SIGN SGD WITH M AJORITY V OTE IS C OMMUNICATION E FFICIENT A ND F AULT T OLERANT
Anonymous authors
Paper under double-blind review
A BSTRACT
Training neural networks on large datasets can be accelerated by distributing the workload over a network of machines.
As datasets grow ever larger, networks of hundreds or thousands of machines become economically viable.
The time cost of communicating gradients limits the effectiveness of using such large machine counts, as may the increased chance of network faults.
We explore a particularly simple algorithm for robust, communication-efﬁcient learning— SIGN SGD.
Workers transmit only the sign of their gradient vector to a server, and the overall update is decided by a majority vote.
This algorithm uses 32 × less communication per iteration than full-precision, distributed SGD.
Under natural conditions veriﬁed by experiment, we prove that SIGN SGD converges in the large and mini-batch settings, establishing convergence for a parameter regime of A DAM as a byproduct.
Aggregating sign gradients by majority vote means that no individual worker has too much power.
We prove that unlike SGD, majority vote is robust when up to 50% of workers behave adversarially.
The class of adversaries we consider includes as special cases those that invert or randomise their gradient estimate.
On the practical side, we built our distributed training system in Pytorch.
Benchmarking against the state of the art collective communications library (NCCL), our framework—with the parameter server housed entirely on one machine—led to a 25% reduction in time for training resnet50 on Imagenet when using AWS p3.2xlarge machines.
I NTRODUCTION
The most powerful supercomputer in the world is currently a cluster of over 27,000 GPUs at Oak
Ridge National Labs (TOP500, 2018).
Distributed algorithms designed for such large-scale systems typically involve both computation and communication: worker nodes compute intermediate results locally, before sharing them with their peers.
When devising new machine learning algorithms for distribution over networks of thousands of workers, we posit the following desiderata:
D1 fast algorithmic convergence; D2 good generalisation performance; D3 communication efﬁciency; D4 robustness to network faults.
When seeking an algorithm that satisﬁes all four desiderata D1–4 , inevitably some tradeoff must be made.
Stochastic gradient descent (SGD) naturally satisﬁes D1–2 , and this has buoyed recent advances in deep learning.
Yet when it comes to large neural network models with hundreds of millions of parameters, distributed SGD can suffer large communication overheads.
To make matters worse, any faulty SGD worker can corrupt the entire model at any time by sending an inﬁnite gradient, meaning that SGD without modiﬁcation is not robust.
A simple algorithm with aspirations towards all desiderata D1–4 is as follows: workers send the sign of their gradient up to the parameter server, which aggregates the signs and sends back only the majority decision.
We refer to this algorithm as SIGN SGD with majority vote.
All communication to and from the parameter server is compressed to one bit, so the algorithm certainly gives us D3 .
What’s more, in deep learning folklore sign based methods are known to perform well, indeed inspiring the popular RMS PROP and A DAM optimisers (Balles & Hennig, 2018), giving hope for [Figure]
D1 .
As far as robustness goes, aggregating gradients by a majority vote denies any individual worker too much power, suggesting it may be a natural way to achieve D4 .
In this work, we make the above aspirations rigorous.
Whilst D3 is immmediate, we provide the ﬁrst convergence guarantees for SIGN SGD in the mini-batch setting, providing theoretical grounds for D1 .
We show how theoretically the behaviour of SIGN SGD changes as gradients move from high to low signal-to-noise ratio.
We also extend the theory of majority vote to show that it achieves a notion of Byzantine fault tolerance.
A distributed algorithm is Byzantine fault tolerant (Blanchard et al., 2017) if its convergence is robust when up to 50% of workers behave adversarially.
The class of adversaries we consider contains interesting special cases, such as robustness to a corrupted worker sending random bits, or a worker that inverts their gradient estimate.
Though our adversarial model is not the most general, it is interesting as a model of network faults, and so gives us D4 .
Next, we embark on a large-scale empirical validation of our theory.
We implement majority vote in the Pytorch deep learning framework, using CUDA kernels to bit pack sign tensors down to one bit.
Our results provide experimental evidence for D1–D4 .
Comparing our framework to NCCL (the state of the art communications library), we were able to speed up Imagenet training by 25% when distributing over 7 to 15 AWS p3.2xlarge machines, albeit at a slight loss in generalisation.
Finally, in an interesting twist, the theoretical tools we develop may be brought to bear on a seemingly unrelated problem in the machine learning literature.
Reddi et al.
(2018) proved that the extremely popular A DAM optimiser in general does not converge in the mini-batch setting.
This result belies the success of the algorithm in a wide variety of practical applications.
SIGN SGD is equivalent to a special case of A DAM , and we establish the convergence rate of mini-batch SIGN SGD for a large class of practically realistic objectives.
Therefore, we expect that these tools should carry over to help understand the success modes of A DAM .
Our insight is that gradient noise distributions in practical problems are often unimodal and symmetric because of the Central Limit Theorem, yet
Reddi et al.
(2018)’s construction relies on bimodal noise distributions.
R ELATED W ORK
For decades, neural network researchers have adapted biologically inspired algorithms for efﬁcient hardware implementation.
Hopﬁeld (1982), for example, considered taking the sign of the synaptic weights of his memory network for readier adaptation into integrated circuits.
This past decade, neural network research has focused on training feedforward networks by gradient descent (LeCun et al., 2015).
It is natural to ask what practical efﬁciency may accompany simply taking the sign of the backpropagated gradient.
In this section, we explore related work pertaining to this question.
Deep learning: whilst stochastic gradient descent (SGD) is the workhorse of machine learning (Robbins & Monro, 1951), algorithms like RMS PROP (Tieleman & Hinton, 2012) and A DAM (Kingma & Ba, 2015) are also extremely popular neural net optimisers.
These algorithms have their roots in the R PROP optimiser (Riedmiller & Braun, 1993), which is a sign-based method similar to
SIGN SGD except for component wise rate
Algorithm 1 S IGNUM with majority vote, the proposed algorithm for distributed optimisation.
Good default settings for the tested machine learning problems are η = 0 .
0001 and β = 0 .
9 , though tuning is recommended.
All operations on vectors are element-wise.
Setting β
=
yields SIGN SGD.
Require:
learning rate η > 0 , momentum constant β ∈ [0 , 1) , weight decay λ ≥ 0 , mini-batch size n , initial point x held by each of M workers, initial momentum v m ← 0 on m th worker repeaton m th worker [Equation] stochasticGradient
[Equation]
(cid:46)
[Equation]
(cid:46) update momentum push sign( v m ) to server (cid:46) send sign momentum on server
[Equation]
(cid:46) aggregate sign momenta push sign( V ) to each worker (cid:46) broadcast majority vote on every worker
[Equation]
(cid:46) update parameters until convergence
Non-convex optimisation: parallel to (and oftentimes in isolation from) advances in deep learning practice, a sophisticated optimisation literature has developed.
Nesterov & Polyak (2006) proposed cubic regularisation as an algorithm that can escape saddle points and provide guaranteed convergence to local minima of non-convex functions.
This has been followed up by more recent works such as N ATASHA (Allen-Zhu, 2017) that use other theoretical tricks to escape saddle points.
It is still unclear how relevant these works are to deep learning, since it is not clear to what extent saddle points are an obstacle in practical problems.
We avoid this issue altogether and satisfy ourselves with establishing convergence to critical points.
Gradient compression: prior work on gradient compression generally falls into two camps.
In the ﬁrst camp, algorithms like QSGD (Alistarh et al., 2017), T ERN G RAD (Wen et al., 2017) and A TOMO (Wang et al., 2018) use stochastic quantisation schemes to ensure that the compressed stochastic gradient remains an unbiased approximation to the true gradient.
These works are therefore able to bootstrap existing SGD convergence theory.
In the second camp, more heuristic algorithms like 1B IT SGD (Seide et al., 2014) and deep gradient compression (Lin et al., 2018) pay less attention to theoretical guarantees and focus more on practical performance.
These algorithms track quantisation errors and feed them back into subsequent updates.
The commonality between the two camps is an effort to, one way or another, correct for bias in the compression.
SIGN SGD with majority vote takes a different approach to these two existing camps.
In directly employing the sign of the stochastic gradient, the algorithm unabashedly uses a biased approximation of the stochastic gradient.
Carlson et al.
and Bernstein et al.
(2018) provide theoretical and empirical evidence that signed gradient schemes can converge well in spite of their biased nature.
Their theory only applies in the large batch setting, meaning the theoretical results are less relevant to deep learning practice.
Still Bernstein et al.
(2018) showed promising experimental results in the mini-batch setting.
An appealing feature of majority vote is that it naturally leads to compression in both directions of communication between workers and parameter server.
As far as we are aware, all existing gradient compression schemes lose compression before scattering results back to workers.
Byzantine fault tolerant optimisation: the problem of modifying SGD to make it Byzantine fault tolerant has recently attracted interest in the literature (Yin et al., 2018).
For example, Blanchard et al.
(2017) proposed K RUM , which operates by detecting and excluding outliers in the gradient aggregation.
Alistarh et al.
propose B YZANTINE SGD which instead focuses on detecting and eliminating adversaries.
Clearly both these strategies incur overheads, and eliminating adversaries precludes the possibility that they might reform.
El Mhamdi et al. (2018) point out that powerful adversaries may be capable of crafting subtle attacks like steering convergence to bad local minimisers. We do not aim to protect against such powerful attacks—we see majority vote as a natural way to protect against less malign faults such as network errors. Thus we satisfy ourselves with convergence guarantees to critical points without placing guarantees on their quality.
[Figure]
T HEORY 3. A SSUMPTIONS
We aim to develop an optimisation theory that is relevant for real problems in deep learning.
For this reason, we are careful about the assumptions we make.
For example, we do not assume convexity because neural network loss functions are typically not convex.
Though we allow our objective function to be non-convex, we insist on a lower bound to enable meaningful convergence results.
Assumption 1 (Lower bound) .
For all x and some constant f ∗ , we have objective value f ( x ) ≥ f ∗ .
Our next two assumptions of Lipschitz smoothness and bounded variance are standard in the stochastic optimisation literature (Allen-Zhu, 2017).
That said, we give them in a component-wise form.
This allows our convergence results to encode information not just about the total noise level and overall smoothness, but also about how these quantities are distributed across dimension.
Assumption 2 (Smooth) .
Let g ( x ) denote the gradient of the objective f ( . ) evaluated at point x .
Then ∀ x, y we require that for some non-negative constant (cid:126)L :=
[ L 1 , ..., L d ] (cid:12)(cid:12)(cid:12) f
( y ) − (cid:2) f
( x ) + g ( x ) T ( y − x ) (cid:3)(cid:12)(cid:12)(cid:12) ≤ 1 2 (cid:88)
[Equation]
Assumption 3 (Variance bound) .
Upon receiving query x ∈ R d , the stochastic gradient oracle gives us an independent, unbiased estimate ˜ g that has coordinate bounded variance:
[Equation]
for a vector of non-negative constants (cid:126)σ :=
[ σ 1 , .., σ d ] .
Our ﬁnal assumption is non-standard.
We assume that the gradient noise is unimodal and symmetric.
Clearly, Gaussian noise is a special case.
Note that even for a moderate mini-batch size, we expect the central limit theorem to kick in rendering typical gradient noise distributions close to Gaussian.
See Figure 2 for noise distributions measured whilst training resnet18 on Cifar-10.
Assumption 4 (Unimodal, symmetric gradient noise) .
At any given point x , each component of the stochastic gradient vector ˜ g ( x ) has a unimodal distribution that is also symmetric about the mean.
Showing how to work with this assumption is a key theoretical contribution of this work.
Combining Assumption 4 with an old tail bound of Gauss (1823) yields Lemma 1, which will be crucial for guaranteeing mini-batch convergence of SIGN SGD.
As will be explained in Section 3.3, this result also constitutes a convergence proof for a parameter regime of A DAM .
This suggests that Assumption 4 may more generally be a theoretical ﬁx for Reddi et al.
(2018)’s non-convergence proof of mini-batch A DAM , a ﬁx which does not involve modifying the A DAM algorithm itself.
[Figure]
At the start of each epoch we compute the SNR for every gradient component.
We plot summary statistics like the mean over weights and the max.
By roughly epoch 40, all gradient components have passed below the critical line (see Theorem 1) and remain there for the rest of training.
3. M INI -B ATCH C ONVERGENCE OF SIGN SGD
With our assumptions in place, we move on to presenting our theoretical results, which are all proved in Appendix B.
Our ﬁrst result establishes the mini-batch convergence behaviour of SIGN SGD.
We will ﬁrst state the result and make some remarks.
We provide intuition for the proof in Section 3.3.
Theorem 1 (Non-convex convergence rate of mini-batch SIGN SGD) .
Run the following algorithm for K iterations under Assumptions 1 to 4: x k +
= x
k − η sign(˜ g k ) .
Set the learning rate, η , and mini-batch size, n , as
[Equation]
Let H k be the set of gradient components at step k with large signal-to-noise ratio S i :=
| g k,i | σ i , i.e. H k :=
(cid:110) i (cid:12)(cid:12)(cid:12) S i > 2 √ 3 (cid:111) .
We refer to 2 √ 3 as the ‘critical SNR’.
Then we have
[Equation]
N
[Equation]
where N = K is the total number of stochastic gradient calls up to step K .
Theorem 1 provides a bound on the average gradient norm.
The right hand side of the bound decays like O (cid:16) 1 √ N (cid:17) , establishing convergence to points of the objective where the gradient vanishes.
Remark 1: mini-batch SIGN SGD attains the same O (cid:16) 1 √ N (cid:17) non-convex convergence rate as SGD.
Remark 2: the gradient appears as a mixed norm : an (cid:96) 1 norm for high SNR components, and a weighted (cid:96) 2 norm for low SNR compoenents.
Remark 3: we wish to understand the dimension dependence of our bound.
We may simplify matters by assuming that, during the entire course of optimisation, every gradient component lies in the low SNR regime.
Figure 3 shows that this is almost true when training a resnet18 model.
In this limit, the bound becomes:
[Equation]
N
[Equation] Further assume that we are in a well-conditioned setting, meaning that the variance is distributed uniformly across dimension ( σ 2 i =
σ 2 d ), and every weight has the same smoothness constant ( L i =
L ).
σ 2 is the total variance bound, and L is the conventional Lipschitz smoothness.
These are the quantities which appear in the standard analysis of SGD.
Then we get
[Equation]
The factors of dimension d have conveniently cancelled.
This illustrates that there are problem geometries where mini-batch SIGN SGD does not pick up an unfavourable dimension dependence.
3. T HE S UBTLETIES OF M INI -B ATCH C ONVERGENCE
Intuitively, the convergence analysis of SIGN SGD depends on the probability that a given bit of the sign stochastic gradient vector is incorrect, or P [sign(˜ g i ) (cid:54)
=
sign( g i )] .
Lemma 1 provides a bound on this quantity under Assumption 4 (unimodal symmetric gradient noise).
Lemma 1 (Bernstein et al.
Let ˜ g
i be an unbiased stochastic approximation to gradient component g i , with variance bounded by σ 2 i .
Further assume that the noise distribution is unimodal and symmetric.
Deﬁne signal-to-noise ratio S i :=
| g i | σ i .
Then we have that
[Equation]
which is in all cases less than or equal to 12 .
The bound characterises how the failure probability of a sign bit depends on the signal-to-noise ratio (SNR) of that gradient component.
Intuitively as the SNR decreases, the quality of the sign estimate should degrade.
The bound is important since it tells us that, under conditions of unimodal symmetric gradient noise, even at extremely low SNR we still have that P [sign(˜ g i ) (cid:54)
=
sign( g i )] ≤ 12 .
This means that even when the gradient is very small compared to the noise, the sign stochastic gradient still tells us, on average, useful information about the true gradient direction, allowing us to guarantee convergence as in Theorem 1.
Without Assumption 4, the mini-batch algorithm may not converge.
This is best appreciated with a simple example.
Consider a stochastic gradient component ˜ g with bimodal noise:
[Equation]
[-There exist noise distributions (violating Assumption 4) for which Cantelli’s equality is tight 
-]The true gradient g = E
[˜ g ] = 4 . 1 is positive.
But the sign gradient sign(˜ g ) is negative with probability 0.9.
Therefore SIGN SGD will tend to move in the wrong direction for this noise distribution.
Note that SIGN SGD is a special case of the A DAM algorithm (Balles & Hennig, 2018).
To see this, set β 1 = β 2 = (cid:15) =
in A DAM , and the A DAM update becomes:
[Equation]
This correspondence suggests that Assumption 4 should be useful for obtaining mini-batch convergence guarantees for A DAM .
Note that when Reddi et al.
(2018) construct toy divergence examples for A DAM , they rely on bimodal noise distributions which violate Assumption 4.
We conclude this section by noting that without Assumption 4, SIGN SGD can still be guaranteed to converge.
The trick is to use a “large” batch size that grows with the number of iterations.
This will ensure that the algorithm stays in the high SNR regime where the failure probability of the sign bit is low.
This is the approach taken by both Carlson et al.
and Bernstein et al.
3. R OBUSTNESS OF CONVERGENCE
We will now study SIGN SGD’s robustness when distributed by majority vote.
We model adversaries as machines that may manipulate their stochastic gradient as follows. Deﬁnition 1 (Blind multiplicative adversary) . A blind multiplicative adversary may manipulate their stochastic gradient estimate ˜ g t at iteration t by element-wise multiplying ˜ g t with any vector v t of their choice. The vector v t must be chosen before observing ˜ g t , so the adversary is ‘blind’.
Some interesting members of this class are:
(i) adversaries that arbitrarily rescale their stochastic gradient estimate; (ii) adversaries that randomise the sign of each coordinate of the stochastic gradient; (iii) adversaries that invert their stochastic gradient estimate. SGD is certainly not robust to rescaling since an adversary could set the gradient to inﬁnity and corrupt the entire model. Our algorithm, on the other hand, is robust to all adversaries in this class.
For ease of analysis, here we derive large batch results.
We make sure to give results in terms of sample complexity N
(and not iteration number K ) to enable fair comparison with other algorithms.
Theorem 2 (Non-convex convergence rate of majority vote with adversarial workers) .
Run algorithm 1 for K iterations under Assumptions 1 to 4.
Switch off momentum and weight decay ( β = λ = 0) .
Set the learning rate, η , and mini-batch size, n , for each worker as
[Equation]
Assume that a fraction α < 12 of the M workers behave adversarially according to Deﬁnition 1.
Then majority vote converges at rate:
[Equation]
where N = K 2 is the total number of stochastic gradient calls per worker up to step K .
The result is intuitive: provided there are more machines sending honest gradients than adversarial gradients, we expect that the majority vote should come out correct on average.
Remark 1: if we switch off adversaries by setting the proportion of adversaries α = 0 , this result reduces to Theorem 2 in (Bernstein et al., 2018).
In this case, we note the 1 √ M variance reduction that majority vote obtains by distributing over M machines, similar to distributed SGD.
Remark 2: the convergence rate degrades as we ramp up α from 0 to 12 .
Remark 3: from an optimisation theory perspective, the large batch size is an advantage.
This is because when using a large batch size, fewer iterations and rounds of communication are theoretically needed to reach a desired accuracy, since only √ N iterations are needed to reach N samples.
But from a practical perspective, workers may be unable to handle such a large batch size in a timely manner.
It should be possible to extend the result to the mini-batch setting by combining the techniques of Theorems 1 and 2, but we leave this for future work.
E XPERIMENTS
For our experiments, we distributed S IGNUM (Algorithm 1) by majority vote.
S IGNUM is the momentum counterpart of SIGN SGD, where each worker maintains a momentum and transmits the sign momentum to the parameter server at each step.
The addition of momentum to SIGN SGD is proposed and studied in (Balles & Hennig, 2018; Bernstein et al., 2018).
We built S IGNUM with majority vote in the Pytorch deep learning framework (Paszke et al., 2017) using the Gloo (2018) communication library.
Unfortunately Pytorch and Gloo do not natively support 1-bit tensors, therefore we wrote our own compression code to bit-pack a sign tensor down to an efﬁcient 1-bit representation.
We obtained a performance boost by fusing together smaller tensors, which saved on compression and communication costs. [Figure] [Figure] [Figure]
[Figure]
Bits sent per iteration
Majority vote O ( Md ) L2 QSGD O ( M 2 √ d log d ) max QSGD O ( Md )
For d weights, M machines.
[Figure]
[Figure]
[Figure]
We test against SGD distributed using the state of the art NCCL (2018) communication library.
NCCL provides an efﬁcient implementation of allreduce.
Our framework is often 4 × faster in communication (including the cost of compression) than NCCL, as can be seen in Figure 4. Further code optimisation should bring the speedup closer to the ideal 32 × .
4. C OMMUNICATION E FFICIENCY
We ﬁrst benchmark majority vote on the Imagenet dataset.
We train a resnet50 model and disitribute learning over 7 to 15 AWS p3.2xlarge machines.
These machines each contain one
Nvidia Tesla V100 GPU, and AWS lists the connection speed between machines as “up to 10 Gbps”.
Results are plotted in Figure 5.
Per epoch , distributing by majority vote is able to attain a similar speedup to distributed SGD.
But per hour majority vote is able to process more epochs than NCCL, meaning it can complete the 80 epoch training job roughly 25% faster.
In terms of overall generalisation, majority vote reaches a slightly degraded test set accuracy.
We hypothesise that this may be ﬁxed by inventing a better regularisation scheme or tuning momentum, which we did not do.
In Figure 6 we compare majority vote to A DAM (distributed by NCCL) for training QRNN (Bradbury et al., 2017). Majority vote completes an epoch roughly 3 times faster than A DAM , but it reaches a degraded accuracy so that the overall test perplexity after 2 hours ends up being similar.
In Figure 7 we show that majority vote has superior convergence to the ‘theory’ version of QSGD that Alistarh et al. (2017) develop. Convergence is similar for the ‘max’ version that Alistarh et al. (2017) use in their experiments.
4. R OBUSTNESS
In this section we test the robustness of S IGNUM with majority vote to Byzantine faults.
Again we run tests on the Imagenet dataset, training resnet50 across 7 AWS p3.2xlarge machines.
Our adversarial workers take the sign of their stochastic gradient calculation, but send the negation to the parameter server.
Our results are plotted in Figure 8.
In the left hand plot, all experiments were carried out using hyperparameters tuned for the 0% adversarial case.
Weight decay was not used in these experiments to simplify matters.
We see that learning is tolerant of up to 43% (3 out of 7) machines behaving adversarially.
The 43% adversarial case was slightly unstable (Figure 8, left), but re-tuning the learning rate for this speciﬁc case stabilised learning (Figure 8, right).
In Figure 9 we compare majority vote to M ULTI -K RUM (Blanchard et al., 2017) with a security level of f = 2 .
When the number of adversaries exceeds f , M ULTI -K RUM fails catastrophically in our experiments, whereas SIGN SGD fails more gracefully.
Note that M ULTI -K RUM requires 2 f + 2 < M , therefore f = 2 is the maximum possible security level for these experiments with
M = 7 workers.
D ISCUSSION AND C ONCLUSION
We have analysed the theoretical and empirical properties of a very simple algorithm for distributed, stochastic optimisation. We have shown that SIGN SGD with majority vote aggregation is robust and communication efﬁcient, whilst its per-iteration convergence rate is competitive with SGD for training large-scale convolutional neural nets on image datasets. We believe that it is important to understand this simple algorithm before going on to devise more complex learning algorithms.
An important takeaway from our theory is that mini-batch SIGN SGD should converge if the gradient noise is Gaussian.
This means that the performance of SIGN SGD may be improved by increasing the per-worker mini-batch size, since this should make the noise ‘more Gaussian’ according to the Central Limit Theorem.
We will now give some possible directions for future work. Our implementation of majority vote may be further optimised by breaking up the parameter server and distributing it across machines. This would prevent a single machine from becoming a communication bottleneck as in our experiments.
Though our framework speeds up Imagenet training, we still have a test set gap.
Future work could attempt to devise new regularisation schemes for signed updates to close this gap. Promising future work could also explore the link between SIGN SGD and model compression. Signed updates force the weights to live on a lattice, facilitating compression of the resulting model. A B ITS S ENT P ER I TERATION : SIGN SGD VS . QSGD
In this section, we perform theoretical calculations of the number of bits sent per iteration in distributed training. We compare SIGN SGD using majority vote aggregation to two forms of QSGD (Alistarh et al., 2017). These calculations give the numbers in the table in Figure 7. The communication cost of SIGN SGD with majority vote is trivially 2 Md bits per iterations, since at each iteration M machines send d -dimensional sign vectors up to the server, and the server sends back one d -dimensional sign vector to all M machines.
There are two variants of QSGD given in (Alistarh et al., 2017). The ﬁrst we refer to as L2 QSGD which is the version developed in the theory section of (Alistarh et al., 2017). The second we refer to as max QSGD which is the version actually used in their experiments.
For each version we compute the number of bits sent for the highest compression version of the algorithm, which is a ternary quantisation (snapping gradient components into { 0 , ± 1 } ). We refer to this as 1-bit QSGD. The higher precision versions of QSGD will send more bits per iteration. 1-bit L2 QSGD takes a gradient vector g and snaps i th coordinate g i to sign( g i ) with probability |
g i | (cid:107) g (cid:107) 2 and sets it to zero otherwise.
Therefore the expected number of bits set to ± 1 is bounded by [Equation] To send a vector compressed in this way, for each non-zero component 1 bit is needed to send the sign and log d bits are needed to send the index. Therefore sending a vector compressed by 1-bit L
QSGD requires at most √ d (1 + log d ) bits. In the experiments in Figure 7 we see that the ‘2-way’ version of 1-bit L2 QSGD (which recompresses the aggregated compressed gradients) converges very poorly. Therefore it makes sense to use the 1-way version where the aggregated compressed gradient is not recompressed. A sensible way to enact this is to have each of the M workers broadcast their compressed gradient vector to all other workers.
This has a cost of ( M − 1)
√ d
1 + log d ) bits for each of the M workers, and from this we get the total cost of O ( M 2 √ d log d ) for 1-bit L2 QSGD. The ﬁnal algorithm to characterise is 1-bit max QSGD. 1-bit max QSGD takes a gradient vector g and snaps i th coordinate g i to sign( g i ) with probability |
g i | (cid:107) g (cid:107) ∞ and sets it to zero otherwise.
As noted in (Alistarh et al., 2017), there are no sparsity guarantees for this algorithm, so compression will generally be much lower than for 1-bit L2 QSGD. It is easy to see that 1-bit max QSGD requires no more than O ( d ) bits to compress a d -dimensional vector, since 2 d bits can always store d numbers in { 0 , ± 1 } . To see that we can’t generally do better than O ( d ) bits, notice that 1-bit max QSGD leaves sign vectors invariant, and thus the compressed form of a sign vector requires exactly d bits. The natural way to enact 1-bit max QSGD is with a two-way compression where the M workers each send an O ( d ) -bit compressed gradient up to the server, and the server sends back an O ( d ) -bit compressed aggregated result back to the M workers.
This gives a number of bits sent per iteration of O ( Md ) .
For very sparse vectors 1-bit max QSGD will compress much better than indicated above.
For a vector g with a single non-zero entry, 1-bit max QSGD will set this entry to 1 and keep the rest zero, thus requiring only log d bits to send the index of the non-zero entry. But it is not clear whether these extremely sparse vectors appear in deep learning problems. In the experiments in Figure 7, 1-bit max QSGD led to compressed vectors that were 5 × more compressed than SIGN SGD—in our experimental setting this additional improvement turned out to be small relative to the time cost of backpropagation. B P ROOFS B. A CCURACY OF THE S IGN S TOCHASTIC G RADIENT
Lemma 1 (Bernstein et al.
Let ˜ g
i be an unbiased stochastic approximation to gradient component g i , with variance bounded by σ 2 i .
Further assume that the noise distribution is unimodal and symmetric.
Deﬁne signal-to-noise ratio S i :=
| g i | σ i .
Then we have that
[Equation]
which is in all cases less than or equal to 12 .
Proof.
Recall Gauss’ inequality for unimodal random variable X with mode ν and expected squared deviation from the mode τ 2 (Gauss, 1823; Pukelsheim, 1994):
[Equation]
By the symmetry assumption, the mode is equal to the mean, so we replace mean µ = ν and variance σ 2 = τ 2 .
[Equation]
Without loss of generality assume that g i is negative.
Then applying symmetry followed by Gauss, the failure probability for the sign bit satisﬁes:
[Equation]
otherwise
M
INI -
G Theorem 1 (Non-convex convergence rate of mini-batch SIGN SGD) .
Run the following algorithm for K iterations under Assumptions 1 to 4: x k +
= x
k − η sign(˜ g k ) .
Set the learning rate, η , and mini-batch size, n , as
[Equation]
Let H k be the set of gradient components at step k with large signal-to-noise ratio S i :=
| g k,i | σ i , i.e. H k :=
(cid:110) i (cid:12)(cid:12)(cid:12) S i > 2 √ 3 (cid:111) .
We refer to 2 √ 3 as the ‘critical SNR’.
Then we have
[Equation]
where N = K is the total number of stochastic gradient calls up to step K .
Proof.
First let’s bound the improvement of the objective during a single step of the algorithm for one instantiation of the noise.
I [ . ] is the indicator function, g k,i denotes the i th component of the true gradient g ( x k ) and ˜ g k is a stochastic sample obeying Assumption 3.
First take Assumption 2, plug in the algorithmic step, and decompose the improvement to expose the stochasticity-induced error:
[Equation]
Next we ﬁnd the expected improvement at time k + 1 conditioned on the previous iterate.
[Equation]
By Assumption 4 and Lemma 1 we have the following bound on the failure probability of the sign:
[Equation]
otherwise
[Equation]
otherwise Substituting this in, we get that
[Equation]
Interestingly a mixture between an (cid:96) 1 and a variance weighted (cid:96) 2 norm has appeared.
Now substitute in the learning rate schedule
[Equation]
and we get:
[Equation]
Now extend the expectation over the randomness in the trajectory and telescope over the iterations:
[Equation]
Finally, rearrange and substitute in N = K to yield the bound
[Equation]
B. R OBUSTNESS OF M AJORITY V OTE
Theorem 2 (Non-convex convergence rate of majority vote with adversarial workers) .
Run algorithm 1 for K iterations under Assumptions 1 to 4.
Switch off momentum and weight decay ( β = λ = 0) .
Set the learning rate, η [Equation] and mini-batch size, n , for each worker as
[Equation]
Assume that a fraction α < 12 of the M workers behave adversarially according to Deﬁnition 1.
Then majority vote converges at rate:
[Equation]
where N = K 2 is the total number of stochastic gradient calls per worker up to step K .
Proof.
We need to bound the failure probability of the vote.
We can then use this bound to derive a convergence rate.
We will begin by showing this bound is worst when the adversary inverts the signs of the sign stochastic gradient.
Given an adversary from the class of blind multiplicative adversaries (Deﬁnition 1), the adversary may manipulate their stochastic gradient estimate ˜ g t into the form v t ⊗ ˜ g t .
Here v t is a vector of the adversary’s choosing, and ⊗ denotes element-wise multiplication.
The sign of this quantity obeys: sign( v t [Equation] ˜ g t [Equation]
= sign( v t )
[Equation]
sign(˜ g t ) .
Therefore, the only thing that matters is the sign of v t , and rescaling attacks are immediately nulliﬁed.
For each component of the stochastic gradient, the adversary must decide (without observing g t , since the adversary is blind) whether or not they would like to invert the sign of that component.
We will now show that the failure probability of the vote is always larger when the adversary decides to invert (by setting every component of sign( v t ) to − 1 ).
Our analysis will then proceed under this worst case.
For a gradient component with true value g , let random variable Z ∈ [0 , M ] denote the number of correct sign bits received by the parameter server.
For a given adversary, we may decompose Z into the contribution from that adversary and a residual term X from the remaining workers (both regular and adversarial):
[Equation]
where ˜ g is the adversary’s stochastic gradient estimate for that component, v is the adversary’s chosen scalar for that component, and I is the 0-1 indicator function.
We are considering Z to be a function of sign( v ) .
But by Assumption 4 and Lemma 1, we see that I [+1 × sign(˜ g ) =
sign(g)] is a Bernoulli random variable with success probability p ≥ 12 .
On the other hand, I [ − 1 × sign(˜ g ) = sign(g)] is a Bernoulli random variable with success probability q = 1 − p ≤ 1 2 .
The essential quantity in our analysis is the probability that more than half the workers provide the correct sign bit.
But from the preceding discussion, this clearly obeys:
[Equation]
As we will see below, the implication of this is that our bounds always worse under the setting v = − 1 , and so we will adopt v = − 1 hereon.
It is worth remarking that blindness in the deﬁnition of blind multiplicative adversaries is important to ensure that I [sign( v )sign(˜ g ) = sign(g)] is indeed a random variable as described above.
Were the adversary not blind, then the adversary could in effect deterministically set sign( v )sign(˜ g ) .
Cooperative adversaries, for example, could use this power to control the vote.
Now we restrict to our worst case blind multiplicative adversaries, that always choose to invert their sign stochastic gradient estimate.
So, we have (1 − α )
M good machines and αM adversaries.
The good workers each compute a stochastic gradient estimate, take its sign and transmit this to the server.
The bad workers follow an identical procedure except they negate their sign bits prior to transmission to the server.
It is intuitive that because the proportion of adversaries α < 12 , the good workers will win the vote on average.
To make this rigorous, we will need Lemma 1 and Cantelli’s inequality.
Cantelli (1928) tells us that for a random variable X with mean µ and variance σ 2 :
[Equation]
For a given gradient component, again let random variable Z ∈ [0 , M ] denote the number of correct sign bits received by the parameter server.
Let random variables G and B denote the number of good and bad workers (respectively) who (possibly inadvertently) sent the correct sign bit.
Then, letting p be the probability that a good worker computed the correct sign bit, q := 1 − p and (cid:15) :
= p − 12 we can decompose Z as follows:
[Equation]
The vote only fails if Z < M 2 which happens with probability
[Equation]
by Cantelli’s inequality
[Equation]
since 1 + x 2 ≥ 2 x
[Equation]
We now need to substitute in a bound on (cid:15) .
Assumption 4 and Lemma 1 tell us that
[Equation]
From this we can derive that 14 (cid:15) 2 − 1 < 4 S 2 as follows.
First take the case S ≤ 2 √ 3 .
Then (cid:15) 2 ≥ S 2and 14 (cid:15) 2 − 1 ≤ 3 S 2 − 1 < 4 S 2 .
Now take the case S > 2 √ 3 .
Then (cid:15) ≥ 12 − 29 1 S 2 and we have 14 (cid:15) 2 − 1 ≤
[Equation] [Equation]
by the condition on S .
We have now completed the ﬁrst part of the proof by showing the key statement that for the i th gradient component with signal to noise ratio S i :=
| g i | σ i , the failure probability of the majority vote is bounded by
[Equation]
The second stage of the proof will proceed by straightforwardly substituting this bound into the convergence analysis of SIGN SGD from Bernstein et al.
First let’s bound the improvement of the objective during a single step of the algorithm for one instantiation of the noise.
I [ . ] is the indicator function, g k,i denotes the i th component of the true gradient g ( x k ) and sign( V k ) is the outcome of the vote at the k th iteration.
First take Assumption 2, plug in the step from Algorithm 1, and decompose the improvement to expose the error induced by stochasticity and adversarial workers:
[Equation]
Next we ﬁnd the expected improvement at time k + 1 conditioned on the previous iterate.
[Equation]
From ( (cid:63) ) , we have that the probability of the vote failing for the i th coordinate is bounded by
[Equation]
where σ k,i refers to the variance of the k th stochastic gradient estimate, computed over a mini-batch of size n .
Therefore, by Assumption 3, we have that σ k,i ≤ σ i / √ n .
We now substitute these results and our learning rate and mini-batch settings into the expected improvement:
[Equation]
Now extend the expectation over randomness in the trajectory, and perform a telescoping sum over the iterations:
[Equation]
We can rearrange this inequality to yield the rate:
[Equation]
Since we are growing our mini-batch size, it will take N =
O ( K 2 ) gradient calls to reach step K .
Substitute this in on the right hand side, square the result, use that 32 < 2 , and we are done:
[Equation]