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

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: Prior work [55, 50] has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape.
Revised: Prior work [55, 50] has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape.

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

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

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

Original: This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process.
Revised: This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process.

Original: We demonstrate how to discretize such processes which gives rise to the new algorithm “fPGD”.
Revised: We demonstrate how to discretize such processes which gives rise to the new 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 [36].
Revised: PGD and Anti-PGD [36].

Original: We study the properties of fPGD both theoretically and empirically, 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 and empirically, 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 ways to exploit noise for training machine learning models.
Revised: These results open the ﬁeld to novel ways to 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 in optimization, dating back to at least [31, 2].
Revised: Injecting random noise to the parameters of a function has been a commonly used technique in optimization, dating back to at least [31, 2].

Original: More recently, [32] demonstrated the empirical benefits of injecting Gaussian noise (a.k.a.
Revised: More recently, [32] demonstrated the empirical beneﬁts of injecting Gaussian noise (a k.a.

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

Original: Injecting Gaussian noise has also been a common technique used to show faster escape from saddle points [15].
Revised: Injecting Gaussian noise has also been a common technique used to show faster escape from saddle points [15].

Original: Importantly, the noise of the optimizer also has an effect on the type of minima that can be reached, raising some important connections to flatness and generalization properties [24].
Revised: Importantly, the noise of the optimizer also has an effect on the type of minima that can be reached, raising some important connections to ﬂatness and generalization properties [24].

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

Original: Despite a large body of work debating the distribution of the intrinsic noise of stochastic gradient descent (SGD) [44, 54, 46, 13], there is a large gap in our knowledge regarding the role of noise to escape undesirable regions.
Revised: Despite a large body of work debating the distribution of the intrinsic noise of stochastic gradient descent (SGD) [44, 54, 46, 13], there is a large gap in our knowledge regarding the role of noise to escape undesirable regions.

Original: In order to address this question, we investigate the behavior of a general type of non-Markovian stochastic process known as fractional Brownian motion (fBM).
Revised: In order to address this question, we investigate the behavior of a general type of non-Markovian stochastic 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 or negatively correlated.
Revised: Brownian motion where the increments need not be independent but can instead be positively or negatively correlated.

Original: Interestingly, the processes that can be constructed from fBM recover known instances in the literature, including perturbed gradient descent (PGD) [55, 22], and anticorrelated
Revised: Interestingly, the processes that can be constructed from fBM recover known instances in the literature, including perturbed gradient descent (PGD) [55, 22], and anticorrelated

Original: PGD (Anti-PGD) [36, 37].
Revised: PGD (Anti-PGD) [36, 37].

Original: As we will see, the correlation between increments has a drastic effect of the behavior 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 the behavior 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 the overall 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 the overall magnitude of the process and its ability to explore the landscape?

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

Original: This can be achieved by bounding its expected supremum, which is connected to the concept of exit time, see Rmk. 2.
Revised: This can 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-time stochastic process with drift known as the fractional Ornstein-Uhlenbeck (fOU) process, which is connected to SGD (see Section 3).
Revised: To conduct such an analysis, we will consider a well-known type of continuous-time stochastic process with drift known as the fractional Ornstein-Uhlenbeck (fOU) process, which is connected to SGD (see Section 3).

Original: We will also demonstrate how to discretize such a process to obtain a practical optimization algorithm that we name fPGD (for fractional PGD).
Revised: We will also demonstrate how to discretize such a process to 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 fOU process.
Revised: Concretely, our main result will be to provide upper and lower bounds on the expected supremum of the fOU process.

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

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

Original: In Section 5.1, we demonstrate that fBM perturbations indeed help optimizers to move faster from one local minimum to another – as suggested by our theory.
Revised: In Section 5.1, we demonstrate that fBM perturbations indeed help 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-Markovian versions of fPGD indeed outperform the known Markovian special cases (PGD and Anti-PGD) – a first 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-Markovian versions of fPGD indeed outperform the known Markovian special cases (PGD and Anti-PGD) – a ﬁrst 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 are known to appear naturally in many situations.
Revised: Yet, non-Markovian processes are known to appear naturally in many situations.

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

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

Original: fBM is a generalization of Brownian motion where the increments need not be independent but can instead be positively or negatively correlated.
Revised: fBM is a generalization of Brownian motion where the increments need not be independent but can instead be positively 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 discussed in this work.
Revised: There are two sets of results that are of relevance to the problem discussed in this work.

Original: The first set of results are escape times for fBM which are studied by [28, 4, 41, 45, 47], but they are restricted to simple 1-dimensional processes, and they are often applicable to specific problems 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 [28, 4, 41, 45, 47], but they are restricted to simple 1-dimensional processes, and they are often applicable to speciﬁc problems such as Kramer’s problem, instead of more general problems of interest.

Original: Extending these results to multiple dimensions is difficult, and we are not aware of any existing result.
Revised: Extending these results 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 expected supremum of "pure" fBM (i.e., without any drift term), including [34, 30, 40].
Revised: There is a large body of work focusing on the expected supremum of "pure" fBM (i.e., without any drift term), including [34, 30, 40].

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. [39] gives an 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. [39] gives an asymptotic result for H 12 , .

Original: In the multi-dimensional case, [26] derive lower and upper bound for the distribution of fBM in the multi-dimensional case, see Rmk. 4.
Revised: In the multi-dimensional case, [26] derive lower and upper bound for the distribution of fBM in the multi-dimensional case, see Rmk. 4.

Original: In this work, our goal will be to provide sharp bounds for stochastic processes with more general drift terms than prior work and in the multi-dimensional case.
Revised: In this work, our goal will be to provide sharp bounds for stochastic processes with more general drift terms than prior work and 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 process is Brownian motion (BM), which is a special case of fBM (corresponding to H = 12 ) where the increments are not correlated.
Revised: A well-studied type of stochastic process is Brownian motion (BM), which is a special case of fBM (corresponding to H = 12 ) where the increments are not correlated.

Original: When also considering a stochastic process with a drift, we obtain the 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, 23] as well as on the expected supremum [16, 21].
Revised: When also considering a stochastic process with a drift, we obtain the 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, 23] as well as on the expected supremum [16, 21].

Original: [Figure]
Revised: [Figure]

Original: Finally, [10] also derived a maximal inequality for a multi-dimensional Ornstein-Uhlenbeck process, although their result is restricted to the radial case.
Revised: Finally, [10] also derived a maximal inequality for a multi-dimensional Ornstein-Uhlenbeck process, although their 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 the literature.
Revised: The distribution of the noise of SGD is often a subject of debate in the literature.

Original: In some cases, the noise of SGD appears to be heavy-tailed [38, 44, 17], which yields faster exit from sharp to flat minima [33].
Revised: In some cases, the noise of SGD appears to be heavy-tailed [38, 44, 17], which yields faster exit from sharp to ﬂat minima [33].

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

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

Original: Multiple PGD methods have been shown to help quickly escape spurious local minima [55] and saddle points [22].
Revised: Multiple PGD methods have been shown to help quickly escape spurious local minima [55] and saddle points [22].

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

Original: Recent work has demonstrated that such perturbations are indeed beneficial for generalization by implicitly regularizing the loss [7, 18, 12].
Revised: Recent work has demonstrated that such perturbations are indeed beneﬁcial for generalization by implicitly regularizing the loss [7, 18, 12].

Original:  Background
Revised:  Background

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

Original: Fractional Brownian motion (fBM) is a generalization of Brownian motion where the increments of the stochastic process need not be independent.
Revised: Fractional Brownian motion (fBM) is a generalization of Brownian motion 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 [25].
Revised: Gaussian processes was developed for some hydrological modelling by [25].

Original: [19] later studied the long term water flow characteristics of the Nile River that was affected by long periods of drought.
Revised: [19] later studied the long term water ﬂow characteristics of the Nile River that was affected by long periods of drought.

Original: In such a scenario, the self similarity of the process is related to a long term dependence in the water levels.
Revised: In such a scenario, the self similarity of the process is related to a long term dependence in the water levels.

Original: Since then, fBM has also found applications in financial modeling [35].
Revised: Since then, fBM has also found applications in ﬁnancial modeling [35].

Original: We now give a formal definition of fBM, and discuss its main properties.
Revised: We now give a formal deﬁnition 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 , then the increments are positively correlated, while they are negatively correlated if H < 1 2 .
Revised: If H > 12 , then the increments are positively correlated, while they are negatively correlated if H < 1 2 .

Original: For H ̸ = 1 2 , the increments of fBM are not independent, although they are stationary, with variance
Revised: For H 6 = 1 2 , 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 illustrated in Fig. 1; see Rmk. 2.
Revised: A crucial property of negatively correlated increments is their ability to ﬁll in the space, as illustrated in 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, 27, 48].
Revised: Ornstein-Uhlenbeck process that has already found many applications in the ﬁeld of machine learning [7, 27, 48].

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 information regarding the fractional Ornstein-Uhlenbeck process, we refer the reader to [11, 29, 3].
Revised: For further information regarding the fractional Ornstein-Uhlenbeck process, we refer the reader to [11, 29, 3].

Original: We will for instance later make use of the covariance of this process that has been derived by [29, 3].
Revised: We will for instance later make use of the covariance of this process that has been derived by [29, 3].

Original: The fractional Ornstein-Uhlenbeck process is relevant for optimization purposes as it corresponds to optimizing 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 to optimizing 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 function r 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 stochastic process reaches stationarity, the stationary distribution of the iterates is known to be approximated by a quadratic function [27, 43].
Revised: We also note that when the stochastic process reaches stationarity, the stationary distribution of the iterates is known to be approximated by a quadratic function [27, 43].

Original: This implies that such processes can be used to analyze the behavior of complex functions at stationarity, but also close to saddle points where quadratic functions often play a crucial role to analyze the escape behavior of gradient descent methods [22].
Revised: This implies that such processes can be used to analyze the behavior of complex functions at stationarity, but also close to saddle points where quadratic functions often play a crucial role to analyze the escape behavior of gradient descent methods [22].

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) [22].
Revised: This method is called perturbed gradient descent (PGD) [22].

Original: Here, by instead discretizing the fractional SDE dX t = −∇ f
Revised: Here, by instead discretizing the fractional SDE dX t = r

Original:  ( X ) dt + σdB Ht for any H ∈ (0 , 1) , we obtain a generalized version of PGD, called fractional PGD (fPGD):
Revised: f ( X ) dt +  dB Ht for any H 2 (0 , 1) , we obtain a generalized 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 (in distribution) as H → 0 ; see Lemma 4.1 by [8] – the method Anti-PGD [36, Section 2] can be considered as the limit case of fPGD as H ↓ 0 .
Revised: Importantly – since B Ht converges to a white-noise process (in distribution) as H ! 0 ; see Lemma 4.1 by [8] – the method Anti-PGD [36, Section 2] can be considered 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 shown to be useful yet, and indeed the below theorems and experiments will indicate that a small H yields quicker escapes from difficult regions such as saddle points and local minima.
Revised: Positive correlations ( H 2 (1 / 2 , 1) ) have not been shown to be useful yet, and indeed the below theorems and experiments will indicate that a small H yields quicker escapes from difﬁcult regions such as saddle points and local minima.

Original: While the next section will analyze the continuous-time model based on fBM, we will return to the optimizer fPGD in the experiments (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 section will analyze the continuous-time model based on fBM, we will return to the optimizer fPGD in the experiments (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 4.1 Preliminaries
Revised:  Analysis 4.1 Preliminaries

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

Original: The main tools we use are Dudley’s inequality and Sudakov’s inequality, which give upper and lower bounds on E [sup t ∈ T X t ] in terms of the metric entropy of T denoted by log N ( T, d, ϵ ) .
Revised: The main tools we use are Dudley’s inequality and Sudakov’s inequality, which give upper and lower bounds on E [sup t 2 T X t ] in terms of the metric entropy 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, we recall 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, we recall the deﬁnition of the sub-Gaussian norm of a random variable X , denoted by k X k 2 , as k 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 stochastic process defined in Eq.
Revised: In this section, we derive both upper and lower bounds on the expected supremum of the stochastic process deﬁned in Eq.

Original: Since the analysis is rather technical, we first present the main ideas in the 1-dimensional case.
Revised: Since the analysis is rather technical, we ﬁrst present the main ideas in the 1-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 3 β and C are positive constants (independent of H ).
Revised: where C  = e 3  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 and can 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 and can not be improved in terms of H .

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

Original: Ht dt + dB Ht , Y H 

Revised: Y H t dt + dB H t , 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 a lower bound in terms of entropy numbers (Thm. 7) or by means of so called Talagrand functionals (see e.g. [ 45 ] ).
Revised: In the general case 14 < |  | < 1 , one gets principally a lower bound in terms of entropy numbers (Thm. 7) or by means of so called Talagrand functionals (see e.g. [ 45 ] ).

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. we however provide experiments that demonstrate the 1 / √ H in this case, for the example of β = 12 .
Revised: In Appendix B. we 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 of fBM, 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 of fBM, namely B Hat d = a H B Ht for a > 0 .

Original: This implies that E (cid:104) sup s ∈ [0 ,t ] B Hs (cid:105) = 
Revised: This implies that E h sup s 2 [0 , [Equation] ] B Hs i = t

Original: t H E (cid:104) sup s ∈ [0 , 1] B Hs (cid:105) .
Revised: [Equation] E h sup s 2 [0 , 1] B Hs i .

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 the expected supremum of the norm, or more precisely E (cid:104) sup t ∈ [0 , 1] ∥ X
Revised: The quantity of interest becomes the expected 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 to the 1-dimensional case, we apply Sudakov’s inequality to the Gaussian component of the multidimensional fOU process
Revised: Similarly to the 1-dimensional case, we apply Sudakov’s inequality to the Gaussian component of the multidimensional fOU process

Original: X Ht 2 .
Revised: X Ht 2 .

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 , . . .

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

Original: [Equation]

Original: j =

Original: [Equation] ∈ [Equation] ) | + 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 do require 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 do require 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 theorem are satisfied.
Revised: If some diagonal element of A is non-negative, then the assumptions of the above theorem are 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: Then min t ∈ [0 , 1] a i [-
-] i 0 ( t ) =
Revised: Then min 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 
  i 0 )

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

Original: Hence, if some diagonal element λ i 0 > − 1 exists, then the conditions of the previous theorem are fulfilled.
Revised: Hence, if some 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 now explain how these are connected to what we call "exploration ability".
Revised: We now explain 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

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: ⇠ + R t

Original: Aϕ ( s ) ds , the quantity a := sup t ∈ [0 , 1] ∥ E [ X Ht ] ∥ < ∞ is independent of H .
Revised: A  ( s ) ds , the quantity a := sup t 2 [0 , 1] k E [ X Ht ] k <is independent of H .

Original: Thus E (cid:104) sup t ∈ [0 , 1] ∥
Revised: Thus E h sup t 2 [0 , 1] k X

Original: X Ht ∥ (cid:105) ≥ ce−(cid:114)
Revised: [Equation] 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: H 1 ∈ (cid:0) 0 , 12 (cid:1) , there is an H < H 1 such that E (cid:104) sup t ∈ [0 , 1] ∥ X Ht ∥
Revised: H 1 0 , , there is an H < H 1 such that E h

Original: [Equation]
Revised: [Equation]

Original: > [Equation] (cid:104) sup
Revised: t 2 [

Original: [Equation]
Revised: [Equation]

Original: ∈

Original: [Equation]
Revised: [Equation]

Original: ∥ [Equation] (cid:105) . 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 H t 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 this sense, can help to escape suboptimal local minima more quickly (experiments in Section 5.1) or to explore the space around a saddle point for an exit (experiments in Section 5.2).
Revised: Better exploration, in this sense, can help to escape suboptimal local minima more quickly (experiments in Section 5.1) or to explore the space around a saddle point for an exit (experiments in Section 5.2).

Original: 2) Exit time: First, note that E (cid:104) sup t ∈ [0 , 1] ∥ X Ht ∥ (cid:105) =
Revised: 2) Exit time: First, note that E h [Equation] ds , therefore

Original: [-(cid:82) ∞ 
 P (cid:16) sup t ∈ [0 , 1] ∥ X Ht ∥ > s (cid:17) ds , therefore (cid:82) ∞ 0 (cid:110) P (cid:16) sup t ∈ [0 , 1] ∥-]
Revised: [-(cid:82) ∞ 
 P (cid:16) sup t ∈ [0 , 1] ∥ X Ht ∥ > s (cid:17) ds , therefore (cid:82) ∞ 0 (cid:110) P (cid:16) sup t ∈ [0 , 1] ∥-]R

Original: X Ht ∥ > s (cid:17) − P (cid:16) sup t ∈ [0 , 1] ∥ X H 1 t ∥ > s (cid:17)(cid:111) ds > 0 .
Revised: [Equation] , 1] k X H 1 t k > s ⌘o ds > 0 .

Original: Thus, we can find a s 0 > 0 such that P (cid:16) supt ∈ [0 , 1] ∥
Revised: Thus, we can ﬁnd a s 0 > 0 such that P ⇣ [Equation] sup

Original: 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 all s ∈{+
+} [ 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: [Equation] ⌘ > 0 and, since probability distributions are right-continuous, there exists ✏ > 0 such that for all s {+
+} [ s 0 , s 0 + ✏ ) , P ⇣ sup t 2 [0 , 1] k X H t k > s ⌘ > P ⇣ sup t 2 [0 , 1] k X H 1 t k > s ⌘ > 0 .

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

Original: P (cid:16) sup t ∈ [0 , 1] ∥ XHt ∥ > s (cid:17) =
Revised: P ⇣ sup t 2 [0 , 1] k X

Original:  P ( τ Hs < 1) where τ Hs denotes the first exit time of X H from the ball B s (0) of radius s .
Revised: Ht k > s ⌘ = P ( ⌧ Hs < 1) where ⌧ Hs denotes the ﬁrst exit time of X H from the ball 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. the probability of an exit before time s of H is larger than that of H 1 .
Revised: We conclude that P ( ⌧ Hs < 1) > P ( ⌧ H 1 s < 1) 8 s 2 [ s 0 , s 0 + ✏ ) , i.e. the probability of an exit before time s of H is larger than that of H 1 .

Original: 3) Space-filling property: We note that fractional Brownian motion with low Hurst parameter H fills the space more densely.
Revised: 3) Space-ﬁlling property: We note that fractional Brownian motion with low Hurst parameter H ﬁlls the space more densely.

Original: This behavior comes from the fact that the graph of the fBM has a Hausdorff dimension and box-counting dimension given by d = 2 − H .
Revised: This behavior comes from the fact that the graph of the fBM has a Hausdorff dimension 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 d for large n , where N ( ε ) denotes the number of boxes with side length ε , which is needed to cover the graph of the fBM.
Revised: Cn d for large n , where N ( " ) denotes the number of boxes with side length " , which is needed to cover the graph of the fBM.

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

Original: See [42].
Revised: See [42].

Original: The above claim about the Hausdorff dimension and the box-filling dimension of fBM is proved by [51].
Revised: The above claim about the Hausdorff dimension and the box-ﬁlling dimension of fBM is proved by [51].

Original: Further related articles are [5] and [52].
Revised: Further related articles are [5] and [52].

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

Original: Let us mention that the results in Theorems 4 and 5 can be also generalized to the case of linear SDE‘s of the form
Revised: Let us mention that the results in Theorems 4 and 5 can be also generalized 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 × d is continuous differentiable and H < 12 .
Revised: where A : [0 , T ] ! R d ⇥ d is locally bounded and measurable, ⌃ : [0 , T ] ! R d ⇥ d is continuous differentiable 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 
Revised: ) B [Equation]

Original: Hs ds (see e.g. [6]).
Revised: ds (see e.g. [6]).

Original: By using the linear transformation Y t := 
Revised: By using the linear transformation Y t := X t  [Equation] ⌃

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

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

Revised: 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 H s .

Original: Denote by Φ( t ) , 0 ≤ t ≤ T the fundamental solution associated with A ( t ) , that is the unique (absolutely continuous) solution to the equation ˙Φ( t ) =
Revised: Denote by  ( t ) , 0  t  T the fundamental solution associated 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 similar arguments as in the proofs of Theorems 4 and 5 and the latter representation of the process to derive a 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 similar arguments as in the proofs of Theorems 4 and 5 and the latter representation of the process to derive a 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 Theoremsand 5 we obtained a lower and upper bound for the expected supremum of the norm of a (centered) multi-dimensional fOU process (without a drift term).
Revised: In Theoremsand 5 we obtained a lower and upper bound for the expected supremum of the norm of a (centered) multi-dimensional fOU process (without a drift term).

Original: In comparison to the latter we also mention the following corresponding result for distributions with respect to the fBm (see Theorem 4.6 by [26]):
Revised: In comparison to the latter we also mention the following corresponding result for distributions with respect to the fBm (see Theorem 4.6 by [26]):

Original: Let B Ht , 0 ≤ t ≤ 1 be a fractional Brownian motion in R d for H ∈ (0 , 1) .
Revised: Let B Ht , 0  t  1 be a fractional Brownian motion in R d for H 2 (0 , 1) .

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

Original: B Ht (cid:13)(cid:13) ( ∥·∥ maximum norm) hold: There exist constants 0 < K 1 ( H ) ≤ K 2 ( H ) < ∞ such that for all 0 < ε ≤ 1 :
Revised: B Ht  ( k · k maximum norm) hold: There exist constants 0 < 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 is the 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 is the most important quantity for the argument made in this paper.

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

Original:  Experiments
Revised:  Experiments

Original: The above analysis reveals some insights into the exploration capability of a stochastic process
Revised: The above analysis reveals some insights into the exploration capability of a stochastic process

Original: X Ht driven by a fracional Brownian motion B Ht with Hurst parameter H .
Revised: X Ht driven by a fractional Brownian motion B Ht with Hurst parameter H .

Original: We have seen that the correlation between increments directly affects the expected supremum of X Ht .
Revised: We have seen that the correlation between increments directly affects the expected supremum of X Ht .

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

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

Original: In Section 5.1, we show that fPGD with small H better explores a non-convex loss landscapes with multiple local minima better, as predicted by our theory (Rmk. 2).
Revised: In Section 5.1, we show that fPGD with small H better explores a non-convex loss landscapes with multiple local minima better, as predicted by our theory (Rmk. 2).

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

Original: In the Appendix B.3, we add experiments on how fPGD behaves on bi-stable optimization landscapes, where there are two adjacent minima of different desirability (also studied by [55, 53]; the results of these experiments confirm that a small H yields faster escapes from local minima.
Revised: In the Appendix B.3, we add experiments on how fPGD behaves on bi-stable optimization landscapes, where there are two adjacent minima of different desirability (also studied by [55, 53]; the results of these experiments conﬁrm that a small H yields faster escapes from local minima.

Original: 5.1 Demonstration of exploration ability on Styblinski–Tang function
Revised: 5.1 Demonstration of exploration ability on Styblinski–Tang function

Original: In this section, we test the ability of fPGD to escape local minima and explore the landscape.
Revised: In this section, we test the ability of fPGD to escape local minima and explore the landscape.

Original: To this end, we consider the Styblinski–Tang function in two dimensions, which is written
Revised: To this end, we consider the Styblinski–Tang function in two dimensions, which is written

Original: [Equation]
Revised: [Equation]

Original: A contour plot of this function is depicted on the upper left of Fig.
Revised: A contour plot of this function is depicted on the upper left of Fig.

Original: This function has four minima: one shallow one (upper right), two medium ones (upper left and lower right), and one global deep minimum (lower left); see upper right subfigure of Fig.
Revised: This function has four minima: one shallow one (upper right), two medium ones (upper left and lower right), and one global deep minimum (lower left); see upper right subﬁgure of Fig.

Original: Since the direct way between the shallow and deep minimum is blocked by a large hill at zero, gradient-based optimizers have to go through the medium ones, to get to the shallow one.
Revised: Since the direct way between the shallow and deep minimum is blocked by a large hill at zero, gradient-based optimizers have to go through the medium ones, to get to the shallow one.

Original: It is therefore a suitable test problem to test the validity of the alleged exploration ability of fPGD with small H (as explained in Remark 2).
Revised: It is therefore a suitable test problem to test the validity of the alleged exploration ability of fPGD with small H (as explained in Remark 2).

Original: We thus run experiments, where fPGD is initialized in the shallow minimum at x 0 = (2 . 7 , 2 . 7) .
Revised: We thus run experiments, where fPGD is initialized in the shallow minimum at x 0 = (2 . 7 , 2 . 7) .

Original: After enough iterations, fPGD will eventually take the following steps: First, it will escape the shallow minimum.
Revised: After enough iterations, fPGD will eventually take the following steps: First, it will escape the shallow minimum.

Original: Second, it will reach either of the medium minima.
Revised: Second, it will reach either of the medium minima.

Original: Third, it will exit the medium minimum.
Revised: Third, it will exit the medium minimum.

Original: Fourth, it will reach the deep minimum.
Revised: Fourth, it will reach the deep minimum.

Original: (This might, of course, take a long time and the budget might not suffice.)
Revised: (This might, of course, take a long time and the budget might not sufﬁce.)

Original: Each of these steps can be interpreted as stopping times whose distribution we can compare for different H to assess how good the exploration is: the lower each of these stopping times, the better the exploration for a given H .
Revised: Each of these steps can be interpreted as stopping times whose distribution we can compare for different H to assess how good the exploration is: the lower each of these stopping times, the better the exploration for a given H .

Original: The precise experimental setup is the following.
Revised: The precise experimental setup is the following.

Original: For H ∈ { 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 } , we run fPGD 10times with a budget of 10 5 steps.
Revised: For H 2 { 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 } , we run fPGD 10times with a budget of 10 5 steps.

Original: We chose a generic step size of η = 0 . 01 .
Revised: We chose a generic step size of ⌘ = 0 . 01 .

Original: To zoom in on the effect, the perturbation scale σ is set to the highest level of noise admissible without divergence on this landscape, i.e. σ = 1 . 25 .
Revised: To zoom in on the effect, the perturbation scale  is set to the highest level of noise admissible without divergence on this landscape, i.e.  = 1 . 25 .

Original: (Note that we here again re-scale the final time to t = 1 so that the final amount of variance is equal for all Hurst parameters.)
Revised: (Note that we here again re-scale the ﬁnal time to t = 1 so that the ﬁnal amount of variance is equal for all Hurst parameters.)

Original: For each run, we determined after each step if fPGD is in the shallow, one of the medium or the deep minimum (or outside of all of them).
Revised: For each run, we determined after each step if fPGD is in the shallow, one of the medium or the deep minimum (or outside of all of them).

Original: From this information, we extracted the below summary statistics to study the exploration ability.
Revised: From this information, we extracted the below summary statistics to study the exploration ability.

Original: As a start, the lower row of Fig.
Revised: As a start, the lower row of Fig.

Original: 2 shows for H ∈ { 0 . 1 , 0 . 5 , 0 . 9 } , where fPGD ends up after 10 4 and after all 10 5 steps.
Revised: 2 shows for H 2 { 0 . 1 , 0 . 5 , 0 . 9 } , where fPGD ends up after 10 4 and after all 10 5 steps.

Original: It is clearly visible that the smaller the Hurst parameter, the faster fPGD moves to better local minima.
Revised: It is clearly visible that the smaller the Hurst parameter, the faster fPGD moves to better local minima.

Original: (For other values of H the same tendency would be visible, as the below figures will confirm.)
Revised: (For other values of H the same tendency would be visible, as the below ﬁgures will conﬁrm.)

Original: To give a more quantitative assessment of this effect, we provide a bar chart of the position of fPGD for all H after 10 4 and 10 5 steps in the upper row of Fig.
Revised: To give a more quantitative assessment of this effect, we provide a bar chart of the position of fPGD for all H after 10 4 and 10 5 steps in the upper row of Fig.

Original: It clearly shows that a small value of H is less likely to remain in the shallow minimum and is more likely to end up in the deep minimum.
Revised: It clearly shows that a small value of H is less likely to remain in the shallow minimum and is more likely to end up in the deep minimum.

Original: This is a strong effect: after 10% of steps only H ∈ { 0 . 1 , 0 . 3 } have exited at all.
Revised: This is a strong effect: after 10% of steps only H 2 { 0 . 1 , 0 . 3 } have exited at all.

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

Original: 1 has a chance of reaching the deep minimum by then.
Revised: 1 has a chance of reaching the deep minimum by then.

Original: While H ∈ { 0 . 3 , 0 . 5 } can eventually reach the deep minimum after an order of magnitude longer, H ∈ { 0 . 7 , 0 . 9 } cannot reach it at all in 10 5 steps.
Revised: While H 2 { 0 . 3 , 0 . 5 } can eventually reach the deep minimum after an order of magnitude longer, H 2 { 0 . 7 , 0 . 9 } cannot reach it at all in 10 5 steps.

Original: The beneficial exploration of H is confirmed by the shape of the cumulative distribution functions (cdfs) in the lower row of Fig.
Revised: The beneﬁcial exploration of H is conﬁrmed by the shape of the cumulative distribution functions (cdfs) in the lower row of Fig.

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 [55] or [22] for
Revised: Both PGD and Anti-PGD have unique advantages; see [55] or [22] for

Original: PGD and [36] for Anti-PGD.
Revised: PGD and [36] for Anti-PGD.

Original: It is thus natural to ask whether there are cases where non-Markovian fPGD ( H ∈ (0 , 1 / 2) ) is beneficial.
Revised: It is thus natural to ask whether there are cases where non-Markovian fPGD ( 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, this landscape has a d -dim.

Original: saddle point at x = 0 .
Revised: saddle point at x = 0 .

Original: If a stochastic optimizer can find the negative eigenvectors, it will reach a global minimum away from zero – while gradient descent remains stuck at zero.
Revised: If a stochastic optimizer can ﬁnd the negative eigenvectors, it will reach a global minimum away from zero – while gradient descent remains stuck at zero.

Original: The problem gets increasingly hard as the number of negative eigendirections decreases [22].
Revised: The problem gets increasingly hard as the number of negative eigendirections decreases [22].

Original: Indeed, [14] showed that vanilla gradient descent takes in the worst case exponential time (in the number of dimensions) to escape such saddle points.
Revised: Indeed, [14] showed that vanilla gradient descent takes in the worst case exponential time (in the number of dimensions) to escape such saddle points.

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

Original: 005 is a good choice for all values of H which we demonstrate in Appendix B.2.)
Revised: 005 is a good choice for all values of H which we demonstrate in Appendix B.2.)

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

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

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

Original: 5 ): While a large Hurst parameter ( H = 0 . 5) moves away from the saddle quickly, the non-Markovian fPGD
Revised: 5 ): While a large Hurst parameter ( H = 0 . 5) moves away from the saddle quickly, the non-Markovian fPGD

Original: [Figure]
Revised: [Figure]

Original: ( H = 0 . 4 ) finds the direction with negative curvature most quickly (in the sense that it gets below zero).
Revised: ( H = 0 . 4 ) ﬁnds the direction with negative curvature most quickly (in the sense that it gets below zero).

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

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

Original: Which H is best, will therefore depend on how many iterations the computational budget allows for.
Revised: Which H is best, will therefore depend on how 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 are beneficial in different situations.
Revised: This suggests that, if all other parameters are the same, different choices of Hurst parameters are beneﬁcial in different situations.

Original: It can thus be advantageous to tune and adapt the correlation of noise by choosing H .
Revised: It can thus be advantageous to tune and adapt the correlation of noise 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 beginning before 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 beginning before exiting the saddle and does not converge to the bottom of the minimum in the end.

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

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

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

Original: We stress, however, that the above experiment is a proof of concept and only shows that there exist landscapes where non-Markovian fPGD performs best.
Revised: We stress, however, that the above experiment is a proof of concept and only shows that there exist landscapes where non-Markovian fPGD performs best.

Original: On many other landscapes, PGD or Anti-PGD will still be better.
Revised: On many other landscapes, PGD or Anti-PGD will still be better.

Original: To make fPGD efficient on general landscapes, further work is needed to better understand the interplay between the scale σ and the correlation parameter H , which interact according to Eq.
Revised: To make fPGD efﬁcient on general landscapes, further work is needed to better understand the interplay between the scale  and the correlation parameter H , which interact according to Eq.

Original: Using this knowledge, one could design an adaptive method that 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 maximize the efﬁciency of perturbed-gradient methods.

Original: For the above experiments, we again emphasize that our constant choice of σ is suitable for all H , as justified in Appendix B.2.
Revised: For the above experiments, we again emphasize that our constant choice of  is suitable for all H , as justiﬁed in Appendix B.2.

Original:  Conclusion
Revised:  Conclusion

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

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

Original: This insight opens the doors to a new area of research where the noise of stochastic processes could be manipulated to speed up the convergence of an 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 of research where the noise of stochastic processes could be manipulated to speed up the convergence of an 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, on some non-convex potentials, that injection of fractional noise with small H speeds up the escape from a suboptimal minimum to a better one.
Revised: We showed, on some non-convex potentials, that injection of fractional noise with small H speeds up the escape from a suboptimal minimum to a better one.

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.

Original: Finally, one direction of interest would be to extend the analysis to more complex objective functions, which could potentially be achieved using the comparison theorem for SDEs [20].
Revised: Finally, one direction of interest would be to extend the analysis to more complex objective functions, which could potentially be achieved using the comparison theorem for SDEs [20].

Original: Acknowledgement We would like to thank Nacira Agram and Bernt Øksendal for some helpful discussions in the early stage of this project, as well as the reviewers for their feedback that greatly helped us improve this manuscript.
Revised: Acknowledgement We would like to thank Nacira Agram and Bernt Øksendal for some helpful discussions in the early stage of this project, as well as the reviewers for their feedback that greatly helped us improve this manuscript.

Original: Frank Proske acknowledges the financial support of the Center for
Revised: Frank Proske acknowledges the ﬁnancial support of the Center for

Original: International Cooperation in Education, project No CPEA-LT-2016/10139, Norway.
Revised: International Cooperation in Education, project No CPEA-LT-2016/10139, Norway.

Original: Hans Kersting and Francis Bach acknowledge support from the French government under the management of the
Revised: Hans Kersting and Francis Bach acknowledge support from the French government under the management of the

Original: Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference
Revised: Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference

Revised: ANR-19-P3IA-0001 (PRAIRIE 3IA Institute), as well as from the European Research Council (grant

Revised: SEQUOIA 724063).

Revised: Aurelien Lucchi acknowledges the ﬁnancial

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Original: We present two well-known inequalities that connect the concept of expected supremum to the metric entropy of T denoted by log N ( T, d, ϵ ) .

Original: First, we state Dudley’s inequality, which as stated by [49] gives the supremum of X t in terms of the metric entropy of T denoted by log N ( T, d, ϵ ) .

Original: Theorem 6 (Dudley’s integral inequality) .

Original: Let ( X t ) t ∈ T be a mean-zero random process on a metric space ( T, d ) with sub-Gaussian increments.

Original: Then, there exists a constant C > 0 such that

Original: [Equation]

Original: We refer the reader to [49] for a detailed discussion and a proof of the theorem.

Original: One can also obtain a lower bound on E [sup t ∈ T X t ] by Sudakov’s inequality (see Theorem 7.4.

Original: by [49]) stated below.

Original: Theorem 7 (Sudakov’s inequality) .

Original: Let ( X t ) t ∈ T be a mean-zero Gaussian process on a metric space ( T, d ) .

Original: Then, for any ϵ > 0 , we have

Original: [Equation]

Original: where d is the canonical metric d ( t, s )

Original: =

Original: ( E ( X t − X s ) 2 ) 1 / 2 .

Original: A.2 One-dimensional case A.2.1 Upper bound

Original: We first derive an upper bound on the expected supremum of X Ht introduced in Eq.

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

Original: Consider the stochastic process

Original: X Ht defined by

Original: [Equation]

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

Original: Then, for t ∈ [0 , 1] ,

Original: [Equation]

Original: where C β = e 3 β and C are positive constants (independent of H ).

Original: Proof.

Original: We will use the bound on the supremum of a stochastic process X t discussed in Theoremthat requires defining a metric δ on [0 , 1] such that

Original: [Equation]

Original: First, we note that Z t,s :=

Original: X Ht − X Hs is known to be randomly distributed [3].

Original: For X 0 , we have E [ Z t,s ] = 0 since E [ X Ht ]

Original: =

Original: X 0 e βt

Original: =

Original: The variance of the increments of the process is derived in [3, 29] as

Original: [Equation]

Original: Next, we upper bound the two integrals.

Original: We assume w.l.o.g. that β is positive.

Original: Else, we can simply exchange the bound on the two integrals (since we have e − βz in one integral and e βz in the other integral).

Original: Then we obtain

Original: [Equation]

Original: where we used e βz ≤ e β in the first inequality since ( t − s ) ≤ 1 (because t ∈ [0 , 1] ) and ¯ C β = e β in the last inequality.

Original: For the other integral, (cid:90)

Original: [Equation]

Original: Therefore, for s ≤ t ,

Original: [Equation]

Original: where we used ¯ C β ≥ 1 and C β :

Original: = e 3 β .

Original: Let Z ∼ N (0 , 1) and f Z is the probability density function of Z .

Original: We have

Original: E

Original: [Equation]

Original: Recall that the integral of the Gaussian function f ( x ) =

Original: e − ax 2 for a > 0 is (cid:82) ∞

Original: −∞ e − ax 2 dx =

Original: (cid:112) π a , therefore

Original: E

Original: [Equation]

Original: [Equation]

Original: where we need (cid:16) 12 − C β σ 2 ( t − s ) 2 H δ 2 ( t,s ) (cid:17) > 0 .

Original: Next, we choose δ such that

Original: [Equation]

Original: [Equation]

Original: Then

Original: [Equation]

Original: for r :=

Original: [Equation]

Original: bracket)

Original: [Equation]

Original: Hence

Original: [Equation]

Original: Then, using Theorem

Original: [Equation]

Original: we obtain that there exists a constant C 2 such that

Original: [Equation]

Original: Finally

Original: [Equation]

Original: we obtain an analytical form for the integral using (cid:82) c 0 (cid:112) log( cx ) dx = c √ π 2 (which can be derived using (cid:82) 1 0 (cid:113) log( 1 x ) dx = √ π 2 combined with a change of variables).

Original: A.2.2 Lower bound

Original: We now turn our attention to a lower bound.

Original: This will demonstrate that the result obtained in the upper bound is actually sharp in terms of the dependency to the Hurst parameter H .

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

Original: Consider the stochastic process

Original: X Ht defined by

Original: [Equation]

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

Original: Then

Original: [Equation]

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

Original: Proof.

Original: We will use Sudakov’s inequality that relies on the canonical metric

Original: [Equation]

Original: In order to get an expression for δ ( t, s )

Original: [Equation]

Original: we first note that Z t,s =

Original: X

Original: Ht − X Hs is known to be randomly distributed [3].

Original: For X 0 , we have E [ Z t,s ] = 0 since E [ X Ht ]

Original: =

Original: X 0 e βt

Original: =

Original: The variance of the increments of the process is given in [3, 29] as

Original: [Equation]

Original: Next, we lower bound the two integrals.

Original: We assume w.l.o.g. that β is positive.

Original: Else, we can simply exchange the bound on the two integrals (since we have e − βz in one integral and e βz in the other integral).

Original: Then we obtain

Original: [Equation]

Original: For the other integral,

Original: [Equation]

Original: Therefore, for s ≤ t ,

Original: [Equation]

Original: where C β = e − 2 | β | .

Original: We therefore obtain the following canonical metric

Original: [Equation]

Original: Then

Original: [Equation]

Original: for r := (cid:16) εσC β (cid:17) 1 H .

Original: So

Original: N δ

Original: [Equation]

Original: ε

Original: [Equation]

Original: By Sudakov’s inequality (Theorem 7), we have

Original: E

Original: [Equation]

Original: From our previous calculation, we have

Original: [Equation]

Original: Since arg max ε ∈ (0 , 1] ε (cid:112) − log( ε ) =

Original: e − 12 , we get

Original: [Equation]

Original: A.3 Multi-dimensional case A.3.1 Lower bound

Original: We first derive a lower bound on the expected supremum of X Ht introduced in Eq.

Original: (2) in the multidimensional case.

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

Original: Consider the d -dimensional stochastic process X H t ∈ R d defined by

Original: [Equation]

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

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

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

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

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

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

Original: Then

Original: [Equation]

Original: where c is a constant independent of H .

Original: Proof.

Original: The solution X t (we omit the superscript H for simplicity) is given by

Original: [Equation]

Original: where ϕ ( t ) is the fundamental solution, that solves the matrix equation

Original: [Equation]

Original: [Equation]

Original: In the case where the matrix A is constant with time, the fundamental solution is given by

Original: [Equation]

Original: By centering X t , we get

Original: [Equation]

Original: Using the stationarity of X t

Original: [Equation]

Original: we find that for t 1 ≤ t 2 ,

Original: [Equation]

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

Original: A .

Original: We have ∥

Original: [Equation]

Original: (cid:88)

Original: j = a ij ( t 2 − t 1 − s ) B H,js ds 

Original:  2 

Original:  + E [( B H,it 2 − t 1 ) 2 ] + 2 (cid:90) t 2 − t d (cid:88)

Original: j = a ij ( t 2 − t 1 − s ) E [ B H,js B H,it 2 − t 1 ] ds (cid:111) ( i ) =

Original: d (cid:88)

Original: i =

Original: (cid:110)

Original: E      (cid:90) t 2 − t d (cid:88)

Original: j = a ij ( t 2 − t 1 − s ) B H,j s ds 

Original:  2    (cid:124) (cid:123)(cid:122) (cid:125) =

Original: A + | t 2 − t 1 | 2 H + (cid:90) t 2 − t d (cid:88)

Original: j = a ij ( t 2 − t 1 − s ) (cid:0) | t 2 − t 1 | 2 H + | s | 2 H − | t 2 − t 1 − s |

Original: 2 H (cid:1) ds (cid:124) (cid:123)(cid:122) (cid:125) =: B (cid:111) ≥ (cid:90) t 2 − t a ii

Original: [Equation]

Original: t

Original: [Equation]

Original: t

Original: [Equation]

Original: t

Original: [Equation]

Original: |

Original: [Equation]

Original: H +

Original: [Equation]

Original: s

Original: [Equation]

Original: H

Original: [Equation]

Original: t

Original: [Equation]

Original: (cid:1) ds

Original: [Equation]

Original: t

Original: [Equation]

Original: − t 1 | 2 H where ( i ) uses

Original: [Equation]

Original: and

Original: [Equation]

Original: =

Original: [Equation]

Original: 2 H (cid:1) .

Original: Next, assume that there exists i 0 ∈ { 1 , . . .

Original: d } such that

Original: [Equation]

Original: where 0 ≤ C 1 ≤ 1 .

Original: Then

Original: [Equation]

Original: Taking δ ( t 2 , t 1 )

Original: =

Original: C 1 | t 2 − t 1 | H as the metric on [0 , 1] , we obtain

Original: [Equation]

Original: for r := (cid:16) εC 1 (cid:17) 1 H .

Original: So

Original: [Equation]

Original: By Sudakov’s inequality (Theorem 7), we have

Original: [Equation]

Original: where C 2 is a constant independent of H .

Original: From our previous calculation, we have

Original: [Equation]

Original: =

Original: [Equation]

Original: [Equation]

Original: ε

Original: [Equation]

Original: Since arg max ε ∈ (0 , 1] ε (cid:112) − log( ε ) =

Original: e − 12 , we get sup 0 <ε ≤ 2 C 1 ε (cid:112) log N δ ( ε ) ≥ e −(cid:115)

Original: [Equation]

Original: (cid:18)

Original: [Equation]

Original: e

Original: [Equation]

Original: (cid:19) .

Original: We conclude that

Original: [Equation]

Original: Alternative proof for lower bound Recall we have shown that

Original: [Equation]

Original: Let’s first focus on the term A :

Original: [Equation]

Original: Then, note that

Original: [Equation]

Original: Therefore

Original: E

Original: [Equation]

Original: For the term B

Original: [Equation]

Original: we have

Original: [Equation]

Original: Combining the lower bounds on A and B

Original: [Equation]

Original: we obtain

Original: [Equation]

Original: Let C 1 = min t ∈ [0 , 1] ( a 1 ( i ) t 2 + a 2 ( i ) t + 1) > 0 .

Original: We can then choose the metric on [0 , 1] as δ ( t 2 , t 1 ) =

Original: C 1 | t 2 − t 1 | H .

Original: The rest of the proof is identical but we obtain a different constant C 1 that covers different scenarios of the entries of the matrix A .

Original: A.3.2 Upper bound

Original: We now turn our attention to an upper bound.

Original: This will demonstrate that the result obtained in the lower bound is actually sharp in terms of the dependency to the Hurst parameter H .

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

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

Original: [Equation]

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

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

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

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

Original: Then

Original: [Equation]

Original: where C 2 = max di,j =1 sup t ∈ [0 , 1] | a ij ( t ) | 2 + max di,j =1 sup

Original: [Equation]

Original: ∈

Original: [Equation]

Original: ] | a ij ( t ) | + 1 and C > 0 is a constant independent of H .

Original: Proof.

Original: The solution X t (we omit the superscript H for simplicity) is given by

Original: [Equation]

Original: where ϕ ( t ) is the fundamental solution, that solves the matrix equation

Original: [Equation]
Revised: [N/A]

Original: In the case where the matrix A is constant with time, the fundamental solution is given by

Original: [Equation]

Original: By centering X t , we get

Original: [Equation]

Original: Using the stationarity of X t

Original: [Equation]

Original: we find that for t 1 ≤ t 2 ,

Original: [Equation]

Original: [Equation]

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

Original: A .

Original: We have

Original: [Equation]

Original: where ( i ) uses

Original: E

Original: [Equation]

Original: and

Original: E

Original: [Equation]

Original: We will once again build on the proof in the 1-dimensional case.

Original: To do so, we first observe that, given an interval T = [0 , 1] ,

Original: [Equation]

Original: thus

Original: [Equation]

Original: Focusing on the i -th coordinate, we first get for the term A that

Original: [Equation]

Original: Then, note that

Original: [Equation]

Original: Therefore

Original: E

Original: [Equation]

Original: For the term B

Original: [Equation]

Original: we have

Original: [Equation]

Original: Combining the bound on the terms A and B, we obtain the following

Original: [Equation]

Original: where

Original: [Equation]

Original: Taking δ 2 ( t 2 , t 1 ) =

Original: dC 2 | t 2 − t 1 | 2 H as the metric on [0 , 1] , we obtain

Original: [Equation]

Original: for r

Original: [Equation]

Original: Hence

Original: [Equation]

Original: Then, using Theorem 6 applied to each centered component process (with positive and negative sign)

Original: [Equation]

Original: we obtain that there exists a constant C such that

Original: [Equation]

Original: Finally, we obtain an analytical form for the integral using (cid:82) c 0 (cid:112) log( cx ) dx = c √ π 2 (which can be derived using (cid:82) 1 0 (cid:113) log( 1 x ) dx = √ π 2 combined with a change of variables).

Original: B Additional experiments B.1 Demonstration of the 1 / √ H scaling from Theorems 1 and

Original: In this section, we demonstrate the 1 / √ H scaling of the expected supremum of the one-dimensional fOUP.

Original: Recall that Theorem 1 proved that

Original: [Equation]

Original: for C > 0 .

Original: Theorem 2 and Corollary 3 also provide a lower bound

Original: [Equation]

Original: although they either require 1) σ > e 2 | β | 12 and | β | < 1 or 2) 0 < σ < e 2 | β |− 12 and | β | ≤ 14 .

Original: Here, we want to verify whether the lower bound E (cid:104) sup t ∈ [0 , 1] ∥

Original: X Ht ∥ (cid:105) ∼ 1 √ H holds generally, even outside the range required for σ and β .

Original: To this end, we ran a Monte Carlo simulation with 10simulations of this expectation and plotted it next to a function of the form f ( x ) =

Original: w 0 + w 1 1 √ H , where we fitted ( w 0 , w 1 ) by ordinary least squares.

Original: For the discretization, we used 10 5 steps (i.e. a step size η of 10 − 5 ).

Original: For H , we chose an evenly spaced grid of mesh size 0 .

Original: from 0 .

Original: 2 to 0 .

Original: We did not include H = 0 .

Original: 1 here because the Monte-Carlo approximation is known not be a good approximation for such small values of H .

Original: This issue is described in more detail in [8] (see e.g. Figure 1 and description in the text).

Original: We remark that we still observe a relatively good

Original: The resulting plots are depicted in Fig.

Original: B.1 for noise scales σ ∈ { 10 − 3 , 10 − 2 , 10 − 1 , 10 0 } .

Original: In each plot, we used β = 1 / 2 such that all values of σ violate both conditions: 1) σ > e 2 | β |− 1 / 2 and | β | <or 2) 0 < σ < e 2 | β |− 12 and | β | ≤ 14 . .

Original: Nonetheless, the fit between the simulation and the theoretical scaling is almost perfect.

Original: This confirms our theoretical findings and shows that the scaling also holds outside the regime of Theorem 2 and Corollary 3.

Original: B.2 Justification for choice of σ in Section 5.

Original: In this section, we justify the use of σ = 0 .

Original: 005 in all experiments from Section 5.2.

Original: There, we used σ = 0 .

Original: 005 for all H ∈ { 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 .

Original: 4 , 0 . 5 } in Figure 4, i.e. in both the Markovian { 0 . 0 , 0 . 5 } and the non-Markovian cases { 0 . 1 , 0 .

Original: Here, we explain why σ is not chosen dependent on H .

Original: Our justification is simple: The choice of σ = 0 .

Original: 005 is a good choice for all H .

Original: To see this, consider

Original: Figure 6.

Original: For a fixed H ∈ { 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 } and all σ ∈ { 0 . 05 , 0 .

Original: } , they depict the performance on the regularized quadratic from Section 5.2, analogous to Figure 4.

Original: We observe that indeed, for all choices of H , the curve for σ = 0 .

Original: 005 is best – in the sense that it exits the saddle fastest (by reaching a negative value) and converges quickly to the final minimum.

Original: Clearly, for all H , there is no point in picking an even smaller or larger order of magnitude for σ .

Original: Hence, we conclude that the choice of σ = 0 .

Original: 005 is close to the optimal choice for all H .

Original: While we acknowledge that grid-searching the optimal σ at a finer scale would maybe uncover small differences in the optimal σ , such precision would be beyond our goal of a proof-of-concept and would distract from the bigger picture.

Original: In fact, the constant σ also ensures that the increments of fBM with different

Original: H are distributed the same (but correlated differently).

Original: Thus, this experimental set-up allows us to zoom in on the effect of the Hurst parameter H – with the result that the non-Markovian H = 0 .performs best.

Original: B.3 Moving to better local minima in bi-stable potentials

Original: We consider the behavior of fPGD in a bi-stable optimization landscape, given by f ( x ) =

Original: [Equation]

Original: (27) where χ is the indicator function and the parameters ( v 0 , v 1 , v 2 , a, c, m ) determine the position and shape of two adjacent local minima.

Original: Similar landscapes have been studied in the context of escaping spurious local minima [55] and in the context of generalization in machine learning by selecting flat minima [33, 53], which are by some believed to generalized better.

Original: We here study both cases.

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

Original: (0 . 0 , 45 . 0 , − 12 .

Original: 5 , 3 . 5 , 13 . 9 , 18 . 0) ; for the corresponding landscape see the upper left subplot of Fig.

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

Original: (0 . 0 , 30 . 0 , 0 . 0 , 1 . 2 , 8 . 7 , 15 . 0) ; for the for the corresponding landscape see the upper left subplot of Fig. 8.

Original: On these landscapes, we run fPGD with different choices of H .

Original: Note that, in the two minima (and the saddle 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 parameter leads to a faster escape from the shallow minimum (left) to the deep minimum (right) in Fig.

Original: 7, or from the sharp to the flat minimum in Fig.

Original: In both cases, we run fPGD for N = 1000 iterations with a fixed step size of η = 1 . 0 , initialised at x = 0 (the bottom of the shallow/sharp 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 σ = 80 . 
-]

Original: · N − H .

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

Original: [-1 distributed like 80 . 
-]
Revised: [-1 distributed like 80 . 
-]

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

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

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

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

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

Original: Figs.

Original: 7 and 8 depict our findings.

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

Original: [Figure]

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

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

Original: We thus escape faster from the spurious shallow/sharp minimum to the desirable deep/flat one.

Original: (Note, however, if the noise is not cooled down, one might oscillate back.)

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

Original: 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.

Original: Note that physicists [45] have also observed faster escapes for smaller Hurst parameters on a related

Original: Kramer-like escape problem.

Original: [Figure]

Original: [Figure]