section, we provide some further extensions before concluding with some open problems. | {
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fa52c60d38db1187398e19192eedad7dbab8ca7e | subsection | 26 | 121 | Vanishing variance | Our previous results show that RandomShuffle converges faster than Sgd after a certain number of epochs. However, one may want to see whether it is possible to show faster convergence of RandomShuffle after only one epoch, or even within one epoch. In this section, we study a specialized class of strongly convex proble... | {
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8144af9e05982d21228d5e0dc690c1c598d27097 | subsection | 27 | 121 | Vanishing variance | For step size \eta \le \min \limits _i\lbrace \frac{2}{L_i+\mu _i}\rbrace and any T\ge 1, there is\max \limits _{P\in \mathcal {P}, x_0 \in D_R(x^*)} \operatorname{\mathrm {\mathbb {E}}}[\left\Vert X_{RS}(T, x_0, \gamma , P) - x^*\right\Vert ^2] \le \max \limits _{P\in \mathcal {P}, x_0 \in D_R(x^*)}\operatorname{\math... | {
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567b390a1374d700ff4255b517eb0e6f3defee59 | subsection | 28 | 121 | Conclusion and open problems | A long-standing problem in the theory of stochastic gradient descent (Sgd) is to prove that RandomShuffle converges faster than the usual with-replacement Sgd. In this paper, we provide the first non-asymptotic convergence rate analysis for RandomShuffle. We show in particular that after \mathcal {O}(\sqrt{n}) epochs, ... | {
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} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
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ddd66e76622526a2f2e271f7f52257c5aa2852a4 | subsection | 29 | 121 | Conclusion and open problems | For one epoch of RandomShuffle, We have the following inequality
\left\Vert x^{t}_n - x^*\right\Vert ^2 &= \left\Vert x^t_0 - x^*\right\Vert ^2 - 2\gamma \left\langle x^t_0 - x^*, \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) \right\rangle + \gamma ^2 \left\Vert \sum _{i=1}^n \nabla f_{\sigma... | {
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"source_ref_id": "ac12aab6cc5b... | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
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a845bb4cc1630854953548a4421d2b498a082fb2 | subsection | 30 | 121 | Conclusion and open problems | Take the expectation of (REF ) over randomness of permutation \sigma _t\left(\cdot \right), we have
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x^t_n - x^*\right\Vert ^2\right] &\le \left(1-2n\gamma \frac{L\mu }{L+\mu }\right)\left\Vert x^t_0 - x^*\right\Vert ^2 -\left(2n\gamma \frac{1}{L+\mu }-2n^2\gamma ^2\r... | {
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681301327b1b30d6ac938950be0beb168b8ef2da | subsection | 31 | 121 | Conclusion and open problems | Firstly, we give a bound on the norm of R^t:
\left\Vert R^t\right\Vert &= \left\Vert \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)}\left(x^t_0\right)\right\Vert \\
&\le \sum _{i=1}^n \left\Vert \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1... | {
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... | |
64a1e98aa57032148613b5ff55c6f4a2d2240d48 | subsection | 32 | 121 | Conclusion and open problems | We begin with the following decomposition:
R^t &= \sum _{i=1}^n \left[\nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \nabla f_{\sigma _t\left(i\right)}\left(x^t_{0}\right)\right] \\
&= \sum _{i=1}^n \left[H_{\sigma _t\left(i\right)}\left(x^t_{i-1} - x^t_0\right)\right] \\
&= \sum _{i=1}^n \left\lbrace H_{... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
0.012390484102070332,
0.022675197571516037,
-0.011604635044932365,
0.007904274389147758,
-0.010696711018681526,
0.007591460831463337,
0.013634110800921917,
0.03357027843594551,
0.005447540432214737,
0.010475452989339828,
-0.03796493262052536,
-0.008850346319377422,
-0.03111354447901249,
0.... | |
c7842c759f7634017820718f6af9bba059e62788 | subsection | 33 | 121 | Conclusion and open problems | Here we define random variables
A^t = -\gamma \sum _{i=1}^n \left[H_{\sigma _t\left(i\right)}\sum _{j=1}^{i-1} \nabla f_{\sigma _t\left(j\right)} \left(x^t_0\right)\right],
B^t = - \gamma \sum _{i=1}^n \left\lbrace H_{\sigma _t\left(i\right)}\sum _{j=1}^{i-1}\left[\nabla f_{\sigma _t\left(j\right)} \left(x^t_{j-1}\righ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.05455244332551956,
-0.0042776502668857574,
-0.019201727584004402,
-0.006784712430089712,
0.0009282234241254628,
-0.01521790400147438,
-0.0022628428414463997,
0.034404367208480835,
-0.004827143624424934,
0.028069933876395226,
-0.05061441287398338,
0.03883083909749985,
-0.000046625817049061... | |
e10ade45b7ea5b57cde1c77149459a4d63e81eae | subsection | 34 | 121 | Conclusion and open problems | Using (REF ) and (REF ), we can decompose the inner product of x^t_0 - x^* and \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] into:
-2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] \right\rangle &= -2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\lef... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.017502684146165848,
0.015023009851574898,
-0.01939486525952816,
0.014595742337405682,
0.002018073108047247,
-0.010544336400926113,
0.008842898532748222,
0.03695858642458916,
0.005981736816465855,
0.03128204122185707,
-0.043062400072813034,
0.023759091272950172,
-0.012344961054623127,
-0... | |
c07e17930e223b70e248f1d558b8734d1106b470 | subsection | 35 | 121 | Conclusion and open problems | For the first term in (REF ), there is
&\ \ \ \ \gamma ^2 n\left(n-1\right) \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i \nabla f_j\left(x^t_0\right)\right\rangle \\
&= \gamma ^2 n\left(n-1\right) \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.056741006672382355,
0.010392853058874607,
-0.0218082033097744,
-0.015932654961943626,
-0.011751096695661545,
-0.007199454121291637,
0.010644662193953991,
0.05432974547147751,
0.018633881583809853,
0.018252352252602577,
-0.0576261542737484,
0.0015881148865446448,
-0.033544037491083145,
-... | |
0fe6e7b7beab2ef9a78c747359b63e11e8dddc97 | subsection | 36 | 121 | Conclusion and open problems | For the second term in (REF ), we use the bound
-2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\left[B^t\right]\right\rangle &\le 2\gamma \left[\frac{\lambda _2}{2}\left\Vert x^t_0- x^*\right\Vert ^2 + \frac{1}{2\lambda _2}\left\Vert \operatorname{\mathrm {\mathbb {E}}}\left[B^t\right]\right\Ve... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.05859992653131485,
0.02411142736673355,
-0.004452221095561981,
-0.02321106381714344,
-0.025835853070020676,
0.006657348480075598,
0.0161149799823761,
0.057073887437582016,
0.027300851419568062,
0.005100787617266178,
-0.028338557109236717,
0.02829277701675892,
-0.019121278077363968,
0.00... | |
1293ebc949d0136b56d8c69bd3e72e56c2ef50d3 | subsection | 37 | 121 | Conclusion and open problems | Toward this end, we use the following important fact:
\left\Vert \Delta \right\Vert &= \left\Vert \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i \nabla f_j\left(x^*\right)\right\Vert \\
&= \left\Vert \frac{1}{n\left(n-1\right)} \sum _{i\ne j} H_i \nabla f_j\left(x^*\right)\right\Vert \\
&= \left\Vert \frac{... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.0626225695014,
0.028778305277228355,
-0.015327575616538525,
0.016281258314847946,
-0.032226819545030594,
-0.016998426988720894,
-0.011436553671956062,
0.02461262419819832,
0.006988581269979477,
0.007572234608232975,
-0.06347706913948059,
0.011390777304768562,
-0.012718302197754383,
0.00... | |
13438f69120773b11054785bafc2a11f1803da99 | subsection | 38 | 121 | Conclusion and open problems | Substituting (REF ) (REF ) back to (REF ) and using (REF ) , we finally get a recursion bound for one epoch:
&\ \ \ \ \operatorname{\mathrm {\mathbb {E}}}\left\Vert x^t_n - x^*\right\Vert ^2 \\
&\le \left(1-2n\gamma \frac{L\mu }{L+\mu } +\frac{1}{2}\gamma \mu \left(n-1\right)\right)\left\Vert x^t_0 - x^*\right\Vert ^2... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.0334845595061779,
0.02286229282617569,
-0.01831425353884697,
-0.01781061291694641,
-0.01639125682413578,
-0.020511964336037636,
-0.0050135268829762936,
0.03699479252099991,
0.011675337329506874,
0.006326048634946346,
-0.03278251364827156,
0.03614012897014618,
-0.026143597438931465,
0.03... | |
7759eb4ac56bd7a9ffb8ec67a3055599b3b26e8b | subsection | 39 | 121 | Conclusion and open problems | Expanding (REF ) over all epochs leads to a final bound of RandomShuffle:
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - x^*\right\Vert ^2\right] \le \left(1-n\gamma \frac{L\mu }{L+\mu }\right)^{\frac{T}{n}}\left\Vert x_0 - x^*\right\Vert ^2 + \frac{T}{n} \left(\gamma ^3nC_1 + \gamma ^{5}n^{5}C_2 + \gamma ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.056337323039770126,
0.02763458527624607,
-0.014725221320986748,
-0.038941726088523865,
-0.011085879057645798,
0.019852343946695328,
-0.0037633241154253483,
0.02031012438237667,
0.020218567922711372,
-0.00421156594529748,
-0.005420865025371313,
0.0006513812113553286,
-0.032532818615436554,... | |
3650462780c1746f740f6dac7eef330ff9a9f4b4 | subsection | 40 | 121 | Conclusion and open problems | Or in the expanding version with constant dependence, we have
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - x^*\right\Vert ^2\right] \le \frac{\left(\log T\right)^2}{T^2}\left(D^2 + 128\frac{L^2G^2}{\mu ^4}\right) + \frac{n^3\left(\log T\right)^4}{T^3}128\frac{L^2G^2}{\mu ^4} + \frac{n^4\left(\log T\right... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.05399071425199509,
0.008973019197583199,
-0.011345985345542431,
0.03222961723804474,
0.004066852852702141,
0.01638949289917946,
-0.020570797845721245,
0.016740478575229645,
0.011376505717635155,
0.018220722675323486,
-0.021013345569372177,
-0.0034678885713219643,
0.01837332360446453,
0.... | |
e57dc4c1915810e81b3a91814c51cfbacd1a04f4 | subsection | 41 | 121 | Conclusion and open problems | Again, define error term
R^t = \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)}\left(x^t_0\right). | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
0.0006196077447384596,
0.01813318580389023,
-0.018484249711036682,
0.02347545512020588,
0.00022036857262719423,
0.012050631456077099,
0.006712178699672222,
0.030756203457713127,
-0.016789987683296204,
-0.015301783569157124,
-0.026024479418992996,
-0.01008925586938858,
-0.027779797092080116,
... | |
cd868780aa7ca4193b7900c874cb402f2cae38fb | subsection | 42 | 121 | Conclusion and open problems | We have the following decomposition for the error term:
R^t &= \sum _{i=1}^n \left[\nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \nabla f_{\sigma _t\left(i\right)}\left(x^t_{0}\right)\right]\\
&= \sum _{i=1}^n \left[\int _{x^t_0}^{x^t_{i-1}} H_{\sigma _t\left(i\right)}\left(x\right) dx\right]\\
&= \sum _... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
0.014037642627954483,
0.005763825494796038,
-0.018081093207001686,
0.01695197820663452,
0.0019339903956279159,
-0.006603032350540161,
0.0014628901844844222,
0.022475486621260643,
-0.0050504994578659534,
0.01001707836985588,
-0.03207296133041382,
-0.005878262687474489,
-0.010902060195803642,
... | |
d719d72593d892e3e2767616793c55f295f0f994 | subsection | 43 | 121 | Conclusion and open problems | Here we define random variables
A^t = -\gamma \sum _{i=1}^n \left[H_{\sigma _t\left(i\right)}\sum _{j=1}^{i-1} \nabla f_{\sigma _t\left(j\right)} \left(x^t_0\right)\right],
B^t = - \gamma \sum _{i=1}^n \left\lbrace H_{\sigma _t\left(i\right)}\sum _{j=1}^{i-1}\left[\nabla f_{\sigma _t\left(j\right)} \left(x^t_{j-1}\righ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.03512922301888466,
-0.01084244716912508,
-0.02678183652460575,
0.00470017408952117,
0.007366912439465523,
-0.0340915210545063,
-0.010575391352176666,
0.030780036002397537,
0.005577641539275646,
0.02484377659857273,
-0.044468529522418976,
0.0207540150731802,
0.005619607400149107,
-0.0093... | |
36983d35f55e541c1ba437e7cd12512c8e8d997b | subsection | 44 | 121 | Conclusion and open problems | \left\Vert C^t\right\Vert &\le \sum _{i=1}^n \left[\int _{0}^{\left\Vert x^t_{i-1}-x^t_0\right\Vert } \left\Vert H_{\sigma _t\left(i\right)}\left(x^t_0+t\frac{x^t_{i-1}-x^t_0}{\left\Vert x^t_{i-1}-x^t_0\right\Vert }\right)-H_{\sigma _t\left(i\right)}\right\Vert dt\right]\\
&\le \sum _{i=1}^n \left[L_H\max \left\lbrace ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.027158524841070175,
0.016600266098976135,
-0.030484680086374283,
-0.01763778366148472,
-0.009185073897242546,
-0.0038105850107967854,
-0.021208060905337334,
0.03844914585351944,
0.003619865048676729,
-0.00912404339760542,
-0.03262074291706085,
0.005713970400393009,
0.0030705914832651615,
... | |
6cf7de84087a875374cf40919e23478366600544 | subsection | 45 | 121 | Conclusion and open problems | Using (REF ) (REF ), we can decompose the innerproduct of x^t_0 - x^* and \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] as following:
-2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] \right\rangle &= -2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\l... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.02541893720626831,
0.014563250355422497,
-0.01617291197180748,
0.007563888095319271,
-0.0036160191521048546,
-0.00850603636354208,
0.021329935640096664,
0.037258729338645935,
0.00494341878220439,
0.04287347570061684,
-0.03844881057739258,
0.018049580976366997,
-0.011092176660895348,
-0.... | |
7ecd9577b06285f3713f0af600f80f06458e3577 | subsection | 46 | 121 | Conclusion and open problems | For the first term in the (REF ), we have further bound:
&\ \ \ \ \gamma ^2 n\left(n-1\right) \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i \nabla f_j\left(x^t_0\right)\right\rangle \\
&= \gamma ^2 n\left(n-1\right)\operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} \left\langle... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.04946403577923775,
0.021222848445177078,
0.0065110353752970695,
-0.007651515770703554,
-0.029431253671646118,
-0.008086347952485085,
-0.009345071390271187,
0.04568023607134819,
0.01280084066092968,
0.01960557885468006,
-0.0411335714161396,
0.01904105953872204,
-0.03469500690698624,
-0.0... | |
78f923f15147b456fe0726c44ad4cb2160fe0c78 | subsection | 47 | 121 | Conclusion and open problems | Note that here H_i\left(x^t_0 - x^*\right) is the matrix H_i(x^*) times vector x^t_0 - x^*, not the Hessian at point x^t_0 - x^*. The last inequality is because of
\left\Vert H_i\left(x^t_0 - x^*\right) - \left(\nabla f_i\left(x^t_0\right) - \nabla f_i\left(x^*\right)\right)\right\Vert &=\left\Vert H_i\left(x^t_0 - x^... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.023717762902379036,
-0.005696688778698444,
-0.014735842123627663,
-0.006448362022638321,
-0.01610182598233223,
0.026816029101610184,
-0.019016943871974945,
0.051831092685461044,
0.023931436240673065,
0.015842366963624954,
-0.03907173126935959,
0.007524360902607441,
0.011958086863160133,
... | |
84d651b79cfc5beabbb367476dc2c2c1e06563d1 | subsection | 48 | 121 | Conclusion and open problems | Substituting (REF ) (REF ) (REF ) back to (REF ), we get
-2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] \right\rangle &\le \gamma ^2n^2 \left\Vert \nabla F\left(x^t_0\right)\right\Vert ^2 + \frac{1}{2}\gamma \mu \left(n-1\right)\left\Vert x^t_0-x^*\right\Vert ^2 + \gamma ^{3} \... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.04845882207155228,
0.01401430368423462,
-0.015395136550068855,
-0.03902949392795563,
-0.023436006158590317,
-0.004588788375258446,
0.009398814290761948,
0.05172399431467056,
0.006312921177595854,
-0.001487637055106461,
-0.03368925675749779,
0.018767112866044044,
-0.037442680448293686,
0... | |
8336cb623d5fe2400be917140e7cdb91a6c73951 | subsection | 49 | 121 | Conclusion and open problems | Now assume
\frac{3}{2}n\gamma \frac{L\mu }{L+\mu }>\frac{1}{2}\gamma \mu \left(n-1\right)+\gamma ^2n^2\left(L_H LD + 3L_H G\right),
and
2n\gamma \frac{1}{L+\mu }-3\gamma ^2n^2>0,
which we call assumption 1 and assumption 2, (REF ) can be further turned into:
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x^t_n -... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.07263966649770737,
0.03683868795633316,
-0.0056654359214007854,
-0.020632106810808182,
-0.014459260739386082,
0.012887435965240002,
-0.012628008611500263,
0.0381816066801548,
0.001049154787324369,
-0.01561905350536108,
-0.021517211571335793,
0.028811698779463768,
-0.02653789333999157,
0... | |
189fd83691a0838f21df2ee7dcfe0561bedb1677 | subsection | 50 | 121 | Conclusion and open problems | Let \gamma = \frac{8\log T}{T\mu }, there is
\operatorname{\mathrm {\mathbb {E}}}\left\Vert x_T - x^*\right\Vert ^2 &\le \left(1-\frac{2n\log T}{T}\right)^{\frac{T}{2n\log T} 2\log T}\left\Vert x_0 - x^*\right\Vert ^2 + \frac{T}{n} \left(\gamma ^3nC_1 + \gamma ^{4}n^{4}C_2 + \gamma ^5n^5C_3\right)\\
&\le \frac{1}{T^2}... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.040472161024808884,
0.016802355647087097,
-0.03482559323310852,
-0.005337533075362444,
-0.00569235160946846,
-0.02395976334810257,
-0.008996357209980488,
0.02229631505906582,
-0.020236080512404442,
0.008836116641759872,
-0.016649747267365456,
-0.0016081273788586259,
-0.03433724120259285,
... | |
c94dc3e3454de0b9450a6161dbb9b35fef2ccdc3 | subsection | 51 | 121 | Conclusion and open problems | Assumption 3 is equivalent to
\frac{T}{\log T} > \frac{8L}{L+\mu }n,
which is satisfied when
\frac{T}{\log T} > 8n.
Since 12\left(1+\frac{L}{\mu }\right)>8, we only need
\frac{T}{\log T} > \max \left\lbrace \frac{32}{\mu ^2}\left(L_H LD + 3L_H G\right)n, 12\left(1+\frac{L}{\mu }\right) n \right\rbrace .
So whenever \fr... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.027503108605742455,
0.0034607823472470045,
-0.025015313178300858,
-0.0021424798760563135,
0.0043727196753025055,
0.019795522093772888,
-0.011965836398303509,
0.0045215291902422905,
-0.021016526967287064,
0.014148380607366562,
0.008195986971259117,
0.023061707615852356,
-0.0213675647974014... | |
8c0410d23535946cc38c87ff268ac6bc8c28bcd5 | subsection | 52 | 121 | Conclusion and open problems | With (REF ) (REF ), we have close-formed expression on the final error:
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - x^*\right\Vert ^2\right] &= \left\Vert \operatorname{\mathrm {\mathbb {E}}}\left[x_T\right] - x^*\right\Vert ^2 + \operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - \operatorname{\... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.008079609833657742,
0.019943438470363617,
-0.025406138971447945,
-0.02381921000778675,
-0.0003204378008376807,
0.010543929412961006,
0.039581697434186935,
0.032532066106796265,
0.01306165475398302,
0.011169546283781528,
-0.017410453408956528,
0.029434500262141228,
0.015197906643152237,
... | |
a43ffdb0943a27fa0f53477f5eab175a4eee0ee5 | subsection | 53 | 121 | Conclusion and open problems | Since \left\langle e_i, e_j\right\rangle = 0 for i\ne j, we can simplify the last term in (REF ):
\left\Vert \sum _{t=1}^T (-1)^{\sigma (t)} \gamma (I-\gamma A)^{T-t} A b\right\Vert ^2 &= \left\Vert \sum _{t=1}^T (-1)^{\sigma (t)} \gamma (I-\gamma A)^{T-t} A (\sum _{i=1}^d b_ie_i)\right\Vert ^2\\
&= \left\Vert \sum _{... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.0420553982257843,
0.03497495502233505,
-0.015747884288430214,
-0.02301144413650036,
0.000318543694447726,
-0.02293514646589756,
0.016251450404524803,
0.04828130826354027,
0.0008578744600526989,
0.01683131605386734,
-0.011292087845504284,
0.03698921948671341,
-0.009819538332521915,
0.000... | |
5da8850efbf01ac6786af715ad6c4bb074f3f12a | subsection | 54 | 121 | Conclusion and open problems | Then (REF ) can simplified as
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - x^*\right\Vert ^2\right] &= \sum _{i=1}^d (1-\gamma \lambda _i)^{2T} a_i^2 + \gamma ^2 \sum _{i=1}^d b_i^2 \lambda _i^2 \operatorname{\mathrm {\mathbb {E}}}\left[\left[ \sum _{t=1}^T (-1)^{\sigma (t)} (1-\gamma \lambda _i)^{T-t}\r... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.05561777204275131,
0.02930382266640663,
-0.028891952708363533,
-0.01845790073275566,
0.020212164148688316,
0.005243721418082714,
0.008389954455196857,
0.03517679125070572,
0.02048674412071705,
0.033864907920360565,
-0.015727350488305092,
0.05314654856920242,
-0.008473854511976242,
0.011... | |
6550ccf93b0e39ed522327ee832ea73d967fce6d | subsection | 55 | 121 | Conclusion and open problems | For contradiction, we assume for any T, there is a \gamma dependent on T such that
\operatorname{\mathrm {\mathbb {E}}}\left[\left\Vert x_T - x^*\right\Vert ^2\right] \le o({1}{T}).
Now we determine the specific requirement of A and b. The only requirement is: A has at least three different positive eigenvalues \lambda... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.03445831686258316,
0.015260547399520874,
-0.032260797917842865,
-0.019655585289001465,
0.021929405629634857,
-0.0110486363992095,
0.018358439207077026,
0.01931985281407833,
0.014192309230566025,
0.012216067872941494,
-0.040043674409389496,
-0.009873573668301105,
-0.009446279145777225,
0... | |
8682fdcf12247823e2cc1b0426036b2a2bb56201 | subsection | 56 | 121 | Conclusion and open problems | Since (REF ), there is (1-\gamma \lambda _i)^{T} = o(1) , so
\gamma ^2\left[- \frac{1}{T-1} \frac{-2(1-\gamma \lambda _i)^T+(1-\gamma \lambda _i)^{2T}}{\gamma ^2\lambda _i^2}\right] = o(\frac{1}{T}).
Again, since |1-\gamma \lambda _1|<1, for i=2,3 there is
|\frac{\gamma ^2}{2\gamma \lambda _i - \gamma ^2\lambda _i^2}|\... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.06084893271327019,
0.03726005181670189,
-0.024290747940540314,
-0.02636583521962166,
0.00568742211908102,
-0.045560408383607864,
0.0065723867155611515,
0.0424172580242157,
-0.006862288806587458,
0.004764312878251076,
-0.04522473365068436,
0.003732489887624979,
-0.0260454174131155,
0.009... | |
32d1defbce188f9d782aa311ff83346d06b83cb4 | subsection | 57 | 121 | Conclusion and open problems | As a result, no step size can leads to convergence of o(\frac{1}{T}).
Proof of Theorem REF
The idea is similar to the proof of theorem REF , with a slightly different analysis on the R^t term adopting the sparsity parameter. For any i, we use H_i to denote H_i(x^*). Again, we have the following decomposition for the e... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.006781651172786951,
0.029262559488415718,
-0.04516015201807022,
0.0002382684324402362,
-0.0035815357696264982,
0.010771307162940502,
-0.013830296695232391,
0.019376147538423538,
0.021878262981772423,
0.014623651280999184,
-0.028835367411375046,
-0.00887183379381895,
0.0012424762826412916,... | |
3e3bf0e436a80349282d81f8495d739b8fe7aef9 | subsection | 58 | 121 | Conclusion and open problems | Here we define random variables
A^t = -\gamma \sum _{i=1}^n \left[H_{\sigma _t(i)}\sum _{j=1}^{i-1} \nabla f_{\sigma _t(j)} (x^t_0)\right],
B^t = - \gamma \sum _{i=1}^n \left\lbrace H_{\sigma _t(i)}\sum _{j=1}^{i-1}\left[\nabla f_{\sigma _t(j)} (x^t_{j-1})-\nabla f_{\sigma _t(j)} (x^t_0)\right]\right\rbrace ,
C^t = \su... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.032189708203077316,
-0.009878121316432953,
-0.02868087776005268,
0.001072672544978559,
-0.008344914764165878,
-0.020747870206832886,
0.00836779922246933,
0.019649453461170197,
-0.007006220053881407,
0.033044032752513885,
-0.04665219411253929,
0.030221713706851006,
0.013394580222666264,
... | |
c90f0a402ca7fb800f828686810fd636866d8f4c | subsection | 59 | 121 | Conclusion and open problems | Here the introduction of \rho in (REF ) is because: if f_{\sigma _t(k)} and f_{\sigma _t(j)} depend on disjoint dimensions of variables and k<j, then there must be \nabla f_{\sigma _t(j)} (x^t_k) = \nabla f_{\sigma _t(j)} (x^t_{k-1}). The introduction of \rho in (REF ) is similar: if f_{\sigma _t(i)} and f_{\sigma _t(j... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.018778612837195396,
0.0007732257945463061,
-0.025902587920427322,
0.02504832111299038,
0.002396904630586505,
0.011830068193376064,
0.04048614203929901,
0.020151542499661446,
-0.0021165984217077494,
0.025841567665338516,
-0.01604801043868065,
0.00964101031422615,
-0.03450627252459526,
0.... | |
2e3598ea9b81db1bf578d8b68cce00b325bff12c | subsection | 60 | 121 | Conclusion and open problems | For the first term in the (REF ), there is
&\ \ \ \ \gamma ^2 n(n-1) \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i \nabla f_j(x^t_0)\right\rangle \\
&= \gamma ^2 n(n-1)\operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} \left\langle H_i (x^t_0 - x^*), \nabla f_j(x^t_0) - \nabla ... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.055429618805646896,
0.018756333738565445,
-0.004586068447679281,
-0.009210290387272835,
-0.02946987748146057,
-0.006470858585089445,
-0.0005880430690012872,
0.04584542289376259,
-0.0013573160395026207,
0.017657507210969925,
-0.0346435122191906,
0.016375545412302017,
-0.0311638992279768,
... | |
9c4f3340de37aad517f238fcb5ff38bda0c864a0 | subsection | 61 | 121 | Conclusion and open problems | Where the last inequality is because of
\left\Vert H_i(x^t_0 - x^*) - (\nabla f_i(x^t_0) - \nabla f_i(x^*))\right\Vert &=\left\Vert H_i(x^t_0 - x^*) - \int _{x^*}^{x^t_0} H_i(x) dx\right\Vert \\
&=\left\Vert \int _{x^*}^{x^t_0} (H_i - H_i(x))dx\right\Vert \\
&\le \int _{0}^{\left\Vert x^t_0-x^*\right\Vert } \left\Vert... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.018445106223225594,
0.003827473847195506,
0.000636482029221952,
-0.023311927914619446,
-0.023800136521458626,
0.029582347720861435,
-0.009367489255964756,
0.04177229106426239,
0.012876483611762524,
-0.00045388081343844533,
-0.040063563734292984,
0.012456930242478848,
-0.007204880937933922... | |
ec05e0081d070d7fae27f0a298d7c2965d1e91ba | subsection | 62 | 121 | Conclusion and open problems | Substituting (REF ) (REF ) (REF ) back to (REF ), we get
-2\gamma \left\langle x^t_0 - x^*, \operatorname{\mathrm {\mathbb {E}}}\left[R^t\right] \right\rangle &\le \gamma ^2n^2 \left\Vert \nabla F(x^t_0)\right\Vert ^2 + \frac{1}{2}\gamma \mu (n-1)\left\Vert x^t_0-x^*\right\Vert ^2 + \gamma ^{3} \mu ^{-1}n^{2}(n-1)\lef... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.04340474680066109,
0.015589995309710503,
-0.020099876448512077,
-0.03556014597415924,
-0.019138379022479057,
-0.014926104806363583,
-0.0033461640123277903,
0.045114073902368546,
0.0024113748222589493,
-0.00186290149576962,
-0.03378976881504059,
0.021473444998264313,
-0.039131421595811844,... | |
2a7269b0559faaeac0aa9b97643760c9d967fc9f | subsection | 63 | 121 | Conclusion and open problems | Here the last inequality is because
\left\Vert R^t\right\Vert &= \left\Vert \sum _{i=1}^n \nabla f_{\sigma _t(i)} (x^t_{i-1}) - \sum _{i=1}^n \nabla f_{\sigma _t(i)}(x^t_0)\right\Vert \\
&\le \sum _{i=1}^n \left\Vert \nabla f_{\sigma _t(i)} (x^t_{i-1}) - \nabla f_{\sigma _t(i)}(x^t_0)\right\Vert \\
&= \sum _{i=1}^n \l... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.05330732837319374,
-0.015127136372029781,
-0.008589589037001133,
-0.009329545311629772,
0.003120019566267729,
0.054710954427719116,
0.0033412433695048094,
0.04290217533707619,
0.004485504701733589,
0.012167313136160374,
-0.02230546995997429,
-0.00034590071300044656,
0.019421931356191635,
... | |
30fcac426276c491886d14bb7a776b7265effb82 | subsection | 64 | 121 | Conclusion and open problems | Again, define error term
R^t = \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \sum _{i=1}^n \nabla f_{\sigma _t\left(i\right)}\left(x^t_0\right).
Assume F^* being the minimum of function F(\cdot ). For one epoch of RandomShuffle, we have
F(x^{t+1}_0) - F^* &\le F(x^t_0) - F^* - \gamma \left\... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.05161808803677559,
0.02029924839735031,
-0.018910352140665054,
-0.020848700776696205,
0.008157855831086636,
0.02594640664756298,
0.019383491948246956,
0.015193911269307137,
-0.004716141149401665,
0.0025965485256165266,
-0.030937274917960167,
-0.005410589277744293,
-0.016269924119114876,
... | |
48ecee59b99c6932edbe444d784366a063786ce3 | subsection | 65 | 121 | Conclusion and open problems | We have the following decomposition for the error term:
R^t &= \sum _{i=1}^n \left[\nabla f_{\sigma _t\left(i\right)} \left(x^t_{i-1}\right) - \nabla f_{\sigma _t\left(i\right)}\left(x^t_{0}\right)\right]\\
&= \sum _{i=1}^n \left[\int _{x^t_0}^{x^t_{i-1}} H_{\sigma _t\left(i\right)}\left(x\right) dx\right]\\
&= \sum _... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
0.011490252800285816,
0.008789356797933578,
-0.01725826784968376,
0.018433233723044395,
-0.002781007206067443,
-0.00651572085916996,
0.004623567685484886,
0.017227748408913612,
0.0016737544210627675,
0.008484170772135258,
-0.03448601812124252,
-0.003318897681310773,
-0.009979581460356712,
... | |
a0918220f3233e0b1cde6a0f466f58b16ce994ac | subsection | 66 | 121 | Conclusion and open problems | Here we define random variables
A^t = -\gamma \sum _{i=1}^n \left[H_{\sigma _t\left(i\right)}(x^t_0)\sum _{j=1}^{i-1} \nabla f_{\sigma _t\left(j\right)} \left(x^t_0\right)\right],
B^t = - \gamma \sum _{i=1}^n \left\lbrace H_{\sigma _t\left(i\right)}(x^t_0)\sum _{j=1}^{i-1}\left[\nabla f_{\sigma _t\left(j\right)} \left(... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.044100336730480194,
-0.0014925827272236347,
-0.021729717031121254,
-0.0047533754259347916,
-0.00006306519935606048,
-0.009438082575798035,
-0.001283716526813805,
0.029496869072318077,
-0.0001350715901935473,
0.023805025964975357,
-0.042360737919807434,
0.028688108548521996,
0.009155779145... | |
f872f45095a0ab080def0a2cc55b341c1a1c9767 | subsection | 67 | 121 | Conclusion and open problems | \left\Vert C^t\right\Vert &\le \sum _{i=1}^n \left[\int _{0}^{\left\Vert x^t_{i-1}-x^t_0\right\Vert } \left\Vert H_{\sigma _t\left(i\right)}\left(x^t_0+t\frac{x^t_{i-1}-x^t_0}{\left\Vert x^t_{i-1}-x^t_0\right\Vert }\right)-H_{\sigma _t\left(i\right)}(x^t_0)\right\Vert dt\right]\\
&\le \sum _{i=1}^n \left[L_H \left\Vert... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.012270919978618622,
0.019917352125048637,
-0.024740250781178474,
0.0009376799571327865,
0.00083609001012519,
-0.005765348672866821,
-0.012339601293206215,
0.022740885615348816,
-0.007428943179547787,
0.028143752366304398,
-0.06074410676956177,
0.005074727814644575,
0.0117214759811759,
-... | |
7e0bfcfd729e1bd86aec80034ae5664fb5419ad2 | subsection | 68 | 121 | Conclusion and open problems | For the first term in the (REF ), we have further bound:
&\ \ \ \ \frac{1}{2}\gamma ^2 n\left(n-1\right) \left\langle \nabla F(x^t_0), \operatorname{\mathrm {\mathbb {E}}}\limits _{i\ne j} H_i(x^t_0) \nabla f_j\left(x^t_0\right)\right\rangle \\
&= \frac{1}{2}\gamma ^2 n^2 \left\langle \nabla F(x^t_0), \operatorname{\m... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.048631586134433746,
0.018168199807405472,
0.004786123055964708,
0.0004216459929011762,
-0.01885465532541275,
-0.025887014344334602,
-0.019434329122304916,
0.03938731551170349,
-0.009587501175701618,
0.007055241148918867,
-0.0578758604824543,
0.005045450758188963,
-0.03307192027568817,
0... | |
301f86d4798c8239d1af74339429579400da998d | subsection | 69 | 121 | Conclusion and open problems | For the third term in (REF ), we use the bound
-\gamma \left\langle \nabla F(x^t_0), \operatorname{\mathrm {\mathbb {E}}}\left[C^t\right]\right\rangle &\le \gamma \left\Vert \nabla F(x^t_0)\right\Vert \cdot \left( n^3\gamma ^2L_H G^2 \right)\\
&= \gamma ^3 n^3 \left\Vert \nabla F(x^t_0)\right\Vert L_H G^2\\
&\le \frac... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
-0.044103704392910004,
0.02721000276505947,
-0.0044904896058142185,
-0.043706923723220825,
-0.013650783337652683,
-0.023226933553814888,
0.010751231573522091,
0.039464421570301056,
0.00040751099004410207,
0.009072544053196907,
-0.04068528488278389,
0.038182515650987625,
-0.009255673736333847... | |
f4ac45c00335d24c1753f0146159b7b162e6f56d | subsection | 70 | 121 | Conclusion and open problems | Now assume
\frac{1}{2}n\mu \gamma > 4L^2n^2\gamma ^2,
which we call assumption 1, (REF ) can be further turned into:
&\ \ \ \ \operatorname{\mathrm {\mathbb {E}}}[F(x^t_0) - F^*] \\
&\le (1-n\mu \gamma )\left[F(x^t_0) - F^* \right] + \gamma ^3 n C_1 + n^4\gamma ^4C_2 + n^5\gamma ^5C_3.
where C_1 =\frac{1}{2} \mu ^{... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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... | |
8c1897a8e6d414e70a1d2e581ec232c0b8d3461a | subsection | 71 | 121 | Conclusion and open problems | What remains to determine is to satisfy the two assumptions: (1) \frac{1}{2}n\mu \gamma > 4L^2n^2\gamma ^2, (2) n\gamma \mu <1.
The first is satisfied when
\frac{T}{\log T} > 16 \frac{L^2}{\mu ^2} n.
The second assumption is satisfied when
\frac{T}{\log T} > 2 n.
Since 2<\frac{L}{\mu }, the theorem is proved.
Proof of... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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... | |
0501852ce870cdae3a6e37eb677241f3cf931d2f | subsection | 72 | 121 | Conclusion and open problems | Therefore, taking expectation of (REF ) leads to:
\operatorname{\mathrm {\mathbb {E}}}[\left\Vert x^{t}_n - x^*\right\Vert ^2] \le \left\Vert x^t_0-x^*\right\Vert ^2 - (2n\gamma -2n^2\gamma ^2L)\left[F(x^t_0) - F(x^*)\right] -2\gamma \operatorname{\mathrm {\mathbb {E}}}\left\langle x^t_0 - x^*, R^t \right\rangle + 2\g... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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0.02... | |
45ca66ca7b74db8580e38a8d82644186383e02b0 | subsection | 73 | 121 | Conclusion and open problems | Then for i+1, there is
\left\Vert \nabla f_{id}(x^t_{i+1}) - \nabla f_{id}(x^t_0)\right\Vert &\le \left\Vert \nabla f_{id}(x^t_{i}) - \nabla f_{id}(x^t_0)\right\Vert + \left\Vert \nabla f_{id}(x^t_{i+1}) - \nabla f_{id}(x^t_i)\right\Vert \\
&\le \left\Vert \nabla f_{id}(x^t_{i}) - \nabla f_{id}(x^t_0)\right\Vert + L\g... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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-0.0542686134576797... | |
ff6ea079ed7cb8c31231179f5c391ee8461bf508 | subsection | 74 | 121 | Conclusion and open problems | Therefore, we have
\left\Vert \nabla f_{id}(x^t_i) - \nabla f_{id}(x^t_0)\right\Vert &\le \left[\sum _{j=0}^{i-1} (1+L\gamma )^j\right] L\gamma (\left\Vert \nabla F(x^t_0)\right\Vert + \delta ) \\
&\le \left[n (1+\frac{1}{n})^n\right] L\gamma (\left\Vert \nabla F(x^t_0)\right\Vert + \delta )\\
&\le 3nL\gamma (\left\Ve... | {
"cite_spans": []
} | 1806.10077 | Random Shuffling Beats SGD after Finite Epochs | [
"Jeff Z. HaoChen",
"Suvrit Sra"
] | [
"math.OC",
"stat.ML"
] | 2,018 | en | Mathematics | [
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