YOLOv8优化策略：Adam该换了！斯坦福最新Sophia优化器，比Adam快2倍 | 2023.5月斯坦福最新成果

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AI小怪兽

发布于 2023-11-03 08:49:34

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发布于 2023-11-03 08:49:34

文章被收录于专栏：YOLO大作战

1.Sophia优化器介绍

斯坦福2023.5月发表的最新研究成果，他们提出了「一种叫Sophia的优化器，相比Adam，它在LLM上能够快2倍，可以大幅降低训练成本」。

论文：https://arxiv.org/pdf/2305.14342.pdf

本文介绍了一种新的模型预训练优化器：Sophia（Second-order Clipped Stochastic Optimization），这是一种轻量级二阶优化器，它使用Hessian对角线的廉价随机估计作为预调节器，并通过限幅机制来控制最坏情况下的更新大小。在GPT-2等预训练语言模型上，Sophia以比Adam少了50%的步骤，且实现了相同的预训练损失。

作者表示 Adam 对于异构曲率（heterogeneous curvatures）的适应性不足。另一方面，vanilla Newton 方法在凸函数中具有最优的 pre-conditioner，但对于负曲率和 Hessian 的快速变化容易受到影响。基于这些见解，该研究设计了一种新的优化器 Sophia，它比 Adam 更适应异构曲率，比 Newton 方法更能抵抗非凸性和 Hessian 的快速变化，并且还使用了成本较低的 pre-conditioner。

研究引入了两个对角 Hessian 估计器，它们的内存和运行时间成本都与计算梯度相似。估计器分别为 Hutchinson 无偏估计器以及 GNB（ Gauss-Newton-Bartlett ）估计器。伪代码如下所示：

比较 wall-clock 时间与计算量。表 1 比较了每一个 step 的总计算量 (TFLOPs) 和 A100 GPU 上的 wall-clock 时间。本文报告了每个 step 的平均时间，Hessian 计算花费的时间的总计算。较小的批量大小，即每 10 个 step 以计算对角 Hessian 估计，Hessian 计算占总计算量的 6%，与 AdamW 相比，整体 wall-clock 时间开销小于 5%。在内存使用方面，优化器 m 和 h 两个状态，这导致了与 AdamW 相同的内存开销。

2.Sophia引入到yolov8

2.1 修改ultralytics/yolo/engine/trainer.py

核心代码：


import torch
from torch.optim.optimizer import Optimizer
import math
from torch import Tensor
from typing import List, Optional


class Sophia(Optimizer):
    def __init__(self, model, input_data, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8, weight_decay=0, k=10,
                 estimator="Hutchinson", rho=1):
        self.model = model
        self.input_data = input_data
        defaults = dict(lr=lr, betas=betas, eps=eps, weight_decay=weight_decay, k=k, estimator=estimator, rho=rho)
        super(Sophia, self).__init__(params, defaults)

    def step(self, closure=None):
        loss = None
        if closure is not None:
            loss = closure()

        for group in self.param_groups:
            for p in group["params"]:
                if p.grad is None:
                    continue
                grad = p.grad.data
                if grad.is_sparse:
                    raise RuntimeError("Sophia does not support sparse gradients")

                state = self.state[p]

                # state init
                if len(state) == 0:
                    state['step'] = 0
                    state['m'] = torch.zeros_like(p.data)
                    state['h'] = torch.zeros_like(p.data)

                m, h = state['m'], state['h']
                beta1, beta2 = group['betas']
                state['step'] += 1

                if group['weight_decay'] != 0:
                    grad = grad.add(group["weight_decay"], p.data)

                # update biased first moment estimate
                m.mul_(beta1).add_(1 - beta1, grad)

                # update hessian estimate
                if state['step'] % group['k'] == 1:
                    if group['estimator'] == "Hutchinson":
                        hessian_estimate = self.hutchinson(p, grad)
                    elif group['estimator'] == "Gauss-Newton-Bartlett":
                        hessian_estimate = self.gauss_newton_bartlett(p, grad)
                    else:
                        raise ValueError("Invalid estimator choice")
                    h.mul_(beta2).add_(1 - beta2, hessian_estimate)

                # update params
                p.data.add_(-group['lr'] * group['weight_decay'], p.data)
                p.data.addcdiv_(-group['lr'], m, h.add(group['eps']).clamp(max=group['rho']))

        return loss

    def hutchinson(self, p, grad):
        u = torch.randn_like(grad)
        grad_dot_u = torch.sum(grad * u)
        hessian_vector_product = torch.autograd.grad(grad_dot_u, p, retain_graph=True)[0]
        return u * hessian_vector_product

    def gauss_newton_bartlett(self, p, grad):
        B = len(self.input_data)
        logits = [self.model(xb) for xb in self.input_data]
        y_hats = [torch.softmax(logit, dim=0) for logit in logits]
        g_hat = \
        torch.autograd.grad(sum([self.loss_function(logit, y_hat) for logit, y_hat in zip(logits, y_hats)]) / B, p,
                            retain_graph=True)[0]
        return B * g_hat * g_hat


class SophiaG(Optimizer):
    def __init__(self, params, lr=1e-4, betas=(0.965, 0.99), rho=0.04,
                 weight_decay=1e-1, *, maximize: bool = False,
                 capturable: bool = False):
        if not 0.0 <= lr:
            raise ValueError("Invalid learning rate: {}".format(lr))
        if not 0.0 <= betas[0] < 1.0:
            raise ValueError("Invalid beta parameter at index 0: {}".format(betas[0]))
        if not 0.0 <= betas[1] < 1.0:
            raise ValueError("Invalid beta parameter at index 1: {}".format(betas[1]))
        if not 0.0 <= rho:
            raise ValueError("Invalid rho parameter at index 1: {}".format(rho))
        if not 0.0 <= weight_decay:
            raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
        defaults = dict(lr=lr, betas=betas, rho=rho,
                        weight_decay=weight_decay,
                        maximize=maximize, capturable=capturable)
        super(SophiaG, self).__init__(params, defaults)

    def __setstate__(self, state):
        super().__setstate__(state)
        for group in self.param_groups:
            group.setdefault('maximize', False)
            group.setdefault('capturable', False)
        state_values = list(self.state.values())
        step_is_tensor = (len(state_values) != 0) and torch.is_tensor(state_values[0]['step'])
        if not step_is_tensor:
            for s in state_values:
                s['step'] = torch.tensor(float(s['step']))

    @torch.no_grad()
    def update_hessian(self):
        for group in self.param_groups:
            beta1, beta2 = group['betas']
            for p in group['params']:
                if p.grad is None:
                    continue
                state = self.state[p]

                if len(state) == 0:
                    state['step'] = torch.zeros((1,), dtype=torch.float, device=p.device) \
                        if self.defaults['capturable'] else torch.tensor(0.)
                    state['exp_avg'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                    state['hessian'] = torch.zeros_like(p, memory_format=torch.preserve_format)

                if 'hessian' not in state.keys():
                    state['hessian'] = torch.zeros_like(p, memory_format=torch.preserve_format)

                state['hessian'].mul_(beta2).addcmul_(p.grad, p.grad, value=1 - beta2)

    @torch.no_grad()
    def step(self, closure=None, bs=5120):
        loss = None
        if closure is not None:
            with torch.enable_grad():
                loss = closure()

        for group in self.param_groups:
            params_with_grad = []
            grads = []
            exp_avgs = []
            state_steps = []
            hessian = []
            beta1, beta2 = group['betas']

            for p in group['params']:
                if p.grad is None:
                    continue
                params_with_grad.append(p)

                if p.grad.is_sparse:
                    raise RuntimeError('Hero does not support sparse gradients')
                grads.append(p.grad)
                state = self.state[p]
                # State initialization
                if len(state) == 0:
                    state['step'] = torch.zeros((1,), dtype=torch.float, device=p.device) \
                        if self.defaults['capturable'] else torch.tensor(0.)
                    state['exp_avg'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                    state['hessian'] = torch.zeros_like(p, memory_format=torch.preserve_format)

                if 'hessian' not in state.keys():
                    state['hessian'] = torch.zeros_like(p, memory_format=torch.preserve_format)

                exp_avgs.append(state['exp_avg'])
                state_steps.append(state['step'])
                hessian.append(state['hessian'])

                if self.defaults['capturable']:
                    bs = torch.ones((1,), dtype=torch.float, device=p.device) * bs

            sophiag(params_with_grad,
                    grads,
                    exp_avgs,
                    hessian,
                    state_steps,
                    bs=bs,
                    beta1=beta1,
                    beta2=beta2,
                    rho=group['rho'],
                    lr=group['lr'],
                    weight_decay=group['weight_decay'],
                    maximize=group['maximize'],
                    capturable=group['capturable'])

        return loss


def sophiag(params: List[Tensor],
            grads: List[Tensor],
            exp_avgs: List[Tensor],
            hessian: List[Tensor],
            state_steps: List[Tensor],
            capturable: bool = False,
            *,
            bs: int,
            beta1: float,
            beta2: float,
            rho: float,
            lr: float,
            weight_decay: float,
            maximize: bool):
    if not all(isinstance(t, torch.Tensor) for t in state_steps):
        raise RuntimeError("API has changed, `state_steps` argument must contain a list of singleton tensors")

    func = _single_tensor_sophiag
    #
    func(params,
         grads,
         exp_avgs,
         hessian,
         state_steps,
         bs=bs,
         beta1=beta1,
         beta2=beta2,
         rho=rho,
         lr=lr,
         weight_decay=weight_decay,
         maximize=maximize,
         capturable=capturable)


def _single_tensor_sophiag(params: List[Tensor],
                           grads: List[Tensor],
                           exp_avgs: List[Tensor],
                           hessian: List[Tensor],
                           state_steps: List[Tensor],
                           *,
                           bs: int,
                           beta1: float,
                           beta2: float,
                           rho: float,
                           lr: float,
                           weight_decay: float,
                           maximize: bool,
                           capturable: bool):
    for i, param in enumerate(params):
        grad = grads[i] if not maximize else -grads[i]
        exp_avg = exp_avgs[i]
        hess = hessian[i]
        step_t = state_steps[i]

        if capturable:
            assert param.is_cuda and step_t.is_cuda and bs.is_cuda

        if torch.is_complex(param):
            grad = torch.view_as_real(grad)
            exp_avg = torch.view_as_real(exp_avg)
            hess = torch.view_as_real(hess)
            param = torch.view_as_real(param)

        # update step
        step_t += 1

        # Perform stepweight decay
        param.mul_(1 - lr * weight_decay)

        # Decay the first and second moment running average coefficient
        exp_avg.mul_(beta1).add_(grad, alpha=1 - beta1)

        if capturable:
            step = step_t
            step_size = lr
            step_size_neg = step_size.neg()

            ratio = (exp_avg.abs() / (rho * bs * hess + 1e-15)).clamp(None, 1)
            param.addcmul_(exp_avg.sign(), ratio, value=step_size_neg)
        else:
            step = step_t.item()
            step_size_neg = - lr

            ratio = (exp_avg.abs() / (rho * bs * hess + 1e-15)).clamp(None, 1)
            param.addcmul_(exp_avg.sign(), ratio, value=step_size_neg)

3.总结

训练稳定性。与 AdamW 和 Lion 相比，Sophia-H 在预训练中具有更好的稳定性。梯度裁剪 (by norm) 是语言模型预训练中的一项重要技术。在实践中，梯度裁剪触发的频率与训练的稳定性有关 —— 如果梯度被频繁裁剪，迭代可能处于非常不稳定的状态。图 7 (a) 比较了 GPT-2 (125M) 触发梯度裁剪的 step 比例。尽管所有方法都使用相同的裁剪阈值 1.0，但 Sophia-H 很少触发梯度裁剪，而 AdamW 和 Lion 在超过 10% 的 step 中触发梯度裁剪。