(xsimpi_theta(cdot|o_{ 代表 当前模型根据前t-1个token采样得到第t个token x

KL3(GRPO使用的无偏，低方差KL1估计) http://joschu.net/blog/kl-approx.html：

[KL(pi_theta||pi_{ref}) = mathbb{E}_{xsimpi_theta(cdot|o_{

• 正向KL：倾向于使模型分布 Q 覆盖目标分布 P 的所有支持点，适合于需要模型分布更广泛覆盖的情况。
• 反向KL：倾向于使模型分布 Q 集中在目标分布 P 的高概率区域，适合于生成任务，能够提高生成样本的质量和稳定性。

因此，在大语言模型和生成任务中，反向KL通常更受青睐。

不同RL算法 loss的计算

对于q的第(i)个sample的第(t)个token的loss: (loss_{i,t}=pg_loss_{i,t}+entropy_loss_{i, t}+kl_loss_{i,t})

再对一个batch中所有的token loss (loss_{i,t})做聚合agg，得到这个batch的整体loss，可用于后续的反向传播和模型更新。

PPO

优化目标：

[J = mathbb{E}_{osimpi_{old}}frac{1}{|o|}sum_{i=1}^{|o|} [min(frac{pi_{theta}(o_i|o_{

优势： GAE
递推公式，t步的累积优势=t步的优势+ t+1步的累积优势=t步及之后 每一步的优势=t步及之后所有的奖励-第t步的预计奖励

[begin{aligned} A_t &= (r_t+gamma V_{t+1}-V_t)+gamma A_{t+1}\ A_t &= sum_{i=t}^T gamma ^{i-t}(r_t+gamma V_{t+1}-V_t)\ A_t &= r_t+gamma r_{t+1}+gamma^2 r_{t+2}+...+gamma^{T-t}r_T-V_t\ end{aligned} ]

奖励：

[r_t=begin{cases}-KL(pi_{old}||pi_{ref}), &tneq T \ -KL(pi_{old}||pi_{ref})+RM(q,o_i), &t=T end{cases} ]

verl/trainer/ppo/ray_trainer.py verl ｜ 如何在奖励中添加KL惩罚项？

###################################################
# 将KL惩罚loss应用到reward中。原始的reward是[0, 0, 0, ..., RM(q,o_i)]
# return KL(pi_old||pi_{ref}) + reward
###################################################
def apply_kl_penalty(data: DataProto, kl_ctrl: core_algos.AdaptiveKLController, kl_penalty="kl"):
    """Apply KL penalty to the token-level rewards.

    This function computes the KL divergence between the reference policy and current policy,
    then applies a penalty to the token-level rewards based on this divergence.

    Args:
        data (DataProto): The data containing batched model outputs and inputs.
        kl_ctrl (core_algos.AdaptiveKLController): Controller for adaptive KL penalty.
        kl_penalty (str, optional): Type of KL penalty to apply. Defaults to "kl".

    Returns:
        tuple: A tuple containing:
            - The updated data with token-level rewards adjusted by KL penalty
            - A dictionary of metrics related to the KL penalty
    """
    response_mask = data.batch["response_mask"]
    token_level_scores = data.batch["token_level_scores"]
    batch_size = data.batch.batch_size[0]

    # compute kl between ref_policy and current policy
    # When apply_kl_penalty, algorithm.use_kl_in_reward=True, so the reference model has been enabled.
    kld = core_algos.kl_penalty(
        data.batch["old_log_probs"], data.batch["ref_log_prob"], kl_penalty=kl_penalty
    )  # (batch_size, response_length)
    kld = kld * response_mask
    beta = kl_ctrl.value

    token_level_rewards = token_level_scores - beta * kld

KL

[KL(pi_{old}||pi_{ref}) = log(frac{pi_{old}(o_t|q, o_{

PPO的KL散度是old到ref的

PPO的代码实现详见下面的Dual-clip PPO（PPO的改进版）

Dual-clip PPO

https://arxiv.org/pdf/1912.09729：对A

论文发现当A是负无穷，这会导致训练不稳定（梯度爆炸）的现象，因此在ppo的clip上，对于A

[mathrm{per token objection} = begin{cases} min(IS*A, clip(IS, 1-epsilon, 1+epsilon)*A), &Ageq0\ max(min(IS*A, clip(IS, 1-epsilon, 1+epsilon)*A), clip_ratio_c*A), &A

代码：

整体的ppo_loss是由pg_loss + kl_loss + entropy_loss构成，不同的RL方法pg_loss, kl_loss的计算方法是不同的。

• pg_loss：具体于verl/trainer/ppo/core_algos.py（我将在dual-clip ppo和gspo部分介绍对应的pg_loss代码）。
• kl_loss：同样位于verl/trainer/ppo/core_algos.py（我将会在grpo部分介绍具体的low_var_kl代码）。

verl/verl/workers/roles/utils/losses.py: ppo_loss的计算

######################################################
# 此函数用于计算整体的actor loss
######################################################
def ppo_loss(config: ActorConfig, model_output, data: TensorDict, dp_group=None):
    log_prob = model_output["log_probs"]
    entropy = model_output.get("entropy", None)

    log_prob = no_padding_2_padding(log_prob, data)  # (bsz, response_length)
    if entropy is not None:
        entropy = no_padding_2_padding(entropy, data)  # (bsz, response_length)

    metrics = {}

    response_mask = data["response_mask"].to(bool)
    # compute policy loss
    old_log_prob = data["old_log_probs"]
    advantages = data["advantages"]

    loss_agg_mode = config.loss_agg_mode

    loss_mode = config.policy_loss.get("loss_mode", "vanilla")

    policy_loss_fn = get_policy_loss_fn(loss_mode)
    # 调用下面的计算pg_loss的代码框
    pg_loss, pg_clipfrac, ppo_kl, pg_clipfrac_lower = policy_loss_fn(
        old_log_prob=old_log_prob,
        log_prob=log_prob,
        advantages=advantages,
        response_mask=response_mask,
        loss_agg_mode=loss_agg_mode,
        config=config,
    )

    metrics.update(
        {
            "pg_loss": pg_loss.detach().item(),
            "pg_clipfrac": pg_clipfrac.detach().item(),
            "ppo_kl": ppo_kl.detach().item(),
            "pg_clipfrac_lower": pg_clipfrac_lower.detach().item(),
        }
    )
    policy_loss = pg_loss

    # 是否使用entropy loss
    # add entropy loss
    if entropy is not None:
        entropy_loss = agg_loss(loss_mat=entropy, loss_mask=response_mask, loss_agg_mode=loss_agg_mode)
        entropy_coeff = config.entropy_coeff
        # token的entropy越大越好，而loss是越小越好，因此是 减去 entropy
        policy_loss -= entropy_coeff * entropy_loss

    # 是否使用KL loss（grpo/gspo使用，ppo/dapo不使用）
    # add kl loss
    if config.use_kl_loss:
        ref_log_prob = data["ref_log_prob"]
        # compute kl loss
        kld = kl_penalty(logprob=log_prob, ref_logprob=ref_log_prob, kl_penalty=config.kl_loss_type)
        kl_loss = agg_loss(loss_mat=kld, loss_mask=response_mask, loss_agg_mode=config.loss_agg_mode)

        policy_loss += kl_loss * config.kl_loss_coef
        metrics["kl_loss"] = kl_loss.detach().item()
        metrics["kl_coef"] = config.kl_loss_coef

    return policy_loss, metrics

verl/trainer/ppo/core_algos.py不同的RL方法计算pg_loss是不同的，这里的是ppo的pg_loss，后面还会介绍gspo的pg_loss的实现。

######################################################
# 此函数用于计算pg_loss，并不计算KL惩罚项
######################################################
@register_policy_loss("vanilla")  # type: ignore[arg-type]
def compute_policy_loss_vanilla(
    old_log_prob: torch.Tensor,
    log_prob: torch.Tensor,
    advantages: torch.Tensor,
    response_mask: torch.Tensor,
    loss_agg_mode: str = "token-mean",
    config: Optional[DictConfig | AlgoConfig] = None,
    rollout_is_weights: torch.Tensor | None = None,
) -> tuple[torch.Tensor, torch.Tensor, torch.Tensor, torch.Tensor]:
    """
    Compute the clipped policy objective and related metrics for PPO.

    Adapted from
    https://github.com/huggingface/trl/blob/main/trl/trainer/ppo_trainer.py#L1122

    Args:
        old_log_prob (torch.Tensor):
            Log-probabilities of actions under the old policy, shape (batch_size, response_length).
        log_prob (torch.Tensor):
            Log-probabilities of actions under the current policy, shape (batch_size, response_length).
        advantages (torch.Tensor):
            Advantage estimates for each action, shape (batch_size, response_length).
        response_mask (torch.Tensor):
            Mask indicating which tokens to include in the loss, shape (batch_size, response_length).
        loss_agg_mode (str, optional):
            Aggregation mode for `agg_loss`. Defaults to "token-mean".
        config: `(verl.trainer.config.ActorConfig)`:
            config for the actor.
        rollout_log_probs: `(torch.Tensor)`:
            log probabilities of actions under the rollout policy, shape (batch_size, response_length).
    """

    assert config is not None
    assert not isinstance(config, AlgoConfig)
    clip_ratio = config.clip_ratio  # Clipping parameter ε for standard PPO. See https://arxiv.org/abs/1707.06347.
    clip_ratio_low = config.clip_ratio_low if config.clip_ratio_low is not None else clip_ratio
    clip_ratio_high = config.clip_ratio_high if config.clip_ratio_high is not None else clip_ratio
    clip_ratio_c = config.get(  # Lower bound of the ratio for dual-clip PPO. See https://arxiv.org/pdf/1912.09729.
        "clip_ratio_c", 3.0
    )

    cliprange = clip_ratio
    cliprange_low = clip_ratio_low
    cliprange_high = clip_ratio_high

    assert clip_ratio_c > 1.0, (
        "The lower bound of the clip_ratio_c for dual-clip PPO should be greater than 1.0,"
        + f" but get the value: {clip_ratio_c}."
    )

    # 计算每一个token的重要性采样的比值的log
    # log(pi_{theta}(o_{i,t}|q,o_{i,0, max(ratio*-A, clip(ratio, 1-epsilon_low, 1+epsilon_high)*-A)
    #A=0时用clip_pg_losses1
    pg_losses = torch.where(advantages 

咱们继续看几种token loss的agg mode。不同RL方法，loss agg mode也是不同的

verl/trainer/ppo/core_algos.py

def agg_loss(loss_mat: torch.Tensor, loss_mask: torch.Tensor, loss_agg_mode: str):
    """
    Aggregate the loss matrix into a scalar.

    Args:
        loss_mat: `(torch.Tensor)`:
            shape: (bs, response_length)
        loss_mask: `(torch.Tensor)`:
            shape: (bs, response_length)
        loss_agg_mode: (str) choices:
            method to aggregate the loss matrix into a scalar.
    Returns:
        loss: `a scalar torch.Tensor`
            aggregated loss
    """
    if loss_agg_mode == "token-mean":
        loss = verl_F.masked_mean(loss_mat, loss_mask)
    elif loss_agg_mode == "seq-mean-token-sum":
        seq_losses = torch.sum(loss_mat * loss_mask, dim=-1)  # token-sum
        loss = torch.mean(seq_losses)  # seq-mean
    elif loss_agg_mode == "seq-mean-token-mean":
        seq_losses = torch.sum(loss_mat * loss_mask, dim=-1) / torch.sum(loss_mask, dim=-1)  # token-mean
        loss = torch.mean(seq_losses)  # seq-mean
    elif loss_agg_mode == "seq-mean-token-sum-norm":
        seq_losses = torch.sum(loss_mat * loss_mask, dim=-1)
        loss = torch.sum(seq_losses) / loss_mask.shape[-1]  # The divisor
        # (loss_mask.shape[-1]) should ideally be constant
        # throughout training to well-replicate the DrGRPO paper.
        # TODO: Perhaps add user-defined normalizer argument to
        # agg_loss to ensure divisor stays constant throughout.
    else:
        raise ValueError(f"Invalid loss_agg_mode: {loss_agg_mode}")

    return loss

GRPO

优化目标：

[J= mathbb{E}_{{o_i}_{i=1}^Gsimpi_{old}(cdot|q)} frac{1}{|G|} sum_{i=1}^{|G|}frac{1}{|o|}sum_{t=1}^{|o_i|}{min[frac{pi_{theta}(o_{i,t}|q, o_{i,

优势：

[A_{i,t} = frac{r_i-mean(r)}{std(r)} ]

KL3

[mathbb{D}_{KL}(pi_{theta}||pi_{ref}) =frac{pi_{ref}(o_{i, t}|q,o_{i,

KL3的方差比KL1小，且是KL1的无偏估计

证明

[begin{aligned} mathbb{D3}_{KL}(P||Q) &= sum_{xsim_{P}}P(x) [frac{Q(x)}{P(x)} - log(frac{P(x)}{Q(x)})-1]\ &= sum_{xsim P}Q(x)+P(x)log(frac{P(x)}{Q(x)})-P(x)\ &=sum_{xsim P}Q(x) -sum_{xsim P}P(x)+mathbb{D1}_{KL}(P||Q) \ &=mathbb{D1}_{KL}(P||Q)+sum_{xsim P}Q(x)-1 当P所有采样在Q中的概率和为1时（vocab一样的话）\ &=mathbb{D_1}_{KL}(P||Q) end{aligned} ]

verl/trainer/ppo/core_algos.py 下面是verl对kl_loss的实现：

def kl_penalty_forward(logprob: torch.FloatTensor, ref_logprob: torch.FloatTensor, kl_penalty) -> torch.FloatTensor:
    """Compute KL divergence given logprob and ref_logprob.
    Copied from https://github.com/huggingface/trl/blob/main/trl/trainer/ppo_trainer.py#L1104
    See more description in http://joschu.net/blog/kl-approx.html

    Args:
        logprob:
        ref_logprob:

    Returns:
        kl_estimate
    """
    if kl_penalty in ("kl", "k1"):
        return logprob - ref_logprob

    if kl_penalty == "abs":
        return (logprob - ref_logprob).abs()

    if kl_penalty in ("mse", "k2"):
        return 0.5 * (logprob - ref_logprob).square()

    ##############################################################
    # 这里的low_var_kl与上述的grpo的KL计算公式相同
    ##############################################################
    # J. Schulman. Approximating kl divergence, 2020.
    # # URL http://joschu.net/blog/kl-approx.html.
    if kl_penalty in ("low_var_kl", "k3"):
        kl = ref_logprob - logprob
        # For numerical stability
        kl = torch.clamp(kl, min=-20, max=20)
        ratio = torch.exp(kl)
        kld = (ratio - kl - 1).contiguous()
        return torch.clamp(kld, min=-10, max=10)

    if kl_penalty == "full":
        # so, here logprob and ref_logprob should contain the logits for every token in vocabulary
        raise NotImplementedError

    raise NotImplementedError

GSPO

seq-level 优化目标：

[J= mathbb{E}_{{o_i}_{i=1}^Gsimpi_{old}(cdot|q)} frac{1}{|G|} sum_{i=1}^{|G|}min[(frac{pi_{theta}(o_{i}|q)}{pi_{old}(o_{i}|q)})^{frac{1}{|o_i|}}A_{i}, clip((frac{pi_{theta}(o_{i}|q)}{pi_{old}(o_{i}|q)})^{frac{1}{|o_i|}}, 1-epsilon, 1+epsilon)A_{i}] ]

[frac{pi_{theta}(o_i|q)}{pi_{old}(o_i|q)} = frac{Pi_{t=1}^{|o_i|} pi_{theta}(o_{i,t}|q, o_{i,

token-level 优化目标：

[J = mathbb{E}_{{o_i}_{i=1}^Gsim pi_{old}(cdot|q)}frac{1}{G}sum_{i=1}^Gfrac{1}{|o_i|}sum_{t=1}^{|o_i|} min(s_{i,t}A_{i,t}, clip(s_{i,t}, 1-epsilon,1+epsilon)A_{i,t})\ hat{s}_{i,t} = sg[(frac{pi_{theta}(o_i|q)}{pi_{old}(o_i|q)})^{frac{1}{|o_i|}}]* frac{pi_{theta}(o_{i,t}|q,o_{i,

可以发现的是 (sg[s_{i,t}]=sg[s_{i}],s_{i}=(frac{pi_{theta}(o_i|q)}{pi_{old}(o_i|q)})^{frac{1}{|o_i|}})，但是在方向上不同

通过证明，可以发现，当(A_{i,t}=A_i)时，seq-level和token-level在前向传播和反向传播上是一样的
token-level 可以更好地扩展 同sample不同token的A的灵活度（每个token的A可以不相同）

verl/trainer/ppo/core_algos.py

##########################################################
# 计算gspo的pg_loss,重点关注IS的计算
##########################################################
@register_policy_loss("gspo")
def compute_policy_loss_gspo(
    old_log_prob: torch.Tensor,
    log_prob: torch.Tensor,
    advantages: torch.Tensor,
    response_mask: torch.Tensor,
    loss_agg_mode: str = "seq-mean-token-mean",
    config: Optional[DictConfig | ActorConfig] = None,
    rollout_is_weights: torch.Tensor | None = None,
) -> tuple[torch.Tensor, torch.Tensor, torch.Tensor, torch.Tensor]:
    """
    Compute the clipped policy objective and related metrics for GSPO.

    See https://arxiv.org/pdf/2507.18071 for more details.

    Args:
        old_log_prob (torch.Tensor):
            Log-probabilities of actions under the old policy, shape (batch_size, response_length).
        log_prob (torch.Tensor):
            Log-probabilities of actions under the current policy, shape (batch_size, response_length).
        advantages (torch.Tensor):
            Advantage estimates for each action, shape (batch_size, response_length).
        response_mask (torch.Tensor):
            Mask indicating which tokens to include in the loss, shape (batch_size, response_length).
        loss_agg_mode (str, optional):
            Aggregation mode for `agg_loss`. For GSPO, it is recommended to use "seq-mean-token-mean".
    """

    assert config is not None
    assert isinstance(config, ActorConfig)
    clip_ratio_low = config.clip_ratio_low if config.clip_ratio_low is not None else config.clip_ratio
    clip_ratio_high = config.clip_ratio_high if config.clip_ratio_high is not None else config.clip_ratio

    negative_approx_kl = log_prob - old_log_prob

    # compute sequence-level importance ratio:
    # si(θ) = (π_θ(yi|x)/π_θold(yi|x))^(1/|yi|) =
    # exp [(1/|y_i|) * Σ_t log(π_θ(y_i,t|x,y_i,

DAPO

优化目标：

[mathcal{J} = mathbb{E}_{(q,a)sim mathcal{D}, {o_i}_{i=1}^Gsim pi_{old}(cdot|q)} [frac{1}{sum_{i=1}^G|o_i|}sum_{i=1}^Gsum_{t=1}^{|o_i|}min(r_{i,t}(theta)A_{i, t}, clip(r_{i,t}(theta),1-epsilon_{low}, 1+epsilon_{high})A_{i,t})]\ s.t. 0

其中

[r_{i,t}(theta)=frac{pi_{theta}(o_{i,t}|q,o_{i,

其loss agg mode是token-mean。