Adaptive learning rates and parallelization for stochastic, sparse,
    non-smooth gradients
Authors: Tom Schaul, Yann LeCun
ICLR 2013

Abstract: Recent work has established an empirically successful framework for adapting learning rates for stochastic gradient descent (SGD). This effectively removes all needs for tuning, while automatically reducing learning rates over time on stationary problems, and permitting learning rates to grow appropriately in non-stationary tasks. Here, we extend the idea in three directions, addressing proper minibatch parallelization, including reweighted updates for sparse or orthogonal gradients, improving robustness on non-smooth loss functions, in the process replacing the diagonal Hessian estimation procedure that may not always be available by a robust finite-difference approximation. The final algorithm integrates all these components, has linear complexity and is hyper-parameter free.

Adaptive learning rates and parallelization for
stochastic, sparse, non-smooth gradients
Tom Schaul
Courant Institute of Mathematical Sciences
New York University, 715 Broadway,
10003, New York
schaul@cims.nyu.edu
Yann LeCun
Courant Institute of Mathematical Sciences
New York University, 715 Broadway,
10003, New York
yann@cims.nyu.edu
Abstract
Recent work has established an empirically successful framework for adapting
learning rates for stochastic gradient descent (SGD). This effectively removes
all needs for tuning, while automatically reducing learning rates over time on
stationary problems, and permitting learning rates to grow appropriately in non-
stationary tasks. Here, we extend the idea in three directions, addressing proper
minibatch parallelization, including reweighted updates for sparse or orthogonal
gradients, improving robustness on non-smooth loss functions, in the process re-
placing the diagonal Hessian estimation procedure that may not always be avail-
able by a robust ﬁnite-difference approximation. The ﬁnal algorithm integrates all
these components, has linear complexity and is hyper-parameter free.
1 Introduction
Many machine learning problems can be framed as minimizing a loss function over a large (maybe
inﬁnite) number of samples. In representation learning, those loss functions are generally built
on top of multiple layers of non-linearities, precluding any direct or closed-form optimization, but
admitting (sample) gradients to guide iterative optimization of the loss.
Stochastic gradient descent (SGD) is among the most broadly applicable and widely-used algo-
rithms for such learning tasks, because of its simplicity, robustness and scalability to arbitrarily
large datasets. Doing many small but noisy updates instead of fewer large ones (as in batch meth-
ods) gives both a speed-up, and makes the learning process less likely to get stuck in sensitive local
optima. In addition, SGD is eminently well-suited for learning in non-stationary environments, e.g.,
when that data stream is generated by a changing environment; but non-stationary adaptivity is use-
ful even on stationary problems, as the initial search phase (before a local optimum is located) of the
learning process can be likened to a non-stationary environment.
Given the increasingly wide adoption of machine learning tools, there is an undoubted beneﬁt to
making learning algorithms, and SGD in particular, easy to use and hyper-parameter free. In recent
work, we made SGD hyper-parameter free by introducing optimal adaptive learning rates that are
based on gradient variance estimates [1]. While broadly successful, the approach was limited to
smooth loss functions, and to minibatch sizes of one. In this paper, we therefore complement that
work, by addressing and resolving the issues of
• minibatches and parallelization,
• sparse gradients, and
• non-smooth loss functions
all while retaining the optimal adaptive learning rates. All of these issues are of practical importance:
minibatch parallelization has strong diminishing returns, but in combination with sparse gradients
and adaptive learning rates, we show how that effect is drastically mitigated. The importance of
1
arXiv:1301.3764v2  [cs.LG]  27 Mar 2013
robustly dealing with non-smooth loss functions is also a very practical concern: a growing num-
ber of learning architectures employ non-smooth nonlinearities, like absolute value normalization or
rectiﬁed-linear units. Our ﬁnal algorithm addresses all of these, while remaining simple to imple-
ment and of linear complexity.
2 Background
There are a number of adaptive settings for SGD learning rates, or equivalently, diagonal precondi-
tioning schemes, to be found in the literature, e.g., [2, 3, 4, 5, 6, 7]. The aim of those is generally
to increase performance on stochastic optimization tasks, a concern complementary to our focus
of producing an algorithm that works robustly without any hyper-parameter tuning. Often those
adaptive schemes produce monotonically decreasing rates, however, which makes them no longer
applicable to non-stationary tasks.
The remainder of this paper build upon the adaptive learning rate scheme of [1], which is not mono-
tonically decreasing, so we recapitulate its main results here. Using an idealized quadratic and
separable loss function, it is possible to derive an optimal learning rate schedule which preserves
the convergence guarantees of SGD. When the problem is approximately separable, the analysis is
simpliﬁed as all quantities are one-dimensional. The analysis also holds as a local approximation in
the non-quadratic but smooth case.
In the idealized case, and for any dimension i, the optimal learning rate can be derived analytically,
and takes the following form
η∗
i = 1
hi
· (θi −θ∗
i)2
(θi −θ∗
i)2 + σ2
i
= 1
hi
·(E[∇θi])2
E[∇2
θi
] (1)
where (θi −θ∗
i) is the distance to the optimal parameter value, and σ2
i and hi are the local sample
variance and curvature, respectively.
We use an exponential moving average with time-constant τ (the approximate number of samples
considered from recent memory) for online estimates of the quantities in equation 1:
gi ← (1 −τ−1
i ) ·gi + τ−1
i ·∇θi
vi ← (1 −τ−1
i ) ·vi + τ−1
i ·(∇θi)2
hi ← (1 −τ−1
i ) ·hi + τ−1
i ·h(bbprop)
i
where the diagonal Hessian entriesh(bbprop)
i are computed using the ‘bbprop’ procedure [8], and the
time-constant (memory) is adapted according to how large a step was taken:
τi(t+ 1) =
(
1 −gi(t)2
vi(t)
)
·τi(t) + 1
The ﬁnal algorithm is called vSGD, and used the learning rates from equation 1 to update the pa-
rameters (element-wise):
θ ← θ −η∗·∇θ
3 Parallelization with minibatches
Compared to the pure online SGD, computation time can be reduced by “minibatch”-parallelization:
nsample-gradients are computed (simultaneously, e.g., on multiple cores) and then a single update
on the resulting averaged minibatch gradient is performed.
∇θ = 1
n
n∑
k=1
∇(k)
θ (2)
While n can be seen as a hyperparameter of the algorithm [9], it is often constrained to a large
extent by the computational hardware, memory requirements and communication bandwidth. A
derivation just like the one that led to equation 1 can be used to determine the optimal learning
2
100 101 102 103 104
minibatch size
10-3
10-2
10-1
100
relative log-loss gain per sample
sparsity=1.0
100 101 102 103 104
minibatch size
10-2
10-1
100
sparsity=0.2
100 101 102 103 104
minibatch size
10-1
100
sparsity=0.05
low noise
med noise
high noise
Figure 1: Diminishing returns of minibatch parallelization. Plotted is the relative log-loss gain (per
number of sample gradients evaluated) of a given minibatch size compared to the gain of the n= 1
case (in the noisy quadratic scenario from section 2, for different noise levels σ, and assuming
optimal learning rates as in equation 4); each ﬁgure corresponds to a different sparsity level. For
example, the ratio is 0.02 for n= 100(left plot, low noise): This means that it takes 50 times more
samples to obtain the same gain in loss than with pure SGD. Those are strongly diminishing returns,
but they are less drastic if the noise level is high (only 5 times more samples in this example). If
the sample gradients are somewhat sparse, however, and we use that fact to increase learning rates
appropriately, then the diminishing returns kick in only for much larger minibatch sizes; see the left
two ﬁgures.
rates automatically, for an arbitrary minibatch size n. The key difference is that the averaging in
equation 2 reduces the effective variance by a factor n, leading to:
η∗
i = 1
hi
· (θi −θ∗
i)2
(θi −θ∗
i)2 + 1
nσ2 = 1
hi
· (E[∇θi])2
1
nE[∇2
θi
] +n−1
n (E[∇θi])2 (3)
This expresses the intuition that using minibatches reduces the sample noise, in turn permitting
larger step sizes: if the noise (or sample diversity) is small, those gains are minimal, if it is large,
they are substantial (see Figure 1, left). Varying minibatch sizes tend to be impractical1 to implement
however, and so common practice is to simply ﬁx a minibatch size, and then re-tune the learning
rates (by a factor between 1 and n). With our adaptive minibatch-aware scheme (equation 3) this is
no longer necessary: in fact, we get an automatic transition from initially small effective minibatches
(by means of the learning rates) to large minibatches toward the end, when the noise level is higher.
4 Sparse gradients
Many common learning architectures (e.g., those using rectiﬁed linear units, or sparsity penalties)
lead to sample gradients that are increasinglysparse, that is, they are non-zero only in small fraction
of the problem dimensions. It is possible to exploit this to speed up learning, by averaging many
sparse gradients in a minibatch, or by doing asynchronous updates [10].
Here, we investigate how to set the learning rates in the presence of sparsity, and our result is simply
based on the observation that doing an update using a set of sparse gradients is equivalent to doing
the same update, but with a smaller effective minibatch size, while ignoring all the zero entries.
We can do this again on an element-by-element basis, where we deﬁne zi to be the number of
non-zero elements in dimension i, within the current minibatch. In each dimension, we rescale the
minibatch gradient accordingly by a factor n/(n−zi), and at the same time reduce the learning
rate to reﬂect the smaller effective minibatch size. Compounding those two effects gives the optimal
learning rate for sparse minibatches (we ignore the case zi = n, when there is no update):
η∗
i = n
n−zi
· 1
hi
· (E[∇θi])2
1
n−zi
E[∇2
θi
] +n−zi−1
n−zi
(E[∇θi])2 (4)
Figure 1 shows how using minibatches with such adaptive learning rates reduces the impact of
diminishing returns if the sample gradients are sparse. In other words, with the right learning rates,
higher sparsity can be directly translated into higher parallelizability.
1If the implementation/computational architecture is ﬂexible enough, the variance-term of the learning rate
can also be used to adapt the minibatch size adaptively to its optimal trade-off.
3
100 101 102 103
10-1
100
rel. gain (sparsity=0.1)
low noise
100 101 102 103
10-1
100
med noise
100 101 102 103
minibatch size
10-1
100
rel. gain (sparsity=0.01)
 100 101 102 103
minibatch size
10-1
100
mb-sparsity
avg-sparsity
Figure 2: Difference between global or instance-based computation of effective minibatch sizes in
the presence of sparse gradients. Our proposed method computes the number of non-zero entries
(n−zi) in the current mini-batch to set the learning rate (green). This involves some additional com-
putation compared to just using the long-term average sparsityp(nz)
i (red), but obtains a substantially
higher relative gain (see ﬁgure 1), especially in the regime where the sparsity level produces mini-
batches with just one or a few non-zero entries (dent in the curve near n = 1/p(nz)
i ). If the noise
level is low (left two ﬁgures), the effect is much more pronounced than if the noise is higher. For
comparison, the performance for 40 different ﬁxed learning-rate SGD settings (between 0.01 and
100) are plotted as yellow dots.
0.0 0.5 1.0 1.50.0
0.5
1.0
Figure 3: Illustrating the effect of reweighting minibatch gradients. Assume the samples are drawn
from 2 different noisy clusters (yellow and light blue vectors), but one of the clusters has a higher
probability of occurrence. The regular minibatch gradient is simply their arithmetic average (red),
dominated by the more common cluster. The reweighted minibatch gradient (blue) does a full step
toward each of the clusters, closely resembling the gradient one would obtain by performing a hard
clustering (difﬁcult in practice) on the samples, in dotted green.
An alternative to computing zi for each minibatch (and each dimension) anew would be to just use
the long-term average sparsity p(nz)
i = E[n−zi] instead. Figure 2 shows that this is suboptimal,
especially if the noise level is small, and in the regime where each minibatch is expected to contain
just a few non-zero entries. This ﬁgure also shows that equation 4 produces a higher relative gain
compared to the outer envelope of the performance of all ﬁxed learning rates.
4
?4 ?2 0 2 4
0
1
2
3
?4 ?2 0 2 4
0
1
2
3
Figure 4: Illustrating the expectation over non-smooth sample losses. In dotted blue, the loss func-
tions for a few individual samples are shown, each a non-smooth function. However, the expectation
over a distribution of such functions is smooth, as shown by the thick magenta curve. Left: absolute
value, right: rectiﬁed linear function; samples are identical but offset by a value drawn fromN(0,1).
4.1 Orthogonal gradients
One reason for the boost in parallelizability if the gradients are sparse comes from the fact that
sparse gradients are mostly orthogonal, allowing independent progress in each direction. But sparse
gradients are in fact a special case of orthogonal gradients, for which we can obtain similar speedups
with a reweighting of the minibatch gradients:
∇θ =
n∑
i=1
1
∑n
j=1
|∇(i)⊤
θ ∇(j)
θ |
∥∇(i)
θ ∥·∥∇(j)
θ ∥
∇(i)
θ (5)
In other words, each sample is weighted by one over the number of times (smoothed) that its gradient
is interfering (non-orthogonal) with another sample’s gradient.
In the limit, this scheme simpliﬁes to the sparse-gradient cases discussed above: if all sample gra-
dients are aligned, they are averaged (reweighted by 1/n, corresponding to the dense case in equa-
tion 2), and if all sample gradients are orthogonal, they are summed (reweighted by 1, corresponding
to the maximally sparse case zi = n−1 in equation 4). See Figure 3 for an illustration.
In practice, this reweighting comes at a certain cost, increasing the computational expense of a single
iteration from O(nd) to O(n2d), where dis the problem dimension. In other words, it is only likely
to be viable if the forward-backward passes of the gradient computation are non-trivial, or if the
minibatch size is small.
5 Non-smooth losses
Many commonly used non-linearities (rectiﬁed linear units, absolute value normalization, etc.) pro-
duce non-smooth sample loss functions. However, when optimizing over a distribution of samples
(or just a large enough dataset), the variability between samples can lead to a smooth expected loss
function, even though each sample has a non-smooth contribution. Figure 4 illustrates this point for
samples that have an absolute value or a rectiﬁed linear contribution to the loss.
It is clear from this observation that it is not possible to reliably estimate the curvature of the true
expected loss function, from the curvature of the individual sample losses (which are all zero in the
two examples above), if the sample losses are non-smooth. This means that our previous approach
of estimating the hi term in the optimal learning rate expression by a moving average of sample
curvatures, as estimated by the “bbprop” procedure [8] (which computes a Gauss-Newton approx-
imation of the diagonal Hessian, at the cost of one additional backward pass) is limited to smooth
sample loss functions, and we need a different approach for the general case2.
2This also alleviates potential implementation effort, e.g., when using third-party software that does not
implement bbprop.
5
Algorithm 1:vSGD-fd: minibatch-SGD with ﬁnite-difference-estimated adaptive learning rates
repeat
draw nsamples, compute the gradients ∇(j)
θ for each sample j
compute the gradients on the same samples, with the parameters shifted by δi = gi
for i∈{1,...,d }do
compute ﬁnite-difference curvatures hfd(j)
i =
⏐⏐⏐⏐
∇(j)
θi
−∇(j)
θi+δi
δi
⏐⏐⏐⏐
if |∇θi −gi|>2
√
vi −gi2 or
⏐⏐⏐hfd
i −hi
fd⏐⏐⏐>2
√
vifd −
(
hi
fd)2
then
increase memory size for outliers τi ←τi + 1
end
update moving averages
gi ← (1 −τ−1
i ) ·gi + τ−1
i ·1
n
∑n
j=1 ∇(j)
θi
vi ← (1 −τ−1
i ) ·vi + τ−1
i ·1
n
∑n
j=1
(
∇(j)
θi
)2
hi
fd
← (1 −τ−1
i ) ·hi
fd
+ τ−1
i ·1
n
∑n
j=1 hfd(j)
i
vifd ← (1 −τ−1
i ) ·vifd + τ−1
i ·1
n
∑n
j=1
(
hfd(j)
i
)2
estimate learning rate η∗
i ←hi
fd
vifd · n·(gi)2
vi + (n−1) ·(gi)2
update memory size τi ←
(
1 −(gi)2
vi
)
·τi + 1
update parameter θi ←θi −η∗
i ·1
n
∑n
j=1 ∇(j)
θi
end
until stopping criterion is met
5.1 Finite-difference curvature
A good estimate of the relevant curvature for our purposes (i.e., for determining a good learning
rate) is to not to compute the true Hessian at the current point, but to take the expectation over noisy
ﬁnite-difference steps, where those steps are on the same scalethan the actually performed update
steps, because this is the regime we care about.
In practice, we obtain this ﬁnite-difference estimates by computing two gradients of the same sample
loss, on points differing by the typical update distance3:
hfd
i =
⏐⏐⏐⏐
∇θi −∇θi+δi
δi
⏐⏐⏐⏐ (6)
where δi = gi. This approach is related to the diagonal Hessian preconditioning in SGD-QN [11],
but the step-difference used is different, and the moving average scheme there is decaying with time,
which thus loses the suitability for non-stationary problems.
5.2 Curvature variability
To further increase robustness, we reuse the same intuition that originally motivated vSGD, and
take into account the variance of the curvature estimates (produced by the ﬁnite-difference method)
to reduce the likelihood of becoming overconﬁdent (underestimating curvature, i.e., overestimating
learning rates) by using a variance-normalization based on the signal-to-noise ratio of the curvature
estimates.
3Of course, this estimate does not need to be computed at every step, which can save computation time.
6
For this purpose we maintain two additional moving averages:
hi
fd
← (1 −τ−1
i ) ·hi
fd
+ τ−1
i ·hfd
i
vi
fd ← (1 −τ−1
i ) ·vi
fd + τ−1
i ·
(
hfd
i
)2
and then compute the curvature term simply as hi = vifd/hi
fd
.
5.3 Outlier detection
If an outlier sample is encountered while the time constants τi is close to one (i.e., the history is
mostly discarded from the moving averages at each update), this has the potential to disrupt the
optimization process. Here, the statistics we keep for the adaptive learning rates have an additional,
unforeseen beneﬁt: they make it trivial to detect outliers.
The outlier’s effect can be mitigated relatively simply by increasing the time-constant τi before
incorporating the sample into the statistics (to make sure old samples are not forgotten), and then
due to the perceived variance shooting up, the learning rate is automatically reduced. If it was not
an outlier, but a genuine change in the data distribution, the algorithm will quickly adapt, increase
the learning rates again.
In practice, we use a detection threshold of two standard deviations, and increase the corresponding
τi by one (see pseudocode).
5.4 Algorithm
Algorithm 1 gives the explicit pseudocode for this ﬁnite-difference estimation, in combination with
the minibatch size-adjusted rates from equation 3, termed “vSGD-fd”. Initialization is akin to the
one of vSGD, in that all moving averages are bootstrapped on a few samples (10) before any updates
are done. It is also wise to add an tiny ϵ= 10−5 term where necessary to avoid divisions by zero.
6 Simulations
An algorithm that has the ambition to work out-of-the-box, without any tuning of hyper-parameters,
must be able to pass a number of elementary tests: those may not be sufﬁcient, but they are necessary.
To that purpose, we set up a collection of elementary (one-dimensional) stochastic optimization test
cases, varying the shape of the loss function, its curvature, and the noise level. The sample loss
functions are
fquad = A·
(
θ−ξ(j)
)2
fabs = A·
⏐⏐⏐θ−ξ(j)
⏐⏐⏐
frectlin =
{
A·(θ−ξ(j)) if θ−ξ(j) >0
0 otherwise
fgauss = A−Ae−(θ−ξ(j))
2
2
where Ais the curvature setting and the ξ(j) are drawn from N(0,σ2). We vary curvature and noise
levels by two orders of magnitude, i.e., A ∈{0.1,1,10}and σ2 ∈{0.1,1,10}, giving us 9x4 test
cases. To visualize the large number of results, we summarize the each test case and algorithm
combination in a concise heatmap square (see Figure 5 for the full explanation).
In Figure 6, we show the results for all test cases on a range of algorithms and minibatch sizes n.
Each square shows the gain in loss for 100 independent runs of 1024 updates each. Each group of
columns corresponds to one of the four functions, with the 9 inner columns using different curvature
and noise level settings. Color scales are identical for all heatmaps within a column, but not across
columns. Each group of rows corresponds to one algorithm, with each row using a different hyper-
parameter setting, namely initial learning rates η0 ∈{0.01,0.1,1,10}(for SGD, A DAGRAD [6]
and the natural gradient [7]) and decay rate γ ∈{0,1}for SGD. All rows come in pairs, with the
upper one using pure SGD (n= 1) and the lower one using minibatches (n= 10).
7
100 101 102 103
iterations
10-4
10-3
10-2
10-1
100
loss
SGD
vSGD
oracle
SGD
vSGD
oracle
Figure 5: Explanation of how to read our concise heatmap performance plots (right), based on the
more common representation as learning curves (left). In the learning curve representation, we plot
one curve for each algorithm and each trial (3x8 total), with a unique color/line-type per algorithm,
and the mean performance per algorithm with more contrast. Performance is measured every power
of 2 iterations. This gives a good idea of the progress, but becomes quickly hard to read. On the right
side, we plot the identical data in heatmap format. Each square corresponds to one algorithm, the
horizontal axis are still the iterations (on log2 scale), and on the vertical axis we arrange (sort) the
performance of the different trials at the given iteration. The color scale is as follows: white is the
initial loss value, the stronger the blue, the lower the loss, and if the color is reddish, the algorithm
overjumped to loss values that are bigger than the initial one. Good algorithm performance is visible
when the square becomes blue on the right side, instability is marked in red, and the variability of
the algorithm across trials is visible by the color range on the vertical axis.
The ﬁndings are clear: in contrast to the other algorithms tested, vSGD-fd does not require any
hyper-parameter tuning to give reliably good performance on the broad range of tests: the learning
rates adapt automatically to different curvatures and noise levels. And in contrast to the predecessor
vSGD, it also deals with non-smooth loss functions appropriately. The learning rates are adjusted
automatically according to the minibatch size, which improves convergence speed on the noisier test
cases (3 left columns), where there is a larger potential gain from minibatches.
The earlier variant (vSGD) was shown to work very robustly on a broad range of real-world bench-
marks and non-convex, deep neural network-based loss functions. We expect those results on smooth
losses to transfer directly to vSGD-fd. This bodes well for future work that will determine its per-
formance on real-world non-smooth problems.
7 Conclusion
We have presented a novel variant of SGD with adaptive learning rates that expands on previous
work in three directions. The adaptive rates properly take into account the minibatch size, which
in combination with sparse gradients drastically alleviates the diminishing returns of parallelization.
Also, the curvature estimation procedure is based on a ﬁnite-difference approach that can deal with
non-smooth sample loss functions. The ﬁnal algorithm integrates these components, has linear
complexity and is hyper-parameter free. Unlike other adaptive schemes, it works on a broad range
of elementary test cases, the necessary condition for an out-of-the-box method.
Future work will investigate how to adjust the presented element-wise approach to highly non-
separable problems (tightly correlated gradient dimensions), potentially relying on a low-rank or
block-decomposed estimate of the gradient covariance matrix, as in TONGA [12].
8
Acknowledgments
The authors want to thank Sixin Zhang, Durk Kingma, Daan Wierstra, Camille Couprie, Cl ´ement
Farabet and Arthur Szlam for helpful discussions. We also thank the reviewers for helpful sugges-
tions, and the ‘Open Reviewing Network’ for perfectly managing the novel open and transparent
reviewing process. This work was funded in part through AFR postdoc grant number 2915104, of
the National Research Fund Luxembourg.
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9
quad
 abs
 rectlin
 gauss
SGD
NaturalGrad
AdaGrad
vSGD
vSGD-fd
Figure 6: Performance comparisons for a number of algorithms (row groups) under different setting
variants (rows) and sample loss functions (columns), the latter grouped by loss function shape. Red
tones indicate a loss value worsening from its initial value, white corresponds to no progress, and
darker blue tones indicate a reduction of loss (in log-scale). For a detailed explanation of how to read
the heatmaps, see Figure 5. The new proposed algorithm vSGD-fd (bottom row group) performs
well across all functions and noise-level settings, namely ﬁxing the vSGD instability on non-smooth
functions like the absolute value. The other algorithms need to have their hyper-parameters tuned to
the task to work well.
10


Peer review:
[Review]:
This is a followup paper for reference [1] which describes a parameter free adaptive method to set learning rates for SGD.  This submission cannot be read without first reading [1].  It expands the work in several directions: the impact of minibatches, the impact of sparsity and gradient orthonormality, and the use of finite difference techniques to approximate curvature. The proposed methods are justified with simple theoretical considerations under simplifying assumptions and with serious empirical studies. I believe that these results are useful.

On the other hand, an opportunity has been lost to write a more substantial self-contained paper. As it stands, the submission reads like three incremental contributions stappled together.