CPT: Efficient Deep Neural Network Training via Cyclic Precision

Hypothesis 1: DNN’s precision has a similar effect as the learning rate. Existing works (Grandvalet et al., 1997; Neelakantan et al., 2015) show that noise can help DNN training theoretically or empirically, motivating us to rethink the role of quantization in DNN training. We conjecture that low precision with large quantization noise helps DNN training exploration with an effect similar to a high learning rate, while high precision with more accurate updates aids model convergence, similar to a low learning rate.

Validating Hypothesis 1. Settings: To empirically justify our hypothesis, we train ResNet-38/74 on the CIFAR-100 dataset for 160 epochs following the basic training setting as in Sec. 4.1. In particular, we divide the training of 160 epochs into three stages: [0-th, 80-th], [80-th,120-th], and [120-th, 160-th]: for the first training stage of [0-th, 80-th], we adopt different learning rates and precisions for the weights and activations, while using full precision for the remaining two stages with a learning rate of 0.01 for the [80-th,120-th] epochs and 0.001 for the [120-th, 160-th] epochs in all the experiments in order to explore the relationship between the learning rate and precision in the first training stage.

Results: As shown in Tab. 1, we can observe that as the learning rate is sufficiently reduced for the first training stage, adopting a lower precision for this stage will lead to a higher accuracy than training with full precision. In particular, with the standard initial learning rate of 0.1, full precision training achieves a 1.00%/0.70% higher accuracy than the 4-bit one on ResNet-38/74, respectively; whereas as the initial learning rate decreases, this accuracy gap gradually narrows and then reverses, e.g., when the initial learning rate becomes 1e-2, training with [0-th, 80-th] of 4-bit achieves a 1.40%/1.57% higher accuracy than the full precision ones.

Insights: This set of experiments show that (1) when the initial learning rate is low, training with lower initial precisions consistently leads to a better accuracy than training with full precision, indicating that lowering the precision introduces a similar effect of favoring exploration as that of a high learning rate; and (2) although a low precision can alleviate the accuracy drop caused by a low learning rate, a high learning rate is in general necessary to maximize the accuracy.

Hypothesis 2: Dynamic precision helps DNN generalization. Recent findings in DNN training have motivated us to better utilize DNN precision to achieve a win-win in both DNN accuracy and efficiency. Specifically, it has been discussed that (1) DNNs learn to fit different patterns at different training stages, e.g., (Rahaman et al., 2019; Xu et al., 2019) reveal that DNN training first learns lower-frequency components and then high-frequency features, with the former being more robust to perturbations and noises; and (2) dynamic learning rate schedules help to improve the optimization in DNN training, e.g.,  (Li et al., 2019) points out that a large initial learning rate helps the model to memorize easier-to-fit and more generalizable patterns while  (Smith, 2017; Loshchilov & Hutter, 2016) show that cyclical learning rate schedules improve DNNs’ classification accuracy. These works inspire us to hypothesize that dynamic precision might help DNNs to reach a better optimum in the optimization landscape, especially considering the similar effect between the learning rate and precision validated in our Hypothesis 1.

Validating Hypothesis 2. Our Hypothesis 2 has been consistently confirmed by various empirical observations. For example, a recent work (Fu et al., 2020) proposes to progressively increase the precision during the training process, and we follow their settings to validate our hypothesis.

Settings: We train a ResNet-74 on CIFAR-100 using the same training setting as (Wang et al., 2018b) except that we quantize the weights, activations, and gradients during training; for the progressive precision case we uniformly increase the precision of weights and activations from 3-bit to 8-bit in the first 80 epochs and adopt static 8-bit gradients, while the static precision baseline uses 8-bit for all the weights/activations/gradients.

Results: Fig. 1 shows that training with progressive precision schedule achieves a slightly higher accuracy (+0.3%) than its static counterpart, while the former can reduce training costs. Furthermore, we visualize the loss landscape (following the method in (Li et al., 2018)) in Fig. 2(b): interestingly the progressive precision schedule helps to converge to a better local minima with wider contours, indicating a lower generalization error (Li et al., 2018) over the static 8-bit baseline in Fig. 2(a).

The progressive precision schedule in (Fu et al., 2020) relies on manual hyper-parameter tuning. As such, a natural following question would be: what kind of dynamic schedules would be effective while being simple to implement for different tasks/models? In this work, we show that a simple cyclic schedule consistently benefits the training convergence while boosting the training efficiency.

The key concept of CPT draws inspiration from (Li et al., 2019) which demonstrates that a large initial learning rate helps the model to learn more generalizable patterns. We thus hypothesize that a lower precision that leads to a short-term poor accuracy might actually help the DNN exploration during training thanks to its associated larger quantization noise, while it is well known that a higher precision enables the learning of higher-complexity, fine-grained patterns that is critical to better convergence. Together, this combination could improve the achieved accuracy as it might better balance coarse-grained exploration and fine-grained optimization during DNN training, which leads to the idea of CPT. Specifically, as shown in Fig. 3, CPT varies the precision cyclically between two bounds instead of fixing the precision during training, letting the models explore the optimization landscape with different granularities.

While CPT can be implemented using different cyclic scheduling methods, here we present as an example an implementation of CPT in a cosine manner:

where $B_{min}^{n}$ and $B_{max}^{n}$ are the lower and upper precision bound, respectively, in the $n$-th cycle of precision schedule, $\lceil\cdot\rfloor$ and $\%$ denote the rounding operation and the remainder operation, respectively, and $B_{t}^{n}$ is the precision at the $t$-th global epoch which falls into the $n$-th cycle with a cycle length of $T_{n}$. Note that the cycle length $T_{n}$ is equal to the total number of training epochs divided by the total number of cycles denoted as $N$, where $N$ is a hyper-parameter of CPT. For example, if $N=2$, then a DNN training with CPT will experience two cycles of cyclic precision schedule during training. As shown in Sec. 4.3, we find that the benefits of CPT are maintained when adopting different total number of cyclic precision schedule cycles during training, i.e., CPT is not sensitive to $N$. A visualization example for the precision schedule can be found in Appendix A. Additionally, we find that CPT is generally effective when using different dynamic precision schedule patterns (i.e., not necessarily the cosine schedule in Eq. (1). We implement CPT following Eq. (1) in this work and discuss the potential variants in Sec. 4.3.

We visualize the training curve of CPT on ResNet-74 with CIFAR-100 in Fig. 1 and find that it achieves a 0.91% higher accuracy paired with a 36.7% reduction in the required training BitOPs (bit operations), as compared to its static fixed precision counterpart. In addition, Fig. 2 (c) visualizes the corresponding loss landscape, showing the effectiveness of CPT, i.e., such a simple and automated precision schedule leads to a better convergence with lower sharpness.

The concept of CPT is simple enough to be plugged into any model or task to boost the training efficiency. One remaining question is how to determine the precision bounds, i.e., $B_{min}^{i}$ and $B_{max}^{i}$ in Eq. (1), which we find can be automatically decided in the first cycle (i.e., $T_{i}=T_{0}$) of the precision schedule using a simple PRT at a negligible computational cost. Specifically, PRT starts from the lowest possible precision, e.g., 2-bit, and gradually increases the precision while monitoring the difference in the training accuracy magnitude averaged over several consecutive iterations; once this training accuracy difference is larger than a preset threshold, indicating that the training can at least partially converge, PRT would claim that the lower bound is identified. While the upper bound can be similarly determined, there exists an alternative which suggests simply adopting the precision of CPT’s static precision counterpart. The remaining cycles use the same precision bounds.

Fig. 4 visualizes the PRT for ResNet-152/MobileNetV2 trained on CIFAR-100. We can see that the lower precision bound identified when the model experiences a notable training accuracy improvement for ResNet-152 is 3-bit while that for MobileNetV2 is 4-bit, aligning with the common observation that ResNet-152 is more robust to quantization than the more compact model MobileNetV2.

Models, datasets and baselines. We consider eleven models (including eight ResNet based models (He et al., 2016), MobileNetV2 (Sandler et al., 2018), Transformer (Vaswani et al., 2017), and LSTM (Hochreiter & Schmidhuber, 1997)) and five tasks (including CIFAR-10/100 (Krizhevsky et al., 2009), ImageNet (Deng et al., 2009), WikiText-103 (Merity et al., 2016), and Penn Treebank (PTB) (Marcus et al., 1993)). Specifically, we follow (Wang et al., 2019b) for implementing MobileNetV2 on CIFAR-10/100. Baselines: We first benchmark CPT over three SOTA static low-precision training techniques: SBM (Banner et al., 2018), DoReFa (Zhou et al., 2016), and WAGEUBN (Yang et al., 2020), each of which adopts a different quantizer. Since SBM is the most competitive baseline among the three based on both their reported and our experiment results, we apply CPT on top of SBM, and all the static precision baselines adopt the SBM quantizer unless specifically stated. Another baseline is the cyclic learning rate (CLR) (Loshchilov & Hutter, 2016) on top of static precision training, and we follow the best setting in (Loshchilov & Hutter, 2016).

Training settings. We follow the standard training setting in all the experiments. In particular, for classification tasks, we follow SOTA settings in (Wang et al., 2018b) for CIFAR-10/100 and (He et al., 2016) for ImageNet experiments, respectively; and for language modeling tasks, we follow (Baevski & Auli, 2018) for Transformer on WikiText-103 and (Merity et al., 2017) for LSTM on PTB.

Precision settings. The lower precision bounds in all the experiments are set using the PRT in Sec.3.3 and the upper bound is the same as the precision of the corresponding static precision baselines. We only apply CPT to the weights and activations (together annotated as FW) and use static precision for the errors and gradients (together annotated as BW), the latter of which is to ensure the stability of the gradients (Wang et al., 2018a) (more discussion in Appendix C). In particular, CPT from 3-bit to 8-bit with 8-bit gradient is annotated as FW(3,8)/BW8. The total number of periodic precision cycles, i.e., $N$ in Sec.3.3, for all the experiments is fixed to be 32 (see the ablation studies in Sec. 4.3).

Hardware settings and metrics. To validate the real hardware efficiency of the proposed CPT, we adopt standard FPGA implementation flows. Specifically, we employ the Vivado HLx design flow to implement FPGA-based accelerators on a Xilinx development board called ZC706 (Xilinx, ). To better evaluate the training cost, we consider both calculated GBitOPs (Giga bit operations) and real-measured latency on the ZC706 FPGA board.

Benchmark on CIFAR-10/100. Benchmark over SOTA quantizers: We benchmark CPT with three SOTA static low-precision training methods, as summarized in Tab. 2, when training ResNet-74/164 and MobileNetV2 on CIFAR-10/100, covering both deep and compact DNNs which are representative difficult cases of low-precision DNN training. Note that the accuracy improvement is the difference between CPT and the strongest baseline under the same setting. Tab. 2 shows that (1) our CPT consistently achieves a win-win with both a higher accuracy (+0.19% $\sim$ +1.25%) and a lower training cost (-21.0% $\sim$ -37.1% computational cost and -14.7% $\sim$ -21.4% latency) under all the cases on CIFAR-10/100, even for the compact model MobileNetV2, and (2) CPT notably outperforms the baselines in terms of accuracy under extremely low precision, which is one of the most useful scenarios for low-precision DNN training. In particular, CPT with periodic precision between 4-bit and 6-bit boosts the accuracy by +1.25% and +1.07% on ResNet-74/164, respectively, as compared to the static 6-bit training on CIFAR-10, verifying that CPT leads to a better convergence.

Benchmark over CLR on top of SBM: We further benchmark CPT with CLR (Loshchilov & Hutter, 2016) (inherit its besting setting on CIFAR-100), with both being applied on top of SBM as it achieves the best performance among SOTA quantizers as shown in Tab. 2, based on more DNN models and precision as shown in Fig. 5. We can see that (1) CPT still consistently outperforms all the baselines with a better accuracy and efficiency trade-off, and (2) CLR on top of SBM leads to a negative effect on the test accuracy, which we conjecture is caused by both the instability of gradients and the sensitivity of gradients to the learning rate under low-precision training, as discussed in (Wang et al., 2018a), showing that CPT is more applicable to low-precision training than CLR.

Loss landscape visualization: To better understand the superior performance achieved by CPT, we visualize the loss landscape following the method in (Li et al., 2018), as shown in Fig. 6 covering both non-compact and compact models and two low-precision settings (i.e., 6-bit and 8-bit) which are bottlenecks for low-precision DNN training. We can observe that, again, both standard and compact DNNs trained with CPT experience wider contours with less sharpness, indicating that CPT helps DNN training optimization to converge to better local optima.

Benchmark on ImageNet. To verify the scalability of CPT on more complex tasks and larger models, we benchmark CPT with the SOTA static precision training method SBM (Banner et al., 2018) on ResNet-18/34/50 with ImageNet, under which it is challenging for low-precision training such as 4-bit to work. As shown in Tab. 3, we can observe that CPT still achieves a reduced computational cost (up to -30.4%) with a comparable accuracy (-0.20% $\sim$ +0.06%). In particular, CPT works well on ResNet-50, leading to both a slightly higher accuracy and a better training efficiency, indicating the scalability of CPT with model complexity, and thus, its potential application in large scale training, in addition to on-device training scenarios.

CPT for boosting accuracy: An important perspective of CPT is its potential to improve training optimality in addition to efficiency. We illustrate CPT’s advantage in improving the final accuracy through training ResNet-18/34 on ImageNet using CPT and static full precision. As shown in Tab. 4, CPT on ResNet-18/34 achieves a 0.91%/0.84% higher accuracy than their full precision counterparts on ImageNet, indicating that CPT can be adopted as a general technique to improve the final accuracy in addition to efficient training.

Benchmark on WikiText-103 and PTB. We also apply CPT on language modeling tasks (including WikiText-103 and PTB) (see Tab. 5) to show that CPT is also applicable to natural language processing models. Tab. 5 shows that (1) CPT again consistently achieves a win-win in terms of accuracy (i.e., perplexity - the lower the better) and training efficiency, and (2) language modeling models/tasks are more sensitive to quantization, especially in LSTM models, as it always adapts to a larger lower precision bound, which is consistent with SOTA observations (Hou et al., 2019).

CPT with different precision ranges. We also evaluate CPT under a wide range of upper precision bounds, which correspond to a different target efficiency, to see if CPT still works well. Fig. 7 plots a boxplot with each experiment being repeated ten times. We can see that (1) regardless of the adopted precision ranges, CPT consistently achieves a win-win (a +0.74% $\sim$ +2.03% higher accuracy and a -18.3% $\sim$ -48.0% reduction in computational cost), especially under lower precision scenarios, and (2) CPT even shrinks the accuracy variance, which better aligns with the practical goal of efficient training.

CPT with different adopted numbers of precision schedule cycles. To evaluate CPT’s sensitivity to the total number of adopted cyclic precision schedule cycles, we apply CPT on ResNet-38/74 and CIFAR-100 when using different numbers of schedule cycles. Fig. 8 plots the mean of the achieved accuracy based on ten repeated experiments. We can see that (1) different choices for the total number of cycles lead to a comparable accuracy (within 0.5% accuracy) on CIFAR-10/100; and (2) CPT with different number of precision schedule cycles consistently outperforms the static baseline SBM in the achieved accuracy. Based on this experiment, we set $N=32$ for simplicity.

CPT with different cyclic precision schedule patterns. We also evaluate CPT using other cyclic precision schedule patterns in addition to the cosine one, including a triangular schedule motivated by (Smith, 2017) and a cosine annealing schedule as the learning rate schedule in (Loshchilov & Hutter, 2016), with all adopting 32 cyclic cycles for fairness. Experiments in Tab. 6 show that (1) CPT with different schedule patterns is consistently effective, and (2) CPT with the other two schedule patterns even surpasses the cosine one in some cases but underperforms on compact models. We leave how to determine the optimal cyclic pattern for a given model and task as a future work.