P ROX Q UANT : Q UANTIZED N EURAL N ETWORKS VIA P ROXIMAL O PERATORS
Anonymous authors
Paper under double-blind review
A BSTRACT
To make deep neural networks feasible in resource-constrained environments (such as mobile devices), it is beneﬁcial to quantize models by using low-precision weights.
One common technique for quantizing neural networks is the straightthrough gradient method, which enables back-propagation through the quantization mapping.
Despite its empirical success, little is understood about why the straight-through gradient method works.
Building upon a novel observation that the straight-through gradient method is in fact identical to Nesterov’s dual-averaging algorithm on a quantization constrained optimization problem, we propose a more principled alternative approach, called P ROX Q UANT , that formulates quantized network training as a regularized learning problem instead and optimizes it via the prox-gradient method.
P ROXQ UANT does back-propagation on the underlying full-precision vector and applies an efﬁcient prox-operator in between stochastic gradient steps to encourage quantizedness.
For quantizing ResNets and LSTMs, P ROX Q UANT outperforms state-of-the-art results on binary quantization and is on par with state-of-the-art on multi-bit quantization.
We further perform theoretical analyses showing that
P ROX Q UANT converges to stationary points under mild smoothness assumptions, whereas variants such as lazy prox-gradient method can fail to converge in the same setting.
I NTRODUCTION
Deep neural networks (DNNs) have achieved impressive results in various machine learning tasks (Goodfellow et al., 2016).
High-performance DNNs typically have over tens of layers and millions of parameters, resulting in a high memory usage and a high computational cost at inference time.
However, these networks are often desired in environments with limited memory and computational power (such as mobile devices), in which case we would like to compress the network into a smaller, faster network with comparable performance.
A popular way of achieving such compression is through quantization – training networks with lowprecision weights and/or activation functions.
In a quantized neural network, each weight and/or activation can be representable in k bits, with a possible codebook of negligible additional size compared to the network itself.
For example, in a binary neural network ( k = 1 ), the weights are restricted to be in {± 1 } .
Compared with a 32-bit single precision ﬂoat, a quantized net reduces the memory usage to k/ 32 of a full-precision net with the same architecture (Han et al., 2015;
Courbariaux et al., 2015; Rastegari et al., 2016;
Hubara et al., 2017; Zhou et al., 2016; Zhu et al., 2016).
In addition, the structuredness of the quantized weight matrix can often enable faster matrixvector product, thereby also accelerating inference (Hubara et al., 2017; Han et al., 2016).
Typically, training a quantized network involves (1) the design of a quantizer q that maps a full-precision parameter to a k -bit quantized parameter, and (2) the straight-through gradient method (Courbariaux et al., 2015) that enables back-propagation from the quantized parameter back onto the original full-precision parameter, which is critical to the success of quantized network training.
With quantizer q , an iterate of the straight-through gradient method (see Figure 1a) proceeds as θ t +
=
θ t − η t (cid:101) ∇ L ( θ ) | θ = q ( θ t ) , and q ( (cid:98) θ ) (for the converged (cid:98) θ ) is taken as the output model.
For training binary networks, choosing q ( · ) = sign( · ) gives the BinaryConnect method (Courbariaux et al., 2015).
Though appealingly simple and empirically effective, it is information-theoretically rather mysterious why the straight-through gradient method works well, at least in the binary case: while the goal is to ﬁnd a parameter θ ∈ {± 1 } d with low loss, the algorithm only has access to stochastic gradients at {± 1 } d .
As this is a discrete set, a priori , gradients in this set do not necessarily contain any information about the function values.
Indeed, a simple one-dimensional example (Figure 1b) shows that BinaryConnect fails to ﬁnd the minimizer of fairly simple convex Lipschitz functions in {± 1 } , due to a lack of gradient information in between.
[Equation]
[Table]
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In this paper we formulate the problem of model quantization as a regularized learning problem and propose to solve it with a proximal gradient method.
Our contributions are summarized as follows.
• We present a uniﬁed framework for deﬁning regularization functionals that encourage binary, ternary, and multi-bit quantized parameters, through penalizing the distance to quantized sets (see Section 3.1).
For binary quantization, the resulting regularizer is a W -shaped non-smooth regularizer, which shrinks parameters towards either − 1 or 1 in the same way that the L 1 norm regularization shrinks parameters towards 0 .
• We propose training quantized networks using P ROX Q UANT (Algorithm 1) — a stochastic proximal gradient method with a homotopy scheme.
Compared with the straightthrough gradient method, P ROX Q UANT has access to additional gradient information at non-quantized points, which avoids the problem in Figure 1b and its homotopy scheme prevents potential overshoot early in the training (Section 3.2).
• We demonstrate the effectiveness and ﬂexibility of P ROX Q UANT through systematic experiments on (1) image classiﬁcation with ResNets (Section 4.1); (2) language modeling with LSTMs (Section 4.2).
The P ROX Q UANT method outperforms the state-of-the-art results on binary quantization and is comparable with the state-of-the-art on ternary and multi-bit quantization.
• We perform a systematic theoretical study of quantization algorithms, showing that our P ROX Q UANT (standard prox-gradient method) converges to stataionary points under mild smoothness assumptions (Section 5.1), where as lazy prox-gradient method such as BinaryRelax (Yin et al., 2018) fails to converge in general (Section 5.2).
Further, we show that BinaryConnect has a very stringent condition to converge to any ﬁxed point (Section 5.3), which we verify through a sign change experiment (Appendix C).
1. P RIOR WORK
Methodologies Han et al. (2015) propose Deep Compression, which compresses a DNN via sparsiﬁcation, nearest-neighbor clustering, and Huffman coding.
This architecture is then made into a specially designed hardware for efﬁcient inference (Han et al., 2016).
In a parallel line of work,
Courbariaux et al.
(2015) propose BinaryConnect that enables the training of binary neural networks, and Li & Liu (2016);
Zhu et al.
(2016) extend this method into ternary quantization.
Training and inference on quantized nets can be made more efﬁcient by also quantizing the activation (Hubara et al., 2017; Rastegari et al., 2016;
Zhou et al., 2016), and such networks have achieved impressive performance on large-scale tasks such as ImageNet classiﬁcation (Rastegari et al., 2016; Zhu et al., 2016) and object detection (Yin et al., 2016).
In the NLP land, quantized language models have been successfully trained using alternating multi-bit quantization (Xu et al., 2018).
Theories Li et al.
(2017) prove the convergence rate of stochastic rounding and BinaryConnect on convex problems and demonstrate the advantage of BinaryConnect over stochastic rounding on non-convex problems.
Anderson & Berg (2017) demonstrate the effectiveness of binary networks through the observation that the angles between high-dimensional vectors are approximately preserved when binarized, and thus high-quality feature extraction with binary weights is possible.
Ding et al.
(2018) show a universal approximation theorem for quantized ReLU networks.
Principled methods Sun & Sun (2018) perform model quantization through a Wasserstein regularization term and minimize via the adversarial representation, similar as in Wasserstein GANs (Arjovsky et al., 2017).
Their method has the potential of generalizing to other generic requirements on the parameter, but might be hard to tune due to the instability of the inner maximization problem.
Prior to our work, a couple of proximal or regularization based quantization algorithms were proposed as alternatives to the straight-through gradient method, which we now brieﬂy review and compare with.
(Yin et al., 2018) propose BinaryRelax, which corresponds to a lazy proximal gradient descent.
(Hou et al., 2017; Hou & Kwok, 2018) propose a proximal Newton method with a diagonal approximate Hessian.
Carreira-Perpin ´ an (2017); Carreira-Perpin ´ an & Idelbayev (2017) formulate quantized network training as a constrained optimization problem and propose to solve them via augmented Lagrangian methods.
Our algorithm is different with all the aformentioned work in using the non-lazy and “soft” proximal gradient descent with a choice of either (cid:96) 1 or (cid:96)regularization, whose advantage over lazy prox-gradient methods is demonstrated both theoretically (Section 5) and experimentally (Section 4.1 and Appendix C).
P RELIMINARIES
The optimization difﬁculty of training quantized models is that they involve a discrete parameter space and hence efﬁcient local-search methods are often prohibitive.
For example, the problem of training a binary neural network is to minimize L ( θ ) for θ ∈ {± 1 } d .
Projected SGD on this set will not move unless with an unreasonably large stepsize (Li et al., 2017), whereas greedy nearestneighbor search requires d forward passes which is intractable for neural networks where d is on the order of millions.
Alternatively, quantized training can also be cast as minimizing L ( q ( θ )) for θ ∈ R d and an appropriate quantizer q that maps a real vector to a nearby quantized vector, but θ (cid:55)→ q ( θ ) is often non-differentiable and piecewise constant (such as the binary case q ( · ) = sign( · ) ), and thus back-propagation through q does not work.
2. T HE STRAIGHT - THROUGH GRADIENT METHOD
The pioneering work of BinaryConnect (Courbariaux et al., 2015) proposes to solve this problem via the straight-through gradient method , that is, propagate the gradient with respect to q ( θ ) unaltered to θ , i.e. to let ∂L ∂θ :=
∂L ∂ q ( θ ) .
One iterate of the straight-through gradient method (with the SGD optimizer) is
[Equation]
This enables the real vector θ to move in the entire Euclidean space, and taking q ( θ ) at the end of training gives a valid quantized model.
Such a customized back-propagation rule yields good empirical performance in training quantized nets and has thus become a standard practice (Courbariaux et al., 2015; Zhu et al., 2016;
Xu et al., 2018).
However, as we have discussed, it is information theoretically unclear how the straight-through method works, and it does fail on very simple convex
Lipschitz functions (Figure 1b).
2. S TRAIGHT - THROUGH GRADIENT AS LAZY PROJECTION
Our ﬁrst observation is that the straight-through gradient method is equivalent to a dual-averaging method, or a lazy projected SGD (Xiao, 2010).
In the binary case, we wish to minimize L ( θ ) over
Q = {± 1 } d , and the lazy projected SGD proceeds as
[Equation]
Written compactly, this is θ t +
=
θ t − η
[Equation]
(cid:101)
[Equation]
L
[Equation]
θ
[Equation]
| θ = q
[Equation]
θ t ) , which is exactly the straight-through gradient method: take the gradient at the quantized vector and perform the update on the original real vector.
2. P ROJECTION AS A LIMITING PROXIMAL OPERATOR
We take a broader point of view that a projection is also a limiting proximal operator with a suitable regularizer, to allow more generality and to motivate our proposed algorithm.
Given any set Q , one could identify a regularizer R : R d → R ≥ 0 such that the following hold:
[Equation]
In the case Q = {± 1 } d for example, one could take
[Equation]
The proximal operator (or prox operator) (Parikh & Boyd, 2014) with respect to R and strength λ > 0 is
[Equation]
(cid:101) θ ∈ R d (cid:26)  (cid:13)(cid:13) (cid:13) (cid:101) θ − θ (cid:13)(cid:13) (cid:13) 2 + λR ( (cid:101) θ ) (cid:27) .
In the limiting case λ = ∞ , the argmin has to satisfy R ( θ ) = 0 , i.e. θ ∈ Q , and the prox operator is to minimize (cid:107) θ − θ 0 (cid:107) 22 over θ ∈ Q , which is the Euclidean projection onto Q .
Hence, projection is also a prox operator with λ = ∞ , and the straight-through gradient estimate is equivalent to a lazy proximal gradient descent with and λ = ∞ .
While the prox operator with λ = ∞ correponds to “hard” projection onto the discrete set Q , when λ < ∞ it becomes a “soft” projection that moves towards Q .
Compared with the hard projection, a ﬁnite λ is less aggressive and has the potential advantage of avoiding overshoot early in training.
Further, as the prox operator does not strictly enforce quantizedness, it is in principle able to query the gradients at every point in the space, and therefore has access to more information than the straight-through gradient method.
Q UANTIZED NET TRAINING VIA REGULARIZED LEARNING
We propose the P ROX Q UANT algorithm, which adds a quantization-inducing regularizer onto the loss and optimizes via the (non-lazy) prox-gradient method with a ﬁnite λ .
The prototypical version of P ROX Q UANT is described in Algorithm 1.
Algorithm for net training
Require:
Regularizer R that induces desired quantizedness, initialization θ 0 , learning rates { η t } t ≥ 0 , regularization strengths { λ t } t ≥while not converged do
Perform the prox-gradient step
[Equation]
The inner SGD step in eq.
(4) can be replaced by any preferred stochastic optimization method such as Momentum SGD or Adam (Kingma & Ba, 2014).
end while
Compared to usual full-precision training, P ROX Q UANT only adds a prox step after each stochastic gradient step, hence can be implemented straightforwardly upon existing full-precision training.
As the prox step does not need to know how the gradient step is performed, our method adapts to other stochastic optimizers as well such as Adam.
In the remainder of this section, we deﬁne a ﬂexible class of quantization-inducing regularizers through “distance to the quantized set”, derive efﬁcient algorithms of their corresponding prox operator, and propose a homotopy method for choosing the regularization strengths.
Our regularization perspective subsumes most existing algorithms for model-quantization (e.g.,(Courbariaux et al., 2015; Han et al., 2015; Xu et al., 2018)) as limits of certain regularizers with strength λ → ∞ .
Our proposed method can be viewed as a principled generalization of these methods to λ < ∞ with a non-lazy prox operator.
3. R EGULARIZATION FOR MODEL QUANTIZATION
Let Q ⊂ R d be a set of quantized parameter vectors.
An ideal regularizer for quantization would be to vanish on Q and reﬂect some type of distance to Q when θ / ∈ Q .
To achieve this, we propose Land L 2 regularizers of the form
[Equation]
This is a highly ﬂexible framework for designing regularizers, as one could specify any Q and choose between L 1 and L 2 .
Speciﬁcally, Q encodes certain desired quantization structure.
By appropriately choosing Q , we can specify which part of the parameter vector to quantize 1 , the number of bits to quantize to, whether we allow adaptively-chosen quantization levels and so on.
The choice between { L 1 , L 2 } will encourage { “hard”,“soft” } quantization respectively, similar as in standard regularized learning (Tibshirani, 1996).
In the following, we present a few examples of regularizers under our framework eq.
(5) which induce binary weights, ternary weights and multi-bit quantization.
We will also derive efﬁcient algorithms (or approximation heuristics) for solving the prox operators corresponding to these regularizers, which generalize the projection operators used in the straight-through gradient algorithms.
Binary neural nets
In a binary neural net, the entries of θ are in {± 1 } .
A natural choice would be taking Q = {− 1 , 1 } d .
The resulting L 1 regularizer is
[Equation]
This is exactly the binary regularizer R bin that we discussed earlier in eq.
Figure 2 plots the
W-shaped one-dimensional component of R bin from which we see its effect for inducing {± 1 } quantization in analog to L 1 regularization for inducing exact sparsity.
[Equation]
[Figure]
[Equation]
[Figure]
The prox operator with respect to R bin , despite being a non-convex optimization problem, admits a simple analytical solution:
[Equation]
We note that the choice of the L 1 version is not unique: the squared
L 2 version works as well, whose prox operator is given by ( θ + λ sign( θ )) /
(1 + λ ) .
See Appendix A.1 for the derivation of these prox operators and the deﬁnition of the soft thresholding operator.
Multi-bit quantization with adaptive levels.
Following (Xu et al., 2018), we consider k -bit quantized parameters with a structured adaptively-chosen set of quantization levels, which translates into
[Equation]
The squared L 2 regularizer for this structure is
[Equation]
which is also the alternating minimization objective in (Xu et al., 2018).
We now derive the prox operator for the regularizer eq.
For any θ , we have
[Equation]
(cid:13)(cid:13)(cid:13)(cid:101)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
[Equation]
(cid:13)(cid:13)(cid:13)(cid:101)
θ
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:27)
[Equation]
(cid:13)(cid:13)(cid:13)(cid:101)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)(cid:101)
[Equation]
(cid:13)(cid:13)(cid:13)
[Equation]
(cid:27)
[Equation]
This is a joint minimization problem in ( (cid:101) θ, B, α ) , and we adopt an alternating minimization schedule to solve it:
(1) Minimize over (cid:101) θ given ( B, α ) , which has a closed-form solution
[Equation]
(2) Minimize over ( B, α ) given (cid:101) θ , which does not depend on θ 0 , and can be done via calling the alternating quantizer of (Xu et al., 2018):
Bα = q alt ( (cid:101) θ ) .
Together, the prox operator generalizes the alternating minimization procedure in (Xu et al., 2018), as λ governs a trade-off between quantization and closeness to θ .
To see that this is a strict generalization, note that for any λ the solution of eq.
(10) will be an interpolation between the input θ and its Euclidean projection to Q .
As λ → + ∞ , the prox operator collapses to the projection.
Ternary quantization Ternary quantization is a variant of 2-bit quantization, in which weights are constrained to be in {− α, 0 , β } for real values α, β > 0 .
We defer the derivation of the ternary prox operator into Appendix A.2.
3. H OMOTOPY METHOD FOR REGULARIZATION STRENGTH
Recall that the larger λ t is, the more aggressive θ t +1 will move towards the quantized set.
An ideal choice would be to (1) force the net to be exactly quantized upon convergence, and (2) not be too aggressive such that the quantized net at convergence is sub-optimal.
We let λ t be a linearly increasing sequence, i.e. λ t := λ · t for some hyper-parameter λ >which we term as the regularization rate .
With this choice, the stochastic gradient steps will start off close to full-precision training and gradually move towards exact quantizedness, hence the name “homotopy method”.
The parameter λ can be tuned by minimizing the validation loss, and controls the aggressiveness of falling onto the quantization constraint.
There is nothing special about the linear increasing scheme, but it is simple enough and works well as we shall see in the experiments.
E XPERIMENTS
We evaluate the performance of P ROX Q UANT on two tasks: image classiﬁcation with ResNets, and language modeling with LSTMs.
On both tasks, we show that the default straight-through gradient method is not the only choice, and our P ROX Q UANT can achieve the same and often better results.
4. I MAGE CLASSIFICATION ON CIFAR-
Problem setup We perform image classiﬁcation on the CIFAR-10 dataset, which containstraining images and 10000 test images of size 32x32.
We apply a commonly used data augmentation strategy (pad by 4 pixels on each side, randomly crop to 32x32, do a horizontal ﬂip with probability 0.5, and normalize).
Our models are ResNets (He et al., 2016) of depth 20, 32, 44, and 56 with weights quantized to binary or ternary.
Method We use P ROX Q UANT with regularizer eq.
(3) in the binary case and eqs.
(15) and (16) in the ternary case, which we respectively denote as PQ-B and PQ-T. We use the homotopy method λ t = λ · t with λ = 10 − 4 as the regularization strength and Adam with constant learning rate 0.
as the optimizer.
We compare with BinaryConnect (BC) for binary nets and Trained Ternary Quantization (TTQ)
(Zhu et al., 2016) for ternary nets.
For BinaryConnect, we train with the recommended Adam optimizer with learning rate decay (Courbariaux et al., 2015) (initial learning rate 0.01, multiply by 0.1 at epoch 81 and 122), which we ﬁnd leads to the best result for BinaryConnect.
For TTQ we compare with the reported results in (Zhu et al., 2016).
For binary quantization, both BC and our P ROX Q UANT are initialized at the same pre-trained fullprecision nets (warm-start) and trained for 300 epochs for fair comparison.
For both methods, we perform a hard quantization θ (cid:55)→ q ( θ ) at epoch 200 and keeps training till the 300-th epoch to stabilize the BatchNorm layers.
We compare in addition the performance drop relative to full precision nets of BinaryConnect, BinaryRelax (Yin et al., 2018), and our P ROX Q UANT .
Result The top-1 classiﬁcation errors for binary quantization are reported in Table 1.
Our P ROXQ UANT consistently yields better results than BinaryConnect.
The performance drop of P ROXQ UANT relative to full-precision nets is about 1% , better than BinaryConnect by 0 .
2% on average and signiﬁcantly better than the reported result of BinaryRelax.
Results and additional details for ternary quantization are deferred to Appendix B.
[Table] [Table]
4. L ANGUAGE MODELING WITH LSTM S
Problem setup We perform language modeling with LSTMs Hochreiter & Schmidhuber (1997) on the Penn Treebank (PTB) dataset (Marcus et al., 1993), which contains 929K training tokens, 73K validation tokens, and 82K test tokens.
Our model is a standard one-hidden-layer LSTM with embedding dimension 300 and hidden dimension 300.
We train quantized LSTMs with the encoder, transition matrix, and the decoder quantized to k -bits for k ∈ { 1 , 2 , 3 } .
The quantization is performed in a row-wise fashion, so that each row of the matrix has its own codebook { α 1 , . . .
, α k } .
Method We compare our multi-bit P ROX Q UANT (eq.
(10)) to the state-of-the-art alternating minimization algorithm with straight-through gradients (Xu et al., 2018).
Training is initialized at a pre-trained full-precision LSTM.
We use the SGD optimizer with initial learning rate 20.0 and decay by a factor of 1.2 when the validation error does not improve over an epoch.
We train forepochs with batch size 20, BPTT 30, dropout with probability 0.5, and clip the gradient norms to 0 .
The regularization rate λ is tuned by ﬁnding the best performance on the validation set.
In addition to multi-bit quantization, we also report the results for binary LSTMs (weights in {± 1 } ), comparing BinaryConnect and our P ROX Q UANT -Binary, where both learning rates are tuned on an exponential grid { 2 . 5 , 5 , 10 , 20 , 40 } .
Result We report the perplexity-per-word (PPW, lower is better) in Table 2.
The performance of P ROX Q UANT is comparable with the Straight-through gradient method.
On Binary LSTMs,
P ROX Q UANT -Binary beats BinaryConnect by a large margin.
These results demonstrate that P ROXQ UANT offers a powerful alternative for training recurrent networks.
[Table]
T HEORETICAL ANALYSIS
In this section, we perform a theoretical study on the convergence of quantization algorithms.
We show in Section 5.1 that our P ROX Q UANT algorithm (i.e. non-lazy prox-gradient method) converges under mild smoothness assumptions on the problem.
In Section 5.2, we provide a simple example showing that the lazy prox-gradient method fails to converge under the same set of assumptions.
In Section 5.3, we show that BinaryConnect has a very stringent condition for converging to a ﬁxed point.
Our theory demonstrates the superiority of our proposed P ROX Q UANT over lazy proxgradient type algorithms such as BinaryConnect and BinaryRelax (Yin et al., 2018).
All missing proofs are deferred to Appendix D.
Prox-gradient algorithms (both lazy and non-lazy) with a ﬁxed λ aim to solve the problem
[Equation]
and BinaryConnect can be seen as the limiting case of the above with λ = ∞ (cf.
Section 2.2).
For simplicity of the presentation, we focus on the full gradient case in our theoretical results.
5. A CONVERGENCE THEOREM FOR P ROX Q UANT
We consider P ROX Q UANT with batch gradient and constant regularization strength λ t ≡ λ :
[Equation]
Theorem 5.1 (Convergence of ProxQuant) .
Assume that the loss L is β -smooth (i.e. has β -Lipschitz gradients) and the regularizer R is differentiable.
Let F λ ( θ ) =
L ( θ ) + λR ( θ ) be the composite objective and assume that it is bounded below by F (cid:63) .
Running ProxQuant with batch gradient ∇ L , constant stepsize η t ≡ η = 12 β and λ t ≡ λ for T steps, we have the convergence guarantee
[Equation]
very clever trick not mentioned in their paper: after computing the alternating quantization q alt ( θ ) , they multiply by a constant 0.3 before taking the gradient; in other words, their quantizer is a rescaled alternating quantizer: θ (cid:55)→ 0 .
3 q alt ( θ ) .
This scaling step gives a signiﬁcant gain in performance – without scaling the PPW is { 116 . 7 , 94 . 3 , 87 . 3 } for { 1 , 2 , 3 } bits.
In contrast, our P ROX Q UANT does not involve a scaling step and achieves better PPW than this unscaled ALT straight-through method.
where C > 0 is a universal constant.
Remark 5.1.
The convergence guarantee requires both the loss and the regularizer to be smooth.
Smoothness of the loss can be satisﬁed if we use a smooth activation function (such as tanh ).
For the regularizer, the quantization-inducing regularizers deﬁned in Section 3.1 (such as the W-shaped regularizer) are non-differentiable.
However, we can use a smoothed version of them that is differentiable and point-wise arbitrarily close to R , which will satisfy the assumptions of Theorem 5.1.
The proof of Theorem 5.1 is deferred to Appendix D.1.
5. N ON - CONVERGENCE OF LAZY PROX - GRADIENT
The lazy prox-gradient algorithm (e.g. BinaryRelax (Yin et al., 2018)) for solving problem eq.
(11) is a variant where the gradients are taken at proximal points but accumulated at the original sequence:
[Equation]
Convergence of the lazy prox-gradient algorithm eq.
(13) is only known to hold for convex problems (Xiao, 2010); on smooth non-convex problems it generally does not converge.
We provide a concrete example that satisﬁes the assumptions in Theorem 5.1 (so that P ROX Q UANT converges) but lazy prox-gradient does not converge.
Theorem 5.2 (Non-convergence of lazy prox-gradient) .
There exists L and R satisfying the assumptions of Theorem 5.1 such that for any constant stepsize η t ≡ η ≤ 12 β , there exists some speciﬁc initialization θ 0 on which the lazy prox-gradient algorithm eq.
(13) oscillates between two non-stataionry points and hence does not converge in the sense of eq.
Remark 5.2.
Our construction is a fairly simple example in one-dimension and not very adversarial: L ( θ ) = 12 θ 2 and R is a smoothed W-shaped regularizer.
See Appendix D.2 for the details.
5. C ONVERGENCE CHARACTERIZATION FOR B INARY C ONNECT
For BinaryConnect, the concept of stataionry points is no longer sensible (as the target points {± 1 } d are isolated and hence every point is stationary).
Here, we consider the alternative deﬁnition of convergence as converging to a ﬁxed point and show that BinaryConnect has a very stringent convergence condition.
Consider the BinaryConnect method with batch gradients:
[Equation]
[Equation]
Deﬁnition 5.1 (Fixed point and convergence) .
We say that s ∈ {± 1 } d is a ﬁxed point of the BinaryConnect algorithm, if s 0 =
s in eq.
(14) implies that s t =
s for all t = 1 , 2 , ... .
We say that
the
BinaryConnect algorithm converges if there exists t < ∞ such that s t is a ﬁxed point.
Theorem 5.3.
Assume that the learning rates satisfy [-(cid:80) ∞ t =
-][Equation]
∞ t =0 η t = ∞ , then s ∈ {± 1 } d is a ﬁxed point for BinaryConnect eq.
(14) if and only if sign( ∇ L ( s )
[ i ])
= − s [ i ] for all i ∈
[ d ] such that ∇ L ( θ )
[ i ] (cid:54) = 0 .
Such a point may not exist, in which case BinaryConnect does not converge for any initialization θ 0 ∈ R d .
Remark 5.3.
Theorem 5.3 is in appearingly a stark contrast with the convergence result for BinaryConnect in (Li et al., 2017) in the convex case, whose bound involves a an additive error O (∆) that does not vanish over iterations, where ∆ is the grid size for quantization.
Hence, their result is only useful when ∆ is small.
In contrast, we consider the original BinaryConnect with ∆ = 1 , in which case the error makes Li et al.
(2017)’s bound vacuous.
The proof of Theorem 5.3 is deferred to Appendix D.3.
Experimental evidence We have already seen that such a ﬁxed point s might not exist in the toy example in Figure 1b.
In Appendix C, we perform a sign change experiment on CIFAR-10, showing that BinaryConnect indeed fails to converge to a ﬁxed sign pattern, corroborating Theorem 5.3.
C ONCLUSION
In this paper, we propose and experiment with the P ROX Q UANT method for training quantized networks.
Our results demonstrate that P ROX Q UANT offers a powerful alternative to the straightthrough gradient method and has theoretically better convergence properties.
For future work, it would be of interest to propose alternative regularizers for ternary and multi-bit P ROX Q UANT and experiment with our method on larger tasks.
A A DDITIONAL RESULTS ON R EGULARIZATION A. P ROX OPERATORS FOR BINARY NETS
Here we derive the prox operators for the binary regularizer eq.
(6) and its squared L 2 variant.
Recall that
[Equation]
By deﬁnition of the prox operator, we have for any θ ∈ R d that
[Equation]
This minimization problem is coordinate-wise separable.
For each (cid:101) θ j , the penalty term remains the same upon ﬂipping the sign, but the quadratic term is smaller when sign( (cid:101) θ j ) =
sign( θ j ) .
Hence, the solution θ (cid:63) to the prox satisﬁes that sign( θ (cid:63)j ) = sign( θ j ) , and the absolute value satisﬁes | θ (cid:63)j | = arg min t ≥
[Equation]
Multiplying by sign( θ (cid:63)j ) = sign( θ j ) , we have θ (cid:63)j = SoftThreshold ( θ j , sign( θ j ) , λ ) , which gives eq.
For the squared L 2 version, by a similar argument, the corresponding regularizer is
[Equation]
For this regularizer we have
[Equation]
Using the same argument as in the L 1 case, the solution θ (cid:63) satisﬁes sign( θ (cid:63)j ) = sign( θ j ) , and
[Equation]
Multiplying by sign( θ (cid:63)j ) = sign( θ j ) gives θ (cid:63)j =
[Equation]
j + λ sign [Equation] j
[Equation]
+ λ
[Equation]
or, in vector form, θ (cid:63) =
( θ + λ sign( θ )) /
(1 + λ ) .
A. P ROX OPERATOR FOR TERNARY QUANTIZATION
For ternary quantization, we use an approximate version of the alternating prox operator eq.
(10): compute (cid:101) θ = prox λR ( θ ) by initializing at (cid:101) θ
= θ and repeating (cid:98)
[Equation] and
[Equation] where q is the ternary quantizer deﬁned as
[Equation]
This is a straightforward extension of the TWN quantizer (Li & Liu, 2016) that allows different levels for positives and negatives.
We ﬁnd that two rounds of alternating computation in eq.
(15) achieves a good performance, which we use in our experiments.
B A DDITIONAL EXPERIMENTAL RESULTS B. T ERNARY QUANTIZATION FOR CIFAR-
Our models are ResNets of depth 20, 32, and 44.
Ternarized training is initialized at pre-trained full-precision nets.
We perform a hard quantization θ (cid:55)→ q ( θ ) at epoch 400 and keeps training till the 600-th epoch to stabilize the BatchNorm layers.
Result The top-1 classiﬁcation errors for ternary quantization are reported in Table 3.
Our results are comparable with the reported results of TTQ, 3 and the best performance of our method overruns (from the same initialization) is slightly better than TTQ.
[Table]
C S IGN CHANGE EXPERIMENT
We experimentally compare the training dynamics of P ROX Q UANT -Binary and BinaryConnect through the sign change metric.
The sign change metric between any θ 1 and θ 2 is the proportion of their different signs, i.e. the (rescaled) Hamming distance:
[Equation]
In R d , the space of all full-precision parameters, the sign change is a natural distance metric that represents the closeness of the binarization of two parameters.
[Figure]
Recall in our CIFAR-10 experiments (Section 4.1), for both BinaryConnect and P ROX Q UANT , we initialize at a good full-precision net θ 0 and stop at a converged binary network (cid:98) θ ∈ {± 1 } d .
We are interested in SignChange ( θ 0 , θ t ) along the training path, as well as SignChange ( θ 0 , (cid:98) θ ) , i.e. the distance of the ﬁnal output model to the initialization.
Our ﬁnding is that P ROX Q UANT produces binary nets with both lower sign changes and higher performances, compared with BinaryConnect.
Put differently, around the warm start, there is a good binary net nearby which can be found by P ROX Q UANT but not BinaryConnect, suggesting that BinaryConnect, and in general the straight-through gradient method, suffers from higher optimization instability than P ROX Q UANT .
This ﬁnding is consistent in all layers, across different warm starts, and across differnent runs from each same warm start (see Figure 3 and Table 4 in Appendix C.1).
This result here is also consistent with Theorem 5.3: the signs in BinaryConnect never stop changing until we manually freeze the signs at epoch 400.
C. R AW DATA FOR
[Table] [Table]
[Table]
D P ROOFS OF THEORETICAL RESULTS D. P ROOF OF T HEOREM 5.
Recall that a function f : R d → R is said to be β
-smooth
if it is differentiable and ∇ f is β -Lipschitz: for all x, y ∈
R
d we have
[Equation]
For any β -smooth function, it satisﬁes the bound
[Equation]
Convergence results of the form of Theorem 5.1 are standard in the literature of proximal algorithms, where we show convergence to stataionarity at some “best” index T best ∈ { 1 , . . . , T } without convexity on either L or R but assuming smoothness (cf. (Ghadimi & Lan, 2013)).
For completeness we provide a proof below.
Proof of Theorem 5.
Recall the ProxQuant iterate
[Equation]
By the fact that θ t +1 minimizes the above objective and applying the smoothness of L , we get that
[Equation]
θ
[Equation]
+
[Equation]
θ
[Equation]
θ
[Equation]
1 ) ≥ L ( θ t +1 )
+
[Equation]
Telescoping the above bound for t = 0 , . .
, T − 1 , we get that
[Equation]
Therefore we have the proximity guarantee
[Equation]
[Equation]
We now turn this into a stationarity guarantee.
The ﬁrst-order optimality condition for θ t +1 gives ∇
[Equation]
Combining the above equality and the smoothness of L , we get
[Equation]
Choosing t =
T best − 1 and applying the proximity guarantee eq.
(17), we get (cid:107)∇
[Equation]
This is the desired bound.
D. P ROOF OF T HEOREM 5.
[Figure]
Let our loss function L : R → R be the quadratic L ( θ ) = 12 θ 2 (so that L is β -smooth with β = 1 ).
Let the regularizer R :
R
→ R be a smoothed version of the W-shaped regularizer in eq.
(3), deﬁned as (for (cid:15) ∈ (0 , 1 / 2] being the smoothing radius)
[Equation]
θ
[Equation] [Equation]
and R ( − θ ) =
R ( θ ) for the negative part.
See Figure 4 for an illustration of the loss L and the regularizer R (with (cid:15)
= 0 . 2 ).
It is straightforward to see that R is piecewise quadratic and differentiable on R by computing the derivatives at (cid:15) and 1 ± (cid:15) .
Further, by elementary calculus, we can evaluate the prox operator in closed form: for all λ ≥ 1 , we have
[Equation]
all | θ | ≤ 1 .
Now, suppose we run the lazy prox-gradient method with constant stepsize η t ≡ η ≤ 12 β = 12 .
For the speciﬁc initialization
[Equation]
we have the equality prox λR
( θ 0 ) = 2 η θ 0 and therefore the next lazy prox-gradient iterate is
[Equation]
As both R and L are even functions, a symmetric argument holds for θ 1 from which we get θ 2 = − θ
= θ 0 .
Therefore the lazy prox-gradient method ends up oscillating between two points:
[Equation]
On the other hand, it is straightforward to check that the only stationary points of L ( θ ) + λR ( θ ) are 0 and ± λ(cid:15) + λ , all not equal to ± θ 0 .
Therefore the sequence { θ t } t ≥ 0 does not have a subsequence with vanishing gradient and thus does not approach stationarity in the ergodic sense.
D. P ROOF OF T HEOREM 5.
We start with the “ ⇒ ” direction.
If s is a ﬁxed point, then by deﬁnition there exists θ 0 ∈ R
d
such that θ t = θ for all t = 0 , 1 , 2 , ... .
By the iterates eq.
[Equation]
Take signs on both sides and apply s t
= s for all t on both sides, we get that
[Equation]
Take the limit T → ∞ and apply the assumption that
[Equation]
= ∞ , we get that for all i ∈
[ d ] such that [ ∇ L ( θ )] i (cid:54) = 0 ,
[Equation]
Now we prove the “ ⇐ ” direction.
If θ obeys that sign( ∇ L ( s )[ i ])
= − s [ i ] for all i ∈
[ d ] such that ∇ L ( s )
[ i ] (cid:54) = 0 , then if we take any θ 0 such that sign( θ 0 )
=
s , θ t will move in a straight line towards the direction of −∇ L ( s ) , which does not change the sign of h 0 .
In other words, s t = sign( θ t )
= sign( θ 0 ) =
s for all t = 0 , 1 , 2 , ... .
Therefore, by deﬁnition, s is a ﬁxed point.