This post walks through how GRPO (Group Relative Policy Optimization) extends the standard Transformer training paradigm, pure next-token prediction via maximum likelihood, into a reinforcement learning (RL) framework. GRPO is the RL algorithm DeepSeek used to train R1’s reasoning behavior. We derive the connections step by step and discuss where the approach can be pushed further.

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Review

1. Why the Naive Transformer Can Learn

We’ll review how all these elements combine so that the Transformer learns

$$p_\theta(x_t \mid x_{<t}),$$

and thus factorizes

$$p_\theta(\mathbf{x}) \;=\; \prod_{t=1}^N p_\theta(x_t \mid x_{<t})$$

for a text sequence $\mathbf{x}=(x_1,\ldots,x_N)$. Finally, we connect it to the negative log-likelihood (NLL) objective.

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1.1 Causal Masking

In a causal language model, position $t$ must only attend to previous positions $1,\dots,t-1$. To enforce this in self-attention, we define a mask $\mathbf{M}$ such that

$$\mathbf{M}_{t,s} \;=\; \begin{cases} 0, & s \le t,\\ -\infty, & s > t. \end{cases}$$

For each self-attention head, the attention weights $\boldsymbol{\alpha}_{t,:}$ are given by a softmax over the scaled dot products plus the mask:

$$\alpha_{t,s} \;=\; \mathrm{softmax}\!\Bigl( \frac{\mathbf{Q}_t \cdot \mathbf{K}_s}{\sqrt{d_k}} \;+\; \mathbf{M}_{t,s} \Bigr),$$

where $\alpha_{t,s}$ is the attention weight of position $t$ on position $s$. Because $\mathbf{M}_{t,s} = -\infty$ for $s>t$, those positions contribute zero after the softmax, which enforces causality.

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1.2 Query, Key, Value Transformations

Each multi-head self-attention block creates queries, keys, and values via learned linear projections of the hidden states $\mathbf{z}_s$. For head $h$ (out of $H$ total heads), we define:

$$\mathbf{Q}_s^{(h)} \;=\; \mathbf{W}_Q^{(h)} \,\mathbf{z}_s, \quad \mathbf{K}_s^{(h)} \;=\; \mathbf{W}_K^{(h)} \,\mathbf{z}_s, \quad \mathbf{V}_s^{(h)} \;=\; \mathbf{W}_V^{(h)} \,\mathbf{z}_s.$$

• $\mathbf{W}_Q^{(h)},\mathbf{W}_K^{(h)},\mathbf{W}_V^{(h)}$ each have dimension $\mathrm{d}_\text{model}\times \mathrm{d}_k$ (or $\mathrm{d}_v$), where typically $\mathrm{d}_k = \mathrm{d}_v = \mathrm{d}_\text{model}/H$.

For position $t$, the attention weights w.r.t. position $s\le t$ become:

$$\alpha_{t,s}^{(h)} \;=\; \mathrm{softmax}\!\Bigl( \frac{\mathbf{Q}_t^{(h)} \cdot \mathbf{K}_s^{(h)}}{\sqrt{d_k}} + \mathbf{M}_{t,s} \Bigr).$$

The context vector for head $h$ at position $t$ is then

$$\mathbf{c}_t^{(h)} \;=\; \sum_{s=1}^{N} \alpha_{t,s}^{(h)} \;\mathbf{V}_s^{(h)},$$

noting that $\alpha_{t,s}^{(h)}=0$ for $s>t$ via the causal mask.

2.1. Multi-Head Concatenation and Linear

If there are $H$ heads, we concatenate their outputs:

$$\mathbf{c}_t \;=\; \mathrm{Concat}\!\bigl[ \mathbf{c}_t^{(1)},\;\mathbf{c}_t^{(2)},\;\dots,\;\mathbf{c}_t^{(H)} \bigr],$$

which is dimension $\mathrm{d}_\text{model} = H\,\mathrm{d}_k$. We then apply another linear projection

$$\mathbf{u}_t \;=\; \mathbf{W}_O\,\mathbf{c}_t \;+\; \mathbf{b}_O,$$

where $\mathbf{W}_O$ has shape $\mathrm{d}_\text{model}\times \mathrm{d}_\text{model}$. The residual connection is typically added next (plus layer normalization, discussed below).

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1.3 Linear Transformations

A linear layer in a Transformer can be written:

$$\mathrm{Linear}(\mathbf{x}) \;=\; \mathbf{W}\,\mathbf{x} \;+\; \mathbf{b},$$

where $\mathbf{W}\in\mathbb{R}^{d_\text{out}\times d_\text{in}}$ and $\mathbf{b}\in \mathbb{R}^{d_\text{out}}$. In the multi-head attention context, we have three such linear layers ($\mathbf{W}_Q,\mathbf{W}_K,\mathbf{W}_V$) to form queries, keys, and values, plus a final one ($\mathbf{W}_O$) to map the concatenated heads back to $\mathrm{d}_\text{model}$.

Additionally, a typical feed-forward sub-layer within the Transformer is also a composition of two linear layers with a nonlinearity $\sigma(\cdot)$ (often ReLU or GELU):

$$\mathrm{FFN}(\mathbf{x}) \;=\; \mathbf{W}_2 \,\sigma\!\bigl(\mathbf{W}_1\,\mathbf{x} + \mathbf{b}_1 \bigr) \;+\; \mathbf{b}_2,$$

where $\mathbf{W}_1\in \mathbb{R}^{d_\text{ff}\times d_\text{model}}$, $\mathbf{W}_2\in \mathbb{R}^{d_\text{model}\times d_\text{ff}}$.

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1.4 Layer Normalization

LayerNorm is applied either before or after the main sub-layer operations (depending on the Transformer variant). The standard formula for a hidden vector $\mathbf{z} = (z_1,\dots,z_d)\in\mathbb{R}^d$ is:

$$\mathrm{LN}(\mathbf{z}) \;=\; \frac{\mathbf{z}\;-\;\mu(\mathbf{z})} {\sqrt{\sigma^2(\mathbf{z}) + \varepsilon}} \;\odot\;\boldsymbol{\gamma} \;+\; \boldsymbol{\beta},$$

where

1. $\mu(\mathbf{z})$ is the mean of the components $(z_1,\dots,z_d)$,
2. $\sigma^2(\mathbf{z})$ is the variance of $(z_1,\dots,z_d)$,
3. $\boldsymbol{\gamma}$ and $\boldsymbol{\beta}$ are learnable parameters of dimension $d$
4. $\varepsilon$ is a small constant (e.g., $10^{-5}$) to avoid division by zero,
5. $\odot$ denotes elementwise multiplication.

Thus, LayerNorm re-centers and re-scales each dimension of $\mathbf{z}$, per sample, improving training stability.

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1.5 Auto-Regressive Next-Token Probability

After $L$ layers of (1) self-attention with causal masking, (2) feed-forward layers, and (3) layer normalization + residual connections, we obtain a final hidden state $\mathbf{h}_t$ for each position $t$. We then project $\mathbf{h}_t$ to vocabulary logits:

$$\mathbf{o}_t \;=\; \mathbf{W}_\text{vocab}\,\mathbf{h}_t \;+\; \mathbf{b}_\text{vocab},$$

where $\mathbf{W}_\text{vocab}$ has size $\lvert\mathcal{V}\rvert\times d_\text{model}$ (assuming a vocabulary $\mathcal{V}$ of size $\lvert \mathcal{V}\rvert$) and $\mathbf{b}_\text{vocab}\in\mathbb{R}^{\lvert \mathcal{V}\rvert}$. A softmax over these logits yields:

$$p_\theta(x_t=w \mid x_{<t}) \;=\; \frac{\exp(\mathbf{o}_t[w])} {\sum_{u\in \mathcal{V}} \exp(\mathbf{o}_t[u])}.$$

This defines the auto-regressive distribution for the next token.

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1.6 Negative Log-Likelihood Objective

With the entire network parameterized by $\theta$, the language modeling likelihood of an observed sequence $\mathbf{x} = (x_1,\dots,x_N)$ factorizes as:

$$p_\theta(\mathbf{x}) \;=\; \prod_{t=1}^{N} p_\theta\bigl(x_t \mid x_{<t}\bigr).$$

The training objective is to minimize the negative log-likelihood over a large corpus $\mathcal{D}$:

$$\mathcal{L}_\text{NLL}(\theta) \;=\; -\,\sum_{\mathbf{x}\in \mathcal{D}} \;\sum_{t=1}^N \log\,p_\theta(x_t \mid x_{<t}),$$

which is equivalent to cross-entropy loss. Concretely, for each token $x_t$, the cross-entropy w.r.t. the model’s predicted distribution is

$$-\log\,p_\theta(x_t \mid x_{<t}) \;=\; -\mathbf{o}_t[x_t] \;+\; \log\!\Bigl( \sum_{u\in \mathcal{V}} \exp(\mathbf{o}_t[u]) \Bigr),$$

and backpropagation through the Q,K,V transformations, the linear layers, and the layer normalization updates all parameters $\theta$ so as to increase the likelihood of the observed tokens.

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1.7 Final Takeaway

• Causal Masking:

  $$\mathbf{M}_{t,s} = \begin{cases} 0, & s\le t\\ -\infty, & \text{otherwise} \end{cases}$$

  ensures that position $t$ never sees positions $s>t$.

• Self-Attention: Uses

  $$\alpha_{t,s} = \mathrm{softmax} \Bigl( \frac{\mathbf{Q}_t \cdot \mathbf{K}_s}{\sqrt{d_k}} + \mathbf{M}_{t,s} \Bigr), \quad \mathbf{c}_t = \sum_{s=1}^{N} \alpha_{t,s}\,\mathbf{V}_s,$$

  to selectively integrate information from previous tokens.

• Linear Layers:

  $$\mathrm{Linear}(\mathbf{x}) = \mathbf{W}\mathbf{x} + \mathbf{b},$$

  transform the embeddings, attention outputs, and feed-forward transformations.

• LayerNorm:

  $$\mathrm{LN}(\mathbf{z}) = \frac{\mathbf{z} - \mu(\mathbf{z})} {\sqrt{\sigma^2(\mathbf{z}) + \varepsilon}} \odot\boldsymbol{\gamma} + \boldsymbol{\beta}.$$

• Vocabulary Projection: A final linear layer $\mathbf{W}_\text{vocab}$ maps the final hidden state $\mathbf{h}_t$ to vocabulary logits $\mathbf{o}_t$, from which we apply softmax.

• Training: Minimizing

  $$\mathcal{L}_\text{NLL}(\theta) \;=\; -\sum_{\mathbf{x}\in \mathcal{D}}\sum_{t=1}^N \log\,p_\theta(x_t \mid x_{<t})$$

  aligns the model parameters $\theta$ to make accurate next-token predictions, purely via supervised next-token prediction.

• Key: Minimizing this Negative Log-Likelihood (NLL) via cross-entropy is equivalent to maximizing the likelihood of the training data. This procedure uses purely supervised learning with no environment-based “reward.”

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Reinforcement Learning in DeepSeek R1

2. Enter GRPO: An RL Overlay on Transformers

When we switch from pure next-token prediction to a reinforcement learning perspective, each generated token (or entire sequence) becomes an “action” that yields a reward. The policy $\pi_\theta$ is still a Transformer, but its training signal changes from log-likelihood of a fixed dataset to expected reward over sequences the model itself generates.

2.1. Group Relative Policy Optimization in a Nutshell

• Group Baseline: Instead of learning a value function as in PPO, GRPO uses a group-level baseline for variance reduction. For a group of $k$ outputs $\{ a^1, a^2, \dots, a^k \}$ from the same prompt $\mathcal{X}$, it computes the group-average reward

  $$b \;=\; \frac{1}{k}\,\sum_{i=1}^{k} r^i,$$

  where $r^i$ is the reward of the $i$-th output.

• Group-Relative Advantage:

  $$A^i \;=\; r^i \;-\; b.$$

  This advantage replaces the role of a learned $V(\cdot)$ function in PPO.

• Policy Ratio: As with PPO, GRPO measures the ratio of new policy to old policy

  $$r_i(\theta) \;=\; \frac{\pi_\theta(a^i \mid \mathcal{X})}{\pi_{\theta_\mathrm{old}}(a^i \mid \mathcal{X})}.$$

• KL Divergence or Clipping: GRPO can control policy updates via a KL penalty or by clipping this ratio in a trust region, similar to PPO.

2.2. Surrogate Objective with Group Averages

If we adopt a clipping-based approach (akin to PPO), the GRPO objective might look like:

$$L^\mathrm{GRPO}(\theta) \;=\; \mathbb{E}_{\mathcal{X},\,a}\! \Bigl[ \min\bigl(r_i(\theta)\,A^i,\; \mathrm{clip}\bigl(r_i(\theta),\,1-\epsilon,\,1+\epsilon\bigr)\,A^i \bigr) \Bigr].$$

Alternatively, if we incorporate a direct KL penalty w.r.t. a reference policy $\pi_{\mathrm{ref}}$, we might write:

$$L^\mathrm{GRPO}(\theta) \;=\; \mathbb{E}_{\mathcal{X},\,a}\! \Bigl[ r_i(\theta)\,A^i \;-\; \beta\,D_\mathrm{KL}\!\Bigl(\pi_\theta(\cdot\mid\mathcal{X})\;\big\|\;\pi_{\mathrm{ref}}(\cdot\mid\mathcal{X})\Bigr) \Bigr].$$

In both cases, the key difference from pure cross-entropy is that we are directly maximizing a reward-based objective rather than maximizing likelihood of tokens in a dataset.

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3. Where They Intersect: The Transformer as a Policy Network

Underneath both pure next-token MLE and GRPO (or PPO), the neural architecture is typically the same Transformer. What changes is the loss function and associated training data:

1. Pure MLE: The training data is a static corpus of text, and the objective is to predict the next token: no explicit reward.

2. GRPO / PPO:

   • We still use the Transformer architecture for $\pi_\theta(a \mid \mathcal{X})$.
   • We gather data (policy outputs) and compute or receive a reward signal $r^i$.
   • We update $\theta$ to improve this reward-based objective.

Thus, the mechanism of self-attention, feed-forward layers, and so on does not change. Instead, the training objective shifts from “match the next token distribution in the dataset” to “output sequences with high reward.”

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4. Potential Improvements: Unifying Likelihood and Reward Signals

A core tension arises in practical LLM training: we want linguistic fluency (which is well-captured by next-token prediction on large corpora) but also want task-specific behaviors (captured by a reward model or user feedback). Some ways forward:

1. Two-Stage Training:

   • First, pre-train the Transformer with standard cross-entropy on a large text corpus (for broad language fluency).
   • Then, fine-tune with GRPO on a smaller dataset or a reward signal. This approach is used in many RLHF (Reinforcement Learning from Human Feedback) pipelines.

2. Hybrid Objective:
   Combine the likelihood term and the reward term in a single objective, for example:

   $$L(\theta) \;=\; \lambda_\mathrm{MLE}\, \Bigl[ -\,\log p_\theta(x) \Bigr] \;+\; \lambda_\mathrm{RL}\, \Bigl[ -\,r_\mathrm{KL}\,D_\mathrm{KL}\bigl(\pi_\theta\|\pi_{\mathrm{ref}}\bigr) \;+\; \dots \Bigr].$$

   Tuning $\lambda_\mathrm{MLE}$ vs. $\lambda_\mathrm{RL}$ balances linguistic correctness with reward maximization.

3. Better Baselines:

   • Instead of using a simple group average or a learned value function, one could integrate more sophisticated baselines (e.g., learned critics that exploit contextual cues).
   • Or incorporate group-level variance reduction plus longer-horizon estimates (GAE-like expansions) for tasks requiring multiple steps.

4. Dynamic Reference Policies:

   • Periodically update $\pi_{\mathrm{ref}}$ to be the current $\pi_\theta$.
   • Use an adaptive schedule for the KL penalty coefficient so that the policy can explore initially but is later constrained when it’s sufficiently good.

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5. Mathematical Summary: The Relation Between MLE and GRPO

We can compare the respective training objectives more formally:

1. Standard Transformer (MLE)

   $$\min_\theta \;\; \Bigl[ -\sum_{\mathbf{x} \in \mathcal{D}} \log p_\theta(\mathbf{x}) \Bigr] \;=\; \min_\theta \;\; \mathcal{L}_\text{NLL}(\theta).$$
   • Data $\mathbf{x}$ is from a static corpus.
   • The “policy” $\pi_\theta$ is trained to predict tokens accurately.

2. GRPO Objective

   $$\max_\theta \;\; \mathbb{E}_{\mathbf{x}\sim \mathcal{X},\,a\sim\pi_\theta} \Bigl[ A^i\,(r_i(\theta)) \;-\; \beta\,D_{\mathrm{KL}}\bigl(\pi_\theta \,\|\, \pi_\mathrm{ref}\bigr) \Bigr],$$
   • Data is generated on the fly by sampling from $\pi_\theta$.
   • Rewards $\{r^i\}$ can come from a reward model or human feedback.
   • The group-based baseline $b$ reduces variance.

Bridging the Two Objectives

In many real-world applications (e.g., RLHF), the pipeline is:

1. Pre-train on MLE (huge corpus).
2. Reward model is trained (from human preferences or other signals).
3. Fine-tune with an RL method like GRPO or PPO to align the model with desired behavior.

This approach leverages both the linguistic understanding from MLE and the targeted reward optimization from RL.

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6. Concluding Thoughts

• GRPO is a natural RL extension of a Transformer that otherwise does pure next-token prediction.
• Transformer architecture stays the same. What changes is the objective and the data generation process, from a static corpus to on-policy sampling.
• Hybrid or multi-stage training preserves fluency while pushing the model toward high-value responses on specific tasks.
• Open problems: more efficient baselines, better reference policy management, and smoother transitions between the MLE and RL phases.

Takeaway: Understanding both the maximum likelihood foundation of Transformer training and how GRPO turns it into a reward-driven RL scheme makes it easier to reason about why RLHF-style pipelines behave the way they do, and where to intervene when they don’t. Large-scale pre-training combined with reward-guided fine-tuning remains the dominant strategy for aligning language models.

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Further Reading & References

• “Attention Is All You Need” by Vaswani et al., 2017, for the original Transformer.
• “Proximal Policy Optimization” by Schulman et al., 2017, for PPO.
• “Learning to Summarize with Human Feedback” by Stiennon et al., 2020, for a practical RLHF application.
• “DeepSeekMath” by Shao et al., 2024, which introduced GRPO.

If you are moving from pure next-token pre-training into an RL fine-tuning loop like this, the tooling below covers the data and observability layer you will need to keep reward signals honest, and the newsletter is where writeups like this one ship first.