DissLiteratur/storage/H7EUCYPR/.zotero-ft-cache

Published as a conference paper at ICLR 2015

arXiv:1412.6980v9 [cs.LG] 30 Jan 2017

ADAM: A METHOD FOR STOCHASTIC OPTIMIZATION

Diederik P. Kingma* University of Amsterdam, OpenAI
dpkingma@openai.com

Jimmy Lei Ba∗ University of Toronto jimmy@psi.utoronto.ca

ABSTRACT
We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.
1 INTRODUCTION
Stochastic gradient-based optimization is of core practical importance in many fields of science and engineering. Many problems in these fields can be cast as the optimization of some scalar parameterized objective function requiring maximization or minimization with respect to its parameters. If the function is differentiable w.r.t. its parameters, gradient descent is a relatively efficient optimization method, since the computation of first-order partial derivatives w.r.t. all the parameters is of the same computational complexity as just evaluating the function. Often, objective functions are stochastic. For example, many objective functions are composed of a sum of subfunctions evaluated at different subsamples of data; in this case optimization can be made more efficient by taking gradient steps w.r.t. individual subfunctions, i.e. stochastic gradient descent (SGD) or ascent. SGD proved itself as an efficient and effective optimization method that was central in many machine learning success stories, such as recent advances in deep learning (Deng et al., 2013; Krizhevsky et al., 2012; Hinton & Salakhutdinov, 2006; Hinton et al., 2012a; Graves et al., 2013). Objectives may also have other sources of noise than data subsampling, such as dropout (Hinton et al., 2012b) regularization. For all such noisy objectives, efficient stochastic optimization techniques are required. The focus of this paper is on the optimization of stochastic objectives with high-dimensional parameters spaces. In these cases, higher-order optimization methods are ill-suited, and discussion in this paper will be restricted to first-order methods. We propose Adam, a method for efficient stochastic optimization that only requires first-order gradients with little memory requirement. The method computes individual adaptive learning rates for different parameters from estimates of first and second moments of the gradients; the name Adam is derived from adaptive moment estimation. Our method is designed to combine the advantages of two recently popular methods: AdaGrad (Duchi et al., 2011), which works well with sparse gradients, and RMSProp (Tieleman & Hinton, 2012), which works well in on-line and non-stationary settings; important connections to these and other stochastic optimization methods are clarified in section 5. Some of Adam’s advantages are that the magnitudes of parameter updates are invariant to rescaling of the gradient, its stepsizes are approximately bounded by the stepsize hyperparameter, it does not require a stationary objective, it works with sparse gradients, and it naturally performs a form of step size annealing.
∗Equal contribution. Author ordering determined by coin flip over a Google Hangout.
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Published as a conference paper at ICLR 2015

Algorithm 1: Adam, our proposed algorithm for stochastic optimization. See section 2 for details,

and for square

a gt

slightly more gt. Good

efficient (but less clear) default settings for the

toersdteerdomf accohminpeutlaetaiornni.nggt2

indicates problems

the are

elementwise α = 0.001,

β1 = 0.9, β2 we denote β1

= 0.999 and and β2 to the

= 10−8. power t.

All

operations

on

vectors

are

element-wise.

With

β1t

and

β2t

Require: α: Stepsize

Require: β1, β2 ∈ [0, 1): Exponential decay rates for the moment estimates Require: f (θ): Stochastic objective function with parameters θ

Remvq0u0←ir←e:00(θI(0nI:nitIiitnaiialtiliziazele2p1nadsrtammmooemmteerennvttevvceteocctrtoorr)) t ← 0 (Initialize timestep)

while θt not converged do

t←t+1

gt ← ∇θft(θt−1) (Get gradients w.r.t. stochastic objective at timestep t) mt ← β1 · mt−1 + (1 − β1) · gt (Update biased first moment estimate) θvmvtttt←←←←θvβmtt2−/·t(1/1v−(t−1−α1−β+·2tβm)1(t(1)tC/(−o(C√moβvpm2tu)pt+·eugtbet2)iab((sUiUa-cpspo-ddcraoartetreecrebtpeciadatersasdeemdficeosrtesnetcdromsrn)aodwmraemwnot mmeseotinmmteaentsetti)emstaitme)ate) end while

return θt (Resulting parameters)

In section 2 we describe the algorithm and the properties of its update rule. Section 3 explains our initialization bias correction technique, and section 4 provides a theoretical analysis of Adam’s convergence in online convex programming. Empirically, our method consistently outperforms other methods for a variety of models and datasets, as shown in section 6. Overall, we show that Adam is a versatile algorithm that scales to large-scale high-dimensional machine learning problems.

2 ALGORITHM

See algorithm 1 for pseudo-code of our proposed algorithm Adam. Let f (θ) be a noisy objective function: a stochastic scalar function that is differentiable w.r.t. parameters θ. We are interested in minimizing the expected value of this function, E[f (θ)] w.r.t. its parameters θ. With f1(θ), ..., , fT (θ) we denote the realisations of the stochastic function at subsequent timesteps 1, ..., T . The stochasticity might come from the evaluation at random subsamples (minibatches) of datapoints, or arise from inherent function noise. With gt = ∇θft(θ) we denote the gradient, i.e. the vector of partial derivatives of ft, w.r.t θ evaluated at timestep t.

The algorithm updates exponential moving averages of the gradient (mt) and the squared gradient

(vt) where averages.

Tthheehmypoevri-npgaraavmeerategressβt1h,eβm2s∈elv[0e,s1a)recoensttrioml athteeseoxfpothneen1tsitalmdoemcaeynrta(ttehseofmtehaens)e

moving and the

2nd raw moment (the uncentered variance) of the gradient. However, these moving averages are

initialized as (vectors of) 0’s, leading to moment estimates that are biased towards zero, especially

during the initial timesteps, and especially when the decay rates are small (i.e. the βs are close to 1).

The good news is that this initialization bias can be easily counteracted, resulting in bias-corrected

estimates mt and vt. See section 3 for more details.

Note that the efficiency of algorithm 1 can, at the expense of clarity, be improved upon by changing

the order αt = α ·

of computation, e.g. by replacing 1 − β2t /(1 − β1t ) and θt ← θt−1

t−heαlat s·tmthtr/e(e√livnte+s inˆ)t.he

loop

with

the

following

lines:

2.1 ADAM’S UPDATE RULE teAwfnfoeicmutipvppeoersrttabenpotutpanrkdoespn:er|i∆tnypto|afr≤aAmdαeatm·e(r’1sspu−apcdβea1ta)e/t r√tuiml1ee−issteβipt2sticniasrte∆hfeutlc=acshαeoi(·c1me−ot/fβ√s1tev)pt>.siTz√ehse1. eA−ffseβscu2tmi,vaiennsgdte|p∆=sitz|0e≤, hthaαes

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Published as a conference paper at ICLR 2015

otherwise. The first case only happens in the most severe case of sparsity: when a gradient has

bwmeioellrnebzceeorsmommaatlolaenlrl.stciWmenehasertineops(s,1we−xecβewp1i)tlla=ht athv√ee1cthu−artrβemn2ttw/ti√emhvetastv≈eep.t±hFa1otrs|imlnecstes/√|sEpva[gtr|s]/e<ca1Ese[tgsh,2et]rh|eef≤oerf1ef.e|c∆Tthitve|ee<sftfeeαpc.stiivzInee magnitude of the steps taken in parameter space at each timestep are approximately bounded by

the stepsize setting α, i.e., |∆t| α. This can be understood as establishing a trust region around the current parameter value, beyond which the current gradient estimate does not provide sufficient

information. This typically makes it relatively easy to know the right scale of α in advance. For

many machine learning models, for instance, we often know in advance that good optima are with

high probability within some set region in parameter space; it is not uncommon, for example, to

have a prior distribution over the parameters. Since α sets (an upper bound of) the magnitude of

steps in parameter space, we can often deduce the right order of magnitude of α such that optima

can be reached we will call the

from ratio

θ0 mt

/w√itvhtinthseosmigennaul-mtob-neroiosef

iterations. With a slight abuse of ratio (SN R). With a smaller SNR

terminology, the effective

stepsize ∆t will be closer to zero. This is a desirable property, since a smaller SNR means that

there is greater uncertainty about whether the direction of mt corresponds to the direction of the true

gradient. For example, the SNR value typically becomes closer to 0 towards an optimum, leading

to smaller effective steps in parameter space: a form of automatic annealing. The effective stepsize

∆t is also invariant to with a factor c and vt

wthiethscaaflaecotofrthce2,gwrahdiicehntcsa;nrceeslcaoluint:g(tch·emgrta)d/i(e√ntcs2g·

wvti)th=famctto/r√c vwti.ll

scale

mt

3 INITIALIZATION BIAS CORRECTION

As explained in section 2, Adam utilizes initialization bias correction terms. We will here derive

the term for the second moment estimate; the derivation for the first moment estimate is completely

analogous. Let g be the gradient of the stochastic objective f , and we wish to estimate its second

raw moment (uncentered variance) using an exponential moving average of the squared gradient,

with decay rate β2. Let g1, ..., gT be the gradients at subsequent timesteps, each a draw from an

underlying gradient distribution gt ∼ p(gt). Let us initialize the exponential moving average as

v0 = 0 (a vector of zeros). First note that the update at timestep t of the exponential moving average

vt = β2 · vt−1 + a function of the

(g1ra−diβen2t)s·agtt2a(lwl phreerveiogut2sintidmiceastteespsth: e

elementwise

square

gt

gt) can be written as

t

vt = (1 − β2) β2t−i · gi2

(1)

i=1

We wish to know how E[vt], the expected value of the exponential moving average at timestep t,

relates Taking

to the true second expectations of the

mleoftm-heanntdEa[ngdt2]r,igsoht-wheancdansidceosrroefcteqfo. r(1t)h:e

discrepancy

between

the

two.

t

E[vt] = E (1 − β2) β2t−i · gi2

(2)

i=1

t

= E[gt2] · (1 − β2) β2t−i + ζ

(3)

i=1

= E[gt2] · (1 − β2t ) + ζ

(4)

wthheeerxepζon=ent0iailfdtehceaytruraeteseβc1oncadnm(aonmdesnhtoEul[dg)i2

] is stationary; be chosen such

otherwise ζ can be kept small since that the exponential moving average

assigns small weights to gradients too far in the past. caused by initializing the running average with zeros.

What is left is In algorithm 1

the we

ttehremref(o1re−dβiv2ti)dewbhyichthiiss

term to correct the initialization bias.

In case of sparse gradients, for a reliable estimate of the second moment one needs to average over many gradients by chosing a small value of β2; however it is exactly this case of small β2 where a lack of initialisation bias correction would lead to initial steps that are much larger.

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Published as a conference paper at ICLR 2015

4 CONVERGENCE ANALYSIS

We analyze the convergence of Adam using the online learning framework proposed in (Zinkevich, 2003). Given an arbitrary, unknown sequence of convex cost functions f1(θ), f2(θ),..., fT (θ). At each time t, our goal is to predict the parameter θt and evaluate it on a previously unknown cost function ft. Since the nature of the sequence is unknown in advance, we evaluate our algorithm using the regret, that is the sum of all the previous difference between the online prediction ft(θt) and the best fixed point parameter ft(θ∗) from a feasible set X for all the previous steps. Concretely, the regret is defined as:

T

R(T ) = [ft(θt) − ft(θ∗)]

(5)

t=1

where in the

aθp∗p=enadrixg.mOinuθr∈rXesultTti=s1cfotm(θp)a.rWabeleshtoowthAe dbaemst

has

√ O( T

)

regret

known bound for

bound and a this general

proof is given convex online

learning problem. We also use some definitions simplify our notation, where gt ∇ft(θt) and gt,i

as the ith element. We define g1:t,i ∈ Rt as a vector that contains the ith dimension of the gradients

over all iterations till t, g1:t,i = [g1,i, g2,i, · · · , gt,i]. Also, we define γ

. √β12
β2

Our following

theorem

holds

when

the

learning

rate

αt

is

decaying at

a

rate

of

t−

1 2

and

first

moment

running

average coefficient β1,t decay exponentially with λ, that is typically close to 1, e.g. 1 − 10−8.

Theorem 4.1. Assume that the function ft has bounded gradients, ∇ft(θ) 2 ≤ G, ∇ft(θ) ∞ ≤

G∞ for all θ ∈ Rd and distance between any θt generated by Adam is bounded, θn − θm 2 ≤ D,

θm − θn

∞

≤ D∞ for any m, n ∈ {1, ..., T }, and β1, β2

∈ [0, 1) satisfy

√β12 β2

< 1.

Let αt =

√α t

and β1,t = β1λt−1, λ ∈ (0, 1). Adam achieves the following guarantee, for all T ≥ 1.

R(T )

≤

D2 2α(1 −

β1)

d i=1

T vT,i+ (1

α(1√+ β1)G∞ − β1) 1 − β2(1 −

γ)2

d i=1

g1:T ,i

d
2+
i=1

D∞ 2

G∞

√ 1

−

β2

2α(1 − β1)(1 − λ)2

Our Theorem 4.1 implies when the data features are sparse and

mation
d i=1

term T vT

can be mu√ch ,i << dG∞ T

smaller than , in particular

its upper if the class

bound

d i=1

of function and

bounded g1:T,i 2

gradients, t√he << dG∞ T

sumand

data features are in the form of

tasoencAitmidoapnmr1o..v2Ienminpe(anDrttuioccvuheliarerO,t ta(hl√e., da2dT0a1)p1ft)io.vreTthhmeeeirnthoroends-u,aldstusacpfhotirvaetshmAe deextahpmoedca.tneDddeAvcadalayugienrEagd[β, 1c,adit=nt1oawchga1ire:dTvs,eizOe2r(]olaoilgssodima√ppTpolr)y-, tant in our theoretical analysis and also matches previous empirical findings, e.g. (Sutskever et al.,

2013) suggests reducing the momentum coefficient in the end of training can improve convergence.

Finally, we can show the average regret of Adam converges,

Corollary 4.2. Assume that the function ft has bounded gradients, ∇ft(θ) 2 ≤ G, ∇ft(θ) ∞ ≤ G∞ for all θ ∈ Rd and distance between any θt generated by Adam is bounded, θn − θm 2 ≤ D, θm − θn ∞ ≤ D∞ for any m, n ∈ {1, ..., T }. Adam achieves the following guarantee, for all T ≥ 1.

R(T ) T

=

O( √1 T

)

This result can be obtained by using Theorem 4.1 and

limT →∞

R(T ) T

=

0.

d i=1

g1:T,i 2

≤

dG∞

√ T

.

Thus,

5 RELATED WORK

Optimization methods bearing a direct relation to Adam are RMSProp (Tieleman & Hinton, 2012; Graves, 2013) and AdaGrad (Duchi et al., 2011); these relationships are discussed below. Other stochastic optimization methods include vSGD (Schaul et al., 2012), AdaDelta (Zeiler, 2012) and the natural Newton method from Roux & Fitzgibbon (2010), all setting stepsizes by estimating curvature

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Published as a conference paper at ICLR 2015

from first-order information. The Sum-of-Functions Optimizer (SFO) (Sohl-Dickstein et al., 2014) is a quasi-Newton method based on minibatches, but (unlike Adam) has memory requirements linear in the number of minibatch partitions of a dataset, which is often infeasible on memory-constrained systems such as a GPU. Like natural gradient descent (NGD) (Amari, 1998), Adam employs a preconditioner that adapts to the geometry of the data, since vt is an approximation to the diagonal of the Fisher information matrix (Pascanu & Bengio, 2013); however, Adam’s preconditioner (like AdaGrad’s) is more conservative in its adaption than vanilla NGD by preconditioning with the square root of the inverse of the diagonal Fisher information matrix approximation.

RMSProp: An optimization method closely related to Adam is RMSProp (Tieleman & Hinton, 2012). A version with momentum has sometimes been used (Graves, 2013). There are a few important differences between RMSProp with momentum and Adam: RMSProp with momentum generates its parameter updates using a momentum on the rescaled gradient, whereas Adam updates are directly estimated using a running average of first and second moment of the gradient. RMSProp also lacks a bias-correction term; this matters most in case of a value of β2 close to 1 (required in case of sparse gradients), since in that case not correcting the bias leads to very large stepsizes and often divergence, as we also empirically demonstrate in section 6.4.

AdaGrad: An algorithm that works well for sparse gradients is AdaGrad (Duchi et al., 2011). Its

basic version updates parameters as θt+1 = θt − α · gt/

t i=1

gt2.

Note

that

if

we

choose

β2

to

be

infinitesimally close to 1 version of Adam with β1

from = 0,

ibnefilonwite, sthimenalli(m1 β−2→β21)vatn=d

at−re1p·lacetim=1engtt2o. fAαdabGyraand

corresponds to a annealed version

αt = α · t−1/2, namely θt − α · t−1/2 · mt/ limβ2→1 vt = θt − α · t−1/2 · gt/ t−1 ·

t i=1

gt2

=

θt − α · gt/

t i=1

gt2.

Note that this direct correspondence between Adam and Adagrad does

not hold when removing the bias-correction terms; without bias correction, like in RMSProp, a β2

infinitesimally close to 1 would lead to infinitely large bias, and infinitely large parameter updates.

6 EXPERIMENTS

To empirically evaluate the proposed method, we investigated different popular machine learning models, including logistic regression, multilayer fully connected neural networks and deep convolutional neural networks. Using large models and datasets, we demonstrate Adam can efficiently solve practical deep learning problems. We use the same parameter initialization when comparing different optimization algorithms. The hyper-parameters, such as learning rate and momentum, are searched over a dense grid and the results are reported using the best hyper-parameter setting.

6.1 EXPERIMENT: LOGISTIC REGRESSION

We evaluate our proposed method on L2-regularized multi-class logistic regression using the MNIST

dataset. Logistic regression has a well-studied convex objective, making it suitable for comparison

irocefagdlriepfsfrseeirdoeinnctteioxoppnteifrmriomimzeenrsstescwitsiiotahndoj4uu.tswtTeodhrerbylyoing1gi/s√atibctordueetgclroaeycs,aslniomanminceillmaysusαimfit e=issstu√hαeets.cthlTaahstsemlsaatbetcpehlseidzsierweαcittihlnyooouunrr

logistic theorat-
the 784

dimension image vectors. We compare Adam to accelerated SGD with Nesterov momentum and

Adagrad using minibatch size of 128. According to Figure 1, we found that the Adam yields similar

convergence as SGD with momentum and both converge faster than Adagrad.

As discussed in (Duchi et al., 2011), Adagrad can efficiently deal with sparse features and gradi-

e1n/t√s tasdeocnaeyoofnitistsmstaeipnstihzeeosrheotiucladl

results whereas SGD is low at learning rare features. Adam with theoratically match the performance of Adagrad. We examine the

sparse feature problem using IMDB movie review dataset from (Maas et al., 2011). We pre-process

the IMDB movie reviews into bag-of-words (BoW) feature vectors including the first 10,000 most

frequent words. The 10,000 dimension BoW feature vector for each review is highly sparse. As sug-

gested in (Wang & Manning, 2013), 50% dropout noise can be applied to the BoW features during

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Published as a conference paper at ICLR 2015

training cost
training cost

0.7

MNIST Logistic Regression

AdaGrad

SGDNesterov

Adam

0.6

0.5

0.4

0.3

0.20 5 10 15 20 25 30 35 40 45 iterations over entire dataset

0.50 IMDB BoW feature Logistic Regression

Adagrad+dropout

0.45

RMSProp+dropout SGDNesterov+dropout

Adam+dropout

0.40

0.35

0.30

0.25

0.200 20 40 60 80 100 120 140 160 iterations over entire dataset

Figure 1: Logistic regression training negative log likelihood on MNIST images and IMDB movie reviews with 10,000 bag-of-words (BoW) feature vectors.

training to prevent over-fitting. In figure 1, Adagrad outperforms SGD with Nesterov momentum by a large margin both with and without dropout noise. Adam converges as fast as Adagrad. The empirical performance of Adam is consistent with our theoretical findings in sections 2 and 4. Similar to Adagrad, Adam can take advantage of sparse features and obtain faster convergence rate than normal SGD with momentum.
6.2 EXPERIMENT: MULTI-LAYER NEURAL NETWORKS Multi-layer neural network are powerful models with non-convex objective functions. Although our convergence analysis does not apply to non-convex problems, we empirically found that Adam often outperforms other methods in such cases. In our experiments, we made model choices that are consistent with previous publications in the area; a neural network model with two fully connected hidden layers with 1000 hidden units each and ReLU activation are used for this experiment with minibatch size of 128. First, we study different optimizers using the standard deterministic cross-entropy objective function with L2 weight decay on the parameters to prevent over-fitting. The sum-of-functions (SFO) method (Sohl-Dickstein et al., 2014) is a recently proposed quasi-Newton method that works with minibatches of data and has shown good performance on optimization of multi-layer neural networks. We used their implementation and compared with Adam to train such models. Figure 2 shows that Adam makes faster progress in terms of both the number of iterations and wall-clock time. Due to the cost of updating curvature information, SFO is 5-10x slower per iteration compared to Adam, and has a memory requirement that is linear in the number minibatches. Stochastic regularization methods, such as dropout, are an effective way to prevent over-fitting and often used in practice due to their simplicity. SFO assumes deterministic subfunctions, and indeed failed to converge on cost functions with stochastic regularization. We compare the effectiveness of Adam to other stochastic first order methods on multi-layer neural networks trained with dropout noise. Figure 2 shows our results; Adam shows better convergence than other methods.
6.3 EXPERIMENT: CONVOLUTIONAL NEURAL NETWORKS Convolutional neural networks (CNNs) with several layers of convolution, pooling and non-linear units have shown considerable success in computer vision tasks. Unlike most fully connected neural nets, weight sharing in CNNs results in vastly different gradients in different layers. A smaller learning rate for the convolution layers is often used in practice when applying SGD. We show the effectiveness of Adam in deep CNNs. Our CNN architecture has three alternating stages of 5x5 convolution filters and 3x3 max pooling with stride of 2 that are followed by a fully connected layer of 1000 rectified linear hidden units (ReLU’s). The input image are pre-processed by whitening, and
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10-1 MNIST Multilayer Neural Network + dropout AdaGrad RMSProp SGDNesterov AdaDelta Adam

training cost

10-2

0

50

100

150

200

iterations over entire dataset

(a)

(b)

Figure 2: Training of multilayer neural networks on MNIST images. (a) Neural networks using dropout stochastic regularization. (b) Neural networks with deterministic cost function. We compare with the sum-of-functions (SFO) optimizer (Sohl-Dickstein et al., 2014)

training cost training cost

3.0

CIFAR10 ConvNet First 3 Epoches

AdaGrad

AdaGrad+dropout

SGDNesterov

2.5

SGDNesterov+dropout

Adam

Adam+dropout

2.0

1.5

1.0

0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

iterations over entire dataset

CIFAR10 ConvNet

102

AdaGrad AdaGrad+dropout

SGDNesterov

101

SGDNesterov+dropout Adam

Adam+dropout

100

10-1

10-2

10-3

10-4 0 5 10 15 20 25 30 35 40 45 iterations over entire dataset

Figure 3: Convolutional neural networks training cost. (left) Training cost for the first three epochs. (right) Training cost over 45 epochs. CIFAR-10 with c64-c64-c128-1000 architecture.

dropout noise is applied to the input layer and fully connected layer. The minibatch size is also set to 128 similar to previous experiments. Interestingly, although both Adam and Adagrad make rapid progress lowering the cost in the initial stage of the training, shown in Figure 3 (left), Adam and SGD eventually converge considerably faster than Adagrad for CNNs shown in Figure 3 (right). We notice the second moment estimate vt vanishes to zeros after a few epochs and is dominated by the in algorithm 1. The second moment estimate is therefore a poor approximation to the geometry of the cost function in CNNs comparing to fully connected network from Section 6.2. Whereas, reducing the minibatch variance through the first moment is more important in CNNs and contributes to the speed-up. As a result, Adagrad converges much slower than others in this particular experiment. Though Adam shows marginal improvement over SGD with momentum, it adapts learning rate scale for different layers instead of hand picking manually as in SGD.
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β1=0

β2=0.99

β2=0.999

β2=0.9999

β2=0.99

β2=0.999

β2=0.9999

Loss

β1=0.9

log10(α) (a) after 10 epochs

(b) after 100 epochs

Figure 4: Effect of bias-correction terms (red line) versus no bias correction terms (green line) after 10 epochs (left) and 100 epochs (right) on the loss (y-axes) when learning a Variational AutoEncoder (VAE) (Kingma & Welling, 2013), for different settings of stepsize α (x-axes) and hyperparameters β1 and β2.

6.4 EXPERIMENT: BIAS-CORRECTION TERM

We also empirically evaluate the effect of the bias correction terms explained in sections 2 and 3.

Discussed in section 5, removal of the bias correction terms results in a version of RMSProp (Tiele-

man & Hinton, 2012) with momentum. We vary the β1 and β2 when training a variational autoencoder (VAE) with the same architecture as in (Kingma & Welling, 2013) with a single hidden

layer with 500 hidden units with softplus nonlinearities and a 50-dimensional spherical Gaussian

latent variable. We iterated over a broad range of hyper-parameter choices, i.e. β1 ∈ [0, 0.9] and

βne2s∈s t[o0.s9p9a,rs0e.9g9r9a,d0ie.9n9ts9,9r]e,saunldtsloing1la0r(gαe)r

∈ [−5, ..., −1]. Values of β2 initialization bias; therefore

close to 1, required we expect the bias

for robustcorrection

term is important in such cases of slow decay, preventing an adverse effect on optimization.

In Figure 4, values β2 close to 1 indeed lead to instabilities in training when no bias correction term was present, especially at first few epochs of the training. The best results were achieved with small values of (1 − β2) and bias correction; this was more apparent towards the end of optimization when gradients tends to become sparser as hidden units specialize to specific patterns. In summary, Adam performed equal or better than RMSProp, regardless of hyper-parameter setting.

7 EXTENSIONS

7.1 ADAMAX

In Adam, the update rule for individual weights is to scale their gradients inversely proportional to a (scaled) L2 norm of their individual current and past gradients. We can generalize the L2 norm based update rule to a Lp norm based update rule. Such variants become numerically unstable for large p. However, in the special case where we let p → ∞, a surprisingly simple and stable algorithm emerges; see algorithm 2. We’ll now derive the algorithm. Let, in case of the Lp norm, the stepsize at time t be inversely proportional to vt1/p, where:

vt = β2pvt−1 + (1 − β2p)|gt|p

(6)

t

= (1 − β2p) β2p(t−i) · |gi|p

(7)

i=1

8

Published as a conference paper at ICLR 2015

Algorithm 2: AdaMax, a variant of Adam based on the infinity norm. See section 7.1 for details.

Good default settings for the tested machine learning problems are α = 0.002, β1 = 0.9 and

βbi2as=-co0r.9re9c9t.ioWn ittehrmβ1tfowrethdeenfiorstet

β1 to the moment.

Apollwoepretr.aHtioenres,o(nαv/e(1cto−rsβa1tr)e)

is the learning element-wise.

rate

with

the

Require: α: Stepsize

Require: β1, β2 ∈ [0, 1): Exponential decay rates Require: f (θ): Stochastic objective function with parameters θ

Require: m0 ←

0θ(0I:nIitniiatliiazlep1asrtammoemterenvtevcteocrtor)

u0 ← 0 (Initialize the exponentially weighted infinity norm) t ← 0 (Initialize timestep)

while θt not converged do

t←t+1

gt ← ∇θft(θt−1) (Get gradients w.r.t. stochastic objective at timestep t)

mt ← β1 · mt−1 + (1 − β1) · gt (Update biased first moment estimate)

ut ← max(β2 · ut−1, |gt|) (Update the exponentially weighted infinity norm)

enθdtw←hilθet−1 − (α/(1 − β1t)) · mt/ut (Update parameters)

return θt (Resulting parameters)

Note that the decay term is here equivalently parameterised as β2p instead of β2. Now let p → ∞, and define ut = limp→∞(vt)1/p, then:

t

1/p

ut

=

pl→im∞(vt)1/p

=

lim
p→∞

(1 − β2p)

β2p(t−i) · |gi|p

(8)

i=1

t

1/p

=

lim (1
p→∞

−

β2p)1/p

β2p(t−i) · |gi|p

(9)

i=1

= lim
p→∞

t

1/p
β2(t−i) · |gi| p

i=1

(10)

= max β2t−1|g1|, β2t−2|g2|, . . . , β2|gt−1|, |gt|

(11)

Which corresponds to the remarkably simple recursive formula:

ut = max(β2 · ut−1, |gt|)

(12)

with initial value u0 = 0. Note that, conveniently enough, we don’t need to correct for initialization bias in this case. Also note that the magnitude of parameter updates has a simpler bound with AdaMax than Adam, namely: |∆t| ≤ α.

7.2 TEMPORAL AVERAGING

Since the last iterate is noisy due to stochastic approximation, better generalization performance is

often achieved by averaging. Previously in Moulines & Bach (2011), Polyak-Ruppert averaging

(Polyak & Juditsky, 1992; Ruppert, 1988) has been shown to improve the convergence of standard

SbeGuDs,ewd,hgeriveiθ¯ntg=hig1ther

nkw=e1igθkh.t

Alternatively, an exponential moving average over the parameters can to more recent parameter values. This can be trivially implemented

by adding one line to the inner loop of algorithms 1 and 2: θ¯t ← β2 · θ¯t−1 + (1 − β2)θt, with θ¯0 = 0.

Initalization bias can again be corrected by the estimator θt = θ¯t/(1 − β2t).

8 CONCLUSION

We have introduced a simple and computationally efficient algorithm for gradient-based optimization of stochastic objective functions. Our method is aimed towards machine learning problems with

9

Published as a conference paper at ICLR 2015
large datasets and/or high-dimensional parameter spaces. The method combines the advantages of two recently popular optimization methods: the ability of AdaGrad to deal with sparse gradients, and the ability of RMSProp to deal with non-stationary objectives. The method is straightforward to implement and requires little memory. The experiments confirm the analysis on the rate of convergence in convex problems. Overall, we found Adam to be robust and well-suited to a wide range of non-convex optimization problems in the field machine learning.
9 ACKNOWLEDGMENTS
This paper would probably not have existed without the support of Google Deepmind. We would like to give special thanks to Ivo Danihelka, and Tom Schaul for coining the name Adam. Thanks to Kai Fan from Duke University for spotting an error in the original AdaMax derivation. Experiments in this work were partly carried out on the Dutch national e-infrastructure with the support of SURF Foundation. Diederik Kingma is supported by the Google European Doctorate Fellowship in Deep Learning.
REFERENCES
Amari, Shun-Ichi. Natural gradient works efficiently in learning. Neural computation, 10(2):251–276, 1998. Deng, Li, Li, Jinyu, Huang, Jui-Ting, Yao, Kaisheng, Yu, Dong, Seide, Frank, Seltzer, Michael, Zweig, Geoff,
He, Xiaodong, Williams, Jason, et al. Recent advances in deep learning for speech research at microsoft. ICASSP 2013, 2013. Duchi, John, Hazan, Elad, and Singer, Yoram. Adaptive subgradient methods for online learning and stochastic optimization. The Journal of Machine Learning Research, 12:2121–2159, 2011. Graves, Alex. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013. Graves, Alex, Mohamed, Abdel-rahman, and Hinton, Geoffrey. Speech recognition with deep recurrent neural networks. In Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on, pp. 6645–6649. IEEE, 2013. Hinton, G.E. and Salakhutdinov, R.R. Reducing the dimensionality of data with neural networks. Science, 313 (5786):504–507, 2006. Hinton, Geoffrey, Deng, Li, Yu, Dong, Dahl, George E, Mohamed, Abdel-rahman, Jaitly, Navdeep, Senior, Andrew, Vanhoucke, Vincent, Nguyen, Patrick, Sainath, Tara N, et al. Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. Signal Processing Magazine, IEEE, 29(6):82–97, 2012a. Hinton, Geoffrey E, Srivastava, Nitish, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan R. Improving neural networks by preventing co-adaptation of feature detectors. arXiv preprint arXiv:1207.0580, 2012b. Kingma, Diederik P and Welling, Max. Auto-Encoding Variational Bayes. In The 2nd International Conference on Learning Representations (ICLR), 2013. Krizhevsky, Alex, Sutskever, Ilya, and Hinton, Geoffrey E. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012. Maas, Andrew L, Daly, Raymond E, Pham, Peter T, Huang, Dan, Ng, Andrew Y, and Potts, Christopher. Learning word vectors for sentiment analysis. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies-Volume 1, pp. 142–150. Association for Computational Linguistics, 2011. Moulines, Eric and Bach, Francis R. Non-asymptotic analysis of stochastic approximation algorithms for machine learning. In Advances in Neural Information Processing Systems, pp. 451–459, 2011. Pascanu, Razvan and Bengio, Yoshua. Revisiting natural gradient for deep networks. arXiv preprint arXiv:1301.3584, 2013. Polyak, Boris T and Juditsky, Anatoli B. Acceleration of stochastic approximation by averaging. SIAM Journal on Control and Optimization, 30(4):838–855, 1992.
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Published as a conference paper at ICLR 2015 Roux, Nicolas L and Fitzgibbon, Andrew W. A fast natural newton method. In Proceedings of the 27th
International Conference on Machine Learning (ICML-10), pp. 623–630, 2010. Ruppert, David. Efficient estimations from a slowly convergent robbins-monro process. Technical report,
Cornell University Operations Research and Industrial Engineering, 1988. Schaul, Tom, Zhang, Sixin, and LeCun, Yann. No more pesky learning rates. arXiv preprint arXiv:1206.1106,
2012. Sohl-Dickstein, Jascha, Poole, Ben, and Ganguli, Surya. Fast large-scale optimization by unifying stochas-
tic gradient and quasi-newton methods. In Proceedings of the 31st International Conference on Machine Learning (ICML-14), pp. 604–612, 2014. Sutskever, Ilya, Martens, James, Dahl, George, and Hinton, Geoffrey. On the importance of initialization and momentum in deep learning. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 1139–1147, 2013. Tieleman, T. and Hinton, G. Lecture 6.5 - RMSProp, COURSERA: Neural Networks for Machine Learning. Technical report, 2012. Wang, Sida and Manning, Christopher. Fast dropout training. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 118–126, 2013. Zeiler, Matthew D. Adadelta: An adaptive learning rate method. arXiv preprint arXiv:1212.5701, 2012. Zinkevich, Martin. Online convex programming and generalized infinitesimal gradient ascent. 2003.
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Published as a conference paper at ICLR 2015

10 APPENDIX

10.1 CONVERGENCE PROOF Definition 10.1. A function f : Rd → R is convex if for all x, y ∈ Rd, for all λ ∈ [0, 1],
λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y)

Also, notice that a convex function can be lower bounded by a hyperplane at its tangent. Lemma 10.2. If a function f : Rd → R is convex, then for all x, y ∈ Rd,
f (y) ≥ f (x) + ∇f (x)T (y − x)

The above lemma can be used to upper bound the regret and our proof for the main theorem is constructed by substituting the hyperplane with the Adam update rules.

The following two lemmas are used to support our main theorem. We also use some definitions sim-

plify our notation, where gt ∇ft(θt) and gt,i as the ith element. We define g1:t,i ∈ Rt as a vector that contains the ith dimension of the gradients over all iterations till t, g1:t,i = [g1,i, g2,i, · · · , gt,i]

Lemma 10.3. Let gt = ∇ft(θt) and g1:t be defined as above and bounded, gt 2 ≤ G, gt ∞ ≤ G∞. Then,

T t=1

gt2,i t

≤ 2G∞

g1:T ,i

2

Proof. We will prove the inequality using induction over T.

The base case for T = 1, we have g12,i ≤ 2G∞ g1,i 2. For the inductive step,

T t=1

gt2,i t

=

T −1 t=1

gt2,i t

+

gT2 ,i T

≤ 2G∞ g1:T −1,i 2 +

gT2 ,i T

= 2G∞

g1:T ,i

2 2

−

gT2

+

gT2 ,i T

From, have,

g1:T ,i

2 2

−

gT2 ,i

+

4

gT4 ,i g1:T ,i

2 2

≥

g1:T ,i

2 2

−

gT2 ,i,

we can take square root of both side and

g1:T ,i

2 2

−

gT2 ,i

≤

≤

g1:T ,i

2− 2

gT2 ,i g1:T ,i

2

g1:T ,i

2− 2

gT2 ,i T G2∞

Rearrange the inequality and substitute the

g1:T ,i

2 2

−

gT2 ,i

term,

G∞

g1:T ,i

2 2

−

gT2

+

gT2 ,i T

≤ 2G∞

g1:T ,i

2

12

Published as a conference paper at ICLR 2015

Lemma 10.4. gt ∞ ≤ G∞,

Let the

γfollow√βiβn122g.

For β1, β2 ∈ [0, inequality holds

1)

that

satisfy

√β12 β2

<

1 and bounded gt,

gt 2 ≤ G,

T t=1

m2t,i tvt,i

≤

2 1−γ

√1 1 − β2

g1:T ,i

2

√

uPsrionogf.thUe nudpedrattheerualsessuimnpAtilogno,ri(t1h−m1−β1t1β)2,t2

≤

1 (1−β1

)2

.

We can expand the last term in the summation

T t=1

m2t,i

T −1
=

tvt,i t=1

m2t,i tvt,i

+

1 − β2T (1 − β1T )2

(

T k=1

(1

−

β1)β1T

−k

gk,i

)2

T

T j=1

(1

−

β2)β2T

−j

gj2,i

T −1
≤
t=1

m2t,i tvt,i

+

(1

1 − β2T − β1T )2

T k=1

T ((1 − β1)β1T −kgk,i)2

T

T j=1

(1

−

β2)β2T

−j gj2,i

T −1
≤
t=1

m2t,i tvt,i

+

(1

1 − β2T − β1T )2

T k=1

T ((1 − β1)β1T −kgk,i)2 T (1 − β2)β2T −kgk2,i

T −1
≤
t=1 T −1
≤
t=1

m2t,i tvt,i

+

(1

1 − β2T − β1T )2

(1 − β1)2

T
T

T (1 − β2) k=1

m2t,i + tvt,i

T

T
γT −k

T (1 − β2) k=1

gk,i

2

√β12 β2

T −k
gk,i 2

Similarly, we can upper bound the rest of the terms in the summation.

T

m2t,i

T
≤

t=1 tvt,i t=1

T
≤

t=1

gt,i 2

T −t
tγj

t(1 − β2) j=0

gt,i 2

T
tγj

t(1 − β2) j=0

For γ < 1, using the upper bound on the arithmetic-geometric series,

t

tγt

<

1 (1−γ)2

:

T
t=1
Apply Lemma 10.3,

gt,i 2 t(1 − β2)

T
tγj
j=0

≤

(1 −

γ)21√1

− β2

T t=1

g√t,i 2 t

T t=1

m2t,i tvt,i

≤

(1 − γ2)G2√∞1 − β2

g1:T ,i

2

To simplify the notation, we define γ

. √β12
β2

Intuitively, our following theorem holds when the

learning

rate

αt

is

decaying

at

a

rate

of

t−

1 2

and

first

moment

running

average

coefficient

β1,t

decay

exponentially with λ, that is typically close to 1, e.g. 1 − 10−8.

Theorem 10.5. Assume that the function ft has bounded gradients, ∇ft(θ) 2 ≤ G, ∇ft(θ) ∞ ≤ G∞ for all θ ∈ Rd and distance between any θt generated by Adam is bounded, θn − θm 2 ≤ D,

13

Published as a conference paper at ICLR 2015

θm − θn

∞

≤ D∞ for any m, n ∈ {1, ..., T }, and β1, β2

∈ [0, 1) satisfy

√β12 β2

< 1.

Let αt =

√α t

and β1,t = β1λt−1, λ ∈ (0, 1). Adam achieves the following guarantee, for all T ≥ 1.

R(T )

≤

D2 2α(1 −

β1)

d i=1

T vT,i+ (1

α(β√1 + 1)G∞ − β1) 1 − β2(1 −

γ)2

d i=1

g1:T ,i

d
2+
i=1

D∞ 2

G∞

√ 1

−

β2

2α(1 − β1)(1 − λ)2

Proof. Using Lemma 10.2, we have,

d
ft(θt) − ft(θ∗) ≤ gtT (θt − θ∗) = gt,i(θt,i − θ,∗i)
i=1

From the update rules presented in algorithm 1,

θt+1 = θt − αtmt/

=

θt

−

1

αt − β1t

vt

√β1,t vt

mt−1

+

(1

−√ β1,t vt

)

gt

We focus on the both sides of the

ith dimension of the parameter above update rule, we have,

vector

θt

∈

Rd.

Subtract

the

scalar

θ,∗i

and

square

(θt+1,i

−

θ,∗i)2

=(θt,i

−

θ,∗i)2

−

1

2αt − β1t

(

β1,t vt,i

mt−1,i

+

(1

−

β1,t) vt,i

gt,i)(θt,i

−

θ,∗i)

+

αt2(

mt,i )2 vt,i

We can rearrange the above equation and use Young’s inequality, ab ≤ a2/2 + b2/2. Also, it can be

shown that vt,i =

t j

=1(1

−

β2)β2t−j

gj2,i/

1 − β2t ≤ g1:t,i 2 and β1,t ≤ β1. Then

gt,i(θt,i

−

θ,∗i)

=

(1 − β1t ) 2αt(1 −

vt,i β1,t)

(θt,i − θ,∗t)2 − (θt+1,i − θ,∗i)2

1

+

(1

β1,t − β1,t)

√vtα4−t1−,i1

(θ,∗i

−

θt,i)√αt−1

mt−1,i
1

vt4−1,i

+

αt(1 − β1t ) 2(1 − β1,t

vt,i )

(

mt,i )2 vt,i

≤

2αt

1 (1 −

β1)

(θt,i − θ,∗t)2 − (θt+1,i − θ,∗i)2

vt,i

+

β1,t 2αt−1(1 −

β1,t) (θ,∗i

−

θt,i)2

+

β1αt−1 2(1 − β1)

m2t−1,i vt−1,i

+

αt 2(1 −

β1)

m2t,i vt,i

vt−1,i

We apply Lemma 10.4 to the above inequality and derive the regret bound by summing across all the dimensions for i ∈ 1, ..., d in the upper bound of ft(θt) − ft(θ∗) and the sequence of convex functions for t ∈ 1, ..., T :

R(T )

≤

d i=1

2α1

1 (1 −

β1)

(θ1,i

− θ,∗i)2

v1,i

+

d i=1

T t=2

2(1

1 −

β1) (θt,i

−

θ,∗i)2(

vt,i αt

−

vt−1,i αt−1

)

+

√β1αG∞ (1 − β1) 1 − β2(1

− γ)2

d i=1

g1:T ,i

2

+

(1

−

√αG∞ β1) 1 − β2(1

−

γ)2

d i=1

g1:T,i 2

+

d i=1

T t=1

β1,t 2αt(1 −

β1,t)

(θ,∗i

−

θt,i)2

vt,i

14

Published as a conference paper at ICLR 2015

From the assumption, θt − θ∗ 2 ≤ D, θm − θn ∞ ≤ D∞, we have:

R(T )

≤

D2 2α(1 −

β1)

d i=1

T vT,i

+

α(1√+ β1)G∞ (1 − β1) 1 − β2(1 − γ)2

d i=1

g1:T ,i

2

+

D∞ 2 2α

dt

β1,t

i=1 t=1 (1 − β1,t)

tvt,i

≤

D2 2α(1 −

β1)

d i=1

T vT,i

+

(1

−

α(1√+ β1) 1

β1)G∞ − β2(1

−

γ

)2

d i=1

+

D∞ 2 G∞

√ 1

−

β2

2α

d i=1

t t=1

(1

β1,t − β1,t)

√ t

g1:T,i 2

We can use arithmetic geometric series upper bound for the last term:

t t=1

(1

β1,t − β1,t

)

√ t

≤

t t=1

(1

1 −

β1)

λt−1

√ t

≤

t t=1

(1

1 −

β1)

λt−1

t

≤

(1

1 − β1)(1 −

λ)2

Therefore, we have the following regret bound:

R(T )

≤

D2 2α(1 −

β1)

d i=1

T vT,i

+

(1

−

α(1√+ β1) 1

β1)G∞ − β2(1

−

γ

)2

d i=1

g1:T ,i

2

+

d i=1

D∞ 2

√ G∞ 1

2αβ1(1 −

− β2 λ)2

15