Nesterov acceleration despite very noisy gradients
Authors: Kanan Gupta
Jonathan W. Siegel
Stephan Wojtowytsch
Venue: NeurIPS 2024 poster (2024)
Keywords: Stochastic optimization
stochastic acceleration
smooth convex optimization
deep learning
accelerated gradient descent
Primary area: optimization

Abstract: We present a generalization of Nesterov's accelerated gradient descent algorithm. Our algorithm (AGNES) provably achieves acceleration for smooth convex and strongly convex minimization tasks with noisy gradient estimates if the noise intensity is proportional to the magnitude of the gradient at every point. Nesterov's method converges at an accelerated rate if the constant of proportionality is below 1, while AGNES accommodates any signal-to-noise ratio. The noise model is motivated by applications in overparametrized machine learning. AGNES requires only two parameters in convex and three in strongly convex minimization tasks, improving on existing methods. We further provide clear geometric interpretations and heuristics for the choice of parameters.

Nesterov acceleration despite very noisy gradients
Kanan Gupta
Jonathan W. Siegel
Stephan Wojtowytsch
Department of Mathematics University of Pittsburgh kanan g@pitt..pittttte
Department of Mathematics
Department of Mathematics University of Pittsburgh s. woj @pi tt.tteee edu
Texas A&M University jwsiegel @tamettaaeeeee edu
Abstract
We present a generalization of Nesterov's accelerated gradient descent algorithm. Our algorithm (AGNES) provably achieves acceleration for smooth convex and strongly convex minimization tasks with noisy gradient estimates the noise intensity is proportional to the magnitude of the gradient every point. Nesterov's method converges at an accelerated rate if the constant of proportionality is below 1, while AGNES accommodates any -ignal-to-noise ratio. The noise model is motivated by applications in overparametrized machine learning. AGNES requires only two parameters in convex and three in strongly convex minimization tasks, improving on existing methods. We further provide clear geometric interpretations and heuristics for the choice of parameters.
Introduction
The recent success of deep learning [LeCun et al., 2015] is built on stochastic first order optimization methods such as stochastic gradient descent [LeCun et al., 1998] and ADAM [Kingma and Ba, 2014], which have enabled the large-scale training of neural networks. While such tasks are generally non-convex. accelerated first order methods for convex optimization have proved practically useful. Specifically, Nesterov [1983]'s 's accelerated gradient descent has become standard training method [Sutskever et al., 2013].
Modern neural networks tend to operate in the overparametrized regime, i.e. the number of model parameters exceeds the number of data points be fit [Belkin, 2021]. In this setting, minibatch gradient estimates are exact (namely, exactly 0) on the set of global minimizers since data can be interpolated exactly. Motivated by such applications, Vaswani et al. [2019] proved that Nesterov [2012]'s s accelerated coordinate descent method (ACDM) achieves acceleration in (strongly) convex optimization with mulriplicative noise, i.e. when assuming stochastic gradient estimates for which the noise intensity scales linearly with the magnitude of the gradient. Conversely, Liu and Belkin [2018] 8] show that the original version of Nesterov [1983]'s 's method generally does not achieve acceleration in this setting.
Another algorithm with similar goal is the continuized Nesterov method (CNM), which has been studied by Even et al. [2021], Berthier et al. [2021] in convex optimization (deterministic or with additive noise) and with multiplicative noise for overparametrized linear least squares regression. For a more extensive discussion of the context of our work in the literature, please see Section 2.
Vaswani al. [2019]'s algorithm is four parameter scheme in the strongly convex case, which reduces to a three parameter scheme the convex case. Liu and Belkin [2018] introduce a simpler three parameter scheme. but only prove that it achieves acceleration for overparametrized linear problems. In this work, we demonstrate that it is possible to achieve the same theoretical guarantees Vaswani al. [2019] with simpler scheme, which can be considered reparametrized version of Liu and Belkin [2018]'s Momentum-Added Stochastic Solver (MaSS) method. More precisely, we prove the following:
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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1. We show that Nesterov's accelerated gradient descent achieves an accelerated convergence rate, but only with noise which strictly smaller than the gradient in the L*-sense. We also show numerically that when the noise is larger than the gradient, the algorithm diverges for a choice of step size for which gradient descent remains convergent.
2. Motivated by this, we introduce a generalization of Nesterov's method, which we call Accelerated Gradient descent with Noisy EStimators (AGNES), which provably achieves acceleration no matter how large the noise is relative to the gradient, both in the convex and strongly convex cases.
3. When moving from NAG to AGNES, the learning rate 'bifurcates' to two parameters in order to accommodate stochastic gradient estimates. The extension requires three hyperparameters in the strongly convex case and two in the convex case.
4. We provide a transparent geometric interpretation of the AGNES parameters in terms of their scaling with problem parameters (Appendix F.3) and the continuum limit models for various scaling regimes (Appendix C).
5. We build strong intuition for the choice of hyperparameters for machine learning applications and empirically demonstrate that AGNES improves the training of CNNs relative to SGD with momentum and Nesterov's accelerated gradient descent.
2 Literature Review
Accelerated first order methods. Accelerated first order methods have been extensively studied in convex optimization. Beginning with the conjugate gradient (CG) algorithm introduced by Hestenes and Stiefel [1952], the Heavy ball method of Polyak [1964], and Nesterov [1983]'s seminal work on accelerated gradient descent, many authors have developed and analyzed accelerated first order methods for convex problems, including Beck and Teboulle [2009], Nesterov [2012, 2013], Chambolle and Dossal [2015], Kim and Fessler [2018] to name just a few.
An important line of research is to gain an understanding of how accelerated methods work. After Polyak [1964] derived the original Heavy ball method discretization of an ordinary differential equation, A Avarez et al. [2002], Su et al. [2014], Wibisono et al. [2016], Zhang et al. 20188. Siegel [2019], Shi al. [2019], Muchlebach and Jordan [2019]. Wilson al. [2021]. Shi et al. [2021], Suh et al. [2022], Attouch et al. [2022], Aujol et al. [2022b,a]. Dambrine et al. [2022] studied accelerated first order methods from the point of view of ODES. This perspective has facilitated the use of Lyapunov functional analysis to quantifyy the convergence properties. We remark that in addition to the intuition provided by differential equations, Joulani et al. [2020] and Gasnikov and Nesterov [2018] have also proposed interesting ideas for explaining and deriving accelerated first-order methods. In addition, there has been large interest in deriving adaptive accelerated first order methods, see for instance Levy et al. [2018) 8), Cutkosky [2019], Kavis et al. [2019].
Stochastic optimization. Robbins and Monro [1951] first introduced optimization algorithms where gradients are only estimated by a stochastic oracle. For convex optimization, Nemirovski et al. [2009], Ghadimi and Lan [2012] obtained minimax-optimal convergence rates with additive stochastic noise.
In deep learning, stochastic algorithms are ubiquitous in the training of deep neural networks, see [LeCun al., 1998, 2015, Goodfellow et al., 2016, Bottou et al., 2018]. Here, the additive noise assumption not usually appropriate. As Wojtowytsch [2023]. Wu et al. [2022a] show, the noise is of low rank and degenerates on the set of global minimizers. Stich [2019]. Stich and Karimireddy [2022], Bassily et al. [2018]. Gower et al. [2019], Damian et al. [2021]. Wojtowytsch [2023], Zhou et al. [2020] consider various non-standard noise models and [Wojtowytsch, 2021, Zhou et al., 2020, Li et al., 2022] study the continuous time limit of stochastic gradient descent. These include noise assumptions for degenerate noise due to Bassily et al. [2018], 8), Damian et al. [2021], I. Wojtowytsch [2023, 2021], low rank noise studied by Damian et al. [2021]. Li et al. [2022] and noise with heavy tails explored by Zhou al. [2020].
Acceleration with stochastic gradients. Kidambi et al. [2018] prove that there are situations in which it is impossible for any first order oracle method to improve upon SGD due to information-

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theoretic lower bounds. More generally, lower bounds in the stochastic first order oracle (SFO) model were presented by Nemirovski al. [2009] (see also [Ghadimi and Lan, 2012]).
partial improvement on the state of the art is given by Jain et al. [2018). who present an accelerated stochastic gradient method motivated by particular low-dimensional and strongly convex problem. Laborde and Oberman [2020] obtain faster convergence of an accelerated method under an additive noise assumption by Lyapunov function analysis. Bollapragada et al. [2022] study an accelerated gradient method for the optimization of strongly convex quadratic objective function with minibatch noise.
Closest to our work are Liu and Belkin [2018], Vaswani et al. [2019], Even et al. [2021], Berthier et al. [2021] who study generalizations of Nesterov's 's method in stochastic optimization. Liu and Belkin [2018], Even et al. [2021] obtain guarantees with noise of approximately multiplicative noise in overparametrized linear least squares problems and for general convex objective functions with additive noise and in deterministic optimization. Vaswani et al. [2019] obtain comparable guarantees for the more complicated method of Nesterov [2013].
3 Algorithm and Convergence Guarantees
3.1 Assumptions
In the remainder of this article, we consider the task of minimizing an objective function ʃ I IR using stochastic gradient estimates g. We assume that J. g and the initial condition xh satisfy:
1. The initial condition zu is a (potentially random) point such that (z)())) + ||= < oc.
2. f is L L-smooth, i.e. V f is L L-Lipschitz continuous with respect to the Euclidean norm.
3. There exists probability space (S2, AA, IP) and a gradient estimator, i.e. measurable function g : _8 x 9 + _ such that for all æ En the properties [x]g(r,w)] = (x) (unbiased gradient oracle) and B. (1g - f o² [+) (multiplicatiive noise scaling) hold.
justification of the multiplicative noise scaling is given in Section 4. In the setting of machine learning, the space Q2 is given by the random subsampling of the dataset. A rigorous discussion of the probabilistic foundations given in Appendix D.
3.2 Nesterov's Method with Multiplicative Noise
First we analyze Nesterov [1983]'s accelerated gradient descent algorithm (NAG) in the setting of multiplicative noise. NAG is given by the initialization
x' and the two-step iteration
1n+1 = 4',-- - -7...
2n+1 = In+1 + (I - -x,) xn + Pm -ng,. -x,) (1)
where g'. =g(2", wx) and the variables wn are iid samples from the probability space Ω2 i.c. g,, is an unbiased estimate (=((). We write p instead of pn. in cases where a dependence on not required. We show that this scheme achieves an D(1/n" convergence rate for convex functions but the best of our knowledge, this analysis optimal.
only in the case that a <
Theorem 1 (NAG, convex case). Suppose that In and x', are generated by the time-stepping scheme (I), f and g satisfy the conditions laid out in Section 3.1, convex, and is a point such that f(x*) = inf f(r). If a < 1 and the parameters are chosen such that
1-
2E [ x x01
0 0<Ns
and Pn. n 3
then
02)+
mm2
The expectation on the right hand side is over the random initialization To-
The proof of Theorem 1 is given in Appendix E. Note that the constant 1/n blows up / 1 and the analysis yields no guarantees for a > This mirrors numerical experiments in Section
Theorem 2 (NAG, strongly convex case). In addition to the assumptions in Theorem suppose that is justtrongyy convex and the parameters are chosen such that
1-02
- ymm then ([f(x)) 2(1 (11 f(x")
n L(I+02) and p
1 pm
3

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Just like in the convex case, the step size decreases to zero as 7 1, and we fail to obtain convergence guarantees for o 2 We argue the proof of Theorem 2, given in Appendix F, that is not possible to modify the Lyapunov sequence analysis to obtain better rate of convergence. This motivates our introduction of the more general AGNES method below.
Notably, there cannot be diverging lower bound for NAG since gradient descent arises in the special case p 0, and gradient descent converges for small stepsize with multiplicative noise [Wojtowytsch, 2023]. On the other hand, Liu and Belkin [2018] show that NAG does not achieve accelerated convergence with multiplicative type noise even for quadratic strongly convex functions.
3.3 AGNES Descent algorithm
The proofs of Theorems and 2 suggest that the momentum step in (1) is quite sensitive to the step size used for the gradient step, which severely restricts the step size 7). We propose the Acceler- ated Gradient descent with Noisy EStimators (AGNES) scheme, which addresses this problem by introducing additional parameter in the momentum step:
To z/:
Tn+1 z ngn. 2 Znt1 + Pm Pn ag'. zn):
(2)
where Sn = g( wn) as before. Equivalently, AGNES can be formulated as three-step scheme with an auxiliary velocity variable on. initialized as v0
J/1 = Fn + a'e,,
In+1 = J/ -19' Un+1 = Pm (v - I.)-
We show that the two formulations of AGNES equivalent in Appendix B.I. However, find (3) more intuitive (see Appendix for a continuous time interpretation) and easier to implement as algorithm without storing past values of In. The pseudocode and set of suggested default parameters are given in Algorithm 1.
Algorithm I: Accelerated Gradient descent with Noisy EStimators (AGNES)
Input: f (objective/loss function), 20 (initial point), = 10-3 (learning rate). 17 10 (correction step size), p 0.99 (momentum), N (number of iterations)
+-0 0 to <-
while < N do
In + V v
// gradieet estimate
Un+1 p(Un 9n)
In+1= f In + Q1p-1 119n
f- +1
end
ON +- (IN(
EN IN 79n
Return: EN
From (1) and (2), we note that NAG is AGNES with the special choice a = n. Allowing a and 7 be different helps AGNES achieve accelerated rate of convergence for both convex and strongly convex functions, no matter how large a is. While for gradient descent, only the product L(1 + o²) has to be considered, this is not the case for momentum-basee schemes. We consider first the convergence rate in the convex case.
Theorem 3 (AGNES, convex case). Suppose that n. and s' are generated by the time-stepping scheme (3), f and g' = gg"", satisfy the conditions laid out in Section 3.1, f is convex, and x² a point such that f(x') inf f(x). the parameters are chosen such that
2(1 - -TL
0 O<T< =(1+02)
a I+ Pn n+1++ 1+1 for ag 2
then
mL((1 0
of E 100 12
E [[f(xn) - f(x*)]
f(x)))
2an
In particular, if T(14264). we may make the universal choice ao = 4,i.e n Only the parameters 77, depend on the specific problem. The proof of Theorem 3 is given in Appendix E.

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There, we also present an alternative version of Theorem 3 for a different choice of parameters
" n + (1 a
+ 74g
1+022 Pr. = + no + 3
for potentially large vio
2n0² 7-a(1+0²) 2 20² The convergence guarantees are similar in both cases.
The benefit of the accelerated scheme is improvement from decay rate of O(1/n) the rate O(1/n²), which is optimal under the given assumptions even in the deterministic case. While the noise be orders of magnitude larger than the quantity we want to estimate, it only affects the constants in the convergence, not the rate. We get an analogous result for strongly convex functions.
Theorem (AGNES, strongly convex case). In addition to the assumptions in Theorem 3, suppose that f is p-strongly convex and the parameters chosen such that
1+o
< 7 S I(I
and
a
then
E + o²
1
V 1402
-((( 2 (1-
B[S(x)) - f(x)))
1+
Choosing 77 too small can be interpreted as overestimating L or a. Choosing a too small (with respect to n) can be interpreted overestimating Since every L-Lipschitz function is L' L'-Lipschitz for L' > L, and since the multiplicative noise bound with constant a implies the same bound with o' a, exponential convergence still holds generally slower rate.
We note that since (=) < (z) inf f) (Lemma in Appendix D), Theorems 3 and lead to analogous convergence results for [(=)( as well. Due to the summability of the sequences n2 and Td for
1, we get not only convergence expectation but also almost sure convergence. The proof is given in Appendix E.
Corollary 5. In the setting of Theorems 3 and 4, f( + inf I with probability 1.
In the deterministic case a = O, we have a = 17 in both Theorems and 4. Theorem 4, the parameters coincide with the usual choice for NAG, while opted for simple statement in Theorem 3 which does not exactly recover the standard choice below easily cover these special cases well. 0 < 1, both AGNES and NAG converge with the same rate n 2 in the convex case, but the constant of NAG is always larger. In the strongly convex case, even the decay rate of NAG is slower than AGNES for (0, 1) since 1- o² (1 [1+07% (2)-1 We see the real power of AGNES in the stochastic setting where it converges for very high values of when Nesterov's 's method may diverge. For the optimal choice of parameters, results in terms of the time-compleses SGD, see Theorems 17 and 22 in Appendices E and F respectively.
1/L. and Pr, = n/ (n + 3). The proofs
summarize the of SGD and AGNES in Figure 1. For the related guarantee for
Remark 6 (Batching). Let us compare AGNES with two families gradient estimators:
1. g/x 9(" wn) studied in Theorems and 4.
2. A gradient estimator g', := 1 Σi= g(2", wn,)) which averages multiple independent my estimates to reduce the variance.
The second gradient estimator falls into the same framework with á Ω2 and 82 o² /ng- Assuming vector additions cost negligible time, optimizer steps are only as expensive as gradient evaluations. In this setting- which is often realistic in deep learning - it is appropriate to compare f (ns n iterations using g') and (X,) (n iterations with g()). For the strongly convex case, we note that
V 1400
51 VL. 1404/a6 and only if
log 1 11
1 I+o
E I+0=/mo
1+
1+ σ²
no 2
Thb + o² Rb.
log 1 (1-
V I+07
T+a7
+ 0² 02/2mb
5

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Time complexity
Convex
p-strongly convex
SGD
o (
0 4 (1 +1)+ log el
(V
o (V 1 (1 (1+ 0²) |1gggl
AGNES
Figure 1: The minimal n for AGNES and SGD such that ([f(z,) inf fl < & when minimizing an L-smooth function with multiplicative noise intensity in the gradient estimates and under convexity assumption. The SGD rate of the j-strongly convex case is achieved more generally under PL. condition with Pl-constant μ.. While SGD requires the optimal choice of one variable to achieve the optimal rate, AGNES requires three (two in the determinstic case).
The approximation is well-justified in the important case that < L. In particular, the upper bound for non-batching AGNES is always favorable compared to the batching version as n, E N21. and the two only match for the optimal batch size ns 1. The optimal batch size for minimizing f is the largest one that can be processed in parallel without increasing the computing time for a single step. A similar argument holds for the convex case.
With slight modification, the proof of Theorem 3 extends to the situation of convex objective functions which do not have minimizers. Such objectives arise for example in linear classification with the popular cross-entropy loss function and linearly separable data.
Theorem 7 (Convexity without minimizers). Let f be convex objective function satisfying the assumptions Section 3.1 and n,, be generated by the time-stepping scheme (3). Assume that 1), a and Pn are as in Theorem 3. Then lim n (z,)] inf f(x).
The proof and more details are given in Appendix E. For completeness, we consider the case of non-convex optimization in Appendix G. As a limitation, we note that multiplicative noise is in machine learning for global minimizers, but not at generic critical points.
3.4 Geometric Interpretation
Let us briefly discuss the parameter choices in Theorem 4. As we consider larger a for fixed and L, the decay factor p moves closer to 1. This slows the 'forgetting' of past gradients in on. allowing us to better average out stochastic noise. The price we pay is computing with more outdated gradients, slowing convergence. Our choice balances these effects.
In AGNES, p inadvertently also governs magnitude of the momentum variable un,, which scales as (1 p)- for objective functions with constant gradient and >> 1. To compensate, choose a smaller compared to 77 when a (and thus (1 p)-1 is large. Nevertheless, the effect of the momentum step does not decrease. For further details, see Appendix F.3.
For further interpretability, we obtain several ODE and SDE continuous time descriptions of AGNES in Appendix C.
4 Motivation for Multiplicative Noise
In supervised learning applications, the learning task often corresponds to minimizing a risk or loss function R(w) = * Di"" ((h(W,hh)))) 2.), =: E. (ww), where h : IRmh x Rd + R&, (w.x) r h(wxr) and é R x R [0, oo) are parametrized function of weights w and data and loss function measuring compliance between h(w, 2r;) and yi respectively.' Safran and Shamir [2018], Chizat and Bach [2018], 8), Du et al. [2018) show that working in the overparametrized regime m >> N simplifies the optimization process and Belkin et al. [2019, 2020] illustrate that facilitates generalization to previously unseen data. Cooper [2019] shows that fitting N constraints with m parameters typically leads to an m -ddimensional submanifold M of the parameter space Rm
Both and R are commonly called a 'loss function' in the literature. To distinguish between the two, we will borrow the terminology of statistics and refer to R as the risk functional and & as the loss function. The notation L, which is often used in place of R, is reserved for the Lipschitz constant in this work.

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Fitting Purge fanctics
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Figure 2: To be able to quantify the gradient noise exactly, we choose relatively small models and data sets. Left: A ReLU network with four hidden layers of width 250 is trained by SGD fit random labels yi (drawn from a 2-dimensional standard Gaussian) at 1, 000 random data points I; (drawn from 500-dimensional standard Gaussian). The variance 6² of the gradient estimators ~ 10 times larger than the loss function and ~ 10 times larger than the parameter gradient. This relationship is stable over approximately ten orders of magnitude. Right: ReLU network with two hidden layers of width 50 is trained by SGD to fit the Runge function 1/(1 + x²) on equispaced data samples in the interval [-8, 8). Also here, the variance in the gradient estimates is proportional to both the loss function and the magnitude of the gradient.
such that all given labels yi are fit exactly by h(w,-) the data points x; for M. R on the smooth set of minimizers M = R ({0)).
If N is large, it is computationally expensive to evaluate the gradient VR(w) = 1 risk function R exactly and we commonly resort to stochastic estimates
N E V4, of the
k (8n, (h(w,,:)) (wkt(w,,;)))
g = Σ v((w) -
Σ
T&h
iEL,
where T C (1...., N) is subsampled collection of ni, data points (a batch or minibatch). Minibatch gradient estimates are very different from the stochasticity we encounter e.g. in statistical mechanies:
1. The covariance matrix Σ = ΣE (ve, VR) & (Ve, VR) of the gradient estimators VG; has low rank N m.
2. Assume specifically that is loss function which satisfies E v) 0 for all y E R& such as the popular f2 -loss function E(h, = ||h 1/ll. Then ((;(w) = 0 for all i (1..... N} and all w E MM- R R-). In particular, minibatch gradient estimates are exact on M.
The following Lemma makes the second observation precise in the overparameterized regime and bounds the stochasticity of mini-bater estimates more generally.
Lemma 8 (Noise intensity). Assume that E(h, y) = |/- - yll and h : RM x Rd - R satisfies ||V h(w,,,;) 112 < C(1 + [|ll])" for some C.p 0 and all w E Rm and = 1,..., N. Then for all w E m
N V4- < 4C² (1 + [[]l)) R(w).
1
N
Lemma 8 is proved in Appendix H. is a modification of [Wojtowytsch, 2023, Lemma 2.] for function models which are locally, but not globally Lipschitz-continoous in the weights such as deep neural networks with smooth activation function. The exponent p may scale with network depth.
Lemma 8 describes the variance of a gradient estimator which uses random index i (L...., N} and the associated gradient Vl, is used to approximate VR. a batch In of n, indices is selected randomly with replacement, then the variance of the estimates scales in the usual way:
E
1
Σ
V4 VR
<
40 + 2 R(w).
(4)
Thb
IET
7

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ACRM
30 0M
30"
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20"0
3010
Figure 3: We plot EE fa(x,)) on a loglog scale for SGD (blue), AGNES (red), NAG (green), ACDM (orange) and CNM (maroon) with d = (left) and = 16 (right) for noise levels a 0 (solid line), 10 (dashed) and a 50 (dotted). The initial condition is zg Means are computed over 200 runs. After an initial plateau, AGNES, CNM and ACDM significantly outperform SGD settings, while NAG (green) diverges a is large. The length of the initial plateau increases with a.
in all simulations.
As noted by Wu al. [2019, 2022b], and |/ often behave similarly in overparametrized deep learning. We illustrate this in Figure 2 together with Lemma 8. Heuristically, therefore replaced (4) by more manageable assumption akin E Σ |V,, VRII²] @R|| in Section 3.1. The setting where the signal-to-noise ratio (the quotient estimate variance and true magnitude) (2(1) is often referred to 'multiplicative noise', it resembles the noise generated by estimates of the form +1+ a VR, where Z ~ N(0,, 1). When the objective function is L-smooth and satisfies PL condition (see e.g. [Karimi al., 2016)), both scaling assumptions are equivalent.
Numerical Experiments
5.1
Convex optimization
We compare the optimization algorithms for the family of objective functions
|30| 11 d(120-11
if [cc|
fa: R.
fa(x)
else
for d 2 2 with gradient estimators (1 oN) where N is unit normal random variable. The functions are convex and their derivatives are Lipschitz-continuuw with L d(d - 1). Various trajectories are compared for different values of and a in Figure 3. We run AGNES with the parameters a = 1 ?) 2[ Ps. xτ5 derived above and SGD with the optimal step size
We present 7 T (see Lemmas 16 and 17). For NAG, we select I(I+0= and p.. n+3
similar experiment in the strongly convex case Appendix
We additionally compare to two other methods of accelerated gradient descent which were recently proposed for multiplicative noise models: The ACDM method of Nesterov [2012], Vaswani et al.
[2019], and the continuized Nesterov method (CNM) Even et al. [2021], Berthier et al. [2021] with the proposed parameters. In this simple setting where all constants known, AGNES, ACDM and CNM perform comparably in the long run and on average.
5.2 Neural network regression
We generated n 100, 000 12-dimensional random vectors. Using fixed, randomly initialized neural network f (with 10 hidden layers, each with width 10, and output dimension 1), we produced labels y/ = f (xx)). The resulting dataset was split into 90% training and 10% testing data. We then trained identically initialized copies of larger neural network (15 hidden layers, each with width 15) using Adam, NAG, SGD with momentum, and AGNES to minimize the mean-squared error (MSE) loss.

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ADAM
30000
3003
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Figure 4: We report the training loss as a running average with decay rate 0.99 (top row) and test loss (bottom row) for batch sizes 100 (left column), 50 (middle column), and 10 (right column) in the setting of Section 5.2. The horizontal axis represents the number of optimizer steps. The performance gap between AGNES and other algorithms widens for smaller batch sizes, where the gradient estimates are more stochastic and the two different parameters a, 17 add the most benefit.
We selected the learning rate 10-3 for Adam as it performed poorly higher or lower rates 10-2 and 10-4. For AGNES, NAG, and SGD, based on initial exploratory experiments, we used learning rate of 10-4, a momentum value of 0.99, and for AGNES, correction step size 17 = 10-3 3 The experiment was repeated 10 times each for batch sizes 100, 50, and 10, and run for 45,000 optimizer steps each time. The average loss and standard deviation for each algorithm are reported in Figure 4. The results show that AGNES performs better than SGD and NAG for all batch sizes. With large batch size, Adam performs well with default hyperparameters. The performance of AGNES relative to other algorithms especially improves as the batch size decreases.
5.3 Image classification
We trained ResNet-34 [He et al., 2016] with batch sizes 50 and 10, and ResNet-50 with batch size 50 on the CIFAR-10 image dataset (Krizhevsky et al., 2009] with standard data augmentation (normalization, random crop, and random flip) using Adam. SGD with momentum, NAG, and AGNES. The model implementations were based on [Liu, 2017]. Each algorithm was provided an identically initialized model and the experiment was repeated 5 times for 50 epochs each. The averages and standard deviations of training loss and test accuracy are reported in Figure 5. We used the same initial learning rate 10-3 for all the algorithms, which was dropped to 10- after 25 epochs. A momentum value of 0.99 was used for SGD, NAG, and AGNES and a constant correction step size n 10 was used for AGNES.
AGNES reliably outperforms SGD and NAG both terms of training loss and test accuracy. The gap in performance appears to increase as model size increases or batch size decreases, suggesting that AGNES primarily excels in situations where gradients are harder to estimate accurately. For the sake of completeness, we include Adam with default hyperparameters as a comparison.
In congruence with convergence guarantees from convex optimization, grid search suggests that is the primary learning rate and 7) should be chosen larger than a. We tried NAG and Adam with higher learning rates 10-2 and 10-1 as well to ensure fair comparison with AGNES, but found that they become unstable or perform worse for larger learning rates in our experiments. The AGNES default parameters a = 10-3, . = 10-2, p = 0.99 in Algorithm 1 give consistently strong performance on different models but can be further tuned to improve performance. While the numerical experiments we performed support our theoretical predictions, we acknowledge that our focus lies on theoretical guarantees and we did not test these predictions over broad set of benchmark problems.

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560.0
200X
20003
25000
20:00
2000
500
MAG
MAG
M6G
5001
10110
0.32
5031
101
633
633
033
6.32
4.33
6.64
0.84
612
6.2
612
302.0
70000
502.0 15:000 1E0E2
72000
Figure 5: We report the training loss as a running average with decay rate 0.99 (top row) and test accuracy (bottom row) for ResNet-34 trained on CIFAR- 10 with batch sizes 50 (left column) and 10 (middle column), and ResNet-50 trained with batch size 50 (right column). The performance of AGNES with the proposed hyperparameters is stable over the changes in model and batch size.
con
901
2006
EACKEL
KAC
GRODKES
Figure 6: We report the average training loss after each epoch for six epochs for training L on MNIST with AGNES for various combinations of the hyperparameters and illustrate that a is the algorithm's 's primary learning rate. Left; For given a (color coded), the difference in the trajectory for the three values of 17 (line style) is marginal. On the other hand, choosing well significantly affects performance. Middle: For any given a, the largest value of r) performs much better than the other three values which have near-identical performance. Nevertheless, the worst performing value of with well chosen a = 5 10-3 performs better than the best performing value of 1 with a= 5 10-4 Right: When a is too large, the loss increases irrespective of the value of r).
We present a more thorough comparison of NAG and AGNES with various parameter selections in Figure 8 in Appendix A. With default parameters minimal parameter tuning. AGNES reliably achieves superior performance compared to NAG (training loss) and smoother curves, suggesting more stable behavior (test accuracy).
5.4 Hyperparameter comparison
We tried various combinations of AGNES hyperparameters a and y to train LeNet-5 on the MNIST dataset to determine which hyperparameter has greater impact on training. With a fixed batch size of 60 and a momentum value = 0.99, we trained independent copies of the model for epochs for each combination of the hyperparameters. The average training loss over the epoch was recorded after each epoch. The results are reported in Figure 6. We see that a has the largest impact on the rate of decay of the loss, which establishes it as the 'primary learning rage'. If is too small, the algorithm converges slowly and if a is too large, it diverges. If a is chosen correctly, a good choice of the correction step size 7 (which can be orders of magnitude larger than a) further accelerates convergence, but y) cannot compensate for a poor choice of a.
10

---

Acknowledgements
Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing. This research was also supported in part by the University of Pittsburgh Center for Research Computing, RRID:SCR 022735, through the resources provided. Specifically, this work used the H2P cluster, which is supported by NSF award number OAC-2117681.
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Appendix
Additional experiments
17
A.I Numerical experiments for AGNES in smooth strongly convex optimization
17
A.2 Extensively comparing against NAG
17
Multiple versions of the AGNES scheme
18
B.1 Equivalence of the two formulations of AGNES
18
B.2 Equivalence of AGNES and MaSS
18
Continuous time interpretation of AGNES
20
D Background material and auxiliary results
22
brief review L-smoothness and (strong) convexity
22
D.2 Stochastic processes, conditional expectations, and a decrease property for SGD
23
Convergence proofs: convex case
25
E.1 Gradient Descent (GD)
25
E.2 AGNES and NAG
26
Convergence proofs: strongly convex case
34
F11 Gradient Descent
34
F2 AGNES and NAG
35
F3 On the role of momentum parameters
40
G AGNES in non-convex optimization
41
H Proof of Lemma 8: Scaling intensity of minibatch noise
42
I Implementation aspects
43
L.1 The last iterate
43
I.2 Weight decay
44
16

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A Additional experiments
A.1 Numerical experiments for AGNES in smooth strongly convex optimization
We compare SGD and AGNES for the family of objective functions
L
: fx,((=)= o
2 22-
We considered several stochastic estimators with multiplicative gradient scaling such as
collinear noise (1 +1++1 V where N is one-dimensional standard normal.
isotropic noise (x=)
[(v]
N, where N d-dimensional unit Gaussian.
vd
Gaussian noise with standard variation o o (x)]] only in the direction orthogonal to V (x:).
Gaussian noise with standard variation or | fx)]] only in the direction of the fixed vector == (1,1)/V2.
Noise of the form Vs(x) + o1 (V N v where (1, 1)/v22222 and variable
o² [ [[]]]]]] each; = 0 otherwise. which takes values or -1 with probability & I+a In this setting, the noise remains macroscopically large at the global minimum, but the probability of encountering noise becomes small.
Numerical results were found to comparable for all settings on a long time-scale, but the geometry of trajectories may change in the early stages of optimization depending the noise structure.
For collinear and isotropic noise, the results obtained for fu.L on R2 were furthermore found comparable (albeit identical) simulations with quadratic form IRd with d 10 and
(d 1) eigenvalues μr and one eigenvalue
(d 1) eigenvalues L and one eigenvalue
eigenvalues equi-spaced between and L.
The evolution of (x,)) for different values of a and 2 / = 1 considered for both SGD and AGNES in Figure 7.
The objective functions μ4 1-convex and L-smooth. We use the optimal parameters a, 77, P derived above for AGNES and the optimal step size 17 4(I+04) for SGD. The mean of the objective value at each iteration is computed over 1,000 samples for each optimizer.
A.2 Extensively comparing against NAG
We ran additional experiments testing a much wider range hyperparameters for NAG for the task of classifying images from CIFAR-10. The results, presented in Figure 8 indicate that AGNES outperforms NAG with little fine-tuning of the hyperparameters.
We trained ResNet-34 using batch size of 50 for 40 epochs using NAG with learning rate in
{8 10-5 10 1,005,104,.2 -0- 10-4,81.80
4,8 10 -4 10 1,--3,2 10 10-3,3 10 8 10-- 10 100-2,22 10-2,5, 10 1,0-1,5-.0 10-1} and momentum value in (0.2, 0.5. 0.8, 0.9, 0.99). These 80 combinations of hyperparameters for NAG were compared against AGNES with the default 10-3 (learning rate), 10-2 (correction step), and p 0.99
10 - -0 1,- 2,
hyperparameters suggested (momentum) as well as AGNES with a slightly smaller learning rate 10-4 (with the other two hyperparameters being the same).
AGNES consistently achieved a lower training loss as well as a better test accuracy faster than any combination of NAG hyperparameters tested. The same random seed was used each time to ensure a fair comparison between the optimizers. Overall, AGNES remained more stable and while other versions of NAG occasionally achieved higher classification accuracy in certain epochs.
17

---

mu = 1, L= 500.0
mu L 10000.0
10
10
20
1077
10
10:7
10
AGNES sigma 6D, Simoo
10.7
6D, sign 10 AGNES, sigms
30
10"4
200000
400000
500030
Figure 7: We compare AGNES (red) and SGD (blue) for the optimization of fu.L with μ 1 and L = 500 (left) / L = 10 (right) for different noise levels o = 0 (solid line), a = 10 (dashed) and 50 (dotted). In all cases, AGNES improves significantly upon SGD. The noise model used is isotropic Gaussian, but comparable results are obtained for different versions of multiplicatively scaling noise.
B Multiple versions of the AGNES scheme
B.1 Equivalence of the two formulations of AGNES
Lemma's 9, The two formulaioons of the AGNES time-stepping scheme (2) and (3) produce the same sequence of points.
Proof. We consider the three-step formulation (3),
Uo = O, I'2 In + an In+11 = x² Tigh. Un+1 = (Un - g/,),
and use it to derive (2) by eliminating the velocity variable un- If Up = 0, then s6 = zu- From the definition zi,, get or x In. Substituting this into the definition of u,+1,
In
g'
un+1 = Pn
Then using this expression for Un+1 to compute 2'4t1.
2n41 Int1 + 0U
1/1 - In In
Ent1 + apn
2n+1 + Pro22, -aghn - xn))
Together with the definition of #n41 and the initialization 26 = xo. this is exactly the two-step
formulation (2) of AGNES.
B.2 Equivalence of AGNES and MaSS
After the completion of this work, we learned of Liu and Belkin [2018]'s Momentum-Adee Stochas- tic Solver (MaSS) method, which generates sequences according to the iteration
Ent1 = x' mi''. 2 = (1 + 7) Trn, + 7229
where 9', is an estimate for fx(2)). This is a version of AGNES with the choice 77 and the momentum step
2/11 In+1 + p(2/ aghn - -xn)
#nti + p(5nt1 +194 agn Fn)
(1 + p) p., + (n a a)
i.c. MaSS coincides with AGNES for 7 p and 172 (1 a)p.
18

---

P-264
P-363
b-de2
AGAS R-564
Rdd with momenium 10, Trainnn Lads
MAG will monoretue 9I Best Arrwarry
3.0
1.0
MA0 with ecteoruu Touring
505 WITh Terennaa Accerwey
6.57-
635-
WS1 1750 14000 we PES Venn 11200 8925
1230 1690
VANT
142300 490S
181
431 220 558 Ata 5420 A691 1430 1302
MAG th nemerium 05 Turring
105 With
633
640
momertam Taiiing
0.0
0.0
6.3
1006 3000 33000 23ice 2900 1990 1900 15000
8.30
12300 19900 20000 25:30 W000 mr2ne 45000
Figure 8: We trained ResNet34 CIFAR-10 with batch size 50 for 40 epochs using NAG. Training losses are reported as a running average with decay rate 0.999 in the left column and test accuracy after every epoch is reported in the right column. Each row represents specific value of momentum used for NAG (from top to bottom: 0.99, 0.9, 0.8, 0.5, and 0.2) with learning rates ranging from 8 10-5 to 0.5. These hyperparameter choices for NAG were compared against AGNES with the default hyperparameters suggested a = 10- (learning rate), 10 (correction step), and p 0.99 (momentum) as well AGNES with slightly smaller learning rate 5-10-4 (with p 0.99, 10 as well). The same two training trajectories with AGNES are shown in all the plots in shades of blue. The horizontal axes represent the number of optimizer steps.
19

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C Continuous time interpretation of AGNES
For better interpretability, we consider the continuous time limit of the AGNES algorithm. Similar ODE analyses of accelerated first order methods have been considered by many authors, including Su al. [2014], Siegel [2019], Wilson al. [2021], Attouch et [2022], Aujol al. [2022a]. Zhang et al. [2018), Dambrine et al. [2022].
Consider the time-stepping scheme
to 0, 2 In + Vn 1n- x/2
199
Un+1 = Pr (29'))
(5)
which reduces to AGNES as in (3) with the choice of parameters 2/ = 1. For the derivation of continuous time dynamics, we show that the same scheme arises with the choice I 7/2 va.
Lemma 10. Let p E (0, and n 0 parameters. Assume that Yı 72 and 51, 72 are parameters such that 71 = Y2- Consider the sequences (Zn, En, Un) and (In, zng vn) generated by the time stepping scheme (5) with parameters (p. 7, 7/1 72) and (p, 1) 71. 72) respectively, zo Do and V1 = 7/1 Uo then In En z In and 71 Up Vi Un for all n E N.
Proof. We proceed by mathematical induction on n. For n = 0, the claim holds by the hypotheses of the lemma. For the inductive hypothesis, we suppose that 2n En and 91 7i0, and prove the claim for n + 1. Note that since znn = 2n + vn automatically follows that x = x,, This implies that
Unt1 = v'n - ngh = H', ngh = In+l-
Considering the velocity term,
V/444 Pm((71 71 (29gg) Pn (8110 on (2gh) PPn( - -2ghh) " 8m
Thus Fn+1 1n+1 and 71 un+1 71 on The induction therefore be continued.
Consider the choice of parameters in Theorem 4 by
= L(I+02) a v Vn/L+
VI(1 -²) V VI(I+ Vp
7 1
ja < L. We denote = (T(++22) and note that
72 Va h
(1 VI
and
Vp D=1-2 V[(1+0*)
VI(1 + VL(1 + +Z) +
-1
2/hh.
Depending on which interpretation of we select, we obtain different continuous time limit. First, consider the deterministic case 0. Then
Int1
In
Un hv (In + -hu))
+
-2 Un (1 / V + lew,))
Un+1
Un
Zn
+ h
Un -2 vf(zn)) + 0(1?)
Un
Keeping f fixed and taking h + 0, this is time-stepping scheme for the coupled ODE system
v - -2- (- v/())
Differentiating the first equation and using the system ODEs to subsequently eliminate v from the expression, observe that
द=Ú= == vf(x)
2 Vf(x),
20

---

Le. we recover the heavy ball ODE. The alternative interpretation = h//VL can be analized equivalently and leads to a system
VT. vf(x)
2 vf(x))
which corresponds second order ODE
1
#
D2
f(x)
VL
1 f(x)
-2V
Vf(x)
VL
1
1 - f²x) VL
-2V
V(()) - Vf(x)
VL
2V
1 VL
Da f())() 2 8-
1 Vf(x).
This continuum limit not simple heavy-ball ODE, but rather a system with adaptive friction modelled by Hessian damping. A related Newton/heavy ball hybrid dynamical system was studied in greater detail by Alvarez et al. [2002]. For L-smooth functions, the er-operaoor norm of D² f(z) satisfies || D2 f())] < L, i.e. the additional friction term can be as large as vT. in directions corresponding high eigenvalues of the Hessian. This provides significant regularization in directions which would otherwise be notably underdamped.
Following Appendix E.3, maintain that the scaling
71 p)
2
V
L
is 'natural' as we vary y. p. The same fixed ratio maintained for the scaling choice = h/ TT as
n - h/VL 2/Th
2
h
Indeed, numerical experiments in Section A suggest that such regularization may be observed in practice as high eigenvalues in the quadratic map do not exhibit 'underdamped" behavior. We therefore believe that Hessian dampening is the potentially more instructive continuum description of AGNES. similar analysis can be conducted in the stochastic case with the scaling
h }
11 = h, ne
p=1- 2
V
for large a. We incorporate noise as
= -1+
and write
In+1 Un+1
En
- -1 (1+03))VV -2 -11 (1 -11-v NNN, V. + thvw)) +
Un
Zn
+
-2/Tu vf(xn) V ay N,Vl()) + +0h))
Un
Un
which can be viewed an approximation the coupled ODE/SDE system
cxr du
(-2 (1-2vu--((()) ((-2//00-/(0) v dt Vf(a)) dt + TTTAB (
= -
under moment bounds on the noise N,. The precise noise type depends the assumptions the covariance structure of N,, noise can point only in gradient direction or be isotropic on the entire space. For small h, the dynamics become deterministic. Again, alternative continuous time limit is
dx dv
(1 ( ( di ooohB v Vf(z))
(v (0- V(-Vf(()/VD) dt + a dB Vf(x) VL
=
21

---

if is scaled towards zero as h/ The first limiting structure is recoverd in the limit L + co. Notably, the noise in the first equation is expected to be non-negligible o > VL. A similar analysis can be conducted in the convex case, noting that
n +
3
3
=
(n + mo 3)/ h
71 + 140 + 3
no +3
where (n + no 3)h roughly corresponds to the time in the continuous time setting,
D Background material and auxiliary results
In this appendix, we gather a few auxiliary results that will be used in the proofs below. We believe that these will be familiar the experts and can be skipped by experienced readers.
D.1 brief review of L-smoothness and (strong) convexity
Recall that if function f L-smooth, then
L
f(y) = (z)+ (y x) 1 yl1
(6)
For convex functions, this in fact equivalent to f being L-Lipschitz.
Lemma 11. If f is convex and differentiable and satisfies (6), then [V]((x) Vf(y)] T|| vll for all x and
Proof. Setting y = fz) in (6) implies that f(x) inf, f(z) t [V Applying
this to the modified function fy (2) = f(x)-Vf(v) (z y), which is still convex and satifies (6), we get
fx(x)= inf f - f(x) Vf(() (x f(v)
1 [V
- [V](x) VS(2)11
1
21
2L
Note that here have used the convexity conclude that inf, (z) fy(v), i.e. that fy is minimized at since by construction (v) = 0 (this is the only place where we use convexity!). !). Swapping the role of x and y. adding these inequalities, and applying Cauchy- Cauchy-Schwartz we get
7 fV))()) vf()) S (Vf())-Vf()) (2 (x- v) S Vf(x) - ff(i)lll yll.
which implies the result.
From the first order strong convexity condition,
f(v) 2 f(=)((()))))))))) + ff(z) (y - x) -1 y11
M
we deduce the more useful formulation V (x) (a v) 2 f(x) - f(y) + $ | - ?l? The convex
case arises the special case f O. We note a special of these conditions when one of the points is a minimizer of f.
Lemma 12. If f an L-smooth function and x² is a point such that f(x*) = inf f(x) then for any R
L
f(x) f(x") S 5 || x01
Similarly, if ff is differentiable and j-strongly convex then for any x E R
x 12- f(²²)
Proof. This follows from the two first order conditions stated above by noting that VS(x*) = 0 minimizer of f-
22

---

Additionally, L-smooth functions which are bounded from below satisfy the inequality
|V - 27. inf f).
Intuitively, if the gradient is large point, then we reduce f quickly by walking in the gradient direction. The L-smoothness condition prevents the gradient from decreasing quickly along our path. Thus if the gradient is larger than threshold at point where f is close inf then the inequality f 2 inf f would be violated.
Let us record a modified gradient descent estimate, which is used only in the non-convex case. The difference to the usual estimate is that the gradient is evaluated the terminal point of the interval rather than the initial point.
Lemma 13. For any n, v and o: f is I-smooth, then
La²
f(x + av) S f(x) aV aV (xr + av) v +
||al|
Note that f is convex, this follows immediately from (6) and the convexity condition (Vf(y) Vf(x)) (y - z) 2 O.
Proof. The proof is essentially identical to the standard decay estimate, We compute
{
a d
f(x) f(x av)
[x + tu)dt
Jo dt
f(x + avv
[V + av) + (vf(c + tv) VV + av))] v d
f(x + 0r) v - L/-
2 f(x + av)
L La - 1)1o|//
La²
= =f + av) - a Vf av) v 2 110112
D.2 Stochastic processes, conditional expectations, and a decrease property for SGD
Now, turn towards very brief review of the stochastic process theory used in the analysis of gradient descent type algorithms. Recall that AA, IP) a probablity space from which we draw elements wn for gradient estimates g( wn) (AGNES) g( w.,) (SGD). We consider zo random variable on Rm with law Q. Let introduce the probability space (Ω, A, PP) where
1. Ω Rd InEN Ω
2. Ā the cylindrical/proddutt o-algebra on Ω and
3. fp Q P.
The product r-algebra and product measure are objects suited to events which are defined using only finitely many variables in the product space. A more detailed introduction can be found in [Klenke, 2013, Example 1.63]. We furthermore define the filtration (FrhneN where F, the r-algebra generated by sets of the form
R B
A
Il Ω KEN
B R. Borel, A, A.
In particular, UNEN Fr (UnEx Á and, examining the time-stepping scheme, it is immediately apparent that In, I',, are F, Feamuauuabble random variables on Ω. In particular, they are A-measurable. Alternatively, we can consider J, as the o-algebra generated by the random variables æ1, ....... In, .Y,, i.e. all the information that is known after intialization and taking gradient steps. All probabilities in the main article are with respect P..
Recall that conditional expectations are a technical tool to capture the stochasticity random variable X which can be predicted from another random quantity Y. This allows to quantify the
23

---

randomness in the gradient estimators 9/ which comes from the fact that Zn is random variable (not known ahead of time) and which randomness comes from the fact that on top of the inherent randomness due to e.g. initialization, we do not compute exact gradients. In particular, even at run time when En is known, there is additional noise in the estimators g our setting due to the selection of wip.
In the next Lemma, we recall two important properties of conditional expectations.
Lemma 14. [Klenke, 2013, Theorem 8. 144 Let and h be A-measurable random variables on probability space (S2, A, 1)) and JF A be a algebra. Then the conditional expectations Egg JF] and / F satisfy the following properties:
1. (linearity) Elag + 3h F
for all B [[[g] + B B
2. (tower identity) E E
Fi]
Eigl
3. If is F-measurable then Eigh F E[h FJ. In particular. E[g F}
For more thorough introduction to filtrations and conditional expectations, c.g. [Klenke, 2013, Chapter 8). Elg' is the mean of 9, if all previous steps are already known. Lemma 15. Suppose g' Fny and x',, sattisfy the assumptions laid out in Section 3.1, then the following statements hold
1. E 6'' [F.] f((())
2. E [llgh Vf(xa))) & o² (IV
3. [[[ll[[] = (1
4. [E(V - -.. ] = V
Proof. First and second claim. This follows from Fubini's theorem.
Third claim. The third result then follows by an application of the tower identity with Fr, expanding the square of the norm dot product, and then using the linearity of conditional expectation:
E 11 ?
E
E
L 11 |F
E
vf(<=))]1 + 2g vf(x,)) [V))(=))) F.,
E
11h v)(a Fr 2B vf(a(,) - |VV [
S V (((h) +
(V 2 -V-
= (1 + o²) [
Fourth claim. For the fourth result, we observe that since f is deterministic function and x² is -e.-maauurable, (x(,)) is also measurable with respect to the o-algebra. Then using the tower identity followed by the third property in Lemma 14,
E (VS((()) s/l
E E (V (a In F.]]
Full = E V ((x vf(a)))
E V ((
Elg
E
As consequence, we note the following decrease estimate.
Lemma 16. Suppose that f. x',, and g', = g(I',, wn) satisfy the conditions laid out in Section 3. then
E [[ - ((9h1l < E f(x,))) - L(1 o²)n
2
24

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Proof. Using L smoothness of f.
f -ngh) S f(=()))))))))))){))))) - n ff(()) + Ln²
2 l1 12
Then taking the expectation and using the results of the previous lemma,
E [ nigh E(S(()) n/ [|V Lof (1 + [(
<B[[(E])] - (1- (
1(1
|(
2
In particular, if S =(1+0)); then
B[f(z', nglh) {(((()) [=
E
Convergence proofs: convex case
E.1
Gradient Descent (GD)
We first present convergence result for stochastic gradient descent for convex functions with multiplicative noise scaling. To the best of our knowledge, convergence proofs for this type of noise which degenerates at the global minimum have been given by Bassily et al. [2018], Wojtowytsch [2023] under allaia-Lojsseewizz (or PL) condition (which holds automatically in the strongly convex case), but not for functions which are merely convex. We note that, much like AGNES, SGD achieves the same rate of convergence in stochastic convex optimization with multiplicative noise as in the deterministic case (albeit with a generally much larger constant). In particular, SGD with multiplicative noise is more similar to deterministic gradient descent than SGD with additive noise in this way.
Analyses of SGD with non-standard noise under various conditions given by Stich and Karim- ireddy [2022], Stich [2019].
Theorem 17 (GD, convex case). Assume that f is convex function and that the assumptions laid in Section 3.1 are satisfied. If the sequence I, generated by the gradient descent scheme
1
9n g(xn; Wn),
Tat1 Fn 199m:
S
L(1 + 02)
then.for any x m and any no 2 1+
B(f(xn) - {(x))} mmo E[ f(x + E x+1
( + no)
In particular, in 1 no 1 0², and x² is point such that f(x") inf f(x), then
L(1 + σ E D0 x= 11
([f(x,) f(a*)]
2(n
+
Proof. Let no 2 and consider the Lyapunov sequence
Ln = 17n + (f) inf f) + || x x11
25

---

We find that
nnn + no + 1)f(2, -99.) inf f} + 1
Lut1
E
||
79n 2²112
2
E
nn + no +1) f(xx)) 2 [x,((==)) -inf -infs}
1
2 Fn x²112 17 (I'z x²)
In +
2 l1 12
1
n(n (( inf f} +
2hn x011 - Vf(zn)
2
1 no)
11² ||
[V )12
T (n n - -1 +2)) B f
S L, + 0
by the convexity of f. The result therefore holds if ni is chosen large since
La nin no) n(n + no)
Lo
mmo moo E[f(z) - f(xx() ) + E [|=0 x²]
B[f(x,) f(x))1
7 no)
If x² is a minimizer of f then the last claim in the theorem follows by using the upper bound f(0) f(=)) S 2 x= I|2 from Lemma 12 and substituting L(1+o)z, + o². L
E.2 AGNES and NAG
The proof's of Theorems and in this section are constructed in analogy to the simplest setting of deterministic continuous-time optimization. As noted by Su et al. [2014], Nesterov's 's time-stepping scheme can be seen as non-standard time discretization of the heavy ball ODE
3 σ
Vf(x) t 0
t = 0
t= 0
with a decaying friction coefficient. The same is true for AGNES, which reduces to Nesterov's 's method in the determinstic case. Taking the derivative and exploiting the first-order convexity condition, we see that the Lyapumov function
1
L(t := f((((())) f(x))) + tt + (=(()) - x²)l"
decreasing in time along the heavy ball ODE, see e.g. [Su al., 2014, Theorem 3]. Here x² is minimizer of the convex function f. In particular
((t) 2
(() t2
1 x01
f(x(t)) f(x))
2
To prove Theorems and 3, we construct analogue L in (7). Note that at,, 2/, In is discrete analogue of the velocity é in the continuous setting. Both the proofs follow the same outline. Since Nesterov's 's algorithm is special case of AGNES, we first prove Theorem 3. We present the Lyapunov sequence in a fairly general form, which allows us to reuse calculations for both proofs and suggests the optimality of our approach for Nesterov's 's original algorithm.
For details on the probalistic set-up and useful properties of gradient estimators, see Appendix D.2. Let us recall the two-step formulation of AGNES, which we use for the proof,
To = %/- Fn+1 = x - ngh.
2/ = In+1 + Pa (a - agg' x..).
(2)
We first prove the alternative version mentioned after Theorem 3 in the main text. Both proofs proceed initially identically and only diverge in Step 3. The reader interested mainly in Theorem 3 is invited to read the first two steps of the proof of Theorem 18 and then skip ahead the proof of Theorem below.
26

---

Theorem 18 (AGNES, convex case, version). Suppose that In and x', are generated by the sime-stepping scheme (3), f and gn = g( satisfy the conditions laid out in Section 3.1, is convex, and x² is point such that [xx") infzER" f(x)). If the parameters are chosen such that
7 as
202n
n1 + no
9 0(1+
no - - a(1 + 03)
Pr. n + ny + 3
1+02
then
B[f(xn) f(xx)) < (amo + no B B[f(xo) inf f] + 100 - x x=1
(n + no)²
In particular, 5 1420 then suffices to choose no 2 2n/ a 2 2(1 20²).
Proof. Set-up. Mimicking the continuous time model (7), we consider the Lyapunov sequence given by
La = P(t)) (1o(m))) [1b(m))( - + aa(n)(z x²)]?
where P(n) some function of n. a(n) ag ain, and b(n) bo bin for some coefficients an: 01, bo, bi. Our goal is to choose these in such a way that Ln is a decreasing sequence.
Step 1. If we denote the first half of the Lyapunov sequence as Lt = P(n) [(x))) then
Le
Leh = (n -(0)
f(x)))
S (P(n + 1) ((n))(
P(n) (x)() f(x")),
where & is positive constant that can be chosen later balance out other terms. Using Lemma 15,
La - -ot S (P(n + 1) + I [(ar)) - Gq,aN. [V..,.lVVV[((1111 [(x)) - P()))([(xn) - [(x))]
PP(n)E (f(,)) - f(x,,) + (P(n + +) P(n)) -(x²)]
- (P(nn + 1) +
(((,))
where Cq.o.L = 17 (1 U(10?))
Using convexity,
2
L0 Lnt1 L6 P(n)E V (x(h)) (I'2 xn)] (P(n+1)++ -
(()))(VV(())) (2/, x')]
(P(n + 1) + ,
(8)
Step 2. We denote
W, = b(n)(z xxn) a(n)(2' x²)
and use the definition of 2//41 from (2),
Wa+11 bbn + 1)(=(441 1n+1) + a(n + 1) - x²)
bbn + 1)pm (2'n -ag' - xn) + a(n + (5n+1 1 + Pm P ash xn) x²)
=(b(n + 1) + a(n + 1))pm (a', ag'n xn) + a(n + 1)(Ex+1 - x²).
We will choose
b(n)
Pro = b(n + 1) + ann
such that the expression becomes
Wa+1 b(n) (4%, ag', - xn) + a(n + 1)(xnt1 x²)
b(n) (I'2 ag'n xn)) + (ap + ain + a1 01)(2' n9h x²)
wn ai (I'n x²) (ab(n) + na(n + 1))9'n
Then
1
122-1
1 2 | un (wn+1 wn) 2 || wnll3
1
2 |
wn (a) (v'n x²) (ab(n) + na(n 1))9k)
+ 2
| - x²) - (ab(n) + n + 1))9²
27

---

We want the terms in this expression to balance the terms in Latt - La, so we choose a1 = 0, i.c. a(n) = og is constant. This implies,
1
E
Wn++1 112
2 Wm 11
E (ab((n) + m)ao)nn In + 2 ((ab(r)) nao))²² "1.11'nl
1 (ab(n) + ? (1 +
v (aa((n) + nao) El vf((a,)) + 2
= (ab(n) + nao (B))) -r.)) Vf(x(,))
x.,)
(ab(n) + n00 (o B(( - - [.(()
+ (ab(n)++
+ ( ([V 2 1700
+
(9)
Step 3. Combining the estimates (8) and (9) from the last two steps,
Ln+1 Ln S (P(n)= - (cu(()) + (oo))(())) [V) (a'a xnn]
+ (P(n + + P(n) (ub(n) + s7o))ao) E(V(()')) (1'2 xx))]
+ (ab(m) + nya0) )2 + 0²) (P(n + 1) k).,.o.. B|| (((")(
Since \V 2
0 and (((a,) -x)= xx)
[(mmm - f(x)) 2 0. we require the coefficients
of these two terms to be non-positive and the coefficient of ( (I/2 Fn) to be zero. That gives us the following system of inequalities,
P(n) = (ab(n) na))b(n)
(10)
P(n + 1) +) P(n)= s (ab(n) mao)ao
(11)
2 (ab(n) + (1 0²) s (P(n 1) + k)n (1 L(1 + +
1
(12)
Step 4. Now we can choose values that will satisfy the above system of inequalities. We substitute
ag 2, b1 6, no. and k 2n- From (10), get P(n) = (a(n + no) 2n) (n no).
Next, we observe that
P(n 1) P(n) 2n<(n + no) 27.
Then (11) holds because
P(n) a + 2a(n + mo) 2n 2n a
= 2(a(n + n4) + 2n)
(ab(n) + nau jag.
We now choose 1) to satisfy v) S I(I+o7)+ which ensures that / S 1) (1 I(I+0")" for (12), it suffices to ensure that
Consequently,
2
(ab(n) + noo) (2 + + (P(n + +1) k)m,
which is equivalent to showing that the polynomial,
<(=) = (az² + 211z + a + 20z + 2n + 2n - a) n
- (a²2²² + 412² 40m²) (1 +0²)
is non-negative for all = 2 no. (z)) simplifies to
<(=) = (1 a11 + 0 o²))=² + 211(1) + a 20 + +0²))= 2 2np00
To guarantee that q is non-negative for
no 2 no. we require that
1. the leading order coefficient is strictly positive? and
2 In principle, would suffice if the quadratic coefficient vanished and the linear coefficient were strictly positive. However, if a(1 +02 = 0, then the coefficient of the linear term is negative since 17+ a 2a(1+ o²) -ao? < O. We therefore require the coefficient of the quadratic term to be positive.
28

---

0² 2. 2 0, ((o) 2 0.
Since (()) < 0 and q quadratic, this suffices to guarantee that q is increasing on [no, oo). The first condition reduces to the fact that
- (11 a= > O.
We can find the minimal admissible value of na by the quadratic formula. We first consider the term outside the square root:
2n(1 + - 2a(11 + +20))
!
91(1
20(1 a(1 + 02))
a aa1+
and thus
17
a a
n
a
4172
1
+
1
01
11
a(1 + 02)
a
"1 a(1 + 02)
a(1 a(1 + o²²)
17
ao²
ax²
ao²
( a
1
+4
a
a(1 02)
1) 1/ + +0²)
11 1 a(1
- aa1 + +
17
a022
17 al1 - 02) )
a
1+ " a(1 + 02)
a
!7
2002
a 1) a(1 aa1+++ 0²)
2no²
1 7-a(1 +()
In particular, in the deterministic case o 0, the choice no is admissible. Notably, we require 74n 2 2o² = 20². Furthermore, if S I+2az then
7
20 11 - a1 + 02)
20
27
ag²
a
so it suffices to choose no 2 2n/a in this case.
Step 5. We have shown that the Lyapunov sequence,
LE 2 [1nn + no) xxn) 2(5², - x² =²)
Ln = ((n+ + mo) [f(x.) f(x))]
monotone decreasing. follows that
Ln Lo
E E[(noc + 2n)mmo (f(20) [(x))) + 2 I|:00 111 (n + no02
-[f(xm)-((=") -fa))-ff('" s P(n) P(n)
2m S ang, we get
20mg E[s(zo) - inf /] + 2 |00 x11
E[f(xn) - f(x*)] <
(n + n0)
Finally, if म.म a
(1+200(14205) and no 2(1 + 20²), then using Lemma 12, the
expression above simplifies to
11112201 202) 5 |1:0 - 2²113)
E [ff(xa) - f(x*){ <
n
Remark 19. Note that SGD arises as special case this analysis if we consider a 0, no 2 2. since P(n) is linear polynomial in this case.
29

---

Remark 20. Note that the proof of Theorem 18 implies more generally that
La+1 S Ln - q(n + no) [[IV)(=)))
even if no is not chosen such that g(n + +no)) 2 for all n. However, g(n + nn) 2 0 for sufficiently large E N, i.e. Ln decreases eventually (assuming that Ln < cc for all finite n). More precisely, for given 1/, a if
nil 2
Ln
no
then L a (n + no)²
a(1 + o²)
Thus poor choice of no will not prevent convergence, but it may delay it.
We now prove the version of this result stated in the main text, for which On #+o0+ with > 2, i.e. with slightly more friction.
Theorem 3 (AGNES, convex case). Suppose that In and x' are generated by the time-stepping scheme (3), f and 9n g( wn) satisfy the conditions laid in Section 3.1, is convex, and x² is a point such that f(x") ) = in f(x). If the parameters are chosen such that
7
2(1 mL) 1,((1 + o2)
イイイ +(1+
1 a
Pn n n +++ for ag 2
then
[ff(xn) f(xx)) f(x*)] )} ag 20 - 1=1
2an²
Proof. The proof for this version of Theorem 3 is identical to the proof of Theorem 18 until Step 3, after which we take alternate approach. Let recall the expression we got in the beginning of Step 3.
Step 3. We want to show that the bound the right hand side of the inequality
Lnt1 Ln < (P(n) (ab(n) + (70)) (
(ar - xnn)
+ (P(n+ +) +) P(m) (ab(n) + 700))a) f (I% - x")]
(13)
+ man (onmm)))+o)))) + o2) - (P(n + + k)..,o.L
E V(()))
non-pon con-posiiive. Using convexity and L-smoothness in the form of [Wojtowytsch. 2023, Lemma B.1], we get the inequality
1
( (()
f(x))
2L |V
which allows us to combine the second and third line in (13), assuming that the coefficient in the second line is non-positive. this the case, then the entire right hand side of (13) bounded from above by
(P(n) (ab(n) + (n))) (s', -x,)]
+ { (P(n + + P(n) (ab(n) nan)a0) 2L
+ (ab(m) + nao) + 02) - (P(n + 1) + ko k).oo..)
Bil (())))))))))))
As E (V(()) (x/, x does not have a sign, we choose to set its coefficient to zero, and we require both the coefficient in the second line of (13) and the coefficient of B[[ (x, in the combined version to be non-positive Noting that Cy,a,L n/2 if S 1/ d²), this leads the system of inequalities
P(n) = (ab(n) + n()))
(14)
P(n + 1) + P(n) < (ab(n) + 1700).0
(15)
P(n + 1)+k Kk P(()) (ab(n) nao)as - ((P(p 1) k)n (ab(n) + 2 o²)).
(16)
30

---

In principle, this approach is more general than that of Theorem 3 as we do not require two terms to be individually non-posiinn con-positeve, but only one of them and their weighted sum. In the proof of Theorem 3, similar role was played by the parameter k, which allowed to shift small positive term between expressions.
Step 4. Now we can choose the parameters and variables so as to satisfy the inequalities above. We begin by setting 1,60 0, k 0, and choosing a, 77, do in the theorem statement. Using (14) as the definition of P(n), we note that
P(n + 1) P(n) 2an + a + 1ja0-
Thus, (15) simplifies to
2on + a + 1/00 I ap an + n00),
which holds since ao and n) 2 a. The right hand side of (16) simplifies
I. [n 1)(a(n 1) nao) I (an + (00) o2))
= (1na- (1 + -2) x²) n² + (2a nao) (1 nep0in-ffae rfag(1 o²) nn(a nao))
= (1n + nao) - 2 + + } n 1/ ag + +²) + n(a + nao))
where the last equality holds since - 1 + ( Thus for (16) to hold, it suffices that
2an + a + 1aa0 ao(an + nao) - L (( + n00) - 2naaa )²) n ai + + + n(a - noo)).
which is equivalent to
{a(2 an) Ln(2a + nan) + 2L o + {a + nawo -an + + a (1+ o²) Ln(a + nao) S 0. (17)
linear polynomial is non-negative for all n 2 0 if and only if both of its coefficients are. The leading order coefficient in (17) is
a(2 ao) LTy(2 + maa) + 2L ( + = a(2 - ao) - Ln(2a + nao) + 2L72aa ao
2a 2n + a Lx)²)
1 1 02 (2 Ln) + ao (n( 1 + +²) 1)),
which is non-positive if and only if ag 2 2(1 2()
1 We remark that it is this part of the computation
that forces us to choose strictly smaller than I[I+0"). In the deterministic case σ 0, we would encounter such limitation as the term would be automatically zero for 1) 1/L. Finally, consider the constant term in (17) and use the fact that 1 < ag and a S
a + 7700 - ain + Lny a1 + o²) Lij(a + naa)
= (a nou) Ln)
(n)n(Ln(1111) + 0f) 1) S 21 bn) + a3 [1 + 0²) 11)
< nau (2 Ln) + ao (n( - - -1))
S 0,
using again that 0. 2 2 This shows that Lnt1 S Lm. Step 5, The conclusion again follows as in the proof of Theorem 18.
1-by
In addition to convergence expectation, we get almost sure convergence as well. Corollary 5. In the setting of Theorems' 3 and 4, f(x.) inf with probability 1.
The same is of course true for Theorem 18.
Proof. The conclusion follows by standard arguments from the fact that the sequence of expectations Bl (x.)) inf f) is summable: By the previous argument, the estimate
C n
E (())( - fx=')] B[f(z.) f(x)){ s
31

---

holds for some C > 0. Since
lim sup f(xx) inf fl > =00 n
lim f(xn) & inf f) P 10000
11 lim lim (f(x) inf fl k n-000
P
(f(xm) inf fl 1
P
km
n2
suffices show that P (lim sup, (f
inf fl 8) for any 0. We further note that
for any N E N we have
P limsup (
- inf fl f|>E)
S P N s.t. (f()) - inf fl E)
n-200
P
(1f(xn) - inf {|
M=N
P (()((T)) inf x)
E B(If(x,)) inf fl]
n=
C
8
by Markov's 's inequality. As the series over n converges, the expression on the right can be made arbitrarily small by choosing N sufficiently large. Thus the quantity on the left must be zero, which concludes the proof. In the strongly convex case, the series E thus the same argument applies there as well.
7 11+0=)
converges and
Next we turn to NAG. Let us recall the statement of Theorem 1.
Theorem 1 (NAG, convex case). Suppose that T. and x', are generated by the time-stepping scheme (I), f and g satisfy the conditions laid out in Section 3.1, is convex, and x" is a point such that f(x*) = inf fIx). If a < 1 and the parameters are chosen such that
1 1-
- f(a")( 2B 2B x 1=
o<ns
L(1 and Pr. n2 3
then Elf(ax)-
nnn
The expectation on the right hand side is over the random initialization xo-
Proof. We consider Lyapunov sequence of the same form before,
1 _ [[b(m)( xn) + a(n)(z) - =")|²"
- f(x")((===>)))> + L. = (()
where P(n) is some function of a(n) ao + a,n, and b(n) bo + 6nn.
Since Nesterov's 's algorithm a special case of AGNES, after substituting a = n, the analysis in steps 1, 2, and of the proof of Theorem 3 remains valid. With that substitution, we get the following system of inequalities corresponding to step 3,
P(n) n7(((n) + an)b(n)
(18)
P(n +1++1)+k-a P < n(b(n)) + ao)an
(19)
+1 < (P(n + k)n (1 (1
L(1 + +
(20)
2
2
32

---

Using the definition of P(n) from (18). (20) is equivalent to
(1 ++0 2bbn + by + bo + ao + ()) + bo) (1 I((++0))
2
(bin + b0 + aa)²
which should still hold in limit as n + 00,
2(b + b1 + bo ao k)(bin 60) (1 L(1+0")n
(1 + of)
lim 22000
(bi + bo + a0)²
L(1+ + +70)
= 2 1
2
This implies
1 1 0² L(1 + 02)
We choose do 2, b(n) n, and & 17. Then (18) implies that P(n) 7mnn + 2). (19) holds because
P(n 1) + k P(n) (2n 4) m7((n)) ag)an
and (20) holds because
(b(n) + a0) 0²) I1 (1) (1+11 +
2
< n((n + 1)(n + + 1)(1 + o²)
2
(P(n + 1) +k) 1 (1- L(1 L(1+02))
We have shown that the Lyapunov sequence
Ln mnnn + 2)k[)(En) f(z") I + = Billn(aa -
xn) + 2(2'/ x²) x²)
1-g² is monotonically decreasing. It follows that where 17 5 M[++ojj.
nnnn + 2) - f(x*)] s Ln S Lo- = -2E[| x=1
We emphasize again that this analysis works only if 1. The condition that S imposed by (20) and does not depend on any specific choice ag, bg. or bj. On the other hand, (18) forces the rate of convergence to be inversely proportional to 1). This means that as approaches 1, the step size y) decreases to zero, and the rate of convergence blows up to infinity. On the other hand, as the proof of Theorem 3 shows, AGNES does not suffer from this problem. Having an additional parameter enables AGNES to converge even the noise a is arbitrarily large.
I(1+o2] is
1-o?
Let us point out how the same techniques used in Theorem 3 can be adapted to prove convergence f*,,) >> inf f. even if global minimizer does not exist. We recall the main statement.
Theorem 7 (Convexity without minimizers). Let f be convex objective function satisfying the assumptions in Section 3.I and &, be generated by the time-stepping scheme (3). Assume that n, a and Pn in Theorem 3. Then lim oo (xx)) infzemm fxx
Proof. The first step follows along the same lines as the proof of Theorem 18 with minor modifica- tions. Note that did not the minimizing property of x² except for Step 5.2. Assume for the moment that inf f -0.
Assume first that and define the Lyapunov sequence L, just as in the proof of Theorem with the selected point xt.
lim inf (zf() inf 0. Select x such that f(x*) < inf ff + e/4
We distinguish between two situations. First, assume that n satisfies E f(f(2() 2 f(x*). In this case find that also [f < [[(()))] s f(x*).
33

---

On the other hand, assume that [x(,) 2 f(=") for n ......N. In that case, the proof of Theo- rem 3 still applies, meaning that T8] [xx) cannot remain larger than f("" + indefinitely. In either case, we find that there exists E N such that E [Zx)] I f(x)) E/2 < lim inf Eff
Note that the proof of Theorem 3 applies with n' 2 no starting point and non-zero initial velocity Vn- The argument therefore shows that, for every n' E N there exists N E N such that [x lim inf (z.)). Inserting the definition of the lower limit, we have reached contradiction.
We conjecture that the statement holds with the limit in place of the lower limit, but that it is impossible to guarantee a rate convergence (nn for any 33 > in this setting. When following this strategy, the key question is how far away the point x² must be chosen. For very flat functions such as
x 11+a - x) xSL. 1,
H> 1
fa R
fa(x)
x² may be very far away from the initial point zo, and the rate of decay can be excrutiatingly slow minimizers do not exist. For easy example, turn the continuous time model. The solution to the heavy ball ODE
x² = - 2/x'== (a( x
1>
t t=
x' 3
t t=
given by
4 (3 +3++a) a(2 + a)²
24a
x(1)
1340
2
for 3
2 2ta a(2+a)
(3++a
Ignoring the complicated constant factor, see that
fa(x(t)) = x(1)
the decay rate can be close zero as desired for a close to zero, and indeed Siegel and Wojtowytsch [2023] show that no rate of decay can be guaranteed even beyond the situation of algebraic rates. For comparison, the solution of the gradient flow equation
{ & = -f,(2) =
t>
is given by =()) = (1 (1+a हों
-fa(((1) ~ 1 I
==
Thus, while both the heavy ball ODE and the gradient flow can be made arbitrarily slow in this setting, the heavy ball remains much faster in comparison.
F Convergence proofs: strongly convex case
F1
Gradient Descent
Bassily et al. [2018], Wojtowytsch [2023] analyze stochastic gradient descent under the PL condition
u(f(z) - inf f) - |V](2))1
VIER
(21)
and the noise scaling assumption
E [1))(
[ <o(f inf
motivated by Lemma 8. The assumption is equivalent to multiplicative noise scaling within constant since every L-smooth function which satisfies a PL condition satisfies
2μ (f(z) - inf f) < [>]())|? < 2L (f(x) - inf f).
For completeness, we provide statement and proof directly in the multiplicative noise scaling regime which attains the optimal constant.
34

---

Additionally, we note that strong convexity implies the PL condition. The PL condition holds in many cases where convexity is false. c.g.
f(z, y) = (y - sin.) The set of minimizers {(r, y)
2
2 1
sin 3 is non-convex, so f cannot be convex. While this result is well-known the experts, have been unable to locate reference and hence provide proof.
Lemma 21. Assume that f : mmm ++ IR is p-strongly convex and Cc-mmoth. Then J satisfies the Pl-condition with constant μ
Proof. Let x, E 1.. Strong convexity combined with the Cauchy-Schwartz inequality means that f(x) - f(v) - (VS()),
x) マ ~1 yll2 &
[V](())]
=11
~リ ||
vll
I max [f(x)]
(22)
2
2p [V((2)11
Since this is true for y x², the result follows.
Several results in this vein are also collected in [Karimi et al., 2016, Theorem 2] together with additional generalizations of convexity, but with suboptimal implication (u-strongly convex & T-smooth) = /1/ T.PP... The additional implication (convexity & PL.) = strong convexity can also be found there.
Theorem 22 (GD, PI. condition). Assume that J satisfies the Pl-condition (21) and that the assump- tions laid out in Section 3.1 are satisfied. Let Fn be the sequence generated by the gradient descent scheme
9n =(IIn Wn),
In+1 In 7: S
L(1+
where W1, w2.... are elements of Q which are drawn independently of each other and the initial condition Tg. Then the estimate
E
[(x))
- inf f(x)) (11- inf f]
2ER"
holds for any n E N. Additionally, the sequence In converges to a limiting random variable Do almost surely and in L such that fx==) inf f almost surely.
Proof. We denote
La = E(s(rr) inf
and compute by Lemma 16 that
Lut1 - E f(x,) [[ inf
$ E (( pm (f(xn) [(x²)) - inf
(1- um) Ly.
The proof of almost sure convergence identical to the corresponding argument in [Wojtowytsch. 2023, Theorem 2.2] and similar in spirit to that of Corollary 5.
As usual, the optimal step-size 1) = I[I+0z as used Figure 1.
F.2 AGNES and NAG
Just like the convex case, we first prove Theorem 4 and set up the Lyapunov sequence with variable coefficients that can be chosen as per the time-stepping scheme. The continuous time analogue in this case is the heavy-ball ODE
f -2 Vf(x))
t> 0
á = 0
t= 0
x Zo
0
35

---

For p-strongly convex f, simple calculation shows that the Lyapunov function
1 + (T((((t) - x² )1*
(t) = f(x(())) - -((()))) +
satisfies 22(t) - - -Vp (t) and thus
((z(t))-f(at S (t) Se Se-vi e ffo)))
1 2 1| x² x²1)))
See for instance [Siegel, 2019, Theorem 1] for details.
Here, we state and prove a slightly generalized version of Theorem 4 in the main text. While we assumed an optimal choice of parameters in the main text, we allow for suboptimal selection here.
Theorem 4 (AGNES, strongly convex case general version). In addition to the assumptions in Theorem suppose that p-sttrongly convex and that
n
1 VTo
v - 1/ a
O<nS =(1+02)'
0 0< S 1+02"
p 1+ Va
- v 1
then
B[((xn) - [xx)] S (1 v) "B [(x)) f(x))
1
| - x 1P|]
+
2
Note that E [=(0) f(x*) 5 - || S 2E (f - f(x")] due to Lemma 12. A discus-
---==111
sion about the set of admissible parameters is provided after the proof. We note several special cases here.
1. If db is selected optimally as vn/(1 0²) for n, the order of decay is 1 v resembling Theorem 3.
1402, strongly
2. If additionally n = 1/(L(1+ + +))) is chosen optimally, then we recover the decay rate Vp/L (11 + d²) claimed in the main text.
3. We recover the gradient descent algorithm with the choice a = 0 which is achieved for +/ ///,. This selection admissible in our analysis since
VIT V 1 Vi+oz = WV S 1+
As expected, the constant of decay is vū μr), achieved in Theorem 22. In this sense, our analysis of AGNES interpolates fully between the optimal AGNES scheme (a NAG-type scheme in the deterministic case) and (stochastic) gradient descent. However, this proof only applies in the strongly convex setting, but not under a mere PL assumption.
4. If μ L i.. if f(z) # A+ μl|z x=1 for some A E R and 2o FT then we can < O. In this case, the gradient step is sufficiently
choose () < sb < /p 17, corresponding strong to compensate for momentum taking us the wrong direction. Needless to say, this is terrible idea and the rate of convergence is worse than that of gradient descent.
Proof. Set-up. Consider the Lyapunov sequence
La = =[f f(x*)] 1 E [|lb(r' I.)) + a(2', x² )?3
for constants b, be chosen later. We want show that there exists some decay factor 0 < 8 < 1 such that Lx+1 < S.Y,.
Step 1. Let consider the first term. Note that
[[f(xn+1)] = [f nghh]]
< [[(()²))] L (VV
where Gg.a. = 1) (1 Lon Im(1+02)
2
2 1//2 if
I(I+oTj
36

---

Step 2. We now turn to the second term and use the definition of 2 from (2),
b( Int1) + a(f - =x") = =bp(sr, - -agh - In) + a(Eat1 + plann - -agn Inn) - x²)
= (b aa)pppzz' - ogn In) + a(z', - -19h - -x")
= (b + aa)(p'p - xn) + a(r'/ x²) - ((b a)per + na)
To simplify notation, we introduce two new dependent variables:
: (6 a)p,
V = (6 + a)pa + bja ac + na.
With these variables, we have
b( In+1) + alv' x²) (3tnn Fn) + alx' x²) - voh.
Taking expectation of the square, we find that
E [|6 xn+1) + +a(x141 - =²))1²
=c² [122,, znll²]] + 2ac E8 [1x2 In) (x// xx)] + [ 22)1 }13}
2e0 [lggh (2r22 wn)] 2nrp E. [gh- (snn x²)] + 4/2
B|lbb11 2
S B - rn]l²]] + 2ac RR [(r1 Fn) (a x²)] + +a2B - xx []]
- -20 (x(h)) (4'a - xnm] 20 E V f((,)) - (2/n - x²) + +1
Step 3. We now use strong convexity to deduce that
E
|16(2'211 - 2n+1) + +a - ==")]²
S o² EE [[10,, - xn] + 2ac 18 [(<', - -xn) (I x=]]] + +02E(l(z', - x x1
200E (((2() - fzxz, + 11 11' - xnll²] 2ar E [f(xh) - f(x²) + 2/t || x² [13]
2
+ 21 + 0²) B[11 f ]]
(6² -c22-ubp -bp) E [||= 2m.11]] + 2ac EE [<= - xx) (2/ ( -z')] + (a² aيوي B[1l=1, x11
-204E (f
- f(x,)] 2ar E [f(2)) +
Step We now add the estimates of Steps 1 and
La+1 = R [(Entt) - [(x²) 2 |b #n+1) al - x²)|²
$ (1 - cbb av) EE (x cΦ E3 [f(n) (1 az)) B[f(x²))
1
+
2 (e² cbj) E || x.n 12 + ac E (I/, - xnn) (a'n - x²)]
1 (a² - E B[1l2% 11 + 2 + 0²)
+
E [[IVS(<)))
2
We require the coefficient of If (I²,)) to be zero, i.c. - ar = cah, so the incquality simplifies to
Lat1 S scv EC)[(((E) - f(x") + - 12- cdju B[llor, - 1m
1
2
+ ac - =xn) (x'n - =x"]] + 1 (a² - anty) E[lla', - x 11
2
1/2 01+ Cg.o.L E [(V{{{'))]
2
The smallest decay factor we can get at this point the coefficient of E( f(x))] So we hope to show that Lx+1 LviL which leads to the following system of inequalities on comparing
37

---

with the coefficients in the upper bound that we obtained in the previous step,
(b + a)p
(23)
v/ ac 1/a
(24)
(c + d)vb = 1
(25)
d² abk & cι²
(26)
ac cbab
(27)
a² abp S ava² 2
(28)
(1 0 2
L(1 + o²)n
2 In 1
(29)
2
Step Now we try to choose constants such that the system of inequalities holds. We assume that
L(I+o")1 for (29) it suffices that (1 + o yo² S n, i.e.
n S I(1o0z). Then since 2 S 17
2
++o+ 02
Note that (27) implies vi = 1/b and substituting that into (25), we get = b - a. Using this, (28) is equivalent to
a² - a2/y S cıaa² = - a)vaa = a
d²
which holds with equality a = v/, (26) holds because
c 1/y4 - a w/ S b 1/b²
if p. 1 > O. Finally (23) implies
p - a
Vpab
sTab
and (24) implies
1/0
1/2 1) T
a
1 Vub
With these choices of parameters, Lnt1 S cv. Ln = (1 ) and thus
B[f(zn) f(x")] < (1 / Lo
f(x²) < 2(1 / [f(o)) f(x'),
(1 - y E [
1 11000 ==]
2
where we have used Lemma 12 for strong convexity in the last step. When the parameters are chosen V 1+o= "I[1+oz)* we get p, a and the convergence rate as
optimally, i.e. 77 = I(1+02) and 1b stated in the theorem.
We focus on the meaningful case in which via 0. As discussed in Section 3, for given f.g g can replace L, a by larger values L', a' and μ by a smaller value μ². Let us briefly explore the effect of these substitutions. The parameter range described in this version of Theorem 4 can be understood as a three parameter family of AGNES parameters a. parametrized by v, p¹ and constraints given by L, p1 a
(m:.b.1))) 0 k I(1 +
0<0S V 2 0 0< k Su
The parameter map is given by
11, 1- V vp n
+ (n.p,a)
I+ 1- v v $5)
38

---

We can converesely obtain vp & from since the function + (1 - 2) 2) is its own inverse and thus V 1b
1-p In particular, in terms of the algorithms parameters, the decay rate is I+p"
1 P 2p
1+p 1+p
Furthermore, we see that
1/2 - 11 Vubb
2/2
77 1+p
1-p
1+ p
1- 11
2p + p
1-1 P
Vu'v
2p
1
16 V
0:
+
11
1 P
I+p
since vb > O. Thus, at the cost of more complicated representation, we could work directly in the parameter variables rather than using the auxiliary quantities v μ². In particular, both the parameter map and its inverse are continuous on D and its image respectively. Hence,despite the rigid appearance of the parameter selection in Theorem 4, there exists an open set of admissible parameters 77, a for which we obtain exponentially fast convergence.
We provide a more general version of Theorem as well. Just as in the convex case, as σ / the step size 7) decreases to zero and the theorem fails to guarantee convergence for a > 1.
Theorem 2 (NAG, strongly convex case). In addition to the assumptions in Theorem I, suppose that is p-strongly convex and the parameters are chosen such that
1 02
1- y
O<ns
o(1+ and p
then [[((=))-((=") s (wn)" (f(o)) fxx
1 + /un
Proof. Consider the Lyapunov sequence
Ln -([=s() - S(x)) + E [Ib(e,, xa) + alxh x²) =)]"]
where a and b are to be determined later. Since NAG is special case of AGNES with 7/ the first four steps are identical to the proof of Theorem 4. We get the following system of inequalities,
c (a + b)p
(30)
v nia + c)
(31)
(c + a)vb =1 1
(32)
2 app < b2
(33)
a² avp S ub
(34)
ac = abea)
(35)
2 + 0²)
Inn(1 + o²)
5n 1
(36)
2
2
Substituting (31) into (32), we get (a
+
= c)²=
and 30- = m1/ /n = /n. Thus (35) simplifies to
0²
Ln( 1 + 02)
< 1
2
2
which is equivalent to
1 1-02
5
L(1 02)
From (35), 6 1/1 = 1/ The rest of the inequalities can be verified to work with c=.c=
b a, = b-a 44ta
This shows that .u I lb (1 Ln. Finally, we get
B(((,M) -(('' < (1 /um)" "F B [3000 [(x") / |1:00 xx 113] < 21 / E (f( fxx")]
2
39

---

F.3 On the role of momentum parameters
Two different AGNES parameters are associated with momentum: a and p. In this section, we disentangle their respective contributions to keeping AGNES stable for highly stochastic noise.
For simplicity, first consider the case f - R. f(z) I and g(z) (1 + oN) f((x) where N is a standard normal random variable. Then
Un+1 p( gn) = =..=
94.
since on = 0. In particular, we note that
Elun+1] = -p Σ
n
'Bg'/lgll = -p Σ
n
n-i
1
pm pa+
p
0
-
f=1
i=1
and
- 1 nut1
)
27
n
27
n
E
Un+1
=
p (gf 1)
0² p
2(n
1 [1g
112]
P
i=0
1 p (
o² p²
(n-1)
0²2
i=0
due to the independence of different gradient estimators between time steps. particular, we see that
1. p becomes closer to 1, the eventual magnitude of the velocity variable increases as lim Bl| T-p-
2. as p becomes closer 1, the eventual variance of the velocity variable increases as limn-to E [[| |d -
3. the noise the normalized velocity estimate asymptotically satisfies
Un E(en]
p1- =021-p 0² p1= = (1- 1 1-p²
lim E
1+p
Thus, if p is closer to 1, both the magnitude and the variance of the velocity variable increase, but the the relative importance of noise approaches zero as p-+ 1. This not surprising - is close the sequence p" decays much slower than if is small. Gradient estimates from different times enter similar scale and cancellations can occur easily. As the influence of past gradients remains large, we say that the momentum variable has 'long memory'.
Of course, when minimizing non- non-linear function f, the gradient is not constant, and we face trade-off:
A long memory allows us to cancel random oscillations in the gradient estimates more easily.
2. A long memory also means compute with more out-of-date gradient estimates from points much further in the past along the trajectory.
Naturally, the relative importance of the first point increases with the stochasticity a of the gradient estimates. Even if the gradient evaluations deterministic, benefit from integrating historic information gained throughout the optimization process, but the rate at which we 'forget' outdated information much higher.
Thus the parameter P corresponds to the rate at which we forget old information. It also impacts the magnitude of the velocity variable. The parameter compensates for the scaling of en with 1/(1 p). We can think of as governing the rate at which we forget past gradients, and a as measure of the confidence with which we integrate past gradient information into time-steps for
40

---

Let us explore this relationship in strongly convex optimization. In Theorem 4, the optimal choice of hyper-parameters is given by 17 I[+oaa and
1 VM/T.
[ VH
2/ VT(1
a = 1- vM/L+ T
p= = VT(1 + +2) VII
=1-
02) + /ii
Let us consider the simplified regime r < L in which
~ 1+02 0= 1 1
a
1-p 2 p/L
In particular, we note: The larger the closer p is to 1, i.e. the longer the memory we keep. The relative importance of the momentum step compared to the gradient step, on the other hand, remains constant, depending only on the 'condition number' L/p. p.
We note that also in the convex casc, high stochasticity forces 7o to be large, meaning that Pn is always close to 1. Notably for generic non-convex objective functions, it is unclear that past gradients along the trajectory would carry useful information, there is discernible geometric relationship between gradients at different points. This mirrors an observation of Appendix G, just after Theorem 23.
G AGNES in non-convex optimization
We consider the case of non-convex optimization. In the deterministic setting, momentum methods for non-convex optimization have recently been studied by Diakonikolas and Jordan [2021]. We note that the algorithm may perform worse than stochastic gradient descent, but that for suitable parameters, the performance is comparable to that of SGD within constant factor.
Theorem 23 (Non-convex case). Assume that f satisfies the assumptions laid out in Section 3.1. Let 7, Ot P be such that
7 3 L(1 + +2))
a 1+ (La + 1)p² S
Then
2E f(xo)) inf f + app 112112 S (1+1) (1 a( + o²))
min
[[IVf(=)))
If Uo = O, the bound is minimal for gradient descent (i.e. 0) since the decay factor = 7 a(1 + o²) maximal.
Proof. Consider
Ln L-= =n=BB -B(OE + 2 11: x
for parameter À > 0 to be fixed later. We have
E [[((na+))] < [[f(x(]]]
" 2 B[IV
E f(xm) + Vf(a()) (r', - xx,) + La
|| 122 2 [V((a))))
2
2
E [[|22 1 = 0² || zn)11 2a (z/2 xn)
S/ + a ||9%
by Lemmas 13 and 16. We deduce that
L + Xp=
Lat1 < {E[[(En)] (1 -xppp))p))))))))))p E(VS(x')) ) (5h-=x))) +
E |0 x. []3]
2
+ Xpza - a(1 + 0²) 1 (
2
Apza ((1+00)) - B B f
< L. +
2
41

---

under the conditions
1 1 Xapx = 0, A S.
The first condition implies that À = (axp²)-1, 1, so the second one reduces to
(1- 1 1- 2 + p- + 12 La)p²
p²
p²a
Finally, we consider the last equation. If
e 1l - Ap a(1 + = 1 a(1 + = >0,
then we find that
E [(xa) +
1
|1001 - inf
2 L2 - Lutt = Σ ( Li+1) 2 2
E [[s{s)) ]
ap2
i=0
and hence
n Σ n
2B [(x0) - inf f - ||00|1
min Osisn
B[IV S [[[]f(=))1
E(n + 1)
H Proof of Lemma 8: Scaling intensity of minibatch noise
In this appendix, we provide theoretical justification for the multiplicative noise scaling regime considered in this article. Recall our main statement:
Lemma 8 (Noise intensity). Assume that E(h, v) ||- yll² and h : Rmm Rd R satisfies ||V x(,,.,)11 < C(1 + [|ll])? for some C.p O and all w E Rm and i = I.....N N. Then for all w E Rm:"
N
V4, VRll z 40 (1 + [| 2D R(w).
Proof. Since VR Ee Vl, observe that
1 n
IV4 - -RI s Σ |ell
as the average of a quantity is the unique value which minimizes the mean square discrepancy:
EX argmin E [ a|2]. We further find by Hölder's inequality that
(w, B1.)) V (wtk
4 n
k
k
(15(11,2)) - -vis)
Σ
|/ vort (w, 2w))
<
4 8
[1(W,
vill V /ww,
< 21 + || 20 1 [kw - vil12
= 41 + ||l|| 2)20 2p R(w).
42

---

I Implementation aspects
We discuss some implementation this section. All the code used for the experiments in the paper has been provided in the supplementary materials. The experiments in section Section 5 and Appendix A were run on Google Colab for compute time less than an hour. The experiments in Section 5.2 on a laptop CPU with compute time less than hour. The experiments in Sections 5.3 and 5.4 were run on single current generation GPU in a local cluster for up to 50 hours. An additional compute no more than 200 hours on a single GPU was used for experiments which were ultimately not used in the submitted version.
1.1 The last iterate
All neural-network based experiments were performed using the PyTorch library. Gradient-biie based optimizers in PyTorch and TensorFlow are implemented in such a way that gradients are computed outside of the optimizer and the point returndd by an optimizer step is the point for the next gradient evaluation. This strategy facilitates the manual manipulation of gradients by scaling, clipping or masking to train only subset of the network parameters.
The approach is theoretically justified for SGD. Guarantees for NAG and AGNES, on the other hand, are given for f(x,,) rather than [(rr,, i.e. not at the point where the gradient is evaluated. A discrepancy arises between theory and practice. In Algorithm 1, this discrepancy is resolved by taking final gradient descent step in the last time step and returning the sequence 2 at intermediate steps. In our numerical experiments, we did not include the final gradient descent step. Skipping the gradient step in particular allows for easier continuation of simulations beyond the initially specified stopping time, if so desired. We do not anticipate major differences under realistic circumstances. This be justified analytically in convex and strongly convex optimization, at least for low learning rate.
1. then Lemma 24. 1f < 3L,
Ei (If(1)))) f(x") 1 - 3Lm
E ([f(k))- - f(x*)] S
Proof. By essentially the same proof as Lemma 16, we have
E
f(
3n [f(()))) 2
< {
S (((()) "[V((=)))
since the correction term to linear approximation bounded by the L-Lipschitz continuity of V both from above and below. Recall furthermore that
|+ s 2L (f(x) f(x"))
for all L-smooth functions. Thus
(1 (13 - [(x*)] & ((1h)) 31 V[(xa)111 s [[[((]+))]]
In particular, if 1f1-3Ln > 0, then
B[S(>") [(x"){ S - 3 E[s(x++1) [(x)))
The condition 17 < 1/LL) is guarantced if the stochastic noise scaling satisfies a > v2 since then 1 -Lmm T For 1] 1/((1 + L.) we than find that
E [[f(2() f(x))] Elf f(x))))
0² +1 -(a))
-
-
I40
3 For instance, the implementations of NAG in PyTorch and Tensorflow return 2', rather than Tn.
43

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1.2 Weight decay
Weight decay machine learning tool which controls the magnitude the coefficients of neural network. In the simplest SGD setting, weight decay takes the form of modified update step
In+1 = (1 - Xn) 119n
for À O. gradient flow is governed by (1) an energy to be minimized and (2) an energy dissipation mechanism [Peletier, 2014]. It is known that different energy/dissipation pairings may induce the same dynamics for instance, Jordan al. [1998] show that the heat equation both the I,2 -.aaddetn flow of the Dirichlet energy and the Wasserstein gradient flow of the entropy function.
In this language, weight decay can be interpreted two different ways:
1. We minimize modified objective function f(x) |a: 22 which includes Tikhonov regularizer. The gradient estimates are stochastic for f and deterministic for the regularizer. This perspective corresponds including weight decay part of the energy.
2. We dynamically include confinement into the optimizer which pushes back against large values of x,, This perspective corresponds to including weight decay as part of the dissipa- tion.
In GD, both perspectives lead to the same optimization algorithm. In advanced minimizers, the two perspectives longer coincide. For Adam, Loshchilov and Hutter [2018. 2019] initiated debate the superior strategy of including weight decay. We note that the two strategies do not coincide for AGNES, but do not comment the superiority of
the other:
1. Treating weight decay dynamic property of the optimizer leads update rule like
2 + 0U
Un+1 p( - g'),, En+1 (1 An)a', -ng..
2. Treating weight decay as a component of the objective function to minimized leads the update rule
, = F. + nu
2a+1 = ( 9'. x',),
Ta+1 = (1 - An))a ng...
In our numerical experiments, we choose the second approach, viewing weight decay as a property of the objective function rather than the dissipation. This coincides with the approach taken by the SGD (and SGD with momentum) optimizer as well as Adam (but not AdamW).
44

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51


Reviews:
[{"review_id": "KVXn8uBhDy", "reviewer": "Reviewer_x6ut", "summary": "This paper introduces ``Accelerated Gradient Descent with Noisy Estimators\" (AGNES), a variant of Nesterov's accelerated gradient descent (NAG), and proves that AGNES achieves an accelerated convergence rate regardless of the noise level relative to the gradient in both convex and strongly convex cases. Additionally, experimental results validate the effectiveness of AGNES.", "questions": "Are Theorem 1 and 2 new theoretical results or just induced by prior works? \n\nI recommend to cite `Continuous-Time Analysis of AGM via Conservation Laws in Dilated Coordinate Systems. J. J. Suh, G. Roh, and E. K. Ryu' as the reference of continuous analysis of NAG type algorithm.\n\nCould the convergence result in Theorem 3 and 4 extend to the expectation of gradient norm?\n\nThe fifth line of the caption in Figure 6 needs spacing.", "limitations": "Yes, the authors addressed the limitations.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "It is impressive that a simple modification of NAG results in a robust theoretical convergence in stochastic settings. The theoretical analysis, including continuous analysis, seems consistent, and the experimental studies are detailed, particularly in the context of deep learning", "weaknesses": "I wonder if AGNES is truly novel. In lines 34 to 36, the authors state, ``In this work, we demonstrate that it is possible to achieve the same theoretical guarantees as Vaswani et al. [2019] with a simpler scheme, which can be considered as a reparametrized version of Liu and Belkin [2018]’s Momentum-Added Stochastic Solver (MaSS) method''. Does this imply that AGNES is essentially the same as MaSS or other previously suggested algorithms? Please clarify this point.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their helpful feedback and comments.\n\n>Does this imply that AGNES is essentially the same as MaSS or other previously suggested algorithms? Please clarify this point.\n\nAGNES is equivalent to MaSS after a reparametrization (shown in Appendix B.2). We derived the algorithm from NAG independently in a way which made the proofs geometrically transparent and allowed us to draw a clear parallel between Theorems 1 and 2 for NAG and Theorems 3 and 4 for AGNES. After the analysis was completed, we realized the equivalence, but decided to work with the version which, in our opinion, allows for a simpler geometric interpretation and more transparent proofs. The results we obtain for AGNES had not been obtained for this time stepping scheme in either parametrization. We will update the introduction of the article to more prominently reflect our contributions and avoid any confusion about the novelty of the scheme. \n\n>Are Theorems 1 and 2 new theoretical results or just induced by prior works?\n\nTo the best of our knowledge, Theorems 1 and 2 are new results that we did not find elsewhere in the literature. \n\n>I recommend to cite `Continuous-Time Analysis of AGM via Conservation Laws in Dilated Coordinate Systems. J. J. Suh, G. Roh, and E. K. Ryu' as the reference of continuous analysis of NAG type algorithm.\n\nWe are grateful to the reviewer for making us aware of this reference, which we will be happy to include in our discussion of continuous-time dynamics.\n\n>Could the convergence result in Theorem 3 and 4 extend to the expectation of gradient norm?\n\nYes, since L-smooth functions satisfy the inequality\n$\\|\\nabla f(x) \\|^2 \\leq 2L (f(x) - \\inf f)$ (see line 616 in Appendix E for context), Theorems 3 and 4 automatically lead to a bound on $\\mathbb E[\\|\\nabla f(x) \\|^2]$ in both convex as well as strongly convex cases. A bound on $\\mathbb E[\\|\\nabla f(x) \\|]$ can then be obtained using Jensen's inequality, i.e. $\\mathbb E[\\|\\nabla f(x) \\|]^2 \\leq \\mathbb E[\\|\\nabla f(x) \\|^2]$. We would be happy to add a remark about this in the final version of the article.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QMDsszRDO6", "reviewer": "Reviewer_2pRK", "summary": "This paper introduces and studies AGNES (Accelerated Gradient Descent with Noisy Estimators), a generalization of Nesterov's accelerated gradient (NAG) descent algorithm. First they show that NAG's guarantees break in the high noise regime. Then they prove that AGNES can accelerate gradient descent at any noise scale in smooth convex and strongly convex minimization tasks. The paper also provides empirical evidence arguing AGNES's improved performance over existing methods in various settings such as ImageNet.", "questions": "1. As the authors mentioned their algorithm is equivalent to a reparameterization of previous algorithms such as that in  Liu and Belkin and also I believe  the one in https://arxiv.org/abs/1803.05591. This should be highlighted more since currently the paper is written as presenting a new algorithm rather than presenting a better analysis. \n2. Could the authors either add more sweeps over hyperparameters or add these limitations to the empirical claims (such as in Contribution 5)?\n\nI would be happy to increase my rating if the authors could answer the above queries. [Addressed]", "limitations": "Yes.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "The strongest aspect of their paper is theoretical result. The fact that they only require two parameters will make it easier to apply to practical applications and also it leads to lesser space usage then Vaswani et al.", "weaknesses": "The optimizer comparison is done on small datasets and even then not done well since the authors either do not sweep over hyperaprameters such as learning rate or sweep with a large factor such as 10. This reduces the certainty of the claim that AGNES outperforms NAG in the high noise regime for deep learning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful to the reviewer for their helpful feedback and will be happy to incorporate it in a revised version.\n\n> This should be highlighted more since currently the paper is written as presenting a new algorithm rather than presenting a better analysis.\n\nWe agree with the point that the novelty of our work lies in the analysis of this algorithm, which is a reparametrization of Liu and Belkin (2018)'s algorithm. We will emphasize this more clearly in the revision and further place our work in the broader context of similar accelerated algorithms. We derived AGNES as a modification of NAG independently in a parametrization particularly suited to proving acceleration in this setting and only realized the equivalence after. We maintain that the 'AGNES' version of the scheme is particularly suited to geometric interpretation and analysis, which is why we retained it.\n\n> This reduces the certainty of the claim that AGNES outperforms NAG in the high noise regime for deep learning...\n>Could the authors either add more sweeps over hyperparameters or add these limitations to the empirical claims (such as in Contribution 5)?\n\nThe work is primarily theoretical and we are happy to acknowledge the limitations of our experimental section. \n\nTo address the question at least on one realistic example, we ran additional experiments testing a much wider range of hyperparameters for NAG for the task of training ResNet34 on classifying images from CIFAR-10. The plots along with the complete details of the experiment are included in the pdf attached to the global rebuttal. The results indicate that AGNES outperforms NAG with little fine-tuning of the hyperparameters.\n\nDetails of the experiment: We trained ResNet34 on CIFAR-10 with a batch size of 50 for 40 epochs using NAG with learning rate in {$8\\cdot 10^{-5}, 10^{-4}, 2\\cdot 10^{-4}, 5\\cdot 10^{-4}, 8\\cdot 10^{-4}, 10^{-3}, 2\\cdot 10^{-3}, 5\\cdot 10^{-3}, 8\\cdot 10^{-3}, 10^{-2}, 2\\cdot 10^{-2}, 5\\cdot 10^{-2}, 8\\cdot 10^{-2}, 10^{-1}, 2 \\cdot 10^{-1}, 5\\cdot 10^{-1}$} and momentum value in {$0.2, 0.5, 0.8, 0.9, 0.99$}.\nThese 80 combinations of hyperparameters for NAG were compared against AGNES with the default hyperparameters suggested $\\alpha = 10^{-3}$ (learning rate), $\\eta = 10^{-2}$ (correction step), and $\\rho = 0.99$ (momentum) as well as AGNES with a slightly smaller learning rate $5\\cdot 10^{-4}$ (with the other two hyperparameters being the same). AGNES consistently achieved a lower training loss as well as a better test accuracy faster than any combination of NAG hyperparameters tested. The same random seed was used each time to ensure a fair comparison between the optimizers. Overall, AGNES remained more stable and while other versions of NAG occasionally achieved a higher classification accuracy in certain epochs, a moving time average of AGNES outperformed every version of NAG (results not plotted due to space constraints, but will be included in the revised manuscript).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RvqABcsJHF", "reviewer": "Reviewer_AMLK", "summary": "This paper proposes an accelerated gradient method which is applicable in stochastic convex optimisation with unbiased but highly noised estimator. The method is analysed in a continuous-time convergence framework. Special attention is paid to the application of the method in machine learning, corresponding numerical experiments were carried out which confirm the advantage of the proposed method over some other ones, including classical Nesterov's accelerated method.", "questions": "Contribution of the paper is not clear. Since we expect that paper contributes either to theory or to practice, I try to find if it does. In theory, it does not make exhaustive literature review (arxiv:2307.01497 is ignored, for example; the only comparision of theoretical guarantees were given on Figure 1 with SGD) which is expectable when yet another accelerated method is proposed, competes with only SGD and NAG, i.e. the simplest methods, does not consider any structured optimisation settings and does not provide some generalizable analysis with any new techniques, but only uses some known tools to show the noise-robustness in the simplest setting, again without comparing their results with the literature on optimisation with state-dependent noise. In practice, it looks better when comparison is carried out on not-toy examples like image classification, but achieved improvement does not look significant in comparison to Adam, and its advantage is still confirmed by few simple problems, which does not convince a practitioner that method worth utilisation. Thus, this method does not seem helpful to the community, at least through its presentation in this text.", "limitations": "The limitations are clear from reading.", "rating": 3, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "The proposed method is indeed a good alternative to a classical NAG method when some simple problem statements are considered, which is the case in the problems chosen for practical evaluation. The analysis in continuous-time fashion seems unnecessary here, yet convenient to proof the results shortly and clearly. Most of minor technical details are clarified in appendices.", "weaknesses": "The language requires proof-reading, at least with an automated tool, and should be made more formal. The same recommendation is regarding the mathematical expressions: text takes several liberties here and there, i.e. uses not introduced notions without purpose, applies same-called functions to differently typed arguments, exploits x* for convex functions which are not guaranteed to reach its inf, etc. which makes an impression of that of informal draft, that I believe can be altered, taking into account that no significant details remain unclarified after reading the appendices. Similarly for numerical experiments: std-shadows are not added to some of the plots in stochastic experiments, axes are not labelled, etc. Besides, text contains many remarks on physical meaning, or on qualitative comparison of the proposed method with classical ones, which are so unclear that contribute nothing to understanding, but shift focus from the factual results. To summarise, text is far from being publishable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful to the reviewer for their helpful feedback.\n\n> The language requires proof-reading [...] and should be made more formal [...] To summarise, text is far from being publishable.\n\nWe understand that our presentation goes against the reviewer's stylistic preferences, and we are happy to integrate feedback into a revised version. To be sure, we would request more specific feedback about alterations the reviewer is suggesting. Since two other reviewers rated the presentation as '4: excellent', we are hesitant to consider changes without further clarification. We understand that this places additional strain on the reviewer's time, and we are very grateful for their service and the feedback they already provided.\n\n> text [...] exploits $x^*$ for convex functions which are not guaranteed to reach its inf\n\nThe existence of a minimizer $x^*$ is stated as part of the assumptions in Theorems 1 and 3. In fact, we study convex objective functions *without* minimizers in Theorem 7 (see also the paragraph above it). We are happy to rephrase the statement to emphasize this in a revised version.\n\nIf a minimizer does not exist, we do not expect that the algorithm would yield quantitative guarantees since even the 'heavy ball ODE' (deterministic, continuous time) can be arbitrarily slow due to Lemma 4.2 in [1] (see also lines 870-883 in Appendix F).\n\n> std-shadows are not added to some of the plots in stochastic experiments\n\nThe std-shadows are included in all experiments involving neural networks (albeit imperceptibly small in some plots compared to the oscillation of the trajectory which obscure them somewhat). They are not included in the plots for convex optimization since they made the plots less clear. Standard deviations were small and comparable for all methods considered. We did not find them to add much to the message: If $\\mathbb E[f(x_n)] < \\varepsilon$ for instance, then $f(x_n)< n\\varepsilon$ with probability at least $1-1/n$ by Chebyshev's inequality, directly yielding bounds on $f(x_n)$ since $\\mathbb E [f(x_n)] \\to 0$ fairly quickly.\n\n> In theory, it does not make exhaustive literature review (arxiv:2307.01497 is ignored, for example\n\nThe literature review is in Appendix C. We would be very happy to move it to the main text in a revised version and dedicate the additional text page to it. In a large and complex field, the review is certainly not complete, but we provide more context with the literature there (see also the next point).\n\nWe were indeed unaware of arxiv:2307.01497 and we will be happy to add it to the literature review.\n\n> the only comparision of theoretical guarantees were given on Figure 1 with SGD)[...], competes with only SGD and NAG, i.e. the simplest methods,\n\nWe are emphasizing the advantages of acceleration in Figure 1, which are shared with other methods such as ACDM and CNM. We present those methods alongside AGNES in Figure 3. We would be happy to replace the label AGNES in Figure 1 by 'AGNES, ACDM'. We conjecture that the same holds for CNM, but the full result is not available in the literature.\n\n> without comparing their results with the literature on optimisation with state-dependent noise\n\nWe would be happy to expand the discussion of the results of Vaswani et. al. and Even et. al. in a revised manuscript and to include the reference suggested above.\n\n> achieved improvement does not look significant in comparison to Adam\n\nWe consider this work a (non-trivial) first step at designing algorithms which combine momentum and adaptivity in a more principled way. The simplicity of the algorithm, e.g. compared to ACDM, is an important part of this. Even in our fairly simple examples, it can be seen that for more stochastic gradient estimates (smaller batch or larger model), AGNES is more stable in its performance compared to Adam. A similar trend would be expected when looking at very diverse datasets (although we do not test this hypothesis here).\n\n[1] Siegel, Wojtowytsch: A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces, *arXiv:2310.17610 [math.OC]*", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WL5KBNd567", "reviewer": "Reviewer_tWyF", "summary": "This paper presents a new accelerated gradient method for smooth (possibly strongly) convex functions. It is assumed that the available gradients are noisy, where the noise level is proportional to the norm of the gradient.\n\nThe authors show that Nesterov's accelerated gradient method (NAG) will converge in this setting when the constant of proportionality for the noise level is less than 1, but will diverge otherwise. To remedy this, the authors present the Accelerated Gradient descent with Noisy EStimators (AGNES) scheme. AGNES employs an additional parameter $\\alpha$ in the momentum step, and show that if $\\eta = \\alpha$ (where $\\eta$ is the step size for NAG), then NAG is equivalent to AGNES.\n\nTheoretical guarantees are presented to show that AGNES converges in the convex case, and at a better rate in the strongly convex case. Due to the noise in the gradients, the theoretical results hold in expectation, and the authors also show almost sure convergence in Corollary 5.\n\nNumerical experiments are also presented to support their findings.", "questions": "N/A", "limitations": "N/A", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Writing. The paper was well written and presented. Clear motivation for the new algorithm was given, as well as a comparison with other results in the literature to put this work in context. I enjoyed reading it.\n\nContribution. Problems where one encounters noisy gradients are abundant in the literature, so new algorithms for such tasks are welcome. The algorithm is conceptually simple, with few parameters to tune, and theoretical results guarantee convergence.\n\nNumerical experiments. The algorithm is supported by numerical experiments, which show the practical behavior of AGNES, and support the findings of the paper.", "weaknesses": "Personally, I would have preferred if the authors had used different notation. For example, in (1), there are two sequences of points, $\\{x_n\\}$ and $\\{x_n'\\}$. But then, the authors use the notation $g_n = g(x_n',\\omega_n)$, i.e., the gradient estimate does not have a prime, but it is associated with the sequence $x_n'$ involving a prime. I found it a bit jarring to read. To be honest, it probably would be even better to have used different letters to denote the two sequences, rather than a prime. \n\nI know it is hard to write in this style where a lot of important information needs to be put off until the appendix. However, in this case, it might have been an idea to have included the Literature Review (Appendix C) in the main body of the paper, and then taken out some of the length in the numerical experiments section (and put it in the appendix) to balance it out to 9 pages.\n\nBoth of these are personal preferences for the authors to take note of (I am not asking them to do anything here).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful to the reviewer for their feedback and encouraging comments.\n\n> I would have preferred if the authors had used different notation. For example, in (1), there are two sequences of points, $x_n$ and $x'_n$. But then, the authors use the notation $g_n=g(x'_n, \\omega_n)$, i.e., the gradient estimate does not have a prime, but it is associated with the sequence involving a prime.\n\nWe thank the reviewer for the good suggestion, and we agree that this would make the notation more consistent. We will be happy to replace $g_n$ with $g'_n$ for NAG and AGNES to make it clearer that those gradients are estimated at $x'_n$.\n\n>it might have been an idea to have included the Literature Review (Appendix C) in the main body of the paper, and then taken out some of the length in the numerical experiments section (and put it in the appendix) to balance it out to 9 pages.\n\nWe agree that the literature review being in the main body would help the readers put the current work in context of prior works. We will be happy use the additional content page that we would get, if accepted for publication, to move the literature review to the main body of the paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MrvwUYbWTT", "reviewer": "Reviewer_e67y", "summary": "The paper studies the accelerated gradient method of Nesterov’s (NAG) for smooth and (strongly-)convex problems under the multiplicative noise model when the variance of the stochastic gradients behave as $O(\\sigma^2 \\| \\nabla f(x) \\|^2)$. The authors identify that the NAG in its original momentum formulation could tolerate multiplicative noise for $\\sigma \\in [0,1)$. They propose a modified version which uses 2 step size sequences for the update and momentum steps, respectively, which are a constant factor apart from each other, and prove that their method achieves the optimal rates for any positive value of $\\sigma$ for general convex and strongly convex problems.", "questions": "1. Could you please comment on other formulations of acceleration such as the optimal method in [V, Eq. (3.11)] and the method of similar triangles [VI]. Specifically, optimal method in [V] has a 2 step size formulation and it might already give the desired result for multiplicative noise.\n\n2.\tIn Figure 3, what happens to NAG when $\\sigma = 0$? Please also include a case where $\\sigma < 1$ to compare NAG and AGNES.\n\n3.\tCould you elaborate on the contributions of your paper in terms of the theoretical analysis w.r.t. prior work?\n\n**References**\\\n[V] Nesterov, Y. (2005). Smooth minimization of non-smooth functions. Mathematical Programming, 103, 127-152.\n\n[VI] Gasnikov, A.V., Nesterov, Y.E. Universal Method for Stochastic Composite Optimization Problems. Comput. Math. and Math. Phys. 58, 48–64 (2018). https://doi.org/10.1134/S0965542518010050", "limitations": "I would suggest the authors to move the Appendix B.2 to the main text and discuss the equivalence to the prior work clearly.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "**Clarity and Presentation**\\\nThe manuscript identifies the key contributions in terms of the algorithmic design and explains how/why the particular parameter choices are necessary for the convergence with optimal rates.\n\n**Strengths**\\\nThe authors identify the limits of NAG and propose an easy but nice fix to remedy the limitations under multiplicative noise.\n\nThey are inspired by the continuous time aspect of NAG, similar to [Even et al., 2021] and [Liu and Belkin., 2018] but their modified construction of the Lyapunov function accommodates the 2 step-size-formulation they have. Looking at the analysis for NAG and AGNES, one could see that the separation in the second term of the Lyapunov function, i.e., $\\| b(n) ( x’_n – x_n ) + a(n) ( x’_n – x^* )  \\|^2$, enables to handle the multiplicative noise and the 2 step size structure at the same time. This is a simple fix, but it proves to be useful.\n\nTo my knowledge, this is the first work that successful studies the general convex and strongly-convex functions under multiplicative noise and proves optimal bounds. I find the results and the simplicity of the proposed method to be important.", "weaknesses": "**Clarity and Presentation**\\\nI think it should be made clearer that the proposed method is as simple, modified version of NAG rather than a new framework as the only difference is using 2 step size sequences, which are only a multiplicative factor of $\\frac{1}{1 + \\sigma^2}$ apart from each other. \n\n**Weaknesses**\nThe equivalence between [Liu & Belkin, 2018] makes me regards this work as a new analysis for an existing framework. They also have overlapping proof ideas with [Even at al., 2021] (use of continuous-time perspective, similar Lyapunov function constructions). Having that said, the authors seem to have identified correct modifications on the existing methodologies. I strongly suggest to include a dedicated section of comparison between the analysis techniques and explain their main strengths/contributions in accordance with the related work as it is not clear from the manuscript.\n\nThe results for the general convex case is not too interesting in my opinion. The main reason is the existing work on adaptive methods, which essentially prove that $\\sum_{i=1}^{n} a_i^2 \\| \\nabla f(x_i) \\|^2 < O(LD^2) $, where $D$ is the diameter of bounded set of iterates. Please refer to [I - IV] for relevant results. I cannot say anything for sure before analyzing in details but looking at their proofs, it should be fairly possible for those methods to tolerate even to unknown values of $\\sigma$ when $\\mu = 0$. Note that most probably this would not be extended to strongly-convex problems.\n\n**References**\n\n[I] Levy, K.Y., Yurtsever, A., & Cevher, V. (2018). Online Adaptive Methods, Universality and Acceleration. ArXiv, abs/1809.02864.\n\n[II] Cutkosky, A.. (2019). Anytime Online-to-Batch, Optimism and Acceleration. *Proceedings of the 36th International Conference on Machine Learning*, in *Proceedings of Machine Learning Research* 97:1446-1454 Available from https://proceedings.mlr.press/v97/cutkosky19a.html.\n\n[III] Kavis, A., Levy, K.Y., Bach, F.R., & Cevher, V. (2019). UniXGrad: A Universal, Adaptive Algorithm with Optimal Guarantees for Constrained Optimization. Neural Information Processing Systems.\n\n[IV] Joulani, P., Raj, A., György, A., & Szepesvari, C. (2020). A simpler approach to accelerated optimization: iterative averaging meets optimism. International Conference on Machine Learning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">I think it should be made clearer that the proposed method is as simple, modified version of NAG rather than a new framework as the only difference is using 2 step size sequences, which are only a multiplicative factor of \n$\\frac{1}{1+\\sigma^2}$ apart from each other.\n\nWe agree that a key point is that our method is a simple modification of NAG. We tried to highlight this in the abstract and will be happy to emphasize it more in the main text of a revised version of the paper. \n\n>The equivalence between [Liu \\& Belkin, 2018] makes me regards this work as a new analysis for an existing framework. They also have overlapping proof ideas with [Even at al., 2021] (use of continuous-time perspective, similar Lyapunov function constructions). Having that said, the authors seem to have identified correct modifications on the existing methodologies. I strongly suggest to include a dedicated section of comparison between the analysis techniques and explain their main strengths/contributions in accordance with the related work as it is not clear from the manuscript.\n\nThank you for this comment. We agree that our work should be regarded as a new analysis (and derivation) of the method of [Liu & Belkin, 2018]. In addition, the ideas behind the proof, in particular Lyapunov function arguments inspired by an analysis of the continuous dynamics, are fairly well-known in the optimization literature. We derived the method from our analysis of NAG independently and found the equivalence after the analysis was completed. The main novelty is a simple parametrization which simplifies the proofs of our main results (which are novel for this method).\n\n>The results for the general convex case is not too interesting in my opinion. The main reason is the existing work on adaptive methods, which essentially prove that \n$$\n\\sum_{i=1}^n a_i^2 |\\nabla f(x_i)|^2 \\leq O(LD^2),\n$$\nwhere $D$ is the diameter of bounded set of iterates. Please refer to [I - IV] for relevant results. I cannot say anything for sure before analyzing in details but looking at their proofs, it should be fairly possible for those methods to tolerate even to unknown values of $\\sigma$ when $\\mu = 0$. Note that most probably this would not be extended to strongly-convex problems.\n\nThank you for pointing out this literature, which we were previously not familiar with, and which we are planning to add to our literature review to include this work. Based upon our reading, this work is very extensive and covers the situations of online optimization and adaptivity with respect to the (additive) noise level $\\sigma$ and the smoothness parameter $L$ (which may even be infinite for some results, assuming that the objective function is Lipschitz-continuous). The second case excludes for instance for strongly convex functions, since we have\n$$\nf(x)\\geq f(x^*) + \\frac\\mu2\\|x-x^*\\|^2,\n$$\ni.e. the function grows quadratically at infinity. If the were Lipschitz-continuous, it could grow at most linearly. Many merely convex functions also grow faster than linearly at infinity.\n\nUnbounded gradients particularly affect the multiplicative noise setting we consider: The multiplicative noise is our 'friend' close to the set of minimizers since it becomes small, but it is our 'enemy' at infinity since it becomes large and may send us in the wrong direction. In particular, we have no a priori bounds on the diameter of the trajectory or on the gradient estimates in $L^\\infty$. The fact that we do not constrain the trajectory also allows us to formulate an result in the setting where the objective function does not have minimizers, in which case the optimizer trajectory *has to* diverge. It is not immediately apparent how the adaptive algorithms would fare in these situations as they have not been studied with multiplicatively scaling noise.\n\nWe agree that this is an important part of the literature and we will include a comparison to it in a revised version of the paper. Still, we believe that we contribute new results even in the convex case here.\n\n>Could you please comment on other formulations of acceleration such as the optimal method in [V, Eq. (3.11)] and the method of similar triangles [VI]. Specifically, optimal method in [V] has a 2 step size formulation and it might already give the desired result for multiplicative noise.\n\nAgain, we thank the reviewer for pointing out this work, which we were previously unaware of. Based upon our reading, the method in reference [V, Eq. (3.11)] is an optimal accelerated method in the plain convex (i.e. not strongly convex) case, which works even with a constraint (the convex set $Q$). Without further analysis, we do not know how this method will perform with multiplicative noise, but we believe that such an analysis will be too extensive to add to this paper and thus leave it to future work. We will certainly mention this work and its contribution to accelerated methods in the convex case in a revised version of the manuscript, however. We were also very intrigued by the method of similar triangles, which gives a nice way of deriving accelerated optimization methods for convex and strongly convex objectives that we were not aware of. However, we do not know how this method will generalize to the case of multiplicative noise without significant further analysis, which we again leave to future work. Of course, we will certainly mention the contributions of this paper in our revision.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We would like to thank all reviewers for their valuable feedback!

The pdf attached contains an additional sweep over hyperparameters to compare AGNES and NAG as suggested by Reviewer 2pRK.

Details of the experiment: We trained ResNet34 on CIFAR-10 with a batch size of 50 for 40 epochs using NAG with learning rate in {$8\cdot 10^{-5}, 10^{-4}, 2\cdot 10^{-4}, 5\cdot 10^{-4}, 8\cdot 10^{-4}, 10^{-3}, 2\cdot 10^{-3}, 5\cdot 10^{-3}, 8\cdot 10^{-3}, 10^{-2}, 2\cdot 10^{-2}, 5\cdot 10^{-2}, 8\cdot 10^{-2}, 10^{-1}, 2 \cdot 10^{-1}, 5\cdot 10^{-1}$} and momentum value in {$0.2, 0.5, 0.8, 0.9, 0.99$}.
These 80 combinations of hyperparameters for NAG were compared against AGNES with the default hyperparameters suggested $\alpha = 10^{-3}$ (learning rate), $\eta = 10^{-2}$ (correction step), and $\rho = 0.99$ (momentum) as well as AGNES with a slightly smaller learning rate $5*10^{-4}$ (with the other two hyperparameters being the same). AGNES consistently achieved a lower training loss as well as a better test accuracy faster than any combination of NAG hyperparameters tested. The same random seed was used each time to ensure a fair comparison between the optimizers. Overall, AGNES remained more stable and while other versions of NAG occasionally achieved a higher classification accuracy in certain epochs, a moving time average of AGNES outperformed every version of NAG (results not plotted due to space constraints, but will be included in the revised manuscript).

Meta-review: The paper studies Nesterov's accelerated gradient descent algorithm for smooth convex and strongly convex problems with noisy gradient estimates, and the noise level is proportional to the norm of the gradient. Acceleration over gradient descent is proved. The theory is solid and four reviewers are positive (two suggest accept and two weak accept). The main question of the reviewer who suggests rejection is about the text, such as the language, the mathematical expressions, and the experimental details. I think these problems can be fixed in the camery ready version. I decide to accept the paper. The AC had a thorough discussion with the SAC and the SAC agreed on the decision. I suggest the authors to fix the problems of presentation pointed out by reviewer AMLK in the camery ready version.
Decision: Accept (poster)