Deep Studying CPU Benchmarks

Deep studying GPU benchmarks has revolutionized the way in which we clear up advanced issues, from picture recognition to pure language processing. Nevertheless, whereas coaching these fashions usually depends on high-performance GPUs, deploying them successfully in resource-constrained environments comparable to edge units or programs with restricted {hardware} presents distinctive challenges. CPUs, being broadly accessible and cost-efficient, usually function the spine for inference in such eventualities. However how can we be sure that fashions deployed on CPUs ship optimum efficiency with out compromising accuracy?

This text dives into the benchmarking of deep studying mannequin inference on CPUs, specializing in three essential metrics: latency, CPU utilization and Reminiscence Utilization. Utilizing a spam classification instance, We discover how fashionable frameworks like PyTorch, TensorFlow, JAX , and ONNX Runtime deal with inference workloads. By the tip, you’ll have a transparent understanding of measure efficiency, optimize deployments, and choose the precise instruments and frameworks for CPU-based inference in resource-constrained environments.

Impression: Optimum inference execution can save a big sum of money and unencumber sources for different workloads.

Studying Goals

  • Perceive the position of Deep Studying GPU benchmarks in assessing {hardware} efficiency for AI mannequin coaching and inference.
  • Learn to make the most of Deep Studying GPU benchmarks to check GPUs and optimize computational effectivity for AI duties.
  • Consider PyTorch, TensorFlow, JAX, ONNX Runtime, and OpenVINO Runtime to decide on the perfect on your wants.
  • Grasp instruments like psutil and time to gather correct efficiency information and optimize inference.
  • Put together fashions, run inference, and measure efficiency, making use of methods to numerous duties like picture classification and NLP.
  • Establish bottlenecks, optimize fashions, and improve efficiency whereas managing sources effectively.

This text was revealed as part of the Knowledge Science Blogathon.

Optimizing Inference with Runtime Acceleration

Inference velocity is important for consumer expertise and operational effectivity in machine studying functions. Runtime optimization performs a key position in enhancing this by streamlining execution. Utilizing hardware-accelerated libraries like ONNX Runtime takes benefit of optimizations tailor-made to particular architectures, lowering latency (time per inference).

Moreover, light-weight mannequin codecs comparable to ONNX decrease overhead, enabling sooner loading and execution. Optimized runtimes leverage parallel processing to distribute computation throughout accessible CPU cores and enhance reminiscence administration, making certain higher efficiency particularly on programs with restricted sources. This method makes fashions sooner and extra environment friendly whereas sustaining accuracy.

Mannequin Inference Efficiency Metrics

To judge the efficiency of our fashions, we concentrate on three key metric:

Latency

  • Definition : Latency refers back to the time it takes for the mannequin to make a prediction after receiving enter. That is usually measured because the time taken from sending the enter information to receiving the output (prediction)
  • Significance : In real-time or near-real-time functions, excessive latency results in delays, which can lead to slower responses.
  • Measurement : Latency is usually measure in milliseconds (ms) or seconds (s). Shorter latency means the system is extra responsive and environment friendly, essential for functions requiring instant decision-making or actions.

CPU Utilization

  • Definition: CPU Utilization is the proportion of the CPU’s processing energy that’s consumed whereas performing inference duties. It tells you the way a lot of the system’s computational sources are getting used throughout mannequin inference.
  • Significance : Excessive CPU utilization implies that the machine would possibly battle to deal with different duties concurrently, resulting in bottlenecks. Environment friendly use of CPU sources ensures that the mannequin inference doesn’t monopolize the system sources.
  • Measurement : It’s sometimes measured as a share (%) of the overall accessible CPU sources. Decrease utilization for a similar workload usually signifies a extra optimized mannequin, using CPU sources extra successfully.

Reminiscence Utilization

  • Definition: Reminiscence utilization refers back to the quantity of RAM utilized by the mannequin in the course of the inference course of. It tracks the reminiscence consumption by the mannequin’s parameters, intermediate computations, and the enter information.
  • Significance : Optimizing reminiscence utilization is particularly essential when deploying fashions to edge units or programs whith restricted reminiscence. Excessive reminiscence consumption may result in reminiscence overfloe, slower processing, or system crashes.
  • Measurement: Reminiscence utilization is measure in megabytes (MB) or gigabytes (GB). Monitoring the reminiscence consumption at completely different levels of inference may help establish reminiscence inefficiencies or reminiscence leaks.

Assumptions and Limitations

To maintain this benchmarking examine targeted and sensible, we made the next assumptions and set a number of boundaries:

  • {Hardware} Constraints: The exams are designed to run on a single machine with restricted CPU cores. Whereas trendy {hardware} is able to dealing with parallel workloads, this setup mirrors the constraints usually seen in edge units or smaller-scale deployments.
  • No Multi-System Parallelization: We didn’t incorporate distributed computing setups or cluster-based options. The benchmarks mirror efficiency standalone circumstances, appropriate for single-node environments with restricted CPU cores and Reminiscence.
  • Scope:The first focus is simply on CPU inference efficiency. Whereas GPU-based inference is a wonderful choice for resource-intensive duties, this benchmarking goals to offer insights into CPU-only setups, that are extra widespread in cost-sensitive or moveable functions.

These assumptions make sure the benchmarks stay related for builders and groups working with resource-constrained {hardware} or who want predictable efficiency with out the added complexity of distributed programs.

We’ll discover the important instruments and frameworks used to benchmark and optimize deep studying mannequin inference on CPUs, offering insights into their capabilities for environment friendly execution in resource-constrained environments.

Profiling Instruments

  • Python Time (time library) : The time library in Python is a light-weight device for measuring the execution time of code blocks. By recording the beginning and finish time stamps, it helps calculate the time taken for operations like mannequin inference or information processing.
  • psutil (CPU, Reminiscence Profiling) : psutil is a Python library for sustem monitoring and profiling. It offers real-time information on CPU utilization, reminiscence consumption, disk I/O and extra, making it ultimate for analyzing utilization throughout mannequin coaching or inference.

Frameworks for Inference

  • TensorFlow : A strong framework for deep studying that’s broadly used for each coaching and inference duties. It affords sturdy assist for varied fashions and deployment methods.
  • PyTorch: Recognized for its ease of use and dynamic computation graphs, PyTorch is a well-liked alternative for analysis and manufacturing deployment.
  • ONNX Runtime: An open-source , cross-platform engine for operating ONXX(Open Neural Community Alternate) fashions, offering environment friendly inference throughout varied {hardware} and frameworks.
  • JAX : A useful framework targeted on high-performance numerical computing and machine studying, providing computerized differentiation and GPU/TPU acceleration.
  • OpenVINO: Optimized for Intel {hardware}, OpenVINO offers instruments for mannequin optimization and deployment on Intel CPUs, GPUs and VPUs.

{Hardware} Specification and Setting

We’re using github codespace (digital machine) with beneath configuration:

  • Specification of Digital Machine: 2 cores, 8 GB RAM, and 32 GB storage
  • Python Model: 3.12.1

Set up Dependencies

The variations of the packages used are as follows and this main embody 5 deep studying inference libraries: Tensorflow, Pytorch, ONNX Runtime, JAX, and OpenVINO:

!pip set up numpy==1.26.4
!pip set up torch==2.2.2
!pip set up tensorflow==2.16.2
!pip set up onnx==1.17.0
!pip set up onnxruntime==1.17.0!pip set up jax==0.4.30
!pip set up jaxlib==0.4.30
!pip set up openvino==2024.6.0
!pip set up matplotlib==3.9.3
!pip set up Matplotlib: 3.4.3
!pip set up Pillow: 8.3.2
!pip set up psutil: 5.8.0

Drawback Assertion and Enter Specification

Since mannequin inference consists of performing a number of matrix operations between community weights and enter information, it doesn’t require mannequin coaching or datasets. For our instance the benchmarking course of, we simulated a regular classification use case. This simulates widespread binary classification duties like spam detection and mortgage utility choices(approval or denial). The binary nature of those issues makes them ultimate for evaluating mannequin efficiency throughout completely different frameworks. This setup displays real-world programs however permits us to concentrate on inference efficiency throughout frameworks without having massive datasets or pre-trained fashions.

Drawback Assertion

The pattern job entails predicting whether or not a given pattern is spam or not (mortgage approval or denial), based mostly on a set of enter options. This binary classification downside is computationally environment friendly, permitting for a targeted evaluation of inference efficiency with out the complexity of multi-class classification duties.

Enter Specification

To simulate real-world electronic mail information, we generated randomly enter. These embeddings mimic the kind of information that could be processed by spam filters however keep away from the necessity for exterior datasets. This simulated enter information permits for benchmarking with out counting on any particular exterior datasets, making it ultimate for testing mannequin inference occasions, reminiscence utilization, and CPU efficiency. Alternatively, you should use picture classification, NLP job or every other deep studying duties to carry out this benchmarking course of.

Fashions Structure and Codecs

Mannequin choice is a essential step in benchmarking because it instantly influences the inference efficiency and insights gained from the profiling course of. As talked about within the earlier part, for this benchmarking examine, we selected a regular Classification use case, which entails figuring out whether or not a given electronic mail is spam or not. This job is a simple two-class classification downside that’s computationally environment friendly but offers significant outcomes for comparability throughout frameworks.

Fashions Structure for Benchmarking

The mannequin for the Classification job is a Feedforward Neural Community (FNN) designed for binary classification (Spam vs. Not Spam). It consists of the next layers:

  • Enter Layer : Accepts a vector of measurement 200(embedding options). We’ve got offered instance of PyTorch, different frameworks observe the very same community configuration
self.fc1 =  torch.nn.Linear(200,128)
  • Hidden Layers : The community has 5 hidden layers, with every successive layer containing fewer models than the earlier one.
self.fc2 = torch.nn.Linear(128, 64)
self.fc3 = torch.nn.Linear(64, 32)
self.fc4 = torch.nn.Linear(32, 16)
self.fc5 = torch.nn.Linear(16, 8)
self.fc6 = torch.nn.Linear(8, 1)
  • Output Layers : A single neuron with a Sigmoid activation perform to output a chance (0 for Not Spam, 1 for Spam). We’ve got utilized sigmoid layer as closing output for binary classification.
self.sigmoid = torch.nn.Sigmoid()

The mannequin is straightforward but efficient for classification job.

The mannequin structure diagram used for benchmarking in our use case is proven beneath:

Neural_Network_Architecture: Deep Learning GPU Benchmarks

Examples of Further Networks for Benchmarking

  • Picture Classification : Fashions like ResNet-50 (medium complexity) and MobileNet (light-weight) could be added to the benchmark suite for duties involving picture recognition. ResNet-50 affords a steadiness between computational complexity and accuracy, whereas MobileNet is optimized for low-resource environments.
  • NLP Duties : DistilBERT: A smaller, sooner variant of the BERT mannequin, fitted to pure language understanding duties.

Mannequin Codecs

  • Native Codecs: Every framework helps its native mannequin codecs, comparable to .pt for PyTorch and .h5 for TensorFlow.
  • Unified Format (ONNX): To make sure compatibility throughout frameworks, We exported the PyTorch mannequin to the ONNX format (mannequin.onnx). ONNX (Open Neural Community Alternate) acts as a bridge, enabling fashions for use in different frameworks like PyTorch, TensorFlow, JAX, or OpenVINO with out important modifications. That is particularly helpful for multi-framework testing and real-world deployment eventualities, the place interoperability is essential.
  • These codecs are optimized for his or her respective frameworks, making them straightforward to avoid wasting, load, and deploy inside these ecosystems.

Benchmarking Workflow

This workflow goals to check the inference efficiency of a number of deep studying frameworks (TensorFlow, PyTorch, ONNX, JAX, and OpenVINO) utilizing the classification job. The duty entails utilizing randomly generated enter information and benchmarking every framework to measure the common time taken for a prediction.

  • Import python packages
  • Disable GPU utilization and suppress Tensorflow Logging
  • Enter information preparation
  • Mannequin Implementations for every framework
  • Benchmarking perform definition
  • Mannequin Inference and Benchmarking execution for every framework
  • Visualization and export of Benchmarking Outcomes

Import Essential Python Packages

To get began with benchmarking deep studying fashions, we first must import the important Python packages that allow seamless integration and efficiency analysis.

import time
import os
import numpy as np
import torch
import tensorflow as tf
from tensorflow.keras import Enter
import onnxruntime as ort
import matplotlib.pyplot as plt
from PIL import Picture
import psutil
import jax
import jax.numpy as jnp
from openvino.runtime import Core
import csv

Disable GPU Utilization and Suppress TensorFlow Logging

os.environ["CUDA_VISIBLE_DEVICES"] = "-1" # Disable GPU
os.environ["TF_CPP_MIN_LOG_LEVEL"] = "3" #Suppress Tensorflow Log

Enter Knowledge Preparation

On this step, we randomly generate enter information for spam classification:

  • Dimensionality of a pattern (200-dimesnional options)
  • The variety of courses (2: Spam or Not Spam)

We generate randome information utilizing NumPy to function enter options for the fashions.

#Generate dummy information
input_data = np.random.rand(1000, 200).astype(np.float32)

Mannequin Definition

On this step, we outline the netwrok structure or setup the mannequin from every deep studying framework( Tensorflow, PyTorch, ONNX, JAX and OpenVINO). Every framework requires a selected strategies for loading fashions and setting them up for inference.

  • PyTorch Mannequin: In PyTorch, we outline a easy neural neural community structure with 5 absolutely linked layers.
  • Tensorflow Mannequin : The TensorFlow mannequin is outlined utilizing the Keras API and consists of a easy feedforward neural community for the classification job.
  • JAX Mannequin: The mannequin is initialized with parameters, and the prediction perform is compiled utilizing JAX’s Simply-in-Time (JIT) compilation for environment friendly execution.
  • ONNX Mannequin: For ONNX, we export a mannequin from PyTorch. After exporting to the ONNX format, we load the mannequin utilizing the onnxruntime. InferenceSession API. This permits us to run inference on the mannequin throughout completely different {hardware} specification.
  • OpenVINO Mannequin: OpenVINO is used for operating optimized and deploying fashions, notably these skilled utilizing different frameworks (like PyTorch or TensorFlow). We load the ONNX mannequin and compile it with OpenVINO’s runtime.

Pytorch

class PyTorchModel(torch.nn.Module):
    def __init__(self):
        tremendous(PyTorchModel, self).__init__()
        self.fc1 = torch.nn.Linear(200, 128)
        self.fc2 = torch.nn.Linear(128, 64)
        self.fc3 = torch.nn.Linear(64, 32)
        self.fc4 = torch.nn.Linear(32, 16)
        self.fc5 = torch.nn.Linear(16, 8)
        self.fc6 = torch.nn.Linear(8, 1)
        self.sigmoid = torch.nn.Sigmoid()

    def ahead(self, x):
        x = torch.relu(self.fc1(x))
        x = torch.relu(self.fc2(x))
        x = torch.relu(self.fc3(x))
        x = torch.relu(self.fc4(x))
        x = torch.relu(self.fc5(x))
        x = self.sigmoid(self.fc6(x))
        return x
        
     # Create PyTorch mannequin
    pytorch_model = PyTorchModel()

TensorFlow

tensorflow_model = tf.keras.Sequential([
    Input(shape=(200,)),
    tf.keras.layers.Dense(128, activation='relu'),
    tf.keras.layers.Dense(64, activation='relu'),
    tf.keras.layers.Dense(32, activation='relu'),
    tf.keras.layers.Dense(16, activation='relu'),
    tf.keras.layers.Dense(8, activation='relu'),
    tf.keras.layers.Dense(1, activation='sigmoid')
])
tensorflow_model.compile()

Jax

def jax_model(x):
    x = jax.nn.relu(jnp.dot(x, jnp.ones((200, 128))))
    x = jax.nn.relu(jnp.dot(x, jnp.ones((128, 64))))
    x = jax.nn.relu(jnp.dot(x, jnp.ones((64, 32))))
    x = jax.nn.relu(jnp.dot(x, jnp.ones((32, 16))))
    x = jax.nn.relu(jnp.dot(x, jnp.ones((16, 8))))
    x = jax.nn.sigmoid(jnp.dot(x, jnp.ones((8, 1))))
    return x

ONNX

# Convert PyTorch mannequin to ONNX
dummy_input = torch.randn(1, 200)
onnx_model_path = "mannequin.onnx"
torch.onnx.export(
    pytorch_model, 
    dummy_input, 
    onnx_model_path, 
    export_params=True, 
    opset_version=11, 
    input_names=['input'], 
    output_names=['output'], 
    dynamic_axes={'enter': {0: 'batch_size'}, 'output': {0: 'batch_size'}}
)

onnx_session = ort.InferenceSession(onnx_model_path)

OpenVINO

# OpenVINO Mannequin Definition
core = Core()
openvino_model = core.read_model(mannequin="mannequin.onnx")
compiled_model = core.compile_model(openvino_model, device_name="CPU")

Benchmarking Operate Definiton

This perform executes benchmarking exams throughout completely different frameworks by taking three arguments: predict_function, input_data, and num_runs. By default, it executes 1,000 occasions however It may be elevated as per necessities.

def benchmark_model(predict_function, input_data, num_runs=1000):
    start_time = time.time()
    course of = psutil.Course of(os.getpid())
    cpu_usage = []
    memory_usage = []
    for _ in vary(num_runs):
        predict_function(input_data)
        cpu_usage.append(course of.cpu_percent())
        memory_usage.append(course of.memory_info().rss)
    end_time = time.time()
    avg_latency = (end_time - start_time) / num_runs
    avg_cpu = np.imply(cpu_usage)
    avg_memory = np.imply(memory_usage) / (1024 * 1024)  # Convert to MB
    return avg_latency, avg_cpu, avg_memory

Mannequin Inference and Carry out Benchmarking for Every Framework

Now that now we have loaded the fashions, it’s time to benchmark the efficiency of every framework. The benchmarking course of carry out inference on the generated enter information.

PyTorch

# Benchmark PyTorch mannequin
def pytorch_predict(input_data):
    pytorch_model(torch.tensor(input_data))

pytorch_latency, pytorch_cpu, pytorch_memory = benchmark_model(lambda x: pytorch_predict(x), input_data)

TensorFlow

# Benchmark TensorFlow mannequin
def tensorflow_predict(input_data):
    tensorflow_model(input_data)

tensorflow_latency, tensorflow_cpu, tensorflow_memory = benchmark_model(lambda x: tensorflow_predict(x), input_data)

JAX

# Benchmark JAX mannequin
def jax_predict(input_data):
    jax_model(jnp.array(input_data))

jax_latency, jax_cpu, jax_memory = benchmark_model(lambda x: jax_predict(x), input_data)

ONNX

# Benchmark ONNX mannequin
def onnx_predict(input_data):
    # Course of inputs in batches
    for i in vary(input_data.form[0]):
        single_input = input_data[i:i+1]  # Extract single enter
        onnx_session.run(None, {onnx_session.get_inputs()[0].identify: single_input})

onnx_latency, onnx_cpu, onnx_memory = benchmark_model(lambda x: onnx_predict(x), input_data)

OpenVINO

# Benchmark OpenVINO mannequin
def openvino_predict(input_data):
    # Course of inputs in batches
    for i in vary(input_data.form[0]):
        single_input = input_data[i:i+1]  # Extract single enter
        compiled_model.infer_new_request({0: single_input})

openvino_latency, openvino_cpu, openvino_memory = benchmark_model(lambda x: openvino_predict(x), input_data)

Outcomes and Dialogue

Right here we focus on the outcomes of efficiency benchmarking of beforehand talked about deep studying frameworks. We examine them on – latency, CPU utilization, and reminiscence utilization. We’ve got included tabular information and plot for fast comparability.

Latency Comparability

Framework Latency (ms) Relative Latency (vs. PyTorch)
PyTorch 1.26 1.0 (baseline)
TensorFlow 6.61 ~5.25×
JAX 3.15 ~2.50×
ONNX 14.75 ~11.72×
OpenVINO 144.84 ~115×

Insights:

  • PyTorch leads because the quickest framework with ~1.26 ms latency.
  • TensorFlow has ~6.61ms latency, about 5.25× PyTorch’s time.
  • JAX sits between PyTorch and TensorFlow in absolute latency.
  • ONNX is comparatively sluggish as nicely, at ~14.75 ms.
  • OpenVINO is the slowest on this experiment, at ~145 ms (115× slower than PyTorch).

CPU Utilization

Framework CPU Utilization (%) Relative CPU Utilization<sup>1</sup>
PyTorch 99.79 ~1.00
TensorFlow 112.26 ~1.13
JAX 130.03 ~1.31
ONNX 99.58 ~1.00
OpenVINO 99.32 1.00 (baseline)

Insights:

  • JAX makes use of essentially the most CPU (~130 %), ~31% greater than OpenVINO.
  • TensorFlow is at ~112 %, greater than PyTorch/ONNX/OpenVINO however nonetheless decrease than JAX.
  • PyTorch, ONNX, and OpenVINO, all have comparable, ~99-100% CPU utilization.

Reminiscence Utilization

Framework Reminiscence (MB) Relative Reminiscence Utilization (vs. PyTorch)
PyTorch ~959.69 1.0 (baseline)
TensorFlow ~969.72 ~1.01×
JAX ~1033.63 ~1.08×
ONNX ~1033.82 ~1.08×
OpenVINO ~1040.80 ~1.08–1.09×

Insights:

  • PyTorch and TensorFlow have comparable reminiscence utilization round ~960-970 MB
  • JAX, ONNX, and OpenVINO use round ~1,030–1,040 MB of reminiscence, roughly 8–9% greater than PyTorch.

Right here is the plot evaluating the Efficiency of Deep Studying Frameworks:

Comparision_of_Deep_Learning_Inference_Framework: Deep Learning GPU Benchmarks

Conclusion

On this article, we introduced a complete benchmarking workflow to guage the inference efficiency of outstanding deep studying frameworks—TensorFlow, PyTorch, ONNX, JAX, and OpenVINO—utilizing a spam classification job as a reference. By analyzing key metrics comparable to latency, CPU utilization and reminiscence consumption, the outcomes highlighted the trade-offs between frameworks and their suitability for various deployment eventualities.

PyTorch demonstrated essentially the most balanced efficiency, excelling in low latency and environment friendly reminiscence utilization, making it ultimate for latency-sensitive functions like real-time predictions and advice programs. TensorFlow offered a middle-ground resolution with reasonably greater useful resource consumption. JAX showcased excessive computational throughput however at the price of elevated CPU utilization, which could be a limiting issue for resource-constrained environments. In the meantime, ONNX and OpenVINO lagged in latency, with OpenVINO’s efficiency notably hindered by the absence of {hardware} acceleration.

These findings underline the significance of aligning framework choice with deployment wants. Whether or not optimizing for velocity, useful resource effectivity, or particular {hardware}, understanding the trade-offs is important for efficient mannequin deployment in real-world environments.

Key Takeaways

  • Deep Studying GPU Benchmarks present essential insights into GPU efficiency, aiding in choosing optimum {hardware} for AI duties.
  • Leveraging Deep Studying GPU Benchmarks ensures environment friendly mannequin coaching and inference by figuring out high-performing GPUs.
  • Achieved the perfect latency (1.26 ms) and maintained environment friendly reminiscence utilization, ultimate for real-time and resource-limited functions.
  • Balanced latency (6.61 ms) with barely greater CPU utilization, appropriate for duties requiring reasonable efficiency compromises.
  • Delivered aggressive latency (3.15 ms) however at the price of extreme CPU utilization (130%), limiting its utility in constrained setups.
  • Confirmed greater latency (14.75 ms), however its cross-platform assist makes it versatile for multi-framework deployments.

Steadily Requested Questions

Q1. Why is PyTorch most popular for real-time functions?

A. PyTorch’s dynamic computation graph and environment friendly execution pipeline enable for low-latency inference (1.26 ms), making it well-suited for functions like advice programs and real-time predictions.

Q2. What affected OpenVINO’s efficiency on this examine?

A. OpenVINO’s optimizations are designed for Intel {hardware}. With out this acceleration, its latency (144.84 ms) and reminiscence utilization (1040.8 MB) have been much less aggressive in comparison with different frameworks.

Q3. How do I select a framework for resource-constrained environments?

A. For CPU-only setups, PyTorch is essentially the most environment friendly. TensorFlow is a robust different for reasonable workloads. Keep away from frameworks like JAX until greater CPU utilization is appropriate.

This autumn. What position does {hardware} play in framework efficiency?

A. Framework efficiency relies upon closely on {hardware} compatibility. For example, OpenVINO excels on Intel CPUs with hardware-specific optimizations, whereas PyTorch and TensorFlow carry out persistently throughout diverse setups.

Q5. Can benchmarking outcomes differ with advanced fashions or duties?

A. Sure, these outcomes mirror a easy binary classification job. Efficiency may fluctuate with advanced architectures like ResNet or duties like NLP or others, the place these frameworks would possibly leverage specialised optimizations.

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