Original: On the Theoretical Properties of Noise Correlation in Stochastic Optimization
Revised: On the Theoretical Properties of Noise Correlation in Stochastic Optimization

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning.
Revised: Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the ﬁeld of machine learning.

Original: Priorwork [51, 46] has shown that the noise of stochastic gradient descent improvesoptimization by overcoming undesirable obstacles in the landscape.
Revised: Priorwork [51, 46] has shown that the noise of stochastic gradient descent improvesoptimization by overcoming undesirable obstacles in the landscape.

Original: Moreover,injecting artificial Gaussian noise has become a popular idea to quickly escapesaddle points.
Revised: Moreover,injecting artiﬁcial Gaussian noise has become a popular idea to quickly escapesaddle points.

Original: Indeed, in the absence of reliable gradient information, the noise isused to explore the landscape, but it is unclear what type of noise is optimal in termsof exploration ability.
Revised: Indeed, in the absence of reliable gradient information, the noise isused to explore the landscape, but it is unclear what type of noise is optimal in termsof exploration ability.

Original: In order to narrow this gap in our knowledge, we study ageneral type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated.
Revised: In order to narrow this gap in our knowledge, we study ageneral type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated.

Original: Thisgeneralizes processes based on Brownian motion, such as the Ornstein-Uhlenbeckprocess.
Revised: Thisgeneralizes processes based on Brownian motion, such as the Ornstein-Uhlenbeckprocess.

Original: We demonstrate how to discretize such processes which gives rise to thenew algorithm “fPGD”.
Revised: We demonstrate how to discretize such processes which gives rise to thenew algorithm “fPGD”.

Original: This method is a generalization of the known algorithms
Revised: This method is a generalization of the known algorithms

Original: PGD and Anti-PGD [33].
Revised: PGD and Anti-PGD [33].

Original: We study the properties of fPGD both theoretically andempirically, demonstrating that it possesses exploration abilities that, in some cases,are favorable over PGD and Anti-PGD.
Revised: We study the properties of fPGD both theoretically andempirically, demonstrating that it possesses exploration abilities that, in some cases,are favorable over PGD and Anti-PGD.

Original: These results open the field to novel waysto exploit noise for training machine learning models.
Revised: These results open the ﬁeld to novel waysto exploit noise for training machine learning models.

Original:  Introduction
Revised:  Introduction

Original: Injecting random noise to the parameters of a function has been a commonly used technique inoptimization, dating back to at least [28, 2].
Revised: Injecting random noise to the parameters of a function has been a commonly used technique inoptimization, dating back to at least [28, 2].

Original: More recently, [29] demonstrated the empirical benefitsof injecting Gaussian noise (a.k.a.
Revised: More recently, [29] demonstrated the empirical beneﬁtsof injecting Gaussian noise (a.k.a.

Original: Brownian motion noise) to the gradients while training deepnetworks.
Revised: Brownian motion noise) to the gradients while training deepnetworks.

Original: Injecting Gaussian noise has also been a common technique used to show faster escapefrom saddle points [14].
Revised: Injecting Gaussian noise has also been a common technique used to show faster escapefrom saddle points [14].

Original: Importantly, the noise of the optimizer also has an effect on the typeof minima that can be reached, raising some important connections to flatness and generalizationproperties [21].
Revised: Importantly, the noise of the optimizer also has an effect on the typeof minima that can be reached, raising some important connections to ﬂatness and generalizationproperties [21].

Original: A distinctive property of noise is its exploration ability which is crucial when thegradient of the objective function is not reliable, either due to saddles, valleys, plateaus, or localminima.
Revised: A distinctive property of noise is its exploration ability which is crucial when thegradient of the objective function is not reliable, either due to saddles, valleys, plateaus, or localminima.

Original: Despite a large body of work debating the distribution of the intrinsic noise of stochasticgradient descent (SGD) [40, 50, 42], there is a large gap in our knowledge regarding the role of noiseto escape undesirable regions.
Revised: Despite a large body of work debating the distribution of the intrinsic noise of stochasticgradient descent (SGD) [40, 50, 42], there is a large gap in our knowledge regarding the role of noiseto escape undesirable regions.

Original: In order to address this question, we investigate the behavior of a general type of non-Markovianstochastic process known as fractional Brownian motion (fBM).
Revised: In order to address this question, we investigate the behavior of a general type of non-Markovianstochastic process known as fractional Brownian motion (fBM).

Original: The latter is a generalization of
Revised: The latter is a generalization of

Original: Brownian motion where the increments need not be independent but can instead be positively ornegatively correlated.
Revised: Brownian motion where the increments need not be independent but can instead be positively ornegatively correlated.

Original: Interestingly, the processes that can be constructed from fBM recover knowninstances in the literature, including perturbed gradient descent (PGD) [51, 20], and anticorrelated
Revised: Interestingly, the processes that can be constructed from fBM recover knowninstances in the literature, including perturbed gradient descent (PGD) [51, 20], and anticorrelated

Original: PGD (Anti-PGD) [33].
Revised: PGD (Anti-PGD) [33].

Original: As we will see, the correlation between increments has a drastic effect of thebehavior of the optimizer, and neither PGD or Anti-PGD are optimal in all cases.
Revised: As we will see, the correlation between increments has a drastic effect of thebehavior of the optimizer, and neither PGD or Anti-PGD are optimal in all cases.

Original: Contribution.
Revised: Contribution.

Original: The first question we address is what is the effect of the noise correlation on theoverall magnitude of the process and its ability to explore the landscape?
Revised: The ﬁrst question we address is what is the effect of the noise correlation on theoverall magnitude of the process and its ability to explore the landscape?

Original: In many applications, thisis typically captured by ensuring a uniform control on the evolution of the random process.
Revised: In many applications, thisis typically captured by ensuring a uniform control on the evolution of the random process.

Original: Thiscan be achieved by bounding its expected supremum, which is connected to the concept of exit time,see Rmk. 2.
Revised: Thiscan be achieved by bounding its expected supremum, which is connected to the concept of exit time,see Rmk. 2.

Original: To conduct such an analysis, we will consider a well-known type of continuous-timestochastic process with drift known as the fractional Ornstein-Uhlenbeck (fOU) process, whichis connected to SGD (see Section 3).
Revised: To conduct such an analysis, we will consider a well-known type of continuous-timestochastic process with drift known as the fractional Ornstein-Uhlenbeck (fOU) process, whichis connected to SGD (see Section 3).

Original: We will also demonstrate how to discretize such a processto obtain a practical optimization algorithm that we name fPGD (for fractional PGD).
Revised: We will also demonstrate how to discretize such a processto obtain a practical optimization algorithm that we name fPGD (for fractional PGD).

Original: Concretely,our main result will be to provide upper and lower bounds on the expected supremum of the fOUprocess.
Revised: Concretely,our main result will be to provide upper and lower bounds on the expected supremum of the fOUprocess.

Original: These bounds will give a precise characterization of the effect of the noise correlation onthe magnitude of the process.
Revised: These bounds will give a precise characterization of the effect of the noise correlation onthe magnitude of the process.

Original: Second, we experimentally investigate the practical behavior of fBMnoise injection:
Revised: Second, we experimentally investigate the practical behavior of fBMnoise injection:

Original: In Section 5.1, we demonstrate on a bi-stable potential that fBM perturbations indeedhelp optimizers to move faster from one local minimum to another – as suggested by our theory.
Revised: In Section 5.1, we demonstrate on a bi-stable potential that fBM perturbations indeedhelp optimizers to move faster from one local minimum to another – as suggested by our theory.

Original: In Section 5.2, we demonstrate the existence of an optimization problem where non-Markovianversions of fPGD indeed outperform the known Markovian special cases (PGD and Anti-PGD) – afirst proof-of-concept for the practical utility of non-Markovian noise injection.
Revised: In Section 5.2, we demonstrate the existence of an optimization problem where non-Markovianversions of fPGD indeed outperform the known Markovian special cases (PGD and Anti-PGD) – aﬁrst proof-of-concept for the practical utility of non-Markovian noise injection.

Original:  Related work
Revised:  Related work

Original: Non-Markovian stochastic processes.
Revised: Non-Markovian stochastic processes.

Original: The vast majority of the work done in stochastic optimization for machine learning focuses on Markovian processes.
Revised: The vast majority of the work done in stochastic optimization for machine learning focuses on Markovian processes.

Original: Yet, non-Markovian processes areknown to appear naturally in many situations.
Revised: Yet, non-Markovian processes areknown to appear naturally in many situations.

Original: They for instance describe the widths of consecutiveannual rings of a tree or the values of the log return of a stock [6].
Revised: They for instance describe the widths of consecutiveannual rings of a tree or the values of the log return of a stock [6].

Original: In this paper, we are especiallyinterested in fractional Brownian Motion (fBM) as an example of a non-Markovian process.
Revised: In this paper, we are especiallyinterested in fractional Brownian Motion (fBM) as an example of a non-Markovian process.

Original: fBM is ageneralization of Brownian motion where the increments need not be independent but can instead bepositively or negatively correlated.
Revised: fBM is ageneralization of Brownian motion where the increments need not be independent but can instead bepositively or negatively correlated.

Original: The amount of correlation is captured by the Hurst parameter
Revised: The amount of correlation is captured by the Hurst parameter

Original: H ∈ (0 , 1) (a formal introduction to fBM is discussed in Section 3).
Revised: H 2 (0 , 1) (a formal introduction to fBM is discussed in Section 3).

Original: Mathematical techniques for non-Markovian processes are largely underdeveloped compared to their
Revised: Mathematical techniques for non-Markovian processes are largely underdeveloped compared to their

Original: Markovian counterparts.
Revised: Markovian counterparts.

Original: There are two sets of results that are of relevance to the problem discussedin this work.
Revised: There are two sets of results that are of relevance to the problem discussedin this work.

Original: The first set of results are escape times for fBM which are studied by [25, 4, 37, 41, 43],but they are restricted to simple 1-dimensional processes, and they are often applicable to specificproblems such as Kramer’s problem, instead of more general problems of interest.
Revised: The ﬁrst set of results are escape times for fBM which are studied by [25, 4, 37, 41, 43],but they are restricted to simple 1-dimensional processes, and they are often applicable to speciﬁcproblems such as Kramer’s problem, instead of more general problems of interest.

Original: Extending theseresults to multiple dimensions is difficult, and we are not aware of any existing result.
Revised: Extending theseresults to multiple dimensions is difﬁcult, and we are not aware of any existing result.

Original: The second results of relevance are bounds on the expected supremum of the stochastic process,sometimes referred to as maximal inequalities.
Revised: The second results of relevance are bounds on the expected supremum of the stochastic process,sometimes referred to as maximal inequalities.

Original: There is a large body of work focusing on the expectedsupremum of "pure" fBM (i.e., without any drift term), including [31, 27, 36].
Revised: There is a large body of work focusing on the expectedsupremum of "pure" fBM (i.e., without any drift term), including [31, 27, 36].

Original: For instance, [8]establish both upper and lower bounds on the expected supremum of fBM scaling as O (cid:16) 1 / √ H (cid:17) .
Revised: For instance, [8]establish both upper and lower bounds on the expected supremum of fBM scaling as O ⇣ 1 / p H ⌘ .

Original: Some limited results exist with time-dependent drift terms in the 1-dimensional case, e.g. [35] givesan asymptotic result for H ∈ (cid:0) 12 , 1 (cid:1) .
Revised: Some limited results exist with time-dependent drift terms in the 1-dimensional case, e.g. [35] givesan asymptotic result for H 12 , .

Original: In the multi-dimensional case, [23] derive lower and upperbound for the distribution of fBM in the multi-dimensional case, see Rmk. 4.
Revised: In the multi-dimensional case, [23] derive lower and upperbound for the distribution of fBM in the multi-dimensional case, see Rmk. 4.

Original: In this work, our goalwill be to provide sharp bounds for stochastic processes with more general drift terms than prior workand in the multi-dimensional case.
Revised: In this work, our goalwill be to provide sharp bounds for stochastic processes with more general drift terms than prior workand in the multi-dimensional case.

Original: Brownian motion and Ornstein-Uhlenbeck process.
Revised: Brownian motion and Ornstein-Uhlenbeck process.

Original: A well-studied type of stochastic processis Brownian motion (BM), which is a special case of fBM (corresponding to H = 12 ) where theincrements are not correlated.
Revised: A well-studied type of stochastic processis Brownian motion (BM), which is a special case of fBM (corresponding to H = 12 ) where theincrements are not correlated.

Original: When also considering a stochastic process with a drift, we obtainthe well-known Ornstein-Uhlenbeck process, which has been the subject of a large body of work,including both results on the exit time [1, 9] as well as on the expected supremum [15, 19].
Revised: When also considering a stochastic process with a drift, we obtainthe well-known Ornstein-Uhlenbeck process, which has been the subject of a large body of work,including both results on the exit time [1, 9] as well as on the expected supremum [15, 19].

Original: Finally,[10] also derived a maximal inequality for a multi-dimensional Ornstein-Uhlenbeck process, althoughtheir result is restricted to the radial case.
Revised: Finally,[10] also derived a maximal inequality for a multi-dimensional Ornstein-Uhlenbeck process, althoughtheir result is restricted to the radial case.

Original: SGD & Noise Injection.
Revised: SGD & Noise Injection.

Original: The distribution of the noise of SGD is often a subject of debate in theliterature.
Revised: The distribution of the noise of SGD is often a subject of debate in theliterature.

Original: In some cases, the noise of SGD appears to be heavy-tailed [34, 40, 16], which yields fasterexit from sharp to flat minima [30].
Revised: In some cases, the noise of SGD appears to be heavy-tailed [34, 40, 16], which yields fasterexit from sharp to ﬂat minima [30].

Original: Noise is also often added to (stochastic) gradient descent.
Revised: Noise is also often added to (stochastic) gradient descent.

Original: Forinstance, Perturbed Gradient Descent (PGD) is a version of GD where artificial noise is added to theparameters after every step.
Revised: Forinstance, Perturbed Gradient Descent (PGD) is a version of GD where artiﬁcial noise is added to theparameters after every step.

Original: Multiple PGD methods have been shown to help quickly escape spuriouslocal minima [51] and saddle points [20].
Revised: Multiple PGD methods have been shown to help quickly escape spuriouslocal minima [51] and saddle points [20].

Original: Another way to add perturbations to SGD is to add noise tothe labels of the data used for training.
Revised: Another way to add perturbations to SGD is to add noise tothe labels of the data used for training.

Original: Recent work has demonstrated that such perturbations areindeed beneficial for generalization by implicitly regularizing the loss [7, 17, 12].
Revised: Recent work has demonstrated that such perturbations areindeed beneﬁcial for generalization by implicitly regularizing the loss [7, 17, 12].

Original: [Figure]
Revised: [Figure]

Original:  Background
Revised:  Background

Original: Fractional Brownian Motion.
Revised: Fractional Brownian Motion.

Original: Fractional Brownian motion (fBM) is a generalization of Brownianmotion where the increments of the stochastic process need not be independent.
Revised: Fractional Brownian motion (fBM) is a generalization of Brownianmotion where the increments of the stochastic process need not be independent.

Original: This family of
Revised: This family of

Original: Gaussian processes was developed for some hydrological modelling by [22].
Revised: Gaussian processes was developed for some hydrological modelling by [22].

Original: [18] later studied thelong term water flow characteristics of the Nile River that was affected by long periods of drought.
Revised: [18] later studied thelong term water ﬂow characteristics of the Nile River that was affected by long periods of drought.

Original: Insuch a scenario, the self similarity of the process is related to a long term dependence in the waterlevels.
Revised: Insuch a scenario, the self similarity of the process is related to a long term dependence in the waterlevels.

Original: Since then, fBM has also found applications in financial modeling [32].
Revised: Since then, fBM has also found applications in ﬁnancial modeling [32].

Original: We now give a formaldefinition of fBM, and discuss its main properties.
Revised: We now give a formaldeﬁnition of fBM, and discuss its main properties.

Original: Definition 1.
Revised: Deﬁnition 1.

Original: Fractional Brownian motion (fBM) is a centered Gaussian process ( B H ( t ))
Revised: Fractional Brownian motion (fBM) is a centered Gaussian process ( B H ( t ))

Original: t ∈ [0 ,T ] for
Revised: t 2 [0 ,T ] for

Original: T > 0 with covariance function
Revised: T > 0 with covariance function

Original: [Equation]
Revised: [Equation]

Original: where H ∈ (0 , 1) is called the Hurst parameter .
Revised: where H 2 (0 , 1) is called the Hurst parameter .

Original: Hence, fBM is a generalization of Brownian motion, which is recovered for H = 12 .
Revised: Hence, fBM is a generalization of Brownian motion, which is recovered for H = 12 .

Original: If H > 12 , thenthe increments are positively correlated, while they are negatively correlated if H < 12 .
Revised: If H > 12 , thenthe increments are positively correlated, while they are negatively correlated if H < 12 .

Original: For H ̸ = 12 ,the increments of fBM are not independent, although they are stationary, with variance
Revised: For H 6 = 12 ,the increments of fBM are not independent, although they are stationary, with variance

Original: [Equation]
Revised: [Equation]

Original: A crucial property of negatively correlated increments is their ability to fill in the space, as illustratedin Fig. 1; see Rmk. 2.
Revised: A crucial property of negatively correlated increments is their ability to ﬁll in the space, as illustratedin Fig. 1; see Rmk. 2.

Original: Fractional Ornstein-Uhlenbeck (OU) process.
Revised: Fractional Ornstein-Uhlenbeck (OU) process.

Original: We conduct our analysis on the fractional
Revised: We conduct our analysis on the fractional

Original: Ornstein-Uhlenbeck process, which can be thought of as a generalization of the well-known
Revised: Ornstein-Uhlenbeck process, which can be thought of as a generalization of the well-known

Original: Ornstein-Uhlenbeck process that has already found many applications in the field of machine learning [7, 24, 44].
Revised: Ornstein-Uhlenbeck process that has already found many applications in the ﬁeld of machine learning [7, 24, 44].

Original: Formally, this process is a d -dimensional stochastic process X Ht ∈ R d defined by
Revised: Formally, this process is a d -dimensional stochastic process X Ht 2 R d deﬁned by

Original: [Equation]
Revised: [Equation]

Original: where A ∈ R d × d and B Ht =
Revised: where A 2 R d ⇥ d and B Ht =

Original: ( B H, 1 t , . . .
Revised: ( B H, 1 t , . . .

Original: B H,dt ) is a fBM with H ∈ (0 , 1) .
Revised: B H,dt ) is a fBM with H 2 (0 , 1) .

Original: For further informationregarding the fractional Ornstein-Uhlenbeck process, we refer the reader to [11, 26, 3].
Revised: For further informationregarding the fractional Ornstein-Uhlenbeck process, we refer the reader to [11, 26, 3].

Original: We will forinstance later make use of the covariance of this process that has been derived by [26, 3].
Revised: We will forinstance later make use of the covariance of this process that has been derived by [26, 3].

Original: The fractional Ornstein-Uhlenbeck process is relevant for optimization purposes as it corresponds tooptimizing a quadratic function of the form f ( x ) = 12 x ⊤ Ax .
Revised: The fractional Ornstein-Uhlenbeck process is relevant for optimization purposes as it corresponds tooptimizing a quadratic function of the form f ( x ) = 12 x > Ax .

Original: Indeed, the gradient of the function∇ f ( x ) =
Revised: Indeed, the gradient of the functionr f ( x ) =

Original: Ax then corresponds to the drift term in Eq.
Revised: Ax then corresponds to the drift term in Eq.

Original: We also note that when the stochasticprocess reaches stationarity, the stationary distribution of the iterates is known to be approximated bya quadratic function [24, 39].
Revised: We also note that when the stochasticprocess reaches stationarity, the stationary distribution of the iterates is known to be approximated bya quadratic function [24, 39].

Original: This implies that such processes can be used to analyze the behavior ofcomplex functions at stationarity, but also close to saddle points where quadratic functions often playa crucial role to analyze the escape behavior of gradient descent methods [20].
Revised: This implies that such processes can be used to analyze the behavior ofcomplex functions at stationarity, but also close to saddle points where quadratic functions often playa crucial role to analyze the escape behavior of gradient descent methods [20].

Original: Discretization by fractional Perturbed Gradient Descent (fPGD).
Revised: Discretization by fractional Perturbed Gradient Descent (fPGD).

Original: For standard Brownian motion
Revised: For standard Brownian motion

Original: B t (i.e. for Hurst parameter H = 1 / 2 ), the discretization of the continuous-time SDE dX t =−∇ f
Revised: B t (i.e. for Hurst parameter H = 1 / 2 ), the discretization of the continuous-time SDE dX t =r

Original:  ( X ) dt + σdB t of any optimization problem argmin x f ( x ) , is written
Revised: f ( X ) dt +  dB t of any optimization problem argmin x f ( x ) , is written

Original: [Equation]
Revised: [Equation]

Original: with step size η > 0 .
Revised: with step size ⌘ > 0 .

Original: This method is called perturbed gradient descent (PGD) [20].
Revised: This method is called perturbed gradient descent (PGD) [20].

Original: Here, byinstead discretizing the fractional SDE dX t = 
Revised: Here, byinstead discretizing the fractional SDE dX t = r

Original: −∇ f ( X ) dt + σdB Ht for any H ∈ (0 , 1) , we obtain ageneralized version of PGD, called fractional PGD (fPGD):
Revised: f ( X ) dt +  dB Ht for any H 2 (0 , 1) , we obtain ageneralized version of PGD, called fractional PGD (fPGD):

Original: [Equation]
Revised: [Equation]

Original: which is equal to PGD if H = 1 / 2 .
Revised: which is equal to PGD if H = 1 / 2 .

Original: The parameter H determines the correlation of noise in fPGD:anticorrelated noise for H ∈ (0 , 1 / 2) , positively correlated noise for H ∈ (1 / 2 , 1) , and uncorrelated(independent) noise for H = 1 / 2 .
Revised: The parameter H determines the correlation of noise in fPGD:anticorrelated noise for H 2 (0 , 1 / 2) , positively correlated noise for H 2 (1 / 2 , 1) , and uncorrelated(independent) noise for H = 1 / 2 .

Original: Importantly – since B Ht converges to a white-noise process (indistribution) as H → 0 ; see Lemma 4.1 by [8] – the method Anti-PGD [33, Section 2] can beconsidered as the limit case of fPGD as H ↓ 0 .
Revised: Importantly – since B Ht converges to a white-noise process (indistribution) as H ! 0 ; see Lemma 4.1 by [8] – the method Anti-PGD [33, Section 2] can beconsidered as the limit case of fPGD as H # 0 .

Original: Hence, fPGD with H ∈ (0 , 1 / 2) interpolates between
Revised: Hence, fPGD with H 2 (0 , 1 / 2) interpolates between

Original: PGD ( H = 1 / 2 ) and Anti-PGD ( H = 0 ).
Revised: PGD ( H = 1 / 2 ) and Anti-PGD ( H = 0 ).

Original: Positive correlations ( H ∈ (1 / 2 , 1) ) have not been shownto be useful yet, and indeed the below theorems and experiments will indicate that a small H yieldsquicker escapes from difficult regions such as saddle points and local minima.
Revised: Positive correlations ( H 2 (1 / 2 , 1) ) have not been shownto be useful yet, and indeed the below theorems and experiments will indicate that a small H yieldsquicker escapes from difﬁcult regions such as saddle points and local minima.

Original: While the next sectionwill analyze the continuous-time model based on fBM, we will return to the optimizer fPGD in theexperiments (Section 5) where we will illustrate how the theoretical properties relate to optimization,and why H might, in some settings, be a useful tuning parameter for perturbed-gradient methods.
Revised: While the next sectionwill analyze the continuous-time model based on fBM, we will return to the optimizer fPGD in theexperiments (Section 5) where we will illustrate how the theoretical properties relate to optimization,and why H might, in some settings, be a useful tuning parameter for perturbed-gradient methods.

Original:  Analysis
Revised:  Analysis

Original: 4.1 Preliminaries
Revised: 4.1 Preliminaries

Original: Our analysis relies on a technique known as chaining that provides a general framework that can beused to prove uniform bounds on a random process.
Revised: Our analysis relies on a technique known as chaining that provides a general framework that can beused to prove uniform bounds on a random process.

Original: The main tools we use are Dudley’s
Revised: The main tools we use are Dudley’s

Original: inequalityand
Revised: inequalityand

Original: Sudakov’s inequality, which give upper and lower bounds on E [sup t ∈ T X t ] in terms of the metricentropy of T denoted by log N ( T, d, ϵ ) .
Revised: Sudakov’s inequality, which give upper and lower bounds on E [sup t 2 T X t ] in terms of the metricentropy of T denoted by log N ( T, d, ✏ ) .

Original: Both inequalities are stated in Appendix A.1.
Revised: Both inequalities are stated in Appendix A.1.

Original: We will consider processes that have sub-Gaussian increments as defined below.
Revised: We will consider processes that have sub-Gaussian increments as deﬁned below.

Original: For that, werecall the definition of the sub-Gaussian norm of a random variable X , denoted by ∥ X ∥ ψ 2 , as∥ X ∥ ψ 2 = inf { t > 0 : E [exp( X 2 /t 2 )] ≤ 2 } .
Revised: For that, werecall the deﬁnition of the sub-Gaussian norm of a random variable X , denoted by k X k 2 , ask X k 2 = inf { t > 0 : E [exp( X 2 /t 2 )]  2 } .

Original: Definition 2.
Revised: Deﬁnition 2.

Original: A random process ( X t ) t ∈ T defined on a metric space ( T, d ) has sub-Gaussian increments if there exists K [-≥ 
-] such that ∥ X t − X s ∥
Revised: A random process ( X t ) t 2 T deﬁned on a metric space ( T, d ) has sub-Gaussian increments if there exists K [-≥ 
-] 0 such that k X t  X s k 2 

Original: ψ 2 ≤ Kd ( t, s ) ∀ t, s ∈ T .
Revised: Kd ( t, s ) 8 t, s 2 T .

Original: 4.2 One-dimensional case
Revised: 4.2 One-dimensional case

Original: In this section, we derive both upper and lower bounds on the expected supremum of the stochasticprocess defined in Eq.
Revised: In this section, we derive both upper and lower bounds on the expected supremum of the stochasticprocess deﬁned in Eq.

Original: Since the analysis is rather technical, we first present the main ideas in the1-dimensional case.
Revised: Since the analysis is rather technical, we ﬁrst present the main ideas in the1-dimensional case.

Original: We then present the general result for the d -dimensional case.
Revised: We then present the general result for the d -dimensional case.

Original: Theorem 1 (Upper bound, 1-D case) .
Revised: Theorem 1 (Upper bound, 1-D case) .

Original: Consider the stochastic process
Revised: Consider the stochastic process

Original: X Ht defined by
Revised: X Ht deﬁned by

Original: [Equation]
Revised: [Equation]

Original: with X 0 = 0 , σ ∈ R and | β | < 1 .
Revised: with X 0 = 0 ,  2 R and |  | < 1 .

Original: Then, for t ∈ [0 , 1] ,
Revised: Then, for t 2 [0 , 1] ,

Original: [Equation]
Revised: [Equation]

Original: where C β = e 2 β 1 − β and C are positive constants (independent of H ).
Revised: where C  = e 2   and C are positive constants (independent of H ).

Original: Theorem 2 (Lower bound, 1-D case) .
Revised: Theorem 2 (Lower bound, 1-D case) .

Original: Consider the stochastic process
Revised: Consider the stochastic process

Original: X Ht defined by
Revised: X Ht deﬁned by

Original: [Equation]
Revised: [Equation]

Original: , with X 0 = 0 , σ ∈ R and | β | < 1 .
Revised: with X 0 = 0 ,  2 R and |  | < 1 .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where C β = e − 2 | β | and c are positive constants (independent of H ).
Revised: where C  = e  2 |  | and c are positive constants (independent of H ).

Original: Note that both upper and lower bounds scale with O (cid:16) 1 √ H (cid:17) , which implies the bound is tight andcan not be improved in terms of H .
Revised: Note that both upper and lower bounds scale with O ⇣ 1 p H ⌘ , which implies the bound is tight andcan not be improved in terms of H .

Original: However, the lower bound requires σ > e 2 | β |− 1 / 2 (large-noiseregime) to be non-vacuous.
Revised: However, the lower bound requires  > e 2 |  |  1 / 2 (large-noiseregime) to be non-vacuous.

Original: We can also derive a lower bound for the small-noise regime as follows.
Revised: We can also derive a lower bound for the small-noise regime as follows.

Original: Corollary 3 (Small noise regime) .
Revised: Corollary 3 (Small noise regime) .

Original: In the case where 0 < σ < e 2 | β |− 12 , we observe that X Ht =
Revised: In the case where 0 <  < e 2 |  |  12 , we observe that X Ht =

Original: σY
Revised:  Y

Original: Ht ,where dY Ht = − βY
Revised: Ht ,where dY Ht =   Y

Original: Ht dt + dB
Revised: Ht dt + dB

Original: Ht , Y H 

Revised: Ht , Y H

Original: = 0 .
Revised: = 0 .

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: We thus obtain a lower bound for | β | ≤ 14 .
Revised: We thus obtain a lower bound for |  |  14 .

Original: In the general case 14 < | β | < 1 , one gets principally alower bound in terms of entropy numbers (Thm. 7) or by means of so called Talagrand functionals (seee.g.
Revised: In the general case 14 < |  | < 1 , one gets principally a 161 lower bound in terms of entropy numbers (Thm. 7) or by means of so called Talagrand functionals (seee.g.

Original: However, the latter quantities cannot be computed explicitly, in general.
Revised: However, the latter quantities cannot be computed explicitly, in general.

Original: In Appendix B.1we however provide experiments that demonstrate the 1 / √ H in this case, for the example of β = 12 .
Revised: In Appendix B.1we however provide experiments that demonstrate the 1 / p H in this case, for the example of  = 12 .

Original: Remark 1 (Extension to any finite interval) .
Revised: Remark 1 (Extension to any ﬁnite interval) .

Original: Although the theorems are stated for the interval
Revised: Although the theorems are stated for the interval

Original: T = [0 , 1] , one can extend the results to an arbitrary finite interval using the self-similarity property offBM, namely B Hat d = a H B Ht for a > 0 .
Revised: T = [0 , 1] , one can extend the results to an arbitrary ﬁnite interval using the self-similarity property offBM, namely B H at d = a H B H t for a > 0 .

Original: This implies that E(cid:104) sup s ∈ [
Revised: This implies that E

Original: [Equation]
Revised: [Equation]

Original: ∈ [0 , 1] B Hs (cid:105) .

Original: In the multi-dimensional case, the same extension works by the independence of the dimensions.
Revised: In the multi-dimensional case, the same extension works by the independence of the dimensions.

Original: 4.3 Multi-dimensional case
Revised: 4.3 Multi-dimensional case

Original: We now generalize the bounds to the multi-dimensional case.
Revised: We now generalize the bounds to the multi-dimensional case.

Original: The quantity of interest becomes theexpected supremum of the norm, or more precisely E (cid:104) sup t ∈ [0 , 1] ∥ X
Revised: The quantity of interest becomes theexpected supremum of the norm, or more precisely E h sup t 2 [0 , 1] k X

Original: Ht − E
Revised: Ht  E

Original: [ X Ht ] ∥ (cid:105) .
Revised: [ X Ht ] k i .

Original: Similarly tothe 1-dimensional case, we apply Sudakov’s inequality to the Gaussian component of the multidimensional fOU process
Revised: Similarly tothe 1-dimensional case, we apply Sudakov’s inequality to the Gaussian component of the multidimensional fOU process

Original: X Ht 1 .
Revised: X Ht 1 .

Original: Theorem 4 (Upper bound, d -dimensional case) .
Revised: Theorem 4 (Upper bound, d -dimensional case) .

Original: Consider the d -dimensional stochastic process X Ht ∈ R d defined by
Revised: Consider the d -dimensional stochastic process X Ht 2 R d deﬁned by

Original: [Equation]
Revised: [Equation]

Original: where A ∈ R d × d and B Ht =
Revised: where A 2 R d ⇥ d and B Ht =

Original: ( B H, 1 t , .. .
Revised: ( B H, 1 t , .

Revised: [Equation]

Original: B H,dt ) is a fBM with H ∈ (0 , 1) .
Revised: B H,dt ) is a fBM with H 2 (0 , 1) .

Original: Denote by a ij ( t ) the matrix entries of exp( tA ) A .
Revised: Denote by a ij ( t ) the matrix entries of exp( tA ) A .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where C 2 = max di,j =1 sup t ∈ [0 , 1] | a ij ( t ) | 2 + max di,j =1 sup t ∈ [0 , 1] | a ij ( t ) | + 1 and C > 0 is a constant independent of H .
Revised: where C 2 = max di,j =1 sup t 2 [0 , 1] | a ij ( t ) | 2 + max di,j =1 sup t 2 [0 , 1] | a ij ( t ) | + 1 and C > 0 is a constant independent of H .

Original: Theorem 5 (Lower bound, d -dimensional case) .
Revised: Theorem 5 (Lower bound, d -dimensional case) .

Original: Consider the d -dimensional stochastic process X Ht ∈ R d defined by
Revised: Consider the d -dimensional stochastic process X Ht 2 R d deﬁned by

Original: [Equation]
Revised: [Equation]

Original: where A ∈ R d × d and B Ht =
Revised: where A 2 R d ⇥ d and B Ht =

Original: ( B H, 1 t , . . .
Revised: ( B H, 1 t , . . .

Original: B H,dt ) is a fBM with H ∈ (0 , 12 ) .
Revised: B H,dt ) is a fBM with H 2 (0 , 12 ) .

Original: Denote by a ij ( t ) the matrix entries of exp( tA )
Revised: Denote by a ij ( t ) the matrix entries of exp( tA )

Original: A and assume that there exists i 0 ∈ { 1 , . . .
Revised: A and assume that there exists i 0 2 { 1 , . . .

Original: d } such that min t ∈ [0 , 1] a i 0 i 0 ( t ) ≥ − 1 + C 1 , where 0 < C 1 < 1 .
Revised: d } such that min t 2 [0 , 1] a i 0 i 0 ( t )   1 + C 1 , where 0 < C 1 < 1 .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where c is a constant independent of H .
Revised: where c is a constant independent of H .

Original: We obtain an upper bound of the form O (cid:16) 1 √ H (cid:17) with a matching lower bound, although we dorequire a condition on the entries of the matrix A , which we discuss further in Example 1 below.
Revised: We obtain an upper bound of the form O ⇣ 1 p H ⌘ with a matching lower bound, although we dorequire a condition on the entries of the matrix A , which we discuss further in Example 1 below.

Original: Example 1 (Assumption lower bound) .
Revised: Example 1 (Assumption lower bound) .

Original: Let A = diag ( λ 1 , ..., λ d ) .
Revised: Let A = diag (  1 , ...,  d ) .

Original: Then e tA A = diag ( e tλ 1 λ 1 , ...,e tλ d λ d ) .
Revised: Then e tA A = diag ( e t  1  1 , ...,e t  d  d ) .

Original: If some diagonal element of A is non-negative, then the assumptions of the above theoremare satisfied.
Revised: If some diagonal element of A is non-negative, then the assumptions of the above theoremare satisﬁed.

Original: Suppose that all entries λ 1 , ..., λ d are negative and that λ i 0 > − 1 , for some i 0 .
Revised: Suppose that all entries  1 , ...,  d are negative and that  i 0 >  1 , for some i 0 .

Original: Thenmin t ∈ [0 , 1] a i 0 i 0 ( t ) =
Revised: Thenmin t 2 [0 , 1] a i 0 i 0 ( t ) =

Original: min t ∈ [0 , 1] ( e tλ i 

Revised: min t 2 [0 , 1] ( e t  i

Original: λ i 0 ) ≥ λ i 0 > − 1 + C 1 for some C 1 ∈ (0 , 1) .
Revised:  i 0 )   i 0 >  1 + C 1 for some C 1 2 (0 , 1) .

Original: Hence, if some 181 diagonal element λ i 0 > − 1 exists, then the conditions of the previous theorem are fulfilled.
Revised: Hence, if some 181 diagonal element  i 0 >  1 exists, then the conditions of the previous theorem are fulﬁlled.

Original: We make a few comments regarding the meaning of the bound, as well as some extensions.
Revised: We make a few comments regarding the meaning of the bound, as well as some extensions.

Original: First of all, the theorems discussed above bound the expected supremum of the process.
Revised: First of all, the theorems discussed above bound the expected supremum of the process.

Original: We nowexplain how these are connected to what we call "exploration ability".
Revised: We nowexplain how these are connected to what we call "exploration ability".

Original: Remark 2 (Exploration ability) .
Revised: Remark 2 (Exploration ability) .

Original: We give three different justifications for the concept of exploration.
Revised: We give three different justiﬁcations for the concept of exploration.

Original: 1) Expected behavior: Based on Theorem 5, we have
Revised: 1) Expected behavior: Based on Theorem 5, we have

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where c, C 1 are constant independent of H .
Revised: where c, C 1 are constant independent of H .

Original: Since E X
Revised: Since E X

Original: Ht solves the ODE ϕ ( t ) =
Revised: Ht solves the ODE  ( t ) =

Original: ξ +[-(cid:82) t 
-]
Revised: ⇠ +[-(cid:82) t 
-]

Original: Aϕ{+R t 
 A +} ( s ) ds , the quantity a :=sup t ∈
Revised: {+R t 
 A +} ( s ) ds , the quantity a :=

Original: [Equation] 0 , 1] ∥ E
Revised: sup t 2 [0 , 1] k E

Original: [ X Ht ] ∥ < ∞is independent of H .
Revised: [ X Ht ] k < 1is independent of H .

Original: Thus E (cid:104) sup t ∈ [0 , 1] ∥
Revised: Thus E h [Equation]

Original: X Ht ∥ (cid:105) ≥ ce−(cid:114)
Revised: ce

Original: H log (cid:16) 2 C 1 e − 12 (cid:17) − a .
Revised: [Equation] e  12 ⌘  a .

Original: Hence, for a given
Revised: Hence, for a given

Original: H1 ∈ (cid:0) 0 , 12 (cid:1)
Revised: H

Original:  , there is an H < H 1 such that E(cid:104) sup t ∈ [0 , 1] ∥
Revised: [Equation]  0 , , there is an H < H 1 such that E

Original: X Ht ∥ (cid:105) > E (cid:104) sup t ∈ [0 , 1] ∥ X H 1 t ∥ (cid:105) .
Revised: [Equation]

Original: We conclude that the average maximal distance of X Ht from the origin is larger than that of X H 1 t .
Revised: We conclude that the average maximal distance of X Ht from the origin is larger than that of X H 1 t .

Original: This is what we mean by the process having a better exploration ability.
Revised: This is what we mean by the process having a better exploration ability.

Original: Better exploration, in thissense, can help to escape suboptimal local minima more quickly (experiments in Section 5.1) or toexplore the space around a saddle point for an exit (experiments in Section 5.2).2)
Revised: Better exploration, in thissense, can help to escape suboptimal local minima more quickly (experiments in Section 5.1) or toexplore the space around a saddle point for an exit (experiments in Section 5.2).2)

Original: Exit time: First, note that E (cid:104) sup
Revised: Exit time: First, note that E [Equation]

Original: t ∈ ∥ X Ht ∥ (cid:105) =
Revised: ⌘ ds , therefore

Original: (cid:82)
Revised: [Equation]

Original: ∞ P (cid:16) sup t ∈ [0 , 1] ∥ X Ht ∥ > s (cid:17) ds , therefore(cid:82) ∞
Revised: Thus, we can ﬁnd a s 0 > 0 such that P

Original: 0 (cid:110)
Revised: [Equation]

Original: P (cid:16) sup t ∈ [0 , 1] ∥ X Ht ∥ >  s (cid:17) − P
Revised: ⌘ >0 and, since probability distributions are right-continuous, there exists ✏ > 0 such that for alls 2 [ s 0 , s 0 + ✏ ) , P

Original: (cid:16) sup t ∈ [0 , 1]∥ X
Revised: [Equation] sup t 2 [0 , 1]

Original: H 1 t ∥ > s (cid:17)(cid:111) ds
Revised: [Equation]

Original: > 0 . Thus, we can find a s 0 > 0 such that P (cid:16) sup t ∈ [0 , 1] ∥ X Ht ∥ > s 0 (cid:17) − P (cid:16) sup t ∈ [0 , 1] ∥ X H 1 t ∥ > s 0 (cid:17) >0 and, since probability distributions are right-continuous, there exists ϵ > 0 such that for alls ∈ [ s 0 , s 0 + ϵ ) , P (cid:16) sup t ∈ [0 , 1] ∥ X Ht ∥ > s (cid:17) > P (cid:16) sup t ∈ [0 , 1] ∥ X H 1 t ∥ > s (cid:17) > 0 .
Revised: ⌘ > 0 .

Original: On the other hand,
Revised: On the other hand,

Original: P (cid:16) sup t ∈ [0 , 1] ∥ X Ht ∥ > s (cid:17) =
Revised: P [Equation]

Original: P ( τ Hs < 1) where τ Hs denotes the first exit time of X H from theball B s (0) of radius s .
Revised: Hs < 1) where ⌧ Hs denotes the ﬁrst exit time of X H from theball B s (0) of radius s .

Original: We conclude that P ( τ Hs < 1) > P τ H 1 s < 1) ∀ s ∈ [ s 0 , s 0 + ϵ ) , i.e. theprobability of an exit before time s of H is larger than that of H 1 .3)
Revised: We conclude that P ( ⌧ Hs < 1) > P ( ⌧ H 1 s < 1) 8 s 2 [ s 0 , s 0 + ✏ ) , i.e. theprobability of an exit before time s of H is larger than that of H 1 .3)

Original: Space-filling property: We note that fractional Brownian motion with low Hurst parameter H fillsthe space more densely.
Revised: Space-ﬁlling property: We note that fractional Brownian motion with low Hurst parameter H ﬁllsthe space more densely.

Original: This behavior comes from the fact that the graph of the fBM has a Hausdorffdimension and box-counting dimension given by d = 2 − H .
Revised: This behavior comes from the fact that the graph of the fBM has a Hausdorffdimension and box-counting dimension given by d = 2  H .

Original: The latter implies that N (1 /n ) ≈
Revised: The latter implies that N (1 /n ) ⇡

Original: Cn dfor large n , where N ( ε ) denotes the number of boxes with side length ε , which is needed to cover thegraph of the fBM.
Revised: Cn dfor large n , where N ( " ) denotes the number of boxes with side length " , which is needed to cover thegraph of the fBM.

Original: Therefore, the graph of a (1-dimensional) fBM "behaves" like a 2 − dimensionalsmooth manifold for small H .
Revised: Therefore, the graph of a (1-dimensional) fBM "behaves" like a 2  dimensionalsmooth manifold for small H .

Original: See [38].
Revised: See [38].

Original: The above claim about the Hausdorff dimension and thebox-filling dimension of fBM is proved by [47].
Revised: The above claim about the Hausdorff dimension and thebox-ﬁlling dimension of fBM is proved by [47].

Original: Further related articles are [5] and [48].
Revised: Further related articles are [5] and [48].

Original: Remark 3 (More general version) .
Revised: Remark 3 (More general version) .

Original: Let us mention that the results in Theorems 5 and 4 can be alsogeneralized to the case of linear SDE‘s of the form
Revised: Let us mention that the results in Theorems 5 and 4 can be alsogeneralized to the case of linear SDE‘s of the form

Original: [Equation]
Revised: [Equation]

Original: where A : [0 , T ] −→ R d × d is locally bounded and measurable, Σ : [0 , T ] −→ R d
Revised: where A : [0 , T ] ! R d ⇥ d is locally bounded and measurable, ⌃ : [0 , T ] !

Original: × d is continuousdifferentiable and H < 12 .
Revised: R d ⇥ d is continuousdifferentiable and H < 12 .

Original: Here the stochastic integral with respect to B H · is given by the Young integral.
Revised: Here the stochastic integral with respect to B H · is given by the Young integral.

Original: In fact, one can show that(cid:82) t 0 Σ( s )
Revised: In fact, one can show that

Original: [-dB Hs = Σ( t ) B Ht − (cid:82) t 
-]
Revised: [-dB Hs = Σ( t ) B Ht − (cid:82) t 
-][Equation]

Original: ∂∂s Σ( s ) B Hs ds (see e.g. [6]).
Revised: s ⌃ ( s ) B Hs ds (see e.g. [6]).

Original: Byusing the linear transformation Y t : =
Revised: Byusing the linear transformation Y t :=

Original: X t − (cid:82) t 0 Σ( s ) dB
Revised: X t  [Equation]

Original:  Hs , we observe that Y t = x + (cid:82) t 0 ( A ( s ) Y s +a ( s ))
Revised: ⌃ ( s ) dB Hs , we observe that Y t = x + R [Equation] 0 ( A ( s ) Y s +a ( s ))

Original: ds , where a ( t ) :
Revised: ds , where a ( t ) :

Original: = A ( t ) (cid:82) t 0 Σ( s ) dB Hs .
Revised: = A ( t ) R t 0 ⌃ ( s ) dB Hs .

Original: Denote by Φ( t ) , 0 ≤ t ≤ T the fundamental solutionassociated with A ( t ) , that is the unique (absolutely continuous) solution to the equation ˙Φ( t ) =
Revised: Denote by  ( t ) , 0  t  T the fundamental solutionassociated with A ( t ) , that is the unique (absolutely continuous) solution to the equation ˙  ( t ) =

Original: A ( t )Φ( t ) , Φ(0) =
Revised: A ( t )  ( t ) ,  (0) =

Original: I d × d , the d × d identity matrix.
Revised: I d ⇥ d , the d ⇥ d identity matrix.

Original: Hence,
Revised: Hence,

Original: [Equation]
Revised: [Equation]

Original: Although the solution X t , 0 ≤ t ≤ T is in general not stationary any longer, we can use similararguments as in the proofs of Theorems 5 and 4 and the latter representation of the process to derivea lower and upper bound for E (cid:2) sup 0 ≤ t ≤ 1 ∥ X t ∥ (cid:3) .
Revised: Although the solution X t , 0  t  T is in general not stationary any longer, we can use similararguments as in the proofs of Theorems 5 and 4 and the latter representation of the process to derivea lower and upper bound for E [Equation]

Original: Remark 4 (Known bounds for distribution of fBM in multi-dimensional case) .
Revised: Remark 4 (Known bounds for distribution of fBM in multi-dimensional case) .

Original: In Theorems 5 and 4we obtained a lower and upper bound for the expected supremum of the norm of a (centered) multidimensional fOU process.
Revised: In Theorems 5 and 4we obtained a lower and upper bound for the expected supremum of the norm of a (centered) multidimensional fOU process.

Original: In comparison to the latter we also mention the following correspondingresult for distributions with respect to the fBm (see Theorem 4.6 by [23]): Let B Ht , 0 ≤ t ≤ 1be a fractional Brownian motion in R d for H ∈ (0 , 1) .
Revised: In comparison to the latter we also mention the following correspondingresult for distributions with respect to the fBm (see Theorem 4.6 by [23]): Let B Ht , 0  t  1be a fractional Brownian motion in R d for H 2 (0 , 1) .

Original: Then the following lower and upperbound of the distribution of sup 0 ≤ t ≤ 1 (cid:13) (cid:13)
Revised: Then the following lower and upperbound of the distribution of sup 0  t 

Original: B Ht (cid:13) (cid:13) ( ∥·∥ maximum norm) hold: There exist constants0 <
Revised: B Ht  ( k · k maximum norm) hold: There exist constants0 <

Original: K 1 ( H ) ≤ K 2 ( H ) < ∞ such that for all 0 < ε ≤ 1 :
Revised: K 1 ( H )  K 2 ( H ) < 1 such that for all 0 < "  1 :

Original: [Equation]
Revised: [Equation]

Original: Remark 5 (Tightness of the bound) .
Revised: Remark 5 (Tightness of the bound) .

Original: As in the 1-D case, the bound is right in terms of H , which isthe most important quantity for the argument made in this paper.
Revised: As in the 1-D case, the bound is right in terms of H , which isthe most important quantity for the argument made in this paper.

Original: Regarding the dimension d , notethat the lower bound does not depend on it (and thus it is not improvable) while the upper boundmight not be tight.
Revised: Regarding the dimension d , notethat the lower bound does not depend on it (and thus it is not improvable) while the upper boundmight not be tight.

Original:  Experiments
Revised:  Experiments

Original: By bounding the expected supremum of the stochastic process X Ht , our analysis reveals some insightsinto its exploration capability.
Revised: By bounding the expected supremum of the stochastic process X Ht , our analysis reveals some insightsinto its exploration capability.

Original: We have seen that the correlation between increments directly affectsthe supremum.
Revised: We have seen that the correlation between increments directly affectsthe supremum.

Original: Concretely, lower values of the Hurst parameters are expected to yield processes withbetter exploration abilities, as we detailed in Rmk. 2.
Revised: Concretely, lower values of the Hurst parameters are expected to yield processes withbetter exploration abilities, as we detailed in Rmk. 2.

Original: To demonstrate the relevance of our theoreticalfindings for optimization, we run a series of experiments with the optimizer fPGD from Eq.
Revised: To demonstrate the relevance of our theoreticalﬁndings for optimization, we run a series of experiments with the optimizer fPGD from Eq.

Original: In
Revised: In

Original: Section 5.1, we will illustrate how fPGD behaves on bi-stable optimization landscapes, where thereare two adjacent minima of different desirability (also studied by [51, 49]).
Revised: Section 5.1, we will illustrate how fPGD behaves on bi-stable optimization landscapes, where thereare two adjacent minima of different desirability (also studied by [51, 49]).

Original: In Section 5.2, we willdemonstrate on an embedded saddle point that different choices of H might be beneficial in differentoptimization phases – thus justifying the introduction of fPGD as a compromise between PGD and
Revised: In Section 5.2, we willdemonstrate on an embedded saddle point that different choices of H might be beneﬁcial in differentoptimization phases – thus justifying the introduction of fPGD as a compromise between PGD and

Original: Anti-PGD.
Revised: Anti-PGD.

Original: In Appendix B.3, we present another experiment on the Styblinski–Tang function in twodimensions which confirms our finding that fPGD with small H has a better exploration ability.
Revised: In Appendix B.3, we present another experiment on the Styblinski–Tang function in twodimensions which conﬁrms our ﬁnding that fPGD with small H has a better exploration ability.

Original: 5.1 Moving to better local minima in non-convex optimization
Revised: 5.1 Moving to better local minima in non-convex optimization

Original: We consider the behavior of fPGD in a bi-stable optimization landscape, given by
Revised: We consider the behavior of fPGD in a bi-stable optimization landscape, given by

Original: [Equation]
Revised: [Equation]

Original: where χ is the indicator function and the parameters ( v 0 , v 1 , v 2 , a, c, m ) determine the position andshape of two adjacent local minima.
Revised: where  is the indicator function and the parameters ( v 0 , v 1 , v 2 , a, c, m ) determine the position andshape of two adjacent local minima.

Original: Similar landscapes have been studied in the context of escapingspurious local minima [51] and in the context of generalization in machine learning by selecting flatminima [30, 49].
Revised: Similar landscapes have been studied in the context of escapingspurious local minima [51] and in the context of generalization in machine learning by selecting ﬂatminima [30, 49].

Original: We here study the former case, but we obtain analogous results in the latter (see
Revised: We here study the former case, but we obtain analogous results in the latter (see

Original: App. B).
Revised: App. B).

Original: For the former case, we set ( v 0 , v 1 , v 2 , a, c, m ) =
Revised: For the former case, we set ( v 0 , v 1 , v 2 , a, c, m ) =

Original: (0 . 0 , 45 . 0 , − 12 . 5 , 3 . 5 , 13 . 9 , 18 . 0) ; forthe corresponding landscape see the upper left subplot of Fig.2.
Revised: (0 . 0 , 45 . 0 ,  12 . 5 , 3 . 5 , 13 . 9 , 18 . 0) ; forthe corresponding landscape see the upper left subplot of Fig.

Original: On this landscape, we run fPGD with different choices of H .
Revised: On this landscape, we run fPGD with different choices of H .

Original: Note that, in the two minima (and thesaddle in between at ( a + c ) / 2 ) fPGD is indeed a discretization of a fractional Ornstein–Uhlenbeck process, like the one considered in Theorems 1 and 2.
Revised: Note that, in the two minima (and thesaddle in between at ( a + c ) / 2 ) fPGD is indeed a discretization of a fractional Ornstein–Uhlenbeck process, like the one considered in Theorems 1 and 2.

Original: We thus expect that a small Hurst parameterleads to a faster escape from the shallow minimum (left) to the deep minimum (right).
Revised: We thus expect that a small Hurst parameterleads to a faster escape from the shallow minimum (left) to the deep minimum (right).

Original: We run fPGDfor N = 1000 iterations with a fixed step size of η = 1 . 0 , initialised at x = 0 (the bottom of theshallow minimum).
Revised: We run fPGDfor N = 1000 iterations with a ﬁxed step size of ⌘ = 1 . 0 , initialised at x = 0 (the bottom of theshallow minimum).

Original: To stay in the setting of Theorems 1 and 2, we set σ = 80 . 

Revised: To stay in the setting of Theorems 1 and 2, we set  = 80 .

Original: · N − H .
Revised: · N  H .

Original: Thus, for all
Revised: Thus, for all

Original: H ∈ (0 , 1) , the employed fBM B Ht , defined on the time interval t ∈ [0 , N ] , is by Rmk.
Revised: H 2 (0 , 1) , the employed fBM B Ht , deﬁned on the time interval t 2 [0 , N ] , is by Rmk.

Original: 1 distributedlike 80 . 

Revised: 1 distributedlike 80 .

Original: · B Ht on t ∈ [0 , 1] – in the sense that the discrete steps { B Hk ; k = 0 , . . . , N } are distributedlike { 80 . 

Revised: · B Ht on t 2 [0 , 1] – in the sense that the discrete steps { B Hk ; k = 0 , . . . , N } are distributedlike { 80 .

Original: · B Hk/N ; k = 0 , . . . , N } .
Revised: · B Hk/N ; k = 0 , . . . , N } .

Original: By this H -dependent choice of σ , we give all fBMs the same finalvariance of 80 . 0 2 which lets us zoom in on the impact of correlation.
Revised: By this H -dependent choice of  , we give all fBMs the same ﬁnalvariance of 80 . 0 2 which lets us zoom in on the impact of correlation.

Original: (Otherwise the perturbationswould have different variances; this way we disentangle the correlation from the variance.)
Revised: (Otherwise the perturbationswould have different variances; this way we disentangle the correlation from the variance.)

Original: We notethat here the experimental outcomes are not sensitive to the precise value of σ .
Revised: We notethat here the experimental outcomes are not sensitive to the precise value of  .

Original: Fig.
Revised: Fig.

Original: 2 depicts our findings.
Revised: 2 depicts our ﬁndings.

Original: We say that fPGD is in the shallow minimum at iteration k if its iterate x kis smaller than a , and in the deep minimum if it is larger than c ; both a and c are plotted as dashedlines in the optimization landscape (top row) and in the trajectories (bottom row).
Revised: We say that fPGD is in the shallow minimum at iteration k if its iterate x kis smaller than a , and in the deep minimum if it is larger than c ; both a and c are plotted as dashedlines in the optimization landscape (top row) and in the trajectories (bottom row).

Original: In the bottom row,for all H , we start below the lower dashed line a (shallow minimum) and then cross above the higherdashed line c (deep minimum).
Revised: In the bottom row,for all H , we start below the lower dashed line a (shallow minimum) and then cross above the higherdashed line c (deep minimum).

Original: It is evident that a smaller H enables faster crossings.
Revised: It is evident that a smaller H enables faster crossings.

Original: We thus escapefaster from the spurious shallow minimum to the desirable deep one.
Revised: We thus escapefaster from the spurious shallow minimum to the desirable deep one.

Original: (Note, however, if the noise isnot cooled down, one might oscillate back.)
Revised: (Note, however, if the noise isnot cooled down, one might oscillate back.)

Original: The upper right plot depicts the cumulative distributionfunction of the first-exit time (i.e. the first time fPGD reaches the deep minimum, above c ).
Revised: The upper right plot depicts the cumulative distributionfunction of the ﬁrst-exit time (i.e. the ﬁrst time fPGD reaches the deep minimum, above c ).

Original: Again,we can observe that for a small Hurst parameter the probability of a fast exit is significantly higher.
Revised: Again,we can observe that for a small Hurst parameter the probability of a fast exit is signiﬁcantly higher.

Original: Overall, these findings for fPGD match the continuous-time result of Theorems 1 and 2.
Revised: Overall, these ﬁndings for fPGD match the continuous-time result of Theorems 1 and 2.

Original: Anotherbi-stable landscape (with a sharp and a flat minimum) is studied in Appendix B with the same findings.
Revised: Anotherbi-stable landscape (with a sharp and a ﬂat minimum) is studied in Appendix B with the same ﬁndings.

Original: Note that physicists [41] have also observed faster escapes for smaller Hurst parameters on a related
Revised: Note that physicists [41] have also observed faster escapes for smaller Hurst parameters on a related

Original: Kramer-like escape problem.
Revised: Kramer-like escape problem.

Original: [Figure]

Original: 5.2 Non-Markovian fPGD as a trade-off between PGD and Anti-PGD
Revised: 5.2 Non-Markovian fPGD as a trade-off between PGD and Anti-PGD

Original: As discussed earlier, fPGD interpolates between two known Markovian methods: PGD ( H = 1 / 2)and Anti-PGD ( H = 0) .
Revised: As discussed earlier, fPGD interpolates between two known Markovian methods: PGD ( H = 1 / 2)and Anti-PGD ( H = 0) .

Original: Both PGD and Anti-PGD have unique advantages; see [51] or [20] for
Revised: Both PGD and Anti-PGD have unique advantages; see [51] or [20] for

Original: PGD and [33] for Anti-PGD.
Revised: PGD and [33] for Anti-PGD.

Original: It is thus natural to ask whether there are cases where non-MarkovianfPGD ( H ∈ (0 , 1 / 2) ) is beneficial.
Revised: It is thus natural to ask whether there are cases where non-MarkovianfPGD ( H 2 (0 , 1 / 2) ) is beneﬁcial.

Original: While we do not have a complete answer to this question,we here present a setting that demonstrates that H ∈ (0 , 1 / 2) can indeed outperform both PGD( H = 1 / 2 ) and Anti-PGD ( H = 0 ) simultaneously.
Revised: While we do not have a complete answer to this question,we here present a setting that demonstrates that H 2 (0 , 1 / 2) can indeed outperform both PGD( H = 1 / 2 ) and Anti-PGD ( H = 0 ) simultaneously.

Original: We consider the following regularized quadratic:f ( x ) =
Revised: We consider the following regularized quadratic:f ( x ) =

Original: 12 x ⊺ Mx + λ (cid:80) di
Revised: 12 x | Mx +

Original:  =1 x 4 i , where x i is the i-th coordinate of x ∈ R d , M ∈ R d × d , d = 400 ,and λ = 0 .0001 .
Revised: P di =1 x 4 i , where x i is the i-th coordinate of x 2 R d , M 2 R d ⇥ d , d = 400 ,and  = 0 .

Original: If M has negative eigenvalues, this landscape has a d -dim.
Revised: If M has negative eigenvalues

Original: saddle point at x = 0 .

Original: If a stochastic optimizer can find the negative eigenvectors, it will reach a global minimum away
Revised: If a stochastic optimizer can ﬁnd the negative eigenvectors, it will reach a global minimum away

Original: [Figure]
Revised: [Figure]

Original: from zero – while gradient descent remains stuck at zero.
Revised: from zero – while gradient descent remains stuck at zero.

Original: The problem gets increasingly hard asthe number of negative eigendirections decreases [20].
Revised: The problem gets increasingly hard asthe number of negative eigendirections decreases [20].

Original: Indeed, [13] showed that vanilla gradientdescent takes in the worst case exponential (in the number of dimensions) time to escape such saddlepoints.
Revised: Indeed, [13] showed that vanilla gradientdescent takes in the worst case exponential (in the number of dimensions) time to escape such saddlepoints.

Original: We choose M as a diagonal matrix whose diagonal entries are uniformly sampled from [0 , 1] ,and – to make it a challenging saddle – we multiply the 10 lowest of these entries by − 1 .
Revised: We choose M as a diagonal matrix whose diagonal entries are uniformly sampled from [0 , 1] ,and – to make it a challenging saddle – we multiply the 10 lowest of these entries by  1 .

Original: We test fPGD on this problem.
Revised: We test fPGD on this problem.

Original: We initialise at the saddle x = 0 and run fPGD with σ = 0 .
Revised: We initialise at the saddle x = 0 and run fPGD with  = 0 .

Original: 005 andη = 0 .
Revised: 005 and⌘ = 0 .

Original: 005 for 10 6 iterations with different Hurst parameters.
Revised: 005 for 10 6 iterations with different Hurst parameters.

Original: (While in general σ and H are tuningparameters that can be jointly tuned, here σ = 0 .
Revised: (While in general  and H are tuningparameters that can be jointly tuned, here  = 0 .

Original: 005 is a good choice for all values of H which wedemonstrate in Appendix B.4.)
Revised: 005 is a good choice for all values of H which wedemonstrate in Appendix B.4.)

Original: The results are presented in Fig.
Revised: The results are presented in Fig.

Original: In this problem, there is indeeda trade-off between uncorrelated ( H = 0 . 5) and anticorrelated perturbations
Revised: In this problem, there is indeeda trade-off between uncorrelated ( H = 0 . 5) and anticorrelated perturbations

Original: ( H < 0 .
Revised: ( H < 0 .

Original: 5 ): While alarge Hurst parameter ( H = 0 . 5) moves away from the saddle quickly, the non-Markovian fPGD( H = 0 . 4 ) finds the direction with negative curvature most quickly (in the sense that it gets belowzero).
Revised: 5 ): While alarge Hurst parameter ( H = 0 . 5) moves away from the saddle quickly, the non-Markovian fPGD( H = 0 . 4 ) ﬁnds the direction with negative curvature most quickly (in the sense that it gets belowzero).

Original: Once they have exited, all H perform similar for a while.
Revised: Once they have exited, all H perform similar for a while.

Original: Eventually, however, smaller Hurstparameters converge better to the bottom of the final minimum – thereby eventually catching up andsurpassing the at-first faster higher Hurst parameters.
Revised: Eventually, however, smaller Hurstparameters converge better to the bottom of the ﬁnal minimum – thereby eventually catching up andsurpassing the at-ﬁrst faster higher Hurst parameters.

Original: Which H is best, will therefore depend onhow many iterations the computational budget allows for.
Revised: Which H is best, will therefore depend onhow many iterations the computational budget allows for.

Original: Here, the non-Markovian H = 0 . 3 and
Revised: Here, the non-Markovian H = 0 . 3 and

Original: H = 0 .
Revised: H = 0 .

Original: 4 yield the best final accuracy.
Revised: 4 yield the best ﬁnal accuracy.

Original: This suggests that, if all other parameters are the same, different choices of Hurst parameters arebeneficial in different situations.
Revised: This suggests that, if all other parameters are the same, different choices of Hurst parameters arebeneﬁcial in different situations.

Original: It can thus be advantageous to tune and adapt the correlation ofnoise by choosing H .
Revised: It can thus be advantageous to tune and adapt the correlation ofnoise by choosing H .

Original: It is especially interesting that both PGD ( H = 1 / 2 ) and Anti-PGD
Revised: It is especially interesting that both PGD ( H = 1 / 2 ) and Anti-PGD

Original: ( H = 0 )have shortcomings in this experiment: PGD has to increase the loss significantly in the beginningbefore exiting the saddle and does not converge to the bottom of the minimum in the end.
Revised: ( H = 0 )have shortcomings in this experiment: PGD has to increase the loss signiﬁcantly in the beginningbefore exiting the saddle and does not converge to the bottom of the minimum in the end.

Original: Anti-PGDtakes a long time to exit the saddle in the beginning.
Revised: Anti-PGDtakes a long time to exit the saddle in the beginning.

Original: In contrast, non-Markovian fPGD (with H = 0 .
Revised: In contrast, non-Markovian fPGD (with H = 0 .

Original: 3or H = 0 . 4 ) stays clear of these two undesirable extremes.
Revised: 3or H = 0 . 4 ) stays clear of these two undesirable extremes.

Revised: We stress, however, that the aboveexperiment is a proof of concept and only shows that there exist landscapes where non-MarkovianfPGD performs best.

Original: We stress, however, that further work is needed to better understand the interplay between the scaleσ and the correlation parameter H , which interact according to Eq.Future work could analyzethe dependency between variance and correlation.
Revised: On many other landscapes, PGD and Anti-PGD will still be best. To make fPGD efﬁcient on general landscapes, further work is needed to better understand theinterplay between the scale  and the correlation parameter H , which interact according to Eq.

Original: Moreover, one could design an adaptive methodthat jointly tunes σ and H to maximize the efficiency of perturbed-gradient methods.
Revised: Using this knowledge, one could design an adaptive method that jointly tunes  and H to maximizethe efﬁciency of perturbed-gradient methods.

Original: For the aboveexperiments, we again emphasize that our constant choice of σ is suitable for all H , as justified in Appendix B.4.
Revised: For the above experiments, we again emphasize thatour constant choice of  is suitable for all H , as justiﬁed in Appendix B.4.

Original:  Conclusion
Revised:  Conclusion

Original: We analysed the behavior of a new type of stochastic noise with correlated increments.
Revised: We analysed the behavior of a new type of stochastic noise with correlated increments.

Original: We foundthat the amount of correlation, captured by the Hurst parameter H , has a direct effect on the overallmagnitude of the process and its ability to explore.
Revised: We foundthat the amount of correlation, captured by the Hurst parameter H , has a direct effect on the overallmagnitude of the process and its ability to explore.

Original: This insight opens the doors to a new area ofresearch where the noise of stochastic processes could be manipulated to speed up the convergence ofan optimization method, as well as target specific minima (e.g., depending on their flatness properties).
Revised: This insight opens the doors to a new area ofresearch where the noise of stochastic processes could be manipulated to speed up the convergence ofan optimization method, as well as target speciﬁc minima (e.g., depending on their ﬂatness properties).

Original: Our experimental results validate our theoretical analysis.
Revised: Our experimental results validate our theoretical analysis.

Original: We showed that, in bi-stable potentials,injection of fractional noise with small H speeds up the escape from a suboptimal minimum to a betterone.
Revised: We showed that, in bi-stable potentials,injection of fractional noise with small H speeds up the escape from a suboptimal minimum to a betterone.

Original: Also, we showed that there are indeed settings where non-Markovian (somewhat anti-correlated)noise outperforms the standard Markovian (uncorrelated or perfectly anti-correlated) noise.
Revised: Also, we showed that there are indeed settings where non-Markovian (somewhat anti-correlated)noise outperforms the standard Markovian (uncorrelated or perfectly anti-correlated) noise.

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